Constructing a counterexample Premises: $p\implies m$, $m\implies t$, $m$.
Conclusion: $m\implies p$
My goal is to provide a counter example for the following problem since it is not true. I am familiar with writing counterexamples but on more concrete proofs. 
 A: There are three variables here; each can take two different values (true or false). That means there are only eight different situations to consider. That's small enough to do the straightforward thing: check 'em all, and see if any are counterexamples.
For example, if $p$, $m$, and $t$ are all "true", then $p \implies m$ is true, $m \implies t$ is true, and $m$ is true; so the premises are true. $m \implies p$ is also true, so this is not a counterexample.
For another example, if $p$ and $t$ are true but $m$ is false, then $p \implies m$ is false, so the premises are not all true; that means that this isn't a counterexample either.
There are six more cases to check - I'll leave those to you.
A: This can be attacked directly: how can $m \implies p$ be false? 
You may find it helpful to covert the implication into a disjunction (that is, express $m \implies p$ using $\lor$ instead of $\implies$) and then negate it. 
A: A way to approach the problem of finding a counterexample is to find the values of the atomic propositions that would make the conclusion false. In this case, that would happen if $m$ were true and $p$ were false.
Knowing that, consider the premises with $m$ replaced by $\top$ and $p$ replaced by $\bot$.  We have:


*

*$p \implies m$ becomes $\bot \implies \top$ is true.

*$m \implies t$ becomes $\top \to t$.

*$m$ becomes $\top$ is true.


We have to make all of the premises true to obtain a counterexample. Premises 1 and 3 are true by the valuations needed to make the conclusion false. If $t$ is true then premise 2 is also true.  
We found a counterexample: $m$ is true; $p$ is false; $t$ is true.
We could use a tree proof generator to find such a counterexample as well:


Tree Proof Generator. https://www.umsu.de/trees/
