Relevance and Dependence

In my youth, I was very interested in relevant logic. With the zeal of the self-righteous, and the enthusiasm for rebellion and novelty that only the jejune can muster, I denied the validity of disjunctive syllogism, endorsed a useful four-valued logic, and all the rest.

Once I got to Princeton, I rejected all this dark heresy. It struck me that the relevantists, my younger self included, were mostly just confused about inference and implication. So it seemed to me that the relevantists objection to p, ¬pq was actually an objection to the argument ‘I have a reason to believe p; I have a reason to believe ¬p; therefore I have a reason to believe q‘. But even the classical logician will concede that second argument is invalid, and obscuring the issue by keeping the ‘I have a reason to believe’ operator implicit helps no one. So those motivations for relevantism based on what it would be good to be able to do with inconsistent or incomplete information sets seem to me to miss the point. They are excellent cases to use in a construction of the logic of the ‘I have a reason to believe that’ operator, and other explicitly epistemic and inferential operators. But there’s no reason to think that the existence of inconsistent bodies of information undermines it being the case that, were those bodies of information correct, every proposition would be true.

It is this last italicised counterfactual that gets at the real relationship that classical logic is concerned with. Counterfactuals, this one included, often express dependence. We apply the term of commendation ‘valid’ to an argument just in case it is good in the sense that its conclusion really does depend on its premises. This claim of dependence supports the normative role for logic in reasoning, since the dependence of conclusion on premises shows that one can’t coherently accept some premises without accepting what turns out to depend on them.

Classical logic gives an account of this specially logical sense of dependence: the truth value assigned by a classical interpretation to the conclusion turns out to depend on the truth value that interpretation assigns to the premises, in the sense that if the premises are all true, that determines the conclusion to be true. It is a small but significant step from here to the idea of consequence as necessary truth preservation; at least if each possible world corresponds to some classical interpretation, then this account will never give any false positives (i.e., saying that an argument is logically valid when the premises don’t depend on the conclusion), though it will give false negatives (e.g., ‘this is an apple; so this is a fruit’: the conclusion depends on the premises, because were this an apple, it would be a fruit is true, but there is a classical interpretation in propositional logic which assigns T to the premise and F to the conclusion).

Recently though I’ve been thinking a bit more about this (prompted by thinking about some work by a student of mine, Neil Dewar). For it is a commonplace that there can be cases of spurious dependence—correlation in truth value without genuine dependence. We can see this in causal cases, where two effects of a common cause might look like they depend on one another and yet do not. And indeed, the relevantist can be conceived of as noting this very phenomenon. For the counterfactuals of dependence are themselves trivialised in the two canonical problem cases for classical consequence:

  • p, ¬p ⊢ q (if the premises had been true, the conclusion would have been true is true, but because of the triviality of the antecedent)
  • p ⊢ q, ¬q (if the premises had been true, the conclusion would have been true is true, but because of the triviality of the consequent)

The fault could be placed directly with the classical consequence relation, or with the counterfactuals it is intended to express. Suppose we go the second route, admitting non-trivial impossible antecedent and necessary consequent counterfactuals, perhaps along the lines Daniel Nolan suggests in ‘Impossible Worlds: a Modest Approach‘—we allow impossible worlds, and a similarity metric that is defined over possible and impossible worlds both, and evaluate counterfactuals according to the standard Lewis-Stalnaker type machinery. This move would reveal that what looks like counterfactual dependence when one attends only to possible worlds is not in fact a case of counterfactual dependence; but what the machinery of impossible worlds shows (assuming one isn’t a modal realist) is rather than the real actual dependency relations are sensitive to hyperintensional differences that can’t be discriminated in standard possible worlds semantics.

If we go with the first option, however, we get something like relevant logic on the Australian plan. We don’t go in for impossible worlds or justifying relevant logic in terms of impossible premise sets. Rather, we set up a semantics that shows up various apparent classical dependencies as spurious, while retaining as much as we can of the orthodox framework:

the Australian plan is strictly 2-valued. Every proposition is, as promised, exactly one of true, false. The same holds, in our Kripke-style semantic modelling, of sentences (at worlds). But there is a price to be paid for the 2-valued clarity of the Australian plan, and it is paid on the truth-condition for DeMorgan ¬. This condition says that ¬A is true at w just in case A is false at the dual world w*. (Meyer and Martin: p. 307)

This approach—that negation is a modal operator—means that A ∧ ¬A can be true at a world even though there are no impossible worlds—it is because that sentence doesn’t express a contradiction. It expresses what we might say by ‘A and it is not the case that A is not denied’—which is different from ‘A and it is not the case that A‘, because asserting A and denying it are not contradictories, unlike asserting A and rejecting it. Now you might not buy the *-semantics, and the associated distinction between denial and rejection. But the idea that English ‘not’ expresses a denial, not a logical rejection, and that this explains the variation between our ordinary opinions about the dependence of conclusions on premises and those predicted by classical logic, is not completely terrible. One way to think about this is that relevant logics, with this account of what a syntactic contradiction involves, avoid having other claims spuriously depend on those contradictions, since—in effect—a contradiction is an ordinary conjunction and like other ordinary conjunctions, directly entails only its conjuncts and what they entail.

Now both of these approaches are puzzling in some sense. For to reveal the difference between the spurious dependencies and the genuine ones that entailment is supposed to reflect, they need to appeal to impossible worlds or a non-standard understanding of contradictoriness. But this I think seems to be a representational artifact; it arises because the basic machinery that these logics share with classical logic is the use of patterns of correlation among truth values to represent dependency. Since the classical assignments don’t reflect the intuitively right dependencies, we need non-classical assignments. This can seem puzzling—relevantists start off objecting to  p, ¬p ⊢ q on the grounds of irrelevance, and end up denying that it is necessarily truth preserving. But that obscures the real lesson, which is rather: actual dependence of conclusion on premises is the essential feature of valid arguments, and these non-standard models are mere tools for helping us see which patterns of inference are—in our classical, bivalent world—those which reflect genuine dependence. It could be done by simply saying that a valid argument is necessarily truth preserving and involves genuine dependency: the approach taken via impossible worlds or reinterpretation of ¬ is supposed to be extensionally equivalent to this proposal, as well as apparently clearer since it doesn’t explicitly involve the notion of dependence directly.

I’m still no relevantist, largely because I see no real role for the notion of dependence in play, especially in light of its ugly consequences: a prominent one being that instances of disjunctive syllogism (q, ¬p ⊢ q) are such that the conclusion turns out not to depend on the premises. This is not correct, as it clearly does. Moreover I’m not convinced that the dependence captured by classical logic, robustly necessary correlation of truth value, really does give rise to spurious dependencies. There may well be other ways to explain our judgements about trivial counterfactuals—e.g., the chunking strategy Lewis uses for inconsistent fictions—that don’t need the apparatus of impossible worlds. But nevertheless, thinking of the relevantist as trying to discriminate genuine from spurious consequence makes for a better rationale for pursuing that project than those explicitly concerned with inference rather than implication.

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