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 premise ...
. When the conditional symbol
is
interpreted as material implication, a formula
is true unless
is true and
is false. Material implication can also be characterized inferentially by
modus ponens,
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
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.
Overview
The assumed antecedent of a conditional proof is called the conditio ...
, and
classical reductio ad absurdum.
Material implication is used in all the basic systems of
classical logic 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 necess ...
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 ''con ...
s in
natural language.
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 prefixed
Polish 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'' and ''q'' is termed 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. 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
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
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
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' (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 died o ...
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 the
binary
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 functional 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 such as the one below.
Truth table
The
truth table 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 d ...
s.
Deductive definition
Material implication can also be characterized
deductively in terms of the following
rules of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
.
#
Modus ponens
#
Conditional proof
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.
Overview
The assumed antecedent of a conditional proof is called the conditio ...
#
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 systems, 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
When
disjunction,
conjunction
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction, a rule of inference of propositional logic
* Conjunction (astronomy), in which two astronomical bodies ...
and
negation are classical, material implication validates the following equivalences:
* Contraposition:
*
Import-Export:
* Negated conditionals:
* Or-and-if:
* Commutativity of antecedents:
*
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 arithmeti ...
:
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:
*
Vacuous conditional:
*
Transitivity:
*
Simplification of disjunctive antecedents:
Tautologies involving material implication include:
*
Reflexivity:
*
Totality:
*
Conditional excluded middle:
Discrepancies with natural language
Material implication does not closely match the usage 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 ''con ...
s in
natural language. 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 "she d ...
, 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." Counterfactua ...
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
In linguistics and philosophy, the denotation of an expression is its literal meaning. For instance, the English word "warm" denotes the property of being warm. Denotation is contrasted with other aspects of meaning including connotation. For in ...
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, ...
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 functional 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,
relevance logic Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but ...
,
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
The Wason selection task (or ''four-card problem'') is a logic puzzle devised by Peter Cathcart Wason in 1966. It is one of the most famous tasks in the study of deductive reasoning. An example of the puzzle is:
A response that identifies a ca ...
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
*
Conditional quantifier
*
Implicational propositional calculus
* ''
Laws of Form
''Laws of Form'' (hereinafter ''LoF'') is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy. ''LoF'' describes three distinct logical systems:
* The "primary arithmetic" (described in C ...
''
*
Logical graph A logical graph is a special type of diagrammatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic.
In his papers on '' qualitative logic'', '' entitative graphs'', and '' existential grap ...
*
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 o ...
*
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." Counterfactua ...
*
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
In logic, the corresponding conditional of an argument (or derivation) is a material conditional whose antecedent is the conjunction of the argument's (or derivation's) premises and whose consequent is the argument's conclusion. An argument is v ...
*
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 necess ...
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, book ...
,
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