Contradiction and Synonymy

Contradiction and Synonymy

This is to be an essay in the philosophy of logic. Regrettably, logic today is taught as mainly formula manipulation with little attention paid to philosophical questions. I will be engaged on foundational questions, not unlike the foundations of physics (crucial but ignored by the mainstream). No doubt this is all about aspiring to be a “science” etc. In any case, my question concerns the meaning of what is called the law of noncontradiction, specifically what this law is really about. Let’s start with an obvious and familiar point: the connection between contradiction and synonymy. A sentence only counts as contradictory if it contains synonyms, as in “This is both red and not red”. The sentence “This is both a bank and not a bank” can be true if we mean different things by each occurrence of “bank”. It is not always easy to tell when contradiction is avoided because of a lack of synonymy: “This is both identical to that and not identical to that” (qualitative and numerical identity); “It is both raining here and not raining here” (uttered when moving about); “This creature is both large and not large” (relative to different classes of animal, e.g., mice or mammals in general). The same words can be used to say different things. So, we must stipulate that contradictory-seeming statements are only contradictory if words are used synonymously in them. But then any difficulties that attach to synonymy carry over to contradiction; and isn’t that a frail reed when trying to formulate the basic laws of logic? What if you are a skeptic about synonymy (like Quine) but a true believer in logic? Is logic only as solid as the concept of “saying the same thing”? If there is no such thing as synonymy, is there no such thing as the law of noncontradiction? Is it possible to detach the two questions, avoiding reliance on the concept of synonymy? It might be thought that it is: why not say something like “This is both red and not that”, where the demonstrative refers anaphorically to the previous utterance of red? Then we can express the law of noncontradiction by saying that nothing of that sort can be true—intuitively, that the same thing cannot be both red and not what I just said. I thus avoid using the word “red” twice and having to answer the question of synonymy; I use it only once and then refer back to that use. Okay, that sounds feasible, but it is contrived and artificial, a mere trick. And I am still using the notion of identity as between things said: first I say something and then I refer to my saying it. The point here is not that such notions are taboo or indefinable; it’s that we shouldn’t have to employ them in order to state the logical law in question. Also, this is surely not what I am thinking when I accept the validity of the law of noncontradiction—I am not thinking about synonymy or what is said. That is not what the law is about, or a presupposition of accepting the law. Really, it might be thought, the law is not about language at all; so why should we get tangled up in questions about meaning? The law isn’t intrinsically meta-linguistic; it’s not about what kinds of statements can be true. The law may have consequences for what statements can be true but it isn’t a law of statements.

So, what is it about? Properties, you might reply: it says that an object cannot both instantiate a property and not instantiate it (at the same time). An object cannot have the property of being red and not have this property. That sounds a lot cleaner; it doesn’t make logic subject to the vagaries of language (saying, meaning, synonymy). As a bonus, we see that the law can apply to things not expressed in language—properties that have no name and may not even be known. The basic notion is simply that an object can’t both have a property and lack it. Doesn’t that sound intuitively correct? But there are two lines of resistance: first, we still have a problem of individuation; second, is the notion of a property broad enough to capture all that we want to capture? Criteria of identity for properties are hard to come by—do we really want to make logic dependent on the success of that search? Objects can clearly have different properties at the same time; it is only identical properties that cannot both be instantiated and not instantiated at the same time. There is, for example, no difficulty in the proposition that an animal can both be a creature with a heart and not a creature with a kidney, these being distinct properties that happen to have the same extension. And don’t we want to extend the law of noncontradiction to “good” and “exists” without supposing them to denote properties in good standing? We don’t seem to be employing an ontology of countable properties in acceding to the law of noncontradiction, and we do seem to be thinking more generally than this notion normally admits.[1] But now we are running out of ideas—we cannot capture what it is that we are thinking when we accept the law in question. The open-ended generality of the law outruns our vocabulary for stating it.

Let’s try a different tack: the idea of the paradigm and its extensions. What really happens in your mind when you are persuaded that the law of noncontradiction holds good? Perhaps something like the following: you gaze at a red object and think “Nothing could be both like that and not like that”. You focus on an example and try to think of the negation of that example existing along with the example. You try to combine perceptual opposites, failing dismally. Then you say to yourself, almost as an afterthought, “And similarly for everything else”, not worrying too much about the exact analysis of that statement. Notice that you don’t even attempt to articulate what kinds things cannot both obtain and not obtain—you don’t think about predicates or senses or properties; you just think demonstratively of what you are gazing at (“like that). You leave the thought in an unarticulated form, relying on ostension. This seems an accurate enough description of the phenomenology of elementary logical thinking, but it leaves much to be desired theoretically. We want to be able to formulate the law more explicitly and rigorously, casting the alleged paradigm aside and spelling out what the respect of similarity is supposed to be between the paradigm case and all conceivable cases. That kind of thinking isn’t very…logical. It isn’t impressive, masterful, exam-ready. But it’s the best we can come up with without falling into muddy waters and limp hand waving. This suggests a scary thought (notice how much I am relying on the vernacular): you don’t really know what you are saying! The law of noncontradiction is true, and you know that it is true, but you can’t say what precisely it is. It is the basis of our reasoning but we can’t formulate it, except obliquely and inadequately—we can only approximate it. We think things like “Nature cannot contradict itself” or “Reality must be consistent”, but we can’t get any further in saying what in nature or reality is incapable of contradiction. We know that something and its opposite cannot both be, but that “something” is left unspecified. Our conviction of the truth of this thought does not depend on our making philosophical sense of synonymy and properties; it is more basic and general than that. It seems anterior to language and ontology—pure logic, as it were. The form of any possible reality. The necessary structure of the world. The way things have to be. Let’s admit it: this is pretty mysterious stuff, mystical even. It transcends what we can properly understand (the “limits of language”). We only partially grasp the meaning of the law of noncontradiction. Not that it betokens the divine, or ushers in the supernatural; but it does indicate the limits of human understanding. We only glimpse the logical truth that we fail to formulate explicitly; we don’t mentally embrace the full import of our words (this too is a puzzling phenomenon). Thus, we feel that the law of noncontradiction expresses a sort of magical exclusion: reality excludes other reality. If one thing is so, then its opposite cannotbe so—reality is necessareily selective. It won’t allow everyone into the club. Once things are thus and so, nothing can contradict how they are. Reality has the power (almost godlike) of suppressing alternatives; it has made up its mind and nothing can change that. If this object is red, then it absolutely cannot (will not) be not red: that is simply out of the question. It is not surprising that some people balk at this putative power, denying that contradictions are absolutely impossible: for what kind of power is it—what kind of brute metaphysical exclusion? It seems like dictatorial annihilation (it crushes the opposition). The opposite could have been so, but for some reason once it isn’t so it cannot get a foot in the door. It cannot exist once reality has come to a decision about its contents. Contradictory statements can coexist—one person can disagree with another—but somehow reality prohibits such largesse. Reality never disagrees with itself. This can seem arbitrary, groundless, narrow-minded, not even clearly stateable, and yet we are told it must be so. But our conceptual and epistemological position makes the situation intelligible: we don’t fully grasp the import of the logical law in question. It is, in short, a mystery, or a partial mystery. It hints at the ineffable.

The language of logic is not in the best shape either. A trip to the dictionary is somewhat disconcerting. For “contradiction” the OED gives us “a combination of statements, ideas, or features that are opposed to one another”; for “contradictory” we have “mutually opposed or inconsistent; containing inconsistent elements”. The word “contradict” comes from the Latin contradicere, meaning “speak against” and dates from the sixteenth century. It is certainly possible to contradict someone else (speak against them) and the law of noncontradiction does not forbid such acts of speech; it says that contradictions cannot be true, or cannot occur in reality. It would be better (and brisker) to use the phrase “the law of consistency” to affirm that statements and states of affairs must be consistent; we don’t want a law prohibiting disagreement between people! And surely the law existed and was recognized long before the word “contradiction” came into use. But what I find most telling here is the disjunction in the definition, especially the “feature” disjunct. Evidently, the word cannot make up its mind as between a de dicto and a de re use: statements and ideas, on the one hand, and “features” on the other. The right thing to say is that statements and ideas should not be contradictory (inconsistent) because reality cannot be contradictory (inconsistent): the de re underlies the de dicto. There cannot be a contradictory combination of features, though clearly words (and ideas) can be contradictory. The world cannot contain contradictions but language can. Use-mention confusion runs through the dictionary definition. But further, the use of “feature” suggests the difficulty I have been alluding to, namely that it is hard to find a word of sufficient generality to cover the case. Is existence or goodness a feature of things, like the contours of a person’s face? Hardly. Yet these also are subject to the law of consistency (nothing can exist and not exist at the same time, or be good and not good at the same time). Must we accept a metaphysics of “features” in order to endorse the law of consistency? Do we have any clear idea of what this word means in the present context? Better to admit that the thought outruns our means of expressing it; and the thought can be true and known to be true without being fully articulated or analyzed. The thought (our thought) is elusive and programmatic, unlike (say) “The cat sat on the mat”. It is schematic not filled in. The OED is straining to catch its generality while conceding its lack of perspicuity. We should accordingly be semantic mysterians about the (so-called) law of noncontradiction.[2]

[1] Certainly, we don’t need to be Platonic realists about universals in order to assert the law of noncontradiction (or excluded middle for that matter), and there is a danger of that if we quantify over properties. Not that we must reject such a theory, but neither should it be a necessary presupposition of basic logic. Aristotle doesn’t need Plato.

[2] I note that the words “object” and “property” are used with incredible promiscuity in philosophy, with little explication; we don’t want our logical laws to participate in such promiscuity, whatever may be said of our metaphysics. Self-evident laws of logic should not depend on dubious metaphysics, or even sound metaphysics. They are pre-metaphysical (also pre-linguistic). We might almost describe them as visceral (instinctive, primordial). They belong in the belly part of our conceptual scheme. The logician speaks from his gut.