Weakening Classical Logic

A perhaps noteworthy aspect of the result that there is a sharp cutoff between the rich and non-rich people is that it is non-constructive. Classical logic tells you there must be some sharp cutoff, but it doesn’t tell you which.

A related feature of classical logic that you might also find puzzling is that it commits you to the law of excluded middle (LEM):

LEM. Either A or it's not the case that A

In the particular case at hand this entails that everyone is either rich or not rich. This is surprising, for let us suppose that Janet has that borderline amount of money in which it is neither clear that she's rich nor clear that she's not rich. The law of excluded middle tells us that even here, either Janet is rich or she isn't. Of course, logic alone doesn't tell us which, but what's surprising is that logic alone delivers disjunctions whose disjuncts seem to be in principle unknowable.

All of this might suggest that it is the non-constructive nature of classical logic that is responsible for this paradox, particularly the law of excluded middle which seems to play a special role in the derivation of the sorites paradoxes.

Indeed, if the dispute just boiled down to the plausibility of the law of excluded middle I think that non-classical responses to the paradoxes would have the clear upper hand. The loss of this non-obvious theorem of classical logic is, to my mind, of little importance when compared to the puzzling phenomenon of cutoff points.

The idea that the law of excluded middle is the culprit is supported by the fact that its presence suffices for a derivation of the existence of cutoff points, against some relatively natural background logic. Given the role of the law of excluded middle in the non-constructive proof of a cutoff point the most obvious alternative to classical logic to consider would be intuitionistic logic. Not only does this logic fail to have the law of excluded middle as a theorem, it has the more general property that a disjunction is only provable if one of the disjuncts is, and an existential formula is provable only if one of its instances is, making it suitable for constructive reasoning in general.

Unfortunately, although excluded middle is one way to prove the existence of a cutoff point it is not the only way. For example, it's been known for a while that one can in a certain sense prove the existence of a cutoff point in intuitionistic logic.^[4] This derivation makes no use of the law of excluded middle since it is not present in intuitionistic logic. Thus the assumptions we must relinquish to avoid the existence of sharp cutoff points must include more than just the assumption that everyone is either rich or not rich.

There are no doubt many intuitive principles that must be given up, but let me focus on a couple that I think are particularly striking:

These are sometimes called conjunctive syllogism and pseudo modus ponens respec­tively. In both cases one can prove from these principles, along with some relatively modest background logic, that there is a sharp cutoff point. For the proof, and a more detailed description of the background logic, see appendix 18.1.4

The kinds of instances of PMP that we need to appeal to can be interpreted as fol­lows. Imagine you have two people, Alice and Bob, with n and n +1 cents respectively. Then to avoid sharp cutoffs the non-classical logician must reject instances of PMP like the following:

(1)   If Alice is rich and, moreover, Bob is rich if Alice is rich, then Bob is rich.

Unlike the claim that either Alice is rich or she isn't, the denial of this principle invites the incredulous stare.^{[5] [6]} IfBob is rich if Alice is rich, and moreover Alice is rich, how on earth could Bob fail to be rich? To deny (1) borders on incoherence.

A similar puzzle can be raised against denials of CS. Imagine now a third character, Charlie, with n — 1 cents. The proof that there are sharp cutoff points now appeals to the following kinds of instances of the principle:

(2)   If Bob is rich if Alice is, and Charlie is rich if Bob is, then Charlie is rich if Alice is.

Once again, I cannot fathom how (2) could be denied, or how one could take the phenomenon of vagueness to motivate the denial of this claim. If Bob's rich if Alice is, and Charlie is rich if Bob is, then how on earth could Charlie fail to be rich if Alice is rich?

Unsurprisingly, these principles are not derivable in the logics that are standardly appealed to in the context of the sorites paradox (this includes the 3-valued Kleene and Lukasiewicz logics, infinite valued Lukasiewicz logic, and various logics that Field has appealed to solve the paradoxes of vagueness).^[7] The above shows that this is not

really an accident, since these principles feature essentially in derivations establishing the existence of sharp cutoff points.

Let me mention there are certain paraconsistent logics that do accept CS and PMP: the paraconsistent logic LP based on the dual of the three valued Kleene logic, for example.^{[8] [9]} Since this logic accepts these principles and the background logic required to derive the existence of sharp cutoffs, these logicians must accept the existence of a last rich person in a sorites for richness.

It should be acknowledged at this juncture that there are non-classical logicians who have made their peace with failures of principles such as PMP, and are quite upfront about it. Field is probably the most prominent example.^{[11] [12]} One interesting thing to note about this is that these theorists are often engaged in a more ambitious project: that of producing an all-purpose non-classical logic that not only deals with the sorites paradoxes but also accommodates the liar paradox. Field, for example, draws a number of parallels between the two paradoxes in Field [54], for example. Given this background project some comfort can be derived from the fact that there are independent reasons to be suspicious of the principle PMP—it is susceptible to a version of the liar paradox known as the ‘Curry paradox’.

Be this as it may, it is worth noting that there is no similar independent argument against CS. There are several consistent approaches to the liar paradox that can recover a significant amount of reasoning about truth without relinquishing CS.¹¹

My main point, which I think most can take on board, is that the costs of the classical/non-classical debate are frequently mischaracterized as a debate about the status of the law of excluded middle.

A final issue worth highlighting is that even if a non-classical logic can give us a satisfactory account of the sorites paradox, there are other puzzles relating to vagueness that need to be addressed before we have a fully adequate theory. One of these further puzzles is the ‘problem of the many' (see Unger [145])—a problem not evidently directly related to the sorites. There are no doubt a number of different issues that need to be addressed here, but here is something that seems initially puzzling about the non-classical theories we have been considering so far. Firstly, it seems as though the property of being at least 29,000ft is a precise property. Although the classical and non-classical logician disagree about much, one might have thought that they could surely agree about the subject matter of their investigations: which properties are vague and precise. Thus, one might have thought that the non-classical logician can agree with the classical logician about the preciseness of this property. Since precise properties satisfy the law of excluded middle we have:

Everything is either at least 29,000ft tall or not at least 29,000ft tall.

Now, all of the logics we have considered so far allow us to infer from a universal generalization, ‘everything is F^,, a specific instance, ‘a is F (indeed many endorse a stronger axiom form of this inference: that if everything is F that a is F). Thus in all of these logics we can infer from the above sentence:

Either Mt Everest is at least 29,000ft tall or Mt Everest is not at least 29,000ft tall.

Where F is being understood as the disjunctive property of either being at least 29,000ft or not at least 29,000ft, and a as Mt Everest. Now evidently the non-classical logician should not accept this conclusion because it is borderline. It follows that the non-classical logician must either make some revisions to the ordinary conception of precision, or make some revisions to the quantified logic beyond those that have to be made to accommodate the sorites paradox (which tend to be modifications to the propositional logic).

1.1