Essentially Negative Concepts
Negation is a basic and ubiquitous element of our conceptual scheme, though it is hard to say anything illuminating about it. Beyond noting that it is a unary truth function we find little to report about the nature of negation. We feel vaguely that it consists in a certain kind of operation (it is described as an “operator”) and hence has an active character, akin perhaps to rejection, but otherwise it strikes us as elusive and puzzling. We wonder what it isexactly. However, there is no denying its central role in language and thought, and that is what I want to explore. My thesis will be that certain concepts are essentially defined by means of negation: this means that our grasp of these concepts embeds a grasp of negation. Negation enters into the analysis of certain concepts, so that these concepts could not exist without negation. In fact, negation crops up surprisingly often in conceptual analysis, structuring a wide variety of concepts (though not all), and sometimes where you would least expect it.
Let me begin with a simple illustrative example: the concept of a bachelor. By definition a bachelor is an unmarried male—that is, a male who is not married. To grasp the concept bachelor you therefore have to grasp the concept of not being a certain way–a negative property. It is the same for spinster and the generic concept single: to be single is to be a person who has the property of not being married. A single person is someone who has not gone through the process of getting married, while a married person is someone who has gone through that process (so this is a positive property). Similarly for the concepts of commission and omission: to commit an act is something positive, while to omit an act is something negative—it is not to do something. Thus the concepts of bachelor and omission are essentially negative concepts, unlike the concepts of being married or a commission. When you employ these negative concepts in thought you are thinking negatively—about what has not happened. You are invoking negation in your mental representation of reality. 
The ubiquity of the negative is apparent in the prefixes and suffixes that decorate the lexicon. In addition to the versatile “non-”, we have “un-“, “dis-”, “ir-“, and “-less” (as in “non-existent”, “unease”, “dislike”, “irrelevant”, and “bottomless”). All of these are easily paraphrased in terms of “not” and clearly express negative concepts: if I assert, “I have a bottomless dislike of the irrelevant and non-existent”, I am having an attack of the negatives. We also have such words as “nothing”, “no-one”, “nowhere”, and “never”—negative quantifiers. We use these to talk about absences, lacks, and emptiness—as in “Colorless nothings never exist” or “Bachelors are never uneasy”. A sentence may contain no explicit occurrence of “not”, but the proposition expressed can be bristling with negativity (“It is notthe case that there is a time at which males that are not married are not at ease”). Negation occurs frequently, and sometimes unobtrusively, in our thoughts and utterances.
But is any of this philosophically significant? Consider the concept of knowledge: it doesn’t on its face appear negative, but if we look more closely a negative element can be discerned, at least according to one standard analysis of the concept. Suppose we define knowledge as “non-accidentally true belief”: then what we have said, clearly, is that knowledge is true belief that is not true by accident—there must not be anything accidental in the way the belief was acquired. This is a natural response to Gettier cases: in addition to the positive conditions of truth, belief, and justification, we need an additional negative condition—that the belief not be true accidentally. Thus knowledge requires that something not be the case as well as that certain things be the case—knowledge is an essentially negative concept. It is like bachelor and not like married, like none and not like some. We might say that the concept is exclusionary in the sense that it rules something out—it insists that something not be the case. It tells us that knowledge must not be a certain way. Not so for belief: this concept does not stipulate that belief must be non-accidental or anything comparable—it is a positive concept. But knowledge requires that certain things notobtain–this is part of its analysis. Negation is internal to the concept of knowledge.
Now consider intention: when you intend to bring something about you believe that it is not already the case. You intend to cut the grass, assuming that it has not already been cut. You can’t intend to do what you know has already been done. Thus intention presupposes a negative judgment—that the intended outcome is not already the case. The process of intending begins with knowledge of what is and is not the case—of what needs to be brought about and what doesn’t. Once the agent has determined what is not the case he can form an intention to make it the case. A doctor can intend to find a cure for cancer only on the assumption that cancer does not yet have a cure; once the cure is discovered the intention withers. The concept of intention is thus an essentially negative concept: intention essentially embeds a judgment with negative content.
Perception also has negation at its heart. When you perceive something you are aware that it is not your perceiving of it: that tree I am seeing in the distance is experienced by me as not being part of my seeing. Perception involves a division into the I and the not-I (even when you are perceiving your own body, since your body is not part of your perceiving either). The object is apprehended as other, but that notion simply is the notion of what is not me. In general, intentionality embeds negation—mental states are directed to something not themselves. When I think of my absent brother I represent him as not being me; and even when I think of myself I represent myself as not being my mere act of thinking. To be conscious of something is to be aware of it as not that very act of consciousness. 
Think now of attitudes towards other people: when I interact with others I think of them as not me. They have minds like me, but their minds are not my mind. Other minds just are minds that are not mine. The problem of other minds is the problem of minds that are not my mind. So our attitudes here involve a negation, which serves to create the right conceptual gulf between oneself and others. All our psychological relations to others presuppose this basic negative judgment: love, hate, fear, sympathy, or indifference. It is because I judge that you are not me that I relate to you in the way I do. Your pain is not my pain, and that is very evident to me. You belong to the great world of that which is not myself—where everything is subject to my distancing negation. I experience reality under the capacious concept Not: the Not-I. Here negation permeates phenomenology, as well as language and thought.
There are many concepts we could consider as candidates for essential negativity, some more controversial than others; I will just mention some of these. The unconscious is what is not conscious; death is the state of not being alive; the future is what is not yet the case; the merely possible or counterfactual is what is not actual; the fictional is what is not factual; ignorance is not knowing; mystery is what is not known or knowable; refutation is showing something not to be the case; error is accepting what is not true; a fallacy is something not valid; an hallucination is a sensory state that is not veridical. I would say that all these are essentially negative concepts. And it is notable that they are all of philosophical interest (as are the other concepts I mentioned—knowledge, intention, perception, the conception of other minds). Are negative concepts characteristically philosophically interesting? Consider the very general and abstract concepts of identity, set, and entailment—all of great philosophical interest. Identity is the relation a thing has to itself and not to anything not that thing (to paraphrase Frege): here we have a double occurrence of negation in the definition. When we think of an object as self-identical we think of it as also not identical to other objects: negation enters conspicuously into our thoughts of identity. Identity, difference, and negation are tightly connected concepts. In the case of sets, we define a set as a collection of some objects and not others: the set of tigers includes all tigers, but it excludes elephants and lions (and indeed anything other than tigers). To be a member of a set an object has to meet a certain condition; anything not meeting that condition fails of membership. So thoughts of sets include thoughts of objects not in those sets—a set is something that rejects certain objects as members. Thus the concept incorporates a negative component—rather like an exclusive club (“Members only”—that is, no non-members allowed). Entailment likewise has an exclusive dimension: a proposition p entails a proposition q but not a proposition r. When we grasp the entailments of a given proposition we grasp something selective: only these propositions are entailments, not all the other propositions that populate logical space. When I survey the entailments of a proposition I recognize what follows and what does not follow—and the latter might not be obvious at first sight. Knowing what does not follow is as important as knowing what does follow—recognizing invalid inferences is as essential to logical understanding as recognizing valid inferences. Again, negation is implicated in our grasp of the concept of entailment. Logic is all about negativity: this follows, but not that. Thus we are thinking negative thoughts whenever we think of identity, sets, or entailment—we have negation on our mind.
A particularly interesting candidate for essential negativity is semantic concepts: truth, falsity, satisfaction, and denotation. It doesn’t take much argument to establish that falsity is bound up with negation: a proposition p is false if and only if not-p. We can even define falsity in terms of negation in Tarski’s style, by simply placing on the right hand side of the biconditional the negation of the sentence mentioned on the left—“Snow is white” is false if and only if it’s not the case that snow is white. The work we do with “false” we could do with “not”. This is not strictly a disquotational theory of falsity, since we don’t just drop the falsity predicate in favor of what it is predicated of; but it is a natural counterpart to the disquotational theory of truth. We could call it “the negational theory of falsity”. According to this theory, falsity is negation, more or less—falsity is what is not so. And what happens if we negate falsity? We get truth—what is not not so, i.e. what is so. The double negation of p entails p. Truth is equivalent to double negation. One might even venture to suggest that double negation provides an analysis of truth, an account of the concept.  Certainly anyone who grasps the concept of truth will understand the equivalence of “it is true that p” and “it is not the case that not-p”. So truth is bound up with negation, as much as falsity is; the three concepts hang intimately together. Truth then is an essentially negative concept in the sense that negation enters its analysis; indeed, it enters twice. Truth amounts to a double dose of negation—negation negated.
Satisfaction follows much the same pattern, being “true of” by another name. When an object satisfies a predicate we can say that it doesn’t not meet a certain condition: x satisfies “white” if and only if it’s not the case that x is not white. Again, this is something anyone who grasps predication (satisfaction) understands—the connection to negation is implicit. So this semantic concept also alludes to negation. When I grasp that an object satisfies the predicate “white” I grasp that the possibility that the object is not white is ruled out—that is, I can reject the proposition that the object is not white. Negation forms the background to my understanding of satisfaction—the family of concepts I bring to bear. Thus “true of” is subject to the same double negation construal as “true”. Even if we decline to analyze satisfaction and truth by means of double negation, we must still accept the conceptual links between these concepts.
What about denotation—does negation also insert itself into the concept of denotation? The following seems like a true thing to say: “Hesperus” denotes Phosphorus if and only if “Hesperus is identical to Phosphorus” is true. Generally: a name “a” denotes an object x if and only if “a = x” is true. Here we have identity employed to define denotation. There is no overt use of negation in this formulation, but negation hovers close by, in the concept of identity. First, to be identical is to be not different from, i.e. it is the negation of difference. Anyone who understands identity understands this: difference is non-identity and identity is non-difference—what could be plainer? Second, as noted earlier, identity leads us to negation via Frege’s dictum that identity is the relation a thing has to itself and to no other thing—with that double use of negation. We grasp the rightness of Frege’s words and negation crops up twice in those words. So the identity clause for denotation leads us quickly to infusions of negation. Denotation falls into line with truth and satisfaction in being an essentially negative concept. Not that the concept is itself a negative concept; rather, it contains negation essentially. This is surely a striking fact: the central semantic concepts are steeped in negation. Negation is in their bones.
It would be nice to have a general theory of why some concepts are essentially negative and some are not—is there something they all have in common? It would also be nice to have a better understanding of negation itself—what it is, how it arose in human thought, how the concept functions in relation to other concepts. We can say with some confidence that it is not a family resemblance concept, that it is univocal and topic-neutral, and that it is in some sense logical. It also seems to belong in a class of its own, like no other concept.  But beyond that negation is hard to pin down, despite its familiarity.
 There is a weak sense in which negation may be said to figure in the mastery of every concept, namely that in grasping any concept we also grasp what it would be for it not to apply. To grasp F you must grasp not-F. But this is not the thesis I am defending; I am defending the thesis that certain concepts—by no means all—implicitly contain negation as part of their analysis. We need to invoke negation to give the content of F itself, not its complementary concept not-F.
 Anyone familiar with Sartre’s Being and Nothingness will recognize this connection: negation is what Sartre calls a “constitutive structure of the for-itself”. Consciousness is nothingness for Sartre; so the Sartrean concept of consciousness is an essentially negative concept in my sense.
 In logic negation is grouped along with “and” and “or” as truth-functional connectives, but its uniqueness is clear: it doesn’t connect propositions; it reverses them. It turns a proposition on its head, converting it into its exact opposite. There is something aggressive and destructive about negation: it doesn’t so much create a new proposition by combination as annihilate the proposition on which it acts. In speech acts, “not” often functions as a device of rejection or prohibition. Is it too much to link negation with death (“To be or not to be”)?