# Logic/math phrasing

Given three statements A B and C, is there any difference at all between an author’s claims:

A and B are necessary and sufficient conditions for C.

and

A and B are individually necessary for C and jointly sufficient.

The only thing I can think is if A and C are equivalent statements and B is necessary for both, the first claim is true, but maybe the second is not because B is not jointly needed given A, but that seems to be stretching the point.

I would read the first claim as: A is a necessary and sufficient condition for C, and also B is a necessary and sufficient condition for C. So if A is true, then C and B must be true as well.

In the second claim, it’s possible that A is true but C and B are false.

The first is a simple AND statement isn’t it? … both A and B must be true for C to be true, if either A or B are false, or both are false, then C is false … A and B are necessary …

The second just seems to be adding words … it’s still A and B are necessary with the words “individually” and “jointly” being redundant …

I would say both statements are logically equivalent …

Symbolic notation is more transparent here; your sentence is equivalent to

A ∧ B ↔ C

That can’t be true unless A and B are the same. If A is a sufficient condition for C, then B can’t be a necessary condition for C.

Well, logically equivalent to each other.

I think there’s more than one possible interpretation (as given by the first two replies), so the statement ought to be reworded for clarity.

I would interpret the two phrasings the same, because while the first statement can be interpreted as meaning that A and B are equivalent, that’d be a really roundabout way of saying that, and so probably not what was meant.

If that’s what is meant, then I would have written the first claim as A and B is a necessary and sufficient condition for C, maybe including parens or quotes like ( A and B ) for clarity. Using “are” as in A and B are … conditions makes me read the and as a normal English and rather than the logical ‎∧, so I parse it as listing two statements that each individually have the given properties. I do see how people can read it the other way though.

As Dr. Strangelove stated earlier, “That can’t be true unless A and B are the same. If A is a sufficient condition for C, then B can’t be a necessary condition for C.”

If the original statement is really so unclear as to leave most readers scratching their heads or misinterpreting it, then maybe it is really a case of bad writing on the author’s part. Hopefully what actually holds true can be figured out from the context.

PS

it could conceivably be that “A and B are necessary and sufficient conditions for C” is a poor way of writing: “the following are equivalent: (1) A; (2) B; (3) C.” I would go back to the original statements and work out for myself what actually holds.

Yes, I understand that. I was talking about the phrasing of the claim, and not evaluating whether it’s a useful or correct statement.

In any case, it’s perfectly reasonable to have three statements A, B, and C, which are all logically equivalent. A and B are then each individually necessary and sufficient conditions for C.

Oh it was clear from the context that what was being stated was

(A ∧ B) <=> C

The second phrasing seem to me to say precisely that in the form

(A ∧ B) => C, C => B, C => A

Someone was arguing that the first phasing meant something different and I couldn’t follow their verbal reasoning. Thanks all.

It’s conceivable, but seems unlikely (I agree with Chronos here). It becomes even less likely as soon as you add a bit of concreteness and wiggle room for implicit conditions.

For example, “chocolate sauce and whipped cream are necessary and sufficient toppings for an ice cream sundae”. It’s possible that I meant:
(chocolate sauce is a necessary and sufficient topping for an ice cream sundae) AND
(whipped cream is a necessary and sufficient topping for an ice cream sundae)

Or expanded out even more:
(chocolate sauce is a necessary topping for an ice cream sundae) AND
(chocolate sauce is a sufficient topping for an ice cream sundae) AND
(whipped cream is a necessary topping for an ice cream sundae) AND
(whipped cream is a sufficient topping for an ice cream sundae)

But that’s just obviously false, because some conditions are mutually exclusive. The much more reasonable interpretation is:
(chocolate sauce AND whipped cream) are necessary and sufficient toppings for an ice cream sundae

That is at least conceivably true. Note that the full conditions for C are inferred and not specified exactly, so we never make a direct logical equivalence between A<=>C or B<=>C. And it’s obvious that A and B are not logically equivalent.

I understood the first to mean that A and B are jointly necessary and sufficient for C and yes the two statements are equivalent. Perhaps the sentence could be interpreted as ambiguous, but the meaning seemed clear enough to me. If you interpret the sentence to mean A is equivalent to C and B is equivalent to C, then all three are equivalent and it is very unlikely you would put it that way, which is one reason that my original interpretation is the default.