In logic, a strict conditional is a material conditional that is acted upon by the necessity operator from modal logic. Hence, for any two propositions p and q, if p -> q says that p materially implies q, then [](p -> q) says that p strictly implies q. Strict conditionals are the result of C. I. Lewis's attempt to find a conditional for logic that can adequately express natural language conditionals. Such a conditional would, for example, avoid the paradoxes of material implication. Consider

(1) If the moon is made of cheese, then Elvis never died.

Clearly (1) is false, most would say, because the moon's chemical makeup doesn't have anything to do with whether Elvis is still around. One could try rendering (1) as

(2) The moon is made of cheese -> Elvis never died.

(2) is true, simply because the antecedent is false. Hence, (2) is not an adequate translation of (1). Suppose that, instead, (1) were rendered as

(3) [] (The moon is made of cheese -> Elvis never died.)

(3) says (roughly) that in every possible world such that the moon is made of cheese, Elvis never died. Since one can easily imagine a world with a cheese moon and a living Elvis, (3) is false. Hence, (3) as a translation of (1) seems to get things right.

Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems. For example, there are paradoxes of strict implication:

(4) If the moon is made of cheese, then 2 + 2 = 4.

Rendered formally with strict implication, (4) becomes

(5) [] (The moon is made of cheese -> 2 + 2 = 4)

(5) says (roughly) that in every possible world where the moon is made of cheese, 2 + 2 = 4. Because it's impossible for there to be a world where 2 + 2 fails to equal 4, (5) is true. 2 + 2 always equals 4, so (4) is true. (See logical conditional.)

To avoid the paradoxes of strict implication, some logician's have created counterfactual conditionals. Others, such as H. P. 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.

Sources and Further Reading

For an introduction to non-classical logics as attempts to find a better translation of the conditional, see

An Introduction to Non-Classical Logics, by Graham Priest, 2001, Cambridge

For an extended philosophical discussion of the issues mentioned in this article see both

Logical Forms, by Mark Sainsbury, 2001, Blackwell Publishers

and

A Philosophical Guide to Conditionals, by Jonathan Bennett, 2003, Oxford