"If Anup is a member of Animal Rights Society, then he opposes horse racing.…

Concept

A conditional “If P, then Q” asserts only that P is sufficient for Q; it never asserts that Q is sufficient for P. When a second premise affirms or denies one of the two parts of that conditional, exactly two inference patterns are valid: affirming the antecedent (P, so Q) and denying the consequent (not-Q, so not-P). The two remaining moves of that kind — affirming the consequent and denying the antecedent — are formal fallacies: a defect in the shape of the argument rather than in the amount or the source of its evidence.

Application

Let P = “Anup is a member of the Animal Rights Society” and Q = “Anup opposes horse racing”. The first premise states “If P, then Q”. The second premise asserts Q, which is the consequent. The conclusion asserts P, which is the antecedent. So the argument affirms the consequent and infers the antecedent from it.

That step is invalid because the conditional leaves other routes to Q open: Anup may oppose horse racing on animal-welfare, religious, or anti-gambling grounds while belonging to no such society. Both premises can therefore be true at the same time as the conclusion is false, which is exactly what an invalid form means. The fallacy committed is affirming the consequent.

Cross-check

Read against this table, the second premise of the argument asserts Q rather than not-P, so the argument has the form that infers P from Q, not the form that infers not-Q from not-P. Hasty generalisation and inappropriate authority are informal fallacies: they concern how much evidence a conclusion rests on and how qualified the cited source is, so neither names a defect in the shape of a conditional argument.