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