# Introduction to Logic

Patrick Hurley, Concise introduction to Logic, Thomson / Wadsworth, 2007[standard Course book] Mendelson, Introduction to Mathematical Logic, pp:1-90 ...

Introduction to Logic A. V. Ravishankar Sarma Indian Institute of Technology Kanpur

July 24, 2008

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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PHI142: Introduction to Logic Answer some frequently asked questions related to course What is Logic? Why study Logic? How logic can be done: What will be taught in the course?

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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PHI142: Introduction to Logic Answer some frequently asked questions related to course What 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 concepts of logic, To discern good reasoning and bad reasoning, when an argument is valid, distinguishing inductive and deductive arguments, identifying fallacies and avoiding them. Objective: Equip oneself with various tools and techniques (Decision procedures) for the validity of a given argument, detecting and avoiding fallacies of a given deductive or inductive argument.

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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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 does some intelligent tasks for you.(Knowledge representation). Engaged in debates, solving puzzles, game like situation. Identify which one is a fallacious argument- what type of fallacy? proving theorems (deduction), is what ever proved is correct, or what ever is obviously true has a proof? Analysing and designing circuits. Some problems concerning the foundations of mathematics (can it be rested on set theory)?

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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Layman’s point of view: Logic (reasoning) argumentation is omnipresent Children arguing with parents (persuasion)-Argumentation Intelligent/bright students exercising their reasoning skills. Politicians trying to woo/persuade their voters by invoking emotionsCaught in a traffic (don’t have license/forgot to take license)- persuasive skills (Informal fallacies) What goes on in solving a mathematical problem (proof, Deduction) Scientist’s tool box: When an Hypothesis is confirmed by evidence? How the computer works? [Logic of [0,1]- Boolean logic]

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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What is Logic? coming up with a definition would be narrowing the scope and subject matter oflogic. Laws of thought (George Boole): The way we think is governed by

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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What is Logic? coming up with a definition would be narrowing the scope and subject matter oflogic. Laws of thought (George Boole): The way we think is governed by The way we think: Law of Identity P = P, Law of excluded middle P ∨ ¬P. Law of non-contradiction¬(P ∧ ¬P). Study of principles of valid reasoning: to discern good reasoning from bad reasoning. Relaxing some of the conditions: Intuitionist logic, Fuzzy logic, Para consistent Logic. Study of connectives ∧, ¬, ∨, →,↔. Study of ∀x , ∃x

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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Examples Example Premises: All IITK students are mortal. Ramu is an IITK student. Conclusion: Ramu is mortal. P: All parallelograms are circles All circles are squares C: All parallelograms are squares. P: Some men are liars and Laloo is a man C: Laloo is a liar First argument is valid and sound. The second argument is unsound but valid.Third argument is invalid. A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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Some Proofs

Example Augustus De Morgan: 2=1 Let x = 1. Then x2 = x. So x2 − 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 a Kheer (Sweet dish)? It would appear that eternal happiness is better, but this is really not so!

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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Some Proofs

Example Augustus De Morgan: 2=1 Let x = 1. Then x2 = x. So x2 − 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 a Kheer (Sweet dish)? It would appear that eternal happiness is better, but this is really not so!After all, nothing is better than eternal happiness, and Kheer is certainly better than nothing. Therefore a Kheer is better than eternal happiness.

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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Proof that I am Dracula

Example P: Everyone is afraid of Dracula. Dracula is afraid of only me. C: Therefore I am Dracula. Since everyone is afraid of Dracula, then Dracula is afraid of Dracula. So Dracula is afraid of Dracula, but also is afraid of no one but me. Therefore I must be Dracula! Example This sentence is false. Is that sentence true or false? If it is false then it is true, and if it is true then it is false.[Liars Paradox]

A. V. Ravishankar Sarma

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Informal Fallacies: Example Abhishek: If someone hits you, you should turn the other cheek. Violence only begets violence, and violence in and of itself is wrong. Aishwarya: Thats a joke. You used to hit people when they picked a fight with you. [Ad hominem- tuquoqueyou 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, it breaks. This egg is broken, so I must have dropped (Fallacy of affirming the consequent) A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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What we can learn from this course? Basic concepts: premise, conclusion, argument, non-argument, deductive and inductive arguments, invalidity of deductive arguments (Counter example method)

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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What we can learn from this course? Basic concepts: premise, conclusion, argument, non-argument, deductive and inductive arguments, invalidity of deductive arguments (Counter example method) Fallacies: Persuasive, and mainly has emotional component–Formal fallacies, Informal fallacies (Relevance, weak induction, presumption.) Classical Aristotle logic: Syllogisms (Euler circles, Venn Diagrams)- Limitations of Aristotle logic, Boole, Propositional Logic (Russell Whitehead/Hilbert Axiomatic systems, proofs of various theorems, Soundness, completeness, satisfiability.

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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What we can learn from this course? Basic concepts: premise, conclusion, argument, non-argument, deductive and inductive arguments, invalidity of deductive arguments (Counter example method) Fallacies: Persuasive, and mainly has emotional component–Formal fallacies, Informal fallacies (Relevance, weak induction, presumption.) Classical Aristotle logic: Syllogisms (Euler circles, Venn Diagrams)- Limitations of Aristotle logic, Boole, Propositional Logic (Russell Whitehead/Hilbert Axiomatic systems, proofs of various theorems, Soundness, completeness, satisfiability. Various techniques: Truth table method (Emilie Post), Indirect truth table(when “n’ is large), Natural Deduction method, Reductio ad absurdum, Refutation tree method. Is propositional Logic complete(Whatever is true has proof and whatever is proved is true) A. V. Ravishankar PHI142:of Introduction to Logic validity, 11/ 16 Logic of quantifiers ∀ Sarma , ∃ . Scope quantifier,

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, oxford university press, pp 1-115[Extra] Bertrand Russell and A N Whitehead, Principia Mathematica, 1910, pp. 89-135 Raymond Smullyan, Forever Undecided: A Puzzle Guide to G¨odel, 1987 Martin Gardner, aha! Insight aha! Gotcha.The mathematical association of America. Lewws Carroll, Symbolic Logic, availale in http://durendal.org:8080/lcsl/ A. V. Ravishankar Sarma

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Futher references to get in touch with logic:

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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Historical Context Indian Logic(640BC). Classical antquity: one of the seven liberal arts: Grammar, Rhetoric, Logic and Arithmetic: Number in itself Geometry: Number in space Music, Harmonics, or Tuning theory (number in time) Astronomy or cosmology- number in space and time. Classical Aristotle logic: Syllogism (1900 years)- Logic as a justificatory tool Medieval Logic: Leibniz, George Boole, Venn- validity Modern Logic: Frege, Betrand Russell, Hilbert-Ackerman (Logicism, formalism, Intuitionism) Propositional, Predicate Logic.

A. V. Ravishankar Sarma

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Russell’s paradox Example There is someone who loves any person only if that person does not does not love themselves. ∃x ∀y [Lxy → ¬Lyy ]. Example In a certain village there is a man, so the paradox runs, who is a barber; this barber shaves all and only those men in the village who do not shave themselves.

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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Russell’s paradox Example There is someone who loves any person only if that person does not does not love themselves. ∃x ∀y [Lxy → ¬Lyy ]. Example In a certain village there is a man, so the paradox runs, who is a barber; this barber shaves all and only those men in the village who do not shave themselves. Query: Does the barber shave himself? Any man in this village is shaved by the barber if and only if he is not shaved by himself. Therefore in particular the barber shaves himself if and only if he does not. We are in trouble if we say the barber shaves himself and we are in trouble if we say he does not. A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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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 the shore. 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, “What did 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 am a Knight.” What are the three inhabitants really?

A. V. Ravishankar Sarma

PHI142: Introduction to Logic

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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 the shore. 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, “What did 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 am a Knight.” What are the three inhabitants really?Answer: The most

basic rule of Knight and Knave island is that no one ever says they are a Knave. Knights always say they are Knights because they tell the truth, Knaves say they are Knights because they always lie. The second must be a Knight, otherwise he would have lied about the firsts response and if so, he must be telling the truth about the first. The third must therefore be a Knave. They are (A)Knight, (B)Knight and (C)Knave. A. V. Ravishankar Sarma

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