Is Reference a Pseudo Relation?

Talk of reference is part of ordinary language, but to talk of a reference relation is to go beyond ordinary language. Is it to go too far? Does it embody a misconception about the nature of reference or the meaning of “refers”? Let us consider some relations in good standing. Spatial and temporal relations have a claim to be paradigms: objects and events clearly stand in spatial and temporal relations. These are real objective relations, not projections of language, observable and measurable. If one object is to the left of another, it is a certain distance to the left. Distance relations can be more or less and they are composed of sub-relations (a mile, say, is composed of shorter distances). One object can be much further away than another. Similarly with temporal relations: before and after, a century ago, in the distant future. These relations have extents and degrees. Thus we have locutions like “very far away”, “close by”, “long ago”, “just now”—as well as precise measurements using scales with units. Is the reference relation (if we are to talk that way) anything like these relations, logically speaking?  

            Reference is not a magnitude, so we cannot speak of amounts or degrees of it. Words do not differ in the quantity of reference they carry, so we cannot say that one word is more referential than another. Objects can be more or less distant from each other, but words cannot differ in how referential they are. We cannot say that one word is “very referential” or another “slightly referential”, as we can speak of objects as very distant or somewhat distant. There are no units of reference, with each relation of reference made up of more or less of these units. Is reference perhaps more like physical relations such as impact or gravity? But again we can speak of almost touching but not “almost referring”, of the degree of impact but not the “degree of referring”, of gravity as more or less intense but not of “intense reference”. Psychological relations are no different: sensations can be more or less intense and they can be measured–but not so for reference relations. Love comes in different strengths, but not reference. When we examine the relations that exist between objects we find that they admit of the kinds of characterizations mentioned, but reference doesn’t—it has no magnitude, direction, or intensity. Words refer or they don’t; there is no more or less. There is no metric of reference in the way there is a metric of space, time, and physical force.

            It is notable that reference has been compared to other relations that do admit of the kind of characterization just mentioned. Thus reference has been compared to perception, acquaintance, grasping, and reaching. But we can speak of these relations as having degrees and gradations: I can see one thing better than another, I can be more intimately acquainted with one person than another, I can grasp things more or less firmly, I can reach further than you can. These are relations that admit of such qualifications. But the idea of more or less successful or intensive reference sounds peculiar: I either do it or I don’t. Words don’t denote with degrees of success or intensity, partially or completely, as if they aim at an ideal that they can fail to meet. I can see an object clearly or obscurely, or grasp an object securely or insecurely, but I can’t refer to an object with these qualifications. I refer to Plato as well or as strongly as I refer to any person closer to me in space and time. Referring is not an act that can vary along a dimension of adequacy. I can see the Empire State building partially or in poor light, but I cannot refer to the Empire State building partially or in poor light: I just refer to it tout court. I can grasp a knife tightly or loosely, but I cannot refer to a knife tightly or loosely. So referring is not analogous to relations that can vary in these ways. Reference is more like identity, which also has no degrees and cannot be measured. It is not like the relation of being older than: one object can be much older than another, but no object is much more identical to an object than other objects are—and no word is much more referential than another. Thus we are inclined to say that reference is a non-natural relation, while those other relations are natural relations. Reference seems not to belong to the empirical world alongside spatial and temporal relations and physical and psychological relations. It seems curiously thin, insubstantial, featureless, a mere shadow or skeleton. We never see reference. It seems like a relation in name only. It cannot be observed, detected, or measured. It has no empirical depth. It lacks a nature.

            So far I have considered reference for singular terms, but there is also satisfaction—the (putative) relation between a predicate and its extension. Each object in a predicate’s extension satisfies it, so there are many relations contained in a single predicate—relations to all the objects of which it is true. The predicate radiates out to every object in its extension, scooping them up, gathering them together—the metaphors come easily. The predicate “planet”, say, encompasses all the celestial bodies that are planets—not just those in our solar system but also unknown planets in deep space. That word I just typed in quotation marks has a semantic relation to all those far-flung objects—they all satisfy it. But it is a funny kind of relation, appearing to span great distances instantaneously and undetectably. Compare the Sun: it stands in various relations to the planets—spatial, gravitational, electromagnetic. The planets vary in their distance from the Sun, moving around it in different ellipses; different degrees of gravitational force reach the different planets; light emanates from the Sun and strikes the different planets with varying intensity. These are real substantial relations—tangible and measurable. But the satisfaction relation is nothing like these: it does not vary in its amount; it cannot be detected by instruments; it does not act across space. There is no semantic force or semantic radiation or semantic separation. Reference seems like a film laid over reality, not part of reality. The Sun has genuine relations to the planets, with a depth and nature, but the word “planet” has nothing like this—it is just blankly “true of” each and every planet. It isn’t even like the members of its extension—a “picture” of them; it merely applies to them, equally and blandly. It has the mere form of a relation with none of the substance. We might call it a “quasi relation” or a “formal relation” or (more brutally) a “pseudo relation”. If it is a relation at all, it is not a relation in the full-blooded way that other relations are relations. Identity has been declared a pseudo relation because of its dissimilarity to other relations, and reference and satisfaction seem to cry out for the same appellation. They count as relations only by courtesy not by right.

            Is there anything we can say positively to capture what is special about reference and satisfaction? Here is one thing we can say: for a predicate to be satisfied by an object is for a certain sentence to be true. For example, for the predicate “planet” to be satisfied by the object Mars is for the sentence “Mars is a planet” to be true: that is, if you substitute the name “Mars” into the open sentence “x is a planet” you get a true closed sentence. This suggests a theory: what it means to say that Mars satisfies “planet” is that “Mars is a planet” is true. Call this the “substitutional theory of satisfaction”. It is modeled on the substitutional theory of quantification: what it means to make a true quantified statement is for substitutions of names into a suitable open sentence to yield a true closed sentence. The putative relation of “quantifying over” is explained in terms of the truth of closed sentences obtained by substitution. There is nothing more to that relation than the truth of such sentences. Similarly, there is nothing more to satisfaction than the truth of certain closed sentences obtained by substitution. The substitution instances are basic, logically and metaphysically. In effect, we do away with satisfaction as a real relation and replace it with the truth of substitution instances. There is no reaching out or radiating; there is just sentential truth, taken as primitive. In the case of “refers” the theory looks like this: for a name to refer to an object is for a certain identity sentence to be true. For example, for “Hesperus” to refer to Phosphorus is for the sentence “Hesperus is identical to Phosphorus” to be true. We don’t explain the truth conditions of the latter sentence by invoking reference; we explain reference by invoking the identity sentence. There is no more to reference and satisfaction than the truth of certain sentences—there are no independent notions of reference and satisfaction. These notions are intra-sentential—defined by reference to the truth of sentences (which is not defined in terms of reference and satisfaction). The substitutional theory is a deflationary theory—an anti-realist theory. Just as truth is declared a pseudo property by the deflationary theory of truth, so reference and satisfaction are declared pseudo relations by the substitutional theory of reference and satisfaction. If this kind of theory is on the right track, we have an explanation for why so-called semantic relations differ from natural relations—they are grounded in quite different kinds of fact. Natural relations are extra-sentential, but semantic relations are intra-sentential. We have a tendency to reify semantic relations, which leads us to strange metaphysical pictures and suspect metaphors, but in fact they amount to nothing more than substitution and truth. As people used to say, semantic relations are “logical constructions” out of sentences and truth; they are not primitive features of the universe.

            The spirit of the substitutional theory can be captured as follows. We have locutions that appear to state relations between whole sentences and worldly entities such as facts or states of affairs or situations. Thus we say that a sentence can “correspond” to such entities or “denote” them or “picture” them. We then wonder what the nature of these alleged relations might be, comparing them to other relations we find in nature; but this leads us to perplexity because they seem non-natural, thin, elusive. We find ourselves contemplating a metaphysics that can tolerate such peculiar relations, or we decide to reject them outright. But there is an alternative: we can explain the meaning of these locutions by reference to the truth of certain sentences. All that it means to say that a sentence corresponds to a fact or denotes an existing state of affairs is that the sentence is true: there is nothing more to the sentence “snow is white” corresponding to the fact that snow is white than that sentence being true. The putative correspondence relation vanishes on analysis into a statement asserting the truth of a sentence. We don’t analyze truth by means of correspondence; we analyze correspondence by means of truth. Thus “corresponds” expresses a pseudo relation: it appears to denote a real relation, but on closer examination it does not. The term is logically misleading, inviting mistaken reification. The fog clears when we recognize that correspondence is nothing more than the truth of a sentence: the relational term disappears under analysis. We can talk in the relational style if we like, but all we mean is captured by the non-relational paraphrase. It sounds agreeably pompous to speak of the sentence “snow is white” designating an existing state of affairs—as if we have got to the metaphysical foundations—but really this is just a misleading way of stating the banal fact that “snow is white” is true. So, at any rate, says the deflationist, and he appears to gain support from the evident oddity of semantic relations in general. Given the deflationary theory, that is just what we should expect—a deep difference between real relations and relations that are mere shadows or projections of language.

            It is not necessarily an error to bring natural relations into the general theory of reference. Reference may well be intertwined with, even dependent upon, such relations as perception, causal chains, and mental imagery. There may be supervenience between such natural relations and semantic relations. The trouble comes when we try to analyze reference in such terms: for then we find ourselves saying strange things about reference derived from truths about the underlying natural relations. If reference is perception, then we will be able to say that reference can be clear or obscure, from this angle and not that, enhanced by a microscope or telescope, etc.  If reference is having a mental image of an object, then we will be able to speak of distorted reference, vivid reference, faded reference, etc. If reference is a causal chain of reference-transmitting links, then we will be able to speak of segments of the reference relation, the history of the reference relation, the amount of energy that went into it, etc. Suppose we divide the chain into links and treat each link as a unit of measurement—call it the “Kripke”. Then we will be able to say that one chain consists of 532 Kripkes while another consists of 1567 Kripkes: we will be able to compare causal chains on a scale, and even speak of the rate of a causal chain—how many Kripkes per hour it can boast. We have a natural empirical relation here, laid out in space and time, capable of deeper characterization. But we surely don’t want to say that the naming relation itself has these kinds of properties. Does my referring to Plato with his name consist of semantic segments or divide into a certain number of Kripke units? Is someone who says “’Plato’ denotes the author of the Republic” really saying that there is a long causal chain of communicative links going all the way back in time to a certain Greek author? If so, some reference relations are longer than others, some slower than others, some more efficient than others. Such facts may be relevant to the existence of reference, but it is surely a category mistake to define reference in these terms. Do we want to say that denotation is composed of carbon atoms just because people are and people make up chains of communication? Such chains involve making noises, but does reference consist of those noises? Causation involves energy transfer, but does reference?

            Compare the relation of quantifying over, in which variables take values instead of names having denotations. This too is a semantic relation–a relation between words and things. In virtue of what does it hold? It may hold in virtue of causal connections between utterances of bound variables and objects in a domain, or maybe mental images of objects occurring simultaneously with such utterances. But do we want to say that quantifying over objects can be analyzed in these terms? If so, we will have to say that quantifying over things can be causally complex, can vary in causal complexity from case to case, can be indistinct or fleeting or fragmentary. These characterizations indeed apply to the alleged basis of the semantic relation of quantifying over, but it looks like a category mistake to apply them to the relation itself. It isn’t a relation of perception or communication or imaging.  [1] These relations have all sorts of properties that quantifying over simply doesn’t have; it abstracts away from all of that. Similarly for singular reference and satisfaction: they stand apart from any psychological or physical properties that might serve as a supervenience base. Seeing an object, say, is a complex psychological and physical relation, which admits of multiple kinds of qualification and evaluation; but referring to something you are seeing does not inherit all of that baggage. We can therefore never identify referring with seeing: seeing is just the wrong kind of relation for referring to be.

            A final point: it has been a longstanding puzzle how words can refer to non-existent objects—for how can a non-existent object stand in a (real) relation to a word? How can “Sherlock Holmes” refer to a certain brilliant but non-existent detective? He isn’t there to be referred to! But if all that it means to say that “Sherlock Holmes” refers to a brilliant detective is that the sentence “Sherlock Holmes is a brilliant detective” is true, then we have reference on the cheap, since that sentence is true. If reference is a pseudo relation, explicable in terms of substitution and truth, then we will be able to tolerate reference to such objects as Sherlock Holmes. Reference just isn’t very demanding of nature, unlike space, time, force, causation, etc. If talk of reference is just projection from talk of sentences and truth, then all that reference requires is that certain sentences be true—not that words somehow reach out with special semantic tentacles into the world outside of them. Real relations like distance and gravity require existent objects related by objective facts of nature, but reference doesn’t require anything so strong—you can happily refer to what does not exist, so long as certain sentences are true. When God created sentences and truth he had already done all he needed to do to create semantic relations; no further creative act was necessary.

Colin McGinn             

  [1] Memory belongs in this list, since it is a genuine relation that admits of the qualifications I have cited—memories can be faded, partial, distorted, and more or less vivid than other memories. The relation thinking of also belongs here. So reference differs from all of these. We can reasonably say that reference isn’t psychological.

                                                Insight and Analysis

Can conceptual analysis provide genuine insight? Can it increase our knowledge of the world? Consider the concept of breathing: how should it be analyzed? Not as absorbing oxygen by means of the lungs—that would rule out organisms that absorb some other gas and don’t have lungs (unless we define lungs as whatever organ processes the gas in question). A better definition would be: absorbing a gas that aids life—including absorbing hydrogen directly into the blood stream via a membrane. There is a generality to the concept of breathing that is not captured simply by the way organisms on earth happen to breathe. This is typical of conceptual analysis: the actual extension of the concept is a subset of its possible extension—the concept covers more than its actual instances. We have developed concepts that are more capacious than what we strictly need to describe the actual world. But the generality is limited: not just anything can be breathed–bread or wine can’t. You can only breathe a gas—but any gas can in principle be breathed. Breathing is definable (more or less) as absorbing a gas for the purposes of life.

            Then what is drinking? Not swallowing H2O down the throat, since an organism can drink other things than water (pure or impure) and doesn’t need a throat (unless we define a throat as whatever aids the intake of what is drunk). You could drink sulfuric acid and not have a long tube like the typical terrestrial throat. Rather, drinking should be defined as absorbing a liquid for the purposes of life (survival): it doesn’t matter what the liquid is and it doesn’t matter what the means of absorption are. An organism could absorb sulfuric acid through its pores and still be a drinker. It is the absorbing of liquid that defines the concept of drinking. And again, we have that combination of generality and limitation: any liquid will do, but you can’t drink bread or stones. The concept is open-ended, but also closed off.

            What about eating? The same pattern applies: it is possible to eat things that no terrestrial organism does, say coal or sand, so long as you have the right kind of digestive system; but you can’t eat gas or liquid. Nor can you eat gravity or electricity or space or time. Eating requires the intake of solids, not gases or liquids. The concept is free to include things not actually eaten, but not so free that it can include anything that exists (numbers, Cartesian souls, etc). So now we have three ordinary concepts lined up next to each other, exhibiting certain similarities and certain differences. All involve the idea of the intake and absorption of some kind of substance for the purposes of life or survival, but the substance in question has to be matter in only one of its three forms—gas, liquid, or solid. Necessarily, organisms breathe gas; necessarily, organisms drink liquid; necessarily, organisms eat solids. The actions in question must have a specific kind of object, with no crossovers allowed. That is the way the concepts are built—what they insist on.

            Given this parity, we can discern a family of concepts here, with a family resemblance (though not in Wittgenstein’s sense). Breathing, drinking, and eating are all alike—they have something in common, namely survival-directed material absorption. They are not as various as we may have initially supposed—so much so that it would be forgivable to say that breathing is a kind of eating, as we more easily say that drinking is a kind of eating. Not that breathing and drinking are literally eating, since you can only eat solids, but that the similarity between them puts breathing and eating in the same conceptual category (as distinct from defecating and fornicating). They all involve the intake of some material substance that aids the processes of life and survival. Breathing and eating may feel very different to us, but from a more abstract perspective they have the same structure—assimilation of the environment. We have achieved this insight (if that is not too strong a word) by means of conceptual analysis, and it is illuminating. We have added something to our understanding of the world and increased our knowledge. And how else could we have done that? Conceptual analysis is necessary to achieve some kinds of knowledge. In a simple case like this we can appreciate its potential. There is a structure to our concepts that invites this kind of investigation.

Implicit Quotation

In ordinary writing people are not fussy about quotation marks. A printed sentence may read, “The word red has three letters”. Logicians and philosophers of language may shudder at such violations of the rules of use and mention, but most people take them in stride. It is not that they are blind to the distinction; they just think it pedantic to insist on adding quotation marks. The same is true for speech: people don’t generally insert explicit quotation indicators in their spoken references to words; they don’t say, “The word, quote, ‘red’, unquote, has three letters”. They often leave the apparatus of quotation implicit, leaving it to the reader or hearer to supply it—and the context is generally sufficient to make it clear that a mention of a bit of language is intended. We can then pedantically step in to make the quotation explicit, thus rendering the true logical form clear. We make explicit what was implicit—we supply a linguistic analysis. That is, there is a good deal of implicit quotation in ordinary language, written and spoken. It is true that confusions can arise because of such sloppiness, and so it is useful to be on the lookout for covert quotation. In an ideal language all quotation would no doubt be explicitly signposted.

How widespread is implicit quotation? Is there perhaps a lot more of it than we realize? Consider this exchange: “I met a man last night, named Jack, who said he was a philosopher”; “This man, Jack, was he interesting?” The trained philosopher of language will spot the missing quotation marks in the first sentence—it should be “named ‘Jack’”, she will insist. But surely much the same should be said about the second sentence: it should really read, “This man, the one named ‘Jack’, was he interesting?” Or consider a baptism ceremony in which these words are pronounced, “I baptize you Seth. Now Seth, live long and prosper”. Clearly the first “Seth” should be in quotation marks, but couldn’t the second occurrence be aptly paraphrased as “child named ‘Seth’”? What about the sentence, “Hi, I’m John—I’m a philosopher”: doesn’t this mean, “Hi, my name is ‘John’—I’m a philosopher”? Notice how the aptness of the paraphrase for the first part of the sentence doesn’t carry over to the second part: it would not do to paraphrase that part as, “I’m called a ‘philosopher’” or “I’m described as a ‘philosopher’” or “I satisfy the predicate ‘philosopher’”. Here the word “philosopher” really is used not mentioned. Another example: “Roses are red”—how can it be paraphrased? We can naturally say, “The flowers called ‘roses’ are red”, but we cannot naturally say, “Roses are called ‘red’”. That is because “rose” is a name for a certain type of flower, but “red” is not a name of anything—it is a descriptive adjective. Nor do we say “Roses have the relation called “are” to the color red”.

            This suggests a hypothesis: all occurrences of names in natural languages implicitly involve quotation. The hypothesis can be applied both to proper names and common names: names of people, places, countries, etc., and names of animal species, chemical kinds, types of drug, etc. That is, we can correctly paraphrase any occurrence of a name that occurs without quotation marks by inserting quotation marks and adding a semantic predicate like “called” or “named”. Thus, “Tabby is a cat” can be paraphrased as, “The animal called ‘Tabby’ belongs to the species called ‘cat’”. We can do this because all uses of names involve implicit quotation. And the intuitive reason for this is simply that names are not descriptive: they are mere labels we stick on things to enable us to pick them out. It is quite otherwise with other words—quantifiers, connectives, predicates of color and shape. These are not names that we attach to things, mere empty labels; they supply substantive concepts—in a clear sense they have meaning. But a name is really meaningless, just a label or tag—that’s the point of name. You don’t learn anything about a person or place just by learning his or its name—all you learn is what we choose to call the person or place in question. But you learn a lot about a person or place by being told his or its size, shape, color, etc.—here you learn objective facts, not linguistic practices. There is a clear difference between names and other bits of language; and that difference is reflected in the fact that names always implicitly involve quotation.  [1]

Names are rather like those numbers you get assigned at the DMV and elsewhere, which function as temporary labels: for the nonce you are labeled, say, 67. There is no descriptive information in that number (it doesn’t correspond to your height or weight)—it is merely a convenient tag. If you say “I am 67” you make no descriptive statement; you simply state what number has been assigned to you. If an official says, “67 is disqualified”, that statement is aptly paraphrased as, “the person assigned the number 67 is disqualified”: the official refers to the number and uses an assignment predicate—as we refer to a name and use a semantic predicate. We could just as well say, “the person assigned the name ‘John Smith’” instead of simply “John Smith”. In the number assignment case we can easily see how the true logical form might become covert: instead of always saying “the person assigned the number N” speakers of the DMV dialect might abbreviate this to simply “N”—as in “67 is disqualified”. That is essentially what we do with names: we abbreviate—hence implicit quotation.

            If our hypothesis is correct, then names also function self-referentially in identity statements: “a = b” means “the designation of ‘a’ = the designation of ‘b’”. This implies that “a” and “b” have different meanings, since the analysis involves reference to two different entities—the names “a” and “b”. Thus we can solve Frege’s problem of informative identities: the names are not distinguished by their reference, nor by some supposed “mode of presentation” of that reference, but by themselves as names. What we learn when we learn that a = b is just that the name “a” refers to the same thing as the name “b”, i.e. that what we call by those two names is the same thing. The case is just like the assigned numbers case: one person may be assigned two different numbers on two different days, so that we can write down the identity sentence “67 = 102”. What this means, when made explicit, is “the person assigned 67 = the person assigned 102”. We don’t need to bring in anything beyond the conventional assignment, such as modes of presentation. Whenever one of these number names occurs it can be translated using “the person assigned number N”. In the case of ordinary proper names the translation is, “the person assigned name N”. We learn something meta-linguistic when we learn that Hesperus is Phosphorus, though the superficial form of the sentence is not meta-linguistic. Perhaps the ease of this solution to Frege’s problem confirms the general correctness of the hypothesis: for that solution follows naturally from the hypothesis.

            There has been much discussion about whether names are “directly referential”. According to the present analysis, the answer is that they are in one way but not in another. The name “a” is equivalent to a definite description “the denotation of ‘a’”, so it is no more directly referential of the bearer of the name than other descriptions. But the description embeds a quotation name of a name—and this name is directly referential. That is, the name directly refers to itself: the proposition expressed by a sentence containing the name thus contains the name (not an individual concept of the name). Names refer to themselves, and they do so directly: the name is part of its own meaning (along with the appended semantic predicate). The name names itself, and the name enters its own meaning. Again, the case is like the number labels: the phrase “the person assigned 67” contains a direct reference to the number 67, though it is not itself a directly referential term (it could be analyzed according to Russell’s theory of descriptions). The propositions expressed by sentences containing names are about names (not so for other words), and those names are constituents of the propositions expressed. Names are accordingly both directly referential and descriptive.

  [1] It might be wondered whether it is possible in principle for implicit quotation to be more widespread: could a language be completely quotational? The answer is no: a sentence consisting only of quoted expressions is not a sentence but a list (at best). The most we can expect is something along these lines: “The denotation of ‘John’ stands in the relation expressed by ‘loves’ to the denotation of ‘Mary’” for “John loves Mary”. Language is essentially non-quotational in its basic nature. Still, there might be a lot more quotation that we generally recognize.

Identity and Difference

Identity has been found problematic; difference, not so much. Identity has been judged a pseudo relation, but no one doubts that difference is a genuine relation between things. We can observe that one thing is different from another, but can we observe that a thing is identical to itself? Difference is essential to counting, but identity never gets us beyond a single entity. Do we really need the concept of identity? As a matter of definition, identity is said to be the relation a thing has to itself and to no other thing; difference is the relation a thing has to everything apart from itself. Everything is either identical to a given thing or different from it. Each thing is identical to itself and different from everything else. What exactly is the logical relationship between the concept of identity and the concept of difference? It tends to be assumed that identity is basic and difference is derivative—difference is simply non-identity or lack of identity—but what about considering it the other way round? What if we take difference as basic?

            If difference is basic, then identity is simply non-difference or lack of difference. We can express the sentence “a is identical to b” by the sentence “a is not different from b”. We start out with the concept of difference, tied to our perception of distinct objects, and then we define identity as simply the opposite of difference: it is the relation a thing x has to a thing y when x and y are not different things. We might initially suppose that everything in the world is different from everything else, but then we make a conceptual discovery and realize that objects are not different from themselves—there is another relation apart from difference, namely identity. So identity is really the absence of difference. We thought that Hesperus was different from Phosphorus, but it turns out that the two are not different—this was an illusory difference. We could express our discovery by saying, “Hesperus and Phosphorus are not different”, but we choose to introduce a shorter form of words and say, “Hesperus and Phosphorus are identical”. We don’t thereby expand expressive power; we had already said what needed to be said by saying that the two are not different.

            Thus we might offer to analyze identity in terms of difference plus negation: “identical” means “not different”. Why is this any less correct than analyzing “different” as “not identical”? In fact, it looks as if difference is more primitive in our system of concepts. Animals and young children surely make judgments of difference, but do they make judgments of identity? Someone could in principle have the concept of difference and never hit on the concept of identity, which would require conjoining difference with negation. But how could someone have the concept of identity and not have the concept of difference? Identity is the absence of difference: it is the relation a thing has to what it is not different from. I realized I was different from everyone else at an early age; it was only later that it dawned on me that I was identical to myself (funny thought). Difference is a given, a datum; but identity is more of a construction or abstraction. Identity is a sophisticated concept; difference is as plain as the nose on your face. A farmer counting his chickens needs the concept of difference; the philosopher explores the concept of identity. To reach the concept of identity you need to combine difference with negation—not a trivial operation.

            The case might be compared to truth and falsehood. Philosophers tend to concentrate on truth, leaving falsehood to take care of itself, but a good case can be made that truth is definable in terms of falsehood and negation.  [1] Thus for a proposition to be true is for it not to be false. This defines truth in terms of falsehood and negation. Similarly, we can define identity as difference plus negation—as not being different. This is a genuine definition and it provides necessary and sufficient conditions. First we master the concept of difference; later we form the concept of identity by combining difference with negation. Identity is what holds when difference doesn’t.

            Nothing in the standard logic of identity will be sacrificed by adopting this position. We will still have reflexivity, symmetry, and transitivity.  We can still distinguish numerical and qualitative identity: some things are numerically different without being qualitatively different, or not different qualitatively while different numerically. Leibniz’ Law can simply be reformulated to read, “If a is not different from b, then a and b have all their properties in common”. It might be a good idea to reform the symbolism we use to express claims of identity and difference, because the standard symbolism makes identity out to be basic with difference coming out as the negation of identity. Thus we now have “=” and that symbol with a slash through it, or modified by “not”. Instead we could have a new sign for difference (say “^”), taken as primitive, and then introduce identity by means of negation. We will then write “Hesperus not-^Phosphorus” to express the fact that Hesperus is identical to Phosphorus, i.e. not different from Phosphorus. We can then add this sign to the usual symbols of the predicate calculus instead of “=”; formulas will then include “a ^ b” and the like. Obviously, this will be equivalent to taking “=” as primitive and defining difference by means of it and negation. But the new formulation is conceptually more perspicuous in the light of the proper order of definition.

            Understanding difference is fundamental to every cognitive relationship to the world and is present in every perception. We see difference everywhere we look. The world is packed with difference. Identity is the exception to this universal rule—not everything is distinct from everything. There is the odd case of things in relation to themselves: here there is not difference. Things assert their difference from other things, but not from themselves. This lack of self-difference is what we name “identity”.

  [1] See my “A Negative Definition of Truth”.