Propositional calculus v 0 0 1

Contents

1 Introduction 11.1 Propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Generic description of a propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . 41.1.5 Example 1. Simple axiom system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.6 Example 2. Natural deduction system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.7 Basic and derived argument forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.8 Proofs in propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.9 Soundness and completeness of the rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.10 Interpretation of a truth-functional propositional calculus . . . . . . . . . . . . . . . . . . 81.1.11 Alternative calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.12 Equivalence to equational logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.13 Graphical calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.14 Other logical calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.15 Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.16 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.17 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.18 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.19 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Chapter I Abduction 132.1 Abductive reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Deduction, induction, and abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Formalizations of abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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2.2 Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Theories of knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.2 Communicating knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.3 Situated knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.4 Partial knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.5 Scientific knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.6 Religious meaning of knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 Understanding as a model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.3 Components of understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.4 Religious perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Certainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 Degrees of certainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.3 Foundational crisis of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.4 Quotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Chapter II Induction 343.1 Inductive reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.2 Inductive vs. deductive reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.3 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.4 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.5 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.6 Inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Formal and informal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Standard types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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3.2.3 Deductive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.4 Inductive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.5 Defeasible arguments and argumentation schemes . . . . . . . . . . . . . . . . . . . . . . 403.2.6 By analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.7 Other kinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.8 Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.9 Fallacies and nonarguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.13 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Chapter III Deduction 454.1 Deductive reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.2 Law of detachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.3 Law of syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.4 Law of contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.1.5 Validity and soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.1.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.1.7 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.1.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.1.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.1 Validity of arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.2 Valid formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.3 Validity of statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.4 Validity and soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.5 Satisfiability and validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.6 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.7 n-Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.1 Of arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.2 sounds and unsounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.3 Relation to completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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4.3.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Index 515.1 List of logic symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Basic logic symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1.2 Advanced and rarely used logical symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Text and image sources, contributors, and licenses 536.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Chapter 1

Introduction

1.1 Propositional calculus

Propositional calculus (also called propositional logic,sentential calculus, or sentential logic) is the branch ofmathematical logic concerned with the study of propo-sitions (whether they are true or false) that are formedby other propositions with the use of logical connectives,and how their value depends on the truth value of theircomponents. Logical connectives are found in naturallanguages. In English for example, some examples are“and” (conjunction), “or” (disjunction), “not” (negation)and “if” (but only when used to denote material condi-tional).The following is an example of a very simple inferencewithin the scope of propositional logic:

Premise 1: If it’s raining then it’s cloudy.Premise 2: It’s raining.Conclusion: It’s cloudy.

Both premises and the conclusion are propositions. Thepremises are taken for granted and then with the applica-tion of modus ponens (an inference rule) the conclusionfollows.As propositional logic is not concerned with the struc-ture of propositions beyond the point where they can't bedecomposed anymore by logical connectives, this infer-ence can be restated replacing those atomic statementswith statement letters, which are interpreted as variablesrepresenting statements:

P → Q

P

Q

The same can be stated succinctly in the following way:

P → Q,P ⊢ Q

When P is interpreted as “It’s raining” and Q as “it’scloudy” the above symbolic expressions can be seen to

exactly correspond with the original expression in natu-ral language. Not only that, but they will also correspondwith any other inference of this form, which will be validon the same basis that this inference is.Propositional logic may be studied through a formalsystem in which formulas of a formal language maybe interpreted to represent propositions. A system ofinference rules and axioms allows certain formulas to bederived. These derived formulas are called theorems andmay be interpreted to be true propositions. A constructedsequence of such formulas is known as a derivation orproof and the last formula of the sequence is the theo-rem. The derivation may be interpreted as proof of theproposition represented by the theorem.When a formal system is used to represent formal logic,only statement letters are represented directly. The nat-ural language propositions that arise when they're inter-preted are outside the scope of the system, and the rela-tion between the formal system and its interpretation islikewise outside the formal system itself.Usually in truth-functional propositional logic, formu-las are interpreted as having either a truth value of trueor a truth value of false. Truth-functional propositionallogic and systems isomorphic to it, are considered to bezeroth-order logic.

1.1.1 History

Main article: History of logic

Although propositional logic (which is interchangeablewith propositional calculus) had been hinted by earlierphilosophers, it was developed into a formal logic byChrysippus in the 3rd century BC[1] and expanded by theStoics. The logic was focused on propositions. This ad-vancement was different from the traditional syllogisticlogic which was focused on terms. However, later in an-tiquity, the propositional logic developed by the Stoicswas no longer understood . Consequently, the systemwas essentially reinvented by Peter Abelard in the 12thcentury.[2]

Propositional logic was eventually refined using symbolic

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2 CHAPTER 1. INTRODUCTION

logic. The 17th/18th-century mathematician GottfriedLeibniz has been credited with being the founder of sym-bolic logic for his work with the calculus ratiocinator. Al-though his work was the first of its kind, it was unknownto the larger logical community. Consequently, many ofthe advances achieved by Leibniz were reachieved by lo-gicians like George Boole and Augustus DeMorgan com-pletely independent of Leibniz.[3]

Just as propositional logic can be considered an advance-ment from the earlier syllogistic logic, Gottlob Frege’spredicate logic was an advancement from the earlierpropositional logic. One author describes predicate logicas combining “the distinctive features of syllogistic logicand propositional logic.”[4] Consequently, predicate logicushered in a new era in logic’s history; however, advancesin propositional logic were still made after Frege, includ-ing Natural Deduction, Truth-Trees and Truth-Tables.Natural deduction was invented by Gerhard Gentzen andJan Łukasiewicz. Truth-Trees were invented by EvertWillem Beth.[5] The invention of truth-tables, however,is of controversial attribution.Within works by Frege[6] and Bertrand Russell,[7] onefinds ideas influential in bringing about the notion of truthtables. The actual 'tabular structure' (being formattedas a table), itself, is generally credited to either LudwigWittgenstein or Emil Post (or both, independently).[6]Besides Frege and Russell, others credited with havingideas preceding truth-tables include Philo, Boole, CharlesSanders Peirce, and Ernst Schröder. Others creditedwith the tabular structure include Łukasiewicz, Schröder,Alfred North Whitehead, William Stanley Jevons, JohnVenn, and Clarence Irving Lewis.[7] Ultimately, somehave concluded, like John Shosky, that “It is far from clearthat any one person should be given the title of 'inventor'of truth-tables.”.[7]

1.1.2 Terminology

In general terms, a calculus is a formal system that con-sists of a set of syntactic expressions (well-formed formu-las), a distinguished subset of these expressions (axioms),plus a set of formal rules that define a specific binary rela-tion, intended to be interpreted to be logical equivalence,on the space of expressions.When the formal system is intended to be a logical sys-tem, the expressions are meant to be interpreted to bestatements, and the rules, known to be inference rules, aretypically intended to be truth-preserving. In this setting,the rules (which may include axioms) can then be usedto derive (“infer”) formulas representing true statementsfrom given formulas representing true statements.The set of axioms may be empty, a nonempty finite set,a countably infinite set, or be given by axiom schemata.A formal grammar recursively defines the expressionsand well-formed formulas of the language. In addition asemanticsmay be givenwhich defines truth and valuations

(or interpretations).The language of a propositional calculus consists of

1. a set of primitive symbols, variously referred to beatomic formulas, placeholders, proposition letters, orvariables, and

2. a set of operator symbols, variously interpreted to belogical operators or logical connectives.

Awell-formed formula is any atomic formula, or any for-mula that can be built up from atomic formulas by meansof operator symbols according to the rules of the gram-mar.Mathematicians sometimes distinguish between proposi-tional constants, propositional variables, and schemata.Propositional constants represent some particular propo-sition, while propositional variables range over the set ofall atomic propositions. Schemata, however, range overall propositions. It is common to represent propositionalconstants by A, B, and C, propositional variables by P, Q,and R, and schematic letters are often Greek letters, mostoften φ, ψ, and χ.

1.1.3 Basic concepts

The following outlines a standard propositional calculus.Many different formulations exist which are all more orless equivalent but differ in the details of:

1. their language, that is, the particular collection ofprimitive symbols and operator symbols,

2. the set of axioms, or distinguished formulas, and

3. the set of inference rules.

Any given proposition may be represented with a lettercalled a 'propositional constant', analogous to represent-ing a number by a letter in mathematics, for instance, a= 5. All propositions require exactly one of two truth-values: true or false. For example, let P be the proposi-tion that it is raining outside. This will be true (P) if it israining outside and false otherwise (¬P).

• We then define truth-functional operators, begin-ning with negation. ¬P represents the negation ofP, which can be thought of as the denial of P. In theexample above, ¬P expresses that it is not rainingoutside, or by a more standard reading: “It is not thecase that it is raining outside.” When P is true, ¬Pis false; and when P is false, ¬P is true. ¬¬P alwayshas the same truth-value as P.

• Conjunction is a truth-functional connective whichforms a proposition out of two simpler propositions,for example, P and Q. The conjunction of P and Q is

1.1. PROPOSITIONAL CALCULUS 3

written P ∧ Q, and expresses that each are true. Weread P ∧ Q for “P and Q”. For any two propositions,there are four possible assignments of truth values:1. P is true and Q is true2. P is true and Q is false3. P is false and Q is true4. P is false and Q is false

The conjunction of P andQ is true in case 1 andis false otherwise. Where P is the propositionthat it is raining outside andQ is the propositionthat a cold-front is over Kansas, P ∧ Q is truewhen it is raining outside and there is a cold-front over Kansas. If it is not raining outside,then P ∧ Q is false; and if there is no cold-frontover Kansas, then P ∧ Q is false.

• Disjunction resembles conjunction in that it formsa proposition out of two simpler propositions. Wewrite it P ∨ Q, and it is read “P or Q”. It expressesthat either P or Q is true. Thus, in the cases listedabove, the disjunction of P and Q is true in all casesexcept 4. Using the example above, the disjunctionexpresses that it is either raining outside or there isa cold front over Kansas. (Note, this use of dis-junction is supposed to resemble the use of the En-glish word “or”. However, it is most like the Englishinclusive “or”, which can be used to express the truthof at least one of two propositions. It is not like theEnglish exclusive “or”, which expresses the truth ofexactly one of two propositions. That is to say, theexclusive “or” is false when both P and Q are true(case 1). An example of the exclusive or is: Youmay have a bagel or a pastry, but not both. Often innatural language, given the appropriate context, theaddendum “but not both” is omitted but implied. Inmathematics, however, “or” is always inclusive or; ifexclusive or is meant it will be specified, possibly by“xor”.)

• Material conditional also joins two simpler propo-sitions, and we write P → Q, which is read “if Pthen Q”. The proposition to the left of the arrowis called the antecedent and the proposition to theright is called the consequent. (There is no such des-ignation for conjunction or disjunction, since theyare commutative operations.) It expresses that Qis true whenever P is true. Thus it is true in everycase above except case 2, because this is the onlycase when P is true but Q is not. Using the exam-ple, if P then Q expresses that if it is raining outsidethen there is a cold-front over Kansas. The materialconditional is often confused with physical causa-tion. The material conditional, however, only relatestwo propositions by their truth-values—which is notthe relation of cause and effect. It is contentious inthe literature whether the material implication rep-resents logical causation.

• Biconditional joins two simpler propositions, and wewrite P ↔ Q, which is read “P if and only if Q”. Itexpresses that P and Q have the same truth-value,thus P if and only if Q is true in cases 1 and 4, andfalse otherwise.

It is extremely helpful to look at the truth tables for thesedifferent operators, as well as the method of analytictableaux.

Closure under operations

Propositional logic is closed under truth-functional con-nectives. That is to say, for any proposition φ, ¬φ is alsoa proposition. Likewise, for any propositions φ and ψ, φ∧ ψ is a proposition, and similarly for disjunction, condi-tional, and biconditional. This implies that, for instance,φ ∧ ψ is a proposition, and so it can be conjoined withanother proposition. In order to represent this, we needto use parentheses to indicate which proposition is con-joined with which. For instance, P ∧ Q ∧ R is not a well-formed formula, because we do not know if we are con-joining P ∧ Q with R or if we are conjoining P with Q ∧R. Thus we must write either (P ∧ Q) ∧ R to represent theformer, or P ∧ (Q ∧ R) to represent the latter. By eval-uating the truth conditions, we see that both expressionshave the same truth conditions (will be true in the samecases), and moreover that any proposition formed by ar-bitrary conjunctions will have the same truth conditions,regardless of the location of the parentheses. This meansthat conjunction is associative, however, one should notassume that parentheses never serve a purpose. For in-stance, the sentence P ∧ (Q ∨ R) does not have the sametruth conditions of (P ∧ Q) ∨ R, so they are different sen-tences distinguished only by the parentheses. One canverify this by the truth-table method referenced above.Note: For any arbitrary number of propositional con-stants, we can form a finite number of cases which listtheir possible truth-values. A simple way to generate thisis by truth-tables, in which one writes P, Q, ..., Z, for anylist of k propositional constants—that is to say, any list ofpropositional constants with k entries. Below this list, onewrites 2k rows, and below P one fills in the first half of therows with true (or T) and the second half with false (or F).Below Q one fills in one-quarter of the rows with T, thenone-quarter with F, then one-quarter with T and the lastquarter with F. The next column alternates between trueand false for each eighth of the rows, then sixteenths, andso on, until the last propositional constant varies betweenT and F for each row. This will give a complete listing ofcases or truth-value assignments possible for those propo-sitional constants.

Argument

The propositional calculus then defines an argument to bea set of propositions. A valid argument is a set of proposi-

4 CHAPTER 1. INTRODUCTION

tions, the last of which follows from—or is implied by—the rest. All other arguments are invalid. The simplestvalid argument is modus ponens, one instance of whichis the following set of propositions:

1. P → Q2. P∴ Q

This is a set of three propositions, each line is a proposi-tion, and the last follows from the rest. The first two linesare called premises, and the last line the conclusion. Wesay that any proposition C follows from any set of propo-sitions (P1, ..., Pn) , if C must be true whenever everymember of the set (P1, ..., Pn) is true. In the argumentabove, for any P and Q, whenever P → Q and P are true,necessarily Q is true. Notice that, when P is true, we can-not consider cases 3 and 4 (from the truth table). When P→ Q is true, we cannot consider case 2. This leaves onlycase 1, in which Q is also true. Thus Q is implied by thepremises.This generalizes schematically. Thus, where φ and ψmaybe any propositions at all,

1. φ→ ψ2. φ∴ ψ

Other argument forms are convenient, but not necessary.Given a complete set of axioms (see below for one suchset), modus ponens is sufficient to prove all other argu-ment forms in propositional logic, thus they may be con-sidered to be a derivative. Note, this is not true of theextension of propositional logic to other logics like first-order logic. First-order logic requires at least one addi-tional rule of inference in order to obtain completeness.The significance of argument in formal logic is that onemay obtain new truths from established truths. In the firstexample above, given the two premises, the truth of Q isnot yet known or stated. After the argument is made, Qis deduced. In this way, we define a deduction systemto be a set of all propositions that may be deduced fromanother set of propositions. For instance, given the set ofpropositionsA = P ∨Q,¬Q∧R, (P ∨Q) → R , wecan define a deduction system, Γ, which is the set of allpropositions which follow from A. Reiteration is alwaysassumed, so P ∨Q,¬Q∧R, (P ∨Q) → R ∈ Γ . Also,from the first element of A, last element, as well as modusponens, R is a consequence, and so R ∈ Γ . Because wehave not included sufficiently complete axioms, though,nothing else may be deduced. Thus, even though mostdeduction systems studied in propositional logic are ableto deduce (P ∨Q) ↔ (¬P → Q) , this one is too weakto prove such a proposition.

1.1.4 Generic description of a proposi-tional calculus

A propositional calculus is a formal system L =L (A, Ω, Z, I) , where:

• The alpha set A is a finite set of elements calledproposition symbols or propositional variables. Syn-tactically speaking, these are the most basic ele-ments of the formal language L , otherwise referredto as atomic formulas or terminal elements. In theexamples to follow, the elements of A are typicallythe letters p, q, r, and so on.

• The omega set Ω is a finite set of elements calledoperator symbols or logical connectives. The set Ω ispartitioned into disjoint subsets as follows:

Ω = Ω0∪Ω1∪. . .∪Ωj∪. . .∪Ωm.

In this partition, Ωj is the set of operator sym-bols of arity j.

In the more familiar propositional calculi, Ω istypically partitioned as follows:

Ω1 = ¬,

Ω2 ⊆ ∧,∨,→,↔.

A frequently adopted convention treats theconstant logical values as operators of arityzero, thus:

Ω0 = 0, 1.

Some writers use the tilde (~), or N, instead of¬; and some use the ampersand (&), the pre-fixed K, or · instead of ∧ . Notation varies evenmore for the set of logical values, with symbolslike false, true, F, T, or ⊥,⊤ all beingseen in various contexts instead of 0, 1.

• The zeta set Z is a finite set of transformation rulesthat are called inference rules when they acquire log-ical applications.

• The iota set I is a finite set of initial points that arecalled axioms when they receive logical interpreta-tions.

1.1. PROPOSITIONAL CALCULUS 5

The language of L , also known as its set of formulas,well-formed formulas, is inductively defined by the fol-lowing rules:

1. Base: Any element of the alpha set A is a formulaof L .

2. If p1, p2, . . . , pj are formulas and f is in Ωj , then(f(p1, p2, . . . , pj)) is a formula.

3. Closed: Nothing else is a formula of L .

Repeated applications of these rules permits the construc-tion of complex formulas. For example:

1. By rule 1, p is a formula.

2. By rule 2, ¬p is a formula.

3. By rule 1, q is a formula.

4. By rule 2, (¬p ∨ q) is a formula.

1.1.5 Example 1. Simple axiom system

Let L1 = L(A,Ω,Z, I) , where A , Ω , Z , I are definedas follows:

• The alpha set A , is a finite set of symbols that is largeenough to supply the needs of a given discussion, forexample:

A = p, q, r, s, t, u.

• Of the three connectives for conjunction, disjunc-tion, and implication ( ∧,∨ , and →), one can betaken as primitive and the other two can be definedin terms of it and negation (¬).[8] Indeed, all of thelogical connectives can be defined in terms of a solesufficient operator. The biconditional (↔) can ofcourse be defined in terms of conjunction and im-plication, with a↔ b defined as (a→ b)∧ (b→ a).

Ω = Ω1 ∪ Ω2

Ω1 = ¬,

Ω2 = →.

• An axiom system discovered by Jan Łukasiewiczformulates a propositional calculus in this languageas follows. The axioms are all substitution instancesof:

• (p→ (q → p))

• ((p → (q → r)) → ((p →q) → (p→ r)))

• ((¬p→ ¬q) → (q → p))

• The rule of inference is modus ponens (i.e., fromp and (p → q) , infer q). Then a ∨ b is definedas ¬a → b , and a ∧ b is defined as ¬(a → ¬b). This system is used in Metamath set.mm formalproof database.

1.1.6 Example 2. Natural deduction sys-tem

Let L2 = L(A,Ω,Z, I) , where A , Ω , Z , I are definedas follows:

• The alpha set A , is a finite set of symbols that is largeenough to supply the needs of a given discussion, forexample:

A = p, q, r, s, t, u.

• The omega set Ω = Ω1 ∪ Ω2 partitions as follows:

Ω1 = ¬,

Ω2 = ∧,∨,→,↔.

In the following example of a propositional calculus, thetransformation rules are intended to be interpreted as theinference rules of a so-called natural deduction system.The particular system presented here has no initial points,whichmeans that its interpretation for logical applicationsderives its theorems from an empty axiom set.

• The set of initial points is empty, that is, I = ∅ .

• The set of transformation rules, Z , is described asfollows:

Our propositional calculus has ten inference rules. Theserules allow us to derive other true formulas given a setof formulas that are assumed to be true. The first ninesimply state that we can infer certain well-formed formu-las from other well-formed formulas. The last rule how-ever uses hypothetical reasoning in the sense that in thepremise of the rule we temporarily assume an (unproven)hypothesis to be part of the set of inferred formulas tosee if we can infer a certain other formula. Since the firstnine rules don't do this they are usually described as non-hypothetical rules, and the last one as a hypothetical rule.In describing the transformation rules, we may introducea metalanguage symbol ⊢ . It is basically a convenient

6 CHAPTER 1. INTRODUCTION

shorthand for saying “infer that”. The format is Γ ⊢ ψ, in which Γ is a (possibly empty) set of formulas calledpremises, andψ is a formula called conclusion. The trans-formation rule Γ ⊢ ψ means that if every proposition inΓ is a theorem (or has the same truth value as the ax-ioms), then ψ is also a theorem. Note that considering thefollowing rule Conjunction introduction, we will knowwhenever Γ has more than one formula, we can alwayssafely reduce it into one formula using conjunction. Sofor short, from that time on we may represent Γ as oneformula instead of a set. Another omission for conve-nience is when Γ is an empty set, in which case Γ maynot appear.

Negation introduction From (p → q) and (p → ¬q) ,infer ¬p .

That is, (p→ q), (p→ ¬q) ⊢ ¬p .Negation elimination From ¬p , infer (p→ r) .That is, ¬p ⊢ (p→ r) .Double negative elimination From ¬¬p , infer p.That is, ¬¬p ⊢ p .Conjunction introduction From p and q, infer (p ∧ q)

.That is, p, q ⊢ (p ∧ q) .Conjunction elimination From (p ∧ q) , infer p.From (p ∧ q) , infer q.That is, (p ∧ q) ⊢ p and (p ∧ q) ⊢ q .Disjunction introduction From p, infer (p ∨ q) .From q, infer (p ∨ q) .That is, p ⊢ (p ∨ q) and q ⊢ (p ∨ q) .Disjunction elimination From (p∨q) and (p→ r) and

(q → r) , infer r.That is, p ∨ q, p→ r, q → r ⊢ r .Biconditional introduction From (p → q) and (q →

p) , infer (p↔ q) .That is, p→ q, q → p ⊢ (p↔ q) .Biconditional elimination From (p ↔ q) , infer (p →

q) .From (p↔ q) , infer (q → p) .That is, (p↔ q) ⊢ (p→ q) and (p↔ q) ⊢ (q → p) .Modus ponens (conditional elimination) From p and

(p→ q) , infer q.That is, p, p→ q ⊢ q .Conditional proof (conditional introduction) From

[accepting p allows a proof of q], infer (p→ q) .That is, (p ⊢ q) ⊢ (p→ q) .

1.1.7 Basic and derived argument forms

1.1.8 Proofs in propositional calculus

One of the main uses of a propositional calculus, wheninterpreted for logical applications, is to determine re-lations of logical equivalence between propositional for-mulas. These relationships are determined by means ofthe available transformation rules, sequences of which arecalled derivations or proofs.In the discussion to follow, a proof is presented as a se-quence of numbered lines, with each line consisting of asingle formula followed by a reason or justification for in-troducing that formula. Each premise of the argument,that is, an assumption introduced as an hypothesis of theargument, is listed at the beginning of the sequence and ismarked as a “premise” in lieu of other justification. Theconclusion is listed on the last line. A proof is complete ifevery line follows from the previous ones by the correctapplication of a transformation rule. (For a contrastingapproach, see proof-trees).

Example of a proof

• To be shown that A→ A.

• One possible proof of this (which, though valid, hap-pens to contain more steps than are necessary) maybe arranged as follows:

InterpretA ⊢ A as “Assuming A, infer A”. Read ⊢ A→A as “Assuming nothing, infer that A implies A”, or “Itis a tautology that A implies A”, or “It is always true thatA implies A”.

1.1.9 Soundness and completeness of therules

The crucial properties of this set of rules are that they aresound and complete. Informally this means that the rulesare correct and that no other rules are required. Theseclaims can be made more formal as follows.We define a truth assignment as a function that mapspropositional variables to true or false. Informally such atruth assignment can be understood as the description ofa possible state of affairs (or possible world) where cer-tain statements are true and others are not. The semanticsof formulas can then be formalized by defining for which“state of affairs” they are considered to be true, which iswhat is done by the following definition.We define when such a truth assignment A satisfies a cer-tain well-formed formula with the following rules:

• A satisfies the propositional variable P if and only ifA(P) = true

1.1. PROPOSITIONAL CALCULUS 7

• A satisfies ¬φ if and only if A does not satisfy φ

• A satisfies (φ ∧ ψ) if and only if A satisfies both φand ψ

• A satisfies (φ ∨ ψ) if and only if A satisfies at leastone of either φ or ψ

• A satisfies (φ → ψ) if and only if it is not the casethat A satisfies φ but not ψ

• A satisfies (φ↔ ψ) if and only if A satisfies both φand ψ or satisfies neither one of them

With this definition we can now formalize what it meansfor a formula φ to be implied by a certain set S of formu-las. Informally this is true if in all worlds that are possi-ble given the set of formulas S the formula φ also holds.This leads to the following formal definition: We say thata set S of well-formed formulas semantically entails (orimplies) a certain well-formed formula φ if all truth as-signments that satisfy all the formulas in S also satisfy φ.Finally we define syntactical entailment such that φ is syn-tactically entailed by S if and only if we can derive it withthe inference rules that were presented above in a finitenumber of steps. This allows us to formulate exactly whatit means for the set of inference rules to be sound andcomplete:Soundness: If the set of well-formed formulas S syntac-tically entails the well-formed formula φ then S semanti-cally entails φ.Completeness: If the set of well-formed formulas S se-mantically entails the well-formed formula φ then S syn-tactically entails φ.For the above set of rules this is indeed the case.

Sketch of a soundness proof

(For most logical systems, this is the comparatively “sim-ple” direction of proof)Notational conventions: Let G be a variable ranging oversets of sentences. Let A, B and C range over sentences.For “G syntactically entails A” we write “G proves A”.For “G semantically entails A” we write “G implies A”.We want to show: (A)(G) (if G proves A, then G impliesA).We note that “G provesA” has an inductive definition, andthat gives us the immediate resources for demonstratingclaims of the form “If G proves A, then ...”. So our proofproceeds by induction.

1. Basis. Show: If A is a member of G, then G impliesA.

2. Basis. Show: If A is an axiom, then G implies A.

3. Inductive step (induction on n, the length of theproof):

(a) Assume for arbitrary G and A that if G provesA in n or fewer steps, then G implies A.

(b) For each possible application of a rule of in-ference at step n + 1, leading to a new theoremB, show that G implies B.

Notice that Basis Step II can be omitted for natural deduc-tion systems because they have no axioms. When used,Step II involves showing that each of the axioms is a (se-mantic) logical truth.The Basis steps demonstrate that the simplest provablesentences from G are also implied by G, for any G. (Theproof is simple, since the semantic fact that a set impliesany of its members, is also trivial.) The Inductive step willsystematically cover all the further sentences that mightbe provable—by considering each case where we mightreach a logical conclusion using an inference rule—andshows that if a new sentence is provable, it is also logicallyimplied. (For example, we might have a rule telling usthat from “A” we can derive “A or B”. In III.aWe assumethat if A is provable it is implied. We also know thatif A is provable then “A or B” is provable. We have toshow that then “A or B” too is implied. We do so byappeal to the semantic definition and the assumption wejust made. A is provable from G, we assume. So it isalso implied by G. So any semantic valuation making allof G true makes A true. But any valuation making Atrue makes “A or B” true, by the defined semantics for“or”. So any valuation which makes all of G true makes“A or B” true. So “A or B” is implied.) Generally, theInductive step will consist of a lengthy but simple case-by-case analysis of all the rules of inference, showing thateach “preserves” semantic implication.By the definition of provability, there are no sentencesprovable other than by being a member of G, an axiom,or following by a rule; so if all of those are semanticallyimplied, the deduction calculus is sound.

Sketch of completeness proof

(This is usually the much harder direction of proof.)We adopt the same notational conventions as above.We want to show: If G implies A, then G proves A. Weproceed by contraposition: We show instead that if Gdoes not prove A then G does not imply A.

1. G does not prove A. (Assumption)

2. If G does not prove A, then we can construct an(infinite) Maximal Set, G∗, which is a superset ofG and which also does not prove A.

8 CHAPTER 1. INTRODUCTION

(a) Place an ordering on all the sentences in thelanguage (e.g., shortest first, and equally longones in extended alphabetical ordering), andnumber them (E1, E2, ...)

(b) Define a series G of sets (G0, G1, ...) induc-tively:

i. G0 = G

ii. If Gk ∪ Ek+1 proves A, then Gk+1 =Gk

iii. If Gk ∪ Ek+1 does not prove A, thenGk+1 = Gk ∪ Ek+1

(c) Define G∗ as the union of all the G . (That is,G∗ is the set of all the sentences that are in anyG .)

(d) It can be easily shown that

i. G∗ contains (is a superset of) G (by (b.i));ii. G∗ does not prove A (because if it proves

A then some sentence was added to someG which caused it to prove A; but thiswas ruled out by definition); and

iii. G∗ is a Maximal Set with respect to A: Ifany more sentences whatever were addedto G∗, it would prove A. (Because if itwere possible to add any more sentences,they should have been added when theywere encountered during the constructionof the G , again by definition)

3. If G∗ is a Maximal Set with respect to A, then it istruth-like. This means that it contains C only if itdoes not contain ¬C; If it contains C and contains“If C then B” then it also contains B; and so forth.

4. If G∗ is truth-like there is a G∗-Canonical valuationof the language: one that makes every sentence inG∗ true and everything outside G∗ false while stillobeying the laws of semantic composition in the lan-guage.

5. A G∗-canonical valuation will make our original setG all true, and make A false.

6. If there is a valuation on which G are true and A isfalse, then G does not (semantically) imply A.

QED

Another outline for a completeness proof

If a formula is a tautology, then there is a truth table forit which shows that each valuation yields the value truefor the formula. Consider such a valuation. By mathe-matical induction on the length of the subformulas, showthat the truth or falsity of the subformula follows from thetruth or falsity (as appropriate for the valuation) of each

propositional variable in the subformula. Then combinethe lines of the truth table together two at a time by using"(P is true implies S) implies ((P is false implies S) im-plies S)". Keep repeating this until all dependencies onpropositional variables have been eliminated. The resultis that we have proved the given tautology. Since everytautology is provable, the logic is complete.

1.1.10 Interpretation of a truth-functionalpropositional calculus

An interpretation of a truth-functional propositionalcalculus P is an assignment to each propositional sym-bol of P of one or the other (but not both) of the truthvalues truth (T) and falsity (F), and an assignment to theconnective symbols of P of their usual truth-functionalmeanings. An interpretation of a truth-functional propo-sitional calculus may also be expressed in terms of truthtables.[10]

For n distinct propositional symbols there are 2n distinctpossible interpretations. For any particular symbol a , forexample, there are 21 = 2 possible interpretations:

1. a is assigned T, or

2. a is assigned F.

For the pair a , b there are 22 = 4 possible interpretations:

1. both are assigned T,

2. both are assigned F,

3. a is assigned T and b is assigned F, or

4. a is assigned F and b is assigned T.[10]

Since P has ℵ0 , that is, denumerably many propositionalsymbols, there are 2ℵ0 = c , and therefore uncountablymany distinct possible interpretations of P .[10]

Interpretation of a sentence of truth-functionalpropositional logic

Main article: Interpretation (logic)

If φ and ψ are formulas of P and I is an interpretationof P then:

• A sentence of propositional logic is true under an in-terpretation I iff I assigns the truth value T to thatsentence. If a sentence is true under an interpreta-tion, then that interpretation is called amodel of thatsentence.

• φ is false under an interpretation I iff φ is not trueunder I .[10]

1.1. PROPOSITIONAL CALCULUS 9

• A sentence of propositional logic is logically valid ifit is true under every interpretation

|= ϕ means that φ is logically valid

• A sentence ψ of propositional logic is a semanticconsequence of a sentence φ iff there is no interpre-tation under which φ is true and ψ is false.

• A sentence of propositional logic is consistent iff itis true under at least one interpretation. It is incon-sistent if it is not consistent.

Some consequences of these definitions:

• For any given interpretation a given formula is eithertrue or false.[10]

• No formula is both true and false under the sameinterpretation.[10]

• φ is false for a given interpretation iff ¬ϕ is true forthat interpretation; and φ is true under an interpre-tation iff ¬ϕ is false under that interpretation.[10]

• If φ and (ϕ→ ψ) are both true under a given inter-pretation, then ψ is true under that interpretation.[10]

• If |=P ϕ and |=P (ϕ→ ψ) , then |=P ψ .[10]

• ¬ϕ is true under I iff φ is not true under I .

• (ϕ→ ψ) is true under I iff either φ is not true underI or ψ is true under I .[10]

• A sentence ψ of propositional logic is a semanticconsequence of a sentence φ iff (ϕ→ ψ) is logicallyvalid, that is, ϕ |=P ψ iff |=P (ϕ→ ψ) .[10]

1.1.11 Alternative calculus

It is possible to define another version of propositionalcalculus, which defines most of the syntax of the logicaloperators by means of axioms, and which uses only oneinference rule.

Axioms

Let φ, χ, and ψ stand for well-formed formulas. (Thewell-formed formulas themselves would not contain anyGreek letters, but only capital Roman letters, connectiveoperators, and parentheses.) Then the axioms are as fol-lows:

• Axiom THEN-2 may be considered to be a “dis-tributive property of implication with respect to im-plication.”

• Axioms AND-1 and AND-2 correspond to “con-junction elimination”. The relation between AND-1and AND-2 reflects the commutativity of the con-junction operator.

• Axiom AND-3 corresponds to “conjunction intro-duction.”

• Axioms OR-1 and OR-2 correspond to “disjunctionintroduction.” The relation between OR-1 and OR-2reflects the commutativity of the disjunction opera-tor.

• Axiom NOT-1 corresponds to “reductio ad absur-dum.”

• Axiom NOT-2 says that “anything can be deducedfrom a contradiction.”

• Axiom NOT-3 is called "tertium non datur" (Latin:“a third is not given”) and reflects the semantic val-uation of propositional formulas: a formula canhave a truth-value of either true or false. Thereis no third truth-value, at least not in classicallogic. Intuitionistic logicians do not accept the ax-iom NOT-3.

Inference rule

The inference rule is modus ponens:

ϕ, ϕ→ χ ⊢ χ

Meta-inference rule

Let a demonstration be represented by a sequence, withhypotheses to the left of the turnstile and the conclusionto the right of the turnstile. Then the deduction theoremcan be stated as follows:

If the sequence

ϕ1, ϕ2, ..., ϕn, χ ⊢ ψ

has been demonstrated, then it is also possible todemonstrate the sequence

ϕ1, ϕ2, ..., ϕn ⊢ χ→ ψ

This deduction theorem (DT) is not itself formulated withpropositional calculus: it is not a theorem of propositionalcalculus, but a theorem about propositional calculus. Inthis sense, it is a meta-theorem, comparable to theoremsabout the soundness or completeness of propositional cal-culus.On the other hand, DT is so useful for simplifying the syn-tactical proof process that it can be considered and used

10 CHAPTER 1. INTRODUCTION

as another inference rule, accompanying modus ponens.In this sense, DT corresponds to the natural conditionalproof inference rule which is part of the first version ofpropositional calculus introduced in this article.The converse of DT is also valid:

If the sequence

ϕ1, ϕ2, ..., ϕn ⊢ χ→ ψ

has been demonstrated, then it is also possible todemonstrate the sequence

ϕ1, ϕ2, ..., ϕn, χ ⊢ ψ

in fact, the validity of the converse of DT is almost trivialcompared to that of DT:

If

ϕ1, ..., ϕn ⊢ χ→ ψ

then

ϕ1, ..., ϕn, χ ⊢ χ→ ψ

ϕ1, ..., ϕn, χ ⊢ χ

and from (1) and (2) can be deduced

ϕ1, ..., ϕn, χ ⊢ ψ

by means of modus ponens, Q.E.D.

The converse of DT has powerful implications: it can beused to convert an axiom into an inference rule. For ex-ample, the axiom AND-1,

⊢ ϕ ∧ χ→ ϕ

can be transformed by means of the converse of the de-duction theorem into the inference rule

ϕ ∧ χ ⊢ ϕ

which is conjunction elimination, one of the ten inferencerules used in the first version (in this article) of the propo-sitional calculus.

Example of a proof

The following is an example of a (syntactical) demonstra-tion, involving only axioms THEN-1 and THEN-2:Prove: A→ A (Reflexivity of implication).Proof:

1. (A → ((B → A) → A)) → ((A → (B →A)) → (A→ A))

ϕ = A,χ = B → A,ψ = A

2. A→ ((B → A) → A)

ϕ = A,χ = B → A

3. (A→ (B → A)) → (A→ A)

From (1) and (2) by modus ponens.

4. A→ (B → A)

ϕ = A,χ = B

5. A→ A

From (3) and (4) by modus ponens.

1.1.12 Equivalence to equational logics

The preceding alternative calculus is an example of aHilbert-style deduction system. In the case of propo-sitional systems the axioms are terms built with logicalconnectives and the only inference rule is modus ponens.Equational logic as standardly used informally in highschool algebra is a different kind of calculus from Hilbertsystems. Its theorems are equations and its inference rulesexpress the properties of equality, namely that it is a con-gruence on terms that admits substitution.Classical propositional calculus as described above isequivalent to Boolean algebra, while intuitionistic propo-sitional calculus is equivalent to Heyting algebra. Theequivalence is shown by translation in each direction ofthe theorems of the respective systems. Theorems ϕ ofclassical or intuitionistic propositional calculus are trans-lated as equations ϕ = 1 of Boolean or Heyting al-gebra respectively. Conversely theorems x = y ofBoolean or Heyting algebra are translated as theorems(x → y) ∧ (y → x) of classical or intuitionistic cal-culus respectively, for which x ≡ y is a standard abbre-viation. In the case of Boolean algebra x = y can also betranslated as (x∧ y)∨ (¬x∧¬y) , but this translation isincorrect intuitionistically.In both Boolean and Heyting algebra, inequality x ≤ ycan be used in place of equality. The equality x = y isexpressible as a pair of inequalities x ≤ y and y ≤ x. Conversely the inequality x ≤ y is expressible as theequality x∧y = x , or as x∨y = y . The significance ofinequality for Hilbert-style systems is that it correspondsto the latter’s deduction or entailment symbol ⊢ . An en-tailment

ϕ1, ϕ2, . . . , ϕn ⊢ ψ

1.1. PROPOSITIONAL CALCULUS 11

is translated in the inequality version of the algebraicframework as

ϕ1 ∧ ϕ2 ∧ . . . ∧ ϕn ≤ ψ

Conversely the algebraic inequality x ≤ y is translated asthe entailment

x ⊢ y

The difference between implication x → y and inequal-ity or entailment x ≤ y or x ⊢ y is that the formeris internal to the logic while the latter is external. Inter-nal implication between two terms is another term of thesame kind. Entailment as external implication betweentwo terms expresses a metatruth outside the language ofthe logic, and is considered part of the metalanguage.Even when the logic under study is intuitionistic, entail-ment is ordinarily understood classically as two-valued:either the left side entails, or is less-or-equal to, the rightside, or it is not.Similar but more complex translations to and from alge-braic logics are possible for natural deduction systems asdescribed above and for the sequent calculus. The entail-ments of the latter can be interpreted as two-valued, but amore insightful interpretation is as a set, the elements ofwhich can be understood as abstract proofs organized asthe morphisms of a category. In this interpretation the cutrule of the sequent calculus corresponds to compositionin the category. Boolean and Heyting algebras enter thispicture as special categories having at most onemorphismper homset, i.e., one proof per entailment, correspondingto the idea that existence of proofs is all that matters: anyproof will do and there is no point in distinguishing them.

1.1.13 Graphical calculi

It is possible to generalize the definition of a formal lan-guage from a set of finite sequences over a finite basisto include many other sets of mathematical structures, solong as they are built up by finitary means from finite ma-terials. What’s more, many of these families of formalstructures are especially well-suited for use in logic.For example, there are many families of graphs thatare close enough analogues of formal languages that theconcept of a calculus is quite easily and naturally ex-tended to them. Indeed, many species of graphs arise asparse graphs in the syntactic analysis of the correspond-ing families of text structures. The exigencies of practi-cal computation on formal languages frequently demandthat text strings be converted into pointer structure ren-ditions of parse graphs, simply as a matter of checking

whether strings are well-formed formulas or not. Oncethis is done, there are many advantages to be gainedfrom developing the graphical analogue of the calculuson strings. The mapping from strings to parse graphsis called parsing and the inverse mapping from parsegraphs to strings is achieved by an operation that is calledtraversing the graph.

1.1.14 Other logical calculi

Propositional calculus is about the simplest kind of logi-cal calculus in current use. It can be extended in severalways. (Aristotelian “syllogistic” calculus, which is largelysupplanted in modern logic, is in some ways simpler – butin other ways more complex – than propositional calcu-lus.) Themost immediate way to develop amore complexlogical calculus is to introduce rules that are sensitive tomore fine-grained details of the sentences being used.First-order logic (a.k.a. first-order predicate logic) re-sults when the “atomic sentences” of propositional logicare broken up into terms, variables, predicates, andquantifiers, all keeping the rules of propositional logicwith some new ones introduced. (For example, from “Alldogs are mammals” we may infer “If Rover is a dog thenRover is a mammal”.) With the tools of first-order logic itis possible to formulate a number of theories, either withexplicit axioms or by rules of inference, that can them-selves be treated as logical calculi. Arithmetic is the bestknown of these; others include set theory and mereology.Second-order logic and other higher-order logics are for-mal extensions of first-order logic. Thus, it makes senseto refer to propositional logic as “zeroth-order logic”, whencomparing it with these logics.Modal logic also offers a variety of inferences that cannotbe captured in propositional calculus. For example, from“Necessarily p” wemay infer that p. From p wemay infer“It is possible that p”. The translation between modal log-ics and algebraic logics concerns classical and intuition-istic logics but with the introduction of a unary opera-tor on Boolean or Heyting algebras, different from theBoolean operations, interpreting the possibility modality,and in the case of Heyting algebra a second operator in-terpreting necessity (for Boolean algebra this is redundantsince necessity is the DeMorgan dual of possibility). Thefirst operator preserves 0 and disjunction while the sec-ond preserves 1 and conjunction.Many-valued logics are those allowing sentences to havevalues other than true and false. (For example, neitherand both are standard “extra values"; “continuum logic”allows each sentence to have any of an infinite numberof “degrees of truth” between true and false.) These log-ics often require calculational devices quite distinct frompropositional calculus. When the values form a Booleanalgebra (which may have more than two or even in-finitely many values), many-valued logic reduces to clas-sical logic; many-valued logics are therefore only of in-

12 CHAPTER 1. INTRODUCTION

dependent interest when the values form an algebra thatis not Boolean.

1.1.15 Solvers

Finding solutions to propositional logic formulas is anNP-complete problem. However, practical methods ex-ist (e.g., DPLL algorithm, 1962; Chaff algorithm, 2001)that are very fast for many useful cases. Recent work hasextended the SAT solver algorithms to work with propo-sitions containing arithmetic expressions; these are theSMT solvers.

1.1.16 See also

Higher logical levels

• First-order logic

• Second-order propositional logic

• Second-order logic

• Higher-order logic

Related topics

1.1.17 References[1] Ancient Logic (Stanford Encyclopedia of Philosophy)

[2] Marenbon, John (2007). Medieval philosophy: an histori-cal and philosophical introduction. Routledge. p. 137.

[3] Leibniz’s Influence on 19th Century Logic

[4] Hurley, Patrick (2007). A Concise Introduction to Logic10th edition. Wadsworth Publishing. p. 392.

[5] Beth, Evert W.; “Semantic entailment and formal deriv-ability”, series: Mededlingen van de Koninklijke Ned-erlandse Akademie van Wetenschappen, Afdeling Let-terkunde, Nieuwe Reeks, vol. 18, no. 13, Noord-Hollandsche Uitg. Mij., Amsterdam, 1955, pp. 309–42. Reprinted in Jaakko Intikka (ed.) The Philosophy ofMathematics, Oxford University Press, 1969

[6] Truth in Frege

[7] Russell’s Use of Truth-Tables

[8] Wernick, William (1942) “Complete Sets of LogicalFunctions,” Transactions of the American MathematicalSociety 51, pp. 117–132.

[9] Toida, Shunichi (2 August 2009). “Proof of Implica-tions”. CS381 Discrete Structures/Discrete MathematicsWeb Course Material. Department Of Computer Science,Old Dominion University. Retrieved 10 March 2010.

[10] Hunter, Geoffrey (1971). Metalogic: An Introduction tothe Metatheory of Standard First-Order Logic. Universityof California Pres. ISBN 0-520-02356-0.

1.1.18 Further reading

• Brown, Frank Markham (2003), Boolean Reason-ing: The Logic of Boolean Equations, 1st edition,Kluwer Academic Publishers, Norwell, MA. 2ndedition, Dover Publications, Mineola, NY.

• Chang, C.C. and Keisler, H.J. (1973), Model The-ory, North-Holland, Amsterdam, Netherlands.

• Kohavi, Zvi (1978), Switching and Finite AutomataTheory, 1st edition, McGraw–Hill, 1970. 2nd edi-tion, McGraw–Hill, 1978.

• Korfhage, Robert R. (1974),Discrete ComputationalStructures, Academic Press, New York, NY.

• Lambek, J. and Scott, P.J. (1986), Introduction toHigher Order Categorical Logic, Cambridge Univer-sity Press, Cambridge, UK.

• Mendelson, Elliot (1964), Introduction toMathemat-ical Logic, D. Van Nostrand Company.

Related works

• Hofstadter, Douglas (1979). Gödel, Escher, Bach:An Eternal Golden Braid. Basic Books. ISBN 978-0-465-02656-2.

1.1.19 External links

• Klement, Kevin C. (2006), “Propositional Logic”,in James Fieser and Bradley Dowden (eds.), InternetEncyclopedia of Philosophy, Eprint.

• Formal Predicate Calculus, contains a systematicformal development along the lines of Alternativecalculus

• forall x: an introduction to formal logic, by P.D.Magnus, covers formal semantics and proof theoryfor sentential logic.

• Category:Propositional Calculus on ProofWiki(GFDLed)

• An Outline of Propositional Logic

Chapter 2

Chapter I Abduction

2.1 Abductive reasoning

“Abductive” redirects here. For other uses, seeAbduction (disambiguation).

Abductive reasoning (also called abduction,[1] abduc-tive inference[2] or retroduction[3]) is a form of logicalinference which goes from an observation to a theorywhich accounts for the observation, ideally seeking to findthe simplest and most likely explanation. In abductivereasoning, unlike in deductive reasoning, the premises donot guarantee the conclusion. One can understand abduc-tive reasoning as “inference to the best explanation”.[4]

The fields of law,[5] computer science, and artificial in-telligence research[6] renewed interest in the subject ofabduction. Diagnostic expert systems frequently employabduction.

2.1.1 History

The American philosopher Charles Sanders Peirce(1839–1914) first introduced the term as “guessing”.[7]Peirce said that to abduce a hypothetical explanation afrom an observed circumstance b is to surmise that amaybe true because then b would be a matter of course.[8]Thus, to abduce a from b involves determining that a issufficient, but not necessary, for b .For example, suppose we observe that the lawn is wet. Ifit rained last night, then it would be unsurprising that thelawn is wet. Therefore, by abductive reasoning, the pos-sibility that it rained last night is reasonable (but note thatPeirce did not remain convinced that a single logical formcovers all abduction);[9]however, some other process mayhave also resulted in a wet lawn, e.g. dew or lawn sprin-klers. Moreover, abducing that it rained last night fromthe observation of a wet lawn can lead to false conclu-sion(s).Peirce argues that good abductive reasoning from P to Qinvolves not simply a determination that Q is sufficient forP, but also that Q is among the most economical explana-tions for P. Simplification and economy both call for that“leap” of abduction.[10]

2.1.2 Deduction, induction, and abduction

Main article: Logical reasoning

Deductive reasoning (deduction) allows deriving bfrom a only where b is a formal logical consequenceof a . In other words, deduction derives the con-sequences of the assumed. Given the truth of theassumptions, a valid deduction guarantees the truthof the conclusion. For example, given that 'Wikiscan be edited by anyone' ( a 1) and 'Wikipedia is awiki' ( a 2), it follows that 'Wikipedia can be editedby anyone' ( b ).

Inductive reasoning (induction) allows inferring bfrom a , where b does not follow necessarily froma . a might give us very good reason to accept b ,but it does not ensure b . For example, if all swansthat we have observed so far are white, we mayinduce that the possibility that all swans are whiteis reasonable. We have good reason to believe theconclusion from the premise, but the truth of theconclusion is not guaranteed. (Indeed, it turns outthat some swans are black.)

Abductive reasoning (abduction) allows inferring aas an explanation of b . Because of this inference,abduction allows the precondition a to be abducedfrom the consequence b . Deductive reasoning andabductive reasoning thus differ in the direction inwhich a rule like " a entails b " is used for inference.As such, abduction is formally equivalent to the log-ical fallacy of affirming the consequent (or Post hocergo propter hoc) because of multiple possible ex-planations for b . For example, in a billiard game,after glancing and seeing the eight ball moving to-wards us, we may abduce that the cue ball struckthe eight ball. The strike of the cue ball would ac-count for the movement of the eight ball. It servesas a hypothesis that explains our observation. Giventhe many possible explanations for the movement ofthe eight ball, our abduction does not leave us cer-tain that the cue ball in fact struck the eight ball, butour abduction, still useful, can serve to orient us in

13

14 CHAPTER 2. CHAPTER I ABDUCTION

our surroundings. Despite many possible explana-tions for any physical process that we observe, wetend to abduce a single explanation (or a few ex-planations) for this process in the expectation thatwe can better orient ourselves in our surroundingsand disregard some possibilities. Properly used, ab-ductive reasoning can be a useful source of priors inBayesian statistics.

2.1.3 Formalizations of abduction

Logic-based abduction

In logic, explanation is done from a logical theory T rep-resenting a domain and a set of observations O . Abduc-tion is the process of deriving a set of explanations of Oaccording to T and picking out one of those explanations.ForE to be an explanation ofO according to T , it shouldsatisfy two conditions:

• O follows from E and T ;

• E is consistent with T .

In formal logic, O and E are assumed to be sets of liter-als. The two conditions for E being an explanation of Oaccording to theory T are formalized as:

T ∪ E |= O

T ∪ E

Among the possible explanations E satisfying these twoconditions, some other condition of minimality is usuallyimposed to avoid irrelevant facts (not contributing to theentailment ofO ) being included in the explanations. Ab-duction is then the process that picks out somemember ofE . Criteria for picking out a member representing “thebest” explanation include the simplicity, the prior proba-bility, or the explanatory power of the explanation.A proof theoretical abduction method for first order clas-sical logic based on the sequent calculus and a dual one,based on semantic tableaux (analytic tableaux) have beenproposed (Cialdea Mayer & Pirri 1993). The methodsare sound and complete and work for full first order logic,without requiring any preliminary reduction of formulaeinto normal forms. These methods have also been ex-tended to modal logic.Abductive logic programming is a computational frame-work that extends normal logic programming with abduc-tion. It separates the theory T into two components, oneof which is a normal logic program, used to generateE bymeans of backward reasoning, the other of which is a setof integrity constraints, used to filter the set of candidateexplanations.

Set-cover abduction

A different formalization of abduction is based on invert-ing the function that calculates the visible effects of thehypotheses. Formally, we are given a set of hypothesesH and a set of manifestationsM ; they are related by thedomain knowledge, represented by a function e that takesas an argument a set of hypotheses and gives as a resultthe corresponding set of manifestations. In other words,for every subset of the hypothesesH ′ ⊆ H , their effectsare known to be e(H ′) .Abduction is performed by finding a set H ′ ⊆ H suchthatM ⊆ e(H ′) . In other words, abduction is performedby finding a set of hypotheses H ′ such that their effectse(H ′) include all observationsM .A common assumption is that the effects of the hypothe-ses are independent, that is, for every H ′ ⊆ H , it holdsthat e(H ′) =

∪h∈H′ e(h) . If this condition is met,

abduction can be seen as a form of set covering.

Abductive validation

Abductive validation is the process of validating a givenhypothesis through abductive reasoning. This can also becalled reasoning through successive approximation. Un-der this principle, an explanation is valid if it is the bestpossible explanation of a set of known data. The best pos-sible explanation is often defined in terms of simplicityand elegance (see Occam’s razor). Abductive validationis common practice in hypothesis formation in science;moreover, Peirce claims that it is a ubiquitous aspect ofthought:

Looking out my window this lovely springmorning, I see an azalea in full bloom. No,no! I don't see that; though that is the only wayI can describe what I see. That is a proposi-tion, a sentence, a fact; but what I perceive isnot proposition, sentence, fact, but only an im-age, which I make intelligible in part by meansof a statement of fact. This statement is ab-stract; but what I see is concrete. I performan abduction when I so much as express in asentence anything I see. The truth is that thewhole fabric of our knowledge is one mattedfelt of pure hypothesis confirmed and refinedby induction. Not the smallest advance can bemade in knowledge beyond the stage of vacantstaring, without making an abduction at everystep.[11]

It was Peirce’s own maxim that “Facts cannot be ex-plained by a hypothesis more extraordinary than thesefacts themselves; and of various hypotheses the least ex-traordinary must be adopted.”[12] After obtaining resultsfrom an inference procedure, we may be left with mul-tiple assumptions, some of which may be contradictory.

2.1. ABDUCTIVE REASONING 15

Abductive validation is a method for identifying the as-sumptions that will lead to your goal.

Probabilistic abduction

Probabilistic abductive reasoning is a form of abductivevalidation, and is used extensively in areas where conclu-sions about possible hypotheses need to be derived, suchas for making diagnoses from medical tests. For exam-ple, a pharmaceutical company that develops a test for aparticular infectious disease will typically determine thereliability of the test by hiring a group of infected and agroup of non-infected people to undergo the test. Assumethe statements x : “Positive test”, x : “Negative test”, y :“Infected”, and y : “Not infected”. The result of these tri-als will then determine the reliability of the test in termsof its sensitivity p(x|y) and false positive rate p(x|y) .The interpretations of the conditionals are: p(x|y) : “Theprobability of positive test given infection”, and p(x|y) :“The probability of positive test in the absence of infec-tion”. The problem with applying these conditionals in apractical setting is that they are expressed in the oppositedirection to what the practitioner needs. The condition-als needed for making the diagnosis are: p(y|x) : “Theprobability of infection given positive test”, and p(y|x): “The probability of infection given negative test”. Theprobability of infection could then have been condition-ally deduced as p(y∥x) = p(x)p(y|x) + p(x)p(y|x) ,where " ∥ " denotes conditional deduction. Unfortunatelythe required conditionals are usually not directly availableto the medical practitioner, but they can be obtained if thebase rate of the infection in the population is known.The required conditionals can be correctly derived byinverting the available conditionals using Bayes rule.The inverted conditionals are obtained as follows:p(x|y) = p(x∧y)

p(y)

p(y|x) = p(x∧y)p(x)

⇒ p(y|x) = p(y)p(x|y)p(x) .

The term p(y) on the right hand side of the equation ex-presses the base rate of the infection in the population.Similarly, the term p(x) expresses the default likelihoodof positive test on a random person in the population. Inthe expressions below a(y) and a(y) = 1− a(y) denotethe base rates of y and its complement y respectively, sothat e.g. p(x) = a(y)p(x|y) + a(y)p(x|y) . The full ex-pression for the required conditionals p(y|x) and p(y|x)are thenp(y|x) = a(y)p(x|y)

a(y)p(x|y)+a(y)p(x|y)p(y|x) = a(y)p(x|y)

a(y)p(x|y)+a(y)p(x|y)

The full expression for the conditionally abduced proba-bility of infection in a tested person, expressed as p(y∥x), given the outcome of the test, the base rate of the infec-tion, as well as the test’s sensitivity and false positive rate,is then given by

p(y∥x) = p(x)(

a(y)p(x|y)a(y)p(x|y)+a(y)p(x|y)

)+

p(x)(

a(y)p(x|y)a(y)p(x|y)+a(y)p(x|y)

).

This further simplifies top(y∥x) = a(y) (p(x|y) + p(x|y)) .Probabilistic abduction can thus be described as a methodfor inverting conditionals in order to apply probabilisticdeduction.A medical test result is typically considered positive ornegative, so when applying the above equation it can beassumed that either p(x) = 1 (positive) or p(x) = 1(negative). In case the patient tests positive, the aboveequation can be simplified to p(y∥x) = p(y|x) whichwill give the correct likelihood that the patient actually isinfected.The Base rate fallacy in medicine,[13] or the Prosecutor’sfallacy[14] in legal reasoning, consists of making the er-roneous assumption that p(y|x) = p(x|y) . While thisreasoning error often can produce a relatively good ap-proximation of the correct hypothesis probability value,it can lead to a completely wrong result and wrong con-clusion in case the base rate is very low and the reliabilityof the test is not perfect. An extreme example of the baserate fallacy is to conclude that a male person is pregnantjust because he tests positive in a pregnancy test. Obvi-ously, the base rate of male pregnancy is zero, and as-suming that the test is not perfect, it would be correct toconclude that the male person is not pregnant.The expression for probabilistic abduction can be gener-alised to multinomial cases,[15] i.e., with a state space Xof multiple xi and a state space Y of multiple states yj .

Subjective logic abduction

Subjective logic generalises probabilistic logic by includ-ing parameters for uncertainty in the input arguments.Abduction in subjective logic is thus similar to probabilis-tic abduction described above.[15] The input arguments insubjective logic are composite functions called subjectiveopinions which can be binomial when the opinion appliesto a single proposition or multinomial when it applies to aset of propositions. Amultinomial opinion thus applies toa frame X (i.e. a state space of exhaustive and mutuallydisjoint propositions xi ), and is denoted by the compos-ite function ωX = (b, u, a) , where b is a vector of beliefmasses over the propositions of X , u is the uncertaintymass, and a is a vector of base rate values over the propo-sitions ofX . These components satisfy u+

∑b(xi) = 1

and∑a(xi) = 1 as well as b(xi), u, a(xi) ∈ [0, 1] .

Assume the frames X and Y , the sets of conditionalopinions ωX|Y and ωX|Y , the opinion ωX on X , andthe base rate function aY on Y . Based on these pa-rameters, subjective logic provides a method for derivingthe set of inverted conditionals ωY |X and ωY |X . Usingthese inverted conditionals, subjective logic also providesa method for deduction. Abduction in subjective logic

16 CHAPTER 2. CHAPTER I ABDUCTION

consists of inverting the conditionals and then applyingdeduction.The symbolic notation for conditional abduction is " ∥", and the operator itself is denoted as ⊚ . The expres-sion for subjective logic abduction is then:[15] ω

Y ∥X =

ωX ⊚ (ωX|Y , ωX|Y , aY ) .The advantage of using subjective logic abduction com-pared to probabilistic abduction is that uncertainty aboutthe probability values of the input arguments can be ex-plicitly expressed and taken into account during the analy-sis. It is thus possible to perform abductive analysis in thepresence of missing or incomplete input evidence, whichnormally results in degrees of uncertainty in the outputconclusions.

2.1.4 History

The philosopher Charles Sanders Peirce (/ˈpɜːrs/; 1839–1914) introduced abduction into modern logic. Over theyears he called such inference hypothesis, abduction, pre-sumption, and retroduction. He considered it a topic inlogic as a normative field in philosophy, not in purely for-mal or mathematical logic, and eventually as a topic alsoin economics of research.As two stages of the development, extension, etc., ofa hypothesis in scientific inquiry, abduction and alsoinduction are often collapsed into one overarching con-cept — the hypothesis. That is why, in the scientificmethod pioneered by Galileo and Bacon, the abductivestage of hypothesis formation is conceptualized simplyas induction. Thus, in the twentieth century this col-lapse was reinforced by Karl Popper's explication of thehypothetico-deductive model, where the hypothesis isconsidered to be just “a guess”[16] (in the spirit of Peirce).However, when the formation of a hypothesis is consid-ered the result of a process it becomes clear that this“guess” has already been tried and made more robust inthought as a necessary stage of its acquiring the statusof hypothesis. Indeed, many abductions are rejected orheavily modified by subsequent abductions before theyever reach this stage.Before 1900, Peirce treated abduction as the use of aknown rule to explain an observation, e.g., it is a knownrule that if it rains the grass is wet; so, to explain the factthat the grass is wet; one infers that it has rained. Thisremains the common use of the term “abduction” in thesocial sciences and in artificial intelligence.Peirce consistently characterized it as the kind of infer-ence that originates a hypothesis by concluding in an ex-planation, though an unassured one, for some very cu-rious or surprising (anomalous) observation stated in apremise. As early as 1865 he wrote that all conceptionsof cause and force are reached through hypothetical infer-ence; in the 1900s he wrote that all explanatory content oftheories is reached through abduction. In other respects

Peirce revised his view of abduction over the years.[17]

In later years his view came to be:

• Abduction is guessing.[7] It is “very little hampered”by rules of logic.[8] Even a well-prepared mind’sindividual guesses are more frequently wrong thanright.[18] But the success of our guesses far exceedsthat of random luck and seems born of attunementto nature by instinct[19] (some speak of intuition insuch contexts[20]).

• Abduction guesses a new or outside idea so as to ac-count in a plausible, instinctive, economical way fora surprising or very complicated phenomenon. Thatis its proximate aim.[19]

• Its longer aim is to economize inquiry itself. Its ra-tionale is inductive: it works often enough, is theonly source of new ideas, and has no substitute inexpediting the discovery of new truths.[21] Its ratio-nale especially involves its role in coordination withother modes of inference in inquiry. It is inferenceto explanatory hypotheses for selection of those bestworth trying.

• Pragmatism is the logic of abduction. Upon the gen-eration of an explanation (which he came to regardas instinctively guided), the pragmatic maxim givesthe necessary and sufficient logical rule to abductionin general. The hypothesis, being insecure, needs tohave conceivable[22] implications for informed prac-tice, so as to be testable[23][24] and, through its trials,to expedite and economize inquiry. The economyof research is what calls for abduction and governsits art.[10]

Writing in 1910, Peirce admits that “in almost everythingI printed before the beginning of this century I more orless mixed up hypothesis and induction” and he tracesthe confusion of these two types of reasoning to logi-cians’ too “narrow and formalistic a conception of infer-ence, as necessarily having formulated judgments fromits premises.”[25]

He started out in the 1860s treating hypothetical inferencein a number of ways which he eventually peeled away asinessential or, in some cases, mistaken:

• as inferring the occurrence of a character (a charac-teristic) from the observed combined occurrence ofmultiple characters which its occurrence would nec-essarily involve;[26] for example, if any occurrenceof A is known to necessitate occurrence of B, C, D,E, then the observation of B, C, D, E suggests by wayof explanation the occurrence of A. (But by 1878 heno longer regarded such multiplicity as common toall hypothetical inference.[27])

• as aiming for a more or less probable hypothesis(in 1867 and 1883 but not in 1878; anyway by

2.1. ABDUCTIVE REASONING 17

1900 the justification is not probability but the lackof alternatives to guessing and the fact that guess-ing is fruitful;[28] by 1903 he speaks of the “likely”in the sense of nearing the truth in an “indefinitesense";[29] by 1908 he discusses plausibility as in-stinctive appeal.[19]) In a paper dated by editorsas circa 1901, he discusses “instinct” and “natural-ness”, along with the kind of considerations (lowcost of testing, logical caution, breadth, and incom-plexity) that he later calls methodeutical.[30]

• as induction from characters (but as early as 1900 hecharacterized abduction as guessing[28])

• as citing a known rule in a premise rather than hy-pothesizing a rule in the conclusion (but by 1903 heallowed either approach[8][31])

• as basically a transformation of a deductive categor-ical syllogism[27] (but in 1903 he offered a variationon modus ponens instead,[8] and by 1911 he was un-convinced that any one form covers all hypotheticalinference[9]).

1867

In 1867, in “The Natural Classification ofArguments”,[26] hypothetical inference always dealswith a cluster of characters (call them P′, P′′, P′′′, etc.)known to occur at least whenever a certain character (M)occurs. Note that categorical syllogisms have elementstraditionally called middles, predicates, and subjects.For example: All men [middle] are mortal [predicate];Socrates [subject] is a man [middle]; ergo Socrates[subject] is mortal [predicate]". Below, 'M' stands for amiddle; 'P' for a predicate; 'S' for a subject. Note alsothat Peirce held that all deduction can be put into theform of the categorical syllogism Barbara (AAA-1).

1878

In 1878, in “Deduction, Induction, and Hypothesis”,[27]there is no longer a need for multiple characters or pred-icates in order for an inference to be hypothetical, al-though it is still helpful. Moreover, Peirce no longer poseshypothetical inference as concluding in a probable hy-pothesis. In the forms themselves, it is understood butnot explicit that induction involves random selection andthat hypothetical inference involves response to a “verycurious circumstance”. The forms instead emphasize themodes of inference as rearrangements of one another’spropositions (without the bracketed hints shown below).

1883

Peirce long treated abduction in terms of induction fromcharacters or traits (weighed, not counted like objects),explicitly so in his influential 1883 “A Theory of ProbableInference”, in which he returns to involving probability inthe hypothetical conclusion.[32] Like “Deduction, Induc-tion, and Hypothesis” in 1878, it was widely read (seethe historical books on statistics by Stephen Stigler), un-like his later amendments of his conception of abduction.Today abduction remains most commonly understood asinduction from characters and extension of a known ruleto cover unexplained circumstances.SherlockHolmes uses this method of reasoning in the sto-ries of Arthur Conan Doyle, although Holmes refers to itas deductive reasoning.

1902 and after

In 1902 Peirce wrote that he now regarded the syllogisti-cal forms and the doctrine of extension and comprehen-sion (i.e., objects and characters as referenced by terms),as being less fundamental than he had earlier thought.[33]In 1903 he offered the following form for abduction:[8]

The surprising fact, C, is observed;But if A were true, C would be amatter of course,Hence, there is reason to suspectthat A is true.

The hypothesis is framed, but not asserted, in a premise,then asserted as rationally suspectable in the conclusion.Thus, as in the earlier categorical syllogistic form, theconclusion is formulated from some premise(s). But allthe same the hypothesis consists more clearly than everin a new or outside idea beyond what is known or ob-served. Induction in a sense goes beyond observationsalready reported in the premises, but it merely amplifiesideas already known to represent occurrences, or tests anidea supplied by hypothesis; either way it requires previ-ous abductions in order to get such ideas in the first place.Induction seeks facts to test a hypothesis; abduction seeksa hypothesis to account for facts.Note that the hypothesis (“A”) could be of a rule. It neednot even be a rule strictly necessitating the surprising ob-servation (“C”), which needs to follow only as a “matterof course"; or the “course” itself could amount to someknown rule, merely alluded to, and also not necessarilya rule of strict necessity. In the same year, Peirce wrotethat reaching a hypothesis may involve placing a surpris-ing observation under either a newly hypothesized rule ora hypothesized combination of a known rule with a pecu-liar state of facts, so that the phenomenon would be notsurprising but instead either necessarily implied or at leastlikely.[31]

18 CHAPTER 2. CHAPTER I ABDUCTION

Peirce did not remain quite convinced about any suchform as the categorical syllogistic form or the 1903 form.In 1911, he wrote, “I do not, at present, feel quite con-vinced that any logical form can be assigned that willcover all 'Retroductions’. For what I mean by a Retroduc-tion is simply a conjecture which arises in the mind.”[9]

Pragmatism

In 1901 Peirce wrote, “There would be no logic in impos-ing rules, and saying that they ought to be followed, un-til it is made out that the purpose of hypothesis requiresthem.”[34] In 1903 Peirce called pragmatism “the logicof abduction” and said that the pragmatic maxim givesthe necessary and sufficient logical rule to abduction ingeneral.[24] The pragmatic maxim is: “Consider what ef-fects, that might conceivably have practical bearings, weconceive the object of our conception to have. Then, ourconception of these effects is the whole of our conceptionof the object.” It is a method for fruitful clarification ofconceptions by equating themeaning of a conception withthe conceivable practical implications of its object’s con-ceived effects. Peirce held that that is precisely tailoredto abduction’s purpose in inquiry, the forming of an ideathat could conceivably shape informed conduct. In vari-ous writings in the 1900s[10][35] he said that the conduct ofabduction (or retroduction) is governed by considerationsof economy, belonging in particular to the economics ofresearch. He regarded economics as a normative sciencewhose analytic portionmight be part of logical methodeu-tic (that is, theory of inquiry).[36]

Three levels of logic about abduction

Peirce came over the years to divide (philosophical) logicinto three departments:

1. Stechiology, or speculative grammar, on the con-ditions for meaningfulness. Classification of signs(semblances, symptoms, symbols, etc.) andtheir combinations (as well as their objects andinterpretants).

2. Logical critic, or logic proper, on validity or justifia-bility of inference, the conditions for true represen-tation. Critique of arguments in their various modes(deduction, induction, abduction).

3. Methodeutic, or speculative rhetoric, on the condi-tions for determination of interpretations. Method-ology of inquiry in its interplay of modes.

Peirce had, from the start, seen the modes of inferenceas being coordinated together in scientific inquiry and,by the 1900s, held that hypothetical inference in par-ticular is inadequately treated at the level of critique ofarguments.[23][24] To increase the assurance of a hypo-thetical conclusion, one needs to deduce implications

about evidence to be found, predictions which inductioncan test through observation so as to evaluate the hypoth-esis. That is Peirce’s outline of the scientific method ofinquiry, as covered in his inquiry methodology, which in-cludes pragmatism or, as he later called it, pragmaticism,the clarification of ideas in terms of their conceivable im-plications regarding informed practice.

Classification of signs As early as 1866,[37] Peirceheld that:1. Hypothesis (abductive inference) is inference throughan icon (also called a likeness).2. Induction is inference through an index (a sign by fac-tual connection); a sample is an index of the totality fromwhich it is drawn.3. Deduction is inference through a symbol (a sign by in-terpretive habit irrespective of resemblance or connectionto its object).In 1902, Peirce wrote that, in abduction: “It is recognizedthat the phenomena are like, i.e. constitute an Icon of, areplica of a general conception, or Symbol.”[38]

Critique of arguments At the critical level Peirce ex-amined the forms of abductive arguments (as discussedabove), and came to hold that the hypothesis should econ-omize explanation for plausibility in terms of the feasibleand natural. In 1908 Peirce described this plausibility insome detail.[19] It involves not likeliness based on obser-vations (which is instead the inductive evaluation of a hy-pothesis), but instead optimal simplicity in the sense ofthe “facile and natural”, as by Galileo’s natural light ofreason and as distinct from “logical simplicity” (Peircedoes not dismiss logical simplicity entirely but sees it in asubordinate role; taken to its logical extreme it would fa-vor adding no explanation to the observation at all). Evena well-prepared mind guesses oftener wrong than right,but our guesses succeed better than random luck at reach-ing the truth or at least advancing the inquiry, and that in-dicates to Peirce that they are based in instinctive attune-ment to nature, an affinity between the mind’s processesand the processes of the real, which would account forwhy appealingly “natural” guesses are the ones that of-tenest (or least seldom) succeed; to which Peirce addedthe argument that such guesses are to be preferred since,without “a natural bent like nature’s”, people would haveno hope of understanding nature. In 1910 Peirce made athree-way distinction between probability, verisimilitude,and plausibility, and defined plausibility with a normative“ought": “By plausibility, I mean the degree to which atheory ought to recommend itself to our belief indepen-dently of any kind of evidence other than our instinct urg-ing us to regard it favorably.”[39] For Peirce, plausibilitydoes not depend on observed frequencies or probabilities,or on verisimilitude, or even on testability, which is nota question of the critique of the hypothetical inferenceas an inference, but rather a question of the hypothesis’s

2.1. ABDUCTIVE REASONING 19

relation to the inquiry process.The phrase “inference to the best explanation” (not usedby Peirce but often applied to hypothetical inference) isnot always understood as referring to the most simpleand natural. However, in other senses of “best”, such as“standing up best to tests”, it is hard to know which is thebest explanation to form, since one has not tested it yet.Still, for Peirce, any justification of an abductive infer-ence as good is not completed upon its formation as anargument (unlike with induction and deduction) and in-stead depends also on its methodological role and promise(such as its testability) in advancing inquiry.[23][24][40]

Methodology of inquiry At the methodeutical levelPeirce held that a hypothesis is judged and selected[23] fortesting because it offers, via its trial, to expedite and econ-omize the inquiry process itself toward new truths, first ofall by being testable and also by further economies,[10] interms of cost, value, and relationships among guesses (hy-potheses). Here, considerations such as probability, ab-sent from the treatment of abduction at the critical level,come into play. For examples:

• Cost: A simple but low-odds guess, if low in costto test for falsity, may belong first in line for testing,to get it out of the way. If surprisingly it stands upto tests, that is worth knowing early in the inquiry,which otherwise might have stayed long on a wrongthough seemingly likelier track.

• Value: A guess is intrinsically worth testing if it hasinstinctual plausibility or reasoned objective proba-bility, while subjective likelihood, though reasoned,can be treacherous.

• Interrelationships: Guesses can be chosen for trialstrategically for their

• caution, for which Peirce gave as example thegame of Twenty Questions,

• breadth of applicability to explain various phe-nomena, and

• incomplexity, that of a hypothesis that seemstoo simple but whose trial “may give a good'leave,' as the billiard-players say”, and be in-structive for the pursuit of various and conflict-ing hypotheses that are less simple.[41]

Other writers

Norwood Russell Hanson, a philosopher of science,wanted to grasp a logic explaining how scientific discov-eries take place. He used Peirce’s notion of abduction forthis.[42]

Further development of the concept can be found in PeterLipton's Inference to the Best Explanation (Lipton, 1991).

2.1.5 Applications

Applications in artificial intelligence include fault diag-nosis, belief revision, and automated planning. The mostdirect application of abduction is that of automaticallydetecting faults in systems: given a theory relating faultswith their effects and a set of observed effects, abductioncan be used to derive sets of faults that are likely to be thecause of the problem.In medicine, abduction can be seen as a component ofclinical evaluation and judgment.[43][44]

Abduction can also be used to model automated plan-ning.[45] Given a logical theory relating action occur-rences with their effects (for example, a formula of theevent calculus), the problem of finding a plan for reach-ing a state can be modeled as the problem of abducting aset of literals implying that the final state is the goal state.In intelligence analysis, Analysis of Competing Hypothe-ses and Bayesian networks, probabilistic abductive rea-soning is used extensively. Similarly in medical diagno-sis and legal reasoning, the same methods are being used,although there have been many examples of errors, espe-cially caused by the base rate fallacy and the prosecutor’sfallacy.Belief revision, the process of adapting beliefs in viewof new information, is another field in which abductionhas been applied. The main problem of belief revisionis that the new information may be inconsistent with thecorpus of beliefs, while the result of the incorporationcannot be inconsistent. This process can be done by theuse of abduction: once an explanation for the observationhas been found, integrating it does not generate inconsis-tency. This use of abduction is not straightforward, asadding propositional formulae to other propositional for-mulae can only make inconsistencies worse. Instead, ab-duction is done at the level of the ordering of preferenceof the possible worlds. Preference models use fuzzy logicor utility models.In the philosophy of science, abduction has been thekey inference method to support scientific realism, andmuch of the debate about scientific realism is focused onwhether abduction is an acceptable method of inference.In historical linguistics, abduction during language acqui-sition is often taken to be an essential part of processesof language change such as reanalysis and analogy.[46]

In anthropology, Alfred Gell in his influential book Artand Agency defined abduction (after Eco[47]) as “a caseof synthetic inference 'where we find some very curiouscircumstances, which would be explained by the suppo-sition that it was a case of some general rule, and there-upon adopt that supposition”.[48] Gell criticizes existing'anthropological' studies of art, for being too preoccupiedwith aesthetic value and not preoccupied enough with thecentral anthropological concern of uncovering 'social re-lationships,' specifically the social contexts in which art-

20 CHAPTER 2. CHAPTER I ABDUCTION

works are produced, circulated, and received.[49] Abduc-tion is used as the mechanism for getting from art toagency. That is, abduction can explain how works ofart inspire a sensus communis: the commonly-held viewsshared by members that characterize a given society.[50]The question Gell asks in the book is, 'how does it ini-tially 'speak' to people?' He answers by saying that “Noreasonable person could suppose that art-like relations be-tween people and things do not involve at least some formof semiosis.”[48] However, he rejects any intimation thatsemiosis can be thought of as a language because then hewould have to admit to some pre-established existence ofthe sensus communis that he wants to claim only emergesafterwards out of art. Abduction is the answer to thisconundrum because the tentative nature of the abduc-tion concept (Peirce likened it to guessing) means thatnot only can it operate outside of any pre-existing frame-work, but moreover, it can actually intimate the existenceof a framework. As Gell reasons in his analysis, the phys-ical existence of the artwork prompts the viewer to per-form an abduction that imbues the artwork with inten-tionality. A statue of a goddess, for example, in somesenses actually becomes the goddess in the mind of thebeholder; and represents not only the form of the deitybut also her intentions (which are adduced from the feel-ing of her very presence). Therefore, through abduction,Gell claims that art can have the kind of agency that plantsthe seeds that grow into cultural myths. The power ofagency is the power to motivate actions and inspire ul-timately the shared understanding that characterizes anygiven society.[50]

2.1.6 See also

• Abductive logic programming

• Analogy

• Analysis of Competing Hypotheses

• Charles Sanders Peirce

• Charles Sanders Peirce bibliography

• Deductive reasoning

• Defeasible reasoning

• Doug Walton

• Gregory Bateson

• Inductive inference

• Inductive probability

• Inductive reasoning

• Inquiry

• List of thinking-related topics

• Practopoiesis

• Logic

• Subjective logic

• Logical reasoning

• Maximum likelihood

• Scientific method

• Sherlock Holmes

• Sign relation

2.1.7 References

• This article is based on material taken from the FreeOn-line Dictionary of Computing prior to 1 Novem-ber 2008 and incorporated under the “relicensing”terms of the GFDL, version 1.3 or later.

• Awbrey, Jon, and Awbrey, Susan (1995), “Inter-pretation as Action: The Risk of Inquiry”, Inquiry:Critical Thinking Across the Disciplines, 15, 40-52.Eprint

• Cialdea Mayer, Marta and Pirri, Fiora (1993)“First order abduction via tableau and sequentcalculi” Logic Jnl IGPL 1993 1: 99-117;doi:10.1093/jigpal/1.1.99. Oxford Journals

• Cialdea Mayer, Marta and Pirri, Fiora(1995) “Propositional Abduction in ModalLogic”, Logic Jnl IGPL 1995 3: 907-919;doi:10.1093/jigpal/3.6.907 Oxford Journals

• Edwards, Paul (1967, eds.), “The Encyclopedia ofPhilosophy,” Macmillan Publishing Co, Inc. & TheFree Press, New York. Collier Macmillan Publish-ers, London.

• Eiter, T., and Gottlob, G. (1995), “The Complex-ity of Logic-Based Abduction, Journal of the ACM,42.1, 3-42.

• Hanson, N. R. (1958). Patterns of Discovery: AnInquiry into the Conceptual Foundations of Science,Cambridge: Cambridge University Press. ISBN978-0-521-09261-6.

• Harman, Gilbert (1965). “The Inference to the BestExplanation”. The Philosophical Review 74 (1): 88–95. doi:10.2307/2183532.

• Josephson, John R., and Josephson, Susan G. (1995,eds.), Abductive Inference: Computation, Philoso-phy, Technology, Cambridge University Press, Cam-bridge, UK.

• Lipton, Peter. (2001). Inference to the Best Expla-nation, London: Routledge. ISBN 0-415-24202-9.

2.1. ABDUCTIVE REASONING 21

• McKaughan, Daniel J. (2008), “From Ugly Duck-ling to Swan: C. S. Peirce, Abduction, and thePursuit of Scientific Theories”, Transactions of theCharles S. Peirce Society, v. 44, no. 3 (summer),446–468. Abstract.

• Menzies, T (1996). “Applications of Abduction:Knowledge-Level Modeling” (PDF). InternationalJournal of Human-Computer Studies 45 (3): 305–335. doi:10.1006/ijhc.1996.0054.

• Queiroz, Joao&Merrell, Floyd (guest eds.). (2005).“Abduction - between subjectivity and objectivity”.(special issue on abductive inference) Semiotica 153(1/4). .

• Santaella, Lucia (1997) “The Development ofPeirce’s Three Types of Reasoning: Abduction, De-duction, and Induction”, 6th Congress of the IASS.Eprint.

• Sebeok, T. (1981) “You Know My Method”. In Se-beok, T. “The Play of Musement”. Indiana. Bloom-ington, IA.

• Yu, Chong Ho (1994), “Is There a Logic of Ex-ploratory Data Analysis?", Annual Meeting of Amer-ican Educational Research Association, New Or-leans, LA, April, 1994. Website of Dr. Chong Ho(Alex) Yu

2.1.8 Notes

[1] • Magnani, L. “Abduction, Reason, and Science:Processes of Discovery and Explanation”. KluwerAcademic Plenum Publishers, New York, 2001. xvii.205 pages. Hard cover, ISBN 0-306-46514-0.

• R. Josephson, J. & G. Josephson, S. “AbductiveInference: Computation, Philosophy, Technology”Cambridge University Press, New York & Cam-bridge (U.K.). viii. 306 pages. Hard cover (1994),ISBN 0-521-43461-0, Paperback (1996), ISBN 0-521-57545-1.

• Bunt, H. & Black, W. “Abduction, Belief and Con-text in Dialogue: Studies in Computational Prag-matics” (Natural Language Processing, 1.) JohnBenjamins, Amsterdam & Philadelphia, 2000. vi.471 pages. Hard cover, ISBN 90-272-4983-0 (Eu-rope), 1-58619-794-2 (U.S.)

[2] R. Josephson, J. &G. Josephson, S. “Abductive Inference:Computation, Philosophy, Technology” Cambridge Uni-versity Press, New York & Cambridge (U.K.). viii. 306pages. Hard cover (1994), ISBN 0-521-43461-0, Paper-back (1996), ISBN 0-521-57545-1.

[3] “Retroduction | Dictionary | Commens”. Commens – Dig-ital Companion to C. S. Peirce. Mats Bergman, SamiPaavola & João Queiroz. Retrieved 2014-08-24.

[4] Sober, Elliot. Core Questions in Philosophy,5th edition.

[5] See, e.g. Analysis of Evidence, 2d ed. by Terence Ander-son (Cambridge University Press, 2005)

[6] For examples, see "Abductive Inference in Reasoningand Perception", John R. Josephson, Laboratory for Ar-tificial Intelligence Research, Ohio State University, andAbduction, Reason, and Science. Processes of Discov-ery and Explanation by Lorenzo Magnani (Kluwer Aca-demic/Plenum Publishers, New York, 2001).

[7] Peirce, C. S.

• “On the Logic of drawing History from AncientDocuments especially from Testimonies” (1901),Collected Papers v. 7, paragraph 219.

• “PAP” ["Prolegomena to an Apology for Pragma-tism"], MS 293 c. 1906, New Elements of Mathe-matics v. 4, pp. 319-320.

• A Letter to F. A. Woods (1913), Collected Papersv. 8, paragraphs 385-388.

(See under "Abduction" and "Retroduction" at CommensDictionary of Peirce’s Terms.)

[8] Peirce, C. S. (1903), Harvard lectures on pragmatism,Collected Papers v. 5, paragraphs 188–189.

[9] A Letter to J. H. Kehler (1911), New Elements of Mathe-matics v. 3, pp. 203–4, see under "Retroduction" at Com-mens Dictionary of Peirce’s Terms.

[10] Peirce, C.S. (1902), application to the Carnegie Institu-tion, see MS L75.329-330, from Draft D of Memoir 27:

Consequently, to discover is simply toexpedite an event that would occur sooneror later, if we had not troubled ourselves tomake the discovery. Consequently, the art ofdiscovery is purely a question of economics.The economics of research is, so far as logicis concerned, the leading doctrine with refer-ence to the art of discovery. Consequently,the conduct of abduction, which is chiefly aquestion of heuristic and is the first questionof heuristic, is to be governed by economicalconsiderations.

[11] Peirce MS. 692, quoted in Sebeok, T. (1981) "You KnowMy Method" in Sebeok, T., The Play of Musement,Bloomington, IA: Indiana, page 24.

[12] Peirce MS. 696, quoted in Sebeok, T. (1981) "You KnowMy Method" in Sebeok, T., The Play of Musement,Bloomington, IA: Indiana, page 31.

[13] Jonathan Koehler. The Base Rate Fallacy Reconsidered:Descriptive, Normative and Methodological Challenges.Behavioral and Brain Sciences. 19, 1996.

[14] Robertson, B., & Vignaux, G. A. (1995). Interpretingevidence: Evaluating forensic evidence in the courtroom.Chichester: John Wiley and Sons.

[15] A. Jøsang. Conditional Reasoning with Subjective Logic.Journal of multiple valued logic and soft computing. 15(1),pp.5-38, 2008.PDF

22 CHAPTER 2. CHAPTER I ABDUCTION

[16] Popper, Karl (2002), Conjectures and Refutations: TheGrowth of Scientific Knowledge, London, UK: Routledge.p 536

[17] See Santaella, Lucia (1997) “The Development of Peirce’sThree Types of Reasoning: Abduction, Deduction, andInduction”, 6th Congress of the IASS. Eprint.

[18] Peirce, C. S. (1908), "A Neglected Argument for the Re-ality of God", Hibbert Journal v. 7, pp. 90–112, see §4.In Collected Papers v. 6, see paragraph 476. In The Es-sential Peirce v. 2, see p. 444.

[19] Peirce, C. S. (1908), "A Neglected Argument for the Re-ality of God", Hibbert Journal v. 7, pp. 90–112. Seeboth part III and part IV. Reprinted, including originallyunpublished portion, in Collected Papers v. 6, paragraphs452–85, Essential Peirce v. 2, pp. 434–50, and elsewhere.

[20] Peirce used the term “intuition” not in the sense of an in-stinctive or anyway half-conscious inference as people of-ten do currently. Instead he used “intuition” usually in thesense of a cognition devoid of logical determination byprevious cognitions. He said, “We have no power of Intu-ition” in that sense. See his “Some Consequences of FourIncapacities” (1868), Eprint.

[21] For a relevant discussion of Peirce and the aims of abduc-tive inference, see McKaughan, Daniel J. (2008), “FromUgly Duckling to Swan: C. S. Peirce, Abduction, andthe Pursuit of Scientific Theories”, Transactions of theCharles S. Peirce Society, v. 44, no. 3 (summer), 446–468.

[22] Peirce means “conceivable” very broadly. See CollectedPapers v. 5, paragraph 196, or Essential Peirce v. 2, p.235, “Pragmatism as the Logic of Abduction” (LectureVII of the 1903 Harvard lectures on pragmatism):

It allows any flight of imagination, pro-vided this imagination ultimately alights upona possible practical effect; and thus many hy-potheses may seem at first glance to be ex-cluded by the pragmatical maxim that are notreally so excluded.

[23] Peirce, C. S., Carnegie Application (L75, 1902, New El-ements of Mathematics v. 4, pp. 37–38. See under"Abduction" at the Commens Dictionary of Peirce’s Terms:

Methodeutic has a special interest in Ab-duction, or the inference which starts a scien-tific hypothesis. For it is not sufficient that ahypothesis should be a justifiable one. Anyhypothesis which explains the facts is jus-tified critically. But among justifiable hy-potheses we have to select that one which issuitable for being tested by experiment.

[24] Peirce, “Pragmatism as the Logic of Abduction” (Lec-ture VII of the 1903 Harvard lectures on pragmatism), seeparts III and IV. Published in part in Collected Papers v.5, paragraphs 180–212 (see 196–200, Eprint and in fullin Essential Peirce v. 2, pp. 226–241 (see sections III andIV).

.... What is good abduction? What shouldan explanatory hypothesis be to be worthyto rank as a hypothesis? Of course, it mustexplain the facts. But what other conditionsought it to fulfill to be good? .... Any hypoth-esis, therefore, may be admissible, in the ab-sence of any special reasons to the contrary,provided it be capable of experimental ver-ification, and only insofar as it is capable ofsuch verification. This is approximately thedoctrine of pragmatism.

[25] Peirce, A Letter to Paul Carus circa 1910, Collected Pa-pers v. 8, paragraphs 227–228. See under "Hypothesis"at the Commens Dictionary of Peirce’s Terms.

[26] (1867), “On the Natural Classification of Arguments”,Proceedings of the American Academy of Arts and Sci-ences v. 7, pp. 261–287. Presented April 9, 1867. Seeespecially starting at p. 284 in Part III §1. Reprinted inCollected Papers v. 2, paragraphs 461–516 and Writingsv. 2, pp. 23–49.

[27] Peirce, C. S. (1878), “Deduction, Induction, and Hypoth-esis”, Popular Science Monthly, v. 13, pp. 470–82, see472. Collected Papers 2.619–44, see 623.

[28] A letter to Langley, 1900, published in Historical Per-spectives on Peirce’s Logic of Science. See excerpts under"Abduction" at the Commens Dictionary of Peirce’s Terms.

[29] “A Syllabus of Certain Topics of Logic'" (1903manuscript), Essential Peirce v. 2, see p. 287. Seeunder "Abduction" at the Commens Dictionary of Peirce’sTerms.

[30] Peirce, C. S., “On the Logic of Drawing History from An-cient Documents”, dated as circa 1901 both by the editorsof Collected Papers (see CP v. 7, bk 2, ch. 3, footnote1) and by those of the Essential Peirce (EP) (Eprint. Thearticle’s discussion of abduction is in CP v. 7, paragraphs218–31 and in EP v. 2, pp. 107–14.

[31] Peirce, C. S., “A Syllabus of Certain Topics of Logic”(1903), Essential Peirce v. 2, p. 287:

The mind seeks to bring the facts, asmodified by the new discovery, into order;that is, to form a general conception embrac-ing them. In some cases, it does this by anact of generalization. In other cases, no newlaw is suggested, but only a peculiar state offacts that will “explain” the surprising phe-nomenon; and a law already known is recog-nized as applicable to the suggested hypoth-esis, so that the phenomenon, under that as-sumption, would not be surprising, but quitelikely, or even would be a necessary result.This synthesis suggesting a new conception orhypothesis, is the Abduction.

[32] Peirce, C. S. (1883), “A Theory of Probable Inference” inStudies in Logic).

[33] In Peirce, C. S., 'Minute Logic' circa 1902, Collected Pa-pers v. 2, paragraph 102. See under "Abduction" at Com-mens Dictionary of Peirce’s Terms.

2.1. ABDUCTIVE REASONING 23

[34] Peirce, “On the Logic of drawing History from AncientDocuments”, 1901 manuscript, Collected Papers v. 7,paragraphs 164–231, see 202, reprinted in Essential Peircev. 2, pp. 75–114, see 95. See under "Abduction" at Com-mens Dictionary of Peirce’s Terms.

[35] Peirce, “On the Logic of Drawing Ancient History fromDocuments”, Essential Peirce v. 2, see pp. 107–9.

[36] Peirce, Carnegie application, L75 (1902), Memoir 28:“On the Economics of Research”, scroll down to DraftE. Eprint.

[37] Peirce, C. S., the 1866 Lowell Lectures on the Logicof Science, Writings of Charles S. Peirce v. 1, p. 485.See under "Hypothesis" atCommens Dictionary of Peirce’sTerms.

[38] Peirce, C. S., “A Syllabus of Certain Topics of Logic”,written 1903. See The Essential Peirce v. 2, p. 287.Quote viewable under "Abduction" at Commens Dictio-nary of Peirce’s Terms.

[39] Peirce, A Letter to Paul Carus 1910, Collected Papers v.8, see paragraph 223.

[40] Peirce, C. S. (1902), Application to the Carnegie Institu-tion, Memoir 27, Eprint: “Of the different classes of ar-guments, abductions are the only ones in which after theyhave been admitted to be just, it still remains to inquirewhether they are advantageous.”

[41] Peirce, “On the Logic of Drawing Ancient History fromDocuments”, Essential Peirce v. 2, see pp. 107–9 and 113.On Twenty Questions, p. 109, Peirce has pointed out thatif each question eliminates half the possibilities, twentyquestions can choose from among 220 or 1,048,576 ob-jects, and goes on to say:

Thus, twenty skillful hypotheses will as-certain what 200,000 stupid ones might failto do. The secret of the business lies in thecaution which breaks a hypothesis up into itssmallest logical components, and only risksone of them at a time.

[42] Schwendtner, Tibor and Ropolyi, László and Kiss, Olga(eds): Hermeneutika és a természettudományok. ÁronKiadó, Budapest, 2001. It is written in Hungarian. Mean-ing of the title: Hermeneutics and the natural sciences.See, e.g., Hanson’s Patterns of Discovery (Hanson, 1958),especially pp. 85-92

[43] Rapezzi, C; Ferrari, R; Branzi, A (24 December 2005).“White coats and fingerprints: diagnostic reasoning inmedicine and investigative methods of fictional detec-tives”. BMJ (Clinical research ed.) 331 (7531): 1491–4.doi:10.1136/bmj.331.7531.1491. PMC 1322237. PMID16373725. Retrieved 17 January 2014.

[44] Rejón Altable, C (October 2012). “Logic structure ofclinical judgment and its relation to medical and psy-chiatric semiology”. Psychopathology 45 (6): 344–51.doi:10.1159/000337968. PMID 22854297. Retrieved 17January 2014.

[45] Kave Eshghi. Abductive planning with the event calcu-lus. In Robert A. Kowalski, Kenneth A. Bowen editors:Logic Programming, Proceedings of the Fifth Interna-tional Conference and Symposium, Seattle, Washington,August 15–19, 1988. MIT Press 1988, ISBN 0-262-61056-6

[46] April M. S. McMahon (1994): Understanding languagechange. Cambridge: Cambridge University Press. ISBN0-521-44665-1

[47] Eco, U. (1976). “A theory of Semiotics”. Bloomington,IA: Indiana. p 131

[48] Gell, A. 1984, Art and Agency. Oxford: Oxford. p 14

[49] Bowden, R. (2004) A critique of Alfred Gell on Art andAgency. Retrieved Sept 2007 from: Find Articles atBNET

[50] Whitney D. (2006) 'Abduction the agency of art.' Re-trieved May 2009 from: University of California, Berke-ley

2.1.9 External links

• Abduction entry by Igor Douven in the Stanford En-cyclopedia of Philosophy

• Abductive reasoning at the Indiana Philosophy On-tology Project

• Abductive reasoning at PhilPapers

• "Abductive Inference" (once there, scroll down),John R. Josephson, Laboratory for Artificial Intel-ligence Research, Ohio State University. (Formerwebpage via the Wayback Machine.)

• "Deduction, Induction, and Abduction", Chapter 3in article "Charles Sanders Peirce" by Robert Burch,2001 and 2006, in the Stanford Encyclopedia of Phi-losophy.

• "Abduction", links to articles and websites on ab-ductive inference, Martin Ryder.

• International Research Group on Abductive Infer-ence, Uwe Wirth and Alexander Roesler, eds. Usesframes. Click on link at bottom of its homepage for English. Wirth moved to U. of Gießen,Germany, and set up Abduktionsforschung, homepage not in English but see Artikel section there.Abduktionsforschunghome page via Google transla-tion.

• "'You Know My Method': A Juxtaposition ofCharles S. Peirce and Sherlock Holmes" (1981), byThomas Sebeok with Jean Umiker-Sebeok, fromThe Play of Musement, Thomas Sebeok, Blooming-ton, Indiana: Indiana University Press, pp. 17–52.

24 CHAPTER 2. CHAPTER I ABDUCTION

• Commens Dictionary of Peirce’s Terms, MatsBergman and Sami Paavola, editors, Helsinki U.Peirce’s own definitions, often many per term acrossthe decades. There, see “Hypothesis [as a formof reasoning]", “Abduction”, “Retroduction”, and“Presumption [as a form of reasoning]".

2.2 Knowledge

For other uses, see Knowledge (disambiguation).

Knowledge is a familiarity, awareness or understand-ing of someone or something, such as facts, information,descriptions, or skills, which is acquired throughexperience or education by perceiving, discovering, orlearning.Knowledge can refer to a theoretical or practical under-standing of a subject. It can be implicit (as with practicalskill or expertise) or explicit (as with the theoretical un-derstanding of a subject); it can be more or less formalor systematic.[1] In philosophy, the study of knowledge iscalled epistemology; the philosopher Plato famously de-fined knowledge as "justified true belief", though “well-justified true belief” is more complete as it accounts forthe Gettier problems. However, several definitions ofknowledge and theories to explain it exist.Knowledge acquisition involves complex cognitive pro-cesses: perception, communication, and reasoning; whileknowledge is also said to be related to the capacity of ac-knowledgment in human beings.[2]

2.2.1 Theories of knowledge

Robert Reid, Knowledge (1896). Thomas Jefferson Building,Washington, D.C.

See also: Epistemology

The eventual demarcation of philosophyfrom science was made possible by the notionthat philosophy’s core was “theory of knowl-edge,” a theory distinct from the sciencesbecause it was their foundation... Without thisidea of a “theory of knowledge,” it is hard toimagine what “philosophy” could have been inthe age of modern science.— Richard Rorty, Philosophy and the Mirrorof Nature

The definition of knowledge is a matter of ongoing debateamong philosophers in the field of epistemology. Theclassical definition, described but not ultimately endorsedby Plato,[3] specifies that a statement must meet threecriteria in order to be considered knowledge: it mustbe justified, true, and believed. Some claim that theseconditions are not sufficient, as Gettier case examplesallegedly demonstrate. There are a number of alter-natives proposed, including Robert Nozick's argumentsfor a requirement that knowledge 'tracks the truth' andSimon Blackburn's additional requirement that we do notwant to say that those who meet any of these condi-tions 'through a defect, flaw, or failure' have knowledge.Richard Kirkham suggests that our definition of knowl-edge requires that the evidence for the belief necessitatesits truth.[4]

In contrast to this approach, Ludwig Wittgenstein ob-served, following Moore’s paradox, that one can say “Hebelieves it, but it isn't so,” but not “He knows it, but itisn't so.”[5] He goes on to argue that these do not corre-spond to distinct mental states, but rather to distinct waysof talking about conviction. What is different here is notthe mental state of the speaker, but the activity in whichthey are engaged. For example, on this account, to knowthat the kettle is boiling is not to be in a particular state ofmind, but to perform a particular task with the statementthat the kettle is boiling. Wittgenstein sought to bypassthe difficulty of definition by looking to the way “knowl-edge” is used in natural languages. He saw knowledgeas a case of a family resemblance. Following this idea,“knowledge” has been reconstructed as a cluster conceptthat points out relevant features but that is not adequatelycaptured by any definition.[6]

2.2.2 Communicating knowledge

Symbolic representations can be used to indicate meaningand can be thought of as a dynamic process. Hence thetransfer of the symbolic representation can be viewed asone ascription process whereby knowledge can be trans-ferred. Other forms of communication include observa-tion and imitation, verbal exchange, and audio and video

2.2. KNOWLEDGE 25

Los portadores de la antorcha (The Torch-Bearers) – Sculp-ture by Anna Hyatt Huntington symbolizing the transmission ofknowledge from one generation to the next (Ciudad Universi-taria, Madrid, Spain)

recordings. Philosophers of language and semioticiansconstruct and analyze theories of knowledge transfer orcommunication.While many would agree that one of the most universaland significant tools for the transfer of knowledge is writ-ing and reading (of many kinds), argument over the use-fulness of the written word exists nonetheless, with somescholars skeptical of its impact on societies. In his col-lection of essays Technopoly, Neil Postman demonstratesthe argument against the use of writing through an ex-cerpt from Plato’s work Phaedrus (Postman, Neil (1992)Technopoly, Vintage, New York, pp 73). In this excerpt,the scholar Socrates recounts the story of Thamus, theEgyptian king and Theuth the inventor of the writtenword. In this story, Theuth presents his new invention“writing” to King Thamus, telling Thamus that his newinvention “will improve both the wisdom and memory ofthe Egyptians” (Postman, Neil (1992) Technopoly, Vin-tage, New York, pp 74). King Thamus is skeptical of thisnew invention and rejects it as a tool of recollection ratherthan retained knowledge. He argues that the written wordwill infect the Egyptian people with fake knowledge asthey will be able to attain facts and stories from an exter-nal source and will no longer be forced to mentally retainlarge quantities of knowledge themselves (Postman, Neil(1992) Technopoly, Vintage, New York,pp 74).Classical early modern theories of knowledge, especiallythose advancing the influential empiricism of the philoso-

pher John Locke, were based implicitly or explicitly on amodel of the mind which likened ideas to words.[7] Thisanalogy between language and thought laid the founda-tion for a graphic conception of knowledge in which themind was treated as a table (a container of content) thathad to be stocked with facts reduced to letters, numbersor symbols. This created a situation in which the spatialalignment of words on the page carried great cognitiveweight, so much so that educators paid very close atten-tion to the visual structure of information on the page andin notebooks.[8]

Media theorists like AndrewRobinson emphasise that thevisual depiction of knowledge in the modern world wasoften seen as being 'truer' than oral knowledge. This playsinto a longstanding analytic notion in the Western intel-lectual tradition in which verbal communication is gen-erally thought to lend itself to the spread of falsehoods asmuch as written communication. It is harder to preserverecords of what was said or who originally said it – usuallyneither the source nor the content can be verified. Gos-sip and rumors are examples prevalent in both media. Asto the value of writing, the extent of human knowledgeis now so great, and the people interested in a piece ofknowledge so separated in time and space, that writing isconsidered central to capturing and sharing it.Major libraries today can have millions of books ofknowledge (in addition to works of fiction). It is onlyrecently that audio and video technology for recordingknowledge have become available and the use of thesestill requires replay equipment and electricity. Verbalteaching and handing down of knowledge is limited tothose who would have contact with the transmitter orsomeone who could interpret written work. Writing isstill the most available and most universal of all formsof recording and transmitting knowledge. It stands un-challenged as mankind’s primary technology of knowl-edge transfer down through the ages and to all culturesand languages of the world.

2.2.3 Situated knowledge

Situated knowledge is knowledge specific to a particularsituation. It is a term coined by Donna Haraway as anextension of the feminist approaches of “successor sci-ence” suggested by Sandra Harding, one which “offers amore adequate, richer, better account of a world, in or-der to live in it well and in critical, reflexive relation toour own as well as others’ practices of domination andthe unequal parts of privilege and oppression that makesup all positions.”[9] This situation partially transforms sci-ence into a narrative, which Arturo Escobar explains as,“neither fictions nor supposed facts.” This narrative of sit-uation is historical textures woven of fact and fiction, andas Escobar explains further, “even the most neutral sci-entific domains are narratives in this sense,” insisting thatrather than a purpose dismissing science as a trivial mat-ter of contingency, “it is to treat (this narrative) in the

26 CHAPTER 2. CHAPTER I ABDUCTION

most serious way, without succumbing to its mystifica-tion as 'the truth' or to the ironic skepticism common tomany critiques.”[10]

Haraway’s argument stems from the limitations of thehuman perception, as well as the overemphasis of thesense of vision in science. According to Haraway, visionin science has been, “used to signify a leap out of themarked body and into a conquering gaze from nowhere.”This is the “gaze that mythically inscribes all the markedbodies, that makes the unmarked category claim thepower to see and not be seen, to represent while escap-ing representation.”[9] This causes a limitation of viewsin the position of science itself as a potential playerin the creation of knowledge, resulting in a position of“modest witness”. This is what Haraway terms a “godtrick”, or the aforementioned representation while escap-ing representation.[11] In order to avoid this, “Harawayperpetuates a tradition of thought which emphasizes theimportance of the subject in terms of both ethical andpolitical accountability”.[12]

Some methods of generating knowledge, such as trial anderror, or learning from experience, tend to create highlysituational knowledge. One of the main attributes of thescientific method is that the theories it generates are muchless situational than knowledge gained by other meth-ods. Situational knowledge is often embedded in lan-guage, culture, or traditions. This integration of situa-tional knowledge is an allusion to the community, and itsattempts at collecting subjective perspectives into an em-bodiment “of views from somewhere.” [9]

Knowledge generated through experience is calledknowledge “a posteriori”, meaning afterwards. The pureexistence of a term like “a posteriori” means this also hasa counterpart. In this case, that is knowledge “a priori”,meaning before. The knowledge prior to any experiencemeans that there are certain “assumptions” that one takesfor granted. For example, if you are being told about achair, it is clear to you that the chair is in space, that it is3D. This knowledge is not knowledge that one can “for-get”, even someone suffering from amnesia experiencesthe world in 3D.Even though Haraway’s arguments are largely based onfeminist studies,[9] this idea of different worlds, as wellas the skeptic stance of situated knowledge is present inthe main arguments of post-structuralism. Fundamen-tally, both argue the contingency of knowledge on thepresence of history; power, and geography, as well asthe rejection of universal rules or laws or elementarystructures; and the idea of power as an inherited trait ofobjectification.[13]

2.2.4 Partial knowledge

One discipline of epistemology focuses on partial knowl-edge. In most cases, it is not possible to understand aninformation domain exhaustively; our knowledge is al-

ways incomplete or partial. Most real problems have tobe solved by taking advantage of a partial understandingof the problem context and problem data, unlike the typ-ical math problems one might solve at school, where alldata is given and one is given a complete understandingof formulas necessary to solve them.This idea is also present in the concept of bounded ra-tionality which assumes that in real life situations peopleoften have a limited amount of information and make de-cisions accordingly.Intuition is the ability to acquire partial knowledge with-out inference or the use of reason.[14] An individual may“know” about a situation and be unable to explain the pro-cess that led to their knowledge.

2.2.5 Scientific knowledge

Sir Francis Bacon, "Knowledge is Power"

The development of the scientific method has made a sig-nificant contribution to how knowledge of the physicalworld and its phenomena is acquired.[15] To be termedscientific, a method of inquiry must be based on gatheringobservable and measurable evidence subject to specificprinciples of reasoning and experimentation.[16] The sci-entific method consists of the collection of data through

2.2. KNOWLEDGE 27

observation and experimentation, and the formulationand testing of hypotheses.[17] Science, and the natureof scientific knowledge have also become the subject ofPhilosophy. As science itself has developed, knowledgehas developed a broader usage which has been develop-ing within biology/psychology—discussed elsewhere asmeta-epistemology, or genetic epistemology, and to someextent related to "theory of cognitive development". Notethat "epistemology" is the study of knowledge and how itis acquired. Science is “the process used everyday to log-ically complete thoughts through inference of facts de-termined by calculated experiments.” Sir Francis Baconwas critical in the historical development of the scientificmethod; his works established and popularized an induc-tive methodology for scientific inquiry. His famous apho-rism, "knowledge is power", is found in the MeditationsSacrae (1597).[18]

Until recent times, at least in the Western tradition, it wassimply taken for granted that knowledge was somethingpossessed only by humans — and probably adult humansat that. Sometimes the notion might stretch to (ii) Society-as-such, as in (e.g.) “the knowledge possessed by theCoptic culture” (as opposed to its individual members),but that was not assured either. Nor was it usual to con-sider unconscious knowledge in any systematic way untilthis approach was popularized by Freud.[19]

Other biological domains where “knowledge” might besaid to reside, include: (iii) the immune system, and(iv) in the DNA of the genetic code. See the list offour “epistemological domains": Popper, (1975);[20] andTraill (2008:[21] Table S, page 31)—also references byboth to Niels Jerne.Such considerations seem to call for a separate definitionof “knowledge” to cover the biological systems. For bi-ologists, knowledge must be usefully available to the sys-tem, though that system need not be conscious. Thus thecriteria seem to be:

• The system should apparently be dynamic and self-organizing (unlike a mere book on its own).

• The knowledge must constitute some sort of repre-sentation of “the outside world”,[22] or ways of deal-ing with it (directly or indirectly).

• Some way must exist for the system to access thisinformation quickly enough for it to be useful.

Scientific knowledge may not involve a claim to certainty,maintaining skepticism means that a scientist will neverbe absolutely certain when they are correct and when theyare not. It is thus an irony of proper scientific method thatone must doubt even when correct, in the hopes that thispractice will lead to greater convergence on the truth ingeneral.[23]

2.2.6 Religious meaning of knowledge

In many expressions of Christianity, such as Catholicismand Anglicanism, knowledge is one of the seven gifts ofthe Holy Spirit.[24]

The Old Testament's tree of the knowledge of good andevil contained the knowledge that separated Man fromGod: “And the LORD God said, Behold, the man is be-come as one of us, to know good and evil...” (Genesis3:22)In Gnosticism, divine knowledge or gnosis is hoped to beattained.विदया दान (Vidya Daan) i.e. knowledge sharing isa major part of Daan, a tenet of all Dharmic Reli-gions.[25] Hindu Scriptures present two kinds of knowl-edge, Paroksh Gyan and Prataksh Gyan. Paroksh Gyan(also spelled Paroksha-Jnana) is secondhand knowledge:knowledge obtained from books, hearsay, etc. PratakshGyan (also spelled Prataksha-Jnana) is the knowledgeborne of direct experience, i.e., knowledge that one dis-covers for oneself.[26] Jnana yoga (“path of knowledge”)is one of three main types of yoga expounded by Krishnain the Bhagavad Gita. (It is compared and contrasted withBhakti Yoga and Karma yoga.)In Islam, knowledge (Arabic: ,علم ʿilm) is given greatsignificance. “The Knowing” (al-ʿAlīm) is one of the 99names reflecting distinct attributes of God. The Qur'anasserts that knowledge comes from God (2:239) andvarious hadith encourage the acquisition of knowledge.Muhammad is reported to have said “Seek knowledgefrom the cradle to the grave” and “Verily the men ofknowledge are the inheritors of the prophets”. Islamicscholars, theologians and jurists are often given the titlealim, meaning “knowledgable”.In Jewish tradition, knowledge (Hebrew: דעת da'ath)is considered one of the most valuable traits a personcan acquire. Observant Jews recite three times a dayin the Amidah “Favor us with knowledge, understand-ing and discretion that come from you. Exalted areyou, Existent-One, the gracious giver of knowledge.” TheTanakh states, “A wise man gains power, and a man ofknowledge maintains power”, and “knowledge is chosenabove gold”.

As a measure of religiosity (in sociology of religion)

According to the sociologist Mervin Verbit, knowledgemay be understood as one of the key components of reli-giosity. Religious knowledge itself may be broken downinto four dimensions:

• content

• frequency

• intensity

28 CHAPTER 2. CHAPTER I ABDUCTION

• centrality

The content of one’s religious knowledge may vary fromperson to person, as will the degree to which it may oc-cupy the person’s mind (frequency), the intensity of theknowledge, and the centrality of the information (in thatreligious tradition, or to that individual).[27][28][29]

2.2.7 See also

• Outline of knowledge – guide to the subject ofknowledge presented as a tree structured list of itssubtopics.

• a priori and a posteriori

• Analytic-synthetic distinction

• Descriptive knowledge

• Epistemic modal logic

• Explicit knowledge

• Figurative system of human knowledge

• Inductive inference

• Inductive probability

• Intelligence

• Knowledge engineering

• Knowledge extraction

• Knowledge management

• Knowledge relativity

• Knowledge representation

• Knowledge retrieval

• Metaknowledge

• Philosophical skepticism

• Procedural knowledge

• Society for the Diffusion of Useful Knowledge

• Tacit knowledge

2.2.8 References

[1] “knowledge: definition of knowledge in Oxford dictionary(American English) (US)". oxforddictionaries.com.

[2] Stanley Cavell, “Knowing and Acknowledging”, Must WeMeanWhatWe Say? (Cambridge University Press, 2002),238–266.

[3] In Plato’s Theaetetus, Socrates and Theaetetus discussthree definitions of knowledge: knowledge as nothingbut perception, knowledge as true judgment, and, finally,knowledge as a true judgment with an account. Each ofthese definitions is shown to be unsatisfactory.

[4] Kirkham, Richard L. (October 1984). “Does the GettierProblem Rest on a Mistake?" (PDF). Mind, New Series.Oxford University Press on behalf of the Mind Associa-tion. pp. 501–513. Retrieved 24 September 2008. jstor(subscription required)

[5] Ludwig Wittgenstein, On Certainty, remark 42

[6] Gottschalk-Mazouz, N. (2008): “Internet and the flowof knowledge,” in: Hrachovec, H.; Pichler, A. (Hg.):Philosophy of the Information Society. Proceedingsof the 30. International Ludwig Wittgenstein Sympo-sium Kirchberg am Wechsel, Austria 2007. Volume2, Frankfurt, Paris, Lancaster, New Brunswik: Ontos,S. 215–232. http://sammelpunkt.philo.at:8080/2022/1/Gottschalk-Mazouz.pdf

[7] Hacking, Ian (1975). Why Does Language Matter to Phi-losophy?. Cambridge: Cambridge University Press.

[8] Eddy, Matthew Daniel (2013). “The Shape of Knowl-edge: Children and the Visual Culture of Literacyand Numeracy”. Science in Context 26: 215–245.doi:10.1017/s0269889713000045.

[9] “Situated Knowledges: The Science Question in Femi-nism and the Privilege of Partial Perspective”. Haraway,Donna. Feminist Studies Vol. 14, No. 3. pp. 575–599.1988.

[10] “Introduction: Development and the Anthropology ofModernity”. Escobar, Arturo. Encountering Develop-ment: The Making and Unmaking of the Third World.

[11] Chapter 1. Haraway, Donna. Mod-est_Witness@Second_Millennium. FemaleMan©Meets_OncoMouse2. Feminism and Technoscience. 1997.

[12] “Posthuman, All Too Human: Towards a New ProcessOntology”. Braidotti, Rosi. Theory Culture Vol. 23. pp.197–208. 2006.

[13] “The Subject and Power”. Foucault, Michel. Critical In-quiry Volume 9, No. 4. pp. 777–795. 1982

[14] Oxford English Dictionary

[15] “Science – Definition of science by Merriam-Webster”.merriam-webster.com.

[16] "[4] Rules for the study of natural philosophy", Newton1999, pp. 794–6, from the General Scholium, which fol-lows Book 3, The System of the World.

[17] scientific method, Merriam-Webster Dictionary.

[18] “Sir Francis Bacon – Quotationspage.com”. Retrieved2009-07-08.

2.3. UNDERSTANDING 29

[19] There is quite a good case for this exclusive specializa-tion used by philosophers, in that it allows for in-depthstudy of logic-procedures and other abstractions whichare not found elsewhere. However this may lead to prob-lems whenever the topic spills over into those excludeddomains—e.g. when Kant (following Newton) dismissedSpace and Time as axiomatically “transcendental” and “apriori” — a claim later disproved by Piaget’s clinical stud-ies. It also seems likely that the vexed problem of "infiniteregress" can be largely (but not completely) solved byproper attention to how unconscious concepts are actu-ally developed, both during infantile learning and as in-herited “pseudo-transcendentals” inherited from the trial-and-error of previous generations. See also "Tacit knowl-edge".

• Piaget, J., and B.Inhelder (1927 / 1969). The child’sconception of time. Routledge & Kegan Paul: Lon-don.

• Piaget, J., and B.Inhelder (1948 / 1956). The child’sconception of space. Routledge&Kegan Paul: Lon-don.

[20] Popper, K.R. (1975). “The rationality of scientific revo-lutions"; in Rom Harré (ed.), Problems of Scientific Revo-lution: Scientific Progress and Obstacles to Progress in theSciences. Clarendon Press: Oxford.

[21] http://www.ondwelle.com/OSM02.pdf

[22] This “outside world” could include other subsystemswithin the same organism—e.g. different “mental levels”corresponding to different Piagetian stages. See Theoryof cognitive development.

[23] “philosophy bites”. philosophybites.com.

[24] “Part Three, No. 1831”. Catechism of the CatholicChurch. Retrieved 2007-04-20.

[25] "विदया दान ही सबस बडा दान : विहिप - Vishva HinduParishad – Official Website”. vhp.org.

[26] Swami Krishnananda. “Chapter 7”. The Philosophy of thePanchadasi. The Divine Life Society. Retrieved 2008-07-05.

[27] Verbit, M. F. (1970). The components and dimensions ofreligious behavior: Toward a reconceptualization of reli-giosity. American mosaic, 24, 39.

[28] Küçükcan, T. (2010). Multidimensional Approach to Re-ligion: a way of looking at religious phenomena. Journalfor the Study of Religions and Ideologies, 4(10), 60–70.

[29] http://www.eskieserler.com/dosyalar/mpdf%20(1135).pdf

2.2.9 External links

• Knowledge at PhilPapers

• Knowledge entry in the Internet Encyclopedia ofPhilosophy

• The Value of Knowledge entry in the Stanford En-cyclopedia of Philosophy

• The Analysis of Knowledge entry in the StanfordEncyclopedia of Philosophy

• Knowledge by Acquaintance vs. Description entryin the Stanford Encyclopedia of Philosophy

• Knowledge at the Indiana Philosophy OntologyProject

2.3 Understanding

This article is about the psychological process. For otheruses, see Understanding (disambiguation).“Understand” redirects here. For other uses, seeUnderstand (disambiguation).

Understanding (also called intellection) is apsychological process related to an abstract or physicalobject, such as a person, situation, or message wherebyone is able to think about it and use concepts to dealadequately with that object. Understanding is a relationbetween the knower and an object of understanding.Understanding implies abilities and dispositions withrespect to an object of knowledge sufficient to supportintelligent behavior.[1]

An understanding is the limit of a conceptualization. Tounderstand something is to have conceptualized it to agiven measure.

2.3.1 Examples

1. One understands the weather if one is able to predictand to give an explanation of some of its features,etc.

2. A psychiatrist understands another person’sanxieties if he/she knows that person’s anxieties,their causes, and can give useful advice on how tocope with the anxiety.

3. A person understands a command if he/she knowswho gave it, what is expected by the issuer, andwhether the command is legitimate, and whetherone understands the speaker (see 4).

4. One understands a reasoning, an argument, or alanguage if one can consciously reproduce the in-formation content conveyed by the message.

5. One understands a mathematical concept if one cansolve problems using it, especially problems that arenot similar to what one has seen before.

30 CHAPTER 2. CHAPTER I ABDUCTION

2.3.2 Understanding as a model

Gregory Chaitin, a noted computer scientist, propoundsa view that comprehension is a kind of data compres-sion.[2] In his essay “The Limits of Reason”, he arguesthat understanding something means being able to figureout a simple set of rules that explains it. For example,we understand why day and night exist because we have asimple model—the rotation of the earth—that explains atremendous amount of data—changes in brightness, tem-perature, and atmospheric composition of the earth. Wehave compressed a large amount of information by usinga simple model that predicts it. Similarly, we understandthe number 0.33333... by thinking of it as one-third. Thefirst way of representing the number requires an infiniteamount of memory; but the second way can produce allthe data of the first representation, but uses much lessinformation. Chaitin argues that comprehension is thisability to compress data.

2.3.3 Components of understanding

Cognition and affect

Main articles: cognition and affect (psychology)

Cognition is the process by which sensory inputs aretransformed. Affect refers to the experience of feelingsor emotions. Cognition and affect constitute understand-ing.

2.3.4 Religious perspectives

In Catholicism and Anglicanism, understanding is one ofthe Seven gifts of the Holy Spirit.

2.3.5 See also

• Active listening

• Awareness

• Binah (Kabbalah)

• Chinese room

• Communication

• Epistemology

• Hermeneutic circle

• Informational listening

• Ishin-denshin

• List of language disorders

• Meaning (linguistics)

• Natural language understanding

• Nous

• Perception

• Thought

2.3.6 References[1] Bereiter, Carl. “Education and mind in the Knowledge

Age”.

[2] Chaitin, Gregory (2006), The Limits Of Reason (PDF)

2.3.7 External links

• Understanding at PhilPapers

2.4 Certainty

For statistical certainty, see Probability. For the film, seeCertainty (film).“Certain” redirects here. For the French footballer, seeFrançois Certain.

Certainty is perfect knowledge that has total securityfrom error, or the mental state of being without doubt.Objectively defined, certainty is total continuity and va-lidity of all foundational inquiry, to the highest degree ofprecision. Something is certain only if no skepticism canoccur. Philosophy (at least, historical Cartesian philoso-phy) seeks this state.It is widely held that certainty about the real world isa failed historical enterprise (that is, beyond deductivetruths, tautology, etc.).[1] This is in large part due to thepower of David Hume's problem of induction. PhysicistCarlo Rovelli adds that certainty, in real life, is uselessor often damaging (the idea is that “total security fromerror” is impossible in practice, and a complete “lack ofdoubt” is undesirable).[2]

2.4.1 History

Pyrrho – ancient Greece

Main article: Pyrrho

Pyrrho is credited as being the first Skeptic philosopher.The main principle of Pyrrho’s thought is expressed bythe word acatalepsia, which denotes the ability to with-hold assent from doctrines regarding the truth of thingsin their own nature; against every statement its contradic-tion may be advanced with equal justification. Secondly,it is necessary in view of this fact to preserve an attitude

2.4. CERTAINTY 31

of intellectual suspense, or, as Timon expressed it, no as-sertion can be known to be better than another.

Al-Ghazali – Islamic theologian

Main article: Al-Ghazali

Al-Ghazali was a professor of philosophy in the 11th cen-tury. His book titled The Incoherence of the Philoso-phers marks a major turn in Islamic epistemology, asGhazali effectively discovered philosophical skepticismthat would not be commonly seen in the West untilAverroes, René Descartes, George Berkeley and DavidHume. He described the necessity of proving the valid-ity of reason—independently from reason. He attemptedthis and failed. The doubt that he introduced to his foun-dation of knowledge could not be reconciled using phi-losophy. Taking this very seriously, he resigned fromhis post at the university, and suffered serious psychoso-matic illness. It was not until he became a religious sufithat he found a solution to his philosophical problems,which are based on Islamic religion; this encounter withskepticism led Ghazali to embrace a form of theologi-cal occasionalism, or the belief that all causal events andinteractions are not the product of material conjunctionsbut rather the immediate and present will of God.

Ibn-Rushd - Averroes

Main article: Ibn Rushd

Latinized name AverroësAverroes was a defender of Aristotelian philosophyagainst Ash'ari theologians led by Al-Ghazali. Aver-roes’ philosophy was considered controversial in Muslimcircles.[3] Averroes had a greater impact on Western Eu-ropean circles and he has been described as the “foundingfather of secular thought in Western Europe”.

Descartes – 17th century

Descartes' Meditations on First Philosophy is a book inwhich Descartes first discards all belief in things whichare not absolutely certain, and then tries to establish whatcan be known for sure. Although the phrase "Cogito, ergosum" is often attributed to Descartes’Meditations on FirstPhilosophy, it is actually put forward in his Discourse onMethod. Due to the implications of inferring the conclu-sion within the predicate, however, he changed the argu-ment to “I think, I exist"; this then became his first cer-tainty.

Ludwig Wittgenstein – 20th century

OnCertainty is a series of notes made by LudwigWittgen-stein just prior to his death. The main theme of the workis that context plays a role in epistemology. Wittgen-stein asserts an anti-foundationalist message throughoutthe work: that every claim can be doubted but certaintyis possible in a framework. “The function [propositions]serve in language is to serve as a kind of frameworkwithin which empirical propositions can make sense”.[4]

2.4.2 Degrees of certainty

See also: Inductive reasoning, Probability interpretationsand Philosophy of statistics

Physicist Lawrence M. Krauss suggests that identifyingdegrees of certainty is under-appreciated in various do-mains, including policy making and the understanding ofscience. This is because different goals require differ-ent degrees of certainty—and politicians are not alwaysaware of (or do not make it clear) how much certainty weare working with.[5]

Rudolf Carnap viewed certainty as a matter of degree(degrees of certainty) which could be objectively mea-sured, with degree one being certainty. Bayesian analy-sis derives degrees of certainty which are interpreted as ameasure of subjective psychological belief.Alternatively, one might use the legal degrees of cer-tainty. These standards of evidence ascend as follows:no credible evidence, some credible evidence, a prepon-derance of evidence, clear and convincing evidence, be-yond reasonable doubt, and beyond any shadow of a doubt(i.e. undoubtable—recognized as an impossible standardto meet—which serves only to terminate the list).

2.4.3 Foundational crisis of mathematics

Main article: Foundations of mathematics

The foundational crisis of mathematics was the early 20thcentury’s term for the search for proper foundations ofmathematics.After several schools of the philosophy of mathematicsran into difficulties one after the other in the 20th cen-tury, the assumption that mathematics had any founda-tion that could be stated within mathematics itself beganto be heavily challenged.One attempt after another to provide unassailable foun-dations for mathematics was found to suffer from var-ious paradoxes (such as Russell’s paradox) and to beinconsistent.Various schools of thought on the right approach to thefoundations of mathematics were fiercely opposing each

32 CHAPTER 2. CHAPTER I ABDUCTION

other. The leading school was that of the formalist ap-proach, of which David Hilbert was the foremost pro-ponent, culminating in what is known as Hilbert’s pro-gram, which sought to ground mathematics on a small ba-sis of a formal system proved sound bymetamathematicalfinitistic means. The main opponent was the intuitionistschool, led by L.E.J. Brouwer, which resolutely discardedformalism as a meaningless game with symbols. The fightwas acrimonious. In 1920 Hilbert succeeded in havingBrouwer, whom he considered a threat to mathematics,removed from the editorial board of Mathematische An-nalen, the leading mathematical journal of the time.Gödel’s incompleteness theorems, proved in 1931,showed that essential aspects of Hilbert’s program couldnot be attained. In Gödel's first result he showed howto construct, for any sufficiently powerful and consistentfinitely axiomatizable system—such as necessary to ax-iomatize the elementary theory of arithmetic—a state-ment that can be shown to be true, but that does not fol-low from the rules of the system. It thus became clearthat the notion of mathematical truth can not be reducedto a purely formal system as envisaged in Hilbert’s pro-gram. In a next result Gödel showed that such a systemwas not powerful enough for proving its own consistency,let alone that a simpler system could do the job. Thisdealt a final blow to the heart of Hilbert’s program, thehope that consistency could be established by finitisticmeans (it was never made clear exactly what axioms werethe “finitistic” ones, but whatever axiomatic system wasbeing referred to, it was a weaker system than the sys-tem whose consistency it was supposed to prove). Mean-while, the intuitionistic school had failed to attract adher-ents among working mathematicians, and floundered dueto the difficulties of doing mathematics under the con-straint of constructivism.In a sense, the crisis has not been resolved, but fadedaway: most mathematicians either do not work from ax-iomatic systems, or if they do, do not doubt the con-sistency of Zermelo–Fraenkel set theory, generally theirpreferred axiomatic system. In most of mathematics as itis practiced, the various logical paradoxes never played arole anyway, and in those branches in which they do (suchas logic and category theory), they may be avoided.

2.4.4 QuotesDoubt is not a pleasant condition, but

certainty is absurd.— Voltaire

In this world nothing can be said to becertain, except death and taxes.— Benjamin Franklin

There is no such thing as absolute cer-

tainty, but there is assurance sufficient for thepurposes of human life.— John Stuart Mill

If you tried to doubt everything you wouldnot get as far as doubting anything. The gameof doubting itself presupposes certainty.— Ludwig Wittgenstein #115 from On Cer-tainty

2.4.5 See also

• Uncertainty

• Almost surely

• Fideism

• Gut feeling

• Infallibility

• Justified true belief

• Neuroethological innate behavior, instinct

• Pascal’s Wager

• Pragmatism

• Skeptical hypothesis

• As contrary concepts

• Fallibilism• Indeterminism• Multiverse

2.4.6 References

[1] Peat, F. David (2002). From Certainty to Uncertainty:The Story of Science and Ideas in the Twentieth Century.National Academies Press. ISBN 978-0-309-09620-1.

[2] edge.org

[3] “Averroës (Ibn Rushd) > By Individual Philosopher > Phi-losophy”. Philosophybasics.com. Retrieved 2012-10-13.

[4] Wittgenstein, Ludwig. “On Certainty”. SparkNotes.

[5] “question center, SHAs – cognitive tools”. edge.com.

2.4. CERTAINTY 33

2.4.7 External links

• “Certitude”. Catholic Encyclopedia. New York:Robert Appleton Company. 1913.

• certainty, The American Heritage Dictionary of theEnglish Language. Bartleby.com

• “certainty vs. doubt”. About.com. Retrieved 2008-02-23.

• Certainty entry by Baron Reed in the Stanford En-cyclopedia of Philosophy

• The certainty of belief - article arguing that beliefrequires certainty

Chapter 3

Chapter II Induction

3.1 Inductive reasoning

“Inductive inference” redirects here. For the techniquein mathematical proof, see Mathematical induction.

Inductive reasoning (as opposed to deductive reasoningor abductive reasoning) is reasoning in which the premisesare viewed as supplying strong evidence for the truth ofthe conclusion. While the conclusion of a deductive argu-ment is certain, the truth of the conclusion of an inductiveargument is probable, based upon the evidence given.[1]

Many dictionaries define inductive reasoning as reasoningthat derives general principles from specific observations,though some sources disagree with this usage.[2]

The philosophical definition of inductive reasoningis more nuanced than simple progression from par-ticular/individual instances to broader generalizations.Rather, the premises of an inductive logical argument in-dicate some degree of support (inductive probability) forthe conclusion but do not entail it; that is, they suggesttruth but do not ensure it. In this manner, there is the pos-sibility of moving from general statements to individualinstances (for example, statistical syllogisms, discussedbelow).

3.1.1 Description

Inductive reasoning is inherently uncertain. It only dealsin degrees to which, given the premises, the conclusionis credible according to some theory of evidence. Exam-ples include a many-valued logic, Dempster–Shafer the-ory, or probability theory with rules for inference such asBayes’ rule. Unlike deductive reasoning, it does not relyon universals holding over a closed domain of discourseto draw conclusions, so it can be applicable even in casesof epistemic uncertainty (technical issues with this mayarise however; for example, the second axiom of proba-bility is a closed-world assumption).[3]

An example of an inductive argument:

90% of biological life forms that we know ofdepend on liquid water to exist.

Therefore, if we discover a new biological lifeform it will probably depend on liquid water toexist.

This argument could have been made every time a newbiological life form was found, and would have been cor-rect every time; however, it is still possible that in thefuture a biological life form not requiring water could bediscovered.As a result, the argument may be stated less formally as:

All biological life forms that we know of de-pend on liquid water to exist.All biological life probably depends on liquidwater to exist.

3.1.2 Inductive vs. deductive reasoning

Unlike deductive arguments, inductive reasoning allowsfor the possibility that the conclusion is false, even ifall of the premises are true.[4] Instead of being valid orinvalid, inductive arguments are either strong or weak,which describes how probable it is that the conclusion istrue.[5] Another crucial difference is that deductive cer-tainty is impossible in non-axiomatic systems, such as re-ality, leaving inductive reasoning as the primary route to(probabilistic) knowledge of such systems.[6]

Given that “if A is true then B, C, and D are true”, anexample of deduction would be "A is true therefore wecan deduce that B, C, and D are true”. An example ofinduction would be "B, C, and D are observed to be truetherefore A may be true”. A is a reasonable explanationfor B, C, and D being true.For example:

A large enough asteroid impact would createa very large crater and cause a severe impactwinter that could drive the non-avian dinosaursto extinction.We observe that there is a very large crater inthe Gulf ofMexico dating to very near the timeof the extinction of the non-avian dinosaurs

34

3.1. INDUCTIVE REASONING 35

Therefore it is possible that this impact couldexplain why the non-avian dinosaurs went ex-tinct.

Note however that this is not necessarily the case. Otherevents also coincide with the extinction of the non-aviandinosaurs. For example, the Deccan Traps in India.A classical example of an incorrect inductive argumentwas presented by John Vickers:

All of the swans we have seen are white.Therefore, all swans are white.

Note that this definition of inductive reasoning excludesmathematical induction, which is a form of deductive rea-soning.

3.1.3 Criticism

Main article: Problem of induction

Inductive reasoning has been criticized by thinkers as di-verse as Sextus Empiricus[7] and Karl Popper.[8]

The classic philosophical treatment of the problem ofinduction was given by the Scottish philosopher DavidHume.[9]

Although the use of inductive reasoning demonstratesconsiderable success, its application has been question-able. Recognizing this, Hume highlighted the fact thatour mind draws uncertain conclusions from relatively lim-ited experiences. In deduction, the truth value of the con-clusion is based on the truth of the premise. In induction,however, the dependence on the premise is always uncer-tain. As an example, let’s assume “all ravens are black.”The fact that there are numerous black ravens supports theassumption. However, the assumption becomes inconsis-tent with the fact that there are white ravens. Therefore,the general rule of “all ravens are black” is inconsistentwith the existence of the white raven. Hume further ar-gued that it is impossible to justify inductive reasoning:specifically, that it cannot be justified deductively, so ouronly option is to justify it inductively. Since this is circu-lar he concluded that our use of induction is unjustifiablewith the help of “Hume’s Fork”.[10]

However, Hume then stated that even if induction wereproved unreliable, we would still have to rely on it. So in-stead of a position of severe skepticism, Hume advocateda practical skepticism based on common sense, where theinevitability of induction is accepted.[11]

Biases

Inductive reasoning is also known as hypothesis construc-tion because any conclusions made are based on cur-

rent knowledge and predictions. As with deductive ar-guments, biases can distort the proper application of in-ductive argument, thereby preventing the reasoner fromforming the most logical conclusion based on the clues.Examples of these biases include the availability heuris-tic, confirmation bias, and the predictable-world bias.The availability heuristic causes the reasoner to dependprimarily upon information that is readily available tohim/her. People have a tendency to rely on informationthat is easily accessible in the world around them. For ex-ample, in surveys, when people are asked to estimate thepercentage of people who died from various causes, mostrespondents would choose the causes that have been mostprevalent in the media such as terrorism, and murders,and airplane accidents rather than causes such as diseaseand traffic accidents, which have been technically “lessaccessible” to the individual since they are not empha-sized as heavily in the world around him/her.The confirmation bias is based on the natural tendencyto confirm rather than to deny a current hypothesis. Re-search has demonstrated that people are inclined to seeksolutions to problems that aremore consistent with knownhypotheses rather than attempt to refute those hypothe-ses. Often, in experiments, subjects will ask questionsthat seek answers that fit established hypotheses, thusconfirming these hypotheses. For example, if it is hy-pothesized that Sally is a sociable individual, subjects willnaturally seek to confirm the premise by asking questionsthat would produce answers confirming that Sally is in facta sociable individual.The predictable-world bias revolves around the inclina-tion to perceive order where it has not been proved toexist, either at all or at a particular level of abstraction.Gambling, for example, is one of the most popular ex-amples of predictable-world bias. Gamblers often beginto think that they see simple and obvious patterns in theoutcomes and, therefore, believe that they are able to pre-dict outcomes based upon what they have witnessed. Inreality, however, the outcomes of these games are diffi-cult to predict and highly complex in nature. However, ingeneral, people tend to seek some type of simplistic orderto explain or justify their beliefs and experiences, and itis often difficult for them to realise that their perceptionsof order may be entirely different from the truth.[12]

3.1.4 Types

Generalization

A generalization (more accurately, an inductive general-ization) proceeds from a premise about a sample to a con-clusion about the population.

The proportion Q of the sample has attributeA.Therefore:

36 CHAPTER 3. CHAPTER II INDUCTION

The proportion Q of the population has at-tribute A.

Example

There are 20 balls—either black or white—in an urn. Toestimate their respective numbers, you draw a sample offour balls and find that three are black and one is white.A good inductive generalization would be that there are15 black and five white balls in the urn.How much the premises support the conclusion dependsupon (a) the number in the sample group, (b) the numberin the population, and (c) the degree to which the sam-ple represents the population (which may be achieved bytaking a random sample). The hasty generalization andthe biased sample are generalization fallacies.

Statistical syllogism

Main article: Statistical syllogism

A statistical syllogism proceeds from a generalization toa conclusion about an individual.

A proportion Q of population P has attributeA.An individual X is a member of P.Therefore:There is a probability which corresponds to Qthat X has A.

The proportion in the first premise would be somethinglike “3/5ths of”, “all”, “few”, etc. Two dicto simpliciterfallacies can occur in statistical syllogisms: "accident" and"converse accident".

Simple induction

Simple induction proceeds from a premise about a samplegroup to a conclusion about another individual.

Proportion Q of the known instances of popu-lation P has attribute A.Individual I is another member of P.Therefore:There is a probability corresponding to Q thatI has A.

This is a combination of a generalization and a statisticalsyllogism, where the conclusion of the generalization isalso the first premise of the statistical syllogism.

Argument from analogy

Main article: Argument from analogy

The process of analogical inference involves noting theshared properties of two or more things, and from this ba-sis inferring that they also share some further property:[13]

P and Q are similar in respect to properties a,b, and c.Object P has been observed to have furtherproperty x.Therefore, Q probably has property x also.

Analogical reasoning is very frequent in common sense,science, philosophy and the humanities, but sometimes itis accepted only as an auxiliary method. A refined ap-proach is case-based reasoning.[14]

Causal inference

A causal inference draws a conclusion about a causal con-nection based on the conditions of the occurrence of aneffect. Premises about the correlation of two things canindicate a causal relationship between them, but addi-tional factors must be confirmed to establish the exactform of the causal relationship.

Prediction

A prediction draws a conclusion about a future individualfrom a past sample.

Proportion Q of observed members of group Ghave had attribute A.Therefore:There is a probability corresponding to Q thatother members of group G will have attributeA when next observed.

3.1.5 Bayesian inference

As a logic of induction rather than a theory of belief,Bayesian inference does not determine which beliefs area priori rational, but rather determines how we should ra-tionally change the beliefs we have when presented withevidence. We begin by committing to a prior probabilityfor a hypothesis based on logic or previous experience,and when faced with evidence, we adjust the strength ofour belief in that hypothesis in a precise manner usingBayesian logic.

3.1. INDUCTIVE REASONING 37

3.1.6 Inductive inference

Around 1960, Ray Solomonoff founded the theory of uni-versal inductive inference, the theory of prediction basedon observations; for example, predicting the next sym-bol based upon a given series of symbols. This is a for-mal inductive framework that combines algorithmic in-formation theory with the Bayesian framework. Univer-sal inductive inference is based on solid philosophicalfoundations,[15] and can be considered as a mathemati-cally formalized Occam’s razor. Fundamental ingredi-ents of the theory are the concepts of algorithmic proba-bility and Kolmogorov complexity.

3.1.7 See also

• Abductive reasoning

• Algorithmic information theory

• Algorithmic probability

• Analogy

• Bayesian probability

• Counterinduction

• Deductive reasoning

• Explanation

• Failure mode and effects analysis

• Falsifiability

• Grammar induction

• Inductive inference

• Inductive logic programming

• Inductive probability

• Inductive programming

• Inductive reasoning aptitude

• Inquiry

• Kolmogorov complexity

• Lateral thinking

• Laurence Jonathan Cohen

• Logic

• Logical positivism

• Machine learning

• Mathematical induction

• Mill’s Methods

• Minimum description length

• Minimum message length

• Open world assumption

• Raven paradox

• Recursive Bayesian estimation

• Retroduction

• Solomonoff’s theory of inductive inference

• Statistical inference

• Stephen Toulmin

• Universal artificial intelligence

3.1.8 References

[1] Copi, I. M.; Cohen, C.; Flage, D. E. (2007). Essentialsof Logic (Second ed.). Upper Saddle River, NJ: PearsonEducation. ISBN 978-0-13-238034-8.

[2] “Deductive and Inductive Arguments”, Internet Encyclo-pedia of Philosophy, Some dictionaries define “deduction”as reasoning from the general to specific and “induction”as reasoning from the specific to the general. While thisusage is still sometimes found even in philosophical andmathematical contexts, for the most part, it is outdated.

[3] Kosko, Bart (1990). “Fuzziness vs. Probability”. In-ternational Journal of General Systems 17 (1): 211–240.doi:10.1080/03081079008935108.

[4] John Vickers. The Problem of Induction. The StanfordEncyclopedia of Philosophy.

[5] Herms, D. “Logical Basis of Hypothesis Testing in Scien-tific Research” (pdf).

[6] “Stanford Encyclopedia of Philosophy : Kant’s account ofreason”.

[7] Sextus Empiricus, Outlines Of Pyrrhonism. Trans.R.G. Bury, Harvard University Press, Cambridge, Mas-sachusetts, 1933, p. 283.

[8] Popper, Karl R.; Miller, David W. (1983). “A proof ofthe impossibility of inductive probability”. Nature 302(5910): 687–688. doi:10.1038/302687a0.

[9] David Hume (1910) [1748]. An Enquiry concerning Hu-man Understanding. P.F. Collier & Son. ISBN 0-19-825060-6.

[10] Vickers, John. “The Problem of Induction” (Section 2).Stanford Encyclopedia of Philosophy. 21 June 2010

[11] Vickers, John. “The Problem of Induction” (Section 2.1).Stanford Encyclopedia of Philosophy. 21 June 2010.

[12] Gray, Peter (2011). Psychology (Sixth ed.). New York:Worth. ISBN 978-1-4292-1947-1.

38 CHAPTER 3. CHAPTER II INDUCTION

[13] Baronett, Stan (2008). Logic. Upper Saddle River, NJ:Pearson Prentice Hall. pp. 321–325.

[14] For more information on inferences by analogy, see Juthe,2005.

[15] Rathmanner, Samuel; Hutter, Marcus (2011). “A Philo-sophical Treatise of Universal Induction”. Entropy 13 (6):1076–1136. doi:10.3390/e13061076.

3.1.9 Further reading

• Cushan, Anna-Marie (1983/2014). Investigationinto Facts and Values: Groundwork for a theoryof moral conflict resolution. [Thesis, MelbourneUniversity], Ondwelle Publications (online): Mel-bourne.

• Herms, D. “Logical Basis of Hypothesis Testing inScientific Research” (PDF).

• Kemerling, G. (27 October 2001). “Causal Reason-ing”.

• Holland, J. H.; Holyoak, K. J.; Nisbett, R. E.; Tha-gard, P. R. (1989). Induction: Processes of Infer-ence, Learning, and Discovery. Cambridge, MA,USA: MIT Press. ISBN 0-262-58096-9.

• Holyoak, K.; Morrison, R. (2005). The Cam-bridge Handbook of Thinking and Reasoning. NewYork: Cambridge University Press. ISBN 978-0-521-82417-0.

3.1.10 External links

• Confirmation and Induction entry in the Internet En-cyclopedia of Philosophy

• Inductive Logic entry in the Stanford Encyclopediaof Philosophy

• Inductive reasoning at PhilPapers

• Inductive reasoning at the Indiana Philosophy On-tology Project

• Four Varieties of Inductive Argument from the De-partment of Philosophy, University of North Car-olina at Greensboro.

• Properties of Inductive Reasoning PDF (166 KiB), apsychological review by Evan Heit of the Universityof California, Merced.

• TheMind, LimberAn article which employs the filmThe Big Lebowski to explain the value of inductivereasoning.

• The Pragmatic Problem of Induction, by ThomasBullemore

3.2 Argument

This article is about the subject as it is studied inlogic and philosophy. For other uses, see Argument(disambiguation).

In logic and philosophy, an argument is a series of state-ments typically used to persuade someone of somethingor to present reasons for accepting a conclusion.[1][2] Thegeneral form of an argument in a natural language isthat of premises (typically in the form of propositions,statements or sentences) in support of a claim: theconclusion.[3][4][5] The structure of some arguments canalso be set out in a formal language, and formally defined“arguments” can be made independently of natural lan-guage arguments, as in math, logic, and computer sci-ence.In a typical deductive argument, the premises are meantto provide a guarantee of the truth of the conclusion,while in an inductive argument, they are thought to pro-vide reasons supporting the conclusion’s probable truth.[6]The standards for evaluating non-deductive argumentsmay rest on different or additional criteria than truth, forexample, the persuasiveness of so-called “indispensabil-ity claims” in transcendental arguments,[7] the quality ofhypotheses in retroduction, or even the disclosure of newpossibilities for thinking and acting.[8]

The standards and criteria used in evaluating argumentsand their forms of reasoning are studied in logic.[9]Ways of formulating arguments effectively are studied inrhetoric (see also: argumentation theory). An argumentin a formal language shows the logical form of the sym-bolically represented or natural language arguments ob-tained by its interpretations.

3.2.1 Formal and informal

Further information: Informal logic and Formal logic

Informal arguments as studied in informal logic, are pre-sented in ordinary language and are intended for every-day discourse. Conversely, formal arguments are studiedin formal logic (historically called symbolic logic, morecommonly referred to as mathematical logic today) andare expressed in a formal language. Informal logic maybe said to emphasize the study of argumentation, whereasformal logic emphasizes implication and inference. Infor-mal arguments are sometimes implicit. That is, the ratio-nal structure –the relationship of claims, premises, war-rants, relations of implication, and conclusion –is not al-ways spelled out and immediately visible and must some-times be made explicit by analysis.

3.2. ARGUMENT 39

3.2.2 Standard types

There are several kinds of arguments in logic, the best-known of which are “deductive” and “inductive.” De-ductive arguments are sometimes referred to as “truth-preserving” arguments, because the truth of the conclu-sion follows given that of the premises. A deductive ar-gument asserts that the truth of the conclusion is a logicalconsequence of the premises. An inductive argument, onthe other hand, asserts that the truth of the conclusion isotherwise supported by the premises. Each premise andthe conclusion are truth bearers or “truth-candidates”, ca-pable of being either true or false (and not both). Whilestatements in an argument are referred to as being eithertrue or false, arguments are referred to as being validor invalid (see logical truth). A deductive argument isvalid if and only if the truth of the conclusion is entailedby (is a logical consequence of) the premises, and itscorresponding conditional is therefore a logical truth. Asound argument is a valid argument with true premises;a valid argument may well have false premises under agiven interpretation, however, the truth value of a con-clusion cannot be determined by an unsound argument.

3.2.3 Deductive

Main article: Deductive argument

A deductive argument is one that, if valid, has a conclu-sion that is entailed by its premises. In other words, thetruth of the conclusion is a logical consequence of thepremises—if the premises are true, then the conclusionmust be true. It would be self-contradictory to assertthe premises and deny the conclusion, because the nega-tion of the conclusion is contradictory to the truth of thepremises.

Validity

Main article: Validity

Deductive arguments may be either valid or invalid. Ifan argument is valid, it is a valid deduction, and if itspremises are true, the conclusion must be true: a validargument cannot have true premises and a false conclu-sion.An argument is formally valid if and only if the denialof the conclusion is incompatible with accepting all thepremises.The validity of an argument depends, however, not on theactual truth or falsity of its premises and conclusion, butsolely on whether or not the argument has a valid logicalform. The validity of an argument is not a guarantee ofthe truth of its conclusion. Under a given interpretation,a valid argument may have false premises that render it

inconclusive: the conclusion of a valid argument with oneor more false premises may be either true or false.Logic seeks to discover the valid forms, the forms thatmake arguments valid. A form of argument is valid ifand only if the conclusion is true under all interpretationsof that argument in which the premises are true. Sincethe validity of an argument depends solely on its form,an argument can be shown to be invalid by showing thatits form is invalid. This can be done by giving a counterexample of the same form of argument with premises thatare true under a given interpretation, but a conclusion thatis false under that interpretation. In informal logic this iscalled a counter argument.The form of argument can be shown by the use of sym-bols. For each argument form, there is a correspondingstatement form, called a corresponding conditional, andan argument form is valid if and only its correspondingconditional is a logical truth. A statement form which islogically true is also said to be a valid statement form.A statement form is a logical truth if it is true under allinterpretations. A statement form can be shown to be alogical truth by either (a) showing that it is a tautology or(b) by means of a proof procedure.The corresponding conditional of a valid argument is anecessary truth (true in all possible worlds) and so theconclusion necessarily follows from the premises, or fol-lows of logical necessity. The conclusion of a valid ar-gument is not necessarily true, it depends on whether thepremises are true. If the conclusion, itself, just so hap-pens to be a necessary truth, it is so without regard to thepremises.Some examples:

• All Greeks are human and all humans are mortal;therefore, all Greeks are mortal. : Valid argument;if the premises are true the conclusion must be true.

• Some Greeks are logicians and some logicians aretiresome; therefore, some Greeks are tiresome. In-valid argument: the tiresome logicians might all beRomans (for example).

• Either we are all doomed or we are all saved; we arenot all saved; therefore, we are all doomed. Validargument; the premises entail the conclusion. (Re-member that this does not mean the conclusion hasto be true; it is only true if the premises are true,which they may not be!)

• Some men are hawkers. Some hawkers are rich.Therefore, some men are rich. Invalid argument.This can be easier seen by giving a counter-examplewith the same argument form:

• Some people are herbivores. Some herbivoresare zebras. Therefore, some people are ze-bras. Invalid argument, as it is possible that thepremises be true and the conclusion false.

40 CHAPTER 3. CHAPTER II INDUCTION

In the above second to last case (Some men are hawk-ers...), the counter-example follows the same logical formas the previous argument, (Premise 1: “Some X are Y.”Premise 2: “Some Y are Z.” Conclusion: “Some X areZ.”) in order to demonstrate that whatever hawkers maybe, they may or may not be rich, in consideration of thepremises as such. (See also, existential import).

The forms of argument that render deductions validare well-established, however some invalid argumentscan also be persuasive depending on their construction(inductive arguments, for example). (See also, formal fal-lacy and informal fallacy).

Soundness

Main article: Soundness

A sound argument is a valid argument whose conclusionfollows from its premise(s), and the premise(s) of whichis/are true.

3.2.4 Inductive

Main article: Inductive argument

Non-deductive logic is reasoning using arguments inwhich the premises support the conclusion but do not en-tail it. Forms of non-deductive logic include the statisticalsyllogism, which argues from generalizations true for themost part, and induction, a form of reasoning that makesgeneralizations based on individual instances. An induc-tive argument is said to be cogent if and only if the truthof the argument’s premises would render the truth of theconclusion probable (i.e., the argument is strong), and theargument’s premises are, in fact, true. Cogency can beconsidered inductive logic's analogue to deductive logic's"soundness.” Despite its name, mathematical induction isnot a form of inductive reasoning. The lack of deductivevalidity is known as the problem of induction.

3.2.5 Defeasible arguments and argumen-tation schemes

In modern argumentation theories, arguments are re-garded as defeasible passages from premises to a conclu-sion. Defeasibility means that when additional informa-tion (new evidence or contrary arguments) is provided,the premises may be no longer lead to the conclusion(non-monotonic reasoning). This type of reasoning is re-ferred to as defeasible reasoning. For instance we con-sider the famous Tweedy example:

Tweedy is a bird.Birds generally fly.Therefore, Tweedy (probably) flies.

This argument is reasonable and the premises support theconclusion unless additional information indicating thatthe case is an exception comes in. If Tweedy is a pen-guin, the inference is no longer justified by the premise.Defeasible arguments are based on generalizations thathold only in the majority of cases, but are subject to ex-ceptions and defaults. In order to represent and assessdefeasible reasoning, it is necessary to combine the logi-cal rules (governing the acceptance of a conclusion basedon the acceptance of its premises) with rules of materialinference, governing how a premise can support a givenconclusion (whether it is reasonable or not to draw a spe-cific conclusion from a specific description of a state ofaffairs). Argumentation schemes have been developed todescribe and assess the acceptability or the fallaciousnessof defeasible arguments. Argumentation schemes arestereotypical patterns of inference, combining semantic-ontological relations with types of reasoning and logicalaxioms and representing the abstract structure of themostcommon types of natural arguments.[10] The argumen-tation schemes provided in (Walton, Reed & Macagno,2008) describe tentatively the patterns of the most typ-ical arguments. However, the two levels of abstractionare not distinguished. For this reason, under the labelof “argumentation schemes” fall indistinctly patterns ofreasoning such as the abductive, analogical, or inductiveones, and types of argument such as the ones from classi-fication or cause to effect. A typical example is the argu-ment from expert opinion, which has two premises and aconclusion.[11]

Each scheme is associated to a set of critical questions,namely criteria for assessing dialectically the reasonable-ness and acceptability of an argument. Thematching crit-ical questions are the standard ways of casting the argu-ment into doubt.If an expert says that a proposition is true, this providesa reason for tentatively accepting it, in the absence ofstronger reasons to doubt it. But suppose that evidenceof financial gain suggests that the expert is biased, for ex-ample by evidence showing that he will gain financiallyfrom his claim.

3.2.6 By analogy

Argument by analogy may be thought of as argumentfrom the particular to particular. An argument by analogymay use a particular truth in a premise to argue towardsa similar particular truth in the conclusion. For example,if A. Plato was mortal, and B. Socrates was like Plato inother respects, then asserting that C. Socrates was mortalis an example of argument by analogy because the rea-soning employed in it proceeds from a particular truth ina premise (Plato was mortal) to a similar particular truthin the conclusion, namely that Socrates was mortal.[12]

3.2. ARGUMENT 41

3.2.7 Other kinds

Other kinds of arguments may have different or addi-tional standards of validity or justification. For exam-ple, Charles Taylor writes that so-called transcendentalarguments are made up of a “chain of indispensabilityclaims” that attempt to show why something is neces-sarily true based on its connection to our experience,[13]while Nikolas Kompridis has suggested that there aretwo types of “fallible” arguments: one based on truthclaims, and the other based on the time-responsive dis-closure of possibility (see world disclosure).[14] The lateFrench philosopher Michel Foucault is said to have beena prominent advocate of this latter form of philosophicalargument.[15]

In informal logic

Argument is an informal calculus, relating an effort to beperformed or sum to be spent, to possible future gain,either economic or moral. In informal logic, an argumentis a connexion between

1. an individual action

2. through which a generally accepted good is obtained.

Ex :

1. (a) You should marry Jane (individual action, in-dividual decision)

(b) because she has the same temper as you. (gen-erally accepted wisdom that marriage is goodin itself, and it is generally accepted that peo-ple with the same character get along well).

2. (a) You should not smoke (individual action, indi-vidual decision)

(b) because smoking is harmful (generally ac-cepted wisdom that health is good).

The argument is neither a) advice nor b) moral or eco-nomical judgement, but the connection between the two.An argument always uses the connective because. Anargument is not an explanation. It does not connect twoevents, cause and effect, which already took place, but apossible individual action and its beneficial outcome. Anargument is not a proof. A proof is a logical and cog-nitive concept; an argument is a praxeologic concept. Aproof changes our knowledge; an argument compels us toact.

Logical status Argument does not belong to logic, be-cause it is connected to a real person, a real event, and areal effort to be made.

1. If you, John, will buy this stock, it will become twiceas valuable in a year.

2. If you, Mary, study dance, you will become a fa-mous ballet dancer.

The value of the argument is connected to the immedi-ate circumstances of the person spoken to. If, in the firstcase,(1) John has no money, or will die the next year, hewill not be interested in buying the stock. If, in the secondcase (2) she is too heavy, or too old, she will not be inter-ested in studying and becoming a dancer. The argumentis not logical, but profitable.

World-disclosing

Main article: World disclosure

World-disclosing arguments are a group of philosophicalarguments that are said to employ a disclosive approach,to reveal features of a wider ontological or cultural-linguistic understanding – a “world,” in a specifically on-tological sense – in order to clarify or transform the back-ground of meaning and “logical space” on which an argu-ment implicitly depends.[16]

3.2.8 Explanations

Main article: Explanation

While arguments attempt to show that something was, is,will be, or should be the case, explanations try to showwhy or how something is or will be. If Fred and Joe ad-dress the issue of whether or not Fred’s cat has fleas, Joemay state: “Fred, your cat has fleas. Observe, the cat isscratching right now.” Joe has made an argument that thecat has fleas. However, if Joe asks Fred, “Why is yourcat scratching itself?" the explanation, "...because it hasfleas.” provides understanding.Both the above argument and explanation require know-ing the generalities that a) fleas often cause itching, and b)that one often scratches to relieve itching. The differenceis in the intent: an argument attempts to settle whether ornot some claim is true, and an explanation attempts toprovide understanding of the event. Note, that by sub-suming the specific event (of Fred’s cat scratching) as aninstance of the general rule that “animals scratch them-selves when they have fleas”, Joe will no longer wonderwhy Fred’s cat is scratching itself. Arguments addressproblems of belief, explanations address problems of un-derstanding. Also note that in the argument above, thestatement, “Fred’s cat has fleas” is up for debate (i.e. isa claim), but in the explanation, the statement, “Fred’scat has fleas” is assumed to be true (unquestioned at thistime) and just needs explaining.[17]

42 CHAPTER 3. CHAPTER II INDUCTION

Arguments and explanations largely resemble each otherin rhetorical use. This is the cause of much difficulty inthinking critically about claims. There are several reasonsfor this difficulty.

• People often are not themselves clear on whetherthey are arguing for or explaining something.

• The same types of words and phrases are used inpresenting explanations and arguments.

• The terms 'explain' or 'explanation,' et cetera are fre-quently used in arguments.

• Explanations are often used within arguments andpresented so as to serve as arguments.[18]

• Likewise, "...arguments are essential to the pro-cess of justifying the validity of any explanation asthere are often multiple explanations for any givenphenomenon.”[17]

Explanations and arguments are often studied in the fieldof Information Systems to help explain user acceptance ofknowledge-based systems. Certain argument types mayfit better with personality traits to enhance acceptance byindividuals.[19]

3.2.9 Fallacies and nonarguments

Main article: Formal fallacy

Fallacies are types of argument or expressions which areheld to be of an invalid form or contain errors in reason-ing. There is not as yet any general theory of fallacy orstrong agreement among researchers of their definitionor potential for application but the term is broadly ap-plicable as a label to certain examples of error, and alsovariously applied to ambiguous candidates.[20]

In Logic types of fallacy are firmly described thus: Firstthe premises and the conclusion must be statements, ca-pable of being true or false. Secondly it must be assertedthat the conclusion follows from the premises. In Englishthe words therefore, so, because and hence typically sep-arate the premises from the conclusion of an argument,but this is not necessarily so. Thus: Socrates is a man, allmen are mortal therefore Socrates is mortal is clearly anargument (a valid one at that), because it is clear it is as-serted that Socrates is mortal follows from the precedingstatements. However I was thirsty and therefore I drank isNOT an argument, despite its appearance. It is not beingclaimed that I drank is logically entailed by I was thirsty.The therefore in this sentence indicates for that reason notit follows that.

Elliptical arguments

Often an argument is invalid because there is a miss-ing premise—the supply of which would render it valid.Speakers and writers will often leave out a strictly neces-sary premise in their reasonings if it is widely acceptedand the writer does not wish to state the blindingly obvi-ous. Example: All metals expand when heated, thereforeiron will expand when heated. (Missing premise: iron isa metal). On the other hand, a seemingly valid argumentmay be found to lack a premise – a 'hidden assumption' –which if highlighted can show a fault in reasoning. Exam-ple: A witness reasoned: Nobody came out the front doorexcept the milkman; therefore the murderer must have leftby the back door. (Hidden assumptions- the milkman wasnot the murderer, and the murderer has left by the frontor back door).

3.2.10 See also

• Abductive reasoning

• Argument map

• Argumentation theory

• Argumentative dialogue

• Belief bias

• Boolean logic

• Deductive reasoning

• Defeasible reasoning

• Evidence

• Evidence-based policy

• Fallacy

• Dialectic

• Formal fallacy

• Inductive reasoning

• Informal fallacy

• Inquiry

• Practical arguments

• Soundness theorem

• Soundness

• Truth

• Validity

3.2. ARGUMENT 43

3.2.11 Notes[1] “Argument”, Internet Encyclopedia of Philosophy.” “In

everyday life, we often use the word “argument” to meana verbal dispute or disagreement. This is not the way thisword is usually used in philosophy. However, the twouses are related. Normally, when two people verbally dis-agree with each other, each person attempts to convincethe other that his/her viewpoint is the right one. Unlesshe or she merely results to name calling or threats, he orshe typically presents an argument for his or her position,in the sense described above. In philosophy, “arguments”are those statements a person makes in the attempt to con-vince someone of something, or present reasons for ac-cepting a given conclusion.”

[2] Ralph H. Johnson, Manifest Rationality: A pragmatic the-ory of argument (New Jersey: Laurence Erlbaum, 2000),46-49.

[3] Ralph H. Johnson, Manifest Rationality: A pragmatic the-ory of argument (New Jersey: Laurence Erlbaum, 2000),46.

[4] The Cambridge Dictionary of Philosophy, 2nd Ed. CUM,1995 “Argument: a sequence of statements such that someof them (the premises) purport to give reason to acceptanother of them, the conclusion”

[5] Stanford Enc. Phil., Classical Logic

[6] “Deductive and Inductive Arguments,” Internet Encyclo-pedia of Philosophy.

[7] hCharles Taylor, “The Validity of Transcendental Argu-ments”, Philosophical Arguments (Harvard, 1995), 20-33."[Transcendental] arguments consist of a string of whatone could call indispensability claims. They move fromtheir starting points to their conclusions by showing thatthe condition stated in the conclusion is indispensable tothe feature identified at the start… Thus we could spellout Kant’s transcendental deduction in the first edition inthree stages: experience must have an object, that is, beof something; for this it must be coherent; and to be co-herent it must be shaped by the understanding through thecategories.”

[8] Kompridis, Nikolas (2006). “World Disclosing Argu-ments?". Critique andDisclosure. Cambridge: MIT Press.pp. 116–124. ISBN 0262277425.

[9] “Argument”, Internet Encyclopedia of Philosophy.”

[10] Macagno, Fabrizio; Walton, Douglas (2015). “Classifyingthe patterns of natural arguments”. Philosophy & Rhetoric. 48 (1): 26–53.

[11] Walton, Douglas; Reed, Chris; Macagno, Fabrizio (2008).Argumentation Schemes. New York: Cambridge Univer-sity Press. p. 310.

[12] Shaw 1922: p. 74.

[13] Charles Taylor, “The Validity of Transcendental Argu-ments”, Philosophical Arguments (Harvard, 1995), 20-33.

[14] Nikolas Kompridis, “Two Kinds of Fallibilism”, Critiqueand Disclosure (Cambridge: MIT Press, 2006), 180-183.

[15] In addition, Foucault said of his own approach that “Myrole ... is to show people that they are much freer thanthey feel, that people accept as truth, as evidence, somethemes which have been built up at a certain moment dur-ing history, and that this so-called evidence can be criti-cized and destroyed.” He also wrote that he was engaged in“the process of putting historico-critical reflection to thetest of concrete practices…I continue to think that this taskrequires work on our limits, that is, a patient labor givingform to our impatience for liberty.” (emphasis added) Hu-bert Dreyfus, "Being and Power: Heidegger and Foucault"and Michel Foucault, “What is Enlightenment?"

[16] Nikolas Kompridis, “World Disclosing Arguments?" inCritique and Disclosure, Cambridge:MIT Press (2006),118-121.

[17] JONATHAN F. OSBORNE, ALEXIS PATTERSONSchool of Education, Stanford University, Stanford, CA94305, USA Received 27 August 2010; revised 22November 2010; accepted 29 November 2010 DOI10.1002/sce.20438 Published online 23 May 2011 in Wi-ley Online Library (wileyonlinelibrary.com)

[18] Critical Thinking, Parker and Moore

[19] Justin Scott Giboney, Susan Brown, and Jay F. Nuna-maker Jr. (2012). “User Acceptance of Knowledge-Based System Recommendations: Explanations, Argu-ments, and Fit” 45th Annual Hawaii International Con-ference on System Sciences, Hawaii, January 5–8.

[20]

3.2.12 References

• Shaw, Warren Choate (1922). The Art of Debate.Allyn and Bacon. p. 74.

• Robert Audi, Epistemology, Routledge, 1998. Par-ticularly relevant is Chapter 6, which explores therelationship between knowledge, inference and ar-gument.

• J. L. Austin How to Do Things With Words, OxfordUniversity Press, 1976.

• H. P. Grice, Logic and Conversation in The Logic ofGrammar, Dickenson, 1975.

• Vincent F. Hendricks, Thought 2 Talk: A CrashCourse in Reflection and Expression, New York: Au-tomatic Press / VIP, 2005, ISBN 87-991013-7-8

• R. A. DeMillo, R. J. Lipton and A. J. Perlis, SocialProcesses and Proofs of Theorems and Programs,Communications of the ACM, Vol. 22, No. 5,1979. A classic article on the social process of ac-ceptance of proofs in mathematics.

• Yu. Manin, A Course in Mathematical Logic,Springer Verlag, 1977. A mathematical view oflogic. This book is different from most books on

44 CHAPTER 3. CHAPTER II INDUCTION

mathematical logic in that it emphasizes the mathe-matics of logic, as opposed to the formal structureof logic.

• Ch. Perelman and L. Olbrechts-Tyteca, The NewRhetoric, Notre Dame, 1970. This classic was orig-inally published in French in 1958.

• Henri Poincaré, Science and Hypothesis, Dover Pub-lications, 1952

• Frans van Eemeren and Rob Grootendorst, SpeechActs in Argumentative Discussions, Foris Publica-tions, 1984.

• K. R. Popper Objective Knowledge; An EvolutionaryApproach, Oxford: Clarendon Press, 1972.

• L. S. Stebbing, A Modern Introduction to Logic,Methuen and Co., 1948. An account of logic thatcovers the classic topics of logic and argument whilecarefully considering modern developments in logic.

• Douglas Walton, Informal Logic: A Handbook forCritical Argumentation, Cambridge, 1998.

• Walton, Douglas; Christopher Reed; FabrizioMacagno, Argumentation Schemes, New York:Cambridge University Press, 2008.

• Carlos Chesñevar, Ana Maguitman and RonaldLoui, Logical Models of Argument, ACM Comput-ing Surveys, vol. 32, num. 4, pp. 337–383, 2000.

• T. Edward Damer. Attacking Faulty Reasoning, 5thEdition, Wadsworth, 2005. ISBN 0-534-60516-8

• Charles Arthur Willard, A Theory of Argumenta-tion. 1989.

• Charles Arthur Willard, Argumentation and the So-cial Grounds of Knowledge. 1982.

3.2.13 Further reading• Salmon, Wesley C. Logic. New Jersey: Prentice-Hall (1963). Library of Congress Catalog Card no.63-10528.

• Aristotle, Prior and Posterior Analytics. Ed. andtrans. John Warrington. London: Dent (1964)

• Mates, Benson. Elementary Logic. NewYork: OUP(1972). Library of Congress Catalog Card no. 74-166004.

• Mendelson, Elliot. Introduction to MathematicalLogic. NewYork: VanNostran Reinholds Company(1964).

• Frege, Gottlob. The Foundations of Arith-metic. Evanston, IL: Northwestern University Press(1980).

• Martin, Brian. The Controversy Manual (Sparsnäs,Sweden: Irene Publishing, 2014).

3.2.14 External links

• Argument at PhilPapers

• Argument at the Indiana Philosophy OntologyProject

• Argument entry in the Internet Encyclopedia of Phi-losophy

Chapter 4

Chapter III Deduction

4.1 Deductive reasoning

Deductive reasoning, also deductive logic, logical de-duction or, informally, "top-down" logic,[1] is the pro-cess of reasoning from one or more statements (premises)to reach a logically certain conclusion.[2] It differs frominductive reasoning or abductive reasoning.Deductive reasoning links premises with conclusions. Ifall premises are true, the terms are clear, and the rules ofdeductive logic are followed, then the conclusion reachedis necessarily true.Deductive reasoning (top-down logic) contrasts withinductive reasoning (bottom-up logic) in the followingway: In deductive reasoning, a conclusion is reachedreductively by applying general rules that hold over theentirety of a closed domain of discourse, narrowing therange under consideration until only the conclusion(s) isleft. In inductive reasoning, the conclusion is reached bygeneralizing or extrapolating from, i.e., there is epistemicuncertainty. However, the inductive reasoning mentionedhere is not the same as induction used in mathematicalproofs – mathematical induction is actually a form of de-ductive reasoning.

4.1.1 Simple example

An example of a deductive argument:

1. All men are mortal.

2. Socrates is a man.

3. Therefore, Socrates is mortal.

The first premise states that all objects classified as “men”have the attribute “mortal”. The second premise statesthat “Socrates” is classified as a “man” – a member ofthe set “men”. The conclusion then states that “Socrates”must be “mortal” because he inherits this attribute fromhis classification as a “man”.

4.1.2 Law of detachment

Main article: Modus ponens

The law of detachment (also known as affirming the an-tecedent andModus ponens) is the first form of deduc-tive reasoning. A single conditional statement is made,and a hypothesis (P) is stated. The conclusion (Q) is thendeduced from the statement and the hypothesis. Themostbasic form is listed below:

1. P → Q (conditional statement)

2. P (hypothesis stated)

3. Q (conclusion deduced)

In deductive reasoning, we can conclude Q from P by us-ing the law of detachment.[3] However, if the conclusion(Q) is given instead of the hypothesis (P) then there is nodefinitive conclusion.The following is an example of an argument using the lawof detachment in the form of an if-then statement:

1. If an angle satisfies 90° < A < 180°, then A is anobtuse angle.

2. A = 120°.

3. A is an obtuse angle.

Since the measurement of angle A is greater than 90° andless than 180°, we can deduce that A is an obtuse angle. Ifhowever, we are given the conclusion that A is an obtuseangle we cannot deduce the premise that A = 120°.

4.1.3 Law of syllogism

The law of syllogism takes two conditional statements andforms a conclusion by combining the hypothesis of onestatement with the conclusion of another. Here is the gen-eral form:

1. P → Q

45

46 CHAPTER 4. CHAPTER III DEDUCTION

2. Q → R

3. Therefore, P → R.

The following is an example:

1. If Larry is sick, then he will be absent.

2. If Larry is absent, then he will miss his classwork.

3. Therefore, if Larry is sick, then he will miss hisclasswork.

We deduced the final statement by combining the hypoth-esis of the first statement with the conclusion of the sec-ond statement. We also allow that this could be a falsestatement. This is an example of the transitive propertyin mathematics. The transitive property is sometimesphrased in this form:

1. A = B.

2. B = C.

3. Therefore, A = C.

4.1.4 Law of contrapositive

Main article: Modus tollens

The law of contrapositive states that, in a conditional, ifthe conclusion is false, then the hypothesis must be falsealso. The general form is the following:

1. P → Q.

2. ~Q.

3. Therefore, we can conclude ~P.

The following are examples:

1. If it is raining, then there are clouds in the sky.

2. There are no clouds in the sky.

3. Thus, it is not raining.

4.1.5 Validity and soundness

Deductive arguments are evaluated in terms of theirvalidity and soundness.An argument is valid if it is impossible for its premisesto be true while its conclusion is false. In other words,the conclusion must be true if the premises are true. Anargument can be valid even though the premises are false.An argument is sound if it is valid and the premises aretrue.

It is possible to have a deductive argument that is logicallyvalid but is not sound. Fallacious arguments often takethat form.The following is an example of an argument that is valid,but not sound:

1. Everyone who eats carrots is a quarterback.

2. John eats carrots.

3. Therefore, John is a quarterback.

The example’s first premise is false – there are people whoeat carrots and are not quarterbacks – but the conclusionmust be true, so long as the premises are true (i.e. it isimpossible for the premises to be true and the conclusionfalse). Therefore, the argument is valid, but not sound.Generalizations are often used tomake invalid arguments,such as “everyone who eats carrots is a quarterback.” Noteveryone who eats carrots is a quarterback, thus provingthe flaw of such arguments.In this example, the first statement uses categorical rea-soning, saying that all carrot-eaters are definitely quarter-backs. This theory of deductive reasoning – also knownas term logic – was developed by Aristotle, but was su-perseded by propositional (sentential) logic and predicatelogic.Deductive reasoning can be contrasted with inductive rea-soning, in regards to validity and soundness. In cases ofinductive reasoning, even though the premises are trueand the argument is “valid”, it is possible for the conclu-sion to be false (determined to be false with a counterex-ample or other means).

4.1.6 History

Aristotle started documenting deductive reasoning in the4th century BC.[4]

4.1.7 Education

Deductive reasoning is generally considered to be a skillthat develops without any formal teaching or training. Asa result of this belief, deductive reasoning skills are nottaught in secondary schools, where students are expectedto use reasoning more often and at a higher level.[5] It isin high school, for example, that students have an abruptintroduction to mathematical proofs – which rely heavilyon deductive reasoning.[5]

4.1.8 See also

• Abductive reasoning

• Analogical reasoning

4.2. VALIDITY 47

• Argument (logic)

• Correspondence theory of truth

• Decision making

• Decision theory

• Defeasible reasoning

• Fallacy

• Fault Tree Analysis

• Geometry

• Hypothetico-deductive method

• Inductive reasoning

• Inference

• Inquiry

• Logic

• Logical consequence

• Mathematical induction

• Mathematical logic

• Natural deduction

• Propositional calculus

• Retroductive reasoning

• Scientific method

• Soundness

• Syllogism

• Theory of justification

4.1.9 References[1] Deduction & Induction, Research Methods Knowledge

Base

[2] Sternberg, R. J. (2009). Cognitive Psychology. Belmont,CA: Wadsworth. p. 578. ISBN 978-0-495-50629-4.

[3] Guide to Logic

[4] Evans, Jonathan St. B. T.; Newstead, Stephen E.; Byrne,Ruth M. J., eds. (1993). Human Reasoning: The Psy-chology of Deduction (Reprint ed.). Psychology Press. p.4. ISBN 9780863773136. Retrieved 2015-01-26. In onesense [...] one can see the psychology of deductive reason-ing as being as old as the study of logic, which originatedin the writings of Aristotle.

[5] Stylianides, G. J.; Stylianides (2008). “A. J.”. Math-ematical Thinking and Learning 10 (2): 103–133.doi:10.1080/10986060701854425.

4.1.10 Further reading

• Vincent F. Hendricks, Thought 2 Talk: A CrashCourse in Reflection and Expression, New York: Au-tomatic Press / VIP, 2005, ISBN 87-991013-7-8

• Philip Johnson-Laird, Ruth M. J. Byrne, Deduction,Psychology Press 1991, ISBN 978-0-86377-149-1

• Zarefsky, David, Argumentation: The Study of Ef-fective Reasoning Parts I and II, The Teaching Com-pany 2002

• Bullemore, Thomas, * The Pragmatic Problem ofInduction.

4.1.11 External links

• Deductive reasoning at PhilPapers

• Deductive reasoning at the Indiana Philosophy On-tology Project

• Deductive reasoning entry in the Internet Encyclope-dia of Philosophy

4.2 Validity

For other uses, see Validity (disambiguation).

In logic, an argument is valid if and only if it takes aform that makes it impossible for the premises to be trueand the conclusion nevertheless to be false.[1] It is not re-quired that a valid argument have premises that are actu-ally true,[2] but to have premises that, if they were true,would guarantee the truth of the argument’s conclusion.A formula is valid if and only if it is true under everyinterpretation, and an argument form (or schema) is validif and only if every argument of that logical form is valid.

4.2.1 Validity of arguments

An argument is valid if and only if the truth of its premisesentails the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid.Under such conditions it would be self-contradictoryto affirm the premises and deny the conclusion. Thecorresponding conditional of a valid argument is a logicaltruth and the negation of its corresponding conditional isa contradiction. The conclusion is a logical consequenceof its premises.An argument that is not valid is said to be “invalid”.An example of a valid argument is given by the followingwell-known syllogism:

48 CHAPTER 4. CHAPTER III DEDUCTION

All men are mortal.Socrates is a man.Therefore, Socrates is mortal.

What makes this a valid argument is not that it has truepremises and a true conclusion, but the logical necessityof the conclusion, given the two premises. The argumentwould be just as valid were the premises and conclusionfalse. The following argument is of the same logical formbut with false premises and a false conclusion, and it isequally valid:

All cups are green.Socrates is a cup.Therefore, Socrates is green.

Nomatter how the universemight be constructed, it couldnever be the case that these arguments should turn out tohave simultaneously true premises but a false conclusion.The above arguments may be contrasted with the follow-ing invalid one:

All men are immortal.Socrates is a man.Therefore, Socrates is mortal.

In this case, the conclusion contradicts the deductive logicof the preceding premises, rather than deriving from it.Therefore, the argument is logically 'invalid', even thoughthe conclusion could be considered 'true' in general terms.The premise 'All men are immortal' would likewise bedeemed false outside of the framework of classical logic.However, within that system 'true' and 'false' essentiallyfunction more like mathematical states such as binary 1sand 0s than the philosophical concepts normally associ-ated with those terms.A standard view is that whether an argument is valid is amatter of the argument’s logical form. Many techniquesare employed by logicians to represent an argument’s log-ical form. A simple example, applied to two of the aboveillustrations, is the following: Let the letters 'P', 'Q', and'S' stand, respectively, for the set of men, the set of mor-tals, and Socrates. Using these symbols, the first argu-ment may be abbreviated as:

All P are Q.S is a P.Therefore, S is a Q.

Similarly, the third argument becomes:

All P are not Q.S is a P.Therefore, S is a Q.

An argument is termed formally valid if it has struc-tural self-consistency, i.e. if when the operands betweenpremises are all true the derived conclusion is always alsotrue. In the third example, the initial premises cannot log-ically result in the conclusion and is therefore categorizedas an invalid argument.

4.2.2 Valid formula

Main article: Well-formed formula

A formula of a formal language is a valid formula if andonly if it is true under every possible interpretation of thelanguage. In propositional logic, they are tautologies.

4.2.3 Validity of statements

A statement can be called valid, i.e. logical truth, if it istrue in all interpretations.

4.2.4 Validity and soundness

Validity of deduction is not affected by the truth of thepremise or the truth of the conclusion. The following de-duction is perfectly valid:

All animals live on Mars.All humans are animals.Therefore, all humans live on Mars.

The problem with the argument is that it is not sound. Inorder for a deductive argument to be sound, the deductionmust be valid and all the premises true.

4.2.5 Satisfiability and validity

Main article: Satisfiability

Model theory analyzes formulae with respect to particularclasses of interpretation in suitable mathematical struc-tures. On this reading, formula is valid if all such inter-pretations make it true. An inference is valid if all inter-pretations that validate the premises validate the conclu-sion. This is known as semantic validity.[3]

4.2.6 Preservation

In truth-preserving validity, the interpretation underwhich all variables are assigned a truth value of 'true' pro-duces a truth value of 'true'.In a false-preserving validity, the interpretation underwhich all variables are assigned a truth value of 'false' pro-duces a truth value of 'false'.[4]

4.3. SOUNDNESS 49

4.2.7 n-Validity

A formula A of a first order language Q is n-valid iff itis true for every interpretation of Q that has a domain ofexactly n members.

ω-Validity

A formula of a first order language is ω-valid if and onlyif it is true for every interpretation of the language and ithas a domain with an infinite number of members.

4.2.8 See also

• Entailment

• Grounds of validity of scientific reasoning

• Reductio ad absurdum

• Mathematical fallacies

• Soundness

4.2.9 References[1] http://www.iep.utm.edu/val-snd/

[2] Beall, Jc and Restall, Greg, “Logical Consequence”, TheStanford Encyclopedia of Philosophy (Fall 2014 Edition),Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2014/entries/logical-consequence/>

[3] L. T. F. Gamut, Logic, Language, and Meaning: Introduc-tion to logic, 1991, p. 115

[4] Robert Cogan,"Critical thinking: step by step”, UniversityPress of America, 1998, p48

• Barwise, Jon; Etchemendy, John. Language, Proofand Logic (1999): 42.

• Beer, Francis A. “Validities: A Political SciencePerspective”, Social Epistemology 7, 1 (1993): 85-105.

4.2.10 External links

4.3 Soundness

In mathematical logic, a logical system has the sound-ness property if and only if its inference rules prove onlyformulas that are valid with respect to its semantics. Inmost cases, this comes down to its rules having the prop-erty of preserving truth, but this is not the case in general.

4.3.1 Of arguments

An argument is sound if and only if1. The argument is valid, and 2. All of its premises aretrue.For instance,

All men are mortal.Socrates is a man.Therefore, Socrates is mortal.

The argument is valid (because the conclusion is truebased on the premises, that is, that the conclusion followsthe premises) and since the premises are in fact true, theargument is sound.The following argument is valid but not sound:

All organisms with wings can fly.Penguins have wings.Therefore, penguins can fly.

Since the first premise is actually false, the argument,though valid, is not sound.

4.3.2 sounds and unsounds

Soundness is among the most fundamental properties ofmathematical logic. The soundness property providesthe initial reason for counting a logical system as desir-able. The completeness property means that every valid-ity (truth) is provable. Together they imply that all andonly validities are provable.Most proofs of soundness are trivial. For example, in anaxiomatic system, proof of soundness amounts to veri-fying the validity of the axioms and that the rules of in-ference preserve validity (or the weaker property, truth).Most axiomatic systems have only the rule of modus po-nens (and sometimes substitution), so it requires only ver-ifying the validity of the axioms and one rule of inference.Soundness properties come in two main varieties: weakand strong soundness, of which the former is a restrictedform of the latter.

Sound

Soundness of a deductive system is the property that anysentence that is provable in that deductive system is alsotrue on all interpretations or structures of the semantictheory for the language upon which that theory is based.In symbols, where S is the deductive system, L the lan-guage together with its semantic theory, and P a sentenceof L: if ⊢S P, then also ⊨L P.

50 CHAPTER 4. CHAPTER III DEDUCTION

un soundness

Strong soundness of a deductive system is the propertythat any sentence P of the language upon which the de-ductive system is based that is derivable from a set Γ ofsentences of that language is also a logical consequence ofthat set, in the sense that any model that makes all mem-bers of Γ true will also make P true. In symbols whereΓ is a set of sentences of L: if Γ ⊢S P, then also Γ ⊨L P.Notice that in the statement of strong soundness, when Γis empty, we have the statement of weak soundness.

Arithmetic soundness

If T is a theory whose objects of discourse can be in-terpreted as natural numbers, we say T is arithmeticallysound if all theorems of T are actually true about the stan-dard mathematical integers. For further information, seeω-consistent theory.

4.3.3 Relation to completeness

The converse of the soundness property is the semanticcompleteness property. A deductive system with a se-mantic theory is strongly complete if every sentence Pthat is a semantic consequence of a set of sentences Γcan be derived in the deduction system from that set. Insymbols: whenever Γ ⊨ P, then also Γ ⊢ P. Complete-ness of first-order logic was first explicitly established byGödel, though some of the main results were containedin earlier work of Skolem.Informally, a soundness theorem for a deductive systemexpresses that all provable sentences are true. Complete-ness states that all true sentences are provable.Gödel’s first incompleteness theorem shows that for lan-guages sufficient for doing a certain amount of arith-metic, there can be no effective deductive system thatis complete with respect to the intended interpretationof the symbolism of that language. Thus, not all sounddeductive systems are complete in this special senseof completeness, in which the class of models (up toisomorphism) is restricted to the intended one. The origi-nal completeness proof applies to all classical models, notsome special proper subclass of intended ones.

4.3.4 See also

• Validity

• Completeness (logic)

4.3.5 References

• Hinman, P. (2005). Fundamentals of MathematicalLogic. A K Peters. ISBN 1-56881-262-0.

• Copi, Irving (1979), Symbolic Logic (5th ed.),Macmillan Publishing Co., ISBN 0-02-324880-7

• Boolos, Burgess, Jeffrey. Computability and Logic,4th Ed, Cambridge, 2002.

4.3.6 External links

• Validity and Soundness in the Internet Encyclopediaof Philosophy.

Chapter 5

Index

5.1 List of logic symbols

In logic, a set of symbols is commonly used to expresslogical representation. As logicians are familiar withthese symbols, they are not explained each time they areused. So, for students of logic, the following table listsmany common symbols together with their name, pro-nunciation, and the related field of mathematics. Addi-tionally, the third column contains an informal defini-tion, the fourth column gives a short example, the fifthand sixth give the unicode location and name for use inHTML documents.[1] The last column provides the La-TeX symbol.It should be noted that, outside logic, different symbolshave the same meaning, and the same symbol has, de-pending on the context, different meanings.

5.1.1 Basic logic symbols

5.1.2 Advanced and rarely used logicalsymbols

These symbols are sorted by their Unicode value:

• U+00B7 · middle dot, an outdated way for denotingAND,[3] still in use in electronics; for example “A·B”is the same as “A&B”

• ·: Center dot with a line above it. Outdated way fordenoting NAND, for example “A·B” is the same as“A NAND B” or “A|B” or "¬(A & B)". See alsoUnicode U+22C5 ⋅ dot operator.

• U+0305 combining overline, used as abbreviationfor standard numerals (Typographical Number The-ory). For example, using HTML style “4" is a short-hand for the standard numeral “SSSS0”.

• Overline, is also a rarely used format for denotingGödel numbers, for example “AVB” says the Gödelnumber of "(AVB)"

• Overline is also an outdated way for denoting nega-tion, still in use in electronics; for example “AVB”is the same as "¬(AVB)"

• U+2191 ↑ upwards arrow or U+007C | vertical line:Sheffer stroke, the sign for the NAND operator.

• U+2201 ∁ complement

• U+2204 ∄ there does not exist: strike out existentialquantifier same as "¬∃"

• U+2234 ∴ therefore

• U+2235 ∵ because

• U+22A7 ⊧ models: is a model of

• U+22A8 ⊨ true: is true of

• U+22AC ⊬ does not prove: negated ⊢, the sign for“does not prove”, for example T ⊬ P says "P is nota theorem of T"

• U+22AD ⊭ not true: is not true of

• U+22BC ⊼ nand: another NAND operator, can alsobe rendered as ∧

• U+22BD ⊽ nor: another NOR operator, can also berendered as V

• U+25C7 white diamond: modal operator for “itis possible that”, “it is not necessarily not” or rarely“it is not provable not” (in most modal logics it isdefined as "¬◻¬")

• U+22C6 ⋆ star operator: usually used for ad-hoc op-erators

• U+22A5 ⊥ up tack or U+2193 ↓ downwards arrow:Webb-operator or Peirce arrow, the sign for NOR.Confusingly, "⊥" is also the sign for contradiction orabsurdity.

• U+2310 reversed not sign

• U+231C top left corner and U+231D top rightcorner: corner quotes, also called “Quine quotes";for quasi-quotation, i.e. quoting specific context ofunspecified (“variable”) expressions;[4] also used fordenoting Gödel number;[5] for example "G" de-notes the Gödel number of G. (Typographical note:although the quotes appears as a “pair” in unicode(231C and 231D), they are not symmetrical in some

51

52 CHAPTER 5. INDEX

fonts. And in some fonts (for example Arial) theyare only symmetrical in certain sizes. Alternativelythe quotes can be rendered as ⌈ and ⌉ (U+2308 andU+2309) or by using a negation symbol and a re-versed negation symbol ¬ in superscript mode. )

• U+25FB ◻ white medium square or U+25A1 white square: modal operator for “it is neces-sary that” (in modal logic), or “it is provable that”(in provability logic), or “it is obligatory that” (indeontic logic), or “it is believed that” (in doxasticlogic).

Note that the following operators are rarely supported bynatively installed fonts. If you wish to use these in a webpage, you should always embed the necessary fonts so thepage viewer can see the web page without having the nec-essary fonts installed in their computer.

• U+27E1 ⟡ white concave-sided diamond

• U+27E2 ⟢ white concave-sided diamond with left-wards tick: modal operator for was never

• U+27E3⟣white concave-sided diamondwith right-wards tick: modal operator for will never be

• U+27E4 ⟤ white square with leftwards tick: modaloperator for was always

• U+27E5 ⟥ white square with rightwards tick:modal operator for will always be

• U+297D right fish tail: sometimes used for “rela-tion”, also used for denoting various ad hoc relations(for example, for denoting “witnessing” in the con-text of Rosser’s trick) The fish hook is also used asstrict implication by C.I.Lewis p q ≡ (p → q), the corresponding LaTeX macro is \strictif. Seehere for an image of glyph. Added to Unicode 3.2.0.

Poland and Germany

As of 2014 in Poland, the universal quantifier is some-times written ∧ and the existential quantifier as ∨ . Thesame applies for Germany.

5.1.3 See also

• Józef Maria Bocheński

• List of notation used in Principia Mathematica

• List of mathematical symbols

• Logic alphabet, a suggested set of logical symbols

• Logical connective

• Mathematical operators and symbols in Unicode

• Polish notation

• Truth function

• Truth table

5.1.4 References[1] “Named character references”. HTML 5.1 Nightly. W3C.

Retrieved 9 September 2015.

[2] Although this character is available in LaTeX, theMediaWiki TeX system doesn't support this character.

[3] Brody, Baruch A. (1973), Logic: theoretical and applied,Prentice-Hall, p. 93, ISBN9780135401460,We turn nowto the second of our connective symbols, the centered dot,which is called the conjunction sign.

[4] Quine, W.V. (1981): Mathematical Logic, §6

[5] Hintikka, Jaakko (1998), The Principles of Mathemat-ics Revisited, Cambridge University Press, p. 113, ISBN9780521624985.

5.1.5 Further reading

Józef Maria Bocheński (1959), A Précis of Mathemati-cal Logic, trans., Otto Bird, from the French and Germaneditions, Dordrecht, South Holland: D. Reidel.

5.1.6 External links

• Named character entities in HTML 4.0

Chapter 6

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54 CHAPTER 6. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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• Understanding Source: https://en.wikipedia.org/wiki/Understanding?oldid=686993826 Contributors: Shii, Patrick, Michael Hardy, Kku,Ixfd64, TonyClarke, Owen, Altenmann, Ancheta Wis, Philwiki, Kenny sh, Bcrazy, Andycjp, Ele~enwiki, Karol Langner, Eep², ChrisHoward, Violetriga, Mr. Billion, Chairboy, Nectarflowed, .:Ajvol:., Runner1928, JaapB, Hookysun, RJFJR, Computerjoe, Velho,Woohookitty, Madmardigan53, Camw, Graham87, BD2412, MikeJ9919, FlaBot, Gareth E. Kegg, Jittat~enwiki, Andrew Eisenberg,999~enwiki, Musicpvm, Fabricationary, Seb35, Introgressive, VinnyCee, DeadEyeArrow, Tomisti, Andrew Lancaster, Arthur Rubin,Meegs, Teryx, SmackBot, Teenwriter, Ohnoitsjamie, Adam M. Gadomski, Mohan1986, Can't sleep, clown will eat me, Xyzzyplugh,Lefteh, Dirgni1986, Wossi, Bcasterline, Poojean, 16@r, Illythr, RichardF, Michaelbusch, CmdrObot, Dycedarg, ShelfSkewed, Neelix,Gregbard, Julian Mendez, Jawns317, PhJ, Storkk, The Transhumanist, .anacondabot, Magioladitis, Falcor84, WIKI-GUY-16, R'n'B, Tad-pole9, J.delanoy, Bylerda, Wtimrock, Maurice Carbonaro, Anonywiki, S (usurped also), Davecrosby uk, James Kidd, Parathopulos, PhilipTrueman, Z.E.R.O., Clarince63, Eubulides, Lova Falk, Newbyguesses, Tiddly Tom, Giorgio1993~enwiki, JSpung, Sanya3, Tikko1, Linfor-est, ClueBot, NickCT, Kl4m, Coldsteel510, Niceguyedc, CohesionBot, PixelBot, Who ordered 137?, Startxxx, Ykhwong, Kittykatsupered-itor, Humanengr, Sapiens23, Burningview, Ryo24, Addbot, Diti lad, Tide rolls, Yobot, Ptbotgourou, Hinio, Raimundo Pastor, BackslashForwardslash, Piano non troppo, Xqbot, Joarsolo, J04n, Aaron Kauppi, Erik9bot, Paine Ellsworth, Sae1962, DrilBot, Tinton5, EmausBot,Nentrex, HiW-Bot, Sahimrobot, DASHBotAV, Remebwiki, ClueBot NG, Jorgecarleitao, MerlIwBot, MusikAnimal, Gibbja, Loriendrew,Klilidiplomus, Acerbicattrition, Lugia2453, Kevin12xd, Tuihjbnb, TheUniversalist, Crizelmarabe and Anonymous: 113

• Certainty Source: https://en.wikipedia.org/wiki/Certainty?oldid=696625095 Contributors: Edward, Michael Hardy, Kku, Ixfd64,JASpencer, Lumos3, Cholling, Dbenbenn, Gyrofrog, Fishal, Andycjp, Sonjaaa, Thorsten1, D6, Zy26, Cherry blossom tree, Wayfarer,Zachlipton, Melaen, DonQuixote, Woohookitty, Jeff3000, Lawrence King, Rjwilmsi, Wragge, Captwheeler, Spencerk, Jayme, YurikBot,

6.1. TEXT 55

Bhny, MightyGiant, Aldux, Josh3580, SmackBot, Elonka, Od Mishehu, NGC6254, Egsan Bacon, RedHillian, 16@r, Macellarius, Clari-tyfiend, DeadCow, George100, Gregbard, Nauticashades, Dr.enh, Mattisse, Teh tennisman, Nimakha, AntiVandalBot, Matthew Fennell,Yahel Guhan, Tsinoyboi, Passanger, Slash, Someman6, Tamabec, STBotD, Squids and Chips, Jeff G., Bovineboy2008, Philogo, Insanity In-carnate, Denisarona, Tautologist, PipepBot, SamuelTheGhost, Jusdafax, S19991002, SchreiberBike, TwiLighT1126, Natdudeuk, Addbot,Elfstones, Lightbot, Kookyunii, Unara, Jonathan321, Captainnipples, Abillionistoomany, ThurstonMoore123, Aaron Kauppi, FrescoBot,Rkr1991, Dawgboy47, Tom.Reding, MastiBot, Yutsi, Redbeanpaste, Lotje, Miracle Pen, Some Wiki Editor, Wiggalama, Tesseract2,Ngc0202, Primefac, AvicBot, Ὁ οἶστρος, ClueBot NG, Amr.rs, Rezabot, BattyBot, AmericanLemming, FrB.TG, Ekoforss and Anony-mous: 76

• Inductive reasoning Source: https://en.wikipedia.org/wiki/Inductive_reasoning?oldid=697029653 Contributors: AxelBoldt, The Cunc-tator, The Anome, Ryguasu, DennisDaniels, Michael Hardy, Earth, Owl, Voidvector, BoNoMoJo (old), Jfitzg, Andres, Evercat, EdH,DesertSteve, Timwi, Trontonian, Bemoeial, Ike9898, Wolfgang Kufner, Radiojon, Markhurd, Peregrine981, Banno, Nufy8, Nurg, Ro-manm, Ojigiri~enwiki, Mikiher, Tobias Bergemann, Filemon, Ancheta Wis, Giftlite, Zigger, Peruvianllama, Bovlb, Jason Quinn, Jackol,Jmeola75, ELApro, Guppyfinsoup, Lucidish, Archer3, Discospinster, Freestylefrappe, Ivan Bajlo, Bender235, Kbh3rd, El C, Aaron-brick, David Crawshaw, Bobo192, I9Q79oL78KiL0QTFHgyc, Flammifer, Espoo, Samohyl Jan, Yuckfoo, Mikeo, Recury, Nightstallion,Voxadam, Kazvorpal, Kenyon, Hq3473, Velho, Mindmatrix, Kzollman, Ruud Koot, Alfakim, Andrea.gf, Rjwilmsi, Jweiss11, Strake,Bryan H Bell, Reinis, Matt Deres, Chris Pressey, Latka, RexNL, Fresheneesz, NotJackhorkheimer, Spencerk, King of Hearts, Chobot,Dresdnhope, YurikBot, RussBot, Gaius Cornelius, Grafen, Holycharly, SAE1962, 24ip, Pkearney, Roy Brumback, Bota47, Shadro,Tomisti, Sethery, Fram, Curpsbot-unicodify, Teply, Infinity0, Bo Jacoby, Jer ome, Sardanaphalus, SmackBot, RedHouse18, David Ker-now, Rtc, McGeddon, Istvan, Eskimbot, Klokie, Yamaguchi , Gilliam, Duke Ganote, Ohnoitsjamie, Betacommand, Bluebot, Anthonzi,LaggedOnUser, DHN-bot~enwiki, DoctorStrangelove, Can't sleep, clown will eat me, Go For It, Avb, Edivorce, Mr.Z-man, Jmnbatista,Richard001, Kalexander, Jon Awbrey, Neshatian, Andeggs, Vina-iwbot~enwiki, Byelf2007, Jonrgrover, Normalityrelief, RichMorin, An-tonielly, Aleenf1, Lukemcgrath, Grumpyyoungman01, Domokato, Levineps, Iridescent, K, Wjejskenewr, FleetCommand, CWY2190, In-digenius, El aprendelenguas, TMN, Gregbard, Slazenger, Peterdjones, Khromatikos, Gogo Dodo, Wikipediarules2221, Miguel de Servet,Letranova, Gacggt, Ucanlookitup, Second Quantization, Danny Reese, Defeatedfear, Fotomatt, AntiVandalBot, Luna Santin, Minhtung91,Spencer, Salgueiro~enwiki, JAnDbot, Davewho2, Dmar198, Coffee2theorems, Magioladitis, Bongwarrior, Equinexus, Hasek is the best,Arno Matthias, Farquaadhnchmn, DAGwyn, Snowded, Moopiefoof, Cathalwoods, MetsBot, Chrisdone, WLU, Mommyzbrat, STBot,Dionysiaca, Pomte, TheSeven, OttoMäkelä, LordAnubisBOT, Mahewa, Touisiau, Chiswick Chap, Heyitspeter, Pianoman55~enwiki,MetsFan76, STBotD, Andy Marchbanks, Straw Cat, Zach425, VolkovBot, Thewolf37, Pasixxxx, Hotfeba, Shinju, Jimmaths, Tiktuk,Philip Trueman, Deleet, Katoa, Jazzwick, Philogo, Abdullais4u, Jackfork, PDFbot, Anarchangel, Saturn star, Jor344, Shifter95, Cnilep,Harmonicemundi, PhysPhD, Jammycaketin, AlleborgoBot, Newbyguesses, Dwandelt, Matthew Yeager, Mark Klamberg, Flyer22 Re-born, Bobklahn, Oxymoron83, Vanished user oij8h435jweih3, MiNombreDeGuerra, Bagatelle, Sunrise, Linkboyz, Melcombe, Onefor-logic, ClueBot, Farras Octara, Eric Schoettle, Niceguyedc, Vandalometer, Rbakels, Excirial, Jusdafax, Kikilamb, Estirabot, ChrisKalt,Hazzzzzz12, Lx 121, XLinkBot, Fastily, Gerhardvalentin, Tegiap, Saeed.Veradi, Skarebo, WikHead, Kwjbot, Kbdankbot, Tayste, Ad-dbot, Tanhabot, Jtradke, Numbo3-bot, Tide rolls, ScienceApe, KUSSOMAK, Legobot, Luckas-bot, Yobot, Oilstone, THEN WHO WASPHONE?, AnomieBOT, Doingmorestuffonline, Vanakaris, Bob Burkhardt, Parthian Scribe, Xqbot, Lord Bane, Hanberke, A157247, F-22Raptored, Omnipaedista, RibotBOT, Delbertpeach, Alialiac, FieldOperative, Paine Ellsworth, SBA1870, Machine Elf 1735, Pinethicket,Kiefer.Wolfowitz, Mavit0, A8UDI, Cleon7177, NeuroBells123, Gamewizard71, TobeBot, Jonkerz, Miracle Pen, Dbmikus, Hyperbytev2,Ripchip Bot, Elspru, NerdyScienceDude, George Richard Leeming, EmausBot, Elanguescence, Grjoni88, T3dkjn89q00vl02Cxp1kqs3x7,Gfoley4, RenamedUser01302013, Mo ainm, ZéroBot, Leminh91, CanonLawJunkie, Oncenawhile, Wagino 20100516, Erianna, EricWes-Brown, L Kensington, Just granpa, 28bot, ClueBot NG, Gareth Griffith-Jones, MelbourneStar, Ek65, Millermk, Schicagos, Tsunamicharlie,Albertttt, Thepigdog, Masssly, Widr, Helpful Pixie Bot, HMSSolent, Curb Chain, BG19bot, Wiki13, MusikAnimal, Jander80, Wandwiki,Blue Mist 1, Will.Oliver, Trailspark, RichardMills65, Ctasa221, Fangli997376557, ChrisGualtieri, Lhu720, Hagrid da fifth, Watchpup32,Neurocitizen, Oligocene, Moonstroller-2, Jochen Burghardt, M strat17, 90b56587, Reid12345, Londomollari42, EMBViki, Cauzality,JustBerry, Aubreybardo, Liz, Logicman2, Hoffoholi, Superploro, Temprack5446, Loraof, Hellerrrr, Isambard Kingdom, Pretty Panther26, LIZSMOBILEONE and Anonymous: 407

• Argument Source: https://en.wikipedia.org/wiki/Argument?oldid=696504847 Contributors: AxelBoldt, Gaurav, Ronz, Markhurd, Hy-acinth, Jjshapiro, Robbot, Michael Devore, Discospinster, Paul August, Bender235, Bobo192, Nsaa, Kata Alreshim, Velella, Rain-bowOfLight, Bjones, BD2412, Nightscream, Koavf, Chobot, Bgwhite, FrankTobia, Stephenb, Kimchi.sg, DRosenbach, Josh3580, Trick-star, SmackBot, Od Mishehu, Jmendez, Yamaguchi , Gilliam, Chris the speller, MartinPoulter, Fuzzform, DHN-bot~enwiki, Hallenrm,Darth Panda, Scwlong, Rrburke, Mr.Z-man, Cybercobra, Dreadstar, Richard001, Trbdavies, Lambiam, Khazar, General Ization, Shlomke,Ckatz, CRGreathouse, Penbat, Gregbard, Gogo Dodo, DumbBOT, Alaibot, Letranova, Northumbrian, JAnDbot, Bongwarrior, VoABotII, Kevinmon, Sonawin, Kateshortforbob, J.delanoy, Rgoodermote, Maurice Carbonaro, Hateloveschool, Johnfos, Jimmaths, TXiKiBoT,Philogo, Go2slash, Graymornings, Synthebot, Lova Falk, Cnilep, Dmcq, Botev, SieBot, Yintan, Enti342, Steven Crossin, Techman224,Correogsk, Msrasnw, Mhnin0, Sokari, Twinsday, ClueBot, The Thing That Should Not Be, Pelle Ohlander, Niceguyedc, Mack-the-random, Cristi215, Leonard^Bloom, Peachypoh, Johnuniq, DumZiBoT, Gerhardvalentin, Little Mountain 5, NellieBly, Rexroad2, Quaintand curious, Addbot, Cxz111, Grayfell, Tcncv, Ronhjones, Download, Favonian, 5 albert square, Bfigura’s puppy, Shulha, Luckas-bot,Yobot, TaBOT-zerem, Suntag, AnomieBOT, Grolltech, Flewis, E235, Citation bot, Maxis ftw, Obersachsebot, Xqbot, Capricorn42, LordBane, The Evil IP address, GrouchoBot, Omnipaedista, Linkubus, Wikiresearchman, Sky Attacker, Steelstring, D'ohBot, Machine Elf1735, Pinethicket, Momergil, Serols, Oshnutz, Cnwilliams, TobeBot, Tangoleader, GregKaye, Walkinxyz, EmausBot, Wikipelli, Zac-chro, Shadfurman, ZéroBot, LiterateTiger, Staszek Lem, TyA, Tijfo098, Considering..., ClueBot NG, Lisnabreeny, Wesniel, Masssly,Widr, Bluemax711, Bigjonstuff, Titodutta, DonBogdan, Jeraphine Gryphon, Yumyum36, Infinite Loop-Maker, Wannabemodel, Teammm,Pratyya Ghosh, ChrisGualtieri, Alanajames, 5000tut, Mogism, Harlequinjelly, Metcalm, MarcusSchmitz, WHAZZUP204, Laleluuuuuuuu,Smr0151993, Thomn12, BubbleFoot, Joseph2302, Miraclexix, Arvidaskzlt, Fabriziomacagno, SparkyMacgillicuddy and Anonymous: 218

• Deductive reasoning Source: https://en.wikipedia.org/wiki/Deductive_reasoning?oldid=697777022 Contributors: The Cunctator, TobyBartels, Youandme, Mrwojo, DennisDaniels, Michael Hardy, TakuyaMurata, BenKovitz, EdH, DesertSteve, Charles Matthews, Dtgm,Hyacinth, Lumos3, Robbot, R3m0t, Romanm, AceMyth, Blainster, Tobias Bergemann, Ancheta Wis, Giftlite, Lethe, Guanaco, Bovlb,Archenzo, Jason Quinn, Piotrus, Karol Langner, Aequo, Stepp-Wulf, EricBright, TedPavlic, Kevinb, Stbalbach, PhilHibbs, Causa sui,Flammifer, Espoo, Jumbuck, Ryanmcdaniel, CyberSkull, Nasukaren, Garrisonroo, SidP, Kenyon, Tariqabjotu, Stephen, Velho, OwenX,Mindmatrix, TheNightFly, Ruud Koot, Jon Harald Søby, ZephyrAnycon, Teemu Leisti, BD2412, Nightscream, Koavf, Gmelli, Jweiss11,Tangotango, YAZASHI, Ggfevans, DirkvdM, FlaBot, Nihiltres, Fresheneesz, Skc0001, Alphachimp, Chobot, YurikBot, Borgx, Erachima,DTRY, Rick Norwood, Holycharly, TriGen, EEMIV, Bota47, Shadro, Mjhy0926, SMcCandlish, Allens, Infinity0, GrinBot~enwiki,DVD R W, Sardanaphalus, SmackBot, Aim Here, KocjoBot~enwiki, Thunderboltz, Stephensuleeman, WookieInHeat, Ieopo, The great

56 CHAPTER 6. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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• Soundness Source: https://en.wikipedia.org/wiki/Soundness?oldid=698487790 Contributors: The Anome, Ixfd64, Eric119, AugPi,Rossami, Hyacinth, Mpost89, Ancheta Wis, Markus Krötzsch, Kpalion, Andycjp, Chalst, Art LaPella, EmilJ, Saturnight, Nortex-oid, Jumbuck, Raboof, Rh~enwiki, Omphaloscope, Oleg Alexandrov, MattGiuca, Justin Custer, Koavf, Mathbot, Kri, Vonkje, Yurik-Bot, Hairy Dude, SmackBot, Gilliam, Skizzik, Bluebot, NYKevin, Mikezhao, Cybercobra, Richard001, Luxgratia, Lambiam, Bjanku-loski06en~enwiki, Jenadeleh, Esurnir, CBM, Simeon, Gregbard, Thijs!bot, Nick Number, Thenub314, Arno Matthias, The dark lord trom-bonator, Deleet, IllaZilla, Tomaxer, Ohiostandard, SieBot, Sullen skies, Kumioko (renamed), DesolateReality, Tomas e, Dhulme, Alexbot,Hans Adler, Addbot, Tide rolls, Citation bot, MauritsBot, Je ne détiens pas la vérité universelle, NoldorinElf, LucienBOT, Amirhoseina-liakbarian, GregKaye, EmausBot, ZéroBot, Josve05a, JonRichfield, ClueBot NG, Helpful Pixie Bot, Epicgenius, Mohamed-Ahmed-FG,JHU1959, Kiwifist and Anonymous: 47

• List of logic symbols Source: https://en.wikipedia.org/wiki/List_of_logic_symbols?oldid=699680468 Contributors: Boud, Jitse Niesen,Hyacinth, Benwing, Ancheta Wis, Petershank, Abdull, Paul August, EmilJ, SpeedyGonsales, PWilkinson, Chira, Dirac1933, Mindma-trix, DePiep, Koavf, R.e.b., YurikBot, Wavelength, KSmrq, KSchutte, Grafen, Trovatore, Mkouklis, Dbmag9, Arthur Rubin, Jbalint,SmackBot, InverseHypercube, Melchoir, Isaac Dupree, Mhss, Bluebot, Da nuke, Pioto, Neo-Jay, Alphathon, DGerman, Cybercobra,DemosDemon, Ihatetoregister, Dqb124, SashatoBot, Lambiam, Eric76, Vanisaac, CBM, Gregbard, Cydebot, Julian Mendez, Carte-sian1, Robocracy, JJ Harrison, David Eppstein, It Is Me Here, Anonymous Dissident, Springbreak04, Bentogoa, Hello71, Kumioko (re-named), Francvs, Leranedo, Dead10ck, Philosophy.dude, Peter.C, Saeed.Veradi, Addbot, The Sage of Stamford, Melab-1, Luckas-bot,GorgeUbuasha, AnomieBOT, Xqbot, Coretheapple, Oscarjquintana, Lagelspeil, Mscdancer, Cailean8, Erelen, George Richard Leeming,EmausBot, ZéroBot, Derekleungtszhei, Quondum, Rfontiveros, Donner60, BrendanLarvor, RockMagnetist, George Makepeace, ClueBotNG, Delusion23, BG19bot, Rm1271, Fylbecatulous, The Mol Man, Tcrosley, TCMemoire, Victor lesyk, Gufosowa, MetazoanMarek,EnRouteAviation and Anonymous: 68

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