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quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle?1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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