1. Rules of inference Section 1.5 Monday, December 02, 2019
MSU/CSE 260 Fall 2009 1
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2. Review: Logical Implications
Monday, December 02, 2019
Review: Logical Implications ? #1 #2 #1 #2
Answer? T
Yes F No
The above tables summarizes the main truth table for #1 and #2, which may consists of many atomic propositions. Every proof technique must directly or indirectly examine all the rows of the main truth table to come to an affirmative answer.
MSU/CSE 260 Fall 2009 2
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3. Monday, December 02, 2019
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)
MSU/CSE 260 Fall 2009 3
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4. Terminology… Theorem: A proposition that can be shown to be true.
Monday, December 02, 2019
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.
MSU/CSE 260 Fall 2009 4
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5. Monday, December 02, 2019
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
The name "modus ponens" comes from the Latin word "modus" meaning method, and the Latin word "ponens" meaning affirming.
MSU/CSE 260 Fall 2009 5
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6. Example
Consider the argument: You have a CSE account if you are taking CSE You are taking CSE Therefore, you have a CSE account.
This argument is an instance of modus ponens: p: You are taking CSE q: You have a CSE account.
Then the argument has the form: p  q p  q
Thus, the argument is valid.
MSU/CSE 260 Fall 2009

7. A Few Tautologies p → (p  q) (p  q) → p p  q → (p  q)
Monday, December 02, 2019
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.
MSU/CSE 260 Fall 2009 7
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8. Rules of Inference p  p  q p logically implies (p  q) p → (p  q)
Monday, December 02, 2019
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
Simplification q
 p  q
p  q logically implies (p  q)
(p  q) → (p  q)
Conjunction
p  q
 q
[p  (p → q)] logically implies q
[p  (p → q)] → q
Modus ponens
MSU/CSE 260 Fall 2009
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9. Rules of Inference p  p  q p logically implies (p  q) p → (p  q)
Monday, December 02, 2019
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
Simplification q
 p  q
p  q logically implies (p  q)
(p  q) → (p  q)
Conjunction
p  q
 q
[p  (p → q)] logically implies q
[p  (p → q)] → q
Modus ponens
MSU/CSE 260 Fall 2009
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10. Rules of Inference p  p  q p logically implies (p  q) p → (p  q)
Monday, December 02, 2019
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
Simplification q
 p  q
p  q logically implies (p  q)
(p  q) → (p  q)
Conjunction
p  q
 q
[p  (p → q)] logically implies q
[p  (p → q)] → q
Modus ponens
MSU/CSE 260 Fall 2009
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11. Rules of Inference p  p  q p logically implies (p  q) p → (p  q)
Monday, December 02, 2019
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
Simplification q
 p  q
p  q logically implies (p  q)
(p  q) → (p  q)
Conjunction
p  q
 q
[p  (p → q)] logically implies q
[p  (p → q)] → q
Modus ponens
MSU/CSE 260 Fall 2009
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12. Rules of Inference Resolution ¬ q p  q  ¬ p
Monday, December 02, 2019
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)
Hypothetical syllogism
p  q ¬p
 q
[(p  q)  ¬p] logically implies q
[(p  q)  ¬p] → q
Disjunctive syllogism
¬ p  r
 q  r
[(p  q)  (¬p  r)] logically implies q  r
[(p  q)  (¬p  r)] → q  r
Resolution
MSU/CSE 260 Fall 2009 12
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13. Rules of Inference Resolution ¬ q p  q  ¬ p
Monday, December 02, 2019
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)
Hypothetical syllogism
p  q ¬p
 q
[(p  q)  ¬p] logically implies q
[(p  q)  ¬p] → q
Disjunctive syllogism
¬ p  r
 q  r
[(p  q)  (¬p  r)] logically implies q  r
[(p  q)  (¬p  r)] → q  r
Resolution
MSU/CSE 260 Fall 2009 13
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14. Rules of Inference Resolution ¬ q p  q  ¬ p
Monday, December 02, 2019
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)
Hypothetical syllogism
p  q ¬p
 q
[(p  q)  ¬p] logically implies q
[(p  q)  ¬p] → q
Disjunctive syllogism
¬ p  r
 q  r
[(p  q)  (¬p  r)] logically implies q  r
[(p  q)  (¬p  r)] → q  r
Resolution
MSU/CSE 260 Fall 2009 14
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15. Rules of Inference Resolution ¬ q p  q  ¬ p
Monday, December 02, 2019
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)
Hypothetical syllogism
p  q ¬p
 q
[(p  q)  ¬p] logically implies q
[(p  q)  ¬p] → q
Disjunctive syllogism
¬ p  r
 q  r
[(p  q)  (¬p  r)] logically implies q  r
[(p  q)  (¬p  r)] → q  r
Resolution
MSU/CSE 260 Fall 2009 15
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16. 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!
MSU/CSE 260 Fall 2009

17. 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.
MSU/CSE 260 Fall 2009

18. 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.
MSU/CSE 260 Fall 2009

19. 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

20. Rules of Inference for Quantifications
Monday, December 02, 2019
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)
Existential Specification/Instantiation (ES) or (EI)
for some specific c (unknown)
 x P(x)
Existential generalization (EG)
Finding one c such that P(c)
MSU/CSE 260 Fall 2009 20
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21. Rules of Inference for Quantifications
Monday, December 02, 2019
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)
Existential Specification/Instantiation (ES) or (EI)
for some specific c (unknown)
 x P(x)
Existential generalization (EG)
Finding one c such that P(c)
MSU/CSE 260 Fall 2009 21
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22. Rules of Inference for Quantifications
Monday, December 02, 2019
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)
Existential Specification/Instantiation (ES) or (EI)
for some specific c (unknown)
 x P(x)
Existential generalization (EG)
Finding one c such that P(c)
MSU/CSE 260 Fall 2009 22
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23. Rules of Inference for Quantifications
Monday, December 02, 2019
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)
Existential Specification/Instantiation (ES) or (EI)
for some specific c (unknown)
 x P(x)
Existential generalization (EG)
Finding one c such that P(c)
MSU/CSE 260 Fall 2009 23
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24. 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
MSU/CSE 260 Fall 2009

25. 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.
MSU/CSE 260 Fall 2009