strict implication

logic

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July 14, 2026

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In logic, an implication describes a relationship where one proposition is a logical consequence of another. The most common form is material implication, symbolized as A → B, read as "If A, then B." This is true unless A is true and B is false. However, this can lead to "paradoxes of material implication," where a false premise can imply any conclusion.

To address this, strict implication was introduced. It states that A strictly implies B if it's impossible for A to be true and B to be false simultaneously. This definition considers the meanings of the propositions, not just their truth values. In intuitionistic logic, implication is defined by the existence of a proof that connects the two propositions.1

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formal logic
Modal logic
…it is said that p strictly implies q. An alternative equivalent way of explaining the notion of strict implication is by saying that p strictly implies q if and only if it is necessary that p materially implies q. “John’s tie is scarlet,” for example, strictly implies “John’s tie is…
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…it is said that p strictly implies q. An alternative equivalent way of explaining the notion of strict implication is by saying that p strictly implies q if and only if it is necessary that p materially implies q. “John’s tie is scarlet,” for example, strictly implies “John’s tie is…
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implication
…in 1932 the notion of strict implication. Strict implication was defined as ∼♦(A·∼B), in which ♦ means “is possible” or “is not self-contradictory.” Thus A strictly implies B if it is impossible for both A and ∼B to be true. This conception of implication is based upon the meanings of…
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…in 1932 the notion of strict implication. Strict implication was defined as ∼♦(A·∼B), in which ♦ means “is possible” or “is not self-contradictory.” Thus A strictly implies B if it is impossible for both A and ∼B to be true. This conception of implication is based upon the meanings of…
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C.I. Lewis
…system of logic based upon strict implication. That is, he rejected systems that do not limit themselves strictly to what is implicit in experience. Because concepts arise from experience, in his system no concept is fixed or indispensable, and the abstract categories of traditional logic are subject to change.
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…system of logic based upon strict implication. That is, he rejected systems that do not limit themselves strictly to what is implicit in experience. Because concepts arise from experience, in his system no concept is fixed or indispensable, and the abstract categories of traditional logic are subject to change.
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