1. Propositional Logic 7/16/20151

2. Propositional Logic A proposition is a statement that is either true or false. We give propositions names such as p, q, and r. E.g. p = “pigs have wings” q = “Obama is U.S. president” In this world anyway, p is false and q is true. 7/16/2015 2

3. Truth Tables  “Not”  ¬p is also written p 7/16/2015 p¬p TF FT − 3

4. Building propositions from propositions “And” So “pigs have wings and Obama is US president” … 7/16/2015 pq p ∧ q TTT TFF FTF FFF 4 Logical Operators/Connectives

5. “Or” So “pigs have wings or Obama is US president” … “Or” is ambiguous in English but not in Propositional Logic E.g., “would you like coffee or ice cream?” 7/16/2015 pq p ∨ q TTT TFT FTT FFF 5 Logical Operators/Connectives

6.  “Exclusive or” 7/16/2015 pq p ⊕ q TTF TFT FTT FFF 6 Logical Operators/Connectives

7. Some expressions can be defined in terms of others  p ⊕ q is equivalent to (p ∨ q) ∧ ¬(p ∧ q)  What does this actually mean?  That the truth value of the formula on the left is the same as the truth value of the formula on the right regardless of the truth values of p and q 7/16/2015 7

8. Propositional Logic 7/16/2015 pq p ⊕ q p v qp ^ q ¬(p ∧ q) (p v q) ^ ¬(p ∧ q) TTFTTFF TFTTFTT FTTTFTT FFFFFTF 8

9. Implies p ⇒ q or p→q or p ⊃ q is equivalent to ¬p ∨ q So “If pigs have wings then Obama is US president” is true in this world In fact a false proposition implies any proposition 7/16/2015 pq p ⇒ q TTT TFF FTT FFT 9

10. Proof by contradiction Recall that in a proof by contradiction we want to prove p and we do so by assuming ¬p and deriving something false Once we have derived any false statement we know we could then derive anything at all! 7/16/2015 10

11. “Implies” Also note that a true proposition is implied by any proposition It is true that “if Goldbach’s conjecture holds then the Pythagorean theorem is true” regardless of whether Goldbach’s conjecture holds or not (we don’t know, and we don’t even need to know what it is!) 7/16/2015 pq p ⇒ q TTT TFF FTT FFT 11

12. “Iff” or logical equivalence p ⇔ q means that p and q have the same truth value Equivalent to (p ⇒ q) ∧ (q ⇒ p) 7/16/2015 pq p ⇔ q TTT TFF FTF FFT 12