Propositional Logic
6.2 Truth Functions
Truth Functions
Truth functions for the tilde:
p ~p
T F
F T
Plug in truth Out comes falsehood
Plug in falsehood
Out comes truth
Truth Functions
Truth functions for the dot:
p q p • q
T T
T F
F T
F F
T
F
F
F
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?
If Jack went but Jill didn’t, what should we say about the sentence, p • q?If Jack didn’t go but Jill did, what should we say about the sentence, p • q?If neither of them went, what
should we say about the sentence, p • q?
Truth Functions
Truth functions for the wedge:
p q p v q
T T
T F
F T
F F
T
T
T
F
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?
If Jack went but Jill didn’t, what should we say about the sentence, p v q?If Jack didn’t go but Jill did, what should we say about the sentence, p v q?If neither of them went, what
should we say about the sentence, p v q?
Truth Functions
Truth functions for the horseshoe (arrow):
p q p → q
T T
T F
F T
F F
T
F
T
T
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?
If Jack went but Jill didn’t, what should we say about the sentence, p → q?If Jack didn’t go but Jill did, what should we say about the sentence, p → q?If neither of them went, what
should we say about the sentence, p → q?
Truth Functions
Truth functions for the triple bar:
p q p Ξ q
T T
T F
F T
F F
T
F
F
T
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?
If Jack went but Jill didn’t, what should we say about the sentence, p Ξ q?If Jack didn’t go but Jill did, what should we say about the sentence, p Ξ q?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