Stewart Shapiro
I have been arguing of late for a kind of relativism or pluralism concerning logic (e.g., Shapiro 2014). The main thesis is that there are different logics for different mathematical structures or, to put it otherwise, there is nothing illegitimate about structures that invoke non-classical logics, and are rendered inconsistent if excluded middle is imposed. The purpose of this chapter is to explore the consequences of this view concerning a core metaphysical issue concerning logic, the extent to which logic is objective. In the philosophical literature, terms like “relativism” and “pluralism” are used in a variety of ways, and at least some of the discussion and debate on the issues appears to be bogged down because the participants do not use the terms the same way. One group of philosophers uses the word “relativism” for what another group calls “contextualism”. So, in order to avoid getting lost in cross-purposes, we need a brief preliminary concern- ing terminology.
The central sense of “relativism” about a given subject matter Φ is given by what Crispin Wright (2008) callsfolk-relativism. The slogan is: “There is no such thing as simply being Φ”. If Φ is relative, in this sense, then in order to get a truth-value for a statement in the form “a is Φ”, one must implicitly or explicitly indicate something else. A major discovery of the early twentieth century is that simultaneity and length are relative, in this sense. To get a truth-value for “a is simultaneous with b”, one needs to indicate a frame of reference. Arguably, so-called predicates of personal taste, such as “tasty” and “fun” are also folk-relative, at least in some uses. To get a truth-value for “p is tasty”, one must indicate a judge, a taster, a standard, or something like that. This folk notion of “relativism” seems to be the one treated in Chris Swoyer’s 2003 article in the Stanford Internet Encyclopedia of Philosophy. Swoyer suggests that discussions of relativism, and relativistic proposals, focus on instances of a “general relativistic schema”:
In other words, in order to formulate a relativistic proposal, one first specifies what one is talking about, the “dependent variable” Y, and then what that is alleged to be relative to, the “independent variable” X. So, according to special relativity, the dependent variable is for simultaneity and other temporal or geometric notions like “occurs before”, and phrases like “has the same length as”. The independent variable is for a reference frame. For predicates of personal taste, the independent variable is for a given taste notion and the dependent variable is for a judge or a standard (depending on the details of the proposal).
The main thesis of Beall and Restall (2006) is an instance of folk- relativism concerning logical validity. They begin with what they call the “Generalised Tarski Thesis” (p. 29):
An argument is validxif and only if, in every casexin which the premises are
true, so is the conclusion.
For Beall and Restall, the variablex ranges over types of “cases”. Classical logic results from the Generalized Tarski Thesis if “cases” are Tarskian models; intuitionistic logic results if “cases” are constructions, or stages in construc- tions (i.e., nodes in Kripke structures); and various relevant and paraconsis- tent logics result if “cases” are situations. So Beall and Restall take logical consequence to be relative to a kind of case, and the General Relativistic Schema is apt. For them, the law of excluded middle is valid relative to Tarskian models, invalid relative to construction stages (Kripke models).
Beall and Restall call their view “pluralism”, eschewing the term “relativism”:
we are notrelativists about logical consequence, or about logic as such. We do not takelogical consequence to be relative to languages, communities of inquiry, contexts, or anything else. (p. 88, emphasis in original)
It seems that Beall and Restall take “relativism” about a given subject matter to be a restriction of what we here call “folk relativism” to those cases in which the “independent variable” ranges over languages, communities of inquiry, or contexts (or something like one of those). Of course, those are the sorts of things that debates concerning, say, morality, knowledge, and modality typically turn on. Here, we do not put any restrictions on the sort of variable that the “independent variable” can range over. However, there is no need to dispute terminology. To keep things as clear as possible, I will usually refer to “folk-relativism” in the present, quasi-technical sense.1
1
John A. Burgess (2010) also attributes a kind of (folk) relativism to Beall and Restall: “For pluralism to be true, one logic must be determinately preferable to another for one clear purpose while determinately inferior to it for another. If so, why then isn’t the notion of consequence simply
I propose below, and elsewhere, a particular kind of folk-relativism for logic. The dependent variableY is for validity or logical consequence, and the independent variableX ranges over mathematical theories or, equiva- lently, structures or types of structures. The claim is that different theories/ structures have different logics.
Once it is agreed that a given word or phrase is relative, in the foregoing, folk sense, then one might want a detailed semantic account that explains this. Are we going to be contextualists, saying that the content of the term shifts in different contexts? Or some sort of full-blown assessment-sensitive relativist (aka MacFarlane (2005), (2009), (2014))? Questions of meaning, our present focus, thus come to the fore, and will be broached below. But, as construed here, folk-relativism, by itself, has no ramifications concern- ing semantics.
Briefly,pluralism about a given subject, such as truth, logic, ethics, or etiquette, is the view that different accounts of the subject are equally correct, or equally good, or equally legitimate, or perhaps even (equally) true (if that makes sense). Arguably, folk-relativism, as the term is used here, usually gives rise to a variety of pluralism, as that term is used here. All we need is that some instances of the “independent variable” in the (GRS) correspond to correct, or good, versions of the dependent variable. Define monism or logical monism to be the opposite of logical relativism/pluralism. The monist holds that there is such a thing as simply being valid – full stop. The slogan of the monist is that there is One True Logic.
1. Relativity to structure
Since the end of the nineteenth century, there has been a trend in mathematics that any consistent axiomatization characterizes a struc- ture, one at least potentially worthy of mathematical study. A key element in the development of that trend was the publication of David Hilbert’s Grundlagen der Geometrie (1899). In that book, Hilbert pro- vided (relative) consistency proofs for his axiomatization, as well as a number of independence proofs, showing that various combinations of axioms are consistent. In a brief, but much-studied correspondence, Gottlob Frege claimed that there is no need to worry about the
purpose relative” (p. 521). Burgess adds, “[p]erhaps pluralism is relativism but relativism of such a harmless kind that to use that word to promote it would dramatise the position too much.” The present label “folk-relativism” is similarly meant to cut down on dramatic effect.
consistency of the axioms of geometry, since the axioms are all true (presumably of space).2 Hilbert replied:
As long as I have been thinking, writing and lecturing on these things, I have been saying the exact reverse: if the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by them exist. This is for me the criterion of truth and existence.
The slogan, then, is that consistency implies existence.
It seems clear, at least by now, that this Hilbertian approach applies, at least approximately, to much of mathematics, if not all of it. Consistency, or some mathematical explication thereof, like satisfiability in set theory, is the only formal criterion for legitimacy – for existence if you will. Of course, one can legitimately dismiss a proposed area of mathematical study as uninteresting, or unfruitful, or inelegant, but if it is consistent, or satisfiable, then there is no further metaphysical, formal, or mathematical hoop the proposed theory must jump through before being legitimate mathematics.
But what of consistency? The crucial observation is that consistency is a matter of logic. In a sense, consistency is (folk) relative to logic: a given theory may be consistent with respect to one logic, and inconsistent with respect to another.
There are a number of interesting and, I think, fruitful theories that invoke intuitionistic logic, and are rendered inconsistent if excluded middle is added. I’ll briefly present one such here, smooth infinitesimal analysis, a sub-theory of its richer cousin, Kock–Lawvere’s synthetic differ- ential geometry (see, for example, John Bell 1998). This is a fascinating theory of infinitesimals, but very different from the standard Robinson- style non-standard analysis (which makes heavy use of classical logic). Smooth infinitesimal analysis is also very different from intuitionistic analysis, both in the mathematics and in the philosophical underpinnings. In the spirit of the Hilbertian perspective, Bell presents the theory axiomatically, albeit informally. Begin with the axioms for a field, and consider the collection of “nilsquares”, numbers n such that n2
= 0. Of course, in both classical and intuitionistic analysis, it is easy to show that 0 is the only nilsquare: if n2
= 0, then n = 0. But not here. Among the new axioms to be added, the most interesting is the principle
2
The correspondence is published in Frege (1976) and translated in Frege (1980). The passage here is in a letter from Hilbert to Frege, dated December 29, 1899.
of micro-affineness, that every function is linear on the nilsquares. Its interesting consequence is this:
Letf be a function and x a number. Then there is a unique number d such that for any nilsquare α,f (x þ α) = f x þ d α.
This number d is the derivative of f at x. As Bell (1998) puts it, the nilsquares constitute an infinitesimal region that can have an orientation, but is too short to be bent.3
It follows from the principle of micro-affineness that 0 is not the only nilsquare:
:ð8αÞðα2 ¼ 0 ! α ¼ 0Þ:
Otherwise, the value d would not be unique, for any function. Recall, however, that in any field, every element distinct from zero has a multi- plicative inverse. It is easy to see that a nilsquare cannot have a multiplica- tive inverse, and so no nilsquare is distinct from zero. In other words, there are no nilsquares other than 0:
ð8αÞ!
α2 ¼ 0 ! ::ðα ¼ 0Þ",which is just ð8αÞ!
α2 ¼ 0 ! :ðα 6¼ 0Þ" :
So, to repeat, zero is not the only nilsquare and no nilsquare is distinct from zero. Of course, all of this would lead to a contradiction if we also had (8x)(x = 0_x 6¼ 0), and so smooth infinitesimal analysis is incon- sistent with classical logic. Indeed, :(8x)(x = 0_x 6¼ 0) is a theorem of the theory (but, since the logic is intuitionist, it does not follow that (9x):(x = 0_x 6¼ 0)).
Smooth infinitesimal analysis is an elegant theory of infinitesimals, showing that at least some of the prejudice against them can be traced to the use of classical logic – Robinson’s non-standard analysis notwithstand- ing. Bell shows how smooth infinitesimal analysis captures a number of intuitions about continuity, many of which are violated in the classical theory of the reals (and also in non-standard analysis). Some of these intuitions have been articulated, and maintained throughout the history of philosophy and science, but have been dropped in the main contem- porary account of continuity, due to Cantor and Dedekind. To take one
3
It follows from the principle of micro-affineness that every function is differentiable everywhere on its domain, and that the derivative is itself differentiable, etc. The slogan is that all functions are smooth. It is perhaps misleading to call the nilsquares a region or an interval, as they have no length.
simple example, a number of historical mathematicians and philosophers followed Aristotle in holding that a continuous substance, such as a line segment, cannot be divided cleanly into two parts, with nothing created or left over. Continua have a sort of unity, or stickiness, or viscosity. This intuition is maintained in smooth infinitesimal analysis (and also in intuitionistic analysis), but not, of course, in classical analysis, which views a continuous substance as a set of points, which can be divided, cleanly, anywhere.
Smooth infinitesimal analysis is an interesting field with the look and feel of mathematics. It has attracted the attention of mainstream mathem- aticians, people whose credentials cannot be questioned. One would think that those folks would recognize their subject when they see it. The theory also seems to be useful in articulating and developing at least some conceptions of the continuum. So one would think smooth infinitesimal analysis should count as mathematics, despite its reliance on intuitionistic logic (see also Hellman 2006).
One reaction to this is to maintain monism, but to insist that intuitionistic logic, or something even weaker, is the One True Logic. Classical theories can be accommodated by adding excluded middle as a (non-logical axiom) when it is needed or wanted. The viability of this would depend on there being no theories that invoke a logic different from those two. Admittedly, I know of no examples that are as compel- ling (at least to me) as the ones that invoke intuitionistic logic. For example, I do not know of any interesting mathematical theories that are consistent with a quantum logic, but become inconsistent if the distributive principle is added. Nevertheless, it does not seem wise to legislate for future generations, telling them what logic they must use, at least not without a compelling argument that only such and such a logic gives rise to legitimate structures. One hard lesson we have learned from history is that it is dangerous to try to provide a priori, armchair arguments concerning what the future of science and mathematics must be.
If a set Γ of sentences entails a contradiction in classical, or intuitionistic, logic, then for every sentence Ψ, Γ entails Ψ. In other words, in classical and intuitionistic logic, any inconsistent theory is trivial. A logic is called paraconsistent if it does not sanction the ill-named inference of ex falso quodlibet. Typical relevance logics are paraconsistent, but there are para- consistent logics that fail the strictures of relevance. The main observation here is that with paraconsistent logics, there are inconsistent, but non- trivial theories.
If we are to countenance paraconsistent logics, then perhaps we should change the Hilbertian slogan from “consistency implies existence” to something like “non-triviality implies existence”. To transpose the themes, on this view, non-triviality is the only formal criterion for mathematical legitimacy. One might dismiss a proposed area of mathematical study as uninteresting, or unfruitful, or inelegant, but if it is non-trivial, then there is no further metaphysical, formal, or mathematical hoop the proposed theory must jump through.
To carry this a small step further, a trivial theory can be dismissed on the pragmatic ground that it is uninteresting and unfruitful (and, indeed, trivial). So the liberal Hilbertian, who countenances paraconsistent logics, might hold that there are no criteria for mathematical legitimacy. There is no metaphysical, formal, or mathematical hoop that a proposed theory must jump through. There are only pragmatic criteria of interest and usefulness.
So are there any interesting and/or fruitful inconsistent mathematical theories, invoking paraconsistent logics of course? There is indeed an indus- try of developing and studying such theories.4It is claimed that such theories may even have applications, perhaps in computer science and psychology. I will not comment here on the viability of this project, nor on how interesting and fruitful the systems may be, nor on their supposed applica- tions. I do wonder, however, what sort of argument one might give to dismiss them out of hand, in advance of seeing what sort of fruit they may bear.
The issues are complex (see Shapiro 2014). For the purposes of this chapter, I propose to simply adopt a Hilbertian perspective – either the original version where consistency is the only formal, mathematical requirement on legitimate theories, or the liberal orientation where there are no formal requirements on legitimacy at all. And let us assume that at least some non-classical theories are legitimate, without specifying which ones those are. I propose to explore the ramifications for what I take to be a longstanding intuition that logic is objective. One would think logichas to be objective, if anything is, since just about any attempt to get at the world, as it is, will depend on, and invoke, logic.
2. What is objectivity?
Intuitively, a stretch of discourse is objective if the propositions (or sentences) in it are true or false independent of human judgment,
4
See, for example, da Costa (1974), Mortensen (1995), (2010), Priest (2006), Brady (2006), Berto (2007), and the papers in Batens et al. (2000). Weber (2009) is an overview of the enterprise.
preferences, and the like. Many of the folk-relative predicates are charac- teristic of paradigm cases of non-objective discourses. Whether something is tasty, it seems, depends on the judge or standard in play at the time. So taste is not objective (or so it seems). Whether something is rude depends on the ambient location, culture, or the like. So etiquette is folk-relative and, it seems, not objective. Etiquette may not be subjective, in the sense that it is not a matter of what an individual thinks, feels, or judges, but, presumably, it is not objective either. It is not independent of human judgment, preferences, and the like.
One would be inclined to think that simultaneity and length are objective, even though both are folk-relative, given relativity. As is the case with much in philosophy (and everywhere else), it depends on what one means by “objective”. We are told that whether two events are simultaneous, and whether two rods are of the same length, depends on the perspective of the observer. Does that undermine at least some of the objectivity? But, vagueness and such aside, time and length do not seem to depend on anyone’s judgment or feelings, or preferences. A given observer can be wrong about whether events are simultaneous, even for events relative to her own reference frame.
One might say that a folk-relative predicate P is objective if, for each valuen of the independent variable, the predicate P-relative-to-n does not depend on anyone’s judgment or feelings. For example, if a given subject can be wrong aboutP-relative-to-n, then the relevant predicate is objective. However, even an established member of a given community can be wrong about what is rude in that community. But one would not think that etiquette is objective, even when restricted to a given community.
Clearly, to get any further on our issue, we do have to better articulate what objectivity is, at least for present purposes. Again, objectivity is tied to independence from human judgment, preferences, and the like. There is a trend to think of objectivity in straightforward metaphysical terms. It must be admitted that this has something going for it. The idea is that some- thing, say a concept, is objective if it is part of the fabric of reality. The metaphor is that the concept cuts nature at its joints, it is fundamental. Theodore Sider (2011) provides a detailed articulation of a view like this,