Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent.

Constructive dilemma is the disjunctive version of modus ponens.

[4] The first to explicitly describe the argument form modus ponens was Theophrastus.

[5] 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.

From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be the case as well.

The argument is only sound on Tuesdays (when John goes to work), but valid on every day of the week.

In single-conclusion sequent calculi, modus ponens is the Cut rule.

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

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 validity of modus ponens in classical two-valued logic can be clearly demonstrated by use of a truth table.

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".

[6] 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[7] or the law of detachment.

[8] Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones",[9] 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".

[11] In mathematical logic, algebraic semantics treats every sentence as a name for an element in an ordered set.

—that is, to affirm modus ponens as valid—is to say that the highest point which lies below both

[a] In the semantics for basic propositional logic, the algebra is Boolean, with

, the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted.

Modus ponens represents an instance of the binomial deduction operator in subjective logic expressed as:

of subjective logic produces an absolute TRUE deduced opinion

Hence, subjective logic deduction represents a generalization of both modus ponens and the Law of total probability.

[13] Philosophers and linguists have identified a variety of cases where modus ponens appears to fail.

[14] 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 is doubtful, since ruling out Shakespeare as the author of Hamlet would leave numerous possible candidates, many of them more plausible alternatives than Hobbes (if the if-thens in the inference are read as material conditionals, the conclusion comes out true simply by virtue of the false antecedent.

The general form of McGee-type counterexamples to modus ponens is simply

[15][16][17] 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.

"[18] 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.

[19] The fallacy of affirming the consequent is a common misinterpretation of the modus ponens.