Logic: How to prove this argument is not valid Good day to all,
I need calcification on how to show this argument is not valid by finding a counterexample, but without using a truth table. Since there are 5 propositions I would need a 32 row truth table. It would be too time consuming to construct.
                    p
                    p v q 
                    p --> (r --> s)
                    t --> r

                   ∴ ~s --> ~t 

Keeping it short I have this in the end.

p is True
q is False
r is True
s if False
t is True

If so, how is that a counterexample? Shouldn't the argument be valid?
Thanks
This is how I worked it out:
My understanding is that for the first four premises we can`t deduce it inference rule as their value could be either true or false which the result will still give me true.
Hence we focus on the conclusion that ~t is "False" which makes ~s "False"
so that the conclusion will be true. Such that we also narrow the input of t--> r . to be True and True. Since negation of t is False.  So on and so forth until we reach the first premise.

The argument is valid.
For the four premises to be all true, we must evaluate (1) $p$ as true, (2) $q$ as either true or false (we cannot infer which), (3) $r$ as false or $s$ as true (since $r\to s$ is an inference from evaluating $p$ being true), and (4) $t$ as false or $r$ as true.
In short, we must resolve to have $s$ as false or $t$ as false.   Therefore $\neg s\to \neg t$ will be evaluated as true when the premises all are so.
