Modus ponens

Formal notation

The modus ponens rule may be written in sequent notation as $${\displaystyle P\to Q,P⊢Q}$$

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.

In classical two-valued logic, modus ponens can be interpreted from the material conditional (implication) operator. The notion of implication as a rigorous mathematical construct can be traced back to around the turn of the 19th century with early works of mathematical logic, such as Begriffsschrift and Principia Mathematica. Given two variables p and q that can either be true or false, implication (p → q) is defined as all cases being true except where p is true and q is false. By the assumptions of modus ponens, p → q and p are true, and by the definition of implication, it can be deduced that 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

Algebraic semantics

In mathematical logic, algebraic semantics treats every sentence as a name for an element in an ordered set. Typically, the set can be visualized as a lattice-like structure with a single element (the "always-true") at the top and another single element (the "always-false") at the bottom. Logical equivalence becomes identity, so that when ${\displaystyle \mathrm{\neg }(P\wedge Q)}$ and ${\displaystyle \mathrm{\neg }P\vee \mathrm{\neg }Q}$, for instance, are equivalent (as is standard), then ${\displaystyle \mathrm{\neg }(P\wedge Q)=\mathrm{\neg }P\vee \mathrm{\neg }Q}$. Logical implication becomes a matter of relative position: ${\displaystyle P}$ logically implies ${\displaystyle Q}$ just in case ${\displaystyle P\le Q}$, i.e., when either ${\displaystyle P=Q}$ or else ${\displaystyle P}$ lies below ${\displaystyle Q}$ and is connected to it by an upward path.

In this context, to say that $P$ and ${\displaystyle P\to Q}$ together imply ${\displaystyle Q}$—that is, to affirm modus ponens as valid—is to say that the highest point which lies below both ${\displaystyle P}$ and ${\displaystyle P\to Q}$ lies below ${\displaystyle Q}$, i.e., that ${\displaystyle P\wedge (P\to Q)\le Q}$. In the semantics for basic propositional logic, the algebra is Boolean, with ${\displaystyle \to }$ construed as the material conditional: ${\displaystyle P\to Q=\mathrm{\neg }P\vee Q}$. Confirming that ${\displaystyle P\wedge (P\to Q)\le Q}$ is then straightforward, because ${\displaystyle P\wedge (P\to Q)=P\wedge Q}$ and ${\displaystyle P\wedge Q\le Q}$. With other treatments of ${\displaystyle \to }$, the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted.

Probability calculus

If ${\displaystyle Pr(P\to Q)=x}$ and ${\displaystyle Pr(P)=y}$, then ${\displaystyle Pr(Q)}$ must lie in the interval ${\displaystyle [x+y-1,x]}$. For the special case ${\displaystyle x=y=1}$, ${\displaystyle Pr(Q)}$ must equal ${\displaystyle 1}$.

Subjective logic

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

$${\displaystyle {\omega }_{Q\Vert P}^A=({\omega }_{Q|P}^A,{\omega }_{Q|\mathrm{\neg }P}^A)⊚{\omega }_P^A,}$$

where ${\displaystyle {\omega }_P^A}$ denotes the subjective opinion about ${\displaystyle P}$ as expressed by source ${\displaystyle A}$, and the conditional opinion ${\displaystyle {\omega }_{Q|P}^A}$ generalizes the logical implication ${\displaystyle P\to Q}$. The deduced marginal opinion about ${\displaystyle Q}$ is denoted by ${\displaystyle {\omega }_{Q\Vert P}^A}$. The case where ${\displaystyle {\omega }_P^A}$ is an absolute TRUE opinion about ${\displaystyle P}$ is equivalent to source ${\displaystyle A}$ saying that ${\displaystyle P}$ is TRUE, and the case where ${\displaystyle {\omega }_P^A}$ is an absolute FALSE opinion about ${\displaystyle P}$ is equivalent to source ${\displaystyle A}$ saying that ${\displaystyle P}$ is FALSE. The deduction operator ${\displaystyle ⊚}$ of subjective logic produces an absolute TRUE deduced opinion ${\displaystyle {\omega }_{Q\Vert P}^A}$ when the conditional opinion ${\displaystyle {\omega }_{Q|P}^A}$ is absolute TRUE and the antecedent opinion ${\displaystyle {\omega }_P^A}$ 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:

1. Either Shakespeare or Hobbes wrote Hamlet.
2. If either Shakespeare or Hobbes wrote Hamlet, then if Shakespeare did not do it, Hobbes did.
3. Therefore, if Shakespeare did not write Hamlet, Hobbes did it.

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. This is one of the paradoxes of material implication).

The general form of McGee-type counterexamples to modus ponens is simply ${\displaystyle P,P\to (Q\to R)}$, therefore, ${\displaystyle Q\to R}$; however, it is not essential that ${\displaystyle P}$ 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, and some authors have rejected McGee's argument.

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 it appears that he is doing exactly what he should, unconditionally, be doing. Here again, modus ponens' failure is not a popular diagnosis and is criticized by some authors, but is sometimes argued for.

Possible fallacies

The fallacy of affirming the consequent (inferring P from if P, then Q and Q) is a common misinterpretation of the modus ponens.