Modus ponens

In propositional logic, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP), also known as modus ponendo ponens (from Latin 'method of putting by placing'),^[1] implication elimination, or affirming the antecedent,^[2] is a deductive argument form and rule of inference.^[3] It can be summarized as "P implies Q. P is true. Therefore, Q must also be true."

"Forward reasoning" redirects here. For other uses, see Forward chaining.

Type

$P$ implies $Q$ . $P$ is true. Therefore, $Q$ must also be true.

$P\to Q,\;P\;\vdash \ Q$

Modus ponens is a mixed hypothetical syllogism and is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of modus ponens.

The history of modus ponens goes back to antiquity.^[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.

Status[edit]

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".^[10]

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".^[10] 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.^[11]

Correspondence to other mathematical frameworks[edit]

Algebraic semantics[edit]

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 $\neg {(P\wedge Q)}$ and $\neg {P}\vee \neg {Q}$ , for instance, are equivalent (as is standard), then $\neg {(P\wedge Q)}=\neg {P}\vee \neg {Q}$ . Logical implication becomes a matter of relative position: $P$ logically implies $Q$ just in case $P\leq Q$ , i.e., when either $P=Q$ or else $P$ lies below $Q$ and is connected to it by an upward path.

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

Probability calculus[edit]

If $\Pr(P\rightarrow Q)=x$ and $\Pr(P)=y$ , then $\Pr(Q)$ must lie in the interval $[x+y-1,x]$ .^[b]^[12] For the special case $x=y=1$ , $\Pr(Q)$ must equal $1$ .

Subjective logic[edit]

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

$\omega _{Q\|P}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})\circledcirc \omega _{P}^{A}\,,$

where $\omega _{P}^{A}$ denotes the subjective opinion about $P$ as expressed by source $A$ , and the conditional opinion $\omega _{Q|P}^{A}$ generalizes the logical implication $P\to Q$ . The deduced marginal opinion about $Q$ is denoted by $\omega _{Q\|P}^{A}$ . The case where $\omega _{P}^{A}$ is an absolute TRUE opinion about $P$ is equivalent to source $A$ saying that $P$ is TRUE, and the case where $\omega _{P}^{A}$ is an absolute FALSE opinion about $P$ is equivalent to source $A$ saying that $P$ is FALSE. The deduction operator $\circledcirc$ of subjective logic produces an absolute TRUE deduced opinion $\omega _{Q\|P}^{A}$ when the conditional opinion $\omega _{Q|P}^{A}$ is absolute TRUE and the antecedent opinion $\omega _{P}^{A}$ is absolute TRUE. Hence, subjective logic deduction represents a generalization of both modus ponens and the Law of total probability.^[13]

Possible fallacies[edit]

The fallacy of affirming the consequent is a common misinterpretation of the modus ponens.^[20]

Condensed detachment

– Principle of classical logic

Import-export (logic)

Latin phrases

– Rule of logical inference

Modus tollens

– Arrangement that allows conflicting parties to coexist in peace

Modus vivendi

– System of propositional logic developed by the Stoic philosophers

Stoic logic

– Allegorical dialogue by Lewis Carroll

What the Tortoise Said to Achilles

Herbert B. Enderton, 2001, A Mathematical Introduction to Logic Second Edition, Harcourt Academic Press, Burlington MA, 978-0-12-238452-3.

ISBN

Audun Jøsang, 2016, Subjective Logic; A formalism for Reasoning Under Uncertainty Springer, Cham, 978-3-319-42337-1

ISBN

and Bertrand Russell 1927 Principia Mathematica to *56 (Second Edition) paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN.

Alfred North Whitehead

1946 Introduction to Logic and to the Methodology of the Deductive Sciences 2nd Edition, reprinted by Dover Publications, Mineola NY. ISBN 0-486-28462-X (pbk).

Alfred Tarski

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Modus ponens"

at PhilPapers

Modus ponens

at Wolfram MathWorld

Modus ponens

Type

Type

Field

Statement

Symbolic statement

Status[edit]

Correspondence to other mathematical frameworks[edit]

Algebraic semantics[edit]

Probability calculus[edit]

Subjective logic[edit]

Possible fallacies[edit]

Condensed detachment

Import-export (logic)

Latin phrases

Modus tollens

Modus vivendi

Stoic logic

What the Tortoise Said to Achilles

ISBN

ISBN

Alfred North Whitehead

Alfred Tarski

"Modus ponens"

Modus ponens

Modus ponens