Material conditional

The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function ⊃ from truth-values to truth-values. In predicate logic, it can be viewed as a subset relation between the extension of (possibly complex) predicates. In symbols, a material conditional is written as one of the following:

The material conditional is false when X is true and Y is false - otherwise, it is true. (Here, X and Y are variables ranging over formulæ of a formal theory.) We call X the antecedent, and Y the consequent. The material conditional is also commonly referred to as material implication with the understanding that the antecedent (X) materially implies the consequent (Y).

A distant approximation to the material conditional is the English construction 'if...then...', where the ellipses are to be filled with English sentences. However, this is the most common reading of the material conditional in English. A closer approximation to X &rarr; Y is 'it's false that X be true while Y false'&mdash;i.e., in symbols, $$\neg(X \and \neg Y)$$. Arguably this is more intuitive than its logically equivalent disjunction &not;X &or; Y.

Definition
Logical implication and the material conditional are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false.

Truth table
The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:

Johnston diagram
The Johnston diagram of "If A then B"



Formal properties
The material conditional is not to be confused with the entailment relation ⊨ (which is used here as a name for itself). But there is a close relationship between the two in most logics, including classical logic which we only consider here. For example, the following principles hold:


 * If $$\Gamma\models\psi$$ then $$\emptyset\models\phi_1\land\dots\land\phi_n\supset\psi$$ for some $$\phi_1,\dots,\phi_n\in\Gamma$$. (This is a particular form of the deduction theorem.)


 * The converse of the above


 * Both ⊃ and ⊨ are monotonic; i.e., if $$\Gamma\models\psi$$ then $$\Delta\cup\Gamma\models\psi$$, and if $$\phi\supset\psi$$ then $$(\phi\land\alpha)\supset\psi$$ 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 logics, nor do they hold in relevance logics.

Other properties of implication:


 * associativity: $$(p \rightarrow (q \rightarrow r)) \rightarrow ((p \rightarrow q) \rightarrow r)$$


 * distributivity: $$s \rightarrow (p \rightarrow q) \rightarrow ((s \rightarrow p) \rightarrow (s \rightarrow q))$$
 * $$p \rightarrow (q \equiv r) \equiv ((p \rightarrow q) \equiv (p \rightarrow r))$$


 * transitivity: ($$a \rightarrow b) \rightarrow ((b \rightarrow c) \rightarrow (a \rightarrow c))$$


 * commutativity: ($$a \rightarrow (b \rightarrow c)) \rightarrow (b \rightarrow (a \rightarrow c))$$


 * idempotency: $$a \rightarrow a$$


 * 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.

Philosophical problems with material conditional
The truth function &sup; does not correspond exactly 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.

Conditionals

 * Counterfactual conditional
 * Indicative conditional
 * Corresponding conditional (logic)
 * Strict conditional
 * Logical implication

Related topics

 * Ampheck
 * Boolean algebra
 * Boolean domain
 * Boolean function


 * Boolean logic
 * Laws of Form
 * Logic gate
 * Logical graph


 * Peirce's law
 * Propositional logic
 * Sole sufficient operator
 * Zeroth order logic
 * Conditional quantifier

Implikace Subjunktion Implico Материјална импликација Logische implicatie 条件文 Subjunksjon (logikk) Wynikanie Condição lógica Logisk implikation เงื่อนไขเชิงตรรกศาสตร์ 实质条件