0
$\begingroup$

Let $a$ be neither tautology nor contradiction, $b$ is tautology and $c$ is contradiction . What is the value of $a\to (b\to c)$?

It's a multiple choice question where there're following options:

  1. the expression is a contradiction

  2. the expression is equivalent to the negation of $a$

The second answer is correct but I don't understand why. $b\to c$ is always false. Therefore $a\to (b\to c)$ is always false. So why isn't it a contradiction?

I derived my conclusion by using the truth table in this Wikipedia article.

$\endgroup$
5
$\begingroup$

$(b→c)$ is FALSE because - being $b$ a tautology, i.e. always TRUE, and $c$ a contradiction, i.e. always FALSE, it is :

(TRUE → FALSE), i.e. FALSE.

But $a$ can be either TRUE or FALSE.

Thus $a→(b→c)$ can be either (TRUE → FALSE) or (FALSE → FALSE).

More specifically, we have that :

$a→(b→c)$ is $a \to$ FALSE.

Thus, when $a$ is TRUE, we have that $(a \to$ FALSE) is FALSE, and when $a$ is FALSE, $(a \to$ FALSE) is TRUE.

Thus :

$a→(b→c)$ is equivalent to $\lnot a$.

$\endgroup$
4
$\begingroup$

Your error is here.

$b\to c$ is always false. CORRECT

Therefore $a\to (b\to c)$ is always false. INCORRECT

$\endgroup$
2
$\begingroup$

$b \to c$ has its truth value specified because $b$ and $c$ each have their truth value specified. By definition $\top \to \bot$ is false, so you have $a \to \bot$ which is by definition equivalent to $\neg a$.

$\endgroup$

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.