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An Introduction to Non-Classical Logic
This revised and considerably expanded edition of An Introduction to
Non-Classical Logic brings together a wide range of topics, including modal,
tense, conditional, intuitionist, many-valued, paraconsistent, relevant and
fuzzy logics. Part I, on propositional logic, is the old Introduction, but
contains much new material. Part II is entirely novel, and covers
quantification and identity for all the logics in Part I. The material is
unified by the underlying theme of world semantics. All of the topics are
explained clearly and accessibly, using devices such as tableau proofs, and
their relations to current philosophical issues and debates is discussed.
Students with a basic understanding of classical logic will find this book an
invaluable introduction to an area that has become of central importance
in both logic and philosophy. It will also interest people working in
mathematics and computer science who wish to know about the area.
graham priest is Boyce Gibson Professor of Philosophy, University of
Melbourne and Arché Professorial Fellow, Departments of Philosophy,
University of St Andrews. His most recent publications include Towards
Non-Being (2005) and Doubt Truth to be a Liar (2006).
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Mathematical Prolegomenon xxvii0.1 Set-theoretic Notation xxvii0.2 Proof by Induction xxix0.3 Equivalence Relations and Equivalence Classes xxx
Part I Propositional Logic 1
1 Classical Logic and the Material Conditional 31.1 Introduction 31.2 The Syntax of the Object Language 41.3 Semantic Validity 51.4 Tableaux 61.5 Counter-models 101.6 Conditionals 111.7 The Material Conditional 121.8 Subjunctive and Counterfactual Conditionals 131.9 More Counter-examples 141.10 Arguments for ⊃ 151.11 ∗Proofs of Theorems 161.12 History 181.13 Further Reading 181.14 Problems 18
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
2.3 Modal Semantics 212.4 Modal Tableaux 242.5 Possible Worlds: Representation 282.6 Modal Realism 282.7 Modal Actualism 292.8 Meinongianism 302.9 *Proofs of Theorems 312.10 History 332.11 Further Reading 342.12 Problems 34
3 Normal Modal Logics 363.1 Introduction 363.2 Semantics for Normal Modal Logics 363.3 Tableaux for Normal Modal Logics 383.4 Infinite Tableaux 423.5 S5 453.6 Which System Represents Necessity? 463.6a The Tense Logic Kt 493.6b Extensions of Kt 513.7 *Proofs of Theorems 563.8 History 603.9 Further Reading 603.10 Problems 60
4 Non-normal Modal Logics; Strict Conditionals 644.1 Introduction 644.2 Non-normal Worlds 644.3 Tableaux for Non-normal Modal Logics 654.4 The Properties of Non-normal Logics 674.4a S0.5 694.5 Strict Conditionals 724.6 The Paradoxes of Strict Implication 724.7 ... and their Problems 734.8 The Explosion of Contradictions 744.9 Lewis’ Argument for Explosion 764.10 *Proofs of Theorems 774.11 History 79
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
5 Conditional Logics 825.1 Introduction 825.2 Some More Problematic Inferences 825.3 Conditional Semantics 845.4 Tableaux for C 865.5 Extensions of C 875.6 Similarity Spheres 905.7 C1 and C2 945.8 Further Philosophical Reflections 975.9 *Proofs of Theorems 985.10 History 1005.11 Further Reading 1015.12 Problems 101
6 Intuitionist Logic 1036.1 Introduction 1036.2 Intuitionism: The Rationale 1036.3 Possible-world Semantics for Intuitionism 1056.4 Tableaux for Intuitionist Logic 1076.5 The Foundations of Intuitionism 1126.6 The Intuitionist Conditional 1136.7 *Proofs of Theorems 1146.8 History 1166.9 Further Reading 1176.10 Problems 117
7 Many-valued Logics 1207.1 Introduction 1207.2 Many-valued Logic: The General Structure 1207.3 The 3-valued Logics of Kleene and Lukasiewicz 1227.4 LP and RM3 1247.5 Many-valued Logics and Conditionals 1257.6 Truth-value Gluts: Inconsistent Laws 1277.7 Truth-value Gluts: Paradoxes of Self-reference 1297.8 Truth-value Gaps: Denotation Failure 1307.9 Truth-value Gaps: Future Contingents 1327.10 Supervaluations, Modality and Many-valued Logic 133
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
7.11 *Proofs of Theorems 1377.12 History 1397.13 Further Reading 1407.14 Problems 140
8 First Degree Entailment 1428.1 Introduction 1428.2 The Semantics of FDE 1428.3 Tableaux for FDE 1448.4 FDE and Many-valued Logics 1468.4a Relational Semantics and Tableaux for L3 and RM3 1498.5 The Routley Star 1518.6 Paraconsistency and the Disjunctive Syllogism 1548.7 *Proofs of Theorems 1558.8 History 1618.9 Further Reading 1618.10 Problems 161
9 Logics with Gaps, Gluts and Worlds 1639.1 Introduction 1639.2 Adding → 1639.3 Tableaux for K4 1649.4 Non-normal Worlds Again 1669.5 Tableaux for N4 1689.6 Star Again 1699.7 Impossible Worlds and Relevant Logic 1719.7a Logics of Constructible Negation 1759.8 *Proofs of Theorems 1799.9 History 1849.10 Further Reading 1859.11 Problems 185
10 Relevant Logics 18810.1 Introduction 18810.2 The Logic B 18810.3 Tableaux for B 19010.4 Extensions of B 19410.4a Content Inclusion 19710.5 The System R 20310.6 The Ternary Relation 206
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
10.7 Ceteris Paribus Enthymemes 20810.8 *Proofs of Theorems 21110.9 History 21610.10 Further Reading 21710.11 Problems 218
11 Fuzzy Logics 22111.1 Introduction 22111.2 Sorites Paradoxes 22111.3 . . . and Responses to Them 22211.4 The Continuum-valued Logic L 22411.5 Axioms for Lℵ 22711.6 Conditionals in L 23011.7 Fuzzy Relevant Logic 23111.7a *Appendix: t-norm Logics 23411.8 History 23711.9 Further Reading 23811.10 Problems 239
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
12.6 Some Philosophical Issues 27512.7 Some Final Technical Comments 27712.8 *Proofs of Theorems 1 27812.9 *Proofs of Theorems 2 28312.10 *Proofs of Theorems 3 28512.11 History 28712.12 Further Reading 28712.13 Problems 288
13 Free Logics 29013.1 Introduction 29013.2 Syntax and Semantics 29013.3 Tableaux 29113.4 Free Logics: Positive, Negative and Neutral 29313.5 Quantification and Existence 29513.6 Identity in Free Logic 29713.7 *Proofs of Theorems 30013.8 History 30413.9 Further Reading 30513.10 Problems 305
14 Constant Domain Modal Logics 30814.1 Introduction 30814.2 Constant Domain K 30814.3 Tableaux for CK 30914.4 Other Normal Modal Logics 31414.5 Modality De Re and De Dicto 31514.6 Tense Logic 31814.7 *Proofs of Theorems 32014.8 History 32514.9 Further Reading 32614.10 Problems 327
15 Variable Domain Modal Logics 32915.1 Introduction 32915.2 Prolegomenon 32915.3 Variable Domain K and its Normal Extensions 33015.4 Tableaux for VK and its Normal Extensions 33115.5 Variable Domain Tense Logic 33515.6 Extensions 336
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
15.7 Existence Across Worlds 33915.8 Existence and Wide-Scope Quantifiers 34115.9 *Proofs of Theorems 34215.10 History 34615.11 Further Reading 34615.12 Problems 347
16 Necessary Identity in Modal Logic 34916.1 Introduction 34916.2 Necessary Identity 35016.3 The Negativity Constraint 35216.4 Rigid and Non-rigid Designators 35416.5 Names and Descriptions 35716.6 *Proofs of Theorems 1 35816.7 *Proofs of Theorems 2 36216.8 History 36416.9 Further Reading 36416.10 Problems 365
17 Contingent Identity in Modal Logic 36717.1 Introduction 36717.2 Contingent Identity 36717.3 SI Again, and the Nature of Avatars 37317.4 *Proofs of Theorems 37617.5 History 38217.6 Further Reading 38217.7 Problems 382
18 Non-normal Modal Logics 38418.1 Introduction 38418.2 Non-normal Modal Logics and Matrices 38418.3 Constant Domain Quantified L 38518.4 Tableaux for Constant Domain L 38618.5 Ringing the Changes 38718.6 Identity 39118.7 *Proofs of Theorems 39318.8 History 39718.9 Further Reading 39718.10 Problems 397
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
21.11 *Proofs of Theorems 47121.12 History 47321.13 Further Reading 47421.14 Problems 474
22 First Degree Entailment 47622.1 Introduction 47622.2 Relational and Many-valued Semantics 47622.3 Tableaux 47922.4 Free Logics with Relational Semantics 48122.5 Semantics with the Routley ∗ 48322.6 Identity 48622.7 *Proofs of Theorems 1 48922.8 *Proofs of Theorems 2 49322.9 *Proofs of Theorems 3 49922.10 History 50222.11 Further Reading 50222.12 Problems 502
23 Logics with Gaps, Gluts and Worlds 50423.1 Introduction 50423.2 Matrix Semantics Again 50523.3 N4 50523.4 N∗ 50823.5 K4 and K∗ 51023.6 Relevant Identity 51223.7 Relevant Predication 51523.8 Logics with Constructible Negation 51723.9 Identity for Logics with Constructible Negation 52123.10 *Proofs of Theorems 1 52323.11 *Proofs of Theorems 2 52723.12 *Proofs of Theorems 3 53023.13 History 53223.14 Further Reading 53223.15 Problems 533
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
24.2 Quantified B 53524.3 Extensions of B 53724.4 Restricted Quantification 54124.5 Semantics vs Proof Theory 54324.6 Identity 54824.7 Properties of Identity 55324.8 *Proofs of Theorems 1 55524.9 *Proofs of Theorems 2 55924.10 History 56124.11 Further Reading 56124.12 Problems 562
25 Fuzzy Logics 56425.1 Introduction 56425.2 Quantified Lukasiewicz Logic 56425.3 Validity in Lℵ 56525.4 Deductions 57025.5 The Sorites Again 57225.6 Fuzzy Identity 57325.7 Vague Objects 57625.8 *Appendix: Quantification and Identity in
t-norm Logics 57825.9 History 58125.10 Further Reading 58225.11 Problems 582
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
〈x1, . . . , xn〉 ∈ R is usually written as Rx1 . . . xn. If n is 3, the relation is a ternary
relation. If n is 2, the relation is a binary relation, and Rx1x2 is usually written
as x1Rx2. A function from X to Y is a binary relation, f , between X and Y , such
that for all x ∈ X there is a unique y ∈ Y such that xfy. More usually, in this
case, we write: f (x) = y.
0.1.11 Examples: 〈2, 3〉 = 〈3, 2〉, since these sets have the same members, but
in a different order. Let N be the set of numbers. Then N × N is the set of
all pairs of the form 〈n,m〉, where n and m are in N. If R = {〈2, 3〉, 〈3, 2〉}then R ⊆ N × N and is a binary relation between N and itself. If f ={〈n, n2〉 : n ∈ N}, then f is a function from numbers to numbers, and
f (n) = n2.
0.2 Proof by Induction
0.2.1 The method of proof by induction (or recursion) on the complexity of
sentences is used heavily in the asterisked sections of the book. It is also used
occasionally in other places, though these can usually be skipped without
loss. What this method comes to is this. Suppose that all of the simplest
formulas of some formal language (that is, those that do not contain any
connectives or quantifiers) have some property, P. (Establishing this fact
is usually called the basis (or base) case.) And suppose that whenever one
constructs a more complex sentence – that is, one with an extra connective
(or quantifier if such things are in the language) – out of formulas that have
property P, the resulting formula also has the property P. (Establishing this
is usually called the induction case.) Then it follows that all the formulas
of the language have the property P. Thus, for example, suppose that the
simple formulas p and q have property P, and that whenever formulas have
that property, so do their negations, conjunctions, etc. Then it follows that
¬p, p ∧ q, ¬p ∧ (p ∧ q), have the property, as do all sentences that we can
construct from p and qusing negation and conjunction.
0.2.2 The proof of the induction case normally breaks down into a number
of different sub-cases, one for each of the connectives (and quantifiers if
present) employed in the construction of more complex formulas. Thus,
we assume that A has the property, then show that ¬A has it; we assume
that A and B have the property, then show that A ∧ B has it; and so on for
every connective (and quantifier). The assumption, in each case, is called
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
as y’ is a relation that partitions them into classes of people with the same
height. Suppose that C is:
a b c
d e f
g h i
and that a, b, d and e, all have the same height, as do c, f and i, as do g and
h. Then the equivalence classes are:
a b
d e
g h
c
f
i
0.3.2 More precisely, if ∼ is a binary relation on a collection of objects, C, it
is an equivalence relation just if it is:
• reflexive: for all x ∈ C, x ∼ x
• symmetric: for all x, y ∈ C, if x ∼ y then y ∼ x
• transitive: for all x, y, z ∈ C, if x ∼ y and y ∼ z then x ∼ z
If x ∈ C, its equivalence class, written [x], is defined as {w ∈ C : w ∼ x}.0.3.3 The fundamental fact about equivalence classes is that every object in
the domain is in exactly one. To see this, note, first, that for any x ∈ C, since
x ∼ x, x ∈ [x] ; so x is in some equivalence class. Now let X = [x] and Y = [y].Suppose that, for some z, z is in both X and Y . Then z ∼ x and z ∼ y . By
symmetry and transitivity, x ∼ y. For any w ∈ X, w ∼ x. Since x ∼ y, w ∼ y.
That is, w ∈ Y . Hence, X ⊆ Y . Similarly, Y ⊆ X. Hence, X = Y .
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information
Cambridge University Press978-0-521-67026-5 - An Introduction to Non-Classical Logic: From If to Is, Second EditionGraham PriestFrontmatterMore information