Modus ponens is a very common rule of inference, and takes the following form:
The modus ponens rule may be written in sequent notation:
or in rule form:
The argument form has two premises. 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; the validity of modus ponens means that the conclusion must be true if all the premises are true. 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. A propositional argument using modus ponens is said to be deductive.
In metalogics, modus ponens is the cut rule. The cut-elimination theorem says that the cut is valid (an admissible rule) in some logical calculus (sequent calculus).
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Modus ponens is a very common rule of inference, and takes the following form:
The modus ponens rule may be written in sequent notation:
or in rule form:
The argument form has two premises. 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; the validity of modus ponens means that the conclusion must be true if all the premises are true. 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. A propositional argument using modus ponens is said to be deductive.
In metalogics, modus ponens is the cut rule. The cut-elimination theorem says that the cut is valid (an admissible rule) in some logical calculus (sequent calculus).
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
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