# Is "sometimes necessary" valid in necessity and sufficiency

I recently learned about this concept and in discussing with a peer, I was wondering, is it ever right to say "x is sometimes necessary but not sufficient for y to be the case". In my mind I feel like `sometimes necessary` is equal to `not necessary`, but wanted to confirm is above is reasonable.

• To say x is sometimes necessary means there are contexts where x is necessary is true while at the same time outside that particular context x is not necessary. You seem to try & oversimplify this so you can memorize it easier. There are contingent truths my friend. Contingent truths are temporary truths that alternate from true to false depending on the details. The weather is a contingent truth. On the other hand somethings are objective truths. Objective truth is constant: either always true or always false; this does not alternate true to false. You can have just sufficiency alone. Commented Jun 27, 2021 at 21:11
• I see, I don't know what I don't know, but this has pointed me to at-least a different frame of mind. I'm guessing the reason this was downvoted is because it is too vague? Commented Jun 28, 2021 at 21:15

If something is only 'sometimes neccessary' then from an atemporal perspective, it is not neccesary at all.

You might want to look at Arthur Prior's temporal logic and how it interacts with a modal logic of neccessity/possibility.

Another angle might be through the temporal logic of Avicenna. According to the SEP, one of his important insights was to wholly temporalise logic:

Avicenna introduces two radical innovations in the Aristotelian analysis of propositions:

1. Temporal & Alethic modalities: Every categorical proposition is modalised, either implicitly or explicitly. The modality may be temporal (e.g. sometimes, always) or alethic (e.g. necessarily or possibly), or a combination of both ...

You have a situation where fulfilling at least one of the conditions A or B is necessary for X to be true. But A isn’t necessary for X, nor is B necessary.

However, if A is not the case then B is necessary, and if B is not the case then A is necessary. So each of A and B is sometimes necessary.

(1) Obviously, if x is necessary for y, then x is sometimes necessary for y.

(2) Necessary conditions are not in general sufficient conditions (e.g. "there exists an animal" is necessary, but not sufficient, for "there exists a dog.")

(3) Therefore, it can be right to say that "x is sometimes necessary but not sufficient for y to be the case."

(In other words, your question boils down to asking, "If x is necessary for y, is x sufficient for y?" The answer to this question is, trivially, "No.")

Formal logic and everyday conversation are different languages and words don't mean exactly the same, which can be confusing.

It seems you are using "necessary" in the colloquial meaning of "something you need to do in order to achieve a goal", as in "sometimes it is necessar to make concessions on salary to get a job". One would expect negotiations to be in favor of the employer most of the time, but it is true, after all, that some jobs can be landed with a satisfying salary.

But formally, "X is necessary for Y" means "if Y is true, X is always true". The notation "Y -> X" is defined by the following truth table:

Y=true , X=true -> true
Y=true , X=false -> false
Y=false , X=true -> true
Y=false , X=false -> true

Which can be read as "if Y is true X is true, if Y is false X can be either true or false. What can't happen is that Y is true and not X".

It is the mirror relation of "sufficient", here we can see Y is sufficient for X because if Y is true X is too, and X can be either true or false if Y is false.

Note that of X were to both "necessary and sufficient" for Y, it would simply mean that X and Y are the same proposition, as the only lines yielding "true" in the tables for X->Y and Y->X are "X=true, Y=true" and "X=false , Y=false".

"X is sometimes necessary for Y" would mean that there are cases where Y is true but not X, which we have seen is prohibited by the truth table defining the necessity relation of X to Y.

In other words, in formal logic "sometimes necessary" means "not necessary at all"