Strict conditional

In logic, a strict conditional is a material conditional that is acted upon by the necessity operator from modal logic. For any two propositions $$p$$ and $$q$$, the formula $$p \rightarrow q$$ says that $$p$$ materially implies $$q$$ while $$\Box (p \rightarrow q)$$ 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. Such a conditional would, for example, avoid the paradoxes of material implication. The following statement, for example, is not correctly formalized by material implication.


 * If Bill Gates had graduated in Medicine, then Elvis never died.

This condition is clearly 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 lead to:


 * Bill Gates graduated in Medicine $$\rightarrow$$ Elvis never died.

This formula is true because a formula $$A \rightarrow B$$ is true whenever the antecedent $$A$$ is false. Hence, this formula is not an adequate translation of the original sentence. Strict conditions are encodings of implications in modal logic attempting A different encoding is:


 * $$\Box$$ (Bill Gates graduated in Medicine $$\rightarrow$$ 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 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. The following sentence, for example, is not correctly formalized by a strict conditional:


 * If Bill Gates graduated in Medicine, then 2 + 2 = 4.

Using strict conditionals, this sentence is expressed as:


 * $$\Box$$ (Bill Gates graduated in Medicine $$\rightarrow$$ 2 + 2 = 4)

In modal logic, this formula means that, in every possible world where Bill Gates graduated in medicine, it holds that 2 + 2 = 4. Since 2 + 2 is equal to 4 in all possible worlds, this formula is true. While it is clearly not the case that 2 + 2 = 4 if Bill Gates graduated in medicine, the corresponding strict material statement is true.

To avoid the paradoxes of strict implication, some logicians have created counterfactual conditionals. Others, such as Paul Grice, have used conversational implicature to argue that, despite apparent difficulties, the material conditional is just fine as a translation for the natural language 'if...then...'. Others still have turned to relevant logic to supply a connection between the antecedent and consequent of provable conditionals.