Contradictory Concepts and Skepticism
Suppose we are engaged on a project of radical interpretation of an alien population. The project is progressing nicely, with many predicates already interpreted, but then we encounter a strange case: there is a predicate “squound” that appears to be an abbreviation of “square and round” and is never assented to in the presence of ordinary square or round things. Further investigation reveals that the natives believe in the existence of round squares in a superlunary realm: their god is deemed capable of all things and one of them is the creation of round squares. There are no round squares here on earth according to this theology, but in heaven there are many splendid round squares, symbolizing that anything is possible in the afterlife. The iconography of these people includes purported pictures of round squares, admitted to be inadequate to their subject (squares with rounded corners usually). They accept that this commits them to contradictions: they believe that the objects of their veneration are both square and not square, as well as round and not round. Queried further about this eccentricity they inform us that their logicians are proponents of paraconsistent logic and dialetheism: they have no trouble reasoning with contradictions and they happily accept that some contradictions are true.  They are relaxed around inconsistency, as they are relaxed around many things; they regard hostility toward contradictions as an expression of a taboo mentality. They speak freely of their liberal ontology, making such remarks as, “When I see my first squound I will be full of joy” or “Nothing is as beautiful as a squound”. In the light of all this we interpret their predicate “squound” in the obvious way: it expresses the concept of a round square. An object satisfies this predicate if and only if it is both round and square—which logically implies that it is both round and not round and square and not square.
It is hard to deny that our natives possess and ascribe a contradictory concept. It is not that “squound” expresses no concept or merely the concept of being either round or square; it expresses precisely the concept of a round square. We may deplore this concept as illogical and wish they would dispense with it (they think we are far too uptight about concepts), but it is a concept they possess and employ. A concept may be a concept of the impossible, but that doesn’t mean it is an impossible concept, i.e. one it is impossible to possess. Thought has the freedom to incorporate concepts of the impossible, and there may be thinkers who tolerate impossibility—as we would describe the situation. For them round squares are possible, because they believe that contradictions can be true, in heaven if not on earth. Don’t we also have the concept of a round square, despite the fact that we think round squares are impossible? It is just that we think the concept doesn’t and can’t apply to anything, while they think it can and does: we would never predicate it of an object, but they don’t hesitate to (“All the squounds in heaven are made of gold”).
Might we possess contradictory concepts that we actually use and value? This has been maintained for certain concepts, namely those that lead to contradiction: the concept of truth and the concept of a set are held to be contradictory concepts. One reaction to this is to repudiate such concepts, scrubbing them from our conceptual scheme; but another reaction is to accept the contradiction as benign, holding that contradictions can be true. We have contradictory concepts and they accordingly give rise to contradictions—we can either lament this fact or learn to live with it. The question I want to raise is whether this might be the situation with respect to the concept of knowledge: is knowledge a contradictory concept? Would a radical interpreter investigating our use of “know” arrive at the conclusion that it expresses a contradictory concept—either deploring our illogicality or applauding our logical tolerance? Why might an interpreter arrive at that conclusion—what evidence might suggest it? She might arrive at it because of skepticism: on the one hand, we confidently assert that people know all sorts of things about the external world; while, on the other hand, we can be induced to accept that we don’t know any of these things. This is how we operate with the word “know”. All sorts of proposals have been made about this conflict of attitudes, but there is a simple proposal that has not (to my knowledge) been considered: that the concept of knowledge is an inconsistent concept. For the concept allows for cases in which something is both known and not known. I know there is a cup on my table, but I also don’t know this—because of the standard skeptical arguments. The sentence “I know there is a cup on my table” is thus both true and false. The skeptic merely exposes the contradictory nature of the concept of knowledge: it both accepts a certain level of justification for attributions of knowledge and also rejects that level of justification. We try to have it both ways with the concept, depending on the context, but the fact is that it leads inexorably to contradiction—because it is contradictory. That is why skepticism is so natural and effective—it simply reveals one aspect of the concept of knowledge and shows that that aspect conflicts with another aspect. I really don’t know there is a cup on the table, the skeptic says; and yet I do, says the common man. Both are right. At any rate that is the diagnosis we are in the process of considering.
As I observed, one can either be tolerant of contradiction or intolerant of it. If we take the latter position, following Aristotelian tradition, then the concept of knowledge should be abandoned as logically defective, contradictions being verboten. But the former, more lenient, approach promises a more ecumenical outcome, since it allows that ordinary attributions of knowledge and skeptical non-attributions are both true. It is true that I know there is a cup on my table and it is true that I don’t know this—the contradictory proposition is a true proposition. All we have to do is accept that contradictions can be true (as paraconsistent dialetheists do) and then we can resolve the troublesome question of skepticism to the satisfaction of all parties. The ordinary man can continue to assert that he knows this or that and the skeptic can insist that he does not: both speak the truth. They contradict each other, to be sure, but so what? Some concepts are inherently contradictory—that’s just the way it is. This looks like a good explanation of why both parties seem right—they are both right. We know and we don’t know. Similarly, certain propositions can be both true and false (according to the dialetheist), namely those that give rise to contradiction (e.g., “This statement is false”). Truth turns out to be a contradictory concept: certain sentences containing the word “true” have both truth-values. A statement can be true and not true simultaneously. If a concept is contradictory, it will give rise to contradiction; but concepts can be contradictory, so we are going to get contradiction. The question is what to do about that—reject the concept or accept it. In the case of knowledge we must either abandon the concept altogether, deeming contradictory concepts unusable, or decide to live with the contradictions, declaring them true not false. The latter approach enables us to retain a perfectly useful concept, despite its contradictoriness. As Wittgenstein might say, contradictory concepts can have a place in our form of life, performing a useful role; there is no more reason to reject them than there is to reject vague concepts, which also fail to live up to a certain logical ideal. The language-game we play with “know” produces contradictions—but that doesn’t stop the game from being played. Our natives play a language-game with “squound” even though the concept squound is contradictory; it has a role in their form of life. Why should contradictoriness put an end to that? We can keep our word “know” while accepting its contradictory character. In practice no confusion results, communication does not break down (similarly for the word “true”).
It is a further question whether there are facts or properties that are inherently contradictory. So far I have spoken only of the concept of knowledge, a certain kind of psychological attribute; I have not said that the propertyor fact of knowledge is contradictory. That is a far more daunting proposition: can reality itself contain contradictions? Our concepts may not map onto reality perfectly; they are sometimes imprecise, confused, or contradictory. The human concept of knowledge is what is contradictory (according to the position we are considering), since it implies that we both know and don’t know certain things. But there may be no objective property or fact corresponding to this concept—it is just a human construction. In fiction contradictions can easily occur; concepts, construed as human construction, can likewise harbor contradictions. If the concept of knowledge is contradictory, we can explain the powerful pull of both common sense and skepticism, whatever may be said of objective reality; but if it is not contradictory, we are left having to reject one or other of common sense or skepticism, neither of which is easy.
To boil it down to basics, the question is whether our ordinary beliefs are justified or not justified: epistemologists have assumed that these are exclusive possibilities, but it may be that both things are true—we are both justified and not justified. The reason the concept of knowledge is contradictory is that the concept of justification is contradictory, allowing a belief to be both justified and unjustified. It is not that the belief is justified with respect to one context but not with respect to another; it is that the belief both satisfies the univocal non-relative concept of justification and does not satisfy it. The concept is inherently and essentially contradictory. That doesn’t make it unusable, since beliefs really do have the attribute of being justified—it is just that they alsohave the attribute of being unjustified. Both attributes can be worth pointing out and they don’t exclude each other. Thus both common sense and skepticism can be true together.
It might be protested that this is a case in which the cure is worse than the disease. In order to save common sense from skepticism, while acknowledging the cogency of skepticism, we give up the law of non-contradiction. Ouch! I can certainly sympathize with that reaction, but I think it is worth adding this position to the range of other (unsatisfactory) positions already available and considering it on its merits. It is at least worth exploring. For anyone who sees a point to tolerating contradiction, at least in special cases, this is a possible application of that kind of logical posture. Any position will have its costs; maybe this position is the least costly, everything considered. It is surely the case that both common sense and skepticism appear true.
 The same can be said of the concept of certainty: this too is a contradictory concept. I am certain that I am sitting at my desk, but I am also not certain. I will say one of these things, but a moment later say the other. But this is not due to ambiguity or relativity or context-dependence; it is just outright logical inconsistency. If we want to keep both statements as true, we need to accept that contradictions can be true. In the case of certainty we seem particularly prone to making contradictory statements, and the solution is to accept that both statements are actually true, despite the fact that one is the negation of the other.