Modus ponens
In propositional logic, modus ponens , also known as modus ponendo ponens or implication elimination, is a deductive argument form and rule of inference. It can be summarized as "P implies Q and P is 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. It, along with modus tollens, is one of the standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal.
Explanation
The form of a modus ponens argument resembles a syllogism, with two premises and a conclusion:The first premise is a conditional claim, namely that P implies Q. The second premise is an assertion that P, the 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:
This argument is valid, but this has no bearing on whether any of the statements in the argument are actually 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 is unsound. The argument is only sound on Tuesdays, but valid on every day of the week. A propositional argument using modus ponens is said to be deductive.
In single-conclusion sequent calculi, modus ponens is the Cut rule. The cut-elimination theorem for a calculus says that every proof involving Cut can be transformed into a proof without Cut, and hence that Cut is 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.
Formal notation
The modus ponens rule may be written in sequent notation aswhere P, Q and P → Q are statements 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.
Justification via truth table
The validity of modus ponens in classical two-valued logic can be clearly demonstrated by use of a truth table.In instances of modus ponens we assume as premises that p → q is true and p is true. Only one line of the truth table—the first—satisfies these two conditions. On this line, q is also true. Therefore, whenever p → q is true and p is true, q must also be true.
Status
While modus ponens is one of the most commonly used argument forms in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution". Modus ponens allows one to eliminate a conditional statement from a logical proof or argument and thereby not carry these antecedents forward in an ever-lengthening string of symbols; for this reason modus ponens is sometimes called the rule of detachment or the law of detachment. Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones", and Russell observes that "the process of the inference cannot be reduced to symbols. Its sole record is the occurrence of ⊦q ... an inference is the dropping of a true premise; it is the dissolution of an implication".A justification for the "trust in inference is the belief that if the two former assertions are not in error, the final assertion is not in error". In other words: if one statement or proposition implies a second one, and the first statement or proposition is true, then the second one is also true. If P implies Q and P is true, then Q is true.
Correspondence to other mathematical frameworks
Probability calculus
Modus ponens represents an instance of the Law of total probability which for a binary variable is expressed as:where e.g. denotes the probability of and the conditional probability generalizes the logical implication. Assume that is equivalent to being TRUE, and that is equivalent to being FALSE. It is then easy to see that when and. Hence, the law of total probability represents a generalization of modus ponens.
Subjective logic
Modus ponens represents an instance of the binomial deduction operator in subjective logic expressed as:where denotes the subjective opinion about as expressed by source, and the conditional opinion generalizes the logical implication. The deduced marginal opinion about is denoted by. The case where is an absolute TRUE opinion about is equivalent to source saying that is TRUE, and the case where is an absolute FALSE opinion about is equivalent to source saying that is FALSE. The deduction operator of subjective logic produces an absolute TRUE deduced opinion when the conditional opinion is absolute TRUE and the antecedent opinion is absolute TRUE. Hence, subjective logic deduction represents a generalization of both modus ponens and the Law of total probability.
Alleged cases of failure
The philosopher and logician Vann McGee has argued that modus ponens can fail to be valid when the consequent is itself a conditional sentence. Here is an example:The first premise seems reasonable enough, because Shakespeare is generally credited with writing Hamlet. The second premise seems reasonable, as well, because with the set of Hamlet''s possible authors limited to just Shakespeare and Hobbes, eliminating one leaves only the other. But the conclusion, considered by itself and with the possible authors not limited to just Shakespeare and Hobbes, is dubious, because if Shakespeare is ruled out as Hamlet's author, there are many more plausible alternatives than Hobbes.
The general form of McGee-type counterexamples to modus ponens is simply, therefore ; it is not essential that be a disjunction, as in the example given. That these kinds of cases constitute failures of modus ponens remains a minority view among logicians, but opinions vary on how the cases should be disposed of.
In deontic logic, some examples of conditional obligation also raise the possibility of modus ponens failure. These are cases where the conditional premise describes an obligation predicated on an immoral or imprudent action, e.g., “If Doe murders his mother, he ought to do so gently,” for which the dubious unconditional conclusion would be "Doe ought to gently murder his mother." It would appear to follow that if Doe is in fact gently murdering his mother, then by modus ponens he is doing exactly what he should, unconditionally, be doing. Here again, modus ponens failure is not a popular diagnosis but is sometimes argued for.