1
$\begingroup$

Accroding to the Truth Table,

If $p$ is TRUE, and $q$ is FALSE, then $p\implies q$ is FALSE.
And the converse, $q \implies p$, is TRUE.

If the conditional statement is

"If two angles are congruent, they are not equal." this is a FALSE statement.

Its converse is "If two angles are not equal, they are congruent." The converse is also FALSE.

Why does this example contradict the Truth Table??

$\endgroup$
  • $\begingroup$ The opposite of "If A then B" isn't "If B then A", it is "A and not B". $\endgroup$ – DanielV Aug 21 '15 at 1:06
3
$\begingroup$

The source of the confusion is that you are not dealing merely with implications in propositional logic, where, as you said, the truth tables ensure that $(p\implies q)\lor(q\implies p)$ is always true. Your statements about angles are universally quantified statements, even though the English language lets you hide the quantifiers. Your first statement really means "For every two angles $x$ and $y$, if $x$ and $y$ are congruent then $x$ and $y$ are not equal." Similarly for the second statement.

So the logical form of these statements is $(\forall x)(\forall y)\,(P(x,y)\implies Q(x,y))$ and $(\forall x)(\forall y)\,(Q(x,y)\implies P(x,y))$. Because of the quantifiers, it is entirely possible for both of these to be false. All that's needed for that to happen is that some particular $x_0$ and $y_0$ satisfy $P$ but not $Q$, while a different pair $x_1$ and $y_1$ satisfy $Q$ but not $P$.

If quantifiers are not involved, for example if you have two particular angles and are making statements about just this pair, not angles in general, then a proposition and its converse will not both be false.

$\endgroup$
  • $\begingroup$ Thank you so much for your detailed answer!! Your answer is very helpful!! Then for the converse statement, if I say "5 degrees is not equal to 10 degrees divided by 2, then the two angles are congruent." So this would be a TRUE(vacuously TRUE?) statement, thus satisfying the Truth Table? Am I correct in using this logic? $\endgroup$ – chris vin Aug 20 '15 at 13:36
-1
$\begingroup$

It doesn't. In your example let $q = $"two angles are not equal" be false so that means they are equal. Hence the statement "If two angles are not equal, they are congruent" is vacuously true.

$\endgroup$
  • $\begingroup$ Thank you so much for your answer!! I posted the same comment, but then for the converse statement, if I say "If 5 degrees is not equal to 10 degrees divided by 2, then the two angles are congruent." So this would be a TRUE(vacuously TRUE?) statement, thus satisfying the Truth Table? Am I correct in using this logic? $\endgroup$ – chris vin Aug 20 '15 at 13:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.