Relevant Logic

One prospect for extending a relative modality treatment beyond alethic modalities is to use a Relevant, rather than a Classical logic. I will close this discussion regarding the scope of the relative modality view by running through this option and some of its benefits and drawbacks.

What is Relevant Logic? Broadly speaking, we might describe differ- ent logical systems—Classical, Intuitionist, Dialethic, Relevant, etc.—to be based on taking different views about what it is for an argument from premises to conclusion to be valid.

The purpose of logical theory is to provide an explanation of the validity and invalidity of argument. The goal is to describe the relation which must hold between premises and conclusion for it to be correct to say that the premises entail the conclusion, that

the conclusion follows from the premises, or that the inference from premises to conclusion is valid. (Read, 1988, p. 19)

The development of Relevant Logic can be understood as a reaction to the Classical validity of certain arguments which intuitively seem invalid, and the associated paradoxes of material and strict implication. These results are achieved by various means. E.g., Read (1988) changes the rules for some of his connectives (most importantly “and” and “if. . . then”), which in turn allows him to give an alternative definition of validity.

In Relevant Logic, Read presents some compelling examples of classically valid but intuitively invalid arguments.

Let us suppose that Roy Dyckhoff has claimed that John Slaney was in Edinburgh on a certain day, and that Crispin Wright has denied it. Consider the following three propositions as they describe this situation.

(1) If John was in Edinburgh, Roy was right. This is clearly true: that’s what Roy claimed.

(2) If Crispin was right, so was Roy.

That is equally obviously false, given the logic of denial. (3) If John was in Edinburgh, Crispin was right.

That too is false, for Crispin denied it. Let us use these proposi- tions to construct an argument, taking as premises (1) together with the denial of (2), and as conclusion (3):

If John was in Edinburgh, then Roy was right.

It’s not the case that if Crispin was right, so was Roy. Hence, if John was in Edinburgh, Crispin was right.

Since (1) is true and (2) and (3) false, this argument, which takes the denial of (2) as its second premise, has true premises and a false conclusion. Hence it is invalid.

Classically, however, the argument is valid. For the sequent P ⊃ Q, ∼ (R ⊃ Q) ` P ⊃ R

which formalizes the argument classically, using ‘⊃’, representing material implication, to capture ‘if’, is (classically) valid. (Read, 1988, pp. 23–4)

Such examples do seem to indicate that something fishy is going on with the Classical view of validity, and the view that ‘if. . . then’ is to be captured by the material conditional (defined as: (p ⊃ q) ⇔ ¬(p & ¬q)).

I do not wish to digress too far by dwelling on the details of Relevant Logic. For present purposes, one important point is that this kind of logic avoids (indeed has been developed in order to avoid) the paradoxes of strict implication.19 These paradoxes include:

• (p & ¬p)  q • p  (q  q) • p  (q ∨ ¬q)

where “p  q” here is short for “(p ⊃ q)”.20

These paradoxes are exactly the kind of results that are problematic for the relative modality treatment of the kinds of modality I have been discussing. Take the following candidate definition of epistemic necessity:

epip ⇔ ∃ϕ(Kϕ & (ϕ → p))

where “K” means “is a conjunction of known truths”, reading “→” as the material conditional. Whenever hpi is a logical truth, the right-hand-side will be satisfied, and so hpi will be epistemically necessary, granted that there is a conjunction of known truths. Note that this includes cases where the second conjunct of the right-hand-side, (ϕ → p), a strict implication, is of the form (ϕ → (q  q)), (where p is (q  q)), or of the form (ϕ → (q ∨ ¬q)), (where p is (q ∨ ¬q)), the paradoxes mentioned above. If the background logic for the definition ruled out its being trivially true that (ϕ → p) for any logically true p, then we would avoid the problem of logical truths always being epistemically necessary. The right-hand-side of the definition would no longer be immediately satisfied whenever hpi is a logical truth, so logical truths are not rendered immediately epistemically necessary in this way.

Take as a second example the following candidate definition of a legal necessity:

legalp ⇔ ∃ϕ(Lϕ & (ϕ → p))

where the operator “L” can be read as “is a conjunction of UK laws”. Suppose that, due to inattention in legislation, this conjunction of laws is in fact contradictory—it contains, for some q, both hqi and h¬qi. According to Classical Logic, anything follows from a contradiction, and so the right- hand-side of the definition will be satisfied. So hpi will be legally necessary, for any p. In a Relevant Logic, the rule of inference that anything follows from a contradiction is eliminated from the system, so according to such a

19_{See Mares (2009)} 20

My own notation. I haved tried to leave the precise interpretation of implication and consequence open elsewhere, i.e. regarding whether “→” is to be read as material implication, or some other kind of implication, and whether “” is to be understood in terms of Classical logical consequence or some other notion of logical consequence.

logic, the right-hand-side of the definition is not guaranteed to be true for any p. Hence the problem of inconsistent conditions for modality can be avoided.

In general, then, the employment of a relevant logic addresses the various problems raised for extending a relative treatment of modality to kinds of modality such as epistemic, doxastic and deontic modalities, by respecting our intuitions that the conjunction of conditions to which the modality is relative should be relevant to the resulting possibilities and necessities. It does this by blocking certain rules for conditionals, such as rules which allow that you can build a necessarily true conditional out of any antecedent and a tautologous consequent. This kind of logic also rejects the rule of inference ex falso quodlibet, providing a solution to problems arising from inconsistent conditions to which a kind of modality is relative.

One drawback to this strategy is that it does not seem appropriate to use Relevant Logic across the board, i.e. to also use it for alethic modalities. E.g., although a given logical truth doesn’t seem to be directly relevant to the laws of physics, we still would not want to say that the negation of a logical truth is therefore physically possible. Physics doesn’t allow for logical impossibility, although our propositional attitudes might. One response here might be to draw a distinction between kinds of modality, between those best treated with a Classical as opposed to a Relevant logic. Given the kinds of modalities for which a Classical logic is problematic, this begins to resemble the distinction made above between veridical and non-veridical modalities. Modalities such as doxastic and deontic are not so much concerned with truth, as they are with specifically what follows from certain propositions. It doesn’t matter if a logical truth is true—it is simply not relevant to this kind of modality. Kinds of necessity which concern ways of being true, alethic necessities, are different. By including some propositions to which necessity is relative, the range of resulting necessities is broadened, but that there should be some logical truths included does not seem problematic. They are still true, and alethic modalities are not in the business of ruling out truth. Epistemic necessity may also imply truth, but it is not so much a way of being true as connected to a propositional attitude which is factive: being known is not a way to be true, but something that might happen to a truth incidentally.

I do not wish to settle the question of how far the relative modality view can or should be extended here. In general, I will focus primarily on alethic modalities, especially logical and metaphysical modality. However, I submit that if the account is to be extended to kinds of modality such as epistemic, deontic and doxastic modalities, then it seems that a promising option for accommodating them requires that we employ a deep distinction between these and other, alethic kinds of modality. This in turn would appear to have consequences for the account given of the modality to which other kinds of modality are relative. Rather than simply logical modality, we would have

both Classical logical modality and Relevant logical modality.21 It remains to be seen whether the two could be reconciled, or whether this reflects a deep and fundamental division between two families of modality.