The material conditional (also known as material implication) is an operation commonly used in

logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...

. When the conditional symbol $\backslash rightarrow$ is interpreted as material implication, a formula $P\; \backslash rightarrow\; Q$ is true unless $P$ is true and $Q$ is false. Material implication can also be characterized inferentially by modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...

, 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'' ...

, conditional proof, and classical reductio ad absurdum.
Material implication is used in all the basic systems of classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...

as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...

s. However, many logics replace material implication with other operators such as the strict conditional In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessi ...

and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentence
Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is ''co ...

s in natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

.
Notation

In logic and related fields, the material conditional is customarily notated with an infix operator →. The material conditional is also notated using the infixes ⊃ and ⇒. In the prefixedPolish notation
Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their operands, in contrast ...

, conditionals are notated as C''pq''. In a conditional formula ''p'' → ''q'', the subformula ''p'' is referred to as the ''antecedent
An antecedent is a preceding event, condition, cause, phrase, or word.
The etymology is from the Latin noun ''antecedentem'' meaning "something preceding", which comes from the preposition ''ante'' ("before") and the verb ''cedere'' ("to go").
...

'' and ''q'' is termed the '' consequent'' of the conditional. Conditional statements may be nested such that the antecedent or the consequent may themselves be conditional statements, as in the formula .
History

In '' Arithmetices Principia: Nova Methodo Exposita'' (1889), Peano expressed the proposition “If A then B” as “A Ɔ B” with the symbol Ɔ, which is the opposite of C. He also expressed the proposition “A ⊂ B” as “A Ɔ B”. Russell followed Peano in his '' Principia Mathematica'' (1910–1913), in which he expressed the proposition “If A then B” as “A ⊃ B”. Following Russell,Gentzen
Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He die ...

expressed the proposition “If A then B” as “A ⊃ B”. Heyting expressed the proposition “If A then B” as “A ⊃ B” at first but later came to express it as “A → B” with a right-pointing arrow.
Definitions

Semantics

From a semantic perspective, material implication is thebinary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...

truth function
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly on ...

al operator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in 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 ...

such as the one below.
Truth table

Thetruth 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 ...

of p → q:
The 3rd and 4th logical cases of this truth table, where the antecedent is false and is true, are called vacuous truth
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...

s.
Deductive definition

Material implication can also be characterized deductively in terms of the following rules of inference. #Modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...

# Conditional proof
# Classical contraposition
# Classical reductio ad absurdum
Unlike the semantic definition, this approach to logical connectives permits the examination of structurally identical propositional forms in various 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 ...

s, where somewhat different properties may be demonstrated. For example, in intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...

, which rejects proofs by contraposition as valid rules of inference, is not a propositional theorem, but the material conditional is used to define negation.
Formal properties

Whendisjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...

, conjunction
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction
Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or ...

and negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...

are classical, material implication validates the following equivalences:
* Contraposition: $P\; \backslash to\; Q\; \backslash equiv\; \backslash neg\; Q\; \backslash to\; \backslash neg\; P$
* Import-Export: $P\; \backslash to\; (Q\; \backslash to\; R)\; \backslash equiv\; (P\; \backslash land\; Q)\; \backslash to\; R$
* Negated conditionals: $\backslash neg(P\; \backslash to\; Q)\; \backslash equiv\; P\; \backslash land\; \backslash neg\; Q$
* Or-and-if: $P\; \backslash to\; Q\; \backslash equiv\; \backslash neg\; P\; \backslash lor\; Q$
* Commutativity of antecedents: $\backslash big(P\; \backslash to\; (Q\; \backslash to\; R)\backslash big)\; \backslash equiv\; \backslash big(Q\; \backslash to\; (P\; \backslash to\; R)\backslash big)$
* Distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...

: $\backslash big(R\; \backslash to\; (P\; \backslash to\; Q)\backslash big)\; \backslash equiv\; \backslash big((R\; \backslash to\; P)\; \backslash to\; (R\; \backslash to\; Q)\backslash big)$
Similarly, on classical interpretations of the other connectives, material implication validates the following entailment
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...

s:
* Antecedent strengthening: $P\; \backslash to\; Q\; \backslash models\; (P\; \backslash land\; R)\; \backslash to\; Q$
* Vacuous conditional: $\backslash neg\; P\; \backslash models\; P\; \backslash to\; Q$
* Transitivity: $(P\; \backslash to\; Q)\; \backslash land\; (Q\; \backslash to\; R)\; \backslash models\; P\; \backslash to\; R$
* Simplification of disjunctive antecedents: $(P\; \backslash lor\; Q)\; \backslash to\; R\; \backslash models\; (P\; \backslash to\; R)\; \backslash land\; (Q\; \backslash to\; R)$
Tautologies involving material implication include:
* Reflexivity: $\backslash models\; P\; \backslash to\; P$
* Totality: $\backslash models\; (P\; \backslash to\; Q)\; \backslash lor\; (Q\; \backslash to\; P)$
* Conditional excluded middle: $\backslash models\; (P\; \backslash to\; Q)\; \backslash lor\; (P\; \backslash to\; \backslash neg\; Q)$
Discrepancies with natural language

Material implication does not closely match the usage ofconditional sentence
Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is ''co ...

s in natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

. For example, even though material conditionals with false antecedents are vacuously true
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "sh ...

, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris is in France". These classic problems have been called the paradoxes of material implication. In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance, counterfactual conditional
Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactual ...

s would all be vacuously true on such an account.
In the mid-20th century, a number of researchers including H. P. Grice and Frank Jackson proposed that pragmatic principles could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionals denote material implication but end up conveying additional information when they interact with conversational norms such as Grice's maxims. Recent work in formal semantics and philosophy of language
In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, t ...

has generally eschewed material implication as an analysis for natural-language conditionals. In particular, such work has often rejected the assumption that natural-language conditionals are truth function
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly on ...

al in the sense that the truth value of "If ''P'', then ''Q''" is determined solely by the truth values of ''P'' and ''Q''. Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...

, relevance logic, probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, and causal models.
Similar discrepancies have been observed by psychologists studying conditional reasoning. For instance, the notorious Wason selection task study, less than 10% of participants reasoned according to the material conditional. Some researchers have interpreted this result as a failure of the participants to confirm to normative laws of reasoning, while others interpret the participants as reasoning normatively according to nonclassical laws.
See also

*Boolean domain
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as ...

* Boolean function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function ...

* Boolean logic
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...

* Conditional quantifier
* Implicational propositional calculus
* '' Laws of Form''
* Logical graph
* Logical equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on ...

* Material implication (rule of inference)
* Peirce's law
* Propositional calculus
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 ...

* Sole sufficient operator
Conditionals

*Counterfactual conditional
Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactual ...

* Indicative conditional
In natural languages, an indicative conditional is a conditional sentence such as "If Leona is at home, she isn't in Paris", whose grammatical form restricts it to discussing what could be true. Indicatives are typically defined in opposition to co ...

* Corresponding conditional
* Strict conditional In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessi ...

Notes

References

Further reading

* Brown, Frank Markham (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition,Kluwer
Wolters Kluwer N.V. () is a Dutch information services company. The company is headquartered in Alphen aan den Rijn, Netherlands (Global) and Philadelphia, United States (corporate). Wolters Kluwer in its current form was founded in 1987 with a m ...

Academic Publishers, Norwell, MA. 2nd edition, Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...

, Mineola, NY, 2003.
* Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), ''The Blackwell Guide to Philosophical Logic'', Blackwell.
* Quine, W.V. (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press
Harvard University Press (HUP) is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing. It is a member of the Association of American University Presses. After the retir ...

, Cambridge
Cambridge ( ) is a College town, university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cam ...

, MA.
* Stalnaker, Robert, "Indicative Conditionals", '' Philosophia'', 5 (1975): 269–286.
External links

* * {{Mathematical logic Logical connectives Conditionals Logical consequence Semantics