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So I’ve previously floated some remarks about Emerson’s famous quote about a foolish consistency being the hobgoblin of a narrow mind. It seems reasonable that there might be something like a foolish consistency, but is that evident reasonableness actually validated in practice? And how might a person with a logical turn of mind validate, in turn, that reasonableness in theory? I’d like to explore this subject a bit by offering some informal remarks on formal logic, the thought being that there are actually reasons for learning the latter, yet intelligent things that can be discovered pursuing the former.

Formal logic, as it is traditionally taught and interpreted, holds that a formal contradiction is the End of Days. It is the thing that causes Jason Voorhees to drop his machete, Freddy Krueger to weep like a baby, and Michael Myers to pee himself hiding under his bed. It is the gate kicked open on the Pit of Hell, the black hole that swallows the universe, the Cubs winning the Series. OK, maybe not the “black hole” thing.

A formal contradiction is basically where a proposition and its denial are simultaneously asserted, something like “P and not-P”, where “P” is an arbitrary proposition. I’ll be both mentioning and using this a bit, so I’ll compress the symbolism a bit more into “P & ~P”. The universe-ending catastrophe that permitting such a contradiction is supposed to rain down upon us is that – again, formally – using the standard rules and axioms of formal logic, any arbitrary proposition whatsoever can be “proven” from such a contradiction. Proving anything true, without even bothering with the inconvenient process of actually looking at the world, might work out for some folks, but most view this “prove anything” possibility as rather dismal with respect to the possibilities of rational inquiry.

One additional note on the proposition (really, more of an abstract possibility of a proposition) “P” above: this “P” can be arbitrarily complex. Which is to say, “P” may not represent any one, simple sentence, but rather a large collection of such sentences. And by “large collection,” I mean entire, elaborate theories. Thus, for example, the theory in physics of quantum mechanics can be thought of as a series of declarations, represented as QM = {p1, p2, p3, … }. But since the declarative sentences within the curly brackets are all being asserted as true together, the commas separating them can be viewed as logical “ands” (the “&” above.) So we can have P = p1 & p2 & p3 & …, and thus have the theory QM represented by the “single” (but rather complex) proposition P. It is still the case that “P & ~P” is a formal contradiction, but now “P” is a very complicated thing indeed.

So, with that under our belts, it is time to revisit the claim that we all probably have contradictory beliefs at work inside our cognitive frameworks; somewhere, inside all of our “heads,” some manner of one or more “P & ~P” is to be found; it may be simple or it may be complex, but it is there. Anyone with an even passing grasp of human psychology, or barely existing sense of honesty when it comes to self-reflection, must surely find this claim extremely probable. (Obviously, it is, for all effective purposes, ultimately impossible to prove.) For reasons that orthodox logicians have never adequately explained, our heads have not collectively exploded despite this logical disaster. Indeed, even amongst those of us who are most overtly self-contradictory, there seems very little inclination to agree that everything is true even though we hold some form of “P & ~P” in our cognitive framework. What is going on here? Why hasn’t the universe torn itself to pieces?

Well, a great part of the problem has to do with the orthodoxy of the orthodox approach to logic. That approach is built around what is known as “proof theory.” I’ll not go into the details here, but I will say that I am no fan of this direction in formal logic. There are vastly better tools, I have argued, including what is known as “model theory.” Thus, using some model-theoretic tools applied within what is known as “modal logic,” Nicholas Rescher and Robert Brandom proposed (back in 1979) a technique whereby inconsistencies (I’ve been using the term “contradictions,” but formally they are the same) can be “quarantined” so that the entire logical framework does not implode. Their book, The Logic of Inconsistency is, sadly, not only a little bit technical, it is a very large bit academic, and hence is all but inaccessible for anyone except those persons with access to a research library, or rather significant amount of disposable income. But the basic idea is that the “world,” or “frame,” or “model” where P is true (call it w1) can be joined with the “world,” or “frame,” or “model” where ~P is true (call it w2) such that the conjunction [P & ~P] is strictly quarantined from both w1 and w2, allowing the overall “meta-world/model” w3 = w1 + w2 to function effectively.

This looks worrisomely like compartmentalization, and one should not take that worry lightly. However, there are real differences between the two. A logical quarantine is not imposed out of ideological commitment, but because there is strong, independent evidence for both the models of w1 and w2, and a working hypothesis that allows their composition in w3 to function. Furthermore, the quarantined contradiction will be explicitly recognized, rather than promiscuously trampled over; the quarantine will itself often function as a spur to further inquiry, whereas in compartmentalization such inquiry is typically brushed aside. Even if one is convinced that there is only a difference of degree, rather than kind, between logical quarantine and illogical compartmentalization, it bears remembering that differences of degree are real differences.

For example, Bryson Brown and Graham Priest have analyzed the approach to the infinitesimal calculus that was employed in the 18th and 19th Centuries in their article “Chunk and Permeate, a paraconsistent inference strategy.” (Again, you need research library access or the monies to get beyond a paywall in order to read it.) Employing some of the formal techniques developed by Rescher and Brandom above, Bryson and Priest show how the contradiction that lay at the very root of the calculus for over 150 years could be both formalized and successfully quarantined, in a manner reflective of how scientists at the time actually operated. The contradiction had to do with the “infinitesimal,” a number that was at once too small to ever effect any “real” numbers, and yet which could accumulate to have real effects on numbers (and the world!), in contradiction with the first claim. Yet despite this contradiction, the calculus was the living heart and soul of Newtonian mechanics, one of the single most successful scientific theories ever developed. After some 150 years or so of use (that is, by the 1850’s – 1860’s) alternative approaches to the calculus had been developed by Cayley and Weierstrass that seemed to eliminate the embarrassing infinitesimal. Rather more embarrassingly, by 1960 Abraham Robinson had developed logical techniques (based upon model theory, it must be emphasized!) that eliminated even the apparent contradictions in the infinitesimal. Of course, Robinson achieved this goal by inventing an entirely new form of arithmetic and analysis, but one shouldn’t cavil.

The main lesson here, is that a foolish insistence on consistency would have crippled the intellectual development of science for centuries. Working around the “contradictions” of the infinitesimal was eminently doable, and the contradictions themselves were a spur to significant scientific and mathematical developments. One cannot casually brush aside contradictions in the manner of ideological compartmentalization, but neither need one dogmatically insist upon the extirpation of such inconsistencies if there are a sufficiently robust, independent reasons for maintaining both of the models that otherwise consistently embrace the elements of the contradictory pair.

But the truly operative term here is foolish – we must not foolishly insist upon consistency. Nevertheless, we can reasonably insist upon it, and whenever we are presented with an overt contradiction, we not only can but must demand to see why this meets the substantial requirements of rational, logical inquiry.