Truth, like gold, is to be obtained not by its growth, but by washing away from it all that is not gold.  ~Leo Tolstoy

We discussed before that equivalent statements have the same truth values in all cases (see last two columns). There are different ways to achieve equivalent statements.

P Q ~P ~Q PQ ~Q~P

T T F F T T

T F F T F F

F T T F T T

F F T T T T

If a conditional is stated as “if p, then q” then here are the representations of the various forms of that conditional:

Conditional pq if p, then q Converse qp if q, then p Inverse ~p~q if not p, then not q Contrapositive ~q~p if not q, then not p

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This is the truth table for all four forms of a given conditional. Notice which columns have the same truth values (and are therefore equivalent).

P Q ~P ~Q PQ QP

~P~Q

~Q~P

T T F F T T T T

T F F T F T T F

F T T F T F F T

F F T T T T T T

In a truth table, a negation is obvious because it has the opposite truth values of the original statement in every case.

It is also true that the negation of a negation would yield the original statement (and its truth values)

~ (~ P)≡P

An if-then statement is only false when you have a true premise, and a false conclusion (p is true, q is false). Therefore, to negate a conditional you would use that statement (P^~Q).

P Q ~Q PQ P^~Q

T T F T F

T F T F T

F T F T F

F F T T F

DeMorgan’s Laws can apply here for creating equivalent statements.

~ (P∧Q) ≡~P∨~Q

~ (P∨Q) ≡~P∧~Q

: (p∧q) =~p∨~q~(p∨q) =~p∧~q

: (p∧q) =~p∨~q~(p∨q) =~p∧~q

In order to create equivalent statements, you may need to make use of the negation, converse, inverse, contrapositive, or DeMorgan’s Laws.

You can always use a truth table to verify if two statements are equivalent or negations of each other.

3.5, p. 136-138; #4-14 even, 24-34 even