You could turn your question into a binary decision diagram (BDD). If you assure
that these BDDs are short and unique for logical equivalence classes, than you
can read-off from the BDDs the following:
1) If the BDD has the form "true", then it is a tautology.
2) If the BDD has the form "false", then its negation is a tautology.
3) In all other cases the BDD nor its negation is a tautology.
In case 2) and 3) there is a model of the BDD that make it false. Let's give it a try,
and let's convert your problem into a BDD via the program in ( * ) and ( ** ). We get the following result:
?- convert((n=>b v s)&(s=>w v a)&(m=>n & w)=>(m=>b v a),X).
X = (a->true;b->true;m->(n->(s->(w->false;true);true);true);true)
So it indeed differs from "true" and should be thus falsifiable. By extracting a CNF from the BDD we can also read off a counter model. Let's also do it for your problem:
L = [not(w),not(s),not(n),not(m),b,a]
So the only counter model is w=1, s=1, n=1, m=1, b=0, a=0.
Proof: If the BDD is not "true", then the CNF will at least contain one row. From this row we can directly construct a model that falsifies the row, for unmentioned propositional variables add either false or true, for mentioned propositional variables add a value that falsifies the literal in the row. Since by construction no
propositional variable occurs twice in a row, the row can be falsified. Since
the row is falsified the whole CNF is falsified.
( * )
Invert, Inter and Union on BDDs with Lexical Variable Order:
( ** )
Convert, DNF and CNF for BDDs: