Is this a valid or invalid inference?

Question

1. $(p \land q ) \iff (r \implies s)$

2. $(s \vee \neg t)$

So, $\neg s \implies ((r \implies \neg p) \vee \neg t)$

My lecturer has written in our notes that this statement is invalid, but I'm not so sure. I've attached my workings and have found there to be a contradiction whilst using the 'no counterexample' method (i.e. assume the premise to be T whilst the conclusion F, if there is a contradiction then the statement is valid) to find the validity of this statement. Thus, I think this inference is valid.

My question is: is the above inference valid or not?

Thank you so much in advanced for your help! :)

Answer 1

We will follow the instructor's notes and assume that the inference is invalid.

This means : assume the premises T and the conclusion F.

We may rewrite the conclusion as :

$s \lor \lnot r \lor \lnot p \lor \lnot t \ $;

in order to have it F we need :

$s=$ F and $r=p=t=$ T.

With this truth-assignment, the second premise : $s \lor \lnot t$ would be F.

Thus, the argument is valid.

---

If instead the sought conclusion is : $¬s \to ((r \to ¬p) \land ¬t)$, with the same approach we have two possibilities in order to "falsify" it; either :

(i) $s=$ F and $t=$ T, or :

(ii) $s=$ F and $r=p=$ T.

In the first case, again, we cannot satisfy the second premise.

In the second case, we have two variables "undefined" : $t$ and $q$.

The two premises are equivalent to :

$($ T $\land q) \leftrightarrow$ F

F $\lor \lnot t$.

Thus, if we set : $q=t=$ F, we can satisfy both premises and we have shown that the argument is invalid.

---

---

Conclusion :

Is this a valid or invalid inference?

It depends on the formula (in this case : the conclusion) ...

Answer 2

If s is not true, then (not t) has to be true, thus the inference is automatically true.

On a side note, I think that you may have made a typo...