Modus ponens
In propositional logic, modus ponens (; MP), also known as modus ponendo ponens (Latin for "method of putting by placing") or 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 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 ("if–then") 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 (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 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 (generally , by a constructive method) 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 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. 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 ("p" and "p" → "q"). 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 (the antecedents) 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 [the consequent] ... 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 [the antecedents] are not in error, the final assertion [the consequent] 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: formula_2, where e.g. formula_3 denotes the probability of formula_4 and the conditional probability formula_5 generalizes the logical implication formula_6. Assume that formula_7 is equivalent to formula_4 being TRUE, and that formula_9 is equivalent to formula_4 being FALSE. It is then easy to see that formula_7 when formula_12 and formula_13. 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: formula_14, where formula_15 denotes the subjective opinion about formula_16 as expressed by source formula_17, and the conditional opinion formula_18 generalizes the logical implication formula_6. The deduced marginal opinion about formula_4 is denoted by formula_21. The case where formula_15 is an absolute TRUE opinion about formula_16 is equivalent to source formula_17 saying that formula_16 is TRUE, and the case where formula_15 is an absolute FALSE opinion about formula_16 is equivalent to source formula_17 saying that formula_16 is FALSE . The deduction operator formula_30 of subjective logic produces an absolute TRUE deduced opinion formula_21 when the conditional opinion formula_18 is absolute TRUE and the antecedent opinion formula_15 is absolute TRUE. Hence, subjective logic deduction represents a generalization of both "modus ponens" and the Law of total probability. Alleged cases of failure. Philosophers and linguists have identified a variety of cases where "modus ponens" appears to fail. Vann McGee, for instance, argued that "modus ponens" can fail for conditionals whose consequents are themselves conditionals. The following is an example: Since Shakespeare did write "Hamlet", the first premise is true. The second  premise is also true, since starting with a set of possible authors limited to just Shakespeare and Hobbes and eliminating one of them leaves only the other. However, the conclusion may seem false, since ruling out Shakespeare as the author of "Hamlet" would leave numerous possible candidates, many of them more plausible alternatives than Hobbes. The general form of McGee-type counterexamples to "modus ponens" is simply formula_34, therefore formula_35; it is not essential that formula_16 be a disjunction, as in the example given. That these kinds of cases constitute failures of "modus ponens" remains a controversial 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. Possible fallacies. The fallacy of affirming the consequent is a common misinterpretation of the "modus ponens".