In
propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo
ponens'' (
Latin
Latin (, or , ) is a classical language belonging to the Italic languages, 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 ...
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
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, i ...
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
Antiquity or Antiquities may refer to:
Historical objects or periods Artifacts
*Antiquities, objects or artifacts surviving from ancient cultures
Eras
Any period before the European Middle Ages (5th to 15th centuries) but still within the histo ...
. 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'', Routle ...
. 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
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be tru ...
, with two premises and a conclusion:
# If ''P'', then ''Q''.
# ''P''.
# Therefore, ''Q''.
The first premise is a
conditional
Conditional (if then) may refer to:
*Causal conditional, if X then Y, where X is a cause of Y
*Conditional probability, the probability of an event A given that another event B has occurred
*Conditional proof, in logic: a proof that asserts a co ...
("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'':
# 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 ...
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
In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abst ...
: 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 ...
, ''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
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 for ...
.
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 arg ...
.
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
and
, for instance, are equivalent (as is standard), then
. Logical implication becomes a matter of relative position:
logically implies
just in case
, i.e., when either
or else
lies below
and is connected to it by an upward path.
In this context, to say that
and
together imply
—that is, to affirm ''modus ponens'' as valid—is to say that
. In the semantics for basic propositional logic, the algebra is
Boolean, with
construed as the
material conditional:
. Confirming that
is then straightforward, because
. With other treatments of
, 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 which for a binary variable is expressed as:
,
where e.g.
denotes the probability of
and the
conditional probability generalizes the logical implication
. Assume that
is equivalent to
being TRUE, and that
is equivalent to
being FALSE. It is then easy to see that
when
and
. 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:
,
where
denotes the subjective opinion about
as expressed by source
, and the conditional opinion
generalizes the logical implication
. The deduced marginal opinion about
is denoted by
. The case where
is an absolute TRUE opinion about
is equivalent to source
saying that
is TRUE, and the case where
is an absolute FALSE opinion about
is equivalent to source
saying that
is FALSE. The deduction operator
of
subjective logic produces an absolute TRUE deduced opinion
when the conditional opinion
is absolute TRUE and the antecedent opinion
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 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 people w ...
, 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 influ ...
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
, therefore
; it is not essential that
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."
[ '']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. E ...
''. 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''.
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 applica ...
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, a ...
1927 ''Principia Mathematica to *56 (Second Edition)'' paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN.
*
Alfred Tarski 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