Strict conditional

Problems

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. 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:

${\displaystyle ◻}$ (Bill Gates graduated in medicine → 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, although it does not seem that the original sentence should be. A similar situation arises with 2 + 2 = 5, which is necessarily false:

If 2 + 2 = 5, then Bill Gates graduated in medicine.

Some logicians view this situation as indicating that the strict conditional is still unsatisfactory. Others have noted that the strict conditional cannot adequately express counterfactual conditionals, and that it does not satisfy certain logical properties. In particular, the strict conditional is transitive, while the counterfactual conditional is not.

Some logicians, 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 relevance logic to supply a connection between the antecedent and consequent of provable conditionals.

Constructive logic

In a constructive setting, the symmetry between ⥽ and ${\displaystyle ◻}$ is broken, and the two connectives can be studied independently. Constructive strict implication can be used to investigate interpretability of Heyting arithmetic and to model arrows and guarded recursion in computer science.