If P and Q can never be true, are they still "necessary" and "sufficient" for each other?

Question

If we are given that propositions P and Q can never be true, is it still accurate to say that P and Q are necessary and sufficient for each other, and why?

I am conflicted here, as the statement P iff Q is true here and I have learned that this also means P and Q are necessary and sufficient for each other (for context, I have learnt that P is necessary for Q if P must be true to conclude Q is true, and that P is sufficient for Q if Q must be true to conclude that P is true); however, it seems counterintuitive to stipulate conditions for propositions being true (by deeming other propositions as necessary or sufficient for them) if these are never relevant to the situation, as these propositions are always false anyway.

I would appreciate an answer grounded in an explanation of necessity and sufficiency.

Answer 1

In mathematics, "P is necessary for Q" is defined as meaning "it is impossible for P to be false and Q to be true." If both P and Q are totally impossible, then it's certainly impossible for P to be false and Q to be true, so we say that P is necessary for Q.

Likewise, in mathematics, "P is sufficient for Q" is defined as meaning "it is impossible for P to be true and Q to be false." If both P and Q are totally impossible, then it's certainly impossible for P to be true and Q to be false, so we say that P is sufficient for Q.

These conclusions may be at odds with the everyday meanings of the words "necessary" and "sufficient." However, when we're writing about math, we use the mathematical definitions of these words, not the everyday meanings.

Answer 2

Clearly, $$(¬P∧¬Q)\implies (\lnot P↔\lnot Q);$$ by contrapositive, $$(¬P↔¬Q)\implies(P↔Q).$$ So, by transitivity, $$(¬P∧¬Q)\implies(P↔Q);$$ that is, $$(¬P∧¬Q)→(P↔Q)$$ is a tautology. After all, P ↔ Q ($P$ and $Q$ necessary and sufficient for each other) just means that $P$ and $Q$ have the same truth value (in this case, 'false').

it seems counterintuitive to stipulate impossible—thus, irrelevant—propositions as conditions for propositions being true

But a contradiction $\bot$ within a conditional statement is not an irrelevant condition, since it is meaningful to truth evaluation:

• $⊥$ is a sufficient condition for $P\quad\equiv\quad\top$
• $⊥$ is a necessary condition for $P\quad\equiv\quad\lnot P;$

in order to truthfully assert that 'pigs can fly' is a necessary condition for $P,$ $P$ must be something false; on the other hand, whatever $P$ stands for, the assertion that 'pigs can fly' is a sufficient condition for $P$ is certainly a redundancy. Like vacuous truth, these assertions are perhaps surprising but not really counterintuitive, so I may be misunderstanding your point.

Answer 3

If $P$ and $Q$ can never be true, are they still "necessary" and "sufficient" for each other?

The "can never be true" is distracting. In propositional logic, we don't make predictions about the future. We talk about what is true (present tense). Change it to:

If $P$ and $Q$ are both false, are they "necessary" and "sufficient" for each other?

Then the answer would be, yes.

Truth Table for $P\iff Q$

See line 4 where $P$ and $Q$ are both false.