Valid and Invalid Arguments - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Valid and Invalid Arguments

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Modus Ponens example

• Assume you are given the following two statements: - “you are in this class” - “if you are in this class, you will get a grade”
• Let p = “you are in this class”
• Let q = “you will get a grade”
• By Modus Ponens, you can conclude that you will get a grade

q

p q

p

∴

→

4

Modus Tollens

• Assume that we know: ¬q and p → q
  • Recall that p → q = ¬q → ¬p
• Thus, we know ¬q and ¬q → ¬p
• We can conclude ¬p

p

p q

q

∴ ¬

→

¬

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Modus Tollens example

• Assume you are given the following two statements: - “you will not get a grade” - “if you are in this class, you will get a grade”
• Let p = “you are in this class”
• Let q = “you will get a grade”
• By Modus Tollens, you can conclude that you are not in this class

p

p q

q

∴ ¬

→

¬

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Example of proof

• We have the hypotheses:
  • “It is not sunny this afternoon and it is colder than yesterday”
  • “We will go swimming only if it is sunny”
  • “If we do not go swimming, then we will take a canoe trip”
  • “If we take a canoe trip, then we will be home by sunset”
• Does this imply that “we will be home by sunset”?

• “It is not sunny this afternoon and it is colder than yesterday”
• “We will go swimming only if it is sunny”
• “If we do not go swimming, then we will take a canoe trip”
• “If we take a canoe trip, then we will be home by sunset”
• Does this imply that “we will be home by sunset”?

• “It is not sunny this afternoon and it is colder than yesterday”
• “We will go swimming only if it is sunny”
• “If we do not go swimming, then we will take a canoe trip”
• “If we take a canoe trip, then we will be home by sunset”
• Does this imply that “we will be home by sunset”?

• “It is not sunny this afternoon and it is colder than yesterday”
• “We will go swimming only if it is sunny”
• “If we do not go swimming, then we will take a canoe trip”
• “If we take a canoe trip, then we will be home by sunset”
• Does this imply that “we will be home by sunset”?

• “It is not sunny this afternoon and it is colder than yesterday”
• “We will go swimming only if it is sunny”
• “If we do not go swimming, then we will take a canoe trip”
• “If we take a canoe trip, then we will be home by sunset”
• Does this imply that “we will be home by sunset”?
• “It is not sunny this afternoon and it is colder than yesterday”
• “We will go swimming only if it is sunny”
• “If we do not go swimming, then we will take a canoe trip”
• “If we take a canoe trip, then we will be home by sunset”
• Does this imply that “we will be home by sunset”?

p

q

r

s

t

¬p ∧ q

r → p

¬r → s

s → t

t

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Example of proof

1. ¬p ∧ q 1 st^ hypothesis
2. ¬p Simplification using step 1
3. r → p 2 nd^ hypothesis
4. ¬r Modus tollens using steps 2 & 3
5. ¬r → s 3 rd^ hypothesis
6. s Modus ponens using steps 4 & 5
7. s → t 4 th^ hypothesis
8. t Modus ponens using steps 6 & 7

p

p q ∴

∧ q

p q

p

∴

→ p

p q

q

∴ ¬

→

¬

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More rules of inference

• Conjunction: if p and q are true separately, then p∧q is true
• Elimination: If p∨q is true, and p is false, then q must be true
• Transitivity: If p→q is true, and q→r is true, then p→r must be true

p q

q

p

∴ ∧

q

p

p q

∴

¬

∨

p r

q r

p q

∴ →

→

→

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Even more rules of inference

• Proof by division into cases: if at least one of p or q is true, then r must be true
• Contradiction rule: If ¬p→c is true, we can conclude p (via the contra-positive)
• Resolution: If p∨q is true, and ¬p∨r is true, then q∨r must be true - Not in the textbook

p q p r q r r

∨ → → ∴

p c p

¬ → ∴

q r

p r

p q

∴ ∨

¬ ∨

∨

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Example of proof

1. ¬t 3 rd^ hypothesis
2. s → t 2 nd^ hypothesis
3. ¬s Modus tollens using steps 2 & 3
4. (¬r∨¬f)→(s∧l) 1 st^ hypothesis
5. ¬(s∧l)→¬(¬r∨¬f) Contrapositive of step 4
6. (¬s∨¬l)→(r∧f) DeMorgan’s law and double negation law
7. ¬s∨¬l Addition from step 3
8. r∧f Modus ponens using steps 6 & 7
9. r Simplification using step 8

p

p q ∴

∧ q

p q

p

∴

→ p

p q

q

∴ ¬

→

¬

p q

p ∴ ∨ Docsity.com

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Modus Badus

• Consider the following:
• Is this true?

p q p→q q∧(p→q)) (q∧(p→q)) → p

T T T T T T F F F T F T T T F F F T F T

Not a

valid

rule!

p

p q

q

∴

→ p

q p

q

∴

¬ → ¬

Fallacy of affirming the conclusion

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Modus Badus

• Consider the following:
• Is this true?

p q p→q ¬p∧(p→q)) (¬p∧(p→q)) → ¬q

T T T F T

T F F F T

F T T T F

F F T T T

Not a

valid

rule!

q

p q

p

∴ ¬

→

¬

Fallacy of denying the hypothesis

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Modus Badus example

• Assume you are given the following two statements: - “you are not in this class” - “if you are in this class, you will get a grade”
• Let p = “you are in this class”
• Let q = “you will get a grade”
• You CANNOT conclude that you will not get a grade - You could be getting a grade for another class

q

p q

p

∴ ¬

→

¬