Propositional Logic 6.2 Truth Functions

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

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?

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?

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?

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?

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)

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 ?

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

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