1. Propositional Logic
6.2 Truth Functions

2. Truth Functions Truth functions for the tilde: Plug in truth p ~p T F
Out comes falsehood
Out comes truth
Plug in falsehood

3. Truth Functions Truth functions for the dot:
Let p = Jack went up the hill.
Let q = Jill went up the hill.
If both of them actually went up the hill, what should we say about the sentence, p • q? p q
p • q T F T
If Jack went but Jill didn’t, what should we say about the sentence, p • q? F
If Jack didn’t go but Jill did, what should we say about the sentence, p • q? F F
If neither of them went, what should we say about the sentence, p • q?

4. Truth Functions Truth functions for the wedge:
Let p = Jack went up the hill.
Let q = Jill went up the hill.
If both of them actually went up the hill, what should we say about the sentence, p v q? p q
p v q T F T
If Jack went but Jill didn’t, what should we say about the sentence, p v q? T
If Jack didn’t go but Jill did, what should we say about the sentence, p v q? T F
If neither of them went, what should we say about the sentence, p v q?

5. Truth Functions Truth functions for the horseshoe (arrow):
Let p = Jack went up the hill.
Let q = Jill went up the hill.
If both of them actually went up the hill, what should we say about the sentence, p → q? p q
p → q T F T
If Jack went but Jill didn’t, what should we say about the sentence, p → q? F
If Jack didn’t go but Jill did, what should we say about the sentence, p → q? T T
If neither of them went, what should we say about the sentence, p → q?

6. Truth Functions Truth functions for the triple bar:
Let p = Jack went up the hill.
Let q = Jill went up the hill.
If both of them actually went up the hill, what should we say about the sentence, p Ξ q? p q
p Ξ q T F T
If Jack went but Jill didn’t, what should we say about the sentence, p Ξ q? F
If Jack didn’t go but Jill did, what should we say about the sentence, p Ξ q? F T
If neither of them went, what should we say about the sentence, p Ξ q?

7. Truth Functions
If the truth table for the horseshoe bothers you, just translate it to this: ~p v q So, saying to a troublemaker in the bar: If you stay, I’ll flatten you (S  F) Is the same as saying Leave or I’ll flatten you (~S v F)

8. Computing Truth Values of Big Propositions
True: A, B, and C False: D, E, and F What’s the truth value of … (A v D)  E ?

9. Computing Truth Values of Big Propositions
True: A, B, and C False: D, E, and F (A v D)  E (T v F)  F (put in the truth values) T  F (simplify from truth table) F

10. Computing Truth Values of Big Propositions
True: A, B, and C False: D, E, and F (B • C)  (E  A) (T • T)  (F  T) (put in the truth values) T  T (simplify from truth table) T