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# Material conditional

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 Title: Material conditional Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Material conditional

Venn diagram of A \rightarrow B.
If a member of the set described by this diagram (the red areas) is a member of A, it is in the intersection of A and B, and it therefore is also in B.

The material conditional (also known as "material implication", "material consequence", or simply "implication", "implies" or "conditional") is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→". The material conditional is used to form statements of the form "pq" (termed a conditional statement) which is read as "if p then q" or "p only if q" and conventionally compared to the English construction "If...then...". But unlike the English construction, the material conditional statement "pq" does not specify a causal relationship between p and q and is to be understood to mean "if p is true, then q is also true" such that the statement "pq" is false only when p is true and q is false.[1] Intuitively, consider that a given p being true and q being false would prove an "if p is true, q is always also true" statement false, even when the "if p then q" does not represent a causal relationship between p and q. Instead, the statement describes p and q as each only being true when the other is true, and makes no claims that p causes q. However, note that such a general and informal way of thinking about the material conditional is not always acceptable, as will be discussed. As such, the material conditional is also to be distinguished from logical consequence.

The material conditional is also symbolized using:

1. p \supset q (Although this symbol may be used for the superset symbol in set theory.);
2. p \Rightarrow q (Although this symbol is often used for logical consequence (i.e. logical implication) rather than for material conditional.)

With respect to the material conditionals above, p is termed the antecedent, and q the consequent of the conditional. Conditional statements may be nested such that either or both of the antecedent or the consequent may themselves be conditional statements. In the example "(pq) → (rs)" both the antecedent and the consequent are conditional statements.

In