Editing Modus ponens (section)

== Explanation ==

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The form of a ''modus ponens'' argument is a mixed [[hypothetical syllogism]], with two premises and a conclusion:

# If ''P'', then ''Q''.
# ''P''.
# Therefore, ''Q''.

The first premise is a [[Material conditional|conditional]] ("if–then") claim, namely that ''P'' implies ''Q''. The second premise is an assertion that ''P'', the [[Antecedent (logic)|antecedent]] of the conditional claim, is the case. From these two premises it can be logically concluded that ''Q'', the [[consequent]] of the conditional claim, must be the case as well.

An example of an argument that fits the form ''modus ponens'':

# If today is Tuesday, then John will go to work.
# Today is Tuesday.
# Therefore, John will go to work.

This argument is [[Validity (logic)|valid]], but this has no bearing on whether any of the statements in the argument are actually [[Truth|true]]; for ''modus ponens'' to be a [[Soundness|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 only sound on Tuesdays (when John goes to work), but valid on every day of the week. A [[propositional calculus|propositional]] argument using ''modus ponens'' is said to be [[Deductive reasoning|deductive]].

In single-conclusion [[sequent calculus|sequent calculi]], ''modus ponens'' is the Cut rule. The [[cut-elimination theorem]] for a calculus says that every proof involving Cut can be transformed (generally, by a constructive method) into a proof without Cut, and hence that Cut is [[admissible rule|admissible]].

The [[Curry–Howard correspondence]] between proofs and programs relates ''modus ponens'' to [[function application]]:  if ''f'' is a function of type ''P'' → ''Q'' and ''x'' is of type ''P'', then ''f x'' is of type ''Q''.

In [[artificial intelligence]], ''modus ponens'' is often called [[forward chaining]].