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September 19

Is this still fallacious in this circumstance?

Is affirming the consequent (and denying the antecedent by extension) still fallacious if the first premise is an “If and only if X is true, then Y is true” statement instead of just “If x is true, then Y is true”? In an “If and only if” statement, the truth value of the antecedent and consequent are dependent on each other and are equal.

Here’s an example.

P1: If and only if someone was born on or before this date in 2002, then they are allowed to buy alcohol in the US.

P2: Jared (a fictional character I created for this example) was born after today’s date in 2002. (If this was just a regular if-then statement, this premise would begin the process of denying the antecedent.)

C: Therefore, Jared is not allowed to buy alcohol in the US.

Would it become valid given the “if and only if” nature of the major premise? Primal Groudon (talk) 04:24, 19 September 2023 (UTC)[reply]

If the first line were merely a simple "if", it would not be clear whether someone born after today in 2002 could buy alcohol. ←Baseball Bugs ^{What's up, Doc?} carrots→ 06:05, 19 September 2023 (UTC)[reply]

The premise can be reformulated as "Y (is true) if and only if X (is true)". The connective "if and only if" indicates the equivalence of two statements. "Y if and only if X" expresses the same as "X if and only if Y", so either of the two can equally function as the antecedent of a material conditional. Therefore these formal fallacies do not apply here. --Lambiam 07:42, 19 September 2023 (UTC)[reply]

It probably does not need pointing out (but I will anyway) that "If X is true, then Y is true" does not preclude Y being true if X is not true.

Example [with simplifications] off the top of my head: "If UK Resident A is an adult citizen of the UK (X), they have the right to register to vote in UK Elections (Y)" – but if UK Resident A is instead a citizen of The Republic of Ireland [and several other jurisdictions] (so X is not true), they also have the right to register to vote in UK Parliamentary Elections (so Y is true). {The poster formerly known as 87.81.230.195} 5.66.229.169 (talk) 10:32, 19 September 2023 (UTC)[reply]

The fallacy of assuming that "If X is true, Y is true" precludes Y being true when X is false is precisely the fallacy of denying the antecedent. The article, also linked to by the OP, contains a simpler example demonstrating its invalidity. --Lambiam 09:33, 20 September 2023 (UTC)[reply]