# How comes that A if and only if B does not exclude A to have other necessary and sufficient conditions?

The fact that B is a necessary condition for A to be true, does not exclude the possibility that another proposition, say C, to be also a necessary condition for A to B true.

In other words, it is not difficult to admit that (A-->B) does not imply that (A-->C) is false.

However, saying that B is not only a necessary condition for A to B true, but also a sufficient condition seems to suggest that any other necessary condition is excluded.

Nevertheless, contrary to what intuition tells me, a truth table test shows that : (A iff B) --> ~ (A iff C) is not valid. Which shows that the fact A has B as necessary and sufficient condition does not logically exclude A to have other necessary and sufficient conditions.

The following dialogue illustrates that this is at leat counterintuitive.

A- I'll give you the keys of this flat iff you officially accept to pay 600 dollars a month.

B - OK, I accept, the lease is signed. Let me have the keys.

A - Wait a minute, there is another necessary condition.

B - But you just told me that if I didn't accept to pay, I would not get the keys, but that, as soon as I would accept, I would get them?

A - That's actually what I said. But now I add that you will get the keys iff you accept to pay an advance of 3 monts rent.

B - How long is this conjunction of biconditionals?

How to explain this apparent disagreement between the truth functional analysis and the intuitive understanding of " necessary and sufficient condition"?

• This is not logic... In term of propositional calculus, at most the dialogue shows that the sentence "I'll give you the keys of this flat iff you officially accept to pay 600 dollars a month" is not TRUE. – Mauro ALLEGRANZA Nov 28 '19 at 12:32
• In modelling "real life" situations with propositional calculus, you have to take into account that the calculus is a very rough "toy model" of natural language, with a simplified syntax and a very elementary semantics : only two "objects" are referred to by formulas of the languge : TRUE and FALSE. Thus, you cannot try to distinguish from the limited point of view of the calculus between statement having the same truth value but different "meaning" : there is no meanong outside of truth value. – Mauro ALLEGRANZA Nov 28 '19 at 12:34
• In the situation you asked about, where B is a necessary and sufficient condition for A but C is also a necessary condition for A, it will be the case that B implies C. (Check that with a truth table.) So once the necessary and sufficient condition B is fulfilled, the "other" necessary condition will automatically have been fulfilled too. C cannot be a genuinely "additional" condition. (Incidentally, everything I wrote here is also true if B is merely a sufficient condition for A, perhaps not necessary.) – Andreas Blass Nov 28 '19 at 16:21

You have that if $$A \leftrightarrow B$$, and $$A \leftrightarrow C$$, then $$B \leftrightarrow C$$. So, if $$B$$ is necessary and sufficient for $$A$$, and if $$C$$ is necessary and sufficient for $$A$$ as well, then $$B$$ and $$C$$ will be necessary and sufficient for each other as well.
So, the fact that $$B$$ and $$C$$ are 'different' is only superficial. Yes, you can always rephrase a set of necessary and sufficient conditions in such a way that it can be said to be a different set of necessary and sufficient conditions, but the 'difference' would be merely syntactical. So yes, $$A \leftrightarrow B$$ does not imply $$A \leftrightarrow C$$, but that is because by using different letters $$B$$ and $$C$$ for the two sets of necessary and sufficient conditions you are simply hiding their actual equivalence behind syntactical symbols. And indeed, if you throw their actual equivalence back in, i.e. If you add $$B \leftrightarrow C$$, then from $$A \leftrightarrow B$$ you now can infer that $$A \leftrightarrow C$$
Consider your dialogue. First, as Mauro points out, if paying an advance of 3 months rent is a necessary condition, then if that is not part of the initial signing of the lease and agreeing to pay $$600$$ dolars per month, then person $$A$$ was simply lying when saying that the intial agreement was a necessary and sufficient condition ... the initial agreement would merely have been a necessary condition, to which one could indeed add any number of others. So, for your story to work, it must have been the case that paying 3 months of rent in advance was simply part and parcel of that initial agreement: presumably it is spelled out in the agreement that the renter has to pay 3 months in advance, and that it is in fact that very part of the initial agreement that, to person A, is necessary and sufficient for B to get the keys. And so, we are not talking about completely sets of necessary and sufficient conditions after all.