Strict conditional

In logic, a strict conditional (symbol: ◻ {displaystyle Box } , 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 ◻ ( p → q ) {displaystyle Box (p ightarrow q)} 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. In logic, a strict conditional (symbol: ◻ {displaystyle Box } , 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 ◻ ( p → q ) {displaystyle Box (p ightarrow q)} 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: 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: 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: