Negation

Almost as soon as we are born, we can use negation, indicating by gesture or other behavior that we reject, exclude, or disagree with something. A few months later, when infants are just learning to talk, their first ten words almost always include a negation operator (Westbury & Nicoladis, 1998). Because it is so common and so easily-mastered, negation may seem to be a simple concept. However, it has bedeviled all efforts to be easily defined and understood. Two researchers who have studied it extensively have described negation as “curiously difficult” (Wilden, 1980) and “far from simple and transparent” (Horn, 1989). One reason for its complexity is that negation serves a wide variety of roles. A logician uses the negation operator in the process of proving a complex logical syllogism. A pre-linguistic uses gestural negation to reject the broccoli being offered her. Do such disparate uses of negation have anything in common? If so, what is it? In trying to formulate an answer to these questions by defining negation, it is useful to consider two approaches to the topic: negation as a technical tool for use in logic, and negation in natural language. We begin with the former.

Negation in logic

Classical (Aristotelean) term logic is the earliest and simplest formal logic. It is limited to single-predicate propositions that are necessarily either true or false. A single-predicate proposition is one like ‘Mary is beautiful’ or ‘Snow is red’, in which one single thing is said (whether rightly or not) to have a single characteristic, or predicate.

Negation of a proposition in term logic may be defined by listing two necessary and sufficient properties of that function with respect to an object or set, X:

i.) X and its complement must include everything, and

ii.) The intersection of X and its negation must be empty.

In simple terms, this means that what a thing is and what it is not together make up everything. Consider, for example, the proposition ‘All men are happy’. This proposition means that the set of all men that are either happy or not-happy (‘X and its complement’) contains all men, and that set of all men that are both happy and not-happy (‘the intersection of X and its negation’) contains nothing. This corresponds to what Kant would later call ‘active negation’, since the use of this form of negation is an active affirmation of the opposite of the negated term.

The astute reader will notice already that there are complications. One complication arises because there are several ways to deny or contradict the truth value of a proposition. In Aristotle’s logic, no proposition is allowed to have more than one negation operator. However, that single negation operator be attached either to the predicate (the characteristic being ascribed) or to its subject (the entity to which the characteristic is ascribed). Thus Aristotle’s term logic recognizes a second form of negation along with the one we have just considered: one can negate the subject term, as in ‘Not-man is happy’, meaning ‘Whatever is not a man is happy’.

Aristotle also recognized that one can negate the predicate term by denying it, without thereby asserting its contrary. For example, one can state ‘Man is not happy’, and mean that ‘Whatever is a man is not happy’, but not that ‘Whatever is a man is unhappy’. As a stranger noted in Plato’s dialog Sophist (§257B), the assertion that something is ‘not big’ does not necessarily mean that it is small. This corresponds to what Kant called ‘passive negation’ (see Elster, 1984), since it does not actively affirm the contrary of the negated term.

Aristotle’s logical definition of negation is further complicated by the fact that he recognized two other ways in which negation could vary: by quantity or by mode. The first distinction (quantity) captures the differences between universal predication (‘All men are not happy’), particular predication (‘Some men are not happy’), singular predication (‘I am not happy’), and indefinite predication (‘At least one man is not happy’). The second distinction (mode) captures differences in the force of the predication, which Aristotle defined as assertoric (‘All men are [or ‘are not’] happy’), apodeictic (‘All men must be [needn’t be] happy’) or problematic (‘All men may be [cannot be] happy’).

As the natural language translations indicate, all of the distinctions recognized by Aristotle can be easily (and, in most cases, are naturally) expressed in ordinary English. Despite this ease of translation, it has long been clear that Aristotle’s logical negation has a different function than natural language negation in English. English allows negation constructions that would be disallowed under the definition of negation given by classical logic (see Horn, 1989; Sharpe et al, 1996, for a detailed discussion of this matter). For example, in English it is not considered to be contradictory or improper to say that an entity is both X and not(X). We can perfectly well understand the sentences “I did and didn’t like my college”. Such contradictions are ruled out in logic, since they allow one to deduce anything at all.

In the propositional logic introduced after Aristotle by the Stoics, logical negation was defined in more powerful and more complex manner. In this propositional logic, the negation operator need not be attached only to the subject or single predicate of a simple proposition. Instead, it can be attached externally to an entire proposition, which may itself contain many predicates. Moreover, in propositional logic subjects and predicates may be quantified, by having descriptors like ‘every’ and ‘a little’ attached to them. These complications unleash the problem that Aristotle tried to control by definitional fiat when he limited negation to subject and predicates in simple propositions: the problem of the scope of negation. This is the problem of deciding which part of a proposition is being negated by any negator.

This complication bedevils ordinary language negation. Consider the denial of the proposition ‘Everybody loves somebody a little bit sometimes’. What exactly is denied is not absolutely clear. Is the denial intended to reflect that there are some people who never love anyone at all? Or that there are some people who only love a lot? Or that some people love all people a little bit all of the time? Or that no one ever loves anyone at all?

This problem of scope of the negation operator over quantified subjects and predicates is “one of the most extensively studied and least understood phenomena within the semantics of negation” (Horn, 1989). Although we cannot hope to clear up this complication here, it is important to address one aspect of it: the claim that the negation of this predicate logic is simply equivalent to assertion of falsity. Many people in both the philosophical and linguistic literatures have adopted such a view at one time. Most notably, it was adopted by Russell and Whitehead (1910) in their Principia Mathematica (for a most explicit statement, see Russell, 1940; others who have advocated a similar position include Apostel, 1972a, Givón, 1979; Pea, 1980a, Strawson, 1952).

Few contemporary logicians would equate negation with the assertion of falsity, for two reasons. One is that there is a well-defined distinction to be drawn between the syntax of negation- how a negator may be properly used and manipulated- and the semantics of negation- what a negator means. Logicians deal mainly with the syntax of a logical symbol, and the specific formal semantics prescribed by that syntax, rendering many issues of interpretation moot.

The second reason that negation cannot be associated with asserion of falsity has to do with logical levels. Russell and Whitehead’s book introduced the distinction between logical levels. It is therefore ironic that Frege (1919), Austin (1950), Quine (1951), and Geach (1980), among others, have all argued that Russell and Whitehead’s view of negation as applying to propositions is an error resulting from a confusion of logical levels. Specifically, the view confuses language with meta-language. Austin wrote that “Affirmation and negation are exactly on a level, in this sense, that no language can exist which does not contain conventions for both and that both refer to the world equally directly, not to statements about the world” (Austin, 1950, p, 128-129, emphasis added). Statements of falsity, in contrast, are necessarily statements about statements- that is, statements in a meta-language. A statement about the truth value of a proposition is therefore not a form of negation at all. It is rather a meta-statement about truth value. Negation is always an assertion about the state of the world. It is never a statement about a proposition.

This assertion is complicated by two facts that lie at the root of the confusion about whether negation is equivalent to an assertion of falsity of a proposition

i.) The fact that negation may be a statement about the act of stating a proposition, since the act of stating a proposition constitutes a factual aspect of the state of the world which may be negated like any other fact about the world, and

ii.) The fact that any proposition about the act of stating a proposition admits of a simple transformation into a statement about the stated proposition itself.

For example, consider the proposition ‘The former President Of The United States did not tell his intern to lie’. That statement is a statement about what a former President said- that it, it is a statement about the empirically-observable physical act of a human being stating a proposition aloud. The error lies in claiming that this sentence is semantically identical to the sentence ‘The proposition ‘The President told his intern to lie’ is false’, which is a proposition about a proposition. The first statement is a statement about what phonemes actually could have been heard to issue from the President’s mouth. The second is a statement about the truth value of a proposition. These cannot be semantically identical, anymore than the set of all English sentences about an elephant could be semantically identical to the elephant. One is a bunch of ordered letters, and the other is a heavy grey mammal.

A second argument against the position that natural-language negation simply negates the proposition to which it applies is given by Horn (1989, p. 58). He points out that the error in equating statements about propositions with statements about the world is very clear when we consider nondeclarative sentences. Consider a cornered criminal who throws down his gun, yelling ‘Don’t shoot!’. It is absurd to argue that this command is identical to the meta-statement ‘Let the statement ‘You shot me’ be false!’.

Quine (1976) gives a third reason that a great deal of ordinary discourse could not admit of negation as a statement of the truth-value of a proposition, in his discussion of what would be required to ‘purify’ ordinary language so that it could be considered equivalent to the formalized language of science. Quine argued that “we may begin by banishing what are known as indicator words (Goodman) or egocentric particulars (Russell): ‘I’, ‘you’, ‘this’, ‘that’, ‘here’, ‘there’, ‘now’, ‘then’ and the like”. He explained this banishment by writing:

“It is only thus…that we come to be able to speak of sentences, i.e. certain linguistic forms, as true and false. As long as indicator words are retained, it is not the sentence but only the several events of its utterance that can be said to be true or false” (p. 222. Emphasis of the final sentence added).

A great deal of ordinary speech contains indicator words of the type Quine was objecting to. Quine is pointing out that these common sentences cannot bear truth values on their own, but only bear truth when they are properly placed in their extra-logical (real world) context.

The point of this discussion is that negation, as defined as a technical tool for logicians, is not the same as the ordinary negation as used in natural language. Some logicians have tried to re-define logical negation in such a way as to capture its uses in natural language. La Palme Reyes et al (1994) defined a non-classical logical model of natural language negation. It includes two negation functions, neither of which is in its most general form equivalent to Aristotelean negation. Those two negation functions take into account the fact that objects to which one might apply negation have a structure whose components may be differentially affected by that negation. The first negation function, which La Palme Reyes et al call ‘heyting’ or strong negation, is used when the negation function applies to all components of its negated object. The second, called ‘co-heyting’ or weak negation, is used when the negation function refers to only some components of the negated object. The formal aspects of this non-classical logic have been worked out under certain highly idealized assumptions (La Palme Reyes et al, 1994). However it is not clear if or how that formal analysis could be widely applied to real life natural language uses of negation, in situations where those assumptions might not or clearly do not hold.

Let us now turn our attention to the development and use of negation in natural language.

Negation in natural language

The reader who has read this far will probably not be surprised to learn that natural language negation is also complicated. There are many apparently different forms of negation in natural languages. Natural language negation words such as the English word ‘not’ can (but need not always) function in a way that is closely analogous to the logical ‘not’ discussed above. Natural language also contains words whose assertion functions as an implicit negation of their opposite, as well as linguistic constructions which do not contain any negation markers, but which can nevertheless function as negations for pragmatic reasons. For example, the positive assertion “What Joe saw was an aircraft glittering in the moonlight” functions as a negation when uttered in response to the claim “Joe saw a UFO!”

Such complex constructions provide new means of using negation, but add no new meanings to negation. For this reason, in this section we will concentrate only upon forms of natural language negation that are explicit in the lexicon. I present six categories of natural language negation, in roughly the order they appear developmentally. Others have proposed distinctions and commonalties that would increase or decrease this number. No definite and universally agreed-upon classification exists.

i.) Negation as rejection/ emphasis of rejection of external entities

The simplest form of negation appearing in the lexicon is the use of the word ‘no’ (or its equivalent in other languages) in what Peirce (Horn, 1989, p.163) called its subjective or pre-logical sense, to reject or to signal displeasure with an undesirable situation or object. This use of negation as ‘affective-volitional function’ was identified in the earliest study of the development of negation (Stern & Stern, 1928) as the first form to appear. It is reliably present by the age of 10-14 months (Pea, 1980b). The production of the word ‘no’ plays roughly the same role for young human infants as do the gestures that often accompany it (and that appear even earlier developmentally; see Pea, 1980b; Ruke-Dravina, 1972), such as pushing away or turning the head away from an undesired object. Such a gesture, either alone or accompanied by non-linguistic verbal productions expressing displeasure, often suffices to communicate the desired message. For this reason, the production of the word ‘no’ in this situation may not necessarily be used as a rejection in itself, but may rather play a role in emphasizing the rejection already being communicated non-linguistically. I will expand on this notion in the next section.

Clearly such negation is very simple. Any animal able to recognize what it does not want- and capable of acting on that recognition- is capable of this first form of negation as a rejection of an undesirable external entity.

ii.) Negation as a statement of refusal to stop or start action

There are two superficially similar forms to negation as a rejection that, however, function pragmatically in a markedly different way from the simple rejection of external entities. Both necessarily involve an element of social manipulation, which can also, but need not necessarily, play a role in object rejection. The first form of such social negation is the use of the word ‘no’ to signal a refusal to comply with a request or command for action or for a cessation of a particular action. Such use is thereby an expression of personal preference (Royce, 1917).

Three requirements must be satisfied for this form of negation to appear. The first is that the negating organism must have the ability to associate a command with a behavior or cessation of a behavior. The second is that the negating organism’s environment must provide the means by which that command is issued in a regular manner. Although the first requirement is common enough among non-humans, the latter is not. The appearance of negation as refusal to comply with a request or command is missing is many mammals because there is a deficit in their natural social environment that makes it unnecessary for them to grasp it. We must therefore include among the necessary functionality for the appearance of these forms of negation a third requirement: the appearance in another of the ability to regularly recognize and enforce codes of behavior in the infant who is developing negation. For these reasons, this form of negation is intimately tied to social organization and environmental structure. Because of its intimate interaction with such external factors, it becomes difficult to say whether it is ‘innate’ or not.

iii) Negation as an imperative

The second of the two forms of negation that differ pragmatically from rejection of an external object is the use of the word ‘no’ as a directive to others to act differently. As well as denying a request or a command to act or cease acting, and refusing objects offered to them, young infants are able to use negation to refuse to accept the actions of others. Such denial often functions pragmatically as a command, denying one action in the hopes of producing an alternate.

iv.) Negation as a comment on one’s own unsuccessful or prohibited action

Gopnik & Meltzoff (1985) identified another form of negation, as the second stage in their three-stage model of negation leading to negation of linguistically-stated propositions. In the first stage infants use negation as a social device to refuse parental requests, as discussed above. In the second stage, a child uses negation to comment on his or her failure to achieve an intended goal. According the Gopnik and Meltzoff, the word ‘no’ becomes a cognitive device for the first time when it is used in such a manner. Many researchers have also noted early uses of negation as self-prohibition, uttered by the child when he or she is about to do something or is doing something that is prohibited. The use of negation in this manner is typically of brief duration (Pea, 1980b).

v.) Negation as scalar predication

Negation may also be used in natural language to compare or quantify scalar values.

Negation is often used for the concept of zero, or non-existence, as when we say ‘there is no way to get there from here’ or an infant notes an unexpected absence by saying ‘no car’. The general case of using negation to mark non-existence includes sub-categories that are sometimes distinguished. For example, Pea (1980b) distinguishes between disappearance negation, which is used to note something that has just disappeared, and unfulfilled expectation negation, which is used to mark the non-existence of an expected entity. Although there are individual differences in the appearance of these subtypes (Pea, 1980b), the appearance of negation as scalar predication appears reliably as the most highly developed (i.e. latest-appearing) form of negation prior to the appearance of negation of linguistic propositions.

The use of negation to mark nonexistence (in the sense of a referent not being manifest in a context where it was expected) appears very early in children’s words. In their study of sententially-expressed negation (i.e. of negation which appears after the one-word stage) McNeill and McNeill (1968) claimed that the first uses of negation among Japanese children were all uses which marked nonexistence. McNeill and McNeill claim that this finding is of particular interest because Japanese has four common forms of negation that are differentiated in the lexicon. One form functions as an assertion of non-existence, another as a denial of a previous predication, a third as an expression of rejection, and a fourth as a denial of a previous predication while implying that the speaker knows something else to be true. Note, however, that there can be no question that these infants were already displaying behavioral forms of negation by the time they put words together to form a sentence.

Negation is not only used to indicate the total absence of a quality, but can also be used to indicate a quantity less or greater than another to which it is compared. For example, to say that something is ‘not bad’ is not to say that it was entirely good, but only that it was ‘less than all bad’. In appropriate circumstances, the negation term may also indicate a greater quantity. Jespersen (1924) identified the pragmatic circumstances that allow the negation operator to function in this way. He noted that the word following ‘not’ must be strongly stressed, and a more exact statement must immediately follow the negated statement, as in the sentence: “He earns not twenty thousand, but thirty thousand dollars per game”.

The use of negation in natural language for scalar predication has a strong constraint on its use, which shows how intimately negation is tied to other cognitive functions: it can only be properly used as an expression of a departure from an expected state of affairs. Neither an infant nor an adult will use negation as a quantifier unless the value expressed thereby is or could be unexpected. As many commentators (e.g. Sigwart, 1895; Bergson, 1911; Baldwin, 1928; Ryle, 1929; Wood, 1933; Strawson, 1952) have pointed out, to assert the negation of a proposition is to imply that there is something surprising or unexpected at the proposition’s negation- to imply that some (imagined or real) interlocutor believes, or might reasonably be expected to believe, the non-negated proposition (see Horn, 1989, S1.2 for a detailed history of this ideas). To use a graphic example suggested by Givón (1979): one cannot deny that ones wife is pregnant without implying that one believes that ones listener has reason to expect that she might be. The reason for this constraint is that “It is no good knowing what something is not unless that helps to eliminate possibilities of what it is.” (Wason, 1959, p. 103). There is no use negating unless the negation is informative. This is a specific case of the more general pragmatic rule that utterances should be (or will be assumed to be) relevant (Grice, 1975; Sperber & Wilson, 1986).

vi.) Negation of stated propositions

No one disputes that negation as denial of a stated utterance is the last form of negation to appear developmentally. Indeed, since it is the only form of negation to require sentence comprehension, it is predictable from its very definition that it is likely to appear later in development than the other forms, which can all be expressed with simpler components of language.

It is remarkable that children are able to negate propositions about as soon as they can produce them. Many studies have estimated that the ability for this form of negation appears between 1.5 years to 2.5 years (Hummer, Wimmer, & Antes, 1993), which is about the same time that children are first able to put two words together.

As discussed above, the ability to negate propositions should not be treated as if it were equivalent to denial of the truth value of propositions. What infants who are just putting together words are able to do is to deny that an actual aspect of the world matches its linguistic description. If the child screams ‘No!’ upon being told that it is bath-time, it is not to deny that the sentence ‘It is bath time’ is a true sentence, nor is it to to assert the proposition ‘The sentence ‘It is bath time’ is false’. What the child is doing is denying that it is in fact a desirable plan to submerge his body in soapy water. To assert otherwise is to impose a post-literate interpretive framework upon a child who is very far from being able to understand such a framework.

Because of these considerations, there are two distinct forms of negation of sentences. The form that an infant exhibits might be termed referential negation, since the child is denying a fact of the world that has been described to him using language. Truth-functional negation – true logical negation- is a learned technical tool for which there is no evidence of innate or inevitably-developing ability. Indeed, the failure rate in college introductory logic classes suggests that truth-functional negation is extremely difficult for most human beings to grasp.

Is there a common meaning to natural language terms of negation?

The plethora of uses might make it seem that natural language negation does not admit of any simple definition that covers all cases. However, numerous philosophers have proposed the same unifying definition, that side steps many of the logical complications discussed above. They have re-cast negation as a positive assertion of the existence of a relevant difference- that is, they have taken negation to mean ‘other than’, to use the pithy expression suggested by Peirce (1869). This expression is similar to that put forth by Plato in Sophist (§257B), in which he insisted that negation was not enantion (contrary) but heteron (other). Hegel also characterized negation in a similar way (though his heavily metaphysical views on negation are unique in other respects) when he interpreted (or perhaps, as Horn, 1989, puts it, “stood on its head”) a dictum stated by Spinoza: Determinatio est negatio [Determination is negation]. Under Hegels’ reading, Spinoza’s dictum was taken as a statement of identity, meaning that every negation is a determination or limitation, and vice versa.

The definition also appears in Brown’s (1969) attempt to give a naturalized, non-mathematical account of Boolean algebra. Brown begins by taking distinction (defined as ‘perfect continence’) as his only primitive. He then proceeds to define negation in terms of distinction. He presents this as an idea he had originated himself.

Wilden (1980) also defined negation as distinction, again without mentioning any earlier proposals to do so. The fact that this principle has apparently been repeatedly independently discovered suggests that it may accurately capture the meaning of negation.

Wilden’s formulation of the definition of negation suggested that negation should be considered as a rule about how to make either/or distinctions. Any expression of negation divides the world into three parts: the negated object or set (say, X), everything else (not-X), and the system which applies the rule for drawing the distinction between X and not-X. That system itself belongs neither to X nor to not-X, but stands outside (at a higher logical level than) both, in virtue of the fact that it defines the composition of those two sets.

In discussing Wilden’s definition of negation, Hoffmeyer (1993) implicitly argues that the act of negation is equivalent to the creation of a sign, as defined by Peirce: something which stands for something to somebody in some respect. In order to assess this claim, it is necessary to understand something of the distinctions Peirce drew between three different forms of representation: iconic, indexical, and symbolic.

Iconic representation, the simplest form, is representation that occurs in virtue of a perceptual similarity between the sign and the signified, as a picture of a bird represents a bird. Indexical representation is representation in which the signifier is associated with what it signifies by correlation in space or time- i.e. in virtue of the fact that the signifier has a quality that is linked with the entity that it signifies by some cognizable relation other than perceptual similarity. Indexical representation is commonly used in animal learning studies, as when a light is paired with punishment. The important defining feature of both iconic and indexical representation is that the connection between the primary sign and the signified exists independently of the representing organism. Simplifying Peirce’s own view somewhat, we may say that the connection is objective, in the sense that an organism or machine with access only to the appropriate sensory and temporal information about the object could in theory learn to connect the signifier with the signified.

This is not the case with the third form of representation, symbolic representation. Symbolic representation is (by definition) independent of the relations that define iconic and indexical representation – similarity, contiguity, and correlation. This means that symbolic representation can be sustained in the absence of any objectively discernible relation between the structure of the sign or its production, and the signifier. Human beings with symbolic representation are able to talk about the dark side of the planet Mercury, Santa Claus’s older sister, or integrity in politics, despite the impossibility of ever having direct sensory acquaintance with these non-existent entities.

One major limitation of iconic and indexical reference is that it is not possible to use them make a statement about any entities that do not have an unambiguously perceptible existence in space and time. Such entities have no perceptible qualities in which their signifier could partake. In particular, therefore, there could be no way to use iconic or indexical reference as scalar negation, to refer to the abstract quality of a particular absence. As Wittgenstein (1953, §446) pointed out “It would be odd to say: ‘A process looks different when it happens from when it doesn’t happen.’ Or ‘A red patch looks different when it is there from when it isn’t there.'” (see also Russell, 1940).

This is why the complex forms of linguistic negation must be fundamentally symbolic. In the complex forms of linguistic negation, the boundary that marks the negated from the unnegated has no perceptible qualities of the kind that are necessary for reference by similarity or spatio-temporal contiguity (by iconic or indexical reference). The lack of relevant perceptible qualities is also what defines a symbol. Viewing a symbol ‘as if’ it stood for something requires that it be dissociated from what it actually is. There are (by definition) no hints from what a symbol is which help one decide to what it stands for (c.f. Premack and Premack’s [1983] definition of a piece of plastic as a word for their monkeys “when the properties ascribed to it are not those of the plastic but of the object it signifies” (p. 32)). Since there can be no linguistic symbolism that is not built upon negation and since negation is itself a form of symbolism, the act of negation must be the first fundamentally linguistic symbolic act. It underlies the ability of language users to use a word to stand in for something that it in no way resembles and with which it it never co-occurs.

Conclusion

It seems simple to ‘just say no’, but negation is in fact astonishingly complicated. In logic the role of negation is so complex as to have defied complete understanding despite over two thousand years of concerted effort. In natural language, negation proves impossible to bound, spilling over to take in constraints at the social and environmental levels, and to be intimately tied to deep and complex issues of memory, expectation, general cognition, and symbolic manipulation that are themselves still largely mysterious. Because of these intimate ties, the function of negation as heteron may be plausibly argued to be a fundamental building block of human language.

In Jonothan Swift’s novel Gulliver’s Travels, the hero reports of meeting, in the grand academy of Lagado, a group of nominalist philosophers. Those men contended that “Since Words are only names for Things, it would be more convenient for all Men to carry about them, such Things as were necessary to express the particular business they are to discourse on.” (Swift, 1735/1977, p. 181). This, of course, proves to be difficult for those who have much to say, since they are obliged to haul a huge bundle of objects everywhere they go. If Swift’s radical nominalists had thought about it a bit longer, they might have arrived at slightly more convenient solution that would still save their lungs from the ‘Diminution by Corrosion’ that they were trying to avoid by not speaking. Instead of carrying the individual objects themselves, they could simply carry around the means to quickly create any object they might need. Perhaps they might carry a block of soft clay with them. By this expedient they could lighten the load they had to carry while greatly extending their possible range of reference. Whoever first began to carry the clay would be capable of astonishing feats of communication, conversing easily about matters of which his fellow philosophers, having failed to load precisely the required the object into their sacks, were forced to remain silent.

The human ability to use symbolic reference differs from animal communication in an analogous fashion to the way that the clay language differs from the object language, and for an analogous reason. Whereas most animals are limited to distinguishing only those dimensions in the world that they are born ‘carrying’ or learned dimensions that have direct biological significance, human beings can construct an infinite number of dimensions. The clay that we use to construct those dimensions is negation as heteron: the ability to formulate rules about how to reliably make either/or distinctions. Although it is clear that many of the distinctions we make are made possible by language, the opposite relation holds true for some early forms of negation. Rather than being made possible by language, those forms of negation make language possible, in virtue of their role as a sine qua non of linguistic reference. Because we can carve up the world in such subtle ways, we humans have mastered our environment in ways no other animal can do. And because we can negate, we can so carve up the world.