Modus Ponens
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In
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
for "method of putting by placing") or implication elimination or affirming the antecedent, is a
deductive Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
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 In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens' ...
''. Both have apparently similar but invalid forms such as
affirming the consequent Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dar ...
,
denying the antecedent Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the inverse from the original statement. It is committed by reasoning in the form: :If ''P'', then ''Q''. :Therefore, if not ...
, and
evidence of absence Evidence of absence is evidence of any kind that suggests something is missing or that it does not exist. What counts as evidence of absence has been a subject of debate between scientists and philosophers. It is often distinguished from absence ...
.
Constructive dilemma Constructive dilemmaCopi and Cohen is a valid rule of inference of propositional logic. It is the inference that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''P'' or ''R'' is true, then either ''Q or S'' has to be true. In sum, ...
is the
disjunctive Disjunctive can refer to: * Disjunctive population, in population ecology, a group of plants or animals disconnected from the rest of its range * Disjunctive pronoun * Disjunctive set * Disjunctive sequence * Logical disjunction In logic, ...
version of ''modus ponens''.
Hypothetical syllogism In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises. An example in English: :If I do not wake up, then I cannot go to work. :If I cannot go to work, then ...
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 Theophrastus (; grc-gre, Θεόφραστος ; c. 371c. 287 BC), a Greek philosopher and the successor to Aristotle in the Peripatetic school. He was a native of Eresos in Lesbos.Gavin Hardy and Laurence Totelin, ''Ancient Botany'', Routledge ...
. It, along with ''
modus tollens In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''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: # If ''P'', then ''Q''. # ''P''. # Therefore, ''Q''. 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 A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
of the conditional claim, must be the case as well. An example of an argument that fits the form ''modus ponens'': # If today is Tuesday, then John will go to work. # Today is Tuesday. # Therefore, John will go to work. This argument is valid, but this has no bearing on whether any of the statements in the argument are actually
true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * ...
; for ''modus ponens'' to be a
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
argument, the premises must be true for any true instances of the conclusion. An
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
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 Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
. In single-conclusion sequent calculi, ''modus ponens'' is the Cut rule. The
cut-elimination theorem The cut-elimination theorem (or Gentzen's ''Hauptsatz'') is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" for ...
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 In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relati ...
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 Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
, ''modus ponens'' is often called
forward chaining Forward chaining (or forward reasoning) is one of the two main methods of reasoning when using an inference engine and can be described logically as repeated application of ''modus ponens''. Forward chaining is a popular implementation strategy ...
.


Formal notation

The ''modus ponens'' rule may be written in
sequent In mathematical logic, a sequent is a very general kind of conditional assertion. : A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n. A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of asse ...
notation as :P \to Q,\; P\;\; \vdash\;\; Q where ''P'', ''Q'' and ''P'' → ''Q'' are statements (or propositions) in a formal language and
In mathematical logic and computer science the symbol \vdash has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" o ...
is a
metalogic Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
al symbol meaning that ''Q'' is a
syntactic consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (lo ...
of ''P'' and ''P'' → ''Q'' in some
logical system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
.


Justification via truth table

The validity of ''modus ponens'' in classical two-valued logic can be clearly demonstrated by use of a
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
. 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 he consequent... an inference is the dropping of a true premise; it is the dissolution of an implication".Whitehead and Russell 1927:9 A justification for the "trust in inference is the belief that if the two former assertions he antecedentsare not in error, the final assertion he consequentis not in error". In other words: if one statement or
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
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 \neg and \neg \vee \neg, for instance, are equivalent (as is standard), then \neg = \neg \vee \neg. 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 P and P \rightarrow Q together imply Q—that is, to affirm ''modus ponens'' as valid—is to say that P \wedge (P \rightarrow Q) \leq Q. In the semantics for basic propositional logic, the algebra is Boolean, with \rightarrow construed as the
material conditional The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q is ...
: P \rightarrow Q = \neg \vee Q. Confirming that P \wedge (P \rightarrow Q) \leq Q is then straightforward, because P \wedge (P \rightarrow Q) = P \wedge 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

''Modus ponens'' represents an instance of the
Law of total probability In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct eve ...
which for a binary variable is expressed as: \Pr(Q)=\Pr(Q\mid P)\Pr(P)+\Pr(Q\mid \lnot P)\Pr(\lnot P)\,, where e.g. \Pr(Q) denotes the probability of Q and the
conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...
\Pr(Q\mid P) generalizes the logical implication P \to Q. Assume that \Pr(Q) = 1 is equivalent to Q being TRUE, and that \Pr(Q) = 0 is equivalent to Q being FALSE. It is then easy to see that \Pr(Q) = 1 when \Pr(Q\mid P) = 1 and \Pr(P) = 1. Hence, the
law of total probability In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct eve ...
represents a generalization of ''modus ponens''.


Subjective logic

''Modus ponens'' represents an instance of the binomial deduction operator in subjective logic expressed as: \omega^_= (\omega^_,\omega^_)\circledcirc \omega^_\,, where \omega^_ denotes the subjective opinion about P as expressed by source A, and the conditional opinion \omega^_ generalizes the logical implication P \to Q. The deduced marginal opinion about Q is denoted by \omega^_. The case where \omega^_ is an absolute TRUE opinion about P is equivalent to source A saying that P is TRUE, and the case where \omega^_ 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^_ when the conditional opinion \omega^_ is absolute TRUE and the antecedent opinion \omega^_ is absolute TRUE. Hence, subjective logic deduction represents a generalization of both ''modus ponens'' and the
Law of total probability In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct eve ...
.


Alleged cases of failure

Philosophers and linguists have identified a variety of cases where ''modus ponens'' appears to fail.
Vann McGee Vann may refer to: * '' Salvadora oleoides'' is a small bushy evergreen tree found in India, Pakistan, and southern Iran * Vann Peak, Marie Byrd Land, Antarctica People with the name * Vann (surname), an English surname (including a list of peopl ...
, for instance, argued that ''modus ponens'' can fail for conditionals whose consequents are themselves conditionals. The following is an example: # Either
Shakespeare William Shakespeare ( 26 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. He is often called England's nation ...
or
Hobbes Thomas Hobbes ( ; 5/15 April 1588 – 4/14 December 1679) was an English philosopher, considered to be one of the founders of modern political philosophy. Hobbes is best known for his 1651 book ''Leviathan'', in which he expounds an influe ...
wrote ''
Hamlet ''The Tragedy of Hamlet, Prince of Denmark'', often shortened to ''Hamlet'' (), is a tragedy written by William Shakespeare sometime between 1599 and 1601. It is Shakespeare's longest play, with 29,551 words. Set in Denmark, the play depicts ...
''. # If either Shakespeare or Hobbes wrote ''Hamlet'', then if Shakespeare didn't do it, Hobbes did. # Therefore, if Shakespeare didn't 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 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 P, P \rightarrow (Q \rightarrow R), therefore Q \rightarrow R; it is not essential that 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. In
deontic logic Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. It ...
, 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." ''
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...
''.
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 Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dar ...
is a common misinterpretation of the ''modus ponens''.


See also

* * * * * * *


References


Sources

*Herbert B. Enderton, 2001, ''A Mathematical Introduction to Logic Second Edition'', Harcourt Academic Press, Burlington MA, . * Audun Jøsang, 2016, ''Subjective Logic; A formalism for Reasoning Under Uncertainty'' Springer, Cham, *
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...
and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
1927 ''Principia Mathematica to *56 (Second Edition)'' paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN. *
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
1946 ''Introduction to Logic and to the Methodology of the Deductive Sciences'' 2nd Edition, reprinted by Dover Publications, Mineola NY. (pbk).


External links

* * *
Modus ponens
' at Wolfram MathWorld {{DEFAULTSORT:Modus Ponens Rules of inference Latin logical phrases Theorems in propositional logic Classical logic