Modus ponens

In propositional logic, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP; also modus ponendo ponens (Latin for 'mode that affirms by affirming') or implication elimination) is a rule of inference. It can be summarized as 'P implies Q and P is asserted to be true, therefore Q must be true.' Modus ponens is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms such as affirming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is the disjunctive version of modus ponens. Hypothetical syllogism is closely related to modus ponens and sometimes thought of as 'double modus ponens.' The history of modus ponens goes back to antiquity. The first to explicitly describe the argument form modus ponens was Theophrastus. The modus ponens rule may be written in sequent notation as where P, Q and P → Q are statements (or propositions) in a formal language and ⊢ is a metalogical symbol meaning that Q is a syntactic consequence of P and P → Q in some logical system. The argument form has two premises (hypothesis). The first premise is the 'if–then' or conditional claim, namely that P implies Q. The second premise is that P, the antecedent of the conditional claim, is true. From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be true as well. In artificial intelligence, modus ponens is often called forward chaining. An example of an argument that fits the form modus ponens: This argument is valid, but this has no bearing on whether any of the statements in the argument are true; for modus ponens to be a sound argument, the premises must be true for any true instances of the conclusion. An argument can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound. For example, John might be going to work on Wednesday. In this case, the reasoning for John's going to work (because it is Wednesday) is unsound. The argument is not only sound on Tuesdays (when John goes to work), but valid on every day of the week. A propositional argument using modus ponens is said to be deductive.