Material conditional
Encyclopedia
The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q (typically read "if p then q" or "p implies q") is logically equivalent to the negative compound: not (p and not q). A material conditional compound itself is often simply called a conditional. By definition of "→", the compound p→q is false if and only if
both p is true and q is false. That is to say that p→q is true if and only if either p is false or q is true (or both). Thus → is a function from pairs of truth values of the components p, q to truth values of the compound p→q, whose truth value is entirely a function of the truth values of the components. Thus p→q is said to be truth-functional. p→q is logically equivalent also to ¬p∨q (either not p, or q (or both)), and to ¬q → ¬p (if not q then not p), but not to ¬p → ¬q. For convenience, p→q is typically read "If p, then q" or "q if p". Saying "It is false that if p then q" does not always sound logically equivalent in everyday English to saying "both p and not q" but, when used in logic, it is taken as logically equivalent. (Other senses of English "if...then..." require other logical forms.) The material implication between two sentences p, q is typically symbolized as
As placed within the material conditionals above, p is known as the antecedent
, and q as the consequent
, of the conditional. One can also use compounds as components, for example pq → (r→s). There, the compound pq (short for "p and q") is the antecedent, and the compound r→s is the consequent, of the larger conditional of which those compounds are components.
The material conditional may also be viewed, not as a truth function, but as a symbol of a formal theory
, taken as a set of sentences, satisfying all the classical inferences involving →, in particular the following characteristic rules:
s, typically the values of two proposition
s, that produces a value of false just in the case when the first operand is true and the second operand is false.
relation
which is typically defined semantically: 
if every interpretation that makes A true also makes B true. However, there is a close relationship between the two in most logics, including classical logic
. For example, the following principles hold:
These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logic
s, nor do they hold in relevance logic
s.
Other properties of implication:
Note that
is logically equivalent
to
; this property is sometimes called currying
. Because of these properties, it is convenient to adopt a right-associative notation for →.
English "if condition then consequence" construction (a kind of conditional sentence
), where condition and consequence are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition (protasis
) and consequence (apodosis
) (see Connexive logic
).
So, although a material conditional from a contradiction is always true, in natural language, "If there are three hydrogen atoms in H2O then the government will lose the next election" is interpreted as false by most speakers, since assertions from chemistry are considered irrelevant conditions for proposing political consequences.
"If P then Q", in natural language, appears to mean "P and Q are connected and P→Q". Just what kind of connection is meant by the natural language is not clearly defined.
When protasis and apodosis are connected, the truth functionality of linguistic and logical conditionals coincide; the distinction is only apparent when the material conditional is true, but its antecedent and consequent are perceived to be unconnected.
The modifier material in material conditional makes the distinction from linguistic conditionals explicit. It isolates the underlying, unambiguous truth functional relationship.
Therefore, exact natural language encapsulation of the material conditional X → Y, in isolation, is seen to be "it's false that X be true while Y false" or "it cannot be that X AND not-Y" — i.e. in symbols,
.
The truth function
corresponds to 'not ... or ...' and does not correspond to the English 'if...then...' construction. For example, any material conditional statement with a false antecedent is true.
So the statement "if 2 is odd then 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "if Pigs fly then Paris is in France" is true. These problems are known as the paradoxes of material implication
, though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions.
There are various kinds of conditionals in English; e.g., there is the indicative conditional
and the subjunctive or counterfactual conditional
. The latter do not have the same truth conditions as the material conditional. For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
both p is true and q is false. That is to say that p→q is true if and only if either p is false or q is true (or both). Thus → is a function from pairs of truth values of the components p, q to truth values of the compound p→q, whose truth value is entirely a function of the truth values of the components. Thus p→q is said to be truth-functional. p→q is logically equivalent also to ¬p∨q (either not p, or q (or both)), and to ¬q → ¬p (if not q then not p), but not to ¬p → ¬q. For convenience, p→q is typically read "If p, then q" or "q if p". Saying "It is false that if p then q" does not always sound logically equivalent in everyday English to saying "both p and not q" but, when used in logic, it is taken as logically equivalent. (Other senses of English "if...then..." require other logical forms.) The material implication between two sentences p, q is typically symbolized as
-
; -
; -
(Typically used for logical implication rather than for material implication.)
As placed within the material conditionals above, p is known as the antecedent
Antecedent (logic)
An antecedent is the first half of a hypothetical proposition.Examples:* If P, then Q.This is a nonlogical formulation of a hypothetical proposition...
, and q as the consequent
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".Examples:* If P, then Q.Q is the consequent of this hypothetical proposition....
, of the conditional. One can also use compounds as components, for example pq → (r→s). There, the compound pq (short for "p and q") is the antecedent, and the compound r→s is the consequent, of the larger conditional of which those compounds are components.
The material conditional may also be viewed, not as a truth function, but as a symbol of a formal theory
Formal theory
Formal theory can refer to:* Another name for a theory which is expressed in formal language.* An axiomatic system, something representable by symbols and its operators...
, taken as a set of sentences, satisfying all the classical inferences involving →, in particular the following characteristic rules:
- Modus ponensModus ponensIn classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
; - Conditional proofConditional proofA 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....
; - Classical contrapositionContrapositionIn traditional logic, contraposition is a form of immediate inference in which from a given proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality . For its symbolic expression in modern logic see the rule...
; - Classical reductioReductio ad absurdumIn logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
.
Definition
The material conditional is associated with an operation on two logical valueLogical value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.In classical logic, with its intended semantics, the truth values are true and false; that is, classical logic is a two-valued logic...
s, typically the values of two proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s, that produces a value of false just in the case when the first operand is true and the second operand is false.
Truth table
The truth table associated with the material conditional not p or q (symbolized as p → q) and the logical implication p implies q (symbolized as p → q, or Cpq) is as follows:| p | q | p → q |
|---|---|---|
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 1 |
Formal properties
When studying logic formally, the material conditional is distinguished from the entailmentEntailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
relation
which is typically defined semantically: if every interpretation that makes A true also makes B true. However, there is a close relationship between the two in most logics, including classical logicClassical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
. For example, the following principles hold:
- If
then 
for some 
. (This is a particular form of the deduction theoremDeduction theoremIn mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then proving B from this assumption. The deduction theorem explains why proofs of conditional...
.)
- The converse of the above
- Both
and ⊨ are monotonic; i.e., if 
then 
, and if 
then 
for any α, Δ. (In terms of structural rules, this is often referred to as weakening or thinning.)
These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logic
Non-monotonic logic
A non-monotonic logic is a formal logic whose consequence relation is not monotonic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences. Intuitively, monotonicity indicates that learning a...
s, nor do they hold in relevance logic
Relevance logic
Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
s.
Other properties of implication:
- distributivityDistributivityIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
:
- transitivityTransitive relationIn mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
:
- idempotency:
- truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of material implication.
- commutativity of antecedents:
Note that
is logically equivalentLogical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
to
; this property is sometimes called curryingCurrying
In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument...
. Because of these properties, it is convenient to adopt a right-associative notation for →.
Philosophical problems with material conditional
The meaning of the material conditional can sometimes be used in the natural languageNatural language
In the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written...
English "if condition then consequence" construction (a kind of conditional sentence
Conditional sentence
In grammar, conditional sentences are sentences discussing factual implications or hypothetical situations and their consequences. Languages use a variety of conditional constructions and verb forms to form such sentences....
), where condition and consequence are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition (protasis
Protasis (linguistics)
In linguistics, a protasis is the subordinate clause in a conditional sentence. For example, in "if X, then Y", the protasis is "if X"...
) and consequence (apodosis
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".Examples:* If P, then Q.Q is the consequent of this hypothetical proposition....
) (see Connexive logic
Connexive logic
Connexive logic names one class of alternative, or non-classical, logics designed to exclude the so-called paradoxes of material implication. The characteristic that separates connexive logic from other non-classical logics is its acceptance of Aristotle's Thesis, i.e...
).
So, although a material conditional from a contradiction is always true, in natural language, "If there are three hydrogen atoms in H2O then the government will lose the next election" is interpreted as false by most speakers, since assertions from chemistry are considered irrelevant conditions for proposing political consequences.
"If P then Q", in natural language, appears to mean "P and Q are connected and P→Q". Just what kind of connection is meant by the natural language is not clearly defined.
- The statement "if (B) all bachelorBachelorA bachelor is a man above the age of majority who has never been married . Unlike his female counterpart, the spinster, a bachelor may have had children...
s are unmarried then (C) the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in a vacuum is constant" may be considered false, because there is no discernible connection between (B) and (C), even though (B)→(C) is true. - The statement "if (S) SocratesSocratesSocrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
was a woman then (T) 1+1=3" may be considered false, for the same reason; even though (S)→(T) is true.
When protasis and apodosis are connected, the truth functionality of linguistic and logical conditionals coincide; the distinction is only apparent when the material conditional is true, but its antecedent and consequent are perceived to be unconnected.
The modifier material in material conditional makes the distinction from linguistic conditionals explicit. It isolates the underlying, unambiguous truth functional relationship.
Therefore, exact natural language encapsulation of the material conditional X → Y, in isolation, is seen to be "it's false that X be true while Y false" or "it cannot be that X AND not-Y" — i.e. in symbols,
.The truth function
corresponds to 'not ... or ...' and does not correspond to the English 'if...then...' construction. For example, any material conditional statement with a false antecedent is true.So the statement "if 2 is odd then 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "if Pigs fly then Paris is in France" is true. These problems are known as the paradoxes of material implication
Paradoxes of material implication
The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic. One of these paradoxes is the paradox of entailment....
, though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions.
There are various kinds of conditionals in English; e.g., there is the indicative conditional
Indicative conditional
In natural languages, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional, an indicative conditional does not have a stipulated definition...
and the subjunctive or counterfactual conditional
Counterfactual conditional
A counterfactual conditional, subjunctive conditional, or remote conditional, abbreviated , is a conditional statement indicating what would be the case if its antecedent were true...
. The latter do not have the same truth conditions as the material conditional. For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.
See also
- Ampheck
- Boolean algebra
- Boolean domainBoolean domainIn mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true...
- Boolean function
- Boolean logicBoolean logicBoolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
- Implicational propositional calculusImplicational propositional calculusIn mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional...
- Laws of FormLaws of FormLaws of Form is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy...
- Logic gateLogic gateA logic gate is an idealized or physical device implementing a Boolean function, that is, it performs a logical operation on one or more logic inputs and produces a single logic output. Depending on the context, the term may refer to an ideal logic gate, one that has for instance zero rise time and...
- Logical graphLogical graphA logical graph is a special type of diagramatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic....
- Paradoxes of material implicationParadoxes of material implicationThe paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic. One of these paradoxes is the paradox of entailment....
- Peirce's lawPeirce's lawIn logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that involves only one sort of connective, namely...
- Propositional logic
- Sole sufficient operator
- Conditional quantifierConditional quantifierIn logic, a conditional quantifier is a kind of Lindström quantifier Q_A that, relative to a classical model A, satisfies some or all of the following conditions :...
Conditionals
- Counterfactual conditionalCounterfactual conditionalA counterfactual conditional, subjunctive conditional, or remote conditional, abbreviated , is a conditional statement indicating what would be the case if its antecedent were true...
- Indicative conditionalIndicative conditionalIn natural languages, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional, an indicative conditional does not have a stipulated definition...
- Corresponding conditional (logic)Corresponding conditional (logic)In logic, the corresponding conditional of an argument is a material conditional whose antecedent is the conjunction of the argument's premises and whose consequent is the argument's conclusion. An argument is valid if and only if its corresponding conditional is a logical truth...
- Strict conditionalStrict conditionalIn logic, a strict conditional is a material conditional that is acted upon by the necessity operator from modal logic. For any two propositions p and q, the formula p \rightarrow q says that p materially implies q while \Box says that p strictly implies q...
- Logical implication