Negation, or denial, is the opposite of affirmation. It may be something that somebody does ("I deny what you have said") or the answer "No" to a question, but its full expression is generally a sentence. One sentence or statement may be the negation or denial of another, or we may call a statement simply a negation, or a negative statement, as opposed to an affirmative one, or affirmation. A negation in the last sense will contain some sign of negation, such as the "not" in "Grass is not pink" or "Not all leaves are green," the "no" in "No Christians are communists," or the phrase "it is not the case that" in "It is not the case that grass is pink." The negation of a sentence may simply be the same sentence with "it is not the case that" prefixed to it, or it may be some simpler form equivalent to this. For example, it might be said that "It is not the case that grass is pink" is negated or denied not only by "It is not the case that it is not the case that grass is pink" but also by the plain "Grass is pink" and that "If he has shut the door, it must have been open" is negated or denied by "He could have shut it even though it was already shut."
Contradictory negation, or contradiction, is the relation between statements that are exact opposites, in the sense that they can be neither true together nor false together—for example, "Some grass is brown" and "No grass is brown." Contrary negation, or contrariety, is the relation between extreme opposites (which may very well both be false)—for example, "No grass is brown" and "All grass is brown." Incompatibility is the relation between statements that cannot both be true, whether or not they stand at opposite ends of a scale ("This is black all over" is incompatible with "This is green all over" as well as with "This is white all over"). Incompatibles imply one another's denials (what is black all over is not green all over or white all over).
Some of these technical expressions apply to terms as well as to statements. The terms black, green, and white, for example, are incompatible; nothing can be more than one of these at once, at least not at the same time, at the same point, from the same angle, and so on. There are also "negative terms," usually formed by prefixing "non" or "not" to the corresponding positive term—for instance, nonred, not-red.
The concept of negation is closely related to that of falsehood, but they are not the same. Sometimes it is the negation that is true and the corresponding affirmation that is false. But in denying a statement, we implicitly or explicitly assert that the statement in question is false, though, of course, the assertion that something is false may itself be true.
There is also a connection between the concept of negation, especially as applied to terms, and that of otherness or diversity. What is not red is other than anything that is red, and what is other than anything that is red is not red. The class of things that are other than all the things included in a given class—that is, whatever exists besides the members of that class—constitutes the remainder or complement of the given class.
Internal and External Negation
When a proposition is complex, it is often important to distinguish the negation of the proposition as a whole ("external" negation) from propositions resulting from the negation of some component or components of it ("internal" negation). The Stoics noted, for example, that the contradictory denial of an implication "If p, then q " should not be formulated as "If p, then not-q " but as "Not (if p, then q )"—"That p does not imply that q. " "If p, then q " and "If p, then not-q " are not even incompatible, although when they are both true, it follows that the component p (since it has contradictory consequences) must be false. Again "Not (p and q )," which is true as long as p and q are not true together, is not to be confused with "Not-p and not-q, " which is true only if p and q are both false and is equivalent to "Neither p nor q "—that is, "Not (p or q )." "Either not-p or not-q " is similarly equivalent to "Not (p and q )." These relations between the internal and external negations of "and" and "or" statements are called De Morgan's laws, although they were well known to the medieval Scholastics long before the birth of the nineteenth-century logician Augustus De Morgan.
Some of the distinctions made in the preceding section are now commonly treated as special cases of external and internal negation. For instance, propositions with negative terms are thought of as involving the negation, not perhaps of internal propositions strictly so called, but of internal "propositional functions" ("open sentences")—for example, "Every non-A is a non-B " may be paraphrased as "For any x, if it is not the case that x is an A, then it is not the case that x is a B "; the difference between "No A is a B, " the contrary opposite of "Every A is a B, " and the contradictory opposite of the latter, "Some A is a B " or "Not every A is a B, " is perhaps simply that between the internally negated form "For every x, if x is an A, then not (x is a B )" and the external negation "Not (for every x, if x is an A, then x is a B )." It is obviously possible to place a sign of negation either inside or outside a variety of other qualifying phrases; for example, we may distinguish "It will be the case that (it is not the case that p )" from "It is not the case that (it will be the case that p )" and "It is thought that (it is not the case that p )" from "It is not the case that (it is thought that p )."
By the use of open sentences all the varieties of negation are reduced to the placing of "not" or "it is not the case that" before some proposition or proposition like expression, the whole being either contained or not contained within some wider propositional context. This reduction assumes that with the basic singular form "x is an A " or "x ϕ 's" there is no real distinction between the internal negation "x is not an A " (or "x is a non-A ") or "x does not ϕ " and the external negation "Not (x is an A )" or "Not (x ϕ 's)." When the subject "x " is a bare "this," such an assumption is plausible, but when it is a singular description like "The present king of France," we must distinguish the internal negation "The present king of France is not bald" (which suggests that there is such a person) from the external negation "It is not the case that the present king of France is bald" (which would be true if there were no such person). The thesis that all forms of negation are reducible to a suitably placed "it is not the case that" can be maintained only if the last two cases have an implicit complexity and may be, respectively, paraphrased as "For some x, x is the sole present king of France, and it is not the case that x is bald" and "It is not the case that (for some x, x is the sole present king of France and is bald)."
It is sometimes held that no negation can be bare or mere negation and that whenever anything is denied, some positive ground of denial is assumed, and something positive is even an intended part of what is asserted. It is trivially true that even in denials, such as that grass is pink, something is made out to be the case—namely, that it is not the case that grass is pink. But something more than this is usually intended by the contention.
One thing that could be meant is that every denial must concern something which, whatever else it is not, is itself and, indeed, simply is (exists). We have seen that some types of denial—"This is not a man" and "The man next door does not smoke" (also "Some men do not lie")—do assert or presuppose the existence of a subject of the denial. But this does not seem to be the case with all forms; for example, no existing subject seems to be involved when we say that there are no fairies. Or if this is taken to mean that among existing things no fairies are to be found (thus presupposing a body of "existing things"—of values for the bound variable x in "For no x is it the case that x is a fairy"), even this positive presupposition seems absent from "There could not be round squares."
It is also sometimes said that in denying that something is red, we at least assume that it is some other color (counting white, black, and gray as colors); in denying that something is square, we assume that it is some other shape. In general (to use the terminology of W. E. Johnson), in denying that something has a "determinate" form of some "determinable" quality, we assume that it has some other determinate form of it. Sometimes a distinction is made at this point between the predication of a negative term and the simple denial of a predication; for example, it is argued that in saying that a thing is nonblue, we do assume that it is some other color but we do not assume this in simply saying that it is not blue. Others contend that we assume that a thing is some other color even in simply denying that it is blue. All denial, it is said, is implicitly restricted to some universe of discourse; if we deny that something is blue or classify it as nonblue, it is assumed that we are considering only colored things.
Against the weaker form of the theory that the predication of a negative term has positive implications which the denial of a predication does not have, it may be objected that there is no more than a verbal difference between "x is a non-B " and "Not (x is a B )." Against the stronger form the objection is that it is perfectly proper to say that virtue is not blue simply on the ground that it is not the kind of thing that could have any color at all. We must always distinguish between what we say and our reasons for saying it (otherwise, there could be no inference at all, as premises and conclusion would coalesce), and there may be diverse reasons for saying exactly the same thing of different subjects—Jones's favorite flower is not blue because it is pink, and virtue is not blue because being an abstraction, it is not colored at all. But it is perfectly true of each of these subjects, and true in the same sense, that it is not blue.
It may be answered that "This flower is blue" and "Virtue is blue" fail to be true in profoundly different ways—the former because it is false, and the latter because it is meaningless, as meaningless as, for example, "Virtue is but" would be—and, further, whereas the denial of a false statement is true, the denial of a meaningless form of words (that is, the result of attaching a negation sign to it) is itself a meaningless form of words. To this, one possible reply (made by J. M. Shorter in "Meaning and Grammar") would be to deny that the negation of a meaningless form of words is meaningless; even "Virtue is not but" might be defended as true precisely because it is not only false, but also meaningless, to say that virtue is but. Less desperately, it could be argued that "Virtue is (is not) blue" is not on a par with "Virtue is (is not) but" since the former is at least a grammatically correct sentence while the latter does not even construe. Perhaps, however, the conception of grammar that suggests this distinction is a rather superficial one. Grammar concerns what words go with what; it is not a set of commands directly fallen from heaven but reflects at least partly the feeling we already have for what does and what does not make sense. Perhaps we need only let this feeling lead us to slightly finer distinctions than the crude one between an adjective and a conjunction to see that "is (is not) blue" no more goes with "virtue" than "is (is not) but" goes with anything.
What is important is the line between falsehood (the negation of which is true) and nonsense (the negation of which is generally agreed to be only further nonsense), wherever this line be drawn. It is also important that what looks like true or false sense may on closer inspection turn out to be nonsense.
Many philosophers who have found negation a metaphysically embarrassing concept have expressed this embarrassment by denying that there are any negative facts. There are obviously negative as well as affirmative statements, but according to these philosophers, it is incredible that the nonlinguistic facts that make our statements true or false should include negative ones. (The linguistic fact that there are negative statements is, of course, not itself a negative, but a positive, fact.)
This question should not be confused with the question of whether there are objective falsehoods—that is, whether the universe contains such objects as the falsehood that Charles I died in his bed even if no one has ever believed or asserted this falsehood (whether there are falsehoods which are, as it were, waiting around to be asserted or believed, or even denied or disbelieved, just as there are facts waiting to be discovered and stated). For such objective falsehoods, if there were any, would not be facts—a fact is what is the case, not what is not the case. The present question is, rather, whether there are special facts that verify true negative statements, whether, for example, there is any such fact as the fact that Charles I did not die in his bed. There is nevertheless some connection between the two questions. For if there is any such language-independent and thought-independent fact as the fact that it is not the case that Charles 1 died in his bed, then, that Charles I died in his bed, which in itself is not a fact but a falsehood, would nevertheless seem to have some kind of existence "out there" as a constituent of this more complex object that is a fact.
In both cases, moreover, what deters the philosophers is partly the multiplicity of the objects involved. They cannot believe that there should be not only the fact that Charles I died on the scaffold but also, over and above that fact, the additional facts that he did not die in his bed, that he was not immortal, that he did not die by drowning, and, furthermore, the facts that he did not die in his bed of appendicitis, that he did not die in his bed of consumption, that he did not die by drowning in six minutes, that he did not die by drowning in six and a half minutes, and so on. This causes an embarrassment of the same sort as the idea that, over and above the fact that he died on the scaffold, there are "out there" the falsehoods that he died in his bed, that he was immortal, that he was drowned in six and a half minutes, and so on.
The most obvious way to reduce this excessive metaphysical population, and the one taken by Raphael Demos (one of the main opponents of negative facts), is to hold that what makes it false to say that Charles I died in his bed and true to say that he did not, false to say that he died by drowning and true to say that he did not, and similarly with all the other alternatives is simply the one positive fact that he died on the scaffold. Against this, however, it may be said that what is asserted by any true statement would seem to be some fact, and the true statement that Charles I did not die in his bed does not assert that he died on the scaffold (even if this is also true). It may be suggested that what the true statement asserts is that Charles died in some positive way that was incompatible with his dying in his bed. This suggestion has the disadvantage (a ) that it only exchanges negative facts for facts that are vague and general in the way that assertions about something or other (but nothing in particular) are always vague and general and that philosophers who are uneasy about the former (because whatever is real must be particular and positive) are likely to be equally uneasy about the latter. The suggestion also presupposes (b ) that there are facts of incompatibility—for example, the fact that Charles I's dying on the scaffold is incompatible with his dying in his bed and that these would seem, like straightforwardly negative facts, to contain objective falsehoods as constituents and would have the same dismaying multiplicity as negative facts or objective falsehoods do.
One way of answering objection (b ) is to argue that the facts of incompatibility which explain the truth of negative statements never concern incompatibilities between propositions but always concern incompatibilities between qualities, like the incompatibility between red and blue or between one way of dying and another. This is to make a certain sort of internal negation the fundamental form in terms of which all other types of negation are to be defined. This eliminates the horde of positive falsehoods that are incompatible with the actual positive facts in favor of a possibly smaller and anyway more acceptable horde of incompatible qualities, each capable in itself of qualifying a real object but unable to do so at the same time as the others. But although there is some plausibility in accounting for simple singular negations in this way (that is, in taking the simple "x is not A " to be true, when it is true, because x is something incompatible with being A ), it is hard to deal similarly with the negations of more complex forms—for example, "Not everything is A " or "It is not the case that if x is A, then y is B. "
Difficulties in dealing with more complex negations also arise with the suggestion that the facts that verify negative statements are facts not so much about incompatibility as about otherness. It is important to note that the otherness account cannot take quite the same form as the incompatibility one; although the fact that x is something incompatible with being red will suffice to verify "x is not red," "x is something other than red" will not, for x may be something other than red (for instance, round) and be red as well. The otherness account would have to claim that what verifies "x is not A " is the fact that x is other than everything that is A. This account, like the preceding one, seems to be applicable only to simple singular negation. However, if the complexities that can arise are capable of being listed, it might be possible to give a separate account of the negation of each kind of complexity. Thus, having said what the simple "x is not A " means, we may say that in forms like "Not (not-p )," "Not (p and q )," "Not (p or q )," "Not (everything ϕ 's)," and "Not (something ϕ 's)" (that is, "Not anything ϕ 's"), the apparently external "not" is to be defined in terms of a comparatively internal "not" as follows:
Not (not-p ) = p,
Not (p or q ) = (Not-p ) and (not-q ),
Not (p and q ) = (Not-p ) or (not-q ),
Not (for every x, x ϕ 's) = For some x, not (x ϕ 's),
Not (for some x, x ϕ 's) = For every x, not (x ϕ 's).
In any given complex formed in these ways the innermost negations—the only ones that remain when all the reductions have been performed—will be simple singular negations explainable as above in terms of otherness or incompatibility.
Negation, Facts, and Falsehood
Another way of eliminating negative facts might be by defining negation in terms of disbelief or falsehood. Affirmative statements, we might say, express beliefs whereas negative ones express disbeliefs. Disbelief, however, is not just the absence of belief, and like belief it must have an object—it must be disbelief in something or disbelief of something—and it must be justified or unjustified; if justified, whatever justifies it must be either a negative fact or whatever we replace negative facts with when using some other and more objective method of dissolving them.
In terms of falsehood we might say that the contradictory negation of a statement is the statement that is true if the given one is false and false if the given one is true. This amounts to defining negation by means of its truth table, a course advocated by Ludwig Wittgenstein in the Tractatus. To this it may be objected that talk of the statement which is true when a given statement is false and false when it is true is legitimate only if we know that there is one and only one statement which meets these conditions, and this seems unlikely; for example, since "Oxford is the capital of Scotland" is false in any case, "Either Oxford is the capital of Scotland or grass is not green" is true if "Grass is green" is false and false if it is true, but what is stated by this complex does not seem to be simply the negation of "Grass is green." It may also be objected that statements are not simply true and false in themselves, as if truth and falsehood were simple properties requiring no further explanation. By the usual definition "Grass is green" is true if grass is green and false if it is not, but to say this is to define falsehood in terms of negation rather than vice versa.
Perhaps the whole problem about negative facts—and the problem about the objective falsehoods that would be parts of such facts if there were any—arise from thinking of facts (and falsehoods) too literally as objects or entities. It is not merely that there are no negative facts but, rather, that there are no facts. That is, expressions of the form "The fact that p " do not name objects, whether or not our "p " is negative in form. The word fact has meaning only as part of the phrase "it is a fact that" (that is, "it is the case that"), and "It is a fact that grass is (or is not) green" is just another way of saying the simple "Grass is (or is not) green." "There are negative facts" is true and, indeed, makes sense only if it means "For some p, it is not the case that p. " But in this sense it is true and metaphysically harmless; it does not mean that there are objects called "That p " which go through a performance called "not being the case," and still less does it mean that there are objects called "The not-being-the-case of that p. "
Even with this caution, however, one can sensibly inquire whether signs of negation are really indispensable—whether what we say when we use them cannot also be said, and more directly, without them—and whether signs of negation are not just convenient abbreviations for complex forms into which no such signs enter. Putting the question in this way, modern logic has evolved other devices for eliminating negation besides the ones thus far mentioned, devices which are worth examining, even though they are a little technical, and which require some preliminary account of negation as the logician sees it.
laws of negation
Negation figures in formal logic primarily as the subject of certain laws, of which the best known are those of contradiction and excluded middle. The law of contradiction asserts that a statement and its direct denial cannot be true together ("Not both p and not-p ") or, as applied to terms, that nothing can both be and not be the same thing at the same time ("Nothing is at once A and not-A "). The law of excluded middle asserts that a statement and its negation exhaust the possibilities—it is either the case that p or not the case that p —or, as applied to terms, that everything either is or is not some given thing—say, A. Each of these laws may be put in the form of an implication, or "if" statement; the law of contradiction then appears as "If p, then not not-p, " and the law of excluded middle as "If not not-p, then p. " Sometimes the combination of these two, "p if and only if not not-p, " is called the law of double negation.
Each of these laws involves a number of derived or related laws. From the law of contradiction it follows that what has contradictory consequences is false; if p implies q and also implies not-q (and so implies "q and not-q "), then not-p. From the law of excluded middle it follows that what is implied by both members of a contradictory pair is true; if p implies q and not-p equally implies q, then q. Again, because of the law of contradiction whatever implies its own denial is false, for if p implies not-p, it implies both p and not-p (since it certainly implies p ) and thus cannot be true. This is the principle of reductio ad absurdum. To take an ancient example, if everything is true, then it is true (among other things) that not everything is true; hence, it cannot be the case that everything is true. Perhaps we can also argue that if it is a fact that there are no negative facts, then that is itself a negative fact; thus, it cannot be that there are no negative facts. Correspondingly, from the law of excluded middle it follows that whatever is implied by its own denial (that is, what we are compelled to affirm even when we try to deny it) is true. (The later Schoolmen called this the consequentia mirabilis. )
Another important law involving negation is the law of contraposition, or transposition, that if p implies q, then the denial of q implies the denial of p or, for terms, if every A is a B, then every non-B is a non-A. If this is combined with the first law of double negation ("If p, then not not-p "), we obtain "If p implies not-q, then q implies not-p "; if it is combined with the second law of double negation ("If not not-p, then p "), we obtain "If not-p implies q, then not-q implies p, " and with both we obtain "If not-p implies not-q, then q implies p. "
Many logicians have questioned the law of excluded middle and the laws associated with it. In particular, the intuitionist logic of L. E. J. Brouwer and Arend Heyting contains none of the laws "Either p or not-p, " "If not not-p, then p, " "If p implies q and not-p also implies q, then q, " "If not-p implies p, then p, " "If not-p implies q (not-q ), then not-q (q ) implies p. "
formal definitions of negation
The laws just discussed and many others figure in modern symbolic calculi as theorems derived by stated rules of inference from given axioms. Some of them, indeed, may themselves appear as axioms, different formulas being taken as axiomatic in different symbolic presentations. The symbols used, moreover, will be divisible into "primitive" symbols that are introduced without explanation and other symbols that are introduced by definition as abridgments of complexes involving other symbols. Which symbols are taken as primitive and which are defined will vary with the particular systematic presentation adopted.
Gottlob Frege, for example, took symbols corresponding to "if" and "not" as undefined and introduced the form "p or q " as a way of writing "If not-p, then q " ("Either I planted peas, or I planted beans" = "If I did not plant peas, I planted beans"). Bertrand Russell at one stage did the same, but he later took "not" and "or" as his primitives, defining "If p, then q " as "Either not-p or q " ("If you smoke, you'll get a cough" = "Either you won't smoke, or you'll get a cough") and "p and q " as "Not either not-p or not-q. " Other writers have defined all the other symbols in terms of "not" and "and." For example, they have defined "If p, then q " as "Not (p without q )"—that is, "Not (p and not-q )" and "p or q " as "Not both not-p and not-q. "
In all these examples the negation sign appears as one of the primitive or undefined symbols, but there are also systems in which this is not the case and in which "not" is defined in terms of something else. For example, Jean Nicod uses a single undefined stroke in such a way that "p | q " amounts to "Not both p and q " and "Not-p " is defined as "p | p " (Not both p and p ). Russell sometimes attempts to avoid even the appearance of complexity in his verbal rendering of Nicod's stroke by reading "p | q " as "p is incompatible with q, " but this would ordinarily be understood as a little stronger than what is intended. We would not normally say that "London is the capital of England" was incompatible with "Berlin is the capital of France," but it is correct to say "London is the capital of England | Berlin is the capital of France," since the two components are not both true.
An earlier and more interesting device was that of C. S. Peirce, who defined negation as the implication of something false. This is not quite a definition of negation in terms of falsehood. Formally, what is meant is that we arbitrarily choose some false proposition—say, "The ancient Romans spoke Polish"—and introduce "Not-p " as an abbreviation for "If p, then the ancient Romans spoke Polish." It is also possible to take as our standard false proposition for this purpose a formula which itself has some logical significance. In his later years Peirce himself liked to use the proposition "For all p, p, " which is, roughly, "Everything is true" (which was shown to be false in the previous section of this entry). In common speech we come close to defining "Not-q " as "If q, then for all p, p " when we say of something we wish to deny, "If you believe that, you would believe anything." A similar definition of "Not-p, " used by Russell in his early writings, is "For all q, if p, then q. " Starting in this way, it is possible to define all the symbols of logic in terms of "if" and the quantifier "for all x. " Certain further technical devices make it possible to define both "if" and "for all x " in terms of a single operator that can be read as "For all x, if …, then …" or "If ever …, then …" (Russell's "formal implication," perhaps better called "universalized implication").
Given definitions of this type, the characteristic laws of negation fall into place as special cases of the characteristic laws of implication or of universality (or both). For instance, the law of transposition, "If (if p, then q ), then (if not-q, then not-p )," expands to "If (if p, then q ) then if (if q ; then anything-at-all), then (if p, then anything-at-all)," which is just a special case of the law of syllogism, "If (if p, then q ), then if (if q, then r ), then (if p, then r )." Moreover, the peculiarities of the intuitionistic negation of Brouwer and Heyting turn out simply to reflect those of intuitionistic implication.
Intuitionistic logic, for example, contains the law "If p implies q, then if p also implies that q implies r, p implies r "; therefore, it contains the special case "If p implies q, then if p also implies that q implies the falsehood, then p implies the falsehood"—that is, "If p implies q, then if p also implies not-q, then not-p. " But it does not contain the law "If p implies r, then if p 's implying q also implies r, then r " (this law, being verified by the usual truth-tables for "if" and "not," does appear in nonintuitionistic or classical implicational logic) and therefore does not contain the law "If p implies r, then if p 's implying the falsehood also implies r, then r " ("If p implies r, then if not-p also implies r, then r ").
It is also possible in both intuitionistic and classical logic to separate those laws of negation which are (or may be represented as) merely special cases of laws of implication (as in the above examples) and those that reflect the special features of what a proposition is being said to imply when we negate it. For example, both versions of logic contain the law (1) "If p, then if also not-p, then anything-at-all." But neither logic contains as a law the implicational formula of which this would be (if they had it) a special case, "If p, then if p implies r, then anything-at-all." However, they do both have, quite naturally, (2) "If p, then if p implies that everything is true, then anything-at-all." To get (1), in other words, it is important not only that we should see "Not-p " as something of the form "If p, then r " but also as this particular thing, "If p, then everything is true." If we drop from intuitionistic logic those laws of negation which require attention to this more special point, we obtain the "minimal" calculus of I. Johannson ("Der Minimalkalkül," Compositio Mathematica, Vol. 4, 119–136).
Technical Eliminations of Negation
Do the developments just sketched mean that we can dispense with negative facts by saying that the facts stated by true negative statements are ones that do not involve any special concept of negation but only (in one version) Nicod's stroke or (in the other) implication and universality? The suggestion, especially in its Peircean form, has its attractions. Peirce's definition would at least explain why negation is a proper subject of study for pure logicians. Logic studies universal rules of implication; even the purest logic must study whatever is involved in the very notions of implication and universality; and what Peirce means by negation is thus involved. Facts as to what is not the case are in this view only an instance of a more general type of complex fact without which logic would be impossible—namely, facts as to what leads to what.
Against this suggestion one might adduce the extreme artificiality and arbitrariness of these symbolic devices. Consider the fact that it is equally possible in a symbolic system to define "and" in terms of "or" and "not" and "or" in terms of "and" and "not." Whatever this fact signifies, it cannot signify that "Not (not-p or not-q )" is the real meaning of "p and q " and that the very form "p or q " that is used in this explanation has for its real meaning "Not (not-p and not-q )." This procedure would obviously be circular, and for this reason we cannot, even symbolically, have both definitions in the same system. It is obvious that the form "or" cannot be both simple and unanalyzable and a complex built up out of "and" and "not"; at least, it can only be this by being used ambiguously and, similarly, mutatis mutandis, with "and." The systems with the different definitions are equivalent in the sense that, given suitably chosen axioms, the same formulas will appear in them as theorems, and the undefined "and" (or "or") and the defined one are equivalent in the sense of having the same truth tables. But if there is an intuitively simple meaning of the form "p and q, " "and" in this sense simply does not appear (is not symbolized) in a system which has only "or" and "not" as its undefined symbols and introduces "p and q " as short for "Not (not-p or not-q )." Primitiveness in a convenient calculus is one thing; intuitive or conceptual simplicity, another. No one symbolic system, we may surmise, can express everything, and in any given system we can take whatever we please as undefined, even if its intuitive meaning is complex.
Turning now to the calculi in which "not" is defined, it is notoriously difficult to explain the meaning of Nicod's stroke except by saying that "p | q " means "Not both p and q " or that it means "Either not-p or not-q "; furthermore, the "not" that is introduced by defining "Not-p " as "p | p " cannot be the "not" which is used in this explanation, though for purposes of logical calculation it may serve just as well. It could similarly be said that the "if" which Peirce uses in his definition of "not" cannot be understood without a more primeval "not" being presupposed. For Peirce did not use "If p, then q " in the familiar sense in which it means that q would be a logical consequence of p ; it is not true that whenever p happens not to be the case, it would logically follow from it that everything whatever is true. Even the colloquial "If you believe that, you would believe anything" is not said of anything we wish to deny but only of particularly outrageous items (things that not only are not, but also could not, be the case). What Peirce meant by "If p, then q, " it might be said, can be explained only by saying that it means "Not at once p and not-q, " and this explanation uses a "not" that cannot be derived from his definition because the definition presupposes that "not."
Additionally, it might be argued that our intuitions as to what is a construction from simpler conceptions and what is itself simple are not very reliable and that if a definition introduces new economies into a calculus and, still more, if it brings a new unity to a whole subject, this may well be a symptom that it also reveals what is conceptually fundamental. The treatment of "not being the case" as an extreme case of implication—as "implying too much," so to speak—does at least reflect something important about the relation between the two concepts. A proposition's implying something, having consequences, is like its taking a risk, and its not being the case is its having too strong consequences.
See also Brouwer, Luitzen Egbertus Jan; Correspondence Theory of Truth; De Morgan, Augustus; Frege, Gottlob; Logic, Traditional; Nothing; Peirce, Charles Sanders; Presupposition; Propositions, Judgments, Sentences, and Statements; Russell, Bertrand Arthur William; Stoicism.
For clear summaries of the stock problems see J. N. Keynes, Formal Logic, 4th ed. (London, 1906), Part I, Ch. 4; Part II, Ch. 3; and Appendix B. See also W. E. Johnson, Logic (Cambridge, U.K.: Cambridge University Press, 1921), Part I, Chs. 5 and 14.
For the special insights of certain major writers see C. S. Peirce, Collected Papers, edited by Charles Hartshorne, Paul Weiss, and Arthur W. Burks, 8 vols. (Cambridge, MA: Harvard University Press, 1931–1958), II.356, 378–380, 550, 593–600; III.381–384, 407–414. See also Peirce's article "Syllogism," in Century Dictionary (New York: Century, 1889–1901); Gottlob Frege, "Negation," in Translations from the Philosophical Writings of Gottlob Frege, edited by P. T. Geach and Max Black (Oxford: Blackwell, 1952); and Bertrand Russell, Introduction to Mathematical Philosophy (London: Allen and Unwin, 1919), Ch. 14; "The Philosophy of Logical Atomism," Lectures I–II, in Logic and Knowledge (London: Allen and Unwin, 1956); An Inquiry into Meaning and Truth (New York: Norton, 1940), Ch. 4; and Human Knowledge, Its Scope and Limits (New York: Simon and Schuster, 1948), Ch. 9.
See also Ludwig Wittgenstein, Tractatus Logico-Philosophicus, translated by D. F. Pears and B. F. McGuinness (London: Routledge and Paul, 1961), 1.12, 2.06, 4.06–4.1, 4.25–4.463, 5.254, 5.43–5.44, 5.451, 5.512, 5.5151, 6.1201–6.1203.
On Wittgenstein (and Russell and Frege) see also G. E. M. Anscombe, An Introduction to Wittgenstein's Tractatus (London: Hutchinson University Library, 1959), Chs. 1–4.
On falsehood and meaninglessness see J. M. Shorter's classic piece, "Meaning and Grammar," in Australasian Journal of Philosophy 34 (1956): 73–91.
Various philosophical problems concerning the nature of negation are discussed in the following works: F. H. Bradley, Principles of Logic (London: K. Paul, Trench, 1883; 2nd ed., London: Oxford University Press, 1922), Vol. I, Book 1, Ch. 3, and Terminal Essay 6; Bernard Bosanquet, Logic or the Morphology of Knowledge, 2 vols. (London: Clarendon Press, 1888), Vol. I, Ch. 7, Secs. 1–3, 5; Raphael Demos, "A Discussion of a Certain Type of Negative Proposition," Mind 24 (1917): 188ff; Ralph M. Eaton, Symbolism and Truth (Cambridge, MA: Harvard University Press, 1925); J. Cook Wilson, Statement and Inference, 2 vols. (Oxford: Clarendon Press, 1926), Vol. I, Ch. 12; J. D. Mabbott, Gilbert Ryle, and H. H. Price, symposium, "Negation," PAS, Supp., 9 (1929); F. P. Ramsey, "Facts and Propositions" (1927), in his Foundations of Mathematics (London: K. Paul, Trench, Trubner, 1931), pp. 138–155; A. J. Ayer, "Negation," Journal of Philosophy 49 (1952): 797–815, reprinted in his Philosophical Essays (London: Macmillan, 1954); Morris Lazerowitz, "Negative Terms," Analysis 12 (1951–1952): 51–66, reprinted in Philosophy and Analysis, edited by Margaret MacDonald (Oxford, 1954); R. L. Cartwright, "Negative Existentials," Journal of Philosophy 57 (1960): 629–639, reprinted in Philosophy and Ordinary Language, edited by Charles E. Caton (Urbana: University of Illinois Press, 1963); and Gerd Buchdahl, "The Problem of Negation," Philosophy and Phenomenological Research 22 (1961): 163–178.
A. N. Prior (1967)
The term negation (Verneinung ) denotes a mental process in which the subject formulates the content of an unconscious wish in a negative form. The content of the wish finds expression in consciousness, yet the subject continues to disown it.
This concept first appeared in Freud's work in connection with the analysis of the "Rat Man" when the patient produced an association having to do with the death of his father but immediately "rejects the idea with energy" (1909d, p. 178). Yet Freud's main discussion of the topic appears in "Negation" (1925h), where he sets forth a theory of negation that is close to being a theory of forms of language or a theory of judgment.
Freud posits two distinct processes of negation: one involves the rejection of a thought, the other the acknowledgment of a disappointed expectation. The first kind of negation, involving rejection, is the kind encountered in the "Rat Man" case. Another example is when a patient refuses to believe that the woman he has just dreamed about is his mother: "You're going to think it was my mother," he will say to the analyst, "but it wasn't my mother." This negation may be interpreted to mean, "I reject the idea that this person could be my mother because I dislike that idea." Negation is a rejection of an unpleasant idea by means of the pleasure principle alone. The process of projection is already at work in the utterance "You're going to think it was my mother," for in this way the patient projects into the mind of the analyst a thought that is in fact the patient's.
Negation as a defense mechanism is more supple than repression in that it preserves the thought content that repression would render unconscious. The defensive aspect of the mechanism is confined to the distancing achieved by means of the negation, which allows the patient to avoid shouldering the disagreeable implications of a thought that has successfully formed.
In addition to this first kind of negation, Freud describes a second type, namely a judgment by a psyche that fails to encounter in the outside world a satisfying mental representation of what it desires. The psyche is then obliged to arrive at the negative conclusion that what it has been seeking in external reality is indeed not present. This type of negation thus amounts to an assertion of absence. In making this assertion, the psyche recognizes the independent existence of the outside world and, thus achieving effective reality-testing.
The idea of negation lies at the center of a very dense conceptual nexus within the Freudian model. Several other terms are closely linked to Verneinung (negation) and overlap with it in meaning to a greater or lesser extent. Occasionally Freud used the Latinate German term die Negation to refer to a basic trait of the "system Ucs.," in which there is "no negation" (1915e, p. 186). This enabled him to define the system of the unconscious as prior to intellectual judgment. By contrast, the Germanic word Verneinung, embracing as it does both negation as a mental process and negation as a grammatical form, presupposes a psychic agency capable of making judgments.
In Freud's usage there is a wider difference in meaning between Verneinung and Verleugnen, which is translated into English as either "denial" or (following the preference of the editors of the Standard Edition ) "disavowal." In the process of disavowal, the subject refuses to embrace the psychic consequences of something perceived. Thus the "Wolf Man" (1918b ) said, in effect, "I see that a woman does not have a penis, but I deny any force to this observation, and what is more, I shall continue to believe that she has a penis." In disavowal, a reality judgment produces a conclusion ("A woman does not have a penis"), but this conclusion is a dead letter having no impact on the psyche. Thus the recognition of a reality ("[I see that] a woman does not have a penis") is juxtaposed to a wish ("[I want] a woman to have a penis") without being integrated together. In both disavowal and negation, the subject avoids responsibility for a disagreeable thought. The two differ, however, in that disavowal is rejection of a disagreeable perception, whereas negation is the acceptance of a wish.
Lastly, the term foreclosure (French forclusion ) was introduced by Jacques Lacan to render Freud's use of the term Verwerfung in connection with the psychotic mechanism of "expulsion of a fundamental 'signifier' " (Laplanche and Pontalis, 1973, p. 166).
The theory of negation is no doubt an area where the psychoanalytic theorization of mental processes comes very close to linguistic concerns, especially to the study of utterances. From a linguistic standpoint, one might say that negation in the sense of rejection is equivalent to a polemical negation (as in, for example, "For me, this woman in my dream is not my mother"), whereas the recognition of absence—a negation that can be expressed as a reality judgment—is equivalent to a "simple" negative report, much like a statement such as "I have not had a dream for a long time."
Clear boundaries need to be drawn between negation, absence, and the idea or representation of absence. These distinctions, in broad outline, are as follows. In all cases, negation is directed at an ideational content. This content, in the context of the theory of the hallucinatory satisfaction of a wish, emerges as a consequence of the absence of the wished-for object. This absence obliges the subject to hallucinate, to represent, the missing object. Absence is thus a precondition of the emergence of the representation. In this regard, absence is distinct from negation, which is an operation affecting an ideational content. As for the representation of absence, it arises only after the capacity for judging reality has been established, when the subject is able to articulate the gap between what he wants and what he sees.
See also: Binding/unbinding of the instincts; Constructions in analysis; Contradiction; Death and psychoanalysis; Death instinct (Thanatos); Defense; Disavowal, denial; Ego and the Mechanisms of Defence, The ; Id; "Negation"; Primitive; Projection; Splitting.
Freud, Sigmund. (1909d). Notes upon a case of obsessional neurosis. SE, 10: 151-318.
——. (1915e). The unconscious. SE, 14: 159-204.
——. (1918b ). From the history of an infantile neurosis. SE, 17: 1-122.
——. (1925h). Negation. SE, 19: 233-239.
Hyppolite, Jean. (1988). A spoken commentary on Freud's Verneinung. In The seminar of Jacques Lacan. Book 1: Freud's papers on technique, 1953-1954 (John Forrester, Trans.). New York: W. W. Norton. (Original work published 1954)
Lacan, Jacques. (1988). Introduction and reply to Jean Hyppolite's presentation of Freud's Verneinung. In The seminar of Jacques Lacan. Book 1: Freud's papers on technique, 1953-1954 (John Forrester, Trans.). New York: W. W. Norton. (Original work published 1954)
Laplanche, Jean, and Pontalis, Jean-Bertrand. (1973). The language of psycho-analysis (Donald Nicholson-Smith, Trans.). London: Hogarth and the Institute of Psycho-Analysis. (Original work published 1967)
Litowitz, Bonnie. (1998). An expanded developmental line for negation. Journal of the American Psychoanalytic Association, 46, 121-148.
Tyson, Robert L. (1994). Neurotic negativism and negation in the psychoanalytic situation. Psychoanalytic Study of the Child, 49, 293-314.
Written by Freud after "A Note upon the 'Mystic Writing-Pad"' (1925a), this article was first published in the review Imago.
Seen in its clinical context, negation dramatizes a situation of interpretative conflict. The patient first produces the interpretation, which he imputes to the psychoanalyst, claiming that it is false. Negation is thus related to a dialogical situation: "Now you'll think I mean . . . but really I've no such intention." (1925h, p. 235).
Returning to a distinction he had established in the analysis of the "Rat Man" ("Notes upon a Case of Obsessional Neurosis," 1909) between the pure "ideational content" ("Notes," p. 176) and the positive or negative judgment in which it is incorporated, Freud says that the contribution of the unconscious to knowledge consists in the access it grants us to the repressed content. This knowledge of the repressed content does not require a lifting of repression. The dichotomy is thus displaced, and the pair of opposites affirmation/negation is eclipsed by the opposition between the affective and the intellectual. "[I]ntellectual acceptance of the repressed" (1925h, p. 236) leaves the process of repression intact, since this latter consists in a process of separation. But isn't the phrase "intellectual acceptance" a contradiction in terms? The compromise in which negation consists thus seems to make it possible, if not to lift the repression, at least to uncover the repressed content.
Over and above registering the content, the function of judgment is, Freud states, to produce "decisions" (Entscheidungen ). Taking up the philosophical distinction between judgments of attribution and judgments of existence, Freud describes two levels of psychic working-through. The first level consists in attributing a good or bad quality to something, and is linked to the instinctual impulses that drive the original pleasure-ego, which "wants to introject into itself everything that is good and to eject from itself everything that is bad" (p. 237). The decision of judgment thus plays the decisive role in operations determined by instinctual factors, swallowing and spitting—operations that establish the first distinction between inside and outside. Freud's insistence here on the instinctual dimension of negation met with a certain reservation on the part of Jean Hyppolite: "There is not yet any judgment in this moment of emergence; there is the first myth of inside and outside." So we might think in terms of a first evaluation that, when it is positive, leads the object to disappear within an enriched pleasure-ego, rather than an operation of judgment bearing on an external object.
In order for the second level of judgment to intervene and decide whether the object actually exists or not, a certain mediation is necessary—that of representation, attesting to the reality of the represented. The definition of the judgment of existence thus ties in with the notion of reality-testing and rests on the occurrence of a cut that is linked with the loss of the primal objects. However, the judgment of existence cannot be defined as a mere process of thought, because it is related to a motor process that "puts an end to the postponement due to thought . . . and leads over from thinking to acting" (p. 238).
In counter-distinction to the theme put forward in "Moses and Monotheism: Three Essays" (1939 [1934-38]), with its emphasis on the cut between the sensory and the intellectual domains, the study of negation invites us to see the processes at work in sense-perception as lying at the very heart of intellectual operations: tactile exploration (tasten ) and the fact of tasting (verkosten ) small perceptual samples, which endows the processes of thought with a two-phase temporality, made up of advances and retreats. Should this participation of two aims be understood in terms of rhythm, or related to the opposition between Eros, which upholds affirmation, and Thanatos, which upholds negation?
The last section of Freud's text returns to the topographical question: Negation, unknown at the level of the unconscious, needs to be situated on a secondary level, and we can gain access to it only by way of the symbol. The study of the interrelation of oral instinctual motions and the establishment of negative and affirmative behavior has been further investigated in the works of René Spitz.
See also: Ego; Purified-pleasure-ego; Negation; Negative, work of; Thought; Working-through.
Freud, Sigmund. (1925h). Die Verneinung. Imago, XI, 217-221; G.W., XIV, p. 11-15; Negation. SE, 19: 233-239.
Freud, Sigmund. (1909d). Notes upon a case of obsessional neurosis. SE, 10: 151-318.
Hyppolite, Jean. (1971). Figures de la pensée philosophique; écrits de Jean Hyppolite (1931-1968). Paris: Presses Universitaires de France.
Kaës, René. (1988). Le pacte dénégatif dans les ensembles transsubjectifs. In A. Missenard (Trans.), Le Négatif. Figures et modalités (pp. 101-136). Paris: Dunod.
Lacan, Jacques. (1966).Écrits. Paris: Le Seuil.
Roustang, François. (1984). Comment devenir un inspire raisonnable. Philosophie 3, 47-66.
Spitz, René. (1957). No and yes; on the genesis of human communication. New York: International Universities Press.
Special cases(1) The verb be is used in the same way when no auxiliary is present: Justin was ill becoming Justin was not/wasn't ill. (2) The verb have allows both alternatives, but in a variety of forms. The negation of Benjamin has his own bedroom can be B. has not his own b. (traditional BrE), B. hasn't his own b. (its informal variant), B. has not got his own b. (a current emphatic, especially BrE usage), B. hasn't got his own b. (its common, informal equivalent), B. does not have his own b. (a widely used formal, especially AmE usage), B. doesn't have his own b. (its common, informal equivalent). (3) See MODAL VERB.
Contracted formsThe contraction n't is typically informal, especially in speech, except when the negation is emphasized, as in a denial of something said before, in which case the full not is used and stressed. With many auxiliaries, there is often also a possibility of auxiliary contraction in informal English: It isn't fair It's not fair (more common); He won't object (more common) or He'll not object; They haven't finished (more common) or They've not finished.
Tag questionsWhen tag questions are used to invite confirmation, positive sentences are normally followed by negative tag questions (David is abroad, isn't he?) and negative sentences by positive tag questions (David isn't abroad, is he?). Positive sentences are sometimes followed by positive tag questions (So David is abroad, is he?), indicating an inference or recollection from what has been said. Occasionally, they suggest suspicion or a challenge: So that's what Doris wants, is it?
Expressions used with negationSome expressions are found exclusively or typically in negative sentences: the not … any relationship in Doris hasn't produced any plays, contrasted with Doris has produced some plays; the not … either relationship in David doesn't smoke a pipe, either (in response to such statements as John doesn't smoke a pipe), contrasted with David smokes a pipe, too (in response to John smokes a pipe).
Negation other than through the verbNo, not, and other negative words may be introduced in order to negate a sentence: Jeremy has no difficulties with this (compare Jeremy hasn't any difficulties with this), Ray said not a word to anybody (compare Ray didn't say a word to anybody); Maurice will never make a fuss, will he? (compare Maurice won't make a fuss, will he?); Nothing surprises them, does it? (compare There isn't anything that surprises them, is there?); Mervyn hardly ever makes a mistake, does he? (compare Mervyn doesn't ever make mistakes, does he?).
Implied contrastsThe negative particle or word extends its scope over the whole or part of the sentence. The extent is manifested when expressions associated with negatives are present, as in the difference between I didn't read some of the papers (that is, I read others) and I didn't read any of the papers (that is, I read none). The focus of the negation (marked intonationally in speech) is the part of the sentence which presents a negative contrast: Ted doesn't teach history may imply that someone else does or that Ted teaches something else.
Double negationPrefixes such as un- and in- make the word negative but not the sentence in which it is used: unhappy in They are unhappy about their new house; insensitively in They spoke rather insensitively to him when he lost his job. Such words may be combined with another negative to cancel out, to a large extent, the force of the negative prefix: Jeremy was not unhappy, meaning that he was fairly happy. See LITOTES. This type of double negation, which results in a positive meaning, is different from the kinds of multiple negation found in both general non-standard English (I didn't see nothing: I didn't see anything) and in some DIALECTS (Glasgow Ah'm no comin neer Ah'm no: I am not coming neither I am not). Such usages are widely stigmatized and equally widely used. See DOUBLE NEGATIVE.
ne·ga·tion / nəˈgāshən/ • n. 1. the contradiction or denial of something: there should be confirmation—or negation—of the findings. ∎ Gram. denial of the truth of a clause or sentence, typically involving the use of a negative word (e.g., not, no, never) or a word or affix with negative force (e.g., nothing, non-). ∎ Logic a proposition whose assertion specifically denies the truth of another proposition: the negation of A is, briefly, “not A.” ∎ Math. inversion: these formulae and their negations. 2. the absence or opposite of something actual or positive: evil is not merely the negation of goodness. DERIVATIVES: neg·a·to·ry / ˈnegəˌtôrē/ adj.
1. In arithmetic, the operation of changing the sign of a nonzero arithmetic quantity; the negation of zero is zero. Negation is usually denoted by the minus sign.
2. In logic, the application of the NOT operation on a statement, truth value, or formula.