# elementary set theory – Are propositions uniquely determined by their non-equivalent implications?

I was reading up on Pragmatism and the pragmatic maxim, which states: “Consider the practical effects of the objects of your conception. Then, your conception of those effects is the whole of your conception of the object”.

I thought that it might be interesting to generalize this to mathematics, with some alterations.

So, here is my question:

“Let $${PROP}$$ be the set of all propositions. Let B be the set of all implications of a proposition $$a$$. Thus, there exists a function such that $$f(a)=B-a.$$ Is this function injective?”