Strict Conditionals

A strict conditional is material conditional with a necessity operator quantifying over it, for example, ⊡(A⊃C). (If you don't see a box symbol in front of (A⊃C), your browser isn't supporting unicode with a sufficient set of characters). The necessity operator quantifies over a specific set of possible worlds. So ⊡(A⊃C) is true in all of the relevant worlds in which A is true, C is true.

There are different types of necessity. Necessity, without any further qualification, indicates logical necessity. ⊡(A⊃C) is true if at all logically possible worlds in which A is true, C is true. That is to say, ⊡(A⊃C) is true if it holds at all worlds. Physical necessity, to take another example, restricts our attention to the class of physically possible worlds. So, considering physical necessity, ⊡(A⊃C) is true if at all physically possible worlds in which A is true, C is true.

A necessity operator comes with an inter-definable possibility operator (for which I am using ◊). We can express the strict conditional in terms of a possibility operator as follows: ¬◊(A&¬C)

For any given necessity operator, possibility operator, or strict conditional, there is generally taken to be some kind of accessibility relation that designates the scope of worlds under consideration.

David Lewis asserts that we can compare the strictness of conditionals. The larger the scope of worlds relevant to the conditional, the stricter the conditional is. That means, as Lewis points out, the strict logical conditional is the strictest conditional.

Strict Conditionals and Counterfactuals

The idea of a strict conditional is integrally tied to Lewis's theory of counterfactuals. Lewis hypothesized that counterfactuals are like strict conditionals based on the comparative similarity of possible worlds. When we asses the truth value of a counterfactual, for instance, "If I had $100,000,000, I would be in the top 1% in value of total assets in America," I certainly want to consider the truth of the consequent at some of the worlds at which the antecedent is true. But, warns Lewis, I ought not consider it at all A-worlds. Worlds in which everyone in America has $100,000,000, for instance, are irrelevant. Those worlds are far too dissimilar to our own.

But, these remarks alone will not do. Consider the following pair of conditionals: "If I were to do my homework tonight, then I would hand it in tomorrow," and "If I were to do my homework tonight and the dog were to eat it, I wouldn't hand it in tomorrow." Considering each counterfactual by itself as a strict conditional within a sphere of relevant similarity is unproblematic. But, considering the two together generates a contradiction. Why? Because all worlds relevant when considering A>C are also going to be relevant to a consideration of A&B>¬C. In other words, the same worlds that were relevant to the supposition that I did my homework are going to be relevant to the supposition that I did my homework and the dog ate it.

In light of these concerns, Lewis concludes that counterfactuals are variably strict conditionals. He states, "Counterfactuals are like strict conditionals but there is no saying how strict they are. Any particular counterfactual is as strict, within limits, as it must be to escape vacuity, and no stricter."