Rules of inference
Section 1.5
MSU/CSE 260 Fall 20091
Review: Logical Implications ? #1 #2
#1 #2 Answer? TTYes TFNo FTYes FFYes
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Terminology
Axiom or Postulate: An underlying assumption; often used to begin a logical argument with. Rules of inference: Rules explaining how conclusions are drawn from axioms/postulates. Proof: A sequence of propositions that forms a valid argument. Fallacy: Incorrect reasoning (invalid argument)
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Terminology…
Theorem: A proposition that can be shown to be true. Lemma: A simple theorem used in the proof of other theorems. Corollary: A fact that can be immediately deduced from a Theorem/Lemma. Conjecture: A proposition whose correctness is unknown.
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Rules of inference
Rules of inference are used to draw conclusions from hypotheses. These are the logical implication questions for which the answer is YES Consider the question: Does [p ∧ (p → q)] logically imply q The answer is YES as [p ∧ (p → q)] → q is a tautology It is the basis of the rule of inference called modus ponens, which can be represented by the symbolic form p p → q ∴ q
which means that whenever p is true and p → q is true we can conclude that q is true. In other words [p ∧ (p → q)] logically implies q
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Example
Consider the argument: You have a CSE account if you are taking CSE 260. You are taking CSE 260. Therefore, you have a CSE account. This argument is an instance of modus ponens: p: You are taking CSE 260. q: You have a CSE account. Then the argument has the form: p → q p ∴ q Thus, the argument is valid.
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- A Few Tautologies
p → (p ∨ q) (p ∧ q) → p p ∧ q → (p ∧ q)
[p ∧ (p → q)] → q [¬ q ∧ (p → q)] → ¬ p [(p → q) ∧ (q → r)] → (p → r)
[(p ∨ q) ∧ ¬p] → q
Each of these tautologies can be formalized as a rule of inference for use in justifying claims in a valid argument.
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Rules of Inference
p p p → (p ∨ q) logically implies Addition ∴ ∨ p q (p ∨ q) is a tautology ∧ p ∧ q (p q) (p ∧ q) → p logically implies Simplification ∴ p p is a tautology p ∧ p q (p ∧ q) → (p ∧ q) q logically implies Conjunction is a tautology ∴ p ∧ q (p ∧ q) p ∧ → [p (p q)] [p ∧ (p → q)] → q p → q logically implies Modus ponens is a tautology ∴ q q
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Rules of Inference
p p p → (p ∨ q) logically implies Addition ∴ ∨ p q (p ∨ q) is a tautology ∧ p ∧ q (p q) (p ∧ q) → p logically implies Simplification ∴ p p is a tautology p ∧ p q (p ∧ q) → (p ∧ q) q logically implies Conjunction is a tautology ∴ p ∧ q (p ∧ q) p ∧ → [p (p q)] [p ∧ (p → q)] → q p → q logically implies Modus ponens is a tautology ∴ q q
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Rules of Inference
p p p → (p ∨ q) logically implies Addition ∴ ∨ p q (p ∨ q) is a tautology ∧ p ∧ q (p q) (p ∧ q) → p logically implies Simplification ∴ p p is a tautology p ∧ p q (p ∧ q) → (p ∧ q) q logically implies Conjunction is a tautology ∴ p ∧ q (p ∧ q) p ∧ → [p (p q)] [p ∧ (p → q)] → q p → q logically implies Modus ponens is a tautology ∴ q q
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Rules of Inference
p p p → (p ∨ q) logically implies Addition ∴ ∨ p q (p ∨ q) is a tautology ∧ p ∧ q (p q) (p ∧ q) → p logically implies Simplification ∴ p p is a tautology p ∧ p q (p ∧ q) → (p ∧ q) q logically implies Conjunction is a tautology ∴ p ∧ q (p ∧ q) p ∧ → [p (p q)] [p ∧ (p → q)] → q p → q logically implies Modus ponens is a tautology ∴ q q
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Rules of Inference
¬ q ∧ → [¬ q (p q)] [¬ q ∧ (p → q)] → ¬ p p → q logically implies Modus tollens is a tautology ∴ ¬ p ¬ p p → q [(p→q) ∧ (q→r)] [(p→q) ∧ (q→r)] → (p→r) Hypothetical q → r logically implies is a tautology syllogism ∴ p → r (p→r) p ∨ q [(p ∨ q) ∧ ¬p] [(p ∨ q) ∧ ¬p] → q Disjunctive ¬p logically implies is a tautology syllogism ∴ q q p ∨ q [(p ∨ q) ∧ (¬p ∨ r)] ¬ p ∨ r logically implies [(p ∨ q) ∧ (¬p ∨ r)] → q ∨ r ∨ Resolution ∴ q ∨ r q r is a tautology
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Rules of Inference
¬ q ∧ → [¬ q (p q)] [¬ q ∧ (p → q)] → ¬ p p → q logically implies Modus tollens is a tautology ∴ ¬ p ¬ p p → q [(p→q) ∧ (q→r)] [(p→q) ∧ (q→r)] → (p→r) Hypothetical q → r logically implies is a tautology syllogism ∴ p → r (p→r) p ∨ q [(p ∨ q) ∧ ¬p] [(p ∨ q) ∧ ¬p] → q Disjunctive ¬p logically implies is a tautology syllogism ∴ q q p ∨ q [(p ∨ q) ∧ (¬p ∨ r)] ¬ p ∨ r logically implies [(p ∨ q) ∧ (¬p ∨ r)] → q ∨ r ∨ Resolution ∴ q ∨ r q r is a tautology
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Rules of Inference
¬ q ∧ → [¬ q (p q)] [¬ q ∧ (p → q)] → ¬ p p → q logically implies Modus tollens is a tautology ∴ ¬ p ¬ p p → q [(p→q) ∧ (q→r)] [(p→q) ∧ (q→r)] → (p→r) Hypothetical q → r logically implies is a tautology syllogism ∴ p → r (p→r) p ∨ q [(p ∨ q) ∧ ¬p] [(p ∨ q) ∧ ¬p] → q Disjunctive ¬p logically implies is a tautology syllogism ∴ q q p ∨ q [(p ∨ q) ∧ (¬p ∨ r)] ¬ p ∨ r logically implies [(p ∨ q) ∧ (¬p ∨ r)] → q ∨ r ∨ Resolution ∴ q ∨ r q r is a tautology
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Rules of Inference
¬ q ∧ → [¬ q (p q)] [¬ q ∧ (p → q)] → ¬ p p → q logically implies Modus tollens is a tautology ∴ ¬ p ¬ p p → q [(p→q) ∧ (q→r)] [(p→q) ∧ (q→r)] → (p→r) Hypothetical q → r logically implies is a tautology syllogism ∴ p → r (p→r) p ∨ q [(p ∨ q) ∧ ¬p] [(p ∨ q) ∧ ¬p] → q Disjunctive ¬p logically implies is a tautology syllogism ∴ q q p ∨ q [(p ∨ q) ∧ (¬p ∨ r)] ¬ p ∨ r logically implies [(p ∨ q) ∧ (¬p ∨ r)] → q ∨ r ∨ Resolution ∴ q ∨ r q r is a tautology
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Example
Is the following argument valid? If the program crashed, an exception was raised. If an exception was raised, someone input a text value for an integer. Therefore, if the program crashed, someone input a text value for an integer. If valid, what rule of inference is used? If not, how do you know it is invalid? p → q p: The program crashed. q → r q: An exception was raised. ∴ p → r r: Someone input a text value for an integer. Hypothetical Syllogism –its valid!
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Example
Is the following argument valid? If the program crashed, an exception was raised. If an exception was raised, someone input a text value for an integer value. Therefore, the program did not crash, If valid, what rule of inference is used? If not, how do you know its invalid? p → q p: The program crashed. q → r q: An exception was raised. ∴¬p r: Someone input a text value for an integer. [(p → q) ∧ (q → r ) → ¬ p] is not a tautology; therefore, the argument is not valid.
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Example
Is the following argument valid? If the program crashed, an exception was raised. If an exception was raised, someone input a text value for an integer value. No one input a text value for an integer. Therefore, the program did not crash, If valid, what rule of inference is used? If not, how do you know its invalid? p → q p: The program crashed. q → r q: An exception was raised. ¬ r r: Someone input a text value for an ∴¬p integer.
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Example
Is the following argument valid? If the program crashed, an exception was raised. If an exception was raised, someone input a text value for an integer value. No one input a text value for an integer. Therefore, the program did not crash, If valid, what rule of inference is used? If not, how do you know its invalid? [ ( (p → q) ∧ (q → r ) ∧¬r ) → ¬ p ] is a tautology; therefore, the argument is valid. We will shortly develop methods of proof so we don’t have to always convert an inference into a formula and demonstrate that the formula is a tautology; but first … . MSU/CSE 260 Fall 2009 19
Rules of Inference for Quantifications
Rule of Name Comments Inference
∀xP(x) Universal ∴ P(c) Specification/Instantiation for any c in the domain (US) or (UI)
P(c) Universal generalization for an arbitrary c, not a ∴ ∀xP(x) (UG) particular one
∃xP(x) Existential ∴ P(c) Specification/Instantiation for some specific c (unknown) (ES) or (EI)
P(c) Existential generalization Finding one c such that P(c) ∴∃xP(x) (EG)
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Rules of Inference for Quantifications
Rule of Name Comments Inference
∀xP(x) Universal ∴ P(c) Specification/Instantiation for any c in the domain (US) or (UI)
P(c) Universal generalization for an arbitrary c, not a ∴ ∀xP(x) (UG) particular one
∃xP(x) Existential ∴ P(c) Specification/Instantiation for some specific c (unknown) (ES) or (EI)
P(c) Existential generalization Finding one c such that P(c) ∴∃xP(x) (EG)
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Rules of Inference for Quantifications
Rule of Name Comments Inference
∀xP(x) Universal ∴ P(c) Specification/Instantiation for any c in the domain (US) or (UI)
P(c) Universal generalization for an arbitrary c, not a ∴ ∀xP(x) (UG) particular one
∃xP(x) Existential ∴ P(c) Specification/Instantiation for some specific c (unknown) (ES) or (EI)
P(c) Existential generalization Finding one c such that P(c) ∴∃xP(x) (EG)
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Rules of Inference for Quantifications
Rule of Name Comments Inference
∀xP(x) Universal ∴ P(c) Specification/Instantiation for any c in the domain (US) or (UI)
P(c) Universal generalization for an arbitrary c, not a ∴ ∀xP(x) (UG) particular one
∃xP(x) Existential ∴ P(c) Specification/Instantiation for some specific c (unknown) (ES) or (EI)
P(c) Existential generalization Finding one c such that P(c) ∴∃xP(x) (EG)
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Example: Socrates is mortal
All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Define M(x): “x is mortal.” Define the universe to be all men. Then the argument being made is:
∀xM(x) ∴ M(Socrates)
which is an example of universal specification
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1- Example:
Show that there is no “largest” integer: That is, show ∀x ∃y P(x, y) where P(x, y) denotes the predicate “y > x” Let x be an arbitrary integer. Then P(x, x+1 ) is true by laws of arithmetic ( x+1 > x ). It follows that ∃y P(x, y) from existential generalization. In turn, it follows that ∀x ∃y P(x, y) from universal generalization.
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© 2006 by A-H. Esfahanian. All Rights Reserved. 1-