Negation is in the first place a phenomenon of semantic opposition. Assuch, negation relates an expression \(e\) to another expression witha meaning that is in some way opposed to the meaning of \(e\). Thisrelation may be realized syntactically and pragmatically in variousways. Moreover, there are different kinds of semantic opposition.Section 1 is concerned mainly with negation and opposition in natural language,both from a historical and a systematic perspective.Section 2 focuses on negation as a unary connective from the point of view ofphilosophical logic. The history of negation is comprehensivelystudied and surveyed in Horn 1989 and Speranza and Horn 2012. Acomprehensive state-of-the-art handbook on negation from a (largely)linguistic perspective is Déprez and Espinal 2020.

• 1. Negation and opposition in natural language
  • 1.1 Introduction
  • 1.2 Negation in natural language: markedness and asymmetry
  • 1.3 Matters of scope
  • 1.4 Contrariety and contradiction
  • 1.5 Negation, presupposition, and singular terms
  • 1.6 From contradiction to contrariety: pragmatic strengthening of negation
  • 1.7 Privation, affixal negation, and the markedness asymmetry
  • 1.8 Double negation
    • 1.8.1 “Logical” double negation
    • 1.8.2 Negative concord and its relations
  • 1.9 Negative polarity
  • 1.10 Metalinguistic negation
• 2. The logic of negation
  • 2.1 Negation as a truth function
  • 2.2 Negation as a modal operator
  • 2.3 Interactions with negation
  • 2.4 Other conceptions of negation as a unary connective
  • 2.5 Negation, rejection, and denial
  • 2.6 Non-trivial negation inconsistent logics
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Negation and opposition in natural language

1.1 Introduction

Negation is asine qua non of every human language, yet isabsent from otherwise complex systems of animal communication.^[1] While animal “languages” are essentially analog systems,it is the digital nature of the natural language negative operator,represented in Stoic and Fregean propositional logic as a one-placesentential connective toggling the truth value of statements betweenT[rue] and F[alse] (or 1 and 0) and applying recursively to its ownoutput, that allows for denial, contradiction, irony, and other keyproperties of human linguistic systems.

The simple syntactic nature of logical negation belies the profoundlycomplex and subtle expression of negation in natural language, asexpressed in linguistically distinct categories and parts of speech(adverbs, verbs, copulas, quantifiers, affixes). As will be partlyexplored here (see also Horn 1989, Ladusaw 1996, Pullum 2002), theinvestigation of the form and meaning of negative expressions inEnglish and other languages and of the interaction of negation withother operators (including multiple iterations of negation itself) isoften far from simple, extending to scope ambiguities (Everybodydidn’t leave), negative incorporation into quantifiers andadverbs (nobody, never, few), neg-raising (I don’twant to go = “I want not to go”), and the widespreadoccurrence of negative polarity items (any, ever, lift afinger) whose distribution is subject to principles of syntax,semantics, and pragmatics. At the core of the mental faculty oflanguage, negation interacts in significant ways with principles ofmorphology, syntax, logical form, and compositional semantics, as wellas with processes of language acquisition and sentence processing,whence the prominent role played by work on negation in thedevelopment of logic, semantics, linguistic theory, cognition, andpsychoanalytic and literary theory.

What sort of operation is negation? In theCategories andDe Interpretatione, Aristotle partitions indicative-mooddeclarative sentences into affirmation and negation/denial(apophasis fromapophanein “deny, sayno”), which respectively affirm or deny something aboutsomething (De Int. 17a25). As a mode of predication, thepredicate denial of Aristotelian term logic, while resulting inwide-scope negation opposed in truth value to the correspondingaffirmative, is syntactically distinct from the unary “it is notthe case that” connective of Stoic and Fregean logic. (See alsothe discussion in Tranchini 2021.)

By combining subject and predicate to form a proposition, thisapproach can be seen as offering a more natural representation ofordinary language negation than the standard iterating operator thatapplies to fully formed propositions (Geach 1972; Englebretsen 1981;Horn 1989, Chap. 7; Sommers and Englebretsen 2000). Thesyncategorematic negation of early Montague Grammar (Montague 1973;cf. the entry onMontague semantics) is itself a means of connecting a term phrase subject with apredicate or IV (intransitive verb) phrase and thus fails to apply toits own output. Cross-linguistically, the structural reflex ofsentence-scope negation may be a free-standing adverb (Germannicht, Englishnot), a bound inflectional form(Japanese-na-, English-n’t), or a verb(Finnishen,ei).^[2]

Where we almost never find negation is in the one place propositionallogic would lead us to look, sentence- or clause-peripheral position,as an external one-place connective interpreted as “it is notthe case that”. (While Horn 1989 takes apparent instances ofexternal negation in English and elsewhere to represent themetalinguistic use of negation, discussed in §1.10 below,Bar-Asher Siegal 2015 presents evidence for the existence of asemantic external negation operator in Jewish Babylonian Aramaic.)Furthermore, unlike speech act types—e.g. interrogatives orexclamatives—negation never seems to be marked in naturallanguage by a global intonation contour. Typically, sentence negationis associated directly on or near the main finite verb or predicateexpression.

1.2 Negation in natural language: markedness and asymmetry

It has often been observed that the logical symmetry of negative andaffirmative propositions in logic belies a fundamental asymmetry innatural language. It was Plato who first posited, inTheSophist, that negative sentences are less valuable thanaffirmative ones, less specific and less informative. The ontological,epistemological, psychological, and grammatical priority ofaffirmatives over negatives is supported by Aristotle:

The affirmative proposition is prior to and better known than thenegative (since affirmation explains denial just as being is prior tonot-being) (Metaphysics 996b14–16)

and St. Thomas Aquinas:

The affirmative enunciation is prior to the negative for threereasons… With respect to vocal sound, affirmative enunciationis prior to negative because it is simpler, for the negativeenunciation adds a negative particle to the affirmative. With respectto thought, the affirmative enunciation, which signifies compositionby the intellect, is prior to the negative, which signifiesdivision… With respect to the thing, the affirmativeenunciation, which signifiesto be, is prior to the negative,which signifiesnot to be, as the having of something isnaturally prior to the privation of it. (St. Thomas, Book I, LessonXIII, cited in Oesterle 1962, 64)

Not only are negative statements (e.g., “Paris isn’t thecapital of Spain”) generally less informative than affirmatives(“Paris is the capital of France”), they aremorphosyntactically more marked (all languages have negative markerswhile few have affirmative markers)^[3] and psychologically more complex and harder to process (seeGivón 1978 and Payne 1985, and empirical studies dating back toJust and Carpenter 1971, 248–9). As Geurts puts it (2003, 46),“negative expressions take longer to process, cause more errors,and are harder to retain than positive ones”. Many philosophers,linguists, and psychologists have situated this asymmetry in logic orsemantics, as in the claim that every negation presupposes acorresponding affirmative but not vice versa.

The strong asymmetricalist position leads to the “paradox ofnegative judgment”: if a positive statement refers orcorresponds to a positive fact, to what state of affairs does anegative statement refer or correspond? What in fact is a negativefact? For Bergson (1911, 289), negation is necessarily “of apedagogical and social nature”; for Wood (1933, 421) it is“infected with error and ignorance”. According toWittgenstein (1953, §447), “the feeling is as if thenegation of a proposition had to make it true in a certain sense inorder to negate it”. Givón (1978, 70) points to thediscourse presuppositionality of utterances like “My wife is notpregnant”. Psycholinguistic studies have shown that negation iseasier to process when the denied proposition, if not already in thediscourse model, is at least a plausible addition to it (e.g.,“The whale is not a fish/?bird”; cf. Wason 1965).

Beyond its marked status, negation has also been analyzed variously asa modality, a propositional attitude, and a speech act. (See Horn1989: §1.2.) The danger is putting the pragmatic cart before thesemantic horse. For example, not every negation is a speaker denialnor is every speaker denial a linguistic negation. Frege (1919, 353)argues that the putative distinction between positive and negativethoughts, as maintained by Kant and others, is unneeded by thelogician; “its ground must be sought outside logic”. Thereare thoughts with negative components, but there is no need tocountenance the existence of negative judgments per se. A negation canoccur in unasserted contexts, e.g. within the antecedent of aconditional, without rendering the containing sentence a denial.Further, identifying “negative thoughts” is not alwaysstraightforward, given lexical and inherent negation. In a paradigmthat includes the sentences “Christ is immortal”,“Christ lives forever”, “Christ is notimmortal”, “Christ is mortal”, “Christ doesnot live forever”, Frege asks rhetorically, “Which of thethoughts we have here is affirmative and which negative”?

Given the repeated attempts over the centuries to liquidate or tameit—negation as positive difference, negation as dissimilarity orincompatibility, negation as falsity, negation as an admission ofepistemic impoverishment, negation as the speech act ofdenial—and its resilience in surviving these attacks, negationqualifies as the Rasputin of the propositional calculus. (For othercomplications of the asymmetry issue, see Tranchini 2021; Ito 2021,2023; and Bonino 2024.)

But the prototypical use of negation is indeed as a denial of aproposition attributable to, or at least considered by, someonerelevant to the discourse context. While affirmation standardlyintroduces a proposition into the discourse model, negation—inits “chief use” (Jespersen 1917, 4), its “mostcommon use” (Ayer 1952, 39), its “standard and primaryuse” (Strawson 1952, 7)—is directed at a proposition thatis already in or that can be accommodated by the discourse model. Inrecent work, Incurvati and Sbardolini (2024) and Sbardolini (2024),drawing on evolutionary game theory, address the question of howadaptive pressures might explain the evolution of expressions fordenial within the social and informational environment.

1.3 Matters of scope

If we think of negation as essentially a means foropposition—the impossibility of simultaneously endorsing twoincompatible options (see the entries oncontradiction and thetraditional square of opposition)—propositional negation is not necessarily privileged. This view is formallyimplemented in the Boolean algebraic model of Keenan and Faltz, onwhich negation is a cross-categorial operation, as are the binaryconnectives:

We can directly interpret conjunctions, disjunctions, and negations inmost categories by taking them to be the appropriate meet, join, andcomplement functions of the interpretations of the expressionsconjoined, disjoined, or negated. The sense in which we have only oneand,or, andnot is explicated on thegrounds that they are always interpreted as the meet, join, andcomplement functions in whatever set we are looking at. (Keenan andFaltz 1985, 6)

Treatments of English and other languages frequently posit negativeoperators whose scope is narrower than the sentence or clause. Thistradition dates back to Aristotle, for whom the predicate termnegation inSocrates is not-wise, affirming that thepredicatenot-wise holds of Socrates, yields a falsestatement if Socrates does not exist, while the predicate denialSocrates isn’t wise denies that the predicatewise holds of Socrates and is true if Socrates does notexist. For Jespersen (1917), the subclausal “special”negation as inNobody came, where “the negativenotion…belong[s] logically to one definite idea”, isopposed to “nexal” negation, applying to “thecombination of two ideas”, typically the subject-predicatenexus. Later linguists usually follow Klima (1964) and Jackendoff(1969) in allowing for constituent negation (e.g., verb phrasenegation inYou can [not go]) alongside sentential negation(You cannot go), utilizing various grammatical and semanticdiagnostics for distinguishing the two varieties. Sandu (1994)provides a formalization of Klima’s and Jackendoff’saccounts of negation, along with arguments for why constituentnegation is not ultimately reducible to contradictory negation (=“it is not so that...”) or vice versa.

A syntactic correlate of the distinction between wide- (sentential)versus narrow-scope (constituent) negation in English is that onlywhen the negative element has clausal scope, as in the (a) examples in(1)-(3) (or in this very sentence), can it trigger negative inversion(Klima 1964). In the corresponding (b) examples, the scope of negationdoes not extend beyond the fronted phrase, whence the exclusion ofever, a satellite of negation (negative polarity item).^[4]

(1)

a.

With no jobwill I be happy. [= I won’t behappy with any job]

b.

With no job I will be happy. [= I will be happy without anyjob]

(2)

a.

In no clothesdoes Robin look good.

b.

In no clothes Robin looks good.

(3)

a.

At no timewere we (ever) alone together in theOval Office.

b.

In no time we were (*ever) alone together in the Oval Office.

Negation also interacts in complicated and often surprising ways withquantification and modality (Fleisher 2020). Perhaps the most analyzedinteraction is with universal quantification. Despite the frequentcondemnation of the wide-scope reading of negation over universals asin thelocus classicus,All that glitters is notgold and similar examples in French, German, and other languages,or in ambiguous sentences likeAll the boys didn’tleave, the availability of such readings (depending on thespeaker, the intonation contour, and the context of utterance) is notas illogical as it may appear (see Tottie and Neukom-Hermann 2010 andwork reviewed therein).

1.4 Contrariety and contradiction

The concept of negation is often semantically restricted tocontradictory opposition between propositions, in which \(\neg A\) canbe paraphrased (if not necessarily syntactically represented) as“it is not the case that \(A\)”. As introduced inAristotle’sCategories (11b17), the genus of opposition(apophasis) is divided into species that include contrarietyand contradiction. Contradictory opposites, whether affirmative andnegative counterparts of a singular predication (Socrates iswise/Socrates isn’t wise) or quantifiedexpressions (All pleasure is good/Some pleasure is notgood), are mutually exhaustive as well as mutually exclusive,while contrary opposites (Socrates is wise/Socrates isunwise;All pleasure is good/No pleasure isgood) do not mutually exhaust their domain. Contraries cannot besimultaneously true, though they may be simultaneously false. Membersof a contradictory pair cannot be trueor falsesimultaneously; contradictories “divide the true and the falsebetween them” (for depictions of the classical neo-Aristoteliansquare see the entries on thetraditional square of opposition andcontradiction).

Contrary terms (enantia) come in two varieties (Cat.11b38ff.). With immediate or logical contraries(odd/even,sick/well), a truemiddle—an entity satisfying the range of the two opposed termsbut falling under neither of them—is excluded, e.g., an integerneither odd nor even. But mediate contrary pairs(black/white,good/bad) allow fora middle—a shade between black and white, a man or an actneither good nor bad. Neither mediate nor immediate contraries fallunder the purview of the Law of Excluded Middle [LEM] (tertium nondatur).

For immediate contraries formed by narrow-scope predicate termnegation, the rendering\(a\) is not-\(F\) in the traditionalquasi-English phrasing corresponds to what Aristotle expresses throughword order, utilizing the distinction between e.g.,einai mêleukon “to be not-white” andmê einaileukon “not to be white” (Prior Analytics I51b10). For Aristotle,\(a\) is neither \(F\) nor not-\(F\)can be true if \(a\) doesn’t exist (Santa is neither whitenor not-white) or isn’t the kind of thing that can beF (The number 7 is neither white nor not-white),given thatnot-\(F\) is taken to affirm the negative propertynon-\(F\)-ness of the subject rather than denying a positiveproperty.

Other cases in which apparent contradictories can be seen ascontraries, and thus immune from any application of LEM, are futurecontingents (There will be/will not be a sea battletomorrow; cf.De Int. Chapter 9) and, in more recentwork (Alxatib and Pelletier 2011, Ripley 2011a), vague predications.Thusa is neither F nor not-F is often judged true whenF is a vague predicate (bald, rich, tall), althoughin the latter case speakers may also be willing to affirm thata is bothF and not-F, which complicatesmatters (see the entries oncontradiction,future contingents, andvagueness).

1.5 Negation, presupposition, and singular terms

In his exposition of sense and reference, Frege (1892) argues thatboth (4a) and its contradictory (4b) presuppose that the nameKepler has a denotation. Every affirmative or negativesentence with a singular subject (name or description) presupposes theexistence of a unique referent for that subject; if the presuppositionfails, no assertion is made in (4a,b).

(4)

a.

Kepler died in misery.

b.

Kepler did not die in misery.

But this presupposition is not part of the content of the expression,and hence (4a) does not entail existence, or the negation of (4a)would not be (4b) butKepler did not die in misery or the name“Kepler” has no reference, an outcome Frege seems tohave taken as an absurdity but one that prefigures the later emergenceof a presupposition-cancelling external or exclusion negation.

Unwilling to countenance the truth-value gaps incurred onFrege’s analysis, Russell (1905, 485) reconsiders the status ofcontradictory negation with vacuous subjects:

By the law of the excluded middle, either “A isB” or “A is notB” mustbe true. Hence either “the present king of France is bald”or “the present king of France is not bald” must be true.Yet if we enumerated the things that are bald and the things that arenot bald, we should not find the king of France on either list.Hegelians, who love a synthesis, will probably conclude that he wearsa wig.

To resolve this (apparent) paradox while preserving a classicalanalysis in which every meaningful sentence is true or false, Russellbanishes singular terms likethe king of France from logicalform, unpacking (5) and (6) as existentially quantified sentencesdespite their superficial subject-predicate syntax.

(5)

The king of France is bald.

(6)

The king of France is not bald.

On Russell’s theory of descriptions, (5) can be represented as(5′), the (false) proposition that there is a unique entity withthe property of being king of France and that this entity is bald,while (6) is ambiguous, depending on the scope of negation.

(5′)

\(\exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedgeBx))\)

(6′)

\(\exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedge \negBx))\)

(6″)

\(\neg \exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedgeBx))\)

(6′), with narrow-scope (“internal”) negation, isthe proposition that there is a unique and hirsute king of France,which is “simply false” in the absence (or oversupply) ofmale French monarchs. In (6″), on the other hand, thedescriptionthe king of France falls within the scope ofexternal negation and yields a true proposition. Unlike (6′),(6″) fails to entail that there is a king of France; indeed, thenon-existence of a king of France guarantees the truth of (6″).This reading is more naturally expressed with the fall-rise contourand continuation characteristic of metalinguistic negation (Horn 1989)as in (7):

(7)

The king of France isn’t \(^{{\rm v}}\)BALD—thereISN’T any king of France!

For Strawson (1950, 1952), negation normally or invariably leaves thesubject “unimpaired”. Strawson tacitly lines up with Fregeand against Russell (and Aristotle) in regarding negative statementslike (4b) and (6) as unambiguous and necessarily presuppositional.Someone who utters (6) does not thereby assert (nor does her statemententail) that there is a king of France. Rather, (6)—along withits affirmative counterpart (5)—presupposes it. If thispresupposition fails, a statement may be made but the question of itstruth value fails to arise.

While many analysts (e.g., Wilson 1975, Atlas 1977, Gazdar 1979, Grice1989) have since followed Russell by preserving a bivalent semanticsand invoking pragmatic explanations of apparent presuppositionaleffects, other linguists and philosophers (e.g., Fodor 1979,Burton-Roberts 1989, von Fintel 2004) have defended and formalizedtheories of semantic presupposition in the Frege-Strawson spirit,allowing for the emergence of truth-value gaps or non-classical truthvalues when presuppositions are not satisfied.

Non-bivalent logics of semantic presupposition, dating back toŁukasiewicz (1930) and Kleene (1952), generally posit (at least)twonot-operators, the distinction arising lexically ratherthan (as for Russell) scopally; see the entry onmany-valued logic andSection 2 below. The ordinary, presupposition-preserving internal or choicenegation is the only one countenanced by Frege and Strawson; on thisreading,Santa is not white, likeSanta is white, isneither true nor false, given that Santa does not exist. Thepresupposition-cancelling or exclusion negation always determines aclassical value. With exclusion negation,Santa is not white(or perhaps more plausiblyIt is not the case that Santa iswhite) is true even if there is no Santa. Thus there is noexcluded middle; any affirmation and its corresponding exclusionnegation are contradictories rather than contraries (see the entry onpresupposition for elaboration and further details).

1.6 From contradiction to contrariety: pragmatic strengthening of negation

In his dictum, “The essence of formal negation is to invest thecontrary with the character of the contradictory”, Bosanquet(1888, 306ff.) encapsulates the widespread tendency for formalcontradictory (wide-scope) negation to be semantically orpragmatically strengthened to a contrary. We can call the tendency forcontradictory (exhaustive) negation to be interpretively strengthenedto a contrary (antonymous) meaningMaxContrary. Onereflex isO> Edrift, whereinexpressions that “ought” to haveOsemantics (as the contradictory ofA) tend tostrengthen to anE-type value, while no there is noI>Adrift corresponding tonegative strengthening on the affirmative side of the Square.

We use \(\copy A\) to represent any contrary of \(A\). Following theAristotelian theory of opposition, the two contradictories \(A\) and\(\neg A\) cannot both be false, just as they cannot both be true,while a given proposition and a contrary of that proposition, \(A\)and \(\copy A\), can both be false, although they cannot both be true.(Others have used \(\kappa\) or R for one-place non-truth-functionalcontrariety connectives; cf. McCall 1967, Humberstone 2005; see alsoBogen 1991 for the distinction between linguistic and metaphysicalcontraries.) It should be noted that while \(\neg\) is an operatorthat takes one proposition into another, © is not, since a givenproposition may have logically distinct contraries, while this is notthe case for contradictories. Geach (1972, 71–73) makes thispoint with the example in (8). While (8a) has two syntacticallydistinct contradictories, e.g.,Not every cat detests everydog andIt’s not every dog that every cat detests,any such co-contradictories of a given proposition will always havethe same truth conditions. But (8a) allows two contraries withdistinct truth conditions, (8b) and (8c).

(8)

a.

Every cat detests every dog.

b.

No cat detests every dog.

c.

There is no dog that every cat detests.

Similarly, (9a) allows three non-identical contraries:

(9)

a.

I believe that you’re telling the truth.

b.

I believe that you’re not telling the truth.

c.

I don’t believe that you’re telling the truth or thatyou’re not; I haven’t made up my mind yet.

d.

I don’t believe that you’re telling the truth or thatyou’re not: I haven’t given the matter any thought.

Thus while we can speak of the contradictory of a proposition, Geachobserves, we cannot (pace McCall 1967) speak ofthecontrary, but only ofa contrary, of a proposition. AsHumberstone (1986, fn. 6) points out in response to Geach’scritique of McCall, however, the lack of uniqueness “does notprevent one from exploring the logical properties of an arbitrarilyselected contrary for a given statement”. For our purposes, thecrucial logical properties of contrariety are that (i) thecontradictory of a proposition \(A\) is not a contrary of \(A\) andthat (ii) contrariety unilaterally entails contradiction:

(10)

a.

\(\copy A \vdash \neg A\)

b.

\(\neg A \not \vdash\copy A\)

For McCall (1967), contrariety is a quasi-modal notion akin to logicalimpossibility, \(\Box \neg\), in that \(\Box \neg A\) entails \(\negA\) but notvice versa, but, as pointed out by an anonymousreviewer, there is no intrinsic modal component of contrariety; allthat is necessary is that contrariety is a non-truth-functionalone-place connective. (See Humberstone 1986, 2003, 2005; Bogen 1991;and Vakarelov 1989a for additional considerations.)

The strengthening of contradictory negation, \(\neg A\), to acontrary, \(\copy A\), typically instantiates the inference schema ofdisjunctive syllogism ormodus tollendo ponens in (11):

(11)

\(\begin{array}{l}A \vee B \\ \neg A \\ \hline B\end{array}\)

While the key disjunctive premise is typically suppressed, the role ofdisjunctive syllogism can be detected in a variety of strengtheningshifts in natural language where the disjunction in question ispragmatically presupposed in relevant contexts. Among theillustrations of this pattern are the following:

• The tendency for negation outside the scope of (certain) negatedpropositional attitude predicates (e.g.,a does not believe that\(p\)) to be interpreted as associated with the embedded clause(e.g.,a believes that not-\(p\)); this is so-called“neg-raising”, to which we return below.

• The tendency for a semantically contradictory negation of an unmarkedpositive value, whether affixal (\(x\)isunfair/unhappy) or clausal (I don’t likehim), to be strengthened (as either an “online” orconventionalized process) to a contrary of the positive predication.As contraries,Chris is happy andChris is unhappyallow an unexcluded middle, since Chris can be neither happy norstrictly unhappy; similarly,I don’t like him isgenerally understood as stronger than a mere assertion that it’snot the case that I like him.

• The strengthening of a negated plural definite (The kidsaren’t sleeping) or bare plural (Beavers don’teat cheese) from a contradictory to a contrary of thecorresponding affirmative. In each case, the negation is understood asinside the scope of the quantified subject.

As stressed by Bartsch 1973 (cf. Horn 1989, Chapter 5; Gajewski 2007),when there are only two alternatives in a given context, as in thecase of neg-raising, the denial of one (I don’t believe itwill rain) amounts to the assertion of the other (I believeit won’t rain). The relevant reasoning is an instance ofthe disjunctive syllogism pattern in (11), as seen in (12), where\(F\) represents a propositional attitude and \(a\) the subject ofthat attitude.

(12)

\(\begin{array}{ll} F (a, p)\vee F (a,\neg p)\, & {\scriptsize\mbox{[the pragmatically assumed disjunction]}}\\ \underline{\neg F(a, p)} & {\scriptsize \mbox{[the sentence explicitly uttered]}}\\F (a, \neg p) & {\scriptsize \mbox{[the stronger negativeproposition conveyed]}} \end{array} \)

The key step is the pragmatically licensed disjunction of contraries:if you assume I’ve made up my mind about the truth value of agiven proposition \(p\) (e.g., “it will rain”) rather thanbeing ignorant or undecided about it, then you will infer that Ibelieve either \(p\) or \(\neg p\), and my denial that I believe theformer (“I don’t think it will rain”) will lead youto conclude that I believe the latter (“I think it won’train”). See Horn 2020b for an overview of this phenomenon;Gajewski 2007 for a neo-Bartschian analysis; and Collins and Postal2014 for a vigorous defense of a grammatical approach to neg-raising.Todd 2020 and De Florio and Frigerio 2024 consider the role ofneg-raising in accounting for the apparent scopelessness of negationin its interaction with futurity (‘will ¬\(p\)’ vs.‘¬ will \(p\)’).

The availability of strengthened contrary readings for apparentcontradictory negation has long been recognized, dating back toclassical rhetoricians of the 4^th century on the figure oflitotes, in which an affirmative is indirectly asserted by negatingits contrary (Hoffmann 1987). Litotic interpretations tend to beasymmetrical: an attribution of “not happy” or “notoptimistic” will tend to convey a contrary (in this case“rather unhappy” or “fairly pessimistic”),while no analogous virtual contrariety is normally signaled by“not sad” or “not pessimistic”, which areusually understood as pure contradictories. This asymmetry isultimately a social fact motivated by politeness, and specifically bythe need to respect negative face, an individual’s desire forfreedom of action and freedom from imposition. (Ducrot 1973, Brown andLevinson 1987, Horn 1989, Ruytenbeek et al. 2017, Gotzner et al.2018).

For Jespersen, the tendency reflected by the neg-raised interpretationofI don’t think that \(p\) not only illustrates thegeneral strengthening to contrariety but also participates in a moregeneral conspiracy in natural language to signal negation as early aspossible. Additional effects of this “neg-first” principle(Horn 1989, 293; after Jespersen 1917, 5) range from diachronic shiftsin the expression of sentential negation (see van der Auwera 2010) andthe fronting and negative inversion in (1a) or (2a) to the emergenceof ambiguities arising in contexts like [neg \(S_1\) because \(S_2\)](Jespersen 1917, 48), as in “She didn’t marry him becausehe’s poor”, where the “illogical” scopereading—on which his poverty was the non-cause of the weddingrather than the cause of the non-wedding—can be rendered more orless accessible by the intonation contour.

The “neg-raised” reading ofI don’t think that\(p\) as “I think that not-\(p\)” has often beendeplored by grammarians or philosophers as an illogical placement ofnegation, an unfortunate ambiguity, or (in Quine’s terms) an“idiosyncratic complication” of one language:

the familiar quirk of English whereby “\(x\) does not believethat \(p\)” is equated to “\(x\) believes that not\(p\)” rather than to “it is not the case that \(x\)believes that \(p\)”. (Quine 1960, 145–6; similar claimsare made by other philosophers)

But this “quirk” has deep roots, dating back to St.Anselm’s 12th century Lambeth fragments (Henry 1967,193–94; Hopkins 1972, 231–32; Horn 1989, §5.2).Anselm points out that “non…omnis qui facit quod nondebet peccat, si proprie consideretur”—not everyonewho does what henon debet (“not-should”) sins,if the matter is considered strictly (with the contradictory readingof negation as the syntax suggests). The problem is thatnondebere peccare is standardly used to convey the contrary meaningdebere non peccare rather than the literal contradictory(“it is not a duty to sin”). It is hard to stipulate e.g.,non debet ducere uxorem (= “a man is free not tomarry”) without seeming to commit oneself to the strongerdebet non ducere uxorem, an injunction to celibacy (Henry1967, 193ff.; Horn 2010a: 201).

For Henry (1967, 193, §6.412), Anselm’s observations onmodal/negative interaction are “complicated by the quirks ofLatin usage”. But far from a Quinean quirk of English and/orLatin usage, “neg-raising”—the lower-clauseunderstanding of negation of abelieve- orought-type predicate—is distributed widely andsystematically across languages and operators.

The raised understanding is always stronger than the contradictory(outer) negation; it applies to a proper subset of the situations towhich the contradictory applies (is true in a proper subset ofpossible worlds). Thus neg-raising, as Anselm recognized, yields avirtual contrariety: the compositional meaning is true but too weak,and the addressee recovers a conversational implicature to “fillin” the stronger proposition.

In some cases, the strengthened or neg-raised contrary reading maybecome salient enough over time to block the literal interpretation,as when FrenchIl ne faut pas partir—literally =“one needn’t leave” (anO vertexmodal)—is now generally used only to express the strongerproposition that one must not-leave (E vertex). Thisis a modal instance of the general phenomenon ofO\(>\)E drift, an upward shift along the right(negative) vertical of the modal square of opposition. Such squareswere constructed by Cajetan, based on Aristotle’sDeInterpretatione 21b10ff. andPrior Analytics32a18–28 (see Oesterle 1962), and by other medievalcommentators.

Figure 1

\(\mathbf{O}>\mathbf{E}\) drift is attested cross-linguistically inthe meaning shift of lexical items like Old Englishnealles(lit. “NEG all”) = “not at all”, Dutchnimmer (lit., “NEG always”) =“never”, or Russiannel’zja (lit.“NEG must”) = “mustn’t”, and in wordslikeunlikely, inadvisable, ordisbelieve, whoseprefixal negations yield only contrary, not contradictory,interpretations. The reverse shift, in which apparentE forms developO meanings, appearsto be unattested (cf. Horn 2015).

In litotes and neg-raising, the interpretation of formalcontradictories as contraries arises from the accessibility of therelevant disjunction, triggering the disjunctive syllogism. Thehomogeneity or all-or-none presupposition (Fodor 1970) applying tobare plurals, plural definites, and mass predications results in acomparable effect; it is natural to strengthen negative statementslikeMammals don’t lay eggs,The childrenaren’t sleeping, orI don’t eat meat toaffirmations of contraries rather than understanding them as simplewide-scope negations of the corresponding positives (Mammals layeggs,The children are sleeping,I eat meat) aswould be the case with overtly quantified universals. The relevantprinciple has been variously formulated:

When a kind is denied to have a generic property P\(_k\), then any ofits individuals cannot have the corresponding individual-levelproperty P\(_i\). (von Fintel 1997, 31)

If the predicate P is false for the NP, its negation not-P is true forthe NP… Whenever a predicate is applied to one of itsarguments, it is true or false of the argument as a whole.(Löbner 2000, 239)

Once again the key step is establishing the relevant disjunction as apragmatically inferred instance of the Law of Excluded Middle, e.g.,“Either mammals lay eggs or mammals don’t lay eggs”.In fact, this practice was first identified by Aristotle (Soph.Elen. 175b40–176a17), who offered an early version of theall-or-none (or both-or-neither) in arguing that a negative answer toa “dialectical” or conjoined question like “AreCoriscus and Callias at home?” would imply that neither is athome, given the default supposition that they are either both in orboth out. Once again LEM applies where it“shouldn’t”; \(A \vee \copy A\) behaves as though itwere an instance of \(A \vee \neg A\), triggering the disjunctivesyllogism:

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\(\begin{array}{l} (Fa \wedge Fb) \vee (\neg Fa \wedge \neg Fb)\\\underline{\neg (Fa \wedge Fb)}\\ (\neg Fa \wedge \neg Fb)\end{array}\)

Other instances of the maximization of contrariety in naturallanguage, in a range of contexts from formal pragmatics to wordlearning, are discussed in Horn 2015.

1.7 Privation, affixal negation, and the markedness asymmetry

For Aristotle, privation is an instance of opposition defined in termsof the absence or presence of a default property for a givensubject:

We say that that which is capable of some particular faculty orpossession has suffered privation [sterêsis] when thefaculty or possession in question is in no way present in that inwhich, and at the time in which, it should be naturally present. We donot call that toothless which has not teeth, or that blind which hasnot sight, but rather that which has not teeth or sight at the timewhen by nature it should. (Categories 12a28–33)

A newborn kitten, while lacking sight, is thus no more“blind” than is a chair, nor is a baby“toothless”.

Privation as the absence of what would be expected by nature to bepresent is revisited in theMetaphysics(1022b23–1023a8), where Aristotle—noting that privationcan range over predictable absence, accidental removal, or deliberate“taking away by force” of the relevantproperty—distinguishes privation “with respect togenus”, as in the blindness of moles, from privation “withrespect to self”, as in the blindness or toothlessness of an oldman. In the end, Aristotle concedes, there may be as many senses ofprivation as there area- prefixed terms in Greek(Met. 1022b33). Indeed, privation may be reanalyzed as theprimary contrariety (1055a34).

In a wide range of languages, affixal negation on simplex basesreflects Aristotelian privation, whence the asymmetry between possibleforms (unhappy, untrue, unkind) and impossible or unlikelyones (unsad, unfalse, uncruel). We can describe a failedcomedy, but not a successful tragedy, asunfunny. AsJespersen (1917, 144) observes, the tendency of semi-productivenegative affixation to be restricted to unmarked or positive basescombines with that of the preference for contrariety reviewedabove:

The modification in sense brought about by the addition of the prefixis generally that of a simple negation:unworthy = “notworthy”, etc… The two terms [X,unX] are thus contradictory terms. But veryoften the prefix produces a “contrary” term or at any ratewhat approaches one:unjust generally implies the opposite ofjust;unwise means more than not wise and approaches foolish,unhappy is not far from miserable, etc.

The counter-expectation property of affixal negation extends even tocontradictory, middle-excluding adjectives likealive/dead; nothing can be both and nothing capableof being either can be “in between”. Butundeadhas been around since Bram Stoker’sDracula (1897) asboth an adjective and a zero-derived occupational noun to describezombies, vampires, and other creatures that are “not quite deadbut not fully alive, dead-and-alive” (OED). Someone or somethingis undead—e.g., a vampire—if it fails to conform toone’s expectation that itshould be dead. Butif something appears to be alive but does not quite fulfill thatexpectation, it is notundead butunalive, e.g.,artificial flowers. Both the undead (but not quite alive) vampire andthe unalive (but not dead) artificial flowers conform toAristotle’s notion of a privative opposite in lacking a propertyassociated by default rules with the respective subject.

The marked status of negative utterances has also been invoked tomotivate a lexicalization asymmetry in the geometry of the Square ofOpposition. Traditionally, the Aristotelian relations ofcontradiction, contrariety, and subalternation are supplemented withan additional relation of subcontrariety, so called because thesubcontraries are located under the contraries. As the contradictoriesof the two contraries, the subcontraries (e.g.,Some pleasure isgood,Some pleasure is not good) can both be true, butcannot both be false. For Aristotle, this was therefore not a trueopposition, since subcontraries are “merely verballyopposed” (Prior Analytics 63b21–30). Withinpragmatic theory, the assertion of one subcontrary (Some men arebald) is not only compatible with, but actually conversationallyimplicates, the other (Some men are not bald), givenGrice’s Maxim of Quantity (“Make your contribution asinformative as is required”; see the entries onPaul Grice,pragmatics, andimplicature). The fact that the two members of a subcontrary pair tend to beequipollent or mutually derivable in a given context helps motivatethe fact that only one of the two subcontraries will lexicalize innatural language, while the markedness of negation explains why thisis always the positive (I vertex, e.g.,some) rather than the negative (O vertex,e.g.,no) value (Horn 2012 and work cited there). Thus,E valuesnone,nor, andnever are possible but the correspondingOvalues *nall (“not all”), *nand(“or not”), and *nalways (“notalways”) are never attested. Similar, if less absolute,asymmetries obtain among non-quantificational and indeed non-logicalvalues (van der Auwera 1996). Various competing explanations to theimplicature-based account have been proposed for motivating theseasymmetries; see, inter alia, Jaspers 2005, Seuren and Jaspers 2014,and references therein.

Recent work (e.g. Enguehard and Spector 2021; Züfle and Katzir2022; Uegaki 2024) utilizes computational models and draws onexperimental studies to support and extend the grounding of theexplanation for the three-cornered square (Horn 1989, 252) and relatedasymmetries of expressibility in communicative efficiency. Asrecognized since classical rhetoric, efficiency reflects the trade-offor tug-of-war between speaker-based economy and hearer-motivatedinformativity (given pragmatic enrichment), shaping both the forms ofexpression and linguistic change: “Languages maximizecommunicative efficiency through a trade-off between informativity andsome notion of lexical complexity” (Enguehard and Spector 2021,16; compare the title of Steinert-Threlkeld 2019, “Quantifiersin natural language optimize the simplicity/informativenesstrade-off”).

1.8 Double negation

1.8.1 “Logical” double negation

Whenduplex negatio affirmat, what exactlydoes the double negation affirm? When a negative termis a contrary rather than a contradictory of the corresponding simpleaffirmative, to deny its application—Socrates isn’t anot-white log—does not result in the mutual annihilation oflogical double negation, any more than does the negation of a mediatecontrary (She’s not unhappy, It isn’t uncommon).While Aristotle countenanced multiple negation, to the extent ofgenerating such unlikely sequences asNot-man is not not-just(De Int. 19b36), each proposition contains only one instanceof negation as wide-scope predicate denial (juxtaposed here with botha negated subject term and a negated predicate term), since eachcategorical statement contains only one predicate.

By contrast, the Stoics defined negation (apophatikon) as aniterating external operator. For Alexander of Aphrodisias,“Not: not: it is day differs fromit is dayonly in manner of speech” (Mates 1953, 126). With theirpropositional connectives and one-place truth/falsity-togglingnegation operator, it is the Stoics rather the Aristotelians whoprefigured modern propositional logic, as well as the precepts oftraditional grammar (“Duplex negatio affirmat”)and the Law of Double Negation.^[5]

Classical Fregean logic allows for but one negative operator, thecontradictory-forming propositional operator applying to a propositionor open sentence, in keeping with “the thesis that all forms ofnegation are reducible to a suitably placed ‘it is not the casethat’” (Prior 2006, 524). Not unexpectedly, Frege (1919,130) proclaims the logical superfluity of double negation:“Wrapping up a thought in double negation does not alter itstruth value”. Within this metaphor, \(\neg\neg A\) is simply away of garbing the thought or proposition \(A\).

But, as noted in §1.1, even a single sentence-external negation(Not: The sun is shining) is a logician’s constructrarely attested in the wild (Geach 1972; Katz 1977):

[P]ropositional negation was as foreign to ordinary Greek as toordinary English, and [Aristotle] never attained to a distinctconception of it. The Stoics did reach such a convention, but in doingso they violated accepted Greek usage; their use of an initialoukhi must have read just as oddly as sentences like“Not: the sun is shining” do in English. (Geach 1972,75)

Further, whether or not we admit the law of double negation in ourlogic, “in ordinary language a doubly negated expression veryseldom, if ever, has the same logical powers as the original unnegatedstatement” (Hintikka 1968, 47).

It is thus worth noting that the system of dual negations described byAristotle inPrior Analytics I, Chapter 46 is both insightfuland internally consistent; its echoes are recognizable inJespersen’s distinction between nexal negation (nothappy) and special negation (unhappy), VonWright’s (1959) distinction between weak (contradictory) versusstrong (contrary) negation, and Jackendoff’s (1969) semanticreanalysis of Klima’s (1964) grammatical categories ofsentential versus constituent negation. In each case, a negativemarker whose scope is narrower than the proposition determines astatement logically distinct from a simple contradictory.

If we represent the narrow-scope contrariety operator ofIt isnot-white as \(\copy A\), its contradictory, \(\neg\copyA\)(It isn’t not-white), does not return us to thesimple positive \(A\). The result, although not the means, is similarto the effect of double negation within intuitionistic logic (Heyting1956). The intuitionistic negation operator does not cancel out, giventhat the Intuitionistic Law of Double Negation is valid in only onedirection, \(A \rightarrow \neg \neg A\), while \(\neg \neg A\rightarrow A\) does not apply (see the entry onintuitionistic logic). Intuitionists posit just one negation operator that sustains doubleintroduction but not double cancellation, while the Aristoteliansystem distinguishes contradictory (sentential) predicate denial fromcontrary (sub-sentential or constituent) term negation.

In ordinary language, semantic double negation (as opposed to negativeconcord as inI ain’t never done nothing to nobody, anagreement phenomenon in which only one semantic negation is expressed,addressed in the next section) tends not to cancel out completely.This is predictably the case when a semantic contrary is negated:not uncommon is weaker thancommon; one can be notunhappy without being happy. (For more on the complex semanticrelations between affixal negation, as represented e.g. byun-,in-,non-, and thecontradictory/contrary distinction, see Horn 1989, Bauer et al. 2013,and Joshi 2020.) But even when an apparently contradictory negation isnegated (from the unexceptionableit’s not impossibleto the more transgressive double-not of Homer Simpson’sconcessiveI’m not NOT licking toads[http://tinyurl.com/34jwhjz]), theduplex negatio of \(A\)doesn’t affirm \(A\), or at least it provides a rhetoricallywelcome concealment, as suggested by Frege’s metaphor of“wrapping up a thought” in double negation. The negationin such cases (impossible,not-licking) is coercedinto a virtual contrary whose negation, \(\neg\copy A\), is weakerthan (is unilaterally entailed by)A; see Krifka2007, Horn 2017, and the entry oncontradiction). But even if \(A\) and¬¬A present the same thoughtin different wrapping, as Frege (1897: 242) observes in a differentconnection, “we are not in a position to say that it is a matterof complete indifference which of these sentences we use”.

Figure 2

1.8.2 Negative concord and its relations

In the previous section it was observed that whenduplex negatioaffirmat, what it affirms is often not simply the doubly negatedproposition but the result of an incomplete cancellation yielded bythe negation of an actual or virtual contrary (not unlikely,not impossible). But a more dramatic problem for the dictumis whenduplex negatio negat, especially in the form ofnegative concord, in which a single logical negation on the main verbspreads to indefinites and adverbs within the same clause (Labov 1972,Zeijlstra 2004, Penka 2011).^[6]

The grammar of negative concord and its relation to double negation isoften complex and subject to a variety of factors (Giannakidou 2020,de Swart 2020). In standard Italian, for example, negative quantifiersfollowing the main verb (whether as objects or postposed subjects)co-occur with mandatory negative marking on the verb to yield a singlenegative meaning, as in (14a). But when a negative quantifier precedesthe verb, negative concord is ruled out, as in (14b).^[7]

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a.

Gianni *(non) ha visto nessuno. “Gianni has seennobody”
*(Non) ha telefonato nessuno. “Nobody has telephoned”
*(Non) ho parlato con nessuno. “I have spoken withnobody”

b.

Nessuno (*non) ha visto Gianni. “Nobody has seenGianni”
Con nessuno (*non) ho parlato. “With nobody have Ispoken”

Negative concord is a feature of many non-standard varieties ofEnglish, especially in informal speech—or lyrics (“Ican’t get no satisfaction”).^[8] The grammar of negative concord in African American VernacularEnglish has been especially well studied; see Green 2002 for aninfluential analysis.

True negative concord within a given clause represents just one kindof hypernegation, the general phenomenon in which a negative markerreinforces rather than cancels the ordinary or canonical marker ofsentence negation. Hypernegation may extend across clause boundariesto result in the occurrence of “pleonastic” or“expletive” negative elements in the scope of inherentlynegative predicates (cf. Espinal 1992, Delfitto 2020). This isexemplified by the negative markers following comparatives,before clauses, or verbs of fearing in French, Russian,Yiddish, and other languages. A standard feature of earlier stages ofEnglish, pleonastic negation persists in colloquial English:

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a.

I miss (not) seeing you around.

b.

Don’t be surprised if it doesn’t rain. [= if itrains]

c.

Not with my wife, you don’t.

d.

The proposal will not be approved, I (don’t) think.

The well-known problems encountered in processing multiple negations,verified in many psycholinguistic studies, are responsible for theappearance of other uninterpreted negations as in (16a), and theconventionalized irony or sarcasm exemplified in (16b,c); see Wason& Reich 1979, Horn 2010b, and the numerous entries and links inthe Language Log “Archive for Misnegation” under“Other Internet Resources”.^[9]

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a.

No head injury is too trivial to ignore.

b.

I could care less.

c.

That’ll teach you to ever mess with me again.

Similarly, in French the expressionVous n’êtes passans ignorer que …, literally “You are not withoutbeing ignorant that …”, is notoriously used in the senseof “You certainly know that …”. Ifduplexnegatio affirmat, thentriplex negatio confundit.

1.9 Negative polarity

Certain linguistic expressions in English and other languages arepolarity sensitive, restricted in their distributions to the scope ofnegation or semantically related contexts, including negativequantifiers, implicitly negative predicates or adverbs, theantecedents of conditionals, comparative clauses, and the restrictorsof universals:

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a.

I {haven’t/*have}ever eatenany kumquatsat all.

b.

{Few/*Many} of the assignments have been turned inyet.

c.

The dean {rarely/*often}lifts a finger to helpstudents on probation.

d

I {doubt/*believe} they’reall that pleasedwith the proposal.

e.

{All/*Many} customers who hadever purchasedany of the affected items were(*ever) contacted.

Negative polarity items (NPIs) like those highlighted in (17) aregenerally restricted to downward entailing or monotone decreasingcontexts, those in which inferences from sets to subsets (but notvice versa) are valid (see Fauconnier 1975; Ladusaw 1980,1996; Peters and Westerståhl 2006; and thegeneralized quantifiers entry). If I’ve eaten kumquats, I’ve eaten fruit, but notnecessarilyvice versa; this is an upward entailing (monotoneincreasing) environment. On the other hand, if I haven’t eatenfruit, I haven’t eaten kumquats, but not necessarilyviceversa; this is a downward entailing (monotone decreasing) environment.^[10] It is just in the latter case that NPIs are licensed.

As (17e) shows, universals likeall orevery licenseNPIs in their restrictor (the relative clause), which is a downwardentailing context (if everyone who knows logic is a vegetarian,everyone who knows classical logic is a vegetarian, but notviceversa). But universals do not license NPIs in their nuclear scopeor predicate expression, which is an upward entailing context (ifeveryone who knows logic is a vegan, everyone who knows logic is avegetarian, but notvice versa). This contrast demonstratesthe insufficiency of an account of polarity licensing that simplymarks a given lexical item as favorable to the occurrence of NPIswithin its scope (Ladusaw 1980).

While downward entailment may be (generally) necessary for thelicensing of NPIs (although there are some thorny issues to resolve;cf. Giannakidou 2011, Israel 2011), it is not necessarily sufficient,depending on the nature of the context and the NPI in question. Forexample, some environments that permit weak NPIs likeany andever fail to license stricter ones likein weeks oruntil midnight.

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a.

{Nobody/Only Chris} hasever provedany of those theorems.

b.

{Nobody/*Only Chris} has been herein weeks.

This has led to the development of more stringent algebraic conditionsthat some polarity items must meet, e.g., anti-additivity (Zwarts1998). The distribution and licensing of polarity items, subject towidespread variation within and across languages, is an important butextremely complex linguistic phenomenon with implications for thearchitecture of grammar and the theory of meaning; see Israel 2011,Giannakidou 2011, Chierchia 2013, Horn 2016, Barker 2018 for extensivediscussion and alternative theoretical approaches. Van der Wouden(1996), Zeijlstra (2013), and Blanchette (2015) provide usefulexaminations of the connections of negative polarity to negativeconcord and expletive negation.

1.10 Metalinguistic negation

In addition to the overlapping dichotomies we have surveyed betweengrammatically and semantically defined varieties of negation within agiven language (wide- vs. narrow-scope, sentential vs. constituent,contradictory vs. contrary, choice vs. exclusion), a “pragmaticambiguity” has been invoked to distinguish ordinary descriptivenegation from a specialized metalinguistic or echoic use (Horn 1989,chapter 6).^[11] In examples like (19), a speaker objects to a previous utterance on avariety of grounds, including its phonetic or grammatical form,register, or associated presuppositions or implicatures:

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a.

Around here we don’t LIKE coffee—we LOVE it.

b.

She doesn’t sell INsurance—she sells inSURance.

c.

It’s not stewed bunny, honey, it’scivet delapin.

d.

I’m not HIS brother—he’s MY brother!

e.

Mozart’s sonatas were for piano and violin, not for violinand piano.

The descriptive/metalinguistic distinction is supported by converginglinguistic diagnostics suggesting that metalinguistic negationoperates on a different level, whence its failure to incorporatemorphologically or license negative polarity items:

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a.

I’m {not happy/*unhappy} with the plan, I’mecstatic!

b.

You didn’t eat {some/*any} of the cookies, you ate themall!

The coherence of the notion of “pragmatic ambiguity” and,more generally, the proper treatment of metalinguistic negation (or,following Carston 1996, echoic negation) have been matters ofconsiderable dispute; for a range of competing views, see McCawley1991, Geurts 1998, Burton-Roberts 1999, and Carston 1999. Pitts 2011,Davis 2016, and Martins 2020 provide excellent overviews of thephenomenon (or phenomena) in question from different perspectives.

The linguistic expression of negation, its interaction with thephenomena of negative polarity, concord, and scope, and the mappingbetween negative form and negative meaning all present complex andimportant problems for syntax, semantics, and pragmatics. (Recentperspectives on a range of issues involved are provided in Atlas 2012,Horn 2018, and work cited therein.) But perhaps the most strikingfeature of negation is the mismatch between the formal anddistributional complexity of negative elements in natural languagesand the simplex character of the one-place negative operator ofpropositional logic. Yet here too the apparent simplicity is deceivingonce we undertake a systematic investigation of the logic of negation.It is to that task that we now turn.

2. The logic of negation

The logic of negation may be presented in quite different ways, byconsidering various styles of proof systems (axiom systems, sequentcalculi, systems of natural deduction, tableaux, etc.) or differentkinds of semantics (algebraic, model-theoretic, proof-theoretic,game-theoretic, etc.). Moreover, in search of characteristics ofnegation as a one-place connective, several dimensions ofclassification are available, depending on the logical vocabulary ofthe language under consideration (propositional, first-order,multi-modal, etc.) and the inferential framework taken into account(single antecedents (i.e., premises) and conclusions, multipleantecedents or multiple conclusions, sets, multisets, or sequences offormulas in antecedent or succedent position).

In a very elementary setting one may consider the interplay betweenjust a single sentential negation, \(\osim\), and the derivabilityrelation, \(\vdash\), as well as single antecedents and singleconclusions. The following inferential principles are stated as properrules with one derivability statement (sequent) or two such statementsas assumption sequent(s) and a single sequent as the conclusion, or asaxiomatic sequents without any assumption sequent:

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\(\begin{align*} A \vdash B \, &/ \, \osim B \vdash \osim A& \mbox{(contraposition)}\\ A &\vdash \osim \osim A &\mbox{(double negation introduction)}\\ \osim \osim A &\vdash A& \mbox{(double negation elimination)}\\ A \vdash B, \; A \vdash\osim B \, &/ \, A \vdash \osim C & (\text{negative} \textit{ex contradictione})\\ A \vdash B, \; A \vdash \osim B \, &/ \, A\vdash C & (\text{unrestricted }\textit{ex contradictione})\\ A\vdash \osim B \, &/\, B \vdash \osim A & \mbox{(constructivecontraposition)}\\ \osim A \vdash B \, &/ \, \osim B \vdash A& \mbox{(classical contraposition) } \end{align*}\)

The first rule, contraposition, for instance, says that if \(B\) isderivable from \(A\), then the negation of \(A\) is derivable from thenegation of \(B\). All these rules and derivability statements arevalid in classical logic (see the entry onclassical logic); classical logic cannot distinguish between them. Some of theseprinciples have been criticized and called into question innon-classical logic. The unrestricted and the negativeexcontradictione rules, for example, introduce an element ofirrelevancy because they allow one to derive a completely arbitraryformula \(C\), respectively a completely arbitrary negated formula\(\osim C\), from an assumption \(A\) if a formula \(B\) as well asits negation \(\osim B\) are derivable from \(A\), see the entries onrelevance logic andparaconsistent logic. Classical contraposition has been criticized because it gives rise tonon-constructive existence proofs in languages containing theexistential quantifier, see the entry onintuitionistic logic. In richer vocabularies, additional negation principles can beformulated, regimenting the interaction between negation and otherlogical operations. Prominent examples are the De Morgan Laws. Inlanguages without implication, one may consider the following sequentsstating De Morgan inference rules:

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\(\begin{align} (\osim A \vee \osim B) &\vdash \osim (A \wedgeB)\\ \osim (A \vee B) &\vdash (\osim A \wedge \osim B)\\ (\osim A\wedge \osim B) &\vdash \osim (A \vee B)\\ \osim (A \wedge B)&\vdash (\osim A \vee \osim B)\\ \end{align}\)

Whereas classical logic validates all of these rules, intuitionisticlogic validates only the first three of them.

The proof-theoretical characterization of negation is important forthe use of negation connectives in derivations. To obtain a morecomprehensive understanding of negation, however, it is instructive tosupplement the proof-theoretic perspective by model-theoreticconsiderations. We first consider truth tables.

2.1 Negation as a truth function

In classical logic, the semantic principle of bivalence is assumed,saying that a formula has exactly one of two semantic values, namelyeither the value T[rue] or the value F[alse] (1 or 0), but not both.Negation, \(\osim \), is semantically characterized by the unaryfunction \(f_{\osim }\) on the set \(\{1, 0\}\), defined by thefollowing truth table:

That is, if \(A\) is a formula, then \(\osim A\) is false if \(A\) istrue, and \(\osim A\) is true if \(A\) is false. The function\(f_{\osim }\) is said to be a truth function because it is a functiondefined on the set of classical truth values \(\{1, 0\}\), see theentry ontruth values.

If negation is meant to express semantic opposition, it is clear thatthe remaining two-valued unary truth functions fail to characterizeany plausible notion of semantic opposition between \(A\) and \(\osimA\):

However, if a distinction is already drawn betweencontradictory-forming and contrary-forming sentential negations, theground is prepared for a pluralism with respect to negation seen as aunary connective. One might think of obtaining different concepts ofnegation by letting the negations interact with other logicaloperations in various ways, but this does not help concerning atomicformulas that do not contain any logical operation.

There are several ways of generalizing the semantics and making roomfor additional sentential negations. One comes with giving upbivalence and admitting sets of truth values (truth degrees) with morethan two elements, see the entry onmany-valued logic. In the so-called Łukasiewicz many-valued logicsŁ\(_{\aleph_{1}}\),Ł\(_{\aleph_{o}}\), andŁ\(_n\), the set of values is either the wholereal unit interval [0,1], or the whole rational unit interval [0,1],or the finite set of rational numbers \(\{ 0, \frac{1}{n-1},\frac{2}{n-1}, \ldots, 1 \}\). These sets include 1 as the designatedvalue representing True, where the designated values of a many-valuedlogic are the values that are preserved in valid inferences.Łukasiewicz negation \(\osim \) is defined by setting \(f_{\osim}(u) = 1- u\). Negation is thus understood in terms of subtractionfrom the numerical representation of True. In so-called Gödelmany-valued logics, the truth function \(f_{\osim }\) for negation\(\osim \) is defined by setting \(f_{\osim }(u) = 1\) if \(u = 0\),and \(f_{\osim }(u) = 0\) if \(u \not = 0\). Here negation isunderstood in terms of the numerical representation of True anddistinctness from the numerical representation of False.

In Kleene’s (strong) three-valued logicK3,with \(i\) as a third value in addition to 0 and 1, the truth function\(f_{\osim }\) for negation \(\osim \) is defined by the same table asthe table for negation inŁ\(_3\), replacing\(\frac{1}{2}\) by \(i\):

InK3 andŁ\(_3\) a formula\(A\) and its negation \(\osim A\) cannot both be true in the sense ofboth taking the designated value 1, but they both fail to be true if\(A\) receives the value \(i\), respectively \(\frac{1}{2}\). If acontrary pair of formulas is defined as a pair of formulas that cannotboth be true but may both fail to be true, then Kleene negation givesrise to contrary pairs.

Also, falsity (understood as receiving the value 0) and non-truth(understood as taking a value different from 1) fall apart inK3. As a result, contraposition fails inK3. Another example of a logic with anon-contraposable negation is Priest’s Logic of Paradox,LP, where negation is interpreted by \(f\osim \) andboth \(i\) and 1 are designated values, see the entry onparaconsistent logic. If inK3 or inLP an implication\((A \supset B)\) is defined as material implication \((\osim A \veeB)\), then contraposition holds in the sense that \((A \supset B)\)entails \((\osim B \supset \osim A)\).

If we think of negation inK3 as representing anatural language negation,K3’s“internal”, presupposition-preserving negation \(\osim \)differs from the external, presupposition-cancelling negation \(\neg\)in Bochvar’s three-valued logicB3 by alwaysreturning a classical value. The truth function \(f_{\neg}\) isdefined by the following table:

It is also possible to use the machinery of possible worlds models tosemantically define various notions of negation. Negation is thentreated as a modal operator.

2.2 Negation as a modal operator

Since modal operators are unary connectives and since there existdifferent notions of alethic necessity (necessary truth) and alethicpossibility (possible truth), a rather natural question then iswhether negations can be analyzed in a revealing way as modaloperators, see the entry onmodal logic.

Very well-known modal logics are thenormal modallogics that have a so-called possible worlds semantics making use of atwo-place relation between possible worlds. Slightly less known aretheclassical (or congruential) modal logics(Segerberg 1971, Chellas 1980, Pacuit 2017). The weakest requirementimposed on a necessity-like modal operator \(\Box\) in systems ofclassical modal logic is the congruence property: \[ \mathord{\vdash A \leftrightarrow B} \slashrel \mathord{\vdash \Box A \leftrightarrow \Box B}\](“if \(A \leftrightarrow B\) is provable, then so is \(\Box A\leftrightarrow \Box B\)”). This property, however, is certainlynot distinctive of negation.

Classical modal logics have a semantics in terms of so-called minimalmodels, also known asneighbourhood models. Aneighbourhood model is a structure \(\cal M\) \(=\) \((W, N, v)\),where \(W\) is a non-empty set of possible worlds, \(N\) is a functionassigning to every \(w\) from \(W\) a set \(N(w)\) of subsets of\(W\), called neighbourhoods of \(w\), and \(v\) is a valuationfunction mapping atomic formulas to the set of worlds where they aretrue. Let \(\llbracket A\rrbracket\) be the set of worlds at whichformula \(A\) is true. Then \( \Box A\) is defined to be true at aworld \(w\) in model \(\cal M\) (in symbols: \({\cal M}, w \models\Box A\)) iff \(\llbracket A\rrbracket \in N(w)\).

In Ripley 2009 it is suggested to use the neighbourhood semantics as ageneral framework for semantically capturing properties characteristicof negation connectives interpreted as a necessity operator \( \Box\), see also Yu 2010. Ripley points out, for example, that thecontraposition rule \[ A \vdash B \slashrel \Box B \vdash \Box A\] is valid in a neighbourhood model\((W, N, v)\) iff for every \(w \in W\), \(N(w)\) is closed undersubsets, i.e., if \(X \in N(w)\) and \(Y \subseteq X\), then \(Y \inN(w)\). It would be nice to have a convincing intuitive understandingof the neighbourhood function \(N\) in terms of a concept thatexplains some core aspects of negation. If \(\llbracket A\rrbracket\in N(w)\) is understood as saying that the proposition expressed by\(A\) is incompatible with world \(w\), then the above constraintemerges as reasonable because it says that if the set of worlds (theproposition) \(X\) is incompatible with \(w\) and proposition \(Y\)implies \(X\), then \(Y\) is incompatible with \(w\) as well. WhereasRipley starts with a positive notion (\({\cal M}, w \models \Box A\)iff \(\llbracket A\rrbracket \in N(w)\)), in order to introduce anegation \(\osim \), one may also stipulate that \({\cal M}, w \models\osim A\) iff \(\llbracket A\rrbracket \not \in N(w)\), so as toobtain a connective that is more overtly a negative impossibilityoperator (although it is the clause for \(\Box A\) that is classicallynegated). The idea is that \(N(w)\) contains the sets of worldscompatible with \(w\), so that \(\llbracket A\rrbracket \not \inN(w)\) indicates that the proposition expressed by \(A\) isincompatible with \(w\). Negation as an “unnecessity”operator \(\neg\) in the sense of (“possibly not”) is thendefined by \({\cal M}, w \models \neg A\) iff \(\overline{\llbracketA\rrbracket} \in N(w)\), where \(\overline{\llbracket A\rrbracket}\)is the complement of \(\llbracket A\rrbracket\) with respect to \(W\).As a result, \(\neg A\) is true at a state \(w\) iff the complement ofthe proposition expressed by \(A\) is compatible with \(w\).

This semantics validates respective versions of congruence(\(\mathord{\vdash A \leftrightarrow B} \slashrel \mathord{\vdash\osim A \leftrightarrow \osim B}\) and \(\mathord{\vdash A\leftrightarrow B} \slashrel \mathord{\vdash \neg A \leftrightarrow\neg B}\)), but it does not yet impose any interesting constraints onnegation. In order to exclude that for some world \(w\) and formula\(A\), both \(w \in \llbracket A\rrbracket\) and \(w \in \llbracket\osim A\rrbracket\), one has to stipulate that for every set of worlds\(X\), if \(w \in X\) then \(X \in N(w)\), which makes sense under thecompatibility reading of the neighbourhood function \(N\) because itsays that if \(X\) is true at \(w\), then \(X\) is compatible with\(w\). In order to validate contraposition, it has to be required thatif \(X \subseteq Y\), then \(\{w\mid Y \not \in N(w)\}\) \(\subseteq\)\(\{w\mid X \not \in N(w)\}\). Under the compatibility reading of\(N\) this condition says that every world \(Y\) is incompatible withis also a world \(X\) is incompatible with, if proposition \(X\)implies proposition \(Y\). Similar conditions can be imposed for thedual negation that requires complementation.

Therelational semantics of normal modal logics, however,does come with a commitment to a substantial property of negationunderstood as impossibility or as unnecessity. The analysis ofnegation as a normal impossibility operator can be traced back to workby Birkhoff and von Neumann (1936) and Goldblatt (1974) on negation inquantum logic. It has been developed by Vakarelov (1977, 1989b) andDošen (1984, 1986, 1999) and has been further investigated inthe algebraic setting of Michael Dunn’s gaggle theory (seeBimbó and Dunn 2008) by Dunn (1993, 1996, 1999) and Dunn andZhou (2005). Arelational model (or Kripke model) isa structure \(\cal M\) \(=\) \((W, R, v)\), where \(W\) is a non-emptyset of information states, \(R\) is a two-place“accessibility” relation on \(W\), and \(v\) is avaluation function. Dunn denotes the accessibility relation by\(\bot\) (pronounced “perp”) and regards it as a relationof incompatibility or orthogonality between states. Negation asimpossibility, denoted by \(\osim \), is then semantically defined bypostulating that \(\osim A\) is true at a state \(w\) in model \(\calM\) iff \(w\) is incompatible with all states \(u\) (from \(W\)) atwhich \(A\) is true: \({\cal M}, w \models \osim A\) iff (for every\(u\): \({\cal M}, u \models A \mbox{ implies } w \bot u).\)Alternatively, the relation \(R\) may be understood as a relation ofcompatibility between states, denoted by \(C\). \({\cal M}, w \models\osim A\) is then defined by requiring that for every \(u\): \(wCu\mbox{ implies } {\cal M}, u \not \models A.\) Negation asunnecessity, denoted by \(\neg\), is accordingly defined by thefollowing clause: \({\cal M}, w \models \neg A\) iff (there exists\(u\) with \(wCu \mbox{ and } {\cal M}, u \not \models A)\).

It proves useful to enrich the above relational semantics by anotherbinary relation \(\leq\) on the set of states \(W\). The relation\(\leq\) is assumed to be a partial order (i.e., it is reflexive,transitive and anti-symmetric), which allows one to think of it as arelation of possible expansion of information states. With such areading it is natural to assume that the truth of atomic formulas\(p\) is persistent with respect to \(\leq\): if \(w \leq u\) and\({\cal M},w \models p\), then \({\cal M},u \models p.\) Theconditions on \(\leq\) and \(C\) and the truth conditions for compoundformulas should then be such that persistence (also called heredity)holds for arbitrary formulas, in particular for negated formulas\(\osim A\) if negation as impossibility is considered. Acompatibility model is a structure \((W,C,\leq, v)\),where \((W,C,v)\) is a Kripke model, \(\leq\) is a partial order on\(W\), and the following condition is satisfied, which guarantees theheredity of negated formulas \(\osim A\): if \(wCu\), \(w' \leq w\),and \(u' \leq u\), then \(w'Cu'\). This condition is a constraint onthecompatibility frame \((W,C,\leq)\) on which amodel \((W,C,\leq,v)\) is based. The condition is not only useful (aswill become clear), but also plausible, because it says that twoinformation states, expansions of which are compatible, are themselvescompatible.

We can now define that a sequent \(A \vdash B\) is valid in acompatibility model iff for every state \(w\) from that model, if\(A\) is true at \(w\), then so is \(B\); \(A \vdash B\) is calledvalid on a compatibility frame iff \(A \vdash B\) is valid in everymodel based on that frame. A rule is valid on a frame iff the validityof the premises sequents on that frame guarantees the validity of theconclusion sequent on the frame. The contraposition rule from the list(21) is valid on any compatibility frame. If the order-inversionexpressed by contraposition is seen as a fundamental property ofnegation, a hierarchy of stronger negations can be obtainedsyntactically by postulating further principles and semantically bycharacterizing these principles by means of conditions oncompatibility frames \((W, C, \leq)\), orrelationalframes \((W, C)\). This line of thought has led from a“kite” of negations in Dunn 1993 to “lopsidedkites” of negations and an extended kite of negations in Shramko2005, Dunn and Zhou 2005, see also Onishi 2015.

A note on terminology. In Dunn 1993, a negationoperation validating the contraposition rule is calledsubminimal. The term “subminimalnegation” had been used by Allen Hazen in an unpublished paperfrom 1992 for a richer language containing negation, conjunction,disjunction, and intuitionistic implication to denote a negation thatfails to validate the intuitionistically valid De Morgan sequent\((\osim A \wedge \osim B) \vdash \osim (A \vee B)\) and theclassically but not intuitionistically valid \(\osim (A \wedge B)\vdash (\osim A \vee \osim B)\). Dunn’s use of the term“subminimal” is thus different from Hazen’s. In Dunnand Zhou 2005 only a single negation as impossibility is used, thevocabulary is enriched by conjunction and disjunction, and in both,the one with negation, conjunction, and disjunction as well as the onewith negation only, subminimal negations are referred to aspreminimal negations. Moreover, the minimal negations fromDunn 1993, 1996 are calledquasi-minimal in Dunn andZhou 2005 because they lack negativeex contradictione, aproperty of negation in what is usually called (Johansson’s)minimal logic, see Kolmogorov 1925 and Johansson 1937. In Vakarelov2005, the term “subminimal negation” is used to denote anegation weaker than Kolmogorov and Johansson’s minimalnegation. Vakarelov introduces his subminimal negation in combinationwith David Nelson’s strong negation (see below) in a languagethat contains a truth constant \(\top\), which is definable in minimallogic. Vakarelov’s subminimal negation, \(\neg\), validates thecontraposition rule but fails to validate \(\neg \neg \top\).Colacito, de Jongh, and Vargas 2017 define a system called“subminimal logic”, see also Niki (2020). Their subminimalnegation is added to the language of positive intuitionistic logic andthus, in particular, contains intuitionistic implication. The basicsubminimal logic is obtained from positive intuitionisticpropositional logic by adding the congruence axiom scheme \((A\leftrightarrow B ) \rightarrow (\neg A \leftrightarrow \neg B)\). Weshall come back to subminimal negation in subminimal logic inSection 2.3.

If the compatibility relation is not assumed to be symmetric (althoughit may be argued that compatibility between statesis asymmetrical relation), then one may distinguish between two negationoperations \(\osim _1\) and \(\osim _2\) such that that \(\osim _1 A\)is true at a state \(w\) just in case \(A\) fails to be true at everystate compatible with \(w\), and \(\osim _2 A\) is true at a state\(w\) just in case \(A\) fails to be true at every state \(w\) iscompatible with:

The two negations form a so-calledGalois connection,which means that \(A \vdash \osim _1 B \mbox{ iff } B \vdash \osim _2A.\) The negations \(\osim _1\) and \(\osim _2\) are called Galoisnegations or split negations; they are both preminimal negations andsatisfy the following interaction principles: \(A \vdash \osim _1\osim _2 A\); \(A \vdash \osim _2 \osim _1 A.\) A discussion of splitnegation can be found in Wansing 2016b.

As noted in Dunn 1993, 1996, if contraposition is assumed, doublenegation introduction \(A \vdash \osim \osim A\) is mutually derivablewith constructive contraposition \(A \vdash \osim B \slashrel B \vdash\osim A\), and if constructive contraposition is assumed, doublenegation elimination is mutually derivable with classicalcontraposition \(\osim A \vdash B \slashrel \osim B \vdash A\). Notealso that in the presence of double negation introduction andelimination, one can derive \(\osim A \vdash \osim B \, / \, B \vdashA\). The demonstrations use only reflexivity and transitivity of thederivability relation \(\vdash\). As a result, the above list ofnegation laws leads to the following unbalanced “kite” ofnegations (cf. Dunn and Zhou 2005):

Figure 3

The graphical arrangement in this diagram is to be understood asfollows: If a sequent or sequent rule is assigned to a node \(n\) andnode \(n'\) is placed below \(n\), then the sequent or sequent ruleassigned to \(n'\) is derivable with the aid of the sequent or sequentrule assigned to \(n\).

Ortho negations satisfy all principles shown in thelopsided kite. An ortho negation in a logic with conjunctiondistributing over disjunction (or, equivalently, disjunctiondistributing over conjunction), is called aBooleanor classical negation. Boolean negation is uniquely determined in thesense that if \(\osim _1\) and \(\osim _2\) are Boolean negations,then \(\osim _1 A\) and \(\osim _2 A\) are interderivable; orthonegation is not uniquely determined, see Restall 2000, and for theuniqueness of connectives Humberstone 2011 and the entry onsentence connectives in formal logic.

The negation principles of Dunn and Zhou’s lopsided kitecorrespond in the sense of modal correspondence theory to propertiesof compatibility frames. A rule \(r\) corresponds to a property \(E\)iff the rule \(r\) is valid on a compatibility frame just in case theframe satisfies \(E\). Greg Restall (2000) observed that doublenegation elimination corresponds to a property of both \(C\)and the relation of possible expansion of informationstates \(\leq\), the other negation principles have been shown tocorrespond to properties only of the compatibility relation \(C\), seeDunn 1996, Dunn and Zhou 2005, Berto 2014. In the following list,“&” denotes conjunction, “\(\Rightarrow\)”denotes Boolean implication, and “\(\forall\)” and“\(\exists\)” refer to universal and existentialquantification, respectively, in the metalanguage:

The following first-order property of \(C\) alone also corresponds todouble negation elimination: \[ \forall x \exists y (xCy \amp \forall z(yCz \Rightarrow (z = x))).\]

We may observe that Dunn and Zhou’s lopsided kite of negationscan be equilibrated, for example, by inserting the sequent schema\(\osim \osim \osim A \vdash \osim A\). This schema corresponds to thefollowing first-order condition on \(C\) (as calculated with the helpof the SQEMA algorithm for computing first-order correspondences inmodal logic due to Georgiev, Tinchev, and Vakarelov (seeOther Internet Resources): \[ \forall x \forall y (xCy \Rightarrow \exists z (xCz \amp \forall u (zCu \Rightarrow uCy))).\]

Figure 4

Negation as unnecessity gives rise to a dual lopsided kite ofnegations that can be combined with the lopsided kite into a“united” kite of negations, see Shramko 2005, Dunn andZhou 2005, andSection 2.4. An even richer inclusion diagram of negations can be found in Ripley2009.

Whilst satisfying the contraposition rule \(A \vdash B \slashrel \osimB \vdash \osim A\) is a basic property of negation as a normalimpossibility operator, there exist unary connectives that arereferred to as negations, although they donotsatisfy contraposition. Prominent examples of logics with anon-contraposable negation in addition to the already mentioned logicsK3,Ł\(_3\), andLP, are Nelson’s logicsN3,N4, andN4\(^{\bot}\) ofconstructive logic with so-called strong negation (see Nelson 1949;Gurevich 1977; Almukdad and Nelson 1984; Wansing 1993; Dunn 2000;Odintsov 2008). These logics contain intuitionistic implication as aprimitive connective. Nelson (1959), however, also considers a variantofN3 with a contraposable strong negation. In thissystemS, the contraction axiom

is replaced by

This replacement avoids a collapse into classical logic. Strongnegation is called “strong” because it captures a notionof negation as definite falsity and because in the systemN3 the strong negation of a formula entails itsintuitionistic negation. The conjunction, disjunction, and strongnegation fragment ofN4 coincides with the logic offirst-degree entailmentFDE, also known as Dunn andBelnap’s useful four-valued logic (Belnap 1977a,b; Dunn 1976;Omori and Wansing 2017). Interestingly, contraposition as stated aboveholds forFDE, whereas it fails inFDE for multiple-premise sequents (see Problem 7,Section 8.10, p. 162 in Priest 2008).

The systemFDE is a well-known system of relevancelogic (see the entry onrelevance logic) and it shares with other relevance logics the property of being aparaconsistent logic, see the entry onparaconsistent logic. Paraconsistent logics fail to satisfy the unrestrictedexcontradictione rule, which is usually presented in amultiple-antecedent framework by the sequent: \[ A, \osim A \vdash B.\]

Logics that do not satisfy the negativeex contradictionerule are paraconsistent in a stricter sense. Double negationelimination and classical contraposition fail to be valid inintuitionistic logic (see the entry onintuitionistic logic); if one of them is added to an axiom system of intuitionistic logic,one obtains a proof system for classical logic.

Now,is negation a normal modal operator of impossibility orunnecessity? According to Berto (2014), the meaning of negation isgrounded in the notion of compatibility, together with its oppositeconcept of incompatibility. Moreover, Berto takes compatibility andincompatibility to be symmetric relations and, as a result, holds thatnothing can be justifiably called a negation if it does not satisfythe contraposition rule and the double negation introduction rule.According to Berto and Restall (2019), “becauseincompatibility is modal, negation is a modal operator aswell.” This view is contentious, in light of non-contraposablenegations in systems such asŁ₃,K3,LP, andN4, andit has been argued by De and Omori 2018 that there is more to negationthan modality.

2.3 Interactions with negation

As already remarked, the classification of unary connectives asnegations may depend on the presence or absence of other logicaloperations. If the propositional language to which a negationoperation is added contains only conjunction and disjunction (andatomic formulas), a natural starting point is to assume that one isdealing with a so-called distributive lattice logic (cf. Dunn and Zhou2005). A distributive lattice logic is a single-antecedent andsingle-conclusion proof system in the language with only conjunction\(\wedge\) and disjunction \(\vee\). In addition to reflexivity andtransitivity of the derivability relation \(\vdash\), the followinginferential schemata are assumed:

• \(A \wedge B \vdash A\), \(A \wedge B \vdash B\),
• \(A \vdash B\), \(A \vdash C\) / \(A \vdash (B \wedge C)\),
• \(A \vdash C\), \(B \vdash C\) / \((A \vee B) \vdash C\),
• \(A \vdash (A \vee B)\), \(B \vdash (A \vee B)\),
• \((A \wedge (B \vee C)) \vdash ((A \wedge B) \vee (A \wedgeC))\).

In this extended vocabulary one may consider further negationprinciples, in particular the De Morgan inference rules from (22):

The first three De Morgan rules are valid on any compatibility frameif the standard evaluation clauses for \(\wedge\) and \(\vee\) areassumed, and they can be proved utilizing standard inference rules for\(\wedge\) and \(\vee\) (cf. Restall 2000). (The first two De Morganrules are valid on any compatibility frame also if negation asimpossibility, \(\osim\), is replaced by negation as unnecessity,\(\neg\).) Whereas the first two De Morgan laws, however, can beproved using only contraposition and inference rules for \(\wedge\)and \(\vee\), a derivation of the third De Morgan law requires theapplication of constructive contraposition:

The proof system for preminimal negation in the extended languagegiven by the rules for distributive lattice logic together withcontraposition is incomplete, and a complete proof system is obtainedif \((\osim A \wedge \osim B) \vdash \osim (A \vee B)\) is added (cf.Dunn and Zhou 2005 for the language with constants \(\top\) and\(\bot\) added). The remaining fourth De Morgan law is provable in thepresence of double negation elimination. The following derivationmakes use of both double negation elimination and classicalcontraposition (cf. Restall 2000):

Restall (2000) showed that \(\osim (A \wedge B) \vdash (\osim A \vee\osim B)\) corresponds to the mixed frame condition \[ \forall x \forall y_1 \forall y_2 ((xCy_1 \amp xCy_2) \Rightarrow \exists z (y_1 \leq z \amp y_ 2 \leq z \amp xCz)).\]

The algorithm SQEMA outputs the following first-order condition for\(\osim (A \wedge B) \vdash (\osim A \vee \osim B)\) in terms of \(C\)alone: \(\forall x \forall y \forall z ((xCy \amp xCz) \Rightarrow ((y=z)\amp xCy))\), which is equivalent to: \[ \forall x \forall y \forall z ((xCy \amp xCz) \Rightarrow (y =z)),\] implying that\(C\) is a function if \(C\) is serial. For functional frames,negation as impossibility and negation as unnecessity coincide.

In the extended language, negativeex contradictione can bestated as \((A \wedge \osim A) \vdash \osim B\) and unrestrictedex contradictione as \((A \wedge \osim A) \vdash B\). It isalso natural to assume a constantly true formula \(\top\) and aconstantly untrue formula \(\bot\), so that the following sequents arevalid: \(A\vdash \top\), \(\bot\vdash A\), and \(\top \vdash \osim\bot\). (With negation as unnecessity, \(\neg \top \vdash \bot\) isvalid.) Whereas these sequents are indeed valid on any compatibilityframe, the equally natural \(\osim \top \vdash \bot\) (as well as\(\top \vdash \neg \bot\)) corresponds with the seriality of \(C\mathrel{:} \forall x \exists y (xCy)\). In the extended vocabulary,unrestrictedex contradictione can be stated as \((A \wedge\osim A) \vdash \bot\), and in this form it is characterized by thereflexivity of the compatibility relation. The law of excluded middle\(\top \vdash (A \vee \osim A)\) corresponds to the mixed condition\(\forall x \forall y(xCy \Rightarrow y \leq x)\) but also to\(\forall x \forall y (xCy \Rightarrow (x =y))\).

The correspondence theory of negation as impossibility and negation asunnecessity in a language with \(\top\), \(\bot\), \(\wedge\), and\(\vee\) has been developed in Lahav, Marcos, and Zohar 2017, usingmethods of the theory of basic sequent systems from Lahav and Avron2013 in order to obtain various cut-admissibility results. Moreover,Lahav and his co-authors also consider the addition of so-calledadjustment operators and the definability of classical negation.

Another interesting classification of negation operators arises if itis assumed that the language under consideration contains a primitiveimplication connective, \(\rightarrow\), that is not defined byputting \((A \rightarrow B) \coloneq (\osim A \vee B)\), or aprimitive so-called co-implication (or subtraction) operation,\(\coimp\), not defined by putting \((A \coimp B) \coloneq (A \wedge\osim B)\), or both. The standard understanding of negatedimplications is conveyed by the equivalence \(\osim (A \rightarrow B)\leftrightarrow (A \wedge \osim B)\). Dually, the classical reading ofnegated co-implications is expressed by \(\osim (A \coimp B)\leftrightarrow (\osim A \vee B)\). Co-implication is the dual ofimplication, insofar as it stands to disjunction as implication standsto conjunction:

(23)

\(\begin{align} (A \wedge B) \vdash C &\mbox{ iff } A \vdash(B\rightarrow C) \mbox{ iff } B \vdash (A \rightarrow C),\\ C \vdash(A \vee B) &\mbox{ iff } (C \coimp B) \vdash A \mbox{ iff } (C\coimp A) \vdash B. \end{align}\)

A formula \((A \coimp B)\) may be read as “\(B\) co-implies\(A\)” or as “\(A\) excludes \(B\)”. If implicationand co-implication are primitive and not defined as in classical logic(and some other logics), further readings of negated implications andco-implications are given by the following equivalences:

(24)

\(\begin{align}\osim (A \rightarrow B) &\leftrightarrow (A\coimp B),\\ \osim (A \coimp B) &\leftrightarrow (A \rightarrowB),\\ \osim (A \rightarrow B) &\leftrightarrow (\osim B \coimp\osim A),\\ \osim (A \coimp B) &\leftrightarrow (\osim B\rightarrow \osim A). \end{align}\)

In the literature, however, one may also find a less-standard readingof negated implications and, consequently, also a correspondingnon-standard understanding of negated co-implications. This reading ofnegated implications is usually tracked back to Aristotle andBoethius, and is referred to as thehyperconnexive reading of(negated) implications (cf. Wansing 2005, McCall 2012, and theconnexive logic entry). Equivalences characteristic of hyperconnexive implication andco-implication are:

(25)

\(\osim (A \rightarrow B) \leftrightarrow (A \rightarrow \osimB),\) \(\osim (A \coimp B) \leftrightarrow (\osim A \coimp B).\)

The preceding typology of negated implications and co-implications hasbeen developed in Wansing 2008, and one might add to this list theequivalences \(\osim (A \rightarrow B) \leftrightarrow (B \rightarrow\osim A)\) and \(\osim (A \coimp B) \leftrightarrow (\osim B \coimpA)\).

The constructive implication of positive intuitionistic logic ispresent in the systems with subminimal negation from Colacito, DeJongh, and Vargas 2017. They also treat negation as a kind of modaloperator by considering frames \((W, R, N)\), where \((W, R)\) is arelational frame, \(R\) is a partial order on \(W\), and \(N\) is aunary function on the set of all upwards closed subsets of \(W\)(i.e., the set of all \(X \subseteq W\) such that if \(w \in X\) and\(wRu\), then \(u \in X\)). A model is obtained by adding to a frame avaluation function that is persistent for atomic formulas. Moreover,the following condition is imposed on the function \(N\):

The truth conditions of negated formulas \(\neg A\) and implications\((A \rightarrow B)\) at a state \(w\) in a model M are then given asfollows:

The basic subminimal logicN characterized by theclass of all frames can be presented as the extension of the standardaxiomatization of positive intuitionistic logic by the congruenceaxiom scheme \( (A \leftrightarrow B ) \rightarrow (\neg A\leftrightarrow \neg B)\). The validity of that axiom (its truth atevery state of every model) is guaranteed by condition (*), but it isnot distinctive of negation. The following additional schematic axiomsare considered in Colacito, De Jongh, and Vargas 2017:

• \((A \rightarrow \neg A) \rightarrow \neg A\),
• \((A \rightarrow B) \rightarrow (\neg B \rightarrow \negA)\),
• \((A \wedge \neg A) \rightarrow \neg B \),
• \(\neg \neg A \rightarrow A \),
• \(\neg (A \wedge B) \rightarrow (\neg A \vee \neg B) \),

and it is shown that the logic obtained by adding the contrapositionaxiom \((A \rightarrow B) \rightarrow (\neg B \rightarrow \neg A)\) ischaracterized by the class of all frames that satisfy for all upwardsclosed sets \(X, X'\): if \(X \subseteq X'\) then \(N(X') \subseteqN(X)\). Moreover, it is shown that the negativeexcontradictione axiom \((A \wedge \neg A) \rightarrow \neg B \)corresponds with the following frame property: \(\forall X \forall X'(X \cap N(X) \subseteq N(X'))\).

Once there is more than just a single negation connective available,the interplay between these operations can be considered. Althoughclassical and intuitionistic logic as well as the familiar systems ofmodal logic comprise only one negation operation, there are also verynaturally arising logical systems with more than just one negation,and the motivation for taking into account multiple negations not onlycomes from natural language semantics but also from the field ofknowledge representation, see, for instance, Wagner 1994.

A well-known example of a logic with two negation operations isHeyting-Brouwer logic, also known as bi-intuitionistic logic, seeRauszer 1980, Goré 2000, Goré and Shillito, 2020. Apropositional logic equivalent to bi-intuitionistic logic was alreadyintroduced by Grigore Moisil (1942), see Drobyshevich et al. 2022. Inaddition to intuitionistic negation, bi-intuitionistic logic containsa so-called co-negation that is in a sense dual to intuitionisticnegation. In bi-intuitionistic logic \(\top\) is definable as \((p\rightarrow p)\) and \(\bot\) as \((p \coimp p)\) for some atomicformula \(p\). The intuitionistic negation \(\osim A\) of \(A\) isthen definable as \((A \rightarrow \bot\)) and the co-negation \(- A\)of \(A\) as \((\top \coimp A)\). Whereas intuitionistic negation is aforward-looking impossibility operation with respect to theinformation order in compatibility frames, i.e., \[ {\cal M}, w \models \osim A \mbox{ iff } \forall u (w\leq u \mbox{ implies } {\cal M}, u \not \models A),\]co-negation is a backward-looking unnecessity operator:\[ {\cal M}, w \models - A \mbox{ iff } \exists u (u \leq w \mbox{ and } {\cal M}, u \not \models A).\]

Another version of bi-intuitionistic logic, called2Int, with a different notion of co-implication andhence co-negation has been developed in Wansing 2016a. In particular,in the semantics of2Int a distinction is drawnbetween a support of truth and a support of falsity relation betweenstates and formulas, and a state supports the truth of \(-A\) iff itsupports the falsity of \(A\).

Other examples of logics with more than just one negation are providedby logics with Galois negations. Moreover, in so-calledtrilattice logics (cf. Shramko and Wansing 2011) adistinction is drawn between a truth negation \(\osim _t\) and afalsity negation \(\osim _f\). Whereas truth negation is interpretedby a unary algebraic operation that inverts a truth order on a set ofgeneralized truth values (see the entry ontruth values), falsity negation is interpreted by an operation inverting a falsityorder on generalized truth values. Furthermore, there is aninformation negation \(\osim _i\) understood as an information orderinversion. The three negations satisfy not only contraposition, butthey are also “period two” (involutive), i.e., theysatisfy the double negation law in both directions. Obviously, in sucha setting various double and triple negation laws may be considered.An in-depth investigation of a hierarchy of double negation principlescan be found in Kamide 2013.

After examining the interplay between negation, implication, andco-implication, recent work on negation as a modal operator in displaycalculi, a generalization of Gentzen’s sequent calculi, can beconsidered. There the above notions of negation as impossibility andunnecessity can be captured by structural sequent rules, see thesupplementSubstructural negations: negation as a modal operator in display calculi.

2.4 Other conceptions of negation as a unary connective

There are several other approaches to negation that build on quitedifferent ideas of expressing semantic opposition. (Some of theconceptions of negation treated in the present entry are also surveyedin Tranchini 2021). A meta-level conception of negation, for example,is the so-callednegation as failure that has beendeveloped in logic programming. The seminal paper Clark 1978 suggeststhe higher-level negation as failure rule (in a slightly irritatingnotation): \(\vdash \osim \vdash p \mbox{ infer } \vdash \osim p\).The idea is that \(\osim p\) may be inferred if the exhaustive searchfor a proof of the atomic statement \(p\) failed.

In Hintikka’s (1973) game-theoretical semantics, negation ismodeled by a role-switch between two players in a semantic game (cf.the entry onlogic and games). A geometrical intuition ofnegation as inversion canbe found in a paper by Ramsey, who suggested that

[w]e might, for instance, express negation not by inserting a word“not”, but by writing what we negate upside down. Such asymbolism is only inconvenient because we are not trained to perceivecomplicated symmetry about a horizontal axis, and if we adopted it, weshould be rid of the redundant “not-not”, for the resultof negating the sentence “\(p\)” twice would be simply thesentence “\(p\)” itself. (F.P. Ramsey 1927,161–2)

The idea of negation as the inversion of arrangements of truth values,such as truth value polygons, has been developed in Varzi and Warglien2003, see also Shramko and Wansing 2011 for negation asorder-inversion in a logic of generalized truth values.

In order to extend Dummett’s verificationism (cf., e.g., Dummett1996) from mathematical to empirical discourse, a notion of“empirical negation” has been suggested (see De 2011,2013, De and Omori 2014). A formula \(\osim A\) is read as“\(A\) is not warranted by our current state of evidence”and it is evaluated with respect to a distinguished base state \(g\)in a model \(\cal M\): \({\cal M}, w \models \osim A\) iff \({\cal M},g \not \models A\).

The supplement document “Additional Conceptions of Negation as a Unary Connective” briefly addresses the following approaches, where negation will bedenoted as \(\neg\) (if not stated otherwise):

Negation as the Routley star

The notion of Routley star negation is more general than thenotion of empirical negation. The Routley star is a unary function\(^*\) on possible worlds that delegates the semantic evaluation of anegated formula \(\neg A\) at a world or state \(w\) to the state\(w^*\): \(\neg A\) is true at a \(w\) in a model \(\cal M\) iff \(A\)is not true at \(w^*\) in \(\cal M\).

Negation as inconsistency

The notion of negation as inconsistency is based on the idea thatthe negation of \(A\) expresses that \(A\) implies (or allows one toderive) something absurd or even something“unwanted”.

Negation as contradictoriness

The idea of negation as contradictoriness is to explicate negationby understanding \(\neg A\) as the contradictory of \(A\), where therelationship of contradiction may be defined in terms of certainlogical laws, such as the Law of Excluded Middle and the Law ofNon-Contradiction.

Negation as falsity

According to negation as falsity, the negation \(\neg A\) of \(A\)expresses that \(A\) is definitely false. This approach to negation isrelated to the view that a proof of \(\neg A\) is a directfalsification of \(A\).

Negation as cancellation

Negation as cancellation develops the idea that the content of theproposition expressed by \(\neg A\) erases or annihilates the contentof the proposition expressed by \(A\).

Negation by iteration

The idea of negation by iteration is to obtain a negation by adouble application of a connective called “demi-negation”in Humberstone 1995, 2000b, or “square root of negation”in quantum computational logic, \(\sqrt{{\tt not}}\), see, for recentreferences, Dalla Chiara, Giuntini, Leporini, and Sergioli 2018, Paoli2019, and the entry onsentence connectives in formal logic. This clearly reminds one of theduplex negatio negat innegative concord, seeSection 1.8.2.

Perfect negation

Perfect negation is a rather restrictive notion of negation thathas been developed by Avron (1999, 2002) in terms of proof-theoreticalas well as semantic conditions.

2.5 Negation, rejection, and denial

As already remarked, negation has been analyzed, for example, as atruth-functional operator, a modal operator, a propositional attitude,and a speech act. The exact relation between negation as a connective,the propositional attitude of rejection and, notably, the speech actof denial is contentious. There is, as a kind of orthodox view, athesis defended by Frege (1919) and Geach (1965), namely that denying\(A\) is the same as asserting \(A\)’s negation. This viewimplies what Ripley (2011b, 623) callsthe denialequivalence:

that to assert the negation of a content \(A\) is equivalent, in itsconversational effects and commitments carried, to denying \(A\).

(Note that Parsons (1984) refers to the claim that denying \(A\) isalways the same as asserting \(\osim A\) as the “EquivalenceThesis.”) There is, however, no clear syntactic restriction onspeech acts of denial, as denials can be realized not only byassertions of negated sentences but, for example, also by means ofirony. Moreover, whereas negated sentences can be embedded intocompound sentences, speech acts cannot be constituents of other speechacts. Therefore, if it is held that to denyis toassert a negation, the idea is that acts of denial can be analyzed asassertions of (propositions expressed by) negated sentences. It may,for instance, be held that it is revealing to understand denials of\(A\) as assertions of \(\copy A\), for some contrary-forming negationoperator \(\copy\).

But there is also a position called “rejectivism” defendedby, for example, Price (1983, 1990), Smiley (1996), and Rumfitt(2000). Lloyd Humberstone (2000a: 331) characterizes rejectivism asfollows:

Whether assent (“acceptance”) and dissent(“rejection”) are thought of as speech acts or aspropositional attitudes, the idea ofrejectivism is that agrasp of the distinction between them is prior to our understanding ofnegation as a sentence, this operator then being explicable asapplying to \(A\) to yield something assent to which is tantamount todissent from \(A\).

At issue is the conceptual priority of the notions of assertion anddenial over the concept of negation. But if the notion of denial isconceptually prior to the concept of negation, one may wonder whynegation is needed at all and how Frege’s argument that anaccount of negation in terms of denial does not make sense of embeddednegations can be met.

As Ripley (2011b) remarks, rejectivists are typically inferentialists,i.e., they hold that the meaning of the logical operations can beexplicated in terms of meaning-conveying rules. If inferentialism isdeveloped in terms of rules for assertingand rulesfor denying compound formulas (as, for example, in Price 1983, 1990;Rumfitt 2000) according to Ripley (2011b), the above questions can beanswered by explaining that negation is a switch between warrantedassertability conditions and warranted deniability conditions. Thisrole of negation is similar to the role strong negation inNelson’s logics plays in turning support of truth conditionsinto support of falsity condition, andvice versa. Price(1990, 225) argues that

if we allow that (an utterance of) \(\osim P\) may properly beregarded both as a denial with content \(P\) and as an assertion withcontent \(\osim P\), then Frege’s argument is powerless; for inthis case the latter reading is available to explain the contributionof \(\osim P\) to complex constructions, in the standard way.

But one may require more from the rejectivist, namely that everyformula is logically equivalent to a formula in what Humberstone(2000a, Footnote 10) calls “Bendall normal form”, namelyto a formula that contains at most one occurrence of the negation signas the principal connective. According to Bendall (1979, 68), theredundancy in this sense of the embedding of a negation operator,

opens the way for an attempt to construe the meaning of negation asderiving from the mental or behavioral phenomena of judgment,disbelief, and denial.

The normal form result holds for classical propositional logic(CPL) in the connectives \(\osim\), \(\wedge\),\(\vee\), and \(\rightarrow\). As Humberstone points out, it fails forthe negation, conjunction, disjunction fragment ofCPL, since in classical propositional logic theBendall normal form (bnf) of a conjunction of atomic formulas \((p\wedge \osim q)\) is \(\osim (p \rightarrow q)\). Moreover, thetranslation of \((\osim p \rightarrow q)\) into bnf is \((p \vee q)\),and the translation of \(\osim \osim p\) is \(p\). These pairs offormulas are not logically equivalent in intuitionistic propositionallogic (IPL). The translation of \((\osim p\rightarrow \osim q)\) into bnf is \((q \rightarrow p)\), which is notlogically equivalent to \((\osim p \rightarrow \osim q)\) inN3 andN4. An in-depth investigationof sequent calculi using formulas prefixed by an assertion sign [+] ora rejection sign [–], understood as non-embeddable forceindicators, can be found in Bendall 1978, Humberstone 2000a.

Humberstone (2000a, 368) challenges rejectivism, by asking therejectivists to “show how the claim for the conceptual priorityof rejection over negation is any more plausible than thecorresponding claim for the conceptual priority of alterjection overdisjunction—or indeed, ambi-assertion over conjunction,”where alterjection (ambi-assertion) is the supposedly primitive speechact the linguistic embodiment of which is disjunction(conjunction).

Another way of repulsing the denial equivalence is to argue thatdenial and rejection are conceptually independent from the concept ofnegation. Such an independence is defended, for example in van derSandt and Maier 2003 (Other Internet Resources) and Priest 2006, Chapter 6. As mentioned in thesupplement, section 3, according to Priest negation is for the main part acontradictory-forming operator. Nevertheless, Priest believes thatthere exist “dialetheia”, sentences \(A\) and \(\osim A\)that are both true. In thedialetheism entry, Berto, Priest, and Weber explain that

a dialetheist manifests her dialetheism in accepting, together withthe LNC [Law of Non-Contradiction], sentences that are inconsistentwith it, that is, true sentences whose negations are true:dialetheias.

This view seems to preclude dialetheists from expressing disagreementconcerning \(A\) as cases in which one person asserts \(A\) andanother person asserts \(\osim A\). A dialetheist might assert \(\osimA\) without disagreeing with \(A\). Therefore, if the assertion of\(\osim A\) is declared to be conceptually independent from the denialof \(A\), disagreement concerning \(A\) can be represented as cases inwhich one person asserts \(A\) and another person denies \(A\). Acritical discussion of this approach may be found in Ripley 2011b.

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• Georgiev, D., T. Tinchev, and D. Vakarelov,SQEMA, Sofia University.
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• Liberman, M., “Archive for Misnegation,” Language Log.
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• Yu, J., 2010, “Negation from the perspective of neighborhood semantics.”

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Acknowledgments

We would like to thank a number of anonymous referees of the StanfordEncyclopedia of Philosophy, as well as KDavid Ripley, JoãoMarcos, and Kensuke Ito for their detailed and thoughtful comments.For the 2025 version of the entry, Heinrich Wansing has been supportedby the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme, grantagreement ERC-2020-ADG, 101018280, ConLog.