Argument form in the context of "Denying the antecedent"

In propositional logic, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP), also known as modus ponendo ponens (from Latin 'mode that by affirming affirms'), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q. P is true. Therefore, Q must also be true."

Modus ponens is a mixed hypothetical syllogism and is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of modus ponens.

In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination) is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It is the inference that if a statement ${\displaystyle P}$ implies a statement ${\displaystyle Q}$ and a statement ${\displaystyle R}$ also implies ${\displaystyle Q}$, then if either ${\displaystyle P}$ or ${\displaystyle R}$ is true, then ${\displaystyle Q}$ has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true.

An example in English: