Introduction to Logic - IITKhome.iitk.ac.in/~avrs/avrs/PHI142/Ravi.pdfMartin Gardner, aha! Insight aha! Gotcha.The mathematical association of America. Lewws Carroll, Symbolic Logic,

Introduction to Logic

A. V. Ravishankar Sarma

Indian Institute of Technology Kanpur

July 24, 2008

A. V. Ravishankar Sarma PHI142: Introduction to Logic 1/ 16

PHI142: Introduction to Logic

Answer some frequently asked questions related to courseWhat is Logic?Why study Logic?How logic can be done: What will be taught in the course?

Aim: To learn principles of valid reasoning, basic conceptsof logic, To discern good reasoning and bad reasoning,when an argument is valid, distinguishing inductive anddeductive arguments, identifying fallacies and avoidingthem.Objective: Equip oneself with various tools andtechniques (Decision procedures) for the validity of agiven argument, detecting and avoiding fallacies of agiven deductive or inductive argument.


PHI142: Introduction to Logic

Answer some frequently asked questions related to courseWhat is Logic?Why study Logic?How logic can be done: What will be taught in the course?Aim: To learn principles of valid reasoning, basic conceptsof logic, To discern good reasoning and bad reasoning,when an argument is valid, distinguishing inductive anddeductive arguments, identifying fallacies and avoidingthem.Objective: Equip oneself with various tools andtechniques (Decision procedures) for the validity of agiven argument, detecting and avoiding fallacies of agiven deductive or inductive argument.


Why study Logic?

Logic deals with what follows from what?(Logical consequence,inference pattern, validating such patterns)

We want the computer to understand our language and doessome intelligent tasks for you.(Knowledge representation).

Engaged in debates, solving puzzles, game like situation.

Identify which one is a fallacious argument- what type offallacy?

proving theorems (deduction), is what ever proved is correct, orwhat ever is obviously true has a proof?

Analysing and designing circuits.

Some problems concerning the foundations of mathematics (canit be rested on set theory)?


Layman’s point of view: Logic (reasoning)argumentation is omnipresent

Children arguing with parents (persuasion)-ArgumentationIntelligent/bright students exercising their reasoning skills.Politicians trying to woo/persuade their voters byinvoking emotions-Caught in a traffic (don’t have license/forgot to takelicense)- persuasive skills (Informal fallacies)What goes on in solving a mathematical problem (proof,Deduction)Scientist’s tool box: When an Hypothesis is confirmed byevidence?How the computer works? [Logic of [0,1]- Boolean logic]


What is Logic?

coming up with a definition would be narrowing the scopeand subject matter oflogic.Laws of thought (George Boole): The way we think isgoverned by

The way we think: Law of Identity P = P, Law of excludedmiddle P ∨ ¬P. Law of non-contradiction¬(P ∧ ¬P).Study of principles of valid reasoning: to discern goodreasoning from bad reasoning.Relaxing some of the conditions: Intuitionist logic, Fuzzylogic, Para consistent Logic.Study of connectives ∧, ¬, ∨,→,↔.Study of ∀x, ∃x


What is Logic?

coming up with a definition would be narrowing the scopeand subject matter oflogic.Laws of thought (George Boole): The way we think isgoverned byThe way we think: Law of Identity P = P, Law of excludedmiddle P ∨ ¬P. Law of non-contradiction¬(P ∧ ¬P).Study of principles of valid reasoning: to discern goodreasoning from bad reasoning.Relaxing some of the conditions: Intuitionist logic, Fuzzylogic, Para consistent Logic.Study of connectives ∧, ¬, ∨,→,↔.Study of ∀x, ∃x


Examples

Example

Premises: All IITK students are mortal.Ramu is an IITK student.Conclusion: Ramu is mortal.

P: All parallelograms are circlesAll circles are squaresC: All parallelograms are squares.

P: Some men are liars and Laloo is a manC: Laloo is a liar

First argument is valid and sound. The second argument isunsound but valid.Third argument is invalid.


Lewis Carrol: Alice in the Wonderland

Example

There are no pencils of mine in this box.[Px → ¬Bx]No sugar-plumbs of mine are cigars.[Sx → ¬Cx]The whole of my property, that is not in the box, consists ofcigars[¬Bx → Cx]

Conclusion: 1-3 All my pencils are cigars[Px → Cx] and finalconclusion is No pencils of mine are sugar-plumbs[Px → ¬Sx]


Some Proofs

Example

Augustus De Morgan: 2=1 Let x = 1. Then x2 = x. Sox2 − 1 = x− 1 Dividing both sides by x -1, we conclude that x +1 = 1; that is, since x = 1, 2 = 1.

Example

Happiness or Kheer? Which is better, eternal happiness or aKheer (Sweet dish)? It would appear that eternal happiness isbetter, but this is really not so!

After all, nothing is better thaneternal happiness, and Kheer is certainly better than nothing.Therefore a Kheer is better than eternal happiness.


Some Proofs

Example

Augustus De Morgan: 2=1 Let x = 1. Then x2 = x. Sox2 − 1 = x− 1 Dividing both sides by x -1, we conclude that x +1 = 1; that is, since x = 1, 2 = 1.

Example

Happiness or Kheer? Which is better, eternal happiness or aKheer (Sweet dish)? It would appear that eternal happiness isbetter, but this is really not so!After all, nothing is better thaneternal happiness, and Kheer is certainly better than nothing.Therefore a Kheer is better than eternal happiness.


Proof that I am Dracula

Example

P: Everyone is afraid of Dracula. Dracula is afraid of onlyme.C: Therefore I am Dracula.

Since everyone is afraid of Dracula, then Dracula is afraid ofDracula. So Dracula is afraid of Dracula, but also is afraid of noone but me. Therefore I must be Dracula!

Example

This sentence is false. Is that sentence true or false? If it is falsethen it is true, and if it is true then it is false.[Liars Paradox]


Informal Fallacies:

Example

Abhishek: If someone hits you, you should turn the othercheek. Violence only begets violence, and violence in and ofitself is wrong. Aishwarya: Thats a joke. You used to hit peoplewhen they picked a fight with you. [Ad hominem- tuquoque-you too]

Example

P:This auditorium is made up of atoms. Atoms are invisible. C:Therefore, the Auditorium is invisible.

Example

Why the argument below is fallacious? If I drop an egg, itbreaks. This egg is broken, so I must have dropped (Fallacy ofaffirming the consequent)


What we can learn from this course?

Basic concepts: premise, conclusion, argument,non-argument, deductive and inductive arguments,invalidity of deductive arguments (Counter examplemethod)

Fallacies: Persuasive, and mainly has emotionalcomponent–Formal fallacies, Informal fallacies (Relevance,weak induction, presumption.)Classical Aristotle logic: Syllogisms (Euler circles, VennDiagrams)- Limitations of Aristotle logic, Boole,Propositional Logic (Russell Whitehead/Hilbert Axiomaticsystems, proofs of various theorems, Soundness,completeness, satisfiability.Various techniques: Truth table method (Emilie Post),Indirect truth table(when “n’ is large), Natural Deductionmethod, Reductio ad absurdum, Refutation tree method. Ispropositional Logic complete(Whatever is true has proofand whatever is proved is true)Logic of quantifiers ∀x, ∃x. Scope of quantifier, validity,satisfiability, definite descriptions (The current king ofIndia is bald), Russell’s paradox.



Basic concepts: premise, conclusion, argument,non-argument, deductive and inductive arguments,invalidity of deductive arguments (Counter examplemethod)Fallacies: Persuasive, and mainly has emotionalcomponent–Formal fallacies, Informal fallacies (Relevance,weak induction, presumption.)Classical Aristotle logic: Syllogisms (Euler circles, VennDiagrams)- Limitations of Aristotle logic, Boole,Propositional Logic (Russell Whitehead/Hilbert Axiomaticsystems, proofs of various theorems, Soundness,completeness, satisfiability.

Various techniques: Truth table method (Emilie Post),Indirect truth table(when “n’ is large), Natural Deductionmethod, Reductio ad absurdum, Refutation tree method. Ispropositional Logic complete(Whatever is true has proofand whatever is proved is true)Logic of quantifiers ∀x, ∃x. Scope of quantifier, validity,satisfiability, definite descriptions (The current king ofIndia is bald), Russell’s paradox.



Basic concepts: premise, conclusion, argument,non-argument, deductive and inductive arguments,invalidity of deductive arguments (Counter examplemethod)Fallacies: Persuasive, and mainly has emotionalcomponent–Formal fallacies, Informal fallacies (Relevance,weak induction, presumption.)Classical Aristotle logic: Syllogisms (Euler circles, VennDiagrams)- Limitations of Aristotle logic, Boole,Propositional Logic (Russell Whitehead/Hilbert Axiomaticsystems, proofs of various theorems, Soundness,completeness, satisfiability.Various techniques: Truth table method (Emilie Post),Indirect truth table(when “n’ is large), Natural Deductionmethod, Reductio ad absurdum, Refutation tree method. Ispropositional Logic complete(Whatever is true has proofand whatever is proved is true)Logic of quantifiers ∀x, ∃x. Scope of quantifier, validity,satisfiability, definite descriptions (The current king ofIndia is bald), Russell’s paradox.


Some good Books

Patrick Hurley, Concise introduction to Logic, Thomson /Wadsworth, 2007[standard Course book]Mendelson, Introduction to Mathematical Logic, pp:1-90[Extra reading]Shawn Hedman, A first course in Mathematical Logic, oxforduniversity press, pp 1-115[Extra]Bertrand Russell and A N Whitehead, PrincipiaMathematica, 1910, pp. 89-135Raymond Smullyan, Forever Undecided: A Puzzle Guide toGodel, 1987Martin Gardner, aha! Insight aha! Gotcha.The mathematicalassociation of America.Lewws Carroll, Symbolic Logic, availale inhttp://durendal.org:8080/lcsl/


Futher references to get in touch with logic:

http://home.iitk.ac.in/ avrs/avrs/PH142/http://groups.google.com/group/sci.logic/topicshttp://www.cs.nyu.edu/pipermail/fom/http://world.logic.at/http://philosophy.lander.edu/logic/links.htmlhttp://groups.google.com/group/sci.mathhttp://sakharov.net/foundation.html [best site]


Historical Context

Indian Logic(640BC). Classical antquity: one of the sevenliberal arts: Grammar, Rhetoric, Logic and

Arithmetic: Number in itselfGeometry: Number in spaceMusic, Harmonics, or Tuning theory (number in time)Astronomy or cosmology- number in space and time.Classical Aristotle logic: Syllogism (1900 years)- Logic asa justificatory toolMedieval Logic: Leibniz, George Boole, Venn- validityModern Logic: Frege, Betrand Russell,Hilbert-Ackerman (Logicism, formalism, Intuitionism)Propositional, Predicate Logic.


Russell’s paradox

Example

There is someone who loves any person only if that persondoes not does not love themselves.∃x∀y[Lxy → ¬Lyy].

Example

In a certain village there is a man, so the paradox runs, who is abarber; this barber shaves all and only those men in the villagewho do not shave themselves.

Query: Does the barber shavehimself? Any man in this village is shaved by the barber if andonly if he is not shaved by himself. Therefore in particular thebarber shaves himself if and only if he does not. We are introuble if we say the barber shaves himself and we are introuble if we say he does not.


Russell’s paradox

Example

There is someone who loves any person only if that persondoes not does not love themselves.∃x∀y[Lxy → ¬Lyy].

Example

In a certain village there is a man, so the paradox runs, who is abarber; this barber shaves all and only those men in the villagewho do not shave themselves. Query: Does the barber shavehimself? Any man in this village is shaved by the barber if andonly if he is not shaved by himself. Therefore in particular thebarber shaves himself if and only if he does not. We are introuble if we say the barber shaves himself and we are introuble if we say he does not.


Knights and Knaves in an Island: Raymond Smullyan

Example

Rules: Knights always tell the truth, and Knaves always lie.As you approach the island, you spot three inhabitants(A, B, C) on theshore. You call out to them, Are you Knights or Knaves? The first (A)says something but you do not hear what he says, so you ask, “Whatdid you say?” The second(B) inhabitant says, “He says he is a Knight,he is and so am I(p).” The third (C)responds, “He is a Knave, but I ama Knight.” What are the three inhabitants really?

Answer: The mostbasic rule of Knight and Knave island is that no one ever saysthey are a Knave. Knights always say they are Knights becausethey tell the truth, Knaves say they are Knights because theyalways lie. The second must be a Knight, otherwise he wouldhave lied about the firsts response and if so, he must be tellingthe truth about the first. The third must therefore be a Knave.They are (A)Knight, (B)Knight and (C)Knave.


Knights and Knaves in an Island: Raymond Smullyan

Example

Rules: Knights always tell the truth, and Knaves always lie.As you approach the island, you spot three inhabitants(A, B, C) on theshore. You call out to them, Are you Knights or Knaves? The first (A)says something but you do not hear what he says, so you ask, “Whatdid you say?” The second(B) inhabitant says, “He says he is a Knight,he is and so am I(p).” The third (C)responds, “He is a Knave, but I ama Knight.” What are the three inhabitants really?Answer: The mostbasic rule of Knight and Knave island is that no one ever saysthey are a Knave. Knights always say they are Knights becausethey tell the truth, Knaves say they are Knights because theyalways lie. The second must be a Knight, otherwise he wouldhave lied about the firsts response and if so, he must be tellingthe truth about the first. The third must therefore be a Knave.They are (A)Knight, (B)Knight and (C)Knave.


Introduction to Logic - IITKhome.iitk.ac.in/~avrs/avrs/PHI142/Ravi.pdfMartin Gardner, aha! Insight aha! Gotcha.The mathematical association of America. Lewws Carroll, Symbolic Logic,

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