material implication

logic

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June 30, 2026

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In logic, material implication describes a relationship between two propositions where the second is a logical consequence of the first. It is often read as "If A, then B" and symbolized as (A \to B). The truth of this compound proposition depends solely on the truth values of A and B, not their meanings. (A \to B) is false only when A is true and B is false; it is true in all other cases. This can be expressed as (\neg A \lor B), meaning "not A or B." This definition can lead to counterintuitive results, such as "If the grass is red, then ice is cold" being considered a true statement.1

To address this, the concept of strict implication was introduced, which considers the meanings of the propositions rather than just their truth values.1

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formal logic
Basic features of PC
… [then] q” or “p [materially] implies q”) is to count as false when p is true and q is false and as true in all other cases; hence it has the same meaning as “either not-p or q” or as “not both p and not-q.” The symbol “⊃” is…
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… [then] q” or “p [materially] implies q”) is to count as false when p is true and q is false and as true in all other cases; hence it has the same meaning as “either not-p or q” or as “not both p and not-q.” The symbol “⊃” is…
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implication
…logic, a broader relationship called material implication is employed, which is read “If A, then B,” and is denoted by A ⊃ B or A → B. The truth or falsity of the compound proposition A ⊃ B depends not on any relationship between the meanings of the propositions but…
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…logic, a broader relationship called material implication is employed, which is read “If A, then B,” and is denoted by A ⊃ B or A → B. The truth or falsity of the compound proposition A ⊃ B depends not on any relationship between the meanings of the propositions but…
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history of logic
The Megarians and the Stoics
…exactly the modern notion of material implication. In Philo’s view, unlike Diodorus’s, conditionals may change their truth value over time.
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…exactly the modern notion of material implication. In Philo’s view, unlike Diodorus’s, conditionals may change their truth value over time.
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formal logic
Modal logic
…it is necessary that p materially implies q. “John’s tie is scarlet,” for example, strictly implies “John’s tie is red,” because it is impossible for John’s tie to be scarlet without being red (or it is necessarily true that, if John’s tie is scarlet, it is red). In general, if…
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…it is necessary that p materially implies q. “John’s tie is scarlet,” for example, strictly implies “John’s tie is red,” because it is impossible for John’s tie to be scarlet without being red (or it is necessarily true that, if John’s tie is scarlet, it is red). In general, if…
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