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Concept# Strict conditional

Summary

In logic, a strict conditional (symbol: , or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For any two propositions p and q, the formula p → q says that p materially implies q while says that p strictly implies q. Strict conditionals are the result of Clarence Irving Lewis's attempt to find a conditional for logic that can adequately express indicative conditionals in natural language. They have also been used in studying Molinist theology.
The strict conditionals may avoid paradoxes of material implication. The following statement, for example, is not correctly formalized by material implication:
If Bill Gates has graduated in Medicine, then Elvis never died.
This condition should clearly be false: the degree of Bill Gates has nothing to do with whether Elvis is still alive. However, the direct encoding of this formula in classical logic using material implication leads to:
Bill Gates graduated in Medicine → Elvis never died.
This formula is true because whenever the antecedent A is false, a formula A → B is true. Hence, this formula is not an adequate translation of the original sentence. An encoding using the strict conditional is:
(Bill Gates graduated in Medicine → Elvis never died.)
In modal logic, this formula means (roughly) that, in every possible world in which Bill Gates graduated in Medicine, Elvis never died. Since one can easily imagine a world where Bill Gates is a Medicine graduate and Elvis is dead, this formula is false. Hence, this formula seems to be a correct translation of the original sentence.
Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems with consequents that are necessarily true (such as 2 + 2 = 4) or antecedents that are necessarily false.

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