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. Symbolically:
The material conditional is false when X is true and Y is false – otherwise, it is true. X and Y, known respectively as the antecedent and consequent, are variables ranging over formulae of a formal theory. The material conditional is also commonly referred to as material implication with the understanding that the antecedent materially implies the consequent.
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.
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) is as follows:
In set theory there is the same difference between the operation and the relation meaning .
On the way from to the difference between logical () and material implication () is demonstrated:
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. For example, the following principles hold:
Other properties of implication:
The meaning of the material conditional can sometimes be used in the natural language 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" — i.e. in symbols, . Arguably this is more intuitive than its logically equivalent disjunction .
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.