It seems like there's been quite a bit of back and forth regarding whether denying the antecedent is always a fallacy. The confusion primarily arises from the distinction between simple conditional statements (P → Q) and biconditional statements (P ↔ Q).

To clarify, the fallacy of denying the antecedent applies strictly to conditional statements (P → Q). That is, from "If P then Q" and "¬P," one cannot validly conclude "¬Q." A classic counterexample:

• P → Q: "If it is raining, then the ground is wet."
• ¬P: "It is not raining."
• ¬Q?: "The ground is not wet." (Invalid inference—something else, like a sprinkler, could have made it wet.)

However, in the case of a biconditional (P ↔ Q), the reasoning changes because P and Q must always share the same truth value. If P is true, Q must be true, and if P is false, Q must also be false. This is captured formally by the tautology:

(A⊃B)⊃(¬A⊃¬B)

This expresses that if A implies B, then ¬A implies ¬B, which holds true only when A ↔ B (i.e., a biconditional). This is why, in a biconditional statement, denying one side allows us to deny the other.

For those who want to see this logically broken down, here's a truth table visualization:

https://truthtablegenerator.org/#(A⊃B)⊃(¬A⊃¬B)

It’s important to be precise when discussing logical fallacies, as they depend on the exact logical structure in question. Hope this helps clarify!