Logical Necessity and Deductive Validity

One of the most helpful and rich discussions of logical necessity in the liter- ature is McFetridge’s “Logical Necessity: Some Issues”. I take McFetridge to have been grappling with two main issues in this paper: (1) If there is such a thing as logical necessity, what is it?; and (2) What is the purpose of beliefs about logical necessity? Much of his discussion focuses on what it takes for someone to count as committed to logical necessity, or what it takes for someone to have a belief in logical necessity. But much of the material is still helpful when it comes to the nature of logical necessity.

The first key point highlighted by McFetridge is a connection between deductive validity in an argument and logical necessity. Deductive validity is a, perhaps the, central notion in logic. We often take the validity of an argu- ment to be a modal matter: we say that in a valid deduction the conclusion

follows of necessity from the premises, or that an argument is valid if it is impossible for the premises to be true and the conclusion false. McFetridge’s point is then that, if any notion of necessity deserves to be called logical ne- cessity, it should be the kind of necessity attaching to deductive validity, if indeed there is any necessity there.

Deductive validity is the central topic of logic. So if, as Aristotle and others have thought, to think of an argument as deductively valid requires us to deploy a notion of necessity, then that notion, if any, will deserve the label ‘logical’ necessity. There will be a legitimate notion of ‘logical’ necessity only if there is a notion of necessity which attaches to the claim, concerning a deduc- tively valid argument, that if the premises are true then so is the conclusion. (McFetridge, 1990, p. 136)

I agree with McFetridge. Indeed, I find it hard to imagine how anyone could argue that, if there is some kind of necessity involved in deductive validity, then this should not be called logical necessity, in favour of giving the title to something else. What could count as more logical than the notion of a deductively valid argument? This is amongst the least controversial of the claims I will be discussing, so I will take this starting point for granted.

Rumfitt (2010) also endorses a connection between logical necessity and logical consequence (the latter being a close relative of deductive validity )1 although not directly. He favours an understanding of logical necessity in terms of logical contradiction—it is logically necessary that p just when it is logically contradictory that ¬p—and demonstrates a connection to logical consequence as a meta-theorem, using classical logic.

What does it mean to say that there is a notion of logical ne- cessity? I mean this: there is a sense of ‘necessary’ for which pIt is necessary that Aq implies and is implied by pIt is logi- cally contradictory that not Aq. If we assume a classical logic . . . we immediately have the following meta-theorem: whenever B follows logically from A1, . . . , An, the statement pIt is logically

necessary that if A1 and . . . and An then Bq is true (where the

conditional is understood to be material). So logical necessity is

1

Deductive validity is a property of an argument. What is an argument? ‘An argument, in the sense that concerns us here, is what a person produces where he or she makes a statement and gives reasons for believing the statement. The statement itself is called the conclusion of the argument. . . ; the stated reasons for believing the conclusion are called the premises’ (Hodges, 1977, p. 36). An argument is deductively valid just when it is impossible for the premises to be true and the conclusion false. Logical consequence is a relation holding between the premises and conclusion of an argument: a conclusion C will be a logical consequence of some premises A1, . . . , An if and only if there is a deductively valid argument from A1, . . . , An to C.

implicated in logical consequence. (Rumfitt, 2010, p. 35)2

Rumfitt is perhaps more precise than McFetridge, but I will use McFetridge’s characterization as the notion of necessity attaching to deductive validity, if any, as my working notion of logical necessity. In invoking logical con- tradiction to characterize logical necessity, and in assuming classical logic, Rumfitt prejudges some important issues. E.g., if logical necessity is in- timately connected with logical consequence and validity, and the source of logical necessity is thereby to be discovered by considering the nature of logic, of which consequence and validity are central notions, then at that level of inquiry it will not be appropriate to have certain prior commitments, either in favour of classical logical systems, or against certain non-classical logics, e.g. paraconsistent and dialetheic logics.

An immediate question arises from McFetridge’s careful way of putting things: how can one be assured that there is indeed any necessity attach- ing to deductive validity? I will therefore consider two arguments which promise to provide some assurance. First, an argument from McFetridge for the conclusion that we are constrained to believe in logical necessity (more exactly, logically necessarily truth-preserving rules of inference). Second, an argument from Rumfitt against what he calls Russell’s Logical Philonianism, a view which explicitly rejects any modal aspect as belonging to deductive validity.