Determine whether the argument is valid or invalid

Question

. Determine whether the following argument is valid: $$\displaylines{ 1:p\cr 2:p ∨ q\cr 3:q → (r → s)\cr 4:t → r\cr ∴ ¬s → ¬t.}$$

Suppose $$\displaylines{¬s → ¬t.}$$ is False, we have s=F; t=T

To make "4" True, r=T

To make "3" True, q=F

To make "1,2" True, p=T

Thus p=T, q=F, r=T, s=F, t=T. <<How is this a counter example?>>

The argument is not valid. <<How did we get this conclusion?>>

Answer 1

In order to show that $\Gamma ∴ \varphi$ is not a valid argument, we have to show that there is a valuation (or truth-values assignment) which satisfy all formula in $\Gamma$ and do not satisfy $\varphi$.

There are formal "procedure" to do this.

In this simple case, we can proceed by trial-and-error.

In order to falsify : $¬s→¬t$, we need a valuation $v$ such that $v(¬s→¬t) = F$, i.e. :

$v(s)=F$ and $v(t)=T$.

We want that all the premises are true; thus, from 4), if we assume $v(t)=T$, we must have also $v(r)=T$.

From 1) we have : $v(p)=T$.

Now we are left with 2) and 3) which, under our previous assumptions, become :

2) $T \lor q$

and :

3) $q → (T → F)$ i.e. $q → F$.

We may satisfy both with $v(q)=F$, because 2) : $v(p \lor q) = T \lor F = T$, by truth-table for $\lor$, and $v(q → (r → s) = F → (T → F) = F → F = T$, by truth-table for $→$.

Conclusion

The valuation $v$ such that :

$v(p)=v(r)=v(t)=T$ and $v(s)=v(q)=F$

is a model of the set of premises (i.e. it satisy all the formulae of the set of premises) but does not satisfy the conclusion.

Thus, the conclusion is not a logical consequence of the premises.

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Note

We can apply the same approach to your previous post.

In that case, we can start trying to falsify the conclusion : $t→w$, i.e. search for a valuation $v$ such that $v(t→w) = F$, and verify that, "propagating" the valuation on the premises, we will end with a contradiction.

Proceeding in this way, we arrive to :

$v(t)=v(u)=v(s)=T$ and $v(w)=v(p)=F$.

Now, in order to satisfy the first premise we must have :

$v(¬p→(r∧¬s))=T$,

which - due to the previous steps - amounts to :

$v(T→(r∧F))=T$

and this is not satisfiable, because we have no way to assign a truth-value to $r$ such that : $v(r \land F)=T$.

Conclusion

Being impossible to find a valuation such that all the premises are satisfied and the conclusion is not, we conclude that this one is a logical consequence of the set of premises.