# Is the following series of equivalences a proof that the "modus tollens" law is a tautology. ( Reducing all tautologies to only one?)

Generally speaking, I'd like to know whether it is possible to reduce using only syntactic means any tautology to some basic tautology, say, for example, the principle of non contradiction. Can this possibility be proved?

More specifically, I'd like to know whether the following transformation of the "modus tollens" law really counts as a proof that this law is a tautology. I think this transformation proves that it has the same form as the tautology ~ (X&~X), but does it actually prove that the modus tollens law is really equivalent to the principle of non-contradiction, and thus , a tautology ?

[ (A--> B) & ~B ] --> ~A

<-> ~ [ (A --> B) & ~B & ~ ~A ]

<-> ~ [ (A --> B) & ~B & A ]

<-> ~ [ ~ (A & ~B) & (A & ~ B) ]

<-> ~ [ (A & ~B ) & ~ (A & ~B) ]

If I set, X = (A& ~B) then the modus ponens law is equivalent to ~ (X & ~ X).

But the equivalence holds apparently under a condition ( namely if X = (A & ~B) ), not absolutely.

Anything of the form $$\neg (X \land \neg X)$$ is a tautology. That is, $$\neg (X \land \neg X)$$ is always true, no matter whether $$X$$ is an atomic variable, or some complex statement like $$(A \land \neg B)$$. This is because any statement $$X$$ (atomic or complex) is true or false, and since $$\neg X$$ has the opposite truth-value, you get that $$X \land \neg X$$ is always false, and thus $$\neg (X \land \neg X)$$ is always true, making it a tautology.
• @EleonoreSaintJames Well, that would be false: $X \lor \neg X$ is not of the form $\neg (X \land \neg X)$, but it is a tautology. Of course, it is true that any tautology is equivalent to a statement of the form $\neg (X \land \neg X)$. Also beware of this: is anything of the form $X$ not a tautology? No. $\neg (A \land \neg A)$ is of the form $X$ (because you can fill in something for $X$ in order to get $\neg (A \land \neg A)$), so there are statements of the form $X$ that are tautologies. Of course, if you think about it: every statement is of the form $X$ ! Commented Apr 11, 2019 at 17:12