CS 220: Discrete Structures and their Applications

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CS 220: Discrete Structures and their Applications Logical inference Section 1.11-1.13 in zybooks

Valid arguments in propositional logic Consider the following argument: You need a current password to access department machines You have a current password Therefore: You can access department machines

Valid arguments in propositional logic We’ll write it in a more formal way: You need a current password to access department machines You have a current password ∴You can access department machines

Valid arguments in propositional logic This is an example of a logical argument that has the form: p → q p ∴q This is an inference rule is called Modus ponens

Valid arguments in propositional logic And more generally: p1 p2 .... pn ∴ c arguments or hypotheses An argument is valid whenever ( p1  p2  ...  pn ) → c is a tautology. conclusion

Verifying argument validity using truth tables Let’s verify the validity of modus ponens: p → q p ∴ q p q p → q T F

Verifying argument validity using truth tables Let’s verify the validity of modus ponens: p → q p ∴ q p q p → q T F

Verifying argument validity using truth tables Consider the following argument: p → q p  q ∴ q p q p  q p → q T F

Verifying argument validity using truth tables Consider the following argument: p → q p  q ∴ q The argument is valid because whenever the hypotheses are true, the conclusion is true as well. p q p  q p → q T F

Verifying argument validity using truth tables Consider the following argument: ¬p p → q ∴ ¬q Is it valid? p q ¬q ¬p p → q T F

Verifying argument validity using truth tables Consider the following argument: ¬p p → q ∴ ¬q Is it valid? p q ¬q ¬p p → q T F

Example Let’s prove the validity of the following argument using inference rules: If it is raining or windy, the game will be cancelled. The game will not be cancelled. Therefore, it is not windy.

Example In propositional logic: w: It is windy r: It is raining c: The game will be cancelled (r  w) → c ¬c ∴ ¬w

Example In propositional logic: w: It is windy r: It is raining c: The game will be cancelled (r  w) → c ¬c ∴ ¬w Proof: 1. (r  w) → c Hypothesis 2. ¬c Hypothesis 3. ¬(r  w) Modus tollens, 1, 2 4. ¬r  ¬w De Morgan's law, 3 5. ¬w Simplification, 4 Modus tollens ¬q p → q ∴ ¬p

Inference rules p p → q Modus ponens ∴ q ¬q p → q Modus tollens ∴ ¬p p Addition ∴ p  q p  q Simplification ∴ p q Conjunction ∴ p  q p → q q → r Hypothetical syllogism ∴ p → r p  q ¬p Disjunctive syllogism ∴ q ¬p  r Resolution ∴ q  r

Validity of inference rules We can prove the validity of inference rules as well. Consider Modus Tollens for example: 1. p → q Hypothesis 2. ¬p  q Conditional expressed with with disjunction 3. ¬q Hypothesis 4. ¬p Disjunctive syllogism, 2, 3