An Amended Minimal Principle of Contradiction

The law of non-contradiction seems self-evidently true, but it has its opponents (or, at least, opponents of its being necessary (de dicto) simpliciter). W.V.O. Quine is perhaps the most well known philosopher to call the principle into question by calling analyticity itself into question in his famous essay “Two Dogmas of Empiricism,” and suggesting that, if we’re to be thoroughgoing empiricists, we ought to adopt a principle of universal revisability (that is to say, we adopt a principle according to which absolutely any of our beliefs, however indubitable to us, should be regarded as revisable in principle, including the principle of revisability). Quine imagined that our beliefs were networked together like parts of a web in that we have beliefs to which we aren’t strongly committed, which we imagine as near the periphery of the web, which are much less costly to change than the beliefs to which we are most strongly committed, which we imagine as near the center of that web. Changing parts of the web nearer to the periphery does less to change the overall structure of the network than changing beliefs at the center of the web. Evolution has, in operating upon our cognitive faculties, selected for our tendency towards epistemic conservatism.

This, he thinks, is why we don’t mind changing our peripheral beliefs (for instance, beliefs about whether there is milk in the fridge or whether a certain economic plan would better conduce to long-term increases in GDP than a competing plan) but we stubbornly hold onto our beliefs about things like mathematics, logic, and even some basic intuitive metaphysical principles (like Parmenides’ ex nihilo nihil fit). Nevertheless, indubitability notwithstanding, if all our knowledge is empirical in principle, then everything we believe is subject to revision, according to Quine. He boldly states:

… no statement is immune to revision. Revision even of the logical law of the excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle?1

This statement is far from short-sighted on Quine’s part. Those who defend his view have suggested that even the law of non-contradiction should be regarded as revisable, especially in light of paraconsistent systems of logic in which the law of non-contradiction is neither axiomatic, nor derivable as a theorem operating within those systems. This is why Chalmers calls attention to the fact that many regard Quine’s essay “as the most important critique of the notion of the a priori, with the potential to undermine the whole program of conceptual analysis.”2 In one fell swoop Quine undermined not only Carnap’s logical positivism, but analyticity itself, and with it a host of philosophical dogmas ranging from the classical theory of concepts to almost every foundationalist epistemological system. The force and scope of his argument was breathtaking, and it continues to plague and perplex philosophers today.

More surprising still is the fact that Quine isn’t alone in thinking that every belief is revisable. Indeed, there is a significant faction of philosophers committed to naturalism and naturalized epistemology, but who think that a fully naturalized epistemology will render all knowledge empirical, and, therefore, subject to revision in principle. Michael Devitt, for instance, defines naturalism epistemologically (rather than metaphysically):

“It is overwhelmingly plausible that some knowledge is empirical, justified by experience. The attractive thesis of naturalism is that all knowledge is; there is only one way of knowing”3

Philosophical attractiveness, I suppose, is in the eye of the beholder. It should be noted, in passing, that metaphysical naturalism and epistemological naturalism are not identical. Metaphysical naturalism does not entail epistemological naturalism, and neither does epistemological naturalism entail metaphysical naturalism. I have argued elsewhere that there may not even be a coherent way to define naturalism, but at least some idea of a naturalized metaphysic can be intuitively extrapolated from science; there is, though, no intuitive way to extrapolate a naturalized epistemology from science. As Putnam puts it:

“The fact that the naturalized epistemologist is trying to reconstruct what he can of an enterprise that few philosophers of any persuasion regard as unflawed is perhaps the explanation of the fact that the naturalistic tendency in epistemology expresses itself in so many incompatible and mutually divergent ways, while the naturalistic tendency in metaphysics appears to be, and regards itself as, a unified movement.”4

Another note in passing; strictly speaking Devitt’s statement could simply entail that we do not ‘know’ any analytic truths (perhaps given some qualified conditions on knowledge), rather than that there are no analytic truths, or even that there are no knowable analytic truths. Quine, I think, is more radical insofar as he seems to suggest that there are no analytic truths at all, and at least suggests that none are possibly known. Devitt’s statement, on the other hand, would be correct even if it just contingently happened to be the case that not a single person satisfied the sufficient conditions for knowing any analytic truth.

Hilary Putnam, unfortunately writing shortly after W.V.O. Quine passed away, provided a principle which is allegedly a priori, and which, it seems, even Quine could not have regarded as revisable. Calling this the minimal principle of contradiction, he states it as:

Not every statement is both true and false”5

Putnam himself thought that this principle establishes that there is at least one incorrigible a priori truth which is believed, if at all, infallibly. Putnam shares in his own intellectual autobiography that he had objected to himself, in his notes, as follows:

“I think it is right to say that, within our present conceptual scheme, the minimal principle of contradiction is so basic that it cannot significantly be ‘explained’ at all. But that does not make it an ‘absolutely a priori truth’ in the sense of an absolutely unrevisable truth. Mathematical intuitionism, for example, represents one proposal for revising the minimal principle of contradiction: not by saying that it is false, but by denying the applicability of the classical concepts of truth and falsity at all. Of course, then there would be a new ‘minimal principle of contradiction’: for example, ‘no statement is both proved and disproved’ (where ‘proof’ is taken to be a concept which does not presuppose the classical notion of truth by the intuitionists); but this is not the minimal principle of contradiction. Every statement is subject to revision; but not in every way.”6

He writes, shortly after recounting this, that he had objected to his own objection by suggesting that “if the classical notions of truth and falsity do not have to be given up, then not every statement is both true and false.”7 This, then, had, he thought, to be absolutely unrevisable.

This minimal principle of contradiction, or some version of it, has seemed, to me, nearly indubitable, and this despite my sincerest philosophical efforts. However, as I was reflecting more deeply upon it recently I realized that it is possible to enunciate an even weaker or more minimalist (that is to say, all things being equal, more indubitable) principle. As a propaedeutic note, I observe that not everyone is agreed upon what the fundamental truth-bearers are (whether propositions, tokens, tokenings, etc.), so one’s statement, ideally, shouldn’t tacitly presuppose any particular view. Putnam’s statement seems non-committal, but I think it is possible to read some relevance into his use of the word ‘statement’ such that the skeptic may quizzaciously opine that the principle isn’t beyond contention after all. In what follows, I will use the term ‘proposition*’ to refer to any truth-bearing element in a system.

Consider that there are fuzzy logics, systems in which bivalence is denied. A fuzzy logic, briefly, is just a system in which propositions are not regarded (necessarily) as straightforwardly true or false, but as what we might think of as ‘true’ to some degree. For instance, what is the degree to which Michael is bald? How many hairs, precisely, does Michael have to have left in order to be considered one hair away from being bald? Well, it seems like for predicates like ‘bald’ there is some ambiguity about their necessary conditions. Fuzzy logic is intended to deal with that fuzziness by allowing us to assign values in a way best illustrated by example: “Michael is 0.78 bald.” That is, it is 0.78 true that Michael is bald (something like 78% true). Obviously we can always ask the fuzzy logician whether her fuzzy statement is 1.0 true (and here she either admits that fuzzy logic is embedded in something like a more conventional bivalent logic, or she winds up stuck with infinite regresses of the partiality of truths), but I digress. Let’s accept, counter-possibly, that fuzzy logics provide a viable way to deny bivalence, and thus allow us to give a principled rejection of Putnam’s principle.

Even so, I think we can amend the principle to make it stronger. Here is my proposal for an amended principle of minimal contradiction:

“Not every single proposition* has every truth value.”

I think that this is as bedrock an analytic statement as one can hope to come by. It is indubitable, incorrigible, indubitably incorrigible, and it holds true across all possible systems/logics/languages. It seems, therefore, as though it is proof-positive of analyticity in an impressively strong sense; namely, in the sense that necessity is not always model-dependent. At least one proposition* is true across all possible systems, so that it is necessary in a stronger sense than something’s merely being necessary as regarded from within some logic or system of analysis.

——

As a post-script, here are some principles I was thinking about as a result of the above lines of thought. First, consider the principle:

At least one proposition* has at least one truth-value.

To deny this is to deny oneself a system altogether. No logic, however esoteric or unconventional or counter-intuitive, can get off the ground without this presupposition.

Consider another one:

For any proposition* P, if we know/assume only about P that it is a proposition*, then P more probably than not has at least one truth-value.

I’m not certain about this last principle, but it does seem intuitive. The way to deny it, I suppose, would be to suggest that even if most propositions* were without truth-values, one could identify a sub-class of propositions with an extremely high probability of having a truth-value, and that will allow one to operate on an alternative assumption.

[Note: some of the following footnotes may be wrong and in need of fixing. Unfortunately I would need several of my books, currently in Oxford with a friend, to adequately check each reference. I usually try to be careful with my references, but here I make special note of my inability to do due diligence.]

1 W.V.O. Quine, “Two Dogmas of Empiricism,” in The Philosophical Review, Vol. 60, No.1 (Jan., 1951), 40.

2 David J. Chalmers, “Revisability and Conceptual Change in “Two Dogmas of Empiricism”.” The Journal of Philosophy 108, no. 8 (2011): 387.

3 Louise Antony, “A Naturalized Approach to the A Priori,Epistemology: An Anthology. Second Edition, Edited by Ernest Sosa, Jaegwon Kim, Jeremy Fantl and Matthew McGrath. (Oxford: Blackwell publishing, 2000), 1.

4 Hilary Putnam, “Why Reason can’t be Naturalized,” Epistemology: An Anthology. Second Edition, Edited by Ernest Sosa, Jaegwon Kim, Jeremy Fantl and Matthew McGrath. (Oxford: Blackwell publishing, 2000), 314.

5 Hilary Putnam, “There is at least one a priori Truth” Epistemology: An Anthology. Second Edition, Edited by Ernest Sosa, Jaegwon Kim, Jeremy Fantl and Matthew McGrath. (Blackwell: 2000): 585-594.

6 Auxier, Randall E., Douglas R. Anderson, and Lewis Edwin Hahn, eds. The Philosophy of Hilary Putnam. Vol. 34. (Open Court, 2015): 71.

7 Auxier, Randall E., Douglas R. Anderson, and Lewis Edwin Hahn, eds. The Philosophy of Hilary Putnam. Vol. 34. (Open Court, 2015): 71.

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Naturalism and Supernaturalism

What, exactly, is Naturalism? The naïve definition would go: Naturalism is the belief that there are no supernatural entities. What, though, are supernatural entities? The go-to example would be God, but that’s an example rather than a definition. As far as definitions go, a typical place to start is to say that a supernatural entity is anything which is empirically undetectable, or not verifiable/falsifiable by the scientific method. However, plenty of unquestionably scientific beliefs are in things which are not strictly falsifiable (such as the existence of our universe), and a ‘scientific’ view of the world often involves commitment to beliefs which aren’t strictly verifiable (such as the legitimacy of inductive reasoning, or the reality of the past). Moreover, this definition entails that moral values, the laws of logic, the fundamental principles of arithmetic (and all mathematics), aesthetic qualities, facts themselves (as model-independent truth-makers), propositions (whether necessary, contingent, or necessarily false), the (noumenal) external world, and even purely mental phenomena (eg. qualia), will all be supernatural. Science itself, it turns out, is replete with presumptions of supernaturalism according to the stipulated definition.

Alvin Plantinga once defined Naturalism as the belief that there is no such being as God, nor anything like God. I used to think that this definition was serviceable, but I have come to see that it invites some of the most egregious difficulties of all. Buddhists and Mormons may qualify as Naturalists on this definition, and mathematical Platonists may not qualify as Naturalists! Surely that can’t be right. A definition of naturalism on which it turns out that Joseph Smith is a naturalist and Frege a supernaturalist cannot be right. The notorious difficulty of defining Naturalism should now be evident. What once looked like a trivially easy task now appears to be a herculean feat; how are we to draw the line between the natural and the supernatural? To echo (mutatis mutandis) a famous saying of St. Augustine: if nobody asks me what Naturalism is, I know, but if you ask me, I do not know.

One could always suggest that the term ‘Naturalism’ has no definition precisely because concepts have no definitions. Wittgenstein’s famous suggestion that concepts like ‘GAME’ have no definition,[1] and Quine’s famous skepticism about analyticity,[2] are just two of many factors which have contributed to the recent retreat from ‘definitions’ in the philosophy of concepts.[3] This trend has led to the wide embrace of prototype theory, theory-theory, and other alternatives to the classical theory of concepts. If we must give up on definitions, it seems to me that we must largely give up on the project of analytic philosophy, and that makes me considerably uneasy; but then, I’ve always been squeamish about anti-rationalist sentiments. It may turn out we can do no better than to say something like that Naturalists adopt belief systems related by a mere family resemblance, but which cannot be neatly subsumed under one definition. I, however, (stubborn rationalist that I am) will not give up on definitions without a fight.

On the other hand, if Naturalism cannot be defined then those of us who wish to remain analytic philosophers can just cut our losses and accuse self-identifying naturalists of having an unintelligible worldview; one the expression of which involves a fundamental theoretical term for which no clear definition can be given. In other words, when somebody claims that Naturalism is true we can simply retort: “I don’t know what that means, and neither do you.” What kind of rejoinder could they give? Either they will provide us with an acceptable definition (so that we’ll have finally teased it out), or they will have to reconsider the philosophical foundations of everything they believe they believe. Win-win by my count.

In the meantime, let’s try on some definitions for size. Here’s one:

P is a naturalist =def. P is an atheist who believes that all that exists is discoverable by the scientific method.

This definition is bad for several reasons. To begin with, it isn’t clear that a Naturalist need be an atheist; why couldn’t they be a verificationist,[4] or a Wittgensteinian? It seems, at first blush, sufficient that one not believe that “God exists” is a metaphysical truth, but then it also seems wrong to say that an agnostic can be a naturalist. An agnostic is agnostic with respect to supernatural entities, but a naturalist is not. So we’re left in a quandary with respect to the first half of our definition.

The second half doesn’t fair much better. Apart from the fact that scientists routinely commit themselves to the reality of entities which are beyond the scope of strictly empirical discoverability (such as the existence of alternative space-times in a multiverse), there is an puzzle involved in stating what, precisely, qualifies as scientifically discoverable. For instance, many of the fundamental entities in particle physics are not directly empirically observable (they are, in fact, often referred to as ‘unobservable entities’), but we have good reasons to think they exist based on the hypothetico-deductive method (i.e., we know what empirical effects they would have if they did exist, and we can verify those). However, that amounts to having good scientific and empirical motivation for believing in unobservable entities. Is it impossible to have good scientific and empirical motivation for believing in ghosts, or numbers, or God? W.V.O. Quine famously stated that if he saw any empirically justifiable motivation for belief in things like God, or the soul, he would happily accept them into his ontology. In fact, in a move motivated by his commitment to his Naturalized Epistemology,[5] Quine did eventually come to accept the existence of certain abstract objects (namely, sets). Quine leaves us with two choices: either we say that even Quine wasn’t really a (metaphysical) Naturalist in the end, or we find a way to allow Naturalists to believe in things like numbers, moral values, aesthetic facts, and other things which we don’t usually think of as ‘Natural’ entities. I suggest we make use of the notion of scientific/empirical motivation; in other words, we should make room for Naturalists to work out an ontology motivated by a scientific view of the world. The only danger I foresee in that move is that if even belief in abstract objects can be scientifically motivated, it seems as though belief in God, or anything, might turn out to be possibly scientifically motivated. Nevertheless, let us consider a second definition:

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue, and that the only entities which exist are the entities to which the acceptance of a literal interpretation of science commits us.

The first half of this definition seems fine to me, so that’s some progress. The second half is problematic because it implies that constructive empiricists, for instance, are not naturalists; the constructive empiricist agrees with the scientific realist that the statements of science should be literally construed/interpreted, but that when we accept a scientific theory we commit ourselves only to (i) the observable entities posited by the theory, and (ii) the empirical adequacy of the theory. Since the constructive empiricist adopts an agnostic attitude towards unobservable entities, none of them would qualify as naturalists on the above definition. In fact, anyone who adopts any version of scientific anti-realism (including the model-dependent realism of Stephen Hawking and Leonard Mlodinow, or even structural realism) will be disqualified from the running for candidate naturalists.

Let’s try a third:

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue, and P believes in some of, and only, the entities to which a literal interpretation of science commits us.

A possible problem with this definition might be that it threatens to include solipsists (though it isn’t clear what in science, interpreted literally, would commit anyone to the existence of persons). Perhaps we should replace “entities to which the acceptance of a literal interpretation of science commits us” with something like “entities to which our best understanding of science commits us.” That might be problematic since what the best understanding of science is seems up for debate. Perhaps it should be changed to: “entities to which a legitimate interpretation of science commits us.”

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue; P believes in some of the entities to which a legitimate understanding of science commits us; P does not believe in any entities belief in which cannot be motivated by a scientific view of the world (with the possible exception of God – caveat in casu necessitas).

This definition isn’t obviously problematic. It looks to be about as good as I can do, off the top of my head. Note that if this definition is successful, then we have also found the definition of supernaturalism, since (obviously) the definition of naturalism and the definition of supernaturalism bear a symmetrical relation of dependence to one another. This still has some notable disadvantages, including that naturalists will not be able to justify believing in moral facts unless they can generate motivation for believing in them given the resources of a scientific worldview. However, those disadvantages may just come with the territory; they may be the disadvantages not of our definition, but of the philosophy of metaphysical naturalism.

One final note; the term ‘supernaturalism’ has a bit of a bad rep because it is popularly associated with things like ghosts, energies, auras, mind-reading, witchcraft and (for better or worse) a variety of religious beliefs. Because of this many philosophers have opted for using synonyms such as ‘ultra-mundane’ to refer to things like moral facts, possible worlds, necessary beings, et alia. I don’t much mind which term is used, but one advantage to retaining the use of the term ‘supernatural’ is that it helps focus our attempt to define ‘natural’ and its cognates. If we had to define the terms ‘natural’ and ‘ultra-mundane’ it might be less apparent that whatever qualifies as unnatural is going to qualify as ultra-mundane, and vice versa.

[1] Ludwig Wittgenstein, Philosophical Investigations second edition, transl. G.E.M. Anscombe (Blackwell Publishers, 1999). http://lab404.com/lang/wittgenstein.pdf

[2] W.V.O. Quine, “Two Dogmas of Empiricism,” in The Philosophical Review vol. 60, no.1 (1951): 20-43.

[3] For more see: Stephen Laurence and Eric Margolis, “Concepts and Cognitive Science,” in Concepts: Core Readings (1999): 3-81.

[4] A verificationist, I mean, ‘about’ Theism.

[5] http://iweb.langara.bc.ca/rjohns/files/2015/03/Quine_selection.pdf