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Chris Westbury [http://www.ualberta.ca/~chrisw/Westbury.html] is an assistant
professor in the Department of Psychology at the University of Alberta in Edmonton,
Alberta, Canada. His research focuses mainly on the functional structure and the
neurological underpinnings of language. He also has active research interests in statistical
representations of language, the nature of meaning, memory processes in lexical access,
and the evolution of language.
For correspondence related to this manuscript:
Chris Westbury
P220 Biological Sciences Bldg.
University Of Alberta
T6G 2E1
E-mail: [email protected]
2
Negation: Short entry
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. 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.
In trying to define 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.
3
This simple definition is fraught with many complications because there are
several ways to deny or contradict the truth value of a proposition. In the propositional
logic introduced after Aristotle by the Stoics, logical negation was defined in more
powerful and more complex manner, by allowing the negation operator to be attached not
only to the subject or single predicate of a simple proposition, but to an entire, possibly
complex, proposition. 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 complex
problem of the scope of negation, or of deciding which part of a proposition is negated.
This problem is stil not fully understood.
Negation in Natural Language
Negation, as defined as a technical tool for logicians, is not the same as the
ordinary negation as used in natural language. However, natural language negation is
also complicated. There are many apparently different forms of negation in natural
languages. Here we consider 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.
The simplest forms of negation appearing in the lexicon is the use of the word
‘no’ (or its equivalent in other languages) in its subjective or pre-logical sense, to reject
or to signal displeasure with an undesirable situation or object.
The second form of 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.
4
The third form of negation 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.
In the fourth form of negation child uses negation to comment on his or her
failure to achieve an intended goal. It has been argued that 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 fifth form of natural language negation uses negation 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 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 last form of negation to appear developmentally is the use of negation to
deny a stated utterance. 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, which is about the same
time that children are first able to put two words together.
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 alluded to above. They have re-cast negation as a positive assertion of the
5
existence of a relevant difference- that is, they have taken negation to mean ‘other than’.
This definition has a long history, and appears to have been independently formulated
many times.
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.
6
Negation: Long entry
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:
7
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.
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
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
16
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
17
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
18
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
19
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
20
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
21
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
22
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.
23
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
24
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.
25
REFERENCES
Apostel, L. (1972). Negation: The Tension Between Ontological Positivity &
Anthropological Negativity. Logique et Analyse, 15, 209-317.
Austin, J.L. (1950). Negation In: Philosophical Papers, J.O. Urmson & G.J. Warnock,
eds. London: Oxford University Press, 117-133.
Baldwin, J.M. (1928). Dictionary Of Philosophy & Psychology, vol. 2, 146-149. New
York: MacMillan.
Bergson, H.. (1911). Creative Evolution. A. Mitchell, Trans. New York: Modern Library.
Bloom, L. (1970). Language Development: Form & Function In Emerging Grammars.
Cambridge, MA: MIT Press.
Boesch, C. (1993). Aspects of transmission of tool-use in wild chimpanzees. In: K.
Gibson & T. Ingold, eds. Tools, Language, and Cognition in Human Evolution.
Cambridge, England: Cambridge University Press, 171-184.
Bowerman, M. (1973). Early Syntactic Development. Cambridge, UK: Cambidge
University Press.
Brown, G. S. (1969). Laws Of Form. Lndon, England: George Allen and Unwin, Ltd.
Diamond, A. (1988). Differences between adult and infant cognition: Is the crucial
variable the presence or absence of language? In: L. Weiskrantz, Ed. (1988). Thought
Without Language. Oxford, Engalnd: Clarendon Press, p. 337-370.
Diamond, A. (1991). The Epigenesis Of Mind: Essays On Biology and Cognition. S.
Carey & R. Gelman, eds. Lawrence Erlbaum, p. 67-110.
26
Efran. J., Lukens, M., & Lukens, R. (1990). Language, Structure, And Change. New
York: W.W. Norton & Company.
Elster, J. (1984). Active And Passive Negation: An Essay In Ibanskian Sociology. In J.
Watzlawick, ed. (1984). The Invented Reality: How Do We Know What We Believe We
Know? (Contributions To Constructivism). New York, NY: W.W. Norton & Company.
Frege, G. (1919). Negation. In Geach & Black, Eds., Translations from the Philosophical
Writings of Gottlob Frege, Oxford: Blackwell, 117-135.
Gärdenfors, P. (1996). Cued And Detached Representations In Animal Cognition.
Behavioral Processes, 35, 263-273.
Geach, P.T. (1980). Logic Matters. Berkeley: University Of California Press.
Gibson, K. R. (1994). Tool use, language, and social behavior in relationships to
information processing capacities. In: K. Gibson & T. Ingold, eds. Tools, Language, and
Cognition in Human Evolution. Cambridge, England: Cambridge University Press, 251-
270.
Givón, T. (1979). On Understanding Grammar. New York: Academic Press.
Goodall, J. (1986). The Chimpanzees Of Gombe: Patterns Of Behavior. Cambrdige, MA:
The Belknap Press.
Gopnik, A. & Meltzoff (1985). From people, to plans, to objects: changes in the meaning
of early words and their relation to cognitive development. Journal Of Pragmatics, 9,
495-512.
Hendriks-Jansen, H. (1996). Catching Ourselves In The Act: Situated Activity, Interactive
Emergence, Evolution, and Human Thought. Cambridge, Mass.: MIT Press.
27
Hoffmeyer, J. (1993). Signs Of Meaning In The Universe. B. Haveland, Trans.
Bloomington, Ind.: Indiana University Press.
Horn, L.R. (1989). A Natural History Of Negation. The University Of Chicago Press:
Chicago, Ill.
Hummer, P., Wimmer, H. & Antes, G. (1993) On The Origins Of Denial Negation.
Journal Of Child Language Development, 20, 607-618.
Jaynes, J. (1976). The Evolution Of Language In The Late Pleistocene. Annals of the
New York Academy Of Sciences, 280, 312-325.
Jespersen, O. (1924). The Philosophy Of Grammar. London: Allen & Unwin.
La Palme Reyes, M., Macnamara, J., Reyes, G. & Zolfaghari, H. (1994). The non-
boolean logic of natural language negation. Philosophia Mathematica, 3, 45-68.
Maturana, H.R. & Varela, F.J. (1987). The Tree Of Knowledge: The Biological Roots Of
Human Understanding. Boston, MA: Shambhala Publications.
McGrew, W.C. (1977). Socialization and object-manipulation of wild chimpanzees. In:
S. Chevalier-Skolkinoff & Poirer (Eds.). Primate Bisocial Development. Garland: New
York.
McNeill, D. and McNeill, N. (1968). What does a child mean when he says ‘no’? In E.
Zale (Ed.) Proceedings Of the Conference On Language And Language Behavior. New
York: Appleton-Century.
Morford, J.P. & Goldin-Meadow, S. (1997). From Here and Now to There and Then: The
development Of Displaced Reference in Homesign and English. Child Development,
68:3, 420-435.
28
Nicoladis, E. (1998). First Clues To The Existence of Two Input Languages: Pragmatic
and Lexical Differentiation in a Bilingual Child. Bilingualism (In press).
Pea, R. (1980a). Logic In Early Child Language. Annals of the New York Academy Of
Sciences, 27-43.
Pea, R. (1980b). The Development Of Negation In Early Child Language. In The Social
Foundations Of Language & Thought: Essays in Honor Of Jerome S. Bruner, D.R.
Olson, Ed. 156-186. New York: W.W. Norton.
Peirce, C. (1869/1984) Grounds Of Validity of the Laws of Logic: Further Consequences
Of Four Incapacities. In: Writings of Charles S. Peirce: A Chronological Edition,
Volume 2, M. H. Fisch & C J. Kloesel, Eds. 211-241. Bloomington, Indiana: Indiana
University Press.
[This is available online at:
http://www.iupui.edu/~peirce/writings/v2/w2/w2_22/v2_22.htm]
Premack, A. & Premack, D. The Mind Of An Ape. New York, NY: W.W. Norton & Co.
Quine, W.V.O. (1981). Mathematical Logic (Revised Edition). New York: Harper &
Row.
Quine, W.V.O. (1976). Scope And Language Of Science. In: The Ways Of Paradox And
Other Essays. Cambridge, Massachusetts: Harvard University Press.
Reynolds, P.C. (1981). On The Evolution Of Human Behavior: The Argument From
Animals To Man. Berkeley, California: University Of California Press.
Royce, J. (1917). Negation. In Encyclopedia Of Religion & Ethics, J. Hastings, ed., Vol.
9, 264-271. New York: Charles Scribner’s Sons.
Ruke-Dravina, V. (1972). The emergence of affirmation and negation in child language:
29
Some universal and languge-restricted characteristics. In K. Ohnesorg (Ed.), Colloquium
Paedolinguisticum, 221-241. The Hague: Mouton.
Ryle, G. (1929). Negation. Aristotelean Society, Supplementary vol. 9, 80-86.
Russell, B. (1940). An Inquiry Into Meaning And Truth. London: George Allen & Unwin.
Russell, B. and Whitehead, A. (1910) Principia Mathematica. Cambridge, England.
Sharpe, D., Eakin, L., Saragovi, C. & Macnamara, J. (1996). Resolving Apparent
Contradictions: Adults and Preschoolers’ Ability To Cope With Non-Classical Negation.
Journal Of Child Language, 23, 675-691.
Shore, B. (1996). Culture In Mind: Cognition, Culture, And The Problem Of Meaning.
New York: Oxford University Press.
Sigwart, C. (1895). Logic, vol. 1, 2nd Ed., H. Dendy, trans. New York: MacMillan.
Stern, C. & Stern, W. (1928/1975). Die Kindersprache. Eine psychologische und
sprachtheoretische. Untersuchung. Darmstadt: Wissenschaftliche Buchgesellschaft.
Strawson, P.F. (1952). Introduction To Logical Theory. London: Methuen.
Swift, J. (1735/1977). Gulliver’s Travels. New York: Oxford Universty Press.
Tomasello. M., & Call, J. (1997). Primate Cognition. New York: Oxford University
Press.
Vauclair, J. (1984). Phylogenetic approach to object manipulation in human and ape
infants. Human Development, 27, 321-328.
Vihman, M.M. (1985). Language differentiation by the bilingual child. Journal Of Child
30
Language, 12, 297-324.
Volterra, V. & Antinucci, F. (1979). Negation In Child Language: A Pragmatic Study. In
Ochs, E. & Schiefflin, B. eds. Developmental Pragmatics. New York: Academic Press.
Wason, P.C. (1959). The Processing Of Positive and Negative Information. Quartlerly
Journal Of Experimental Psychology, 11:92-107.
Westbury, C. & Nicoladis, E. (1998). Meaning in children's first words: Implications for
a theory of lexical ontology. In: The proceedings of the 22nd Boston University
Conference On Language Development, p. 768-778.
Wilden, A. (1980). Analog And Digital Communication: On Negation, Signification,
And Meaning. In: Wilden, A. System And Structure: Essays In Communication And
Exchange. 2nd Edition. London, England: Tavistock Publications.
Wimmer, H., & Perner, J. (1983). Beliefs about beliefs: Representation and constraining
function of wrong beliefs in young children's understanding of deception. Cognition, 13,
103-128.
Wittgenstein, L. (1953). Philosophical Investigations. Oxford: Blackwell
Wood, L. (1933). The Paradox Of Negative Judgment. Philosophical Review, 42: 412-
423.