Rules of inference

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Rules of inference

• Section 1.5

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Review Logical Implications
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1 2 Answer?
T T Yes
T F No
F T Yes
F F Yes
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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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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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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 ? q p logically implies (p ? q) p ? (p ? q) is a tautology Addition
p ? q ? p (p ? q) logically implies p (p ? q) ? p is a tautology Simplification
p q ? p ? q p ? q logically implies (p ? q) (p ? q) ? (p ? q) is a tautology Conjunction
p p ? q ? q p ? (p ? q) logically implies q p ? (p ? q) ? q is a tautology Modus ponens
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Rules of Inference
p ? p ? q p logically implies (p ? q) p ? (p ? q) is a tautology Addition
p ? q ? p (p ? q) logically implies p (p ? q) ? p is a tautology Simplification
p q ? p ? q p ? q logically implies (p ? q) (p ? q) ? (p ? q) is a tautology Conjunction
p p ? q ? q p ? (p ? q) logically implies q p ? (p ? q) ? q is a tautology Modus ponens
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Rules of Inference
p ? p ? q p logically implies (p ? q) p ? (p ? q) is a tautology Addition
p ? q ? p (p ? q) logically implies p (p ? q) ? p is a tautology Simplification
p q ? p ? q p ? q logically implies (p ? q) (p ? q) ? (p ? q) is a tautology Conjunction
p p ? q ? q p ? (p ? q) logically implies q p ? (p ? q) ? q is a tautology Modus ponens
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Rules of Inference
p ? p ? q p logically implies (p ? q) p ? (p ? q) is a tautology Addition
p ? q ? p (p ? q) logically implies p (p ? q) ? p is a tautology Simplification
p q ? p ? q p ? q logically implies (p ? q) (p ? q) ? (p ? q) is a tautology Conjunction
p p ? q ? q p ? (p ? q) logically implies q p ? (p ? q) ? q is a tautology Modus ponens
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Rules of Inference
q p ? q ? p q ? (p ? q) logically implies p q ? (p ? q) ? p is a tautology Modus tollens
p ? q q ? r ? p ? r (p?q) ? (q?r) logically implies (p?r) (p?q) ? (q?r) ? (p?r) is a tautology Hypothetical syllogism
p ? q p ? q (p ? q) ? p logically implies q (p ? q) ? p ? q is a tautology Disjunctive syllogism
p ? q p ? r ? q ? r (p ? q) ? (p ? r) logically implies q ? r (p ? q) ? (p ? r) ? q ? r is a tautology Resolution
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Rules of Inference
q p ? q ? p q ? (p ? q) logically implies p q ? (p ? q) ? p is a tautology Modus tollens
p ? q q ? r ? p ? r (p?q) ? (q?r) logically implies (p?r) (p?q) ? (q?r) ? (p?r) is a tautology Hypothetical syllogism
p ? q p ? q (p ? q) ? p logically implies q (p ? q) ? p ? q is a tautology Disjunctive syllogism
p ? q p ? r ? q ? r (p ? q) ? (p ? r) logically implies q ? r (p ? q) ? (p ? r) ? q ? r is a tautology Resolution
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Rules of Inference
q p ? q ? p q ? (p ? q) logically implies p q ? (p ? q) ? p is a tautology Modus tollens
p ? q q ? r ? p ? r (p?q) ? (q?r) logically implies (p?r) (p?q) ? (q?r) ? (p?r) is a tautology Hypothetical syllogism
p ? q p ? q (p ? q) ? p logically implies q (p ? q) ? p ? q is a tautology Disjunctive syllogism
p ? q p ? r ? q ? r (p ? q) ? (p ? r) logically implies q ? r (p ? q) ? (p ? r) ? q ? r is a tautology Resolution
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Rules of Inference
q p ? q ? p q ? (p ? q) logically implies p q ? (p ? q) ? p is a tautology Modus tollens
p ? q q ? r ? p ? r (p?q) ? (q?r) logically implies (p?r) (p?q) ? (q?r) ? (p?r) is a tautology Hypothetical syllogism
p ? q p ? q (p ? q) ? p logically implies q (p ? q) ? p ? q is a tautology Disjunctive syllogism
p ? q p ? r ? q ? r (p ? q) ? (p ? r) logically implies q ? r (p ? q) ? (p ? r) ? q ? r is a tautology Resolution
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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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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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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
  dont have to always convert an inference into a
  formula and demonstrate that the formula is a
  tautology but first
• .

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Rules of Inference for Quantifications
Rule of Inference Name Comments
?x P(x) ? P(c) Universal Specification/Instantiation (US) or (UI) for any c in the domain
P(c) ? ?x P(x) Universal generalization (UG) for an arbitrary c, not a particular one
?x P(x) ? P(c) Existential Specification/Instantiation (ES) or (EI) for some specific c (unknown)
P(c) ? ?x P(x) Existential generalization (EG) Finding one c such that P(c)
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Rules of Inference for Quantifications
Rule of Inference Name Comments
?x P(x) ? P(c) Universal Specification/Instantiation (US) or (UI) for any c in the domain
P(c) ? ?x P(x) Universal generalization (UG) for an arbitrary c, not a particular one
?x P(x) ? P(c) Existential Specification/Instantiation (ES) or (EI) for some specific c (unknown)
P(c) ? ?x P(x) Existential generalization (EG) Finding one c such that P(c)
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Rules of Inference for Quantifications
Rule of Inference Name Comments
?x P(x) ? P(c) Universal Specification/Instantiation (US) or (UI) for any c in the domain
P(c) ? ?x P(x) Universal generalization (UG) for an arbitrary c, not a particular one
?x P(x) ? P(c) Existential Specification/Instantiation (ES) or (EI) for some specific c (unknown)
P(c) ? ?x P(x) Existential generalization (EG) Finding one c such that P(c)
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Rules of Inference for Quantifications
Rule of Inference Name Comments
?x P(x) ? P(c) Universal Specification/Instantiation (US) or (UI) for any c in the domain
P(c) ? ?x P(x) Universal generalization (UG) for an arbitrary c, not a particular one
?x P(x) ? P(c) Existential Specification/Instantiation (ES) or (EI) for some specific c (unknown)
P(c) ? ?x P(x) Existential generalization (EG) Finding one c such that P(c)
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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
• ?x M(x)
• ? M(Socrates) which is an
  example of universal specification

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Example

• Show that there is no largest integer
• That is, show ?x ?y P(x, y) where P(x, y)
  denotes the predicate y gt x
• Let x be an arbitrary integer.
• Then P(x, x1 ) is true by laws of arithmetic (
  x1 gt x ).
• It follows that ?y P(x, y) from existential
  generalization.
• In turn, it follows that ?x ?y P(x, y) from
  universal generalization.