## 1. Negation and opposition in natural language


### 1.1 Introduction



Negation is a *sine qua non* of every human language, yet is
absent 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-place
sentential connective toggling the truth value of statements between
T[rue] and F[alse] (or 1 and 0) and applying recursively to its own
output, that allows for denial, contradiction, and other key
properties of human linguistic systems.



The simple syntactic nature of logical negation belies the profoundly
complex and subtle expression of negation in natural language, as
expressed in linguistically distinct categories and parts of speech
(adverbs, verbs, copulas, quantifiers, affixes). As will be partly
explored here (see also Horn 1989, Ladusaw 1996, Pullum 2002), the
investigation of the form and meaning of negative expressions in
English and other languages and of the interaction of negation with
other operators (including multiple iterations of negation itself) is
often far from simple, extending to scope ambiguities (*Everybody
didn't leave*), negative incorporation into quantifiers and
adverbs (*nobody, never, few*), neg-raising (*I don't
want to go* = "I want not to go"), and the widespread
occurrence of negative polarity items (*any, ever, lift a
finger*) whose distribution is subject to principles of syntax,
semantics, and pragmatics. At the core of the mental faculty of
language, negation interacts in significant ways with principles of
morphology, syntax, logical form, and compositional semantics, as well
as with processes of language acquisition and sentence processing,
whence the prominent role played by work on negation in the
development of logic, semantics, linguistic theory, cognition, and
psychoanalytic and literary theory.



What sort of operation is negation? In the *Categories* and
*De Interpretatione*, Aristotle partitions indicative-mood
declarative sentences into affirmation and negation/denial
(*apophasis* from *apophanein* "deny, say
no"), which respectively affirm or deny something about
something (*De Int.* 17a25). As a mode of predication, the
predicate denial of Aristotelian term logic, while resulting in
wide-scope negation opposed in truth value to the corresponding
affirmative, is syntactically distinct from the unary "it is not
the case that" connective of Stoic and Fregean logic.



By combining subject and predicate to form a proposition, this
approach can be seen as offering a more natural representation of
ordinary language negation than the standard iterating operator that
applies to fully formed propositions (Geach 1972; Englebretsen 1981;
Horn 1989, Chap. 7; Sommers and Englebretsen 2000). Indeed, the
syncategorematic negation of Montague Grammar (Montague 1973; cf. the
entry on
 Montague semantics)
 is itself a means of connecting a term phrase subject with a
predicate or IV (intransitive verb) phrase and thus fails to apply to
its own output (see Horn 1989, SS7.2 on "Aristotle as a
Montague grammarian"). Cross-linguistically, the structural
reflex of sentence-scope negation may be a free-standing adverb
(German *nicht*, English *not*), a bound inflectional
form (Japanese *-na-*, English *-n't*), or a verb
(Finnish *en*,
 *ei*).[2]



Where we do not find negation is in the one place propositional logic
would lead us to look, sentence- or clause-peripheral position, as an
external one-place connective interpreted as "it is not the case
that". (Horn 1989 takes apparent instances of external negation
in English and elsewhere to represent the metalinguistic use of
negation, discussed in SS1.10 below, while Bar-Asher Siegal 2015
presents evidence for the existence of a semantic external negation
operator in Jewish Babylonian Aramaic.) Furthermore, unlike speech act
types--e.g. interrogatives or exclamatives--negation never
seems to be marked in natural language by a global intonation contour.
Typically, sentence negation is associated directly on or near the
main finite verb or predicate expression.


### 1.2 Negation in natural language: markedness and asymmetry



It has often been observed that the logical symmetry of negative and
affirmative propositions in logic belies a fundamental asymmetry in
natural language. It was Plato who first observed, in *The
Sophist*, that negative sentences are less valuable than
affirmative ones, less specific and less informative. The ontological,
epistemological, psychological, and grammatical priority of
affirmatives over negatives is supported by Aristotle:



> 
> 
> The affirmative proposition is prior to and better known than the
> negative (since affirmation explains denial just as being is prior to
> not-being) (*Metaphysics* 996b14-16)
> 
> 
> 



and St. Thomas Aquinas:



> 
> 
> The affirmative enunciation is prior to the negative for three
> reasons... With respect to vocal sound, affirmative enunciation
> is prior to negative because it is simpler, for the negative
> enunciation adds a negative particle to the affirmative. With respect
> to thought, the affirmative enunciation, which signifies composition
> by the intellect, is prior to the negative, which signifies
> division... With respect to the thing, the affirmative
> enunciation, which signifies *to be*, is prior to the negative,
> which signifies *not to be*, as the having of something is
> naturally prior to the privation of it. (St. Thomas, Book I, Lesson
> XIII, cited in Oesterle 1962, 64)
> 
> 
> 



Not only are negative statements (e.g., "Paris isn't the
capital of Spain") generally less informative than affirmatives
("Paris is the capital of France"), they are
morphosyntactically more marked (all languages have negative markers
while few have affirmative
 markers)[3]
 and psychologically more complex and harder to process (see Just and
Carpenter 1971, 248-9; and other work reviewed in Horn 1989,
Chapter 3). Many philosophers, linguists, and psychologists have
situated this asymmetry in logic or semantics, as in the claim that
every negation presupposes a corresponding affirmative but not vice
versa.



The strong asymmetricalist position leads to the "paradox of
negative judgment": if a positive statement refers or
corresponds to a positive fact, to what state of affairs does a
negative statement refer or correspond? What in fact is a negative
fact? For Bergson (1911, 289), negation is necessarily "of a
pedagogical and social nature"; for Wood (1933, 421) it is
"infected with error and ignorance". According to
Wittgenstein (1953, SS447), "the feeling is as if the
negation of a proposition had to make it true in a certain sense in
order to negate it". Givon (1978, 70) points to the
discourse presuppositionality of utterances like "My wife is not
pregnant". Psycholinguistic studies have shown that negation is
easier to process when the denied proposition, if not already in the
discourse model, is at least a plausible addition to it (e.g.,
"The whale is not a fish/?bird"; cf. Wason 1965; Horn
1989, Chapter 3).



Beyond its marked status, negation has also been analyzed variously as
a modality, a propositional attitude, and a speech act. The danger
here is putting the pragmatic cart before the semantic horse. For
example, not every negation is a speaker denial (in making this point,
Frege points to the non-denial nature of embedded negation as in
"If not-\(p\) then \(q\)"), nor is every speaker denial a
linguistic negation. Given the repeated attempts over the centuries to
liquidate or tame it--negation as positive difference, as
dissimilarity or incompatibility, as falsity, as an admission of
epistemic impoverishment, as the speech act of denial--and its
resilience in surviving these attacks, negation qualifies as the
Rasputin of the propositional calculus.



But the prototypical use of negation is indeed as a denial of a
proposition attributable to, or at least considered by, someone
relevant to the discourse context. While affirmation standardly
introduces a proposition into the discourse model, negation--in
its "chief use" (Jespersen 1917, 4), its "most
common use" (Ayer 1952, 39), its "standard and primary
use" (Strawson 1952, 7)--is directed at a proposition that
is already in or that can be accommodated by the discourse model.


### 1.3 Matters of scope



If we think of negation as essentially a means for
opposition--the impossibility of simultaneously endorsing two
incompatible options (see the entries on
 contradiction
 and the
 traditional square of opposition)--propositional
 negation is not necessarily privileged. This view is formally
implemented in the Boolean algebraic model of Keenan and Faltz, on
which negation is a cross-categorial operation, as are the binary
connectives:



> 
> 
> We can directly interpret conjunctions, disjunctions, and negations in
> most categories by taking them to be the appropriate meet, join, and
> complement functions of the interpretations of the expressions
> conjoined, disjoined, or negated. The sense in which we have only one
> *and*, *or*, and *not* is explicated on the
> grounds that they are always interpreted as the meet, join, and
> complement functions in whatever set we are looking at. (Keenan and
> Faltz 1985, 6)
> 
> 
> 



Treatments of English and other languages frequently posit negative
operators whose scope is narrower than the sentence or clause. This
tradition dates back to Aristotle, for whom the predicate term
negation in *Socrates is not-wise*, affirming that the
predicate *not-wise* holds of Socrates, yields a false
statement if Socrates does not exist, while the predicate denial
*Socrates isn't wise* denies that the predicate
*wise* holds of Socrates and is true if Socrates does not
exist. For Jespersen (1917), the subclausal "special"
negation as in *Nobody came*, where "the negative
notion...belong[s] logically to one definite idea", is
opposed to "nexal" negation, applying to "the
combination of two ideas", typically the subject-predicate
nexus. Later linguists usually follow Klima (1964) and Jackendoff
(1969) in allowing for constituent negation (e.g., verb phrase
negation in *You can [not go]*) alongside sentential negation
(*You cannot go*), utilizing various grammatical and semantic
diagnostics for distinguishing the two varieties.



A syntactic correlate of the distinction between wide- (sentential)
versus narrow-scope (constituent) negation in English is that only
when 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 negation
does not extend beyond the fronted phrase, whence the exclusion of
*ever*, a satellite of negation (negative polarity
 item).[4]



(1)


a.
With no job **will I** be happy. [= I won't be
happy with any job]
b.
With no job I will be happy. [= I will be happy without any
job]
 
(2)


a.
In no clothes **does Robin** look good.
b.
In no clothes Robin looks good.
 
(3)


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


Negation also interacts in complicated and often surprising ways with
quantification and modality. Perhaps the most analyzed interaction is
with universal quantification. Despite the frequent condemnation of
the wide-scope reading of negation over universals as in the *locus
classicus,* *All that glitters is not gold* and similar
examples in French, German, and other languages, or in ambiguous
sentences like *All the boys didn't leave*, the
availability of such readings (depending on the speaker, the
intonation contour, and the context of utterance) is not as illogical
as it may appear (Horn 1989, SS3.4; Tottie and Neukom-Hermann
2010).


### 1.4 Contrariety and contradiction



The concept of negation is often semantically restricted to
contradictory opposition between propositions, in which \(\neg A\) can
be paraphrased (if not necessarily syntactically represented) as
"it is not the case that \(A\)". As introduced in
Aristotle's *Categories* (11b17), the genus of opposition
(*apophasis*) is divided into species that include contrariety
and contradiction. Contradictory opposites, whether affirmative and
negative counterparts of a singular predication (*Socrates is
wise*/*Socrates isn't wise*) or quantified
expressions (*All pleasure is good*/*Some pleasure is not
good*), are mutually exhaustive as well as mutually exclusive,
while contrary opposites (*Socrates is wise*/*Socrates is
unwise*; *All pleasure is good*/*No pleasure is
good*) do not mutually exhaust their domain. Contraries cannot be
simultaneously true, though they may be simultaneously false. Members
of a contradictory pair cannot be true **or** false
simultaneously; contradictories "divide the true and the false
between them" (see the entries on
 contradiction
 and the
 traditional square of opposition).



Contrary terms (*enantia*) come in two varieties (*Cat.*
11b38ff.). In immediate or logical contraries
(*odd*/*even*, *sick*/*well*), a true
middle--an entity satisfying the range of the two opposed terms
but falling under neither of them--is excluded, e.g., an integer
neither odd nor even. But mediate contrary pairs
(*black*/*white*, *good*/*bad*) allow for
a middle--a shade between black and white, a man or an act
neither good nor bad. Neither mediate nor immediate contraries fall
under the purview of the Law of Excluded Middle [LEM] (*tertium non
datur*).



For immediate contraries formed by narrow-scope predicate term
negation, the rendering *\(a\) is not-\(F\)* in the traditional
quasi-English phrasing corresponds to what Aristotle expresses through
word order, utilizing the distinction between e.g., *einai me
leukon* "to be not-white" and *me einai
leukon* "not to be white" (*Prior Analytics* I
51b10). For Aristotle, *\(a\) is neither \(F\) nor not-\(F\)*
can be true if \(a\) doesn't exist (*Santa is neither white
nor not-white*) or isn't the kind of thing that can be
*F* (*The number 7 is neither white nor not-white*),
given that *not-\(F\)* is taken to affirm the negative property
*non-\(F\)-ness* of the subject rather than denying a positive
property.



Other cases in which apparent contradictories can be seen as
contraries, and thus immune from any application of LEM, are future
contingents (*There will be*/*will not be a sea battle
tomorrow*; cf. *De Int.* Chapter 9) and, in more recent
work (Alxatib and Pelletier 2011, Ripley 2011a), vague predications.
Thus *a is neither F nor not-F* is often judged true when
*F* is a vague predicate (*bald, rich, tall*), although
in the latter case speakers may also be willing to affirm that
*a* is both *F* and not-*F*, which complicates
matters (see the entries on
 contradiction,
 future contingents, and
 vagueness).


### 1.5 Negation, presupposition, and singular terms



In his exposition of sense and reference, Frege (1892) argues that
both (4a) and its contradictory (4b) presuppose that the name
*Kepler* has a denotation. Every affirmative or negative
sentence with a singular subject (name or description) presupposes the
existence of a unique referent for that subject; if the presupposition
fails, 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) but *Kepler did not die in misery or the name
"Kepler" has no reference*, an outcome Frege seems to
have taken as an absurdity but one that prefigures the later emergence
of a presupposition-cancelling external or exclusion negation.



Unwilling to countenance the truth-value gaps incurred on
Frege's analysis, Russell (1905, 485) reconsiders the status of
contradictory negation with vacuous subjects:



> 
> 
> By the law of the excluded middle, either "*A* is
> *B*" or "*A* is not *B*" must
> be 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 are
> not bald, we should not find the king of France on either list.
> Hegelians, who love a synthesis, will probably conclude that he wears
> a wig.
> 
> 
> 



To resolve this (apparent) paradox while preserving a classical
analysis in which every meaningful sentence is true or false, Russell
banishes singular terms like *the king of France* from logical
form, unpacking (5) and (6) as existentially quantified sentences
despite 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 with
the property of being king of France and that this entity is bald,
while (6) is ambiguous, depending on the scope of negation.



(5')
\(\exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedge
Bx))\)
(6')
\(\exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedge \neg
Bx))\)
(6'')
\(\neg \exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedge
Bx))\)


(6'), with narrow-scope ("internal") negation, is
the proposition that there is a unique and hirsute king of France,
which is "simply false" in the absence (or oversupply) of
male French monarchs. In (6''), on the other hand, the
description *the king of France* falls within the scope of
external negation and yields a true proposition. Unlike (6'),
(6'') fails to entail that there is a king of France; indeed, the
non-existence of a king of France guarantees the truth of (6'').
This reading is more naturally expressed with the fall-rise contour
and continuation characteristic of metalinguistic negation (Horn 1989)
as in (7):



(7)
The king of France isn't \(^{{\rm v}}\)BALD--there
ISN'T any king of France!


For Strawson (1950, 1952), negation normally or invariably leaves the
subject "unimpaired". Strawson tacitly lines up with Frege
and against Russell (and Aristotle) in regarding negative statements
like (4b) and (6) as unambiguous and necessarily presuppositional.
Someone who utters (6) does not thereby assert (nor does her statement
entail) that there is a king of France. Rather, (6)--along with
its affirmative counterpart (5)--presupposes it. If this
presupposition fails, a statement may be made but the question of its
truth value fails to arise.



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



Non-bivalent logics of semantic presupposition, dating back to
Lukasiewicz (1930) and Kleene (1952), generally posit (at least)
two *not*-operators, the distinction arising lexically rather
than (as for Russell) scopally; see the entry on
 many-valued logic
 and
 Section 2
 below. The ordinary, presupposition-preserving internal or choice
negation is the only one countenanced by Frege and Strawson; on this
reading, *Santa is not white*, like *Santa is white*, is
neither true nor false, given that Santa does not exist. The
presupposition-cancelling or exclusion negation always determines a
classical value. With exclusion negation, *Santa is not white*
(or perhaps more plausibly *It is not the case that Santa is
white*) is true even if there is no Santa. Thus there is no
excluded middle; any affirmation and its corresponding exclusion
negation are contradictories rather than contraries (see the entry on
 presupposition
 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 the
contrary with the character of the contradictory", Bosanquet
(1888) encapsulates the widespread tendency for formal contradictory
(wide-scope) negation to be semantically or pragmatically strengthened
to a contrary.



We use \(\copy A\) to represent any contrary of \(A\). Following the
Aristotelian 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-functional
contrariety connectives; cf. McCall 1967, Humberstone 2005; see also
Bogen 1991 for the distinction between linguistic and metaphysical
contraries.) It should be noted that while \(\neg\) is an operator
that takes one proposition into another, (c) is not, since a given
proposition may have logically distinct contraries, while this is not
the case for contradictories. Geach (1972, 71-73) makes this
point with the example in (8). While (8a) has two syntactically
distinct contradictories, e.g., *Not every cat detests every
dog* and *It's not every dog that every cat detests*,
any such co-contradictories of a given proposition will always have
the same truth conditions. But (8a) allows two contraries with
distinct 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 that
you're not; I haven't made up my mind yet.
d.
I don't believe that you're telling the truth or that
you're not: I haven't given the matter any thought.
 


Thus while we can speak of the contradictory of a proposition, Geach
observes, we cannot (*pace* McCall 1967) speak of *the*
contrary, but only of *a* contrary, of a proposition. As
Humberstone (1986, fn. 6) points out in response to Geach's
critique of McCall, however, the lack of uniqueness "does not
prevent one from exploring the logical properties of an arbitrarily
selected contrary for a given statement". For our purposes, the
crucial logical properties of contrariety are that (i) the
contradictory of a proposition \(A\) is not a contrary of \(A\) and
that (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 logical
impossibility, \(\Box \neg\), in that \(\Box \neg A\) entails \(\neg
A\) but not *vice versa*, but as pointed out by an anonymous
reviewer, there is no intrinsic modal component of contrariety; all
that is necessary is that contrariety is a non-truth-functional
one-place connective. (See Humberstone 1986, 2003, 2005; Bogen 1991;
and Vakarelov 1989a for additional considerations.)



The strengthening of contradictory negation, \(\neg A\), to a
contrary, \(\copy A\), typically instantiates the inference schema of
disjunctive syllogism or *modus 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 of
disjunctive syllogism can be detected in a variety of strengthening
shifts in natural language where the disjunction in question is
pragmatically presupposed in relevant contexts. Among the
illustrations of this pattern are the following:


* The tendency for negation outside the scope of (certain) negated
propositional 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 unmarked
positive value, whether affixal (\(x\) *is
unfair*/*unhappy*) or clausal (*I don't like
him*), to be strengthened (as either an "online" or
conventionalized process) to a contrary of the positive predication.
As contraries, *Chris is happy* and *Chris is unhappy*
allow an unexcluded middle, since Chris can be neither happy nor
strictly unhappy; similarly, *I don't like him* is
generally understood as stronger than a mere assertion that it's
not the case that I like him.
* The strengthening of a negated plural definite (*The kids
aren't sleeping*) or bare plural (*Beavers don't
eat cheese*) from a contradictory to a contrary of the
corresponding affirmative. In each case, the negation is understood as
inside the scope of the quantified subject.



As stressed by Bartsch 1973 (cf. Horn 1978; Horn 1989, Chapter 5),
when there are only two alternatives in a given context, as in the
case of neg-raising, the denial of one (*I don't believe it
will rain*) amounts to the assertion of the other (*I believe
it won't rain*). The relevant reasoning is an instance of
the disjunctive syllogism pattern in (11), as seen in (12), where
\(F\) represents a propositional attitude and \(a\) the subject of
that attitude.



(12)
\(\begin{array}{ll} F (a, p)\vee F (a,\neg p)\, & {\scriptsize
\mbox{[the pragmatically assumed disjunction]}}\\ \underline{\neg F
(a, p)} & {\scriptsize \mbox{[the sentence explicitly uttered]}}\\
F (a, \neg p) & {\scriptsize \mbox{[the stronger negative
proposition conveyed]}} \end{array} \)


The key step is the pragmatically licensed disjunction of contraries:
if you assume I've made up my mind about the truth value of a
given proposition \(p\) (e.g., "it will rain") rather than
being ignorant or undecided about it, then you will infer that I
believe either \(p\) or \(\neg p\), and my denial that I believe the
former ("I don't think it will rain") will lead you
to conclude that I believe the latter ("I think it won't
rain"). (See Horn 1989, Chapter 5 for more on this phenomenon;
Gajewski 2007 for a neo-Bartschian analysis; and Collins and Postal
2014 for a vigorous defense of a grammatical approach to
neg-raising).



The availability of strengthened contrary readings for apparent
contradictory negation has long been recognized, dating back to
classical rhetoricians of the 4th century on the figure of
litotes, in which an affirmative is indirectly asserted by negating
its contrary (Hoffmann 1987). Litotic interpretations tend to be
asymmetrical: an attribution of "not happy" or "not
optimistic" 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 are
usually understood as pure contradictories. This asymmetry is
ultimately a social fact arising from the desire to respect negative
face (Ducrot 1973, Brown and Levinson 1987, Horn 1989).



For Jespersen, the tendency reflected by the neg-raised interpretation
of *I don't think that \(p\)* not only illustrates the
general strengthening to contrariety but also participates in a more
general conspiracy in natural language to signal negation as early as
possible. Additional effects of this "neg-first" principle
(Horn 1989, 293; after Jespersen 1917, 5) range from diachronic shifts
in the expression of sentential negation (see van der Auwera 2010) and
the fronting and negative inversion in (1a) or (2a) to the emergence
of ambiguities arising in contexts like [neg \(S\_1\) because \(S\_2\)]
(Jespersen 1917, 48), as in "She didn't marry him because
he's poor", where the "illogical" scope
reading--on which his poverty was the non-cause of the wedding
rather than the cause of the non-wedding--can be rendered more or
less accessible by the intonation contour.



The "neg-raised" reading of *I don't think that
\(p\)* as "I think that not-\(p\)" has often been
deplored by grammarians or philosophers as an illogical placement of
negation, an unfortunate ambiguity, or (in Quine's terms) an
"idiosyncratic complication" of one language:



> 
> 
> the familiar quirk of English whereby "\(x\) does not believe
> that \(p\)" 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 claims
> are 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, SS5.2).
Anselm points out that "*non...omnis qui facit quod non
debet peccat, si proprie consideretur*"--not everyone
who does what he *non debet* ("not-should") sins,
if the matter is considered strictly (with the contradictory reading
of negation as the syntax suggests). The problem is that *non
debere peccare* is standardly used to convey the contrary meaning
*debere 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 to
marry") without seeming to commit oneself to the stronger
*debet non ducere uxorem*, an injunction to celibacy (Henry
1967, 193ff.; Horn 1978, 200).



For Henry (1967, 193, SS6.412), Anselm's observations on
modal/negative interaction are "complicated by the quirks of
Latin usage". But far from a Quinean quirk of English and/or
Latin usage, "neg-raising"--the lower-clause
understanding of negation of a *believe*- or
*ought*-type predicate--is distributed widely and
systematically across languages and operators.



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



In some cases, the strengthened or neg-raised contrary reading may
become salient enough over time to block the literal interpretation,
as when French *Il ne faut pas partir*--literally =
"one needn't leave" (an **O** vertex
modal)--is now generally used only to express the stronger
proposition that one must not-leave (**E** vertex). This
is a modal instance of the general phenomenon of **O**
\(>\) **E** drift (Horn 1989), an upward shift along
the right (negative) vertical of the modal square of opposition. Such
squares were constructed by Cajetan, based on Aristotle's *De
Interpretatione* 21b10ff. and *Prior Analytics*
32a18-28 (see Oesterle 1962), and by other medieval
commentators.



![[A square diagram, upper left vertex is 'A' and labeled 'box phi'; upper right vertex is 'E' and labeled 'box not phi equivalent not diamond phi'; lower left vertex is 'I' and labeled 'diamond phi'; lower right vertex is 'O' and labeled 'not box phi equivalent diamond not phi'. 'A' and 'E' are connected by a double headed arrow labeled 'contraries'; 'A' and 'I' are connect by an arrow pointing to 'I' and labeled 'entails'. 'E' connects to 'O' with an arrow pointing to 'O' labeled 'entails'. 'I' and 'O' are connected by a double headed arrow labeled 'subcontraries'. In the center of the square is a label, 'contradictories' with lines from all four vertices to it. The line from 'A' is labeled 'a must F'; the line from 'E' is labeled 'a can't F'; the line from 'I' is labeled 'a may F' and the line from 'O' is labeled 'a needn't F']](figure1-newsquare.png)

Figure 1




\(\mathbf{O}>\mathbf{E}\) drift is attested cross-linguistically in
the meaning shift of lexical items like Old English *nealles*
(lit. "NEG all") = "not at all", Dutch
*nimmer* (lit., "NEG always") =
"never", or Russian *nel'zja* (lit.
"NEG must") = "mustn't", and in words
like *unlikely, inadvisable*, or *disbelieve*, whose
prefixal negations yield only contrary, not contradictory,
interpretations. The reverse shift, in which apparent
**E** forms develop **O** meanings, appears
to be unattested (cf. Horn 2012, 2015).



In litotes and neg-raising, the interpretation of formal
contradictories as contraries arises from the accessibility of the
relevant disjunction, triggering the disjunctive syllogism. The
homogeneity or all-or-none presupposition (Fodor 1970) applying to
bare plurals, plural definites, and mass predications results in a
comparable effect; it is natural to strengthen negative statements
like *Mammals don't lay eggs*, *The children
aren't sleeping*, or *I don't eat meat* to
affirmations of contraries rather than understanding them as simple
wide-scope negations of the corresponding positives (*Mammals lay
eggs*, *The children are sleeping*, *I eat meat*) as
would be the case with overtly quantified universals. The relevant
principle has been variously formulated:



> 
> 
> When a kind is denied to have a generic property P\(\_k\), then any of
> its individuals cannot have the corresponding individual-level
> property P\(\_i\). (von Fintel 1997, 31)
> 
> 
> 
> If the predicate P is false for the NP, its negation not-P is true for
> the NP... Whenever a predicate is applied to one of its
> arguments, it is true or false of the argument as a whole.
> (Lobner 2000, 239)
> 
> 
> 



Once again the key step is establishing the relevant disjunction as a
pragmatically 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 the
all-or-none (or both-or-neither) in arguing that a negative answer to
a "dialectical" or conjoined question like "Are
Coriscus and Callias at home?" would imply that neither is at
home, given the default supposition that they are either both in or
both out. Once again LEM applies where it
"shouldn't"; \(A \vee (c) A\) behaves as though it
were an instance of \(A \vee \neg A\), triggering the disjunctive
syllogism:



(13)
\(\begin{array}{l} (Fa \wedge Fb) \vee (\neg Fa \wedge \neg Fb)\\
\underline{\neg (Fa \wedge Fb)}\\ (\neg Fa \wedge \neg Fb)
\end{array}\)


Other instances of the maximization of contrariety in natural
language, in a range of contexts from formal pragmatics to word
learning, are discussed in Horn 2015.


### 1.7 Privation, affixal negation, and the markedness asymmetry



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



> 
> 
> We say that that which is capable of some particular faculty or
> possession has suffered privation [*steresis*] when the
> faculty or possession in question is in no way present in that in
> which, and at the time in which, it should be naturally present. We do
> not call that toothless which has not teeth, or that blind which has
> not sight, but rather that which has not teeth or sight at the time
> when 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 be
present is revisited in the *Metaphysics*
(1022b23-1023a8), where Aristotle--noting that privation
can range over predictable absence, accidental removal, or deliberate
"taking away by force" of the relevant
property--distinguishes privation "with respect to
genus", as in the blindness of moles, from privation "with
respect to self", as in the blindness or toothlessness of an old
man. In the end, Aristotle concedes, there may be as many senses of
privation as there are *a*- prefixed terms in Greek
(*Met.* 1022b33). Indeed, privation may be reanalyzed as the
primary contrariety (1055a34).



In a wide range of languages, affixal negation on simplex bases
reflects Aristotelian privation, whence the asymmetry between possible
forms (*unhappy, untrue, unkind*) and impossible or unlikely
ones (*unsad, unfalse, uncruel*). We can describe a failed
comedy, but not a successful tragedy, as *unfunny*. As
Jespersen (1917, 144) observes, the tendency of semi-productive
negative affixation to be restricted to unmarked or positive bases
combines with that of the preference for contrariety reviewed
above:



> 
> 
> The modification in sense brought about by the addition of the prefix
> is generally that of a simple negation: *unworthy* = "not
> worthy", etc... The two terms [***X**,
> un**X***] are thus contradictory terms. But very
> often the prefix produces a "contrary" term or at any rate
> what approaches one: *unjust* generally implies the opposite of
> just; *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 to
contradictory, middle-excluding adjectives like
*alive*/*dead*; nothing can be both and nothing capable
of being either can be "in between". But *undead*
has been around since Bram Stoker's *Dracula* (1897) as
both an adjective and a zero-derived occupational noun to describe
zombies, vampires, and other creatures that are "not quite dead
but not fully alive, dead-and-alive" (OED). Someone or something
is undead--e.g., a vampire--if it fails to conform to
one's expectation that it **should** be dead. But
if something appears to be alive but does not quite fulfill that
expectation, it is not *undead* but *unalive*, e.g.,
artificial flowers. Both the undead (but not quite alive) vampire and
the unalive (but not dead) artificial flowers conform to
Aristotle's notion of a privative opposite in lacking a property
associated by default rules with the respective subject.



The marked status of negative utterances has also been invoked to
motivate an asymmetry in the geometry of lexicalization. Within the
Square of Opposition, the Aristotelian relations of contradiction,
contrariety, and subalternation are supplemented with an additional
relation of subcontrariety, so called because the subcontraries are
located under the contraries. As the contradictories of the two
contraries, the subcontraries (e.g., *Some pleasure is good*,
*Some pleasure is not good*) can both be true, but cannot both
be false. For Aristotle, this was therefore not a true opposition,
since subcontraries are "merely verbally opposed"
(*Prior Analytics* 63b21-30). In pragmatic terms, the
assertion of one subcontrary (*Some men are bald*) is not only
compatible with, but actually conversationally implicates, the other
(*Some men are not bald*), given Grice's Maxim of
Quantity ("Make your contribution as informative as is
required"; see the entries on
 Paul Grice,
 pragmatics, and
 implicature).
 The fact that the two members of a subcontrary pair tend to be
equipollent or mutually derivable in a given context may explain the
fact that only one of the two subcontraries will lexicalize in natural
language, and the markedness of negation explains why this is always
the positive (**I** vertex, e.g., *some*) rather
than the negative (**O** vertex, e.g., *no*) value
(Horn 1989, 2012). Thus the **E** values *none*,
*nor*, and *never* are possible but the corresponding
**O** values \**nall* ("not all"),
\**nand* ("or not"), and \**nalways*
("not always") are never attested. Similar, if less
absolute, asymmetries obtain among non-quantificational and indeed
non-logical values (van der Auwera 1996, Horn 2012). Various competing
explanations to the pragmatic account of Horn (1989, 2012) have been
proposed for motivating these asymmetries; see, inter alia, Jaspers
2005, Seuren and Jaspers 2014, and references therein.


### 1.8 Double negation


#### 1.8.1 "Logical" double negation



When *duplex negatio affirmat*, what exactly
**does** the double negation affirm? When a negative term
is a contrary rather than a contradictory of the corresponding simple
affirmative, to deny its application--*Socrates isn't a
not-white log*--does not result in the mutual annihilation of
logical double negation, any more than does the negation of a mediate
contrary (*She's not unhappy, It isn't uncommon*).
While Aristotle countenanced multiple negation, to the extent of
generating such unlikely sequences as *Not-man is not not-just*
(*De Int.* 19b36), each proposition contains only one instance
of negation as wide-scope predicate denial (juxtaposed here with both
a negated subject term and a negated predicate term), since each
categorical statement contains only one predicate.



By contrast, the Stoics defined negation (*apophatikon*) as an
iterating external operator. For Alexander of Aphrodisias,
"*Not: not: it is day* differs from *it is day*
only in manner of speech" (Mates 1953, 126). With their
propositional connectives and one-place truth/falsity-toggling
negation operator, it is the Stoics rather the Aristotelians who
prefigured modern propositional logic, as well as the precepts of
traditional grammar ("*Duplex negatio affirmat*")
and the Law of Double
 Negation.[5]



Classical Fregean logic allows for but one negative operator, the
contradictory-forming propositional operator applying to a proposition
or open sentence, in keeping with "the thesis that all forms of
negation are reducible to a suitably placed 'it is not the case
that'" (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 its
truth value". Within this metaphor, \(\neg\neg A\) is simply a
way of garbing the thought or proposition \(A\).



But, as noted in SS1.1, even a single sentence-external negation
(*Not: The sun is shining*) is a logician's construct
rarely attested in the wild (Geach 1972; Katz 1977):



> 
> 
> [P]ropositional negation was as foreign to ordinary Greek as to
> ordinary English, and [Aristotle] never attained to a distinct
> conception of it. The Stoics did reach such a convention, but in doing
> so they violated accepted Greek usage; their use of an initial
> *oukhi* 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 our
logic,



> 
> 
> in ordinary language a doubly negated expression very seldom, if ever,
> has the same logical powers as the original unnegated statement.
> (Hintikka 1968, 47)
> 
> 
> 



It is thus worth noting that the system of dual negations described by
Aristotle in *Prior Analytics* I, Chapter 46 is both insightful
and internally consistent; its echoes are recognizable in
Jespersen's distinction between nexal negation (*not
happy*) and special negation (*unhappy*), Von
Wright's (1959) distinction between weak (contradictory) versus
strong (contrary) negation, and Jackendoff's (1969) semantic
reanalysis of Klima's (1964) grammatical categories of
sentential versus constituent negation. In each case, a negative
marker whose scope is narrower than the proposition determines a
statement logically distinct from a simple contradictory.



If we represent the narrow-scope contrariety operator of *It is
not-white* as \(\copy A\), its contradictory, \(\neg\copy
A\)(*It isn't not-white*), does not return us to the
simple positive \(A\). The result, if not the means, is similar to
that in intuitionistic logic (Heyting 1956). The intuitionistic
negation operator does not cancel out, given that the intuitionistic
Law of Double Negation is valid in only one direction, \(A \rightarrow
\neg \neg A\), while \(\neg \neg A \rightarrow A\) does not apply (see
the entry on
 intuitionistic logic).
 The intuitionists posit just one negation operator that sustains
double introduction but not double cancellation, while the
Aristotelian system distinguishes contradictory (sentential) predicate
denial from contrary (constituent) predicate term negation.



In ordinary language, semantic double negation (as opposed to negative
concord as in *I ain't never done nothing to nobody*, an
agreement 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 than *common*; one can be not
unhappy without being happy. But even when an apparently contradictory
negation is negated (from the unexceptionable *it's not
impossible* to the more unusual double-*not* of Homer
Simpson's concessive *I'm not not licking toads*
[http://tinyurl.com/34jwhjz]), the *duplex negatio* of \(A\)
doesn't affirm \(A\), or at least it provides a rhetorically
welcome concealment, as suggested by Frege's metaphor of
"wrapping up a thought" in double negation. The negation
in such cases (*impossible*, *not-licking*) is coerced
into a virtual contrary whose negation, \(\neg\copy A\), is weaker
than (is unilaterally entailed by) \(A\) (see Horn 2017 and the entry
on
 contradiction):



![[a square with diagonal lines and the vertices labeled: top left as 'A' and 'it's possible that A'; top right with 'E' and 'it's impossible that A' and 'copyright sign [it's possible that A]'; bottom left with 'I' and 'it isn't impossible that A' and 'negation sign copyright sign[it's possible that A]'; bottom right with 'O' and 'it isn't possible that A' and 'negation sign[it's possible that A].']](figure2-square.png)

Figure 2



#### 1.8.2 Negative concord and its relations



In the previous section it was observed that when *duplex negatio
affirmat*, what it affirms is often not simply the doubly negated
proposition but the result of an incomplete cancellation yielded by
the negation of an actual or virtual contrary (*not unlikely*,
*not impossible*). But a more dramatic problem for the dictum
is when *duplex negatio negat*, especially in the form of
negative concord, in which a single logical negation on the main verb
spreads to indefinites and adverbs within the same clause (Labov 1972,
Zeijlstra 2004, Penka
 2011).[6]



The grammar of negative concord is often complex and subject to a
variety of factors. In standard Italian, for example, negative
quantifiers following the main verb (whether as objects or postposed
subjects) co-occur with mandatory negative marking on the verb to
yield a single negative meaning, as in (14a). But when a negative
quantifier precedes the verb, negative concord is ruled out, as in
 (14b).[7]



(14)


a.
Gianni \*(non) ha visto nessuno. "Gianni has seen
nobody"
  

\*(Non) ha telefonato nessuno. "Nobody has telephoned"
  

\*(Non) ho parlato con nessuno. "I have spoken with
nobody"
b.
Nessuno (\*non) ha visto Gianni. "Nobody has seen
Gianni"
  

Con nessuno (\*non) ho parlato. "With nobody have I
spoken"
 


Negative concord is a feature of many non-standard varieties of
English, especially in informal speech--or lyrics ("I
can't get no
 satisfaction").[8]
 The grammar of negative concord in African American Vernacular
English has been especially well studied; see Green 2002 for an
influential analysis.



True negative concord within a given clause represents just one kind
of hypernegation, the general phenomenon in which a negative marker
reinforces rather than cancels the ordinary or canonical marker of
sentence negation (Horn 2010a). Hypernegation may extend across clause
boundaries to result in the occurrence of "pleonastic" or
"expletive" negative elements in the scope of inherently
negative predicates (cf. Espinal 1992, Horn 2010a). This is
exemplified 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 of
English, pleonastic negation persists in colloquial English:



(15)


a.
I miss (not) seeing you around.
b.
Don't be surprised if it doesn't rain. [= if it
rains]
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 the
appearance of other uninterpreted negations as in (16a), and the
conventionalized irony or sarcasm exemplified in
 (16b,c):[9]



(16)


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 expression *Vous n'etes pas
sans ignorer que ...*, literally "You are not without
being ignorant that ...", is notoriously used in the sense
of "You certainly know that ...". If *duplex
negatio affirmat*, then *triplex negatio confundit*.


### 1.9 Negative polarity



Certain linguistic expressions in English and other languages are
polarity sensitive, restricted in their distributions to the scope of
negation or semantically related contexts, including negative
quantifiers, implicitly negative predicates or adverbs, the
antecedents of conditionals, comparative clauses, and the restrictors
of universals:



(17)


a.
I {haven't/\*have} **ever** eaten
**any** kumquats **at all**.
b.
{Few/\*Many} of the assignments have been turned in
**yet**.
c.
The dean {rarely/\*often} **lifts a finger** to help
students on probation.
d
I {doubt/\*believe} they're **all that** pleased
with the proposal.
e.
{All/\*Many} customers who had **ever** purchased
**any** of the affected items were
(\***ever**) contacted.
 


Negative polarity items (NPIs) like those highlighted in (17) are
generally restricted to downward entailing or monotone decreasing
contexts, those in which inferences from sets to subsets (but not
*vice versa*) are valid (see Fauconnier 1975; Ladusaw 1980,
1996; Peters and Westerstahl 2006; and the
 generalized quantifiers
 entry). If I've eaten kumquats, I've eaten fruit, but not
necessarily *vice versa*; this is an upward entailing (monotone
increasing) environment. On the other hand, if I haven't eaten
fruit, I haven't eaten kumquats, but not necessarily *vice
versa*; this is a downward entailing (monotone decreasing)
 environment.[10]
 It is just in the latter case that NPIs are licensed.



As (17e) shows, universals like *all* or *every* license
NPIs in their restrictor (the relative clause), which is a downward
entailing context (if everyone who knows logic is a vegetarian,
everyone who knows classical logic is a vegetarian, but not *vice
versa*). But universals do not license NPIs in their nuclear scope
or predicate expression, which is an upward entailing context (if
everyone who knows logic is a vegan, everyone who knows logic is a
vegetarian, but not *vice versa*). This contrast demonstrates
the insufficiency of an account of polarity licensing that simply
marks a given lexical item as favorable to the occurrence of NPIs
within its scope (Ladusaw 1980).



While downward entailment may be (generally) necessary for the
licensing 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. For
example, some environments that permit weak NPIs like *any* and
*ever* fail to license stricter ones like *in weeks* or
*until midnight*.



(18)


a.
{Nobody/Only Chris} has **ever** proved
**any** of those theorems.
b.
{Nobody/\*Only Chris} has been here **in weeks**.
 


This has led to the development of more stringent algebraic conditions
that some polarity items must meet, e.g., anti-additivity (Zwarts
1998). The distribution and licensing of polarity items, subject to
widespread variation within and across languages, is an important but
extremely complex linguistic phenomenon with implications for the
architecture of grammar and the theory of meaning; see Israel 2011,
Giannakidou 2011, Chierchia 2013, Horn 2016, and Barker 2018 for
extensive discussion and alternative theoretical approaches. Van der
Wouden (1996) and Blanchette (2015) provide useful examinations of the
connections of negative polarity to negative concord and expletive
negation.


### 1.10 Metalinguistic negation



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



(19)


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's *civet de
lapin*.
d.
I'm not HIS brother--he's MY brother!
e.
Mozart's sonatas were for piano and violin, not for violin
and piano.
 


The descriptive/metalinguistic distinction is supported by converging
linguistic diagnostics suggesting that metalinguistic negation
operates on a different level, whence its failure to incorporate
morphologically or license negative polarity items:



(20)


a.
I'm {not happy/\*unhappy} with the plan, I'm
ecstatic!
b.
You didn't eat {some/\*any} of the cookies, you ate them
all!
 


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 of
considerable dispute; for a range of competing views, see McCawley
1991, Geurts 1998, Burton-Roberts 1999, and Carston 1999. Pitts 2011
provides an excellent overview of the phenomenon (or phenomena) in
question.



The linguistic expression of negation, its interaction with the
phenomena of negative polarity, concord, and scope, and the
mapping between negative form and negative meaning all present
complex and important problems for syntax, semantics, and
pragmatics. (Recent perspectives on a range of issues involved
are provided in Atlas 2012, Horn 2018, and work cited
therein.)  But perhaps the most striking feature of
negation is the mismatch between the formal and distributional
complexity of negative elements in natural languages and the
simplex character of the one-place negative operator of
propositional logic. Yet here too the apparent simplicity is
deceiving once 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, by
considering various styles of proof systems (axiom systems, sequent
calculi, systems of natural deduction, tableaux, etc.) or different
kinds of semantics (algebraic, model-theoretic, proof-theoretic,
game-theoretic, etc.). Moreover, in search of characteristics of
negation as a one-place connective, several dimensions of
classification are available, depending on the logical vocabulary of
the language under consideration (propositional, first-order,
multi-modal, etc.) and the inferential framework taken into account
(single antecedents (i.e., premises) and conclusions, multiple
antecedents or multiple conclusions, sets, multisets, or sequences of
formulas in antecedent or succedent position).



In a very elementary setting one may consider the interplay between
just a single sentential negation, \(\osim\), and the derivability
relation, \(\vdash\), as well as single antecedents and single
conclusions. The following inferential principles are stated as proper
rules with one derivability statement (sequent) or two such statements
as assumption sequent(s) and a single sequent as the conclusion, or as
axiomatic sequents without any assumption sequent:



(21)
\(\begin{align\*} A \vdash B \, &/ \, \osim B \vdash \osim A
& \mbox{(contraposition)}\\ A &\vdash \osim \osim A &
\mbox{(double negation introduction)}\\ \osim \osim A &\vdash A
& \mbox{(double negation elimination)}\\ A \vdash B, \; A \vdash
\osim B \, &/ \, A \vdash \osim C & (\text{negative} \textit{
ex contradictione})\\ A \vdash B, \; A \vdash \osim B \, &/ \, A
\vdash C & (\text{unrestricted }\textit{ex contradictione})\\ A
\vdash \osim B \, &/\, B \vdash \osim A & \mbox{(constructive
contraposition)}\\ \osim A \vdash B \, &/ \, \osim B \vdash A
& \mbox{(classical contraposition) } \end{align\*}\)


The first rule, contraposition, for instance, says that if \(B\) is
derivable from \(A\), then the negation of \(A\) is derivable from the
negation of \(B\). All these rules and derivability statements are
valid in classical logic (see the entry on
 classical logic);
 classical logic cannot distinguish between them. Some of these
principles have been criticized and called into question in
non-classical logic. The unrestricted and the negative *ex
contradictione* rules, for example, introduce an element of
irrelevancy because they allow to derive a completely arbitrary
formula \(C\), respectively a completely arbitrary negated formula
\(\osim C\), from an assumption \(A\) if a formula \(B\) as well as
its negation \(\osim B\) are derivable from \(A\), see the entries on
 relevance logic
 and
 paraconsistent logic.
 Classical contraposition has been criticized because it gives rise to
non-constructive existence proofs in languages containing the
existential quantifier, see the entry on
 intuitionistic logic.
 In richer vocabularies, additional negation principles can be
formulated, regimenting the interaction between negation and other
logical operations. Prominent examples are the De Morgan Laws. In
languages without implication, one may consider the following sequents
stating De Morgan inference rules:



(22)
\(\begin{align} (\osim A \vee \osim B) &\vdash \osim (A \wedge
B)\\ \osim (A \vee B) &\vdash (\osim A \wedge \osim B)\\ (\osim A
\wedge \osim B) &\vdash \osim (A \vee B)\\ \osim (A \wedge B)
&\vdash (\osim A \vee \osim B)\\ \end{align}\)


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



The proof-theoretical characterization of negation is important for
the use of negation connectives in derivations. To obtain a more
comprehensive understanding of negation, however, the proof theory has
to be supplemented by a semantics. 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, namely
either the value T[rue] or the value F[alse] (1 or 0), but not both.
Negation, \(\osim \), is semantically characterized by the unary
function \(f\_{\osim }\) on the set \(\{1, 0\}\), defined by the
following truth table:

 
\[ \begin{array}{c|c}
f\_{\osim } & \\
 \hline 1 & 0 \\
 0 & 1 
\end{array} \]


That is, if \(A\) is a formula, then \(\osim A\) is false if \(A\) is
true, and \(\osim A\) is true if \(A\) is false. The function
\(f\_{\osim }\) is said to be a truth function because it is a function
defined on the set of classical truth values \(\{1, 0\}\), see the
entry on
 truth values.



If negation is meant to express semantic opposition, it is clear that
the remaining two-valued unary truth functions fail to characterize
any plausible notion of semantic opposition between \(A\) and \(\osim
A\):

 
\[ \begin{array}{ccc}
\begin{array}{c|c}
f\_{\textit{id}} & \\
 \hline 1 & 1 \\
 0 & 0 
\end{array} & \quad \begin{array}{c|c}
f\_{\top} & \\
 \hline 1 & 1 \\
 0 & 1 \\
 
\end{array} & \quad \begin{array}{c|c}
f\_{\perp} & \\
 \hline 1 & 0 \\
 0 & 0 
\end{array} 
\end{array} \]


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



There are several ways of generalizing the semantics and making room
for additional sentential negations. One comes with giving up
bivalence and admitting sets of truth values (truth degrees) with more
than two elements, see the entry on
 many-valued logic.
 In the so-called Lukasiewicz many-valued logics
**L\(\_{\aleph\_{1}}\)**,
**L\(\_{\aleph\_{o}}\)**, and
**L\(\_n\)**, the set of values is either the whole
real 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 designated
value representing True, where the designated values of a many-valued
logic are the values that are preserved in valid inferences.
Lukasiewicz negation \(\osim \) is defined by setting \(f\_{\osim
}(u) = 1- u\). Negation is thus understood in terms of subtraction
from the numerical representation of True. In so-called Godel
many-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 is
understood in terms of the numerical representation of True and
distinctness from the numerical representation of False.



In Kleene's (strong) three-valued logic **K3**,
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 as
the table for negation in **L\(\_3\)**, replacing
\(\frac{1}{2}\) by \(i\):

 
\[ \begin{array}{c|c}
f\osim & \\
 \hline 1 & 0 \\
 i & i \\
 0 & 1 \\
 
\end{array} \]


In **K3** and **L\(\_3\)** a formula
\(A\) and its negation \(\osim A\) cannot both be true in the sense of
both taking the designated value 1, but they both fail to be true if
\(A\) receives the value \(i\), respectively \(\frac{1}{2}\). If a
contrary pair of formulas is defined as a pair of formulas that cannot
both be true but may both fail to be true, then Kleene negation gives
rise to contrary pairs.



Also, falsity (understood as receiving the value 0) and non-truth
(understood as taking a value different from 1) fall apart in
**K3**. As a result, contraposition fails in
**K3**. Another example of a logic with a
non-contraposable negation is Priest's Logic of Paradox,
**LP**, where negation is interpreted by \(f\osim \) and
both \(i\) and 1 are designated values, see the entry on
 paraconsistent logic.
 If in **K3** or in **LP** an implication
\((A \supset B)\) is defined as material implication \((\osim A \vee
B)\), then contraposition holds in the sense that \((A \supset B)\)
entails \((\osim B \supset \osim A)\).



If we think of negation in **K3** as representing a
natural language negation, **K3**'s
"internal", presupposition-preserving negation \(\osim \)
differs from the external, presupposition-cancelling negation \(\neg\)
in Bochvar's three-valued logic **B3** by always
returning a classical value. The truth function \(f\_{\neg}\) is
defined by the following table:

 
\[ \begin{array}{c|c}
f\_{\neg} & \\
 \hline 1 & 0 \\
 i & 1 \\
 0 & 1 \\
 
\end{array} \]


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


### 2.2 Negation as a modal operator



Since modal operators are unary connectives and since there exist
different notions of alethic necessity (necessary truth) and alethic
possibility (possible truth), a rather natural question then is
whether negations can be analyzed in a revealing way as modal
operators, see the entry on
 modal logic.



Very well-known modal logics are the **normal** modal
logics that have a so-called possible worlds semantics making use of a
two-place relation between possible worlds. Slightly less known are
the **classical** (or congruential) modal logics
(Segerberg 1971, Chellas 1980, Pacuit 2017). The weakest requirement
imposed on a necessity-like modal operator \(\Box\) in systems of
classical 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 certainly
not distinctive of negation.



Classical modal logics have a semantics in terms of so-called minimal
models, also known as **neighbourhood models**. A
neighbourhood model is a structure \(\cal M\) \(=\) \((W, N, v)\),
where \(W\) is a non-empty set of possible worlds, \(N\) is a function
assigning to every \(w\) from \(W\) a set \(N(w)\) of subsets of
\(W\), called neighbourhoods of \(w\), and \(v\) is a valuation
function mapping atomic formulas to the set of worlds where they are
true. Let \(\llbracket A\rrbracket\) be the set of worlds at which
formula \(A\) is true. Then \( \Box A\) is defined to be true at a
world \(w\) 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 a
general framework for semantically capturing properties characteristic
of negation connectives interpreted as a necessity operator \( \Box
\), see also Yu 2010. Ripley points out, for example, that the
contraposition 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 under
subsets, i.e., if \(X \in N(w)\) and \(Y \subseteq X\), then \(Y \in
N(w)\). It would be nice to have a convincing intuitive understanding
of the neighbourhood function \(N\) in terms of a concept that
explains 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 constraint
emerges as reasonable because it says that if the set of worlds (the
proposition) \(X\) is incompatible with \(w\) and proposition \(Y\)
implies \(X\), then \(Y\) is incompatible with \(w\) as well. Whereas
Ripley starts with a positive notion (\({\cal M}, w \models \Box A\)
iff \(\llbracket A\rrbracket \in N(w)\)), in order to introduce a
negation \(\osim \), one may also stipulate that \({\cal M}, w \models
\osim A\) iff \(\llbracket A\rrbracket \not \in N(w)\), so as to
obtain a connective that is more overtly a negative impossibility
operator (although it is the clause for \(\Box A\) that is classically
negated). The idea is that \(N(w)\) contains the sets of worlds
compatible with \(w\), so that \(\llbracket A\rrbracket \not \in
N(w)\) indicates that the proposition expressed by \(A\) is
incompatible with \(w\). Negation as an "unnecessity"
operator \(\neg\) in the sense of ("possibly not") is then
defined by \({\cal M}, w \models \neg A\) iff \(\overline{\llbracket
A\rrbracket} \in N(w)\), 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 of
the 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 on
negation. 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 the
compatibility reading of the neighbourhood function \(N\) because it
says that if \(X\) is true at \(w\), then \(X\) is compatible with
\(w\). In order to validate contraposition, it has to be required that
if \(X \subseteq Y\), 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 with
is also a world \(X\) is incompatible with, if proposition \(X\)
implies proposition \(Y\). Similar conditions can be imposed for the
dual negation that requires complementation.



The *relational* semantics of normal modal logics, however,
does come with a commitment to a substantial property of negation
understood as impossibility or as unnecessity. The analysis of
negation as a normal impossibility operator can be traced back to work
by Birkhoff and von Neumann (1936) and Goldblatt (1974) on negation in
quantum logic. It has been developed by Vakarelov (1977, 1989b) and
Dosen (1984, 1986, 1999) and has been further investigated in
the algebraic setting of Michael Dunn's gaggle theory (see
Bimbo and Dunn 2008) by Dunn (1993, 1996, 1999) and Dunn and
Zhou (2005). A **relational model** (or Kripke model) is
a structure \(\cal M\) \(=\) \((W, R, v)\), where \(W\) is a non-empty
set of information states, \(R\) is a two-place
"accessibility" relation on \(W\), and \(v\) is a
valuation function. Dunn denotes the accessibility relation by
\(\bot\) (pronounced "perp") and regards it as a relation
of incompatibility or orthogonality between states. Negation as
impossibility, denoted by \(\osim \), is then semantically defined by
postulating that \(\osim A\) is true at a state \(w\) in model \(\cal
M\) iff \(w\) is incompatible with all states \(u\) (from \(W\)) at
which \(A\) 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 of
compatibility 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 as
unnecessity, denoted by \(\neg\), is accordingly defined by the
following 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 another
binary 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 a
relation of possible expansion of information states. With such a
reading 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 \({\cal M},u \models p.\) The
conditions on \(\leq\) and \(C\) and the truth conditions for compound
formulas 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. A
**compatibility 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 the
heredity of negated formulas \(\osim A\): if \(wCu\), \(w' \leq w\),
and \(u' \leq u\), then \(w'Cu'\). This condition is a constraint on
the **compatibility frame** \((W,C,\leq)\) on which a
model \((W,C,\leq,v)\) is based. The condition is not only useful (as
will become clear), but also plausible, because it says that two
information states, expansions of which are compatible, are themselves
compatible.



We can now define that a sequent \(A \vdash B\) is valid in a
compatibility model iff for every state \(w\) from that model, if
\(A\) is true at \(w\), then so is \(B\); \(A \vdash B\) is called
valid on a compatibility frame iff \(A \vdash B\) is valid in every
model based on that frame. A rule is valid on a frame iff the validity
of the premises sequents on that frame guarantees the validity of the
conclusion sequent on the frame. The contraposition rule from the list
(21) is valid on any compatibility frame. If the order-inversion
expressed by contraposition is seen as a fundamental property of
negation, a hierarchy of stronger negations can be obtained
syntactically by postulating further principles and semantically by
characterizing these principles by means of conditions on
compatibility frames \((W, C, \leq)\), or **relational
frames** \((W, C)\). This line of thought has led from a
"kite" of negations in Dunn 1993 to "lopsided
kites" of negations and an extended kite of negations in Shramko
2005, Dunn and Zhou 2005, see also Onishi 2015.



**A note on terminology**. In Dunn 1993, a negation
operation validating the contraposition rule is called
**subminimal**. The term "subminimal
negation" had been used by Allen Hazen in an unpublished paper
from 1992 for a richer language containing negation, conjunction,
disjunction, and intuitionistic implication to denote a negation that
fails to validate the intuitionistically valid De Morgan sequent
\((\osim A \wedge \osim B) \vdash \osim (A \vee B)\) and the
classically 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 Dunn
and Zhou 2005 only a single negation as impossibility is used, the
vocabulary is enriched by conjunction and disjunction, and in both,
the one with negation, conjunction, and disjunction as well as the one
with negation only, subminimal negations are referred to as
*preminimal* negations. Moreover, the minimal negations from
Dunn 1993, 1996 are called **quasi-minimal** in Dunn and
Zhou 2005 because they lack negative *ex contradictione*, a
property of negation in what is usually called (Johansson's)
minimal logic, see Kolmogorov 1925 and Johansson 1937. In Vakarelov
2005, the term "subminimal negation" is used to denote a
negation weaker than Kolmogorov and Johansson's minimal
negation. Vakarelov introduces his subminimal negation in combination
with David Nelson's strong negation (see below) in a language
that contains a truth constant \(\top\), which is definable in minimal
logic. Vakarelov's subminimal negation, \(\neg\), validates the
contraposition rule but fails to validate \(\neg \neg \top\).
Colacito, de Jongh, and Vargas 2017 define a system called
"subminimal logic", see also Niki (forthcoming). Their
subminimal negation is added to the language of positive
intuitionistic logic and thus, in particular, contains intuitionistic
implication. The basic subminimal logic is obtained from positive
intuitionistic propositional logic by adding the congruence axiom
scheme \((A \leftrightarrow B ) \rightarrow (\neg A \leftrightarrow
\neg B)\). We shall come back to subminimal negation in subminimal
logic in
 Section 2.3.



If the compatibility relation is not assumed to be symmetric (although
it may be argued that compatibility between states *is* a
symmetrical relation), then one may distinguish between two negation
operations \(\osim \_1\) and \(\osim \_2\) such that that \(\osim \_1 A\)
is true at a state \(w\) just in case \(A\) fails to be true at every
state compatible with \(w\), and \(\osim \_2 A\) is true at a state
\(w\) just in case \(A\) fails to be true at every state \(w\) is
compatible with:

 
\[ \begin{align}{\cal
M}, w &\models \osim \_1 A \mbox{ iff } \forall u (wCu \mbox{ implies } {\cal M}, u \not \models A);\\
 {\cal M}, w &\models \osim \_2 A \mbox{ iff } \forall u (uCw \mbox{ implies } {\cal M}, u \not \models A). 
\end{align}\]


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



As noted in Dunn 1993, 1996, if contraposition is assumed, double
negation introduction \(A \vdash \osim \osim A\) is mutually derivable
with constructive contraposition \(A \vdash \osim B \slashrel B \vdash
\osim A\), and if constructive contraposition is assumed, double
negation elimination is mutually derivable with classical
contraposition \(\osim A \vdash B \slashrel \osim B \vdash A\). Note
also that in the presence of double negation introduction and
elimination, one can derive \(\osim A \vdash \osim B \, / \, B \vdash
A\). The demonstrations use only reflexivity and transitivity of the
derivability relation \(\vdash\). As a result, the above list of
negation laws leads to the following unbalanced "kite" of
negations (cf. Dunn and Zhou 2005):



![[a diagram of 6 points, 5 connected with lines in a loose pentagon and one below connected by a line to the pentagon. The points are labeled in clockwise order: 'De Morgan \sim \sim A \vdash A', 'ortho', 'intuitionistic A \vdash B, A \vdash \sim B/A \vdash C', 'minimal A \vdash B, A \vdash \sim B/A \vdash \sim C', 'quasi-minimal A \vdash \sim B/B \vdash \sim A', and the solo point connected to the last is labeled 'preminimal A \vdash B/\sim B \vdash \sim A'.]](figure3-kite.png)

Figure 3




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



**Ortho negations** satisfy all principles shown in the
lopsided kite. An ortho negation in a logic with conjunction
distributing over disjunction (or, equivalently, disjunction
distributing over conjunction), is called a **Boolean**
or classical negation. Boolean negation is uniquely determined in the
sense that if \(\osim \_1\) and \(\osim \_2\) are Boolean negations,
then \(\osim \_1 A\) and \(\osim \_2 A\) are interderivable; ortho
negation is not uniquely determined, see Restall 2000, and for the
uniqueness of connectives Humberstone 2011 and the entry on
 sentence connectives in formal logic.



The negation principles of Dunn and Zhou's lopsided kite
correspond in the sense of modal correspondence theory to properties
of compatibility frames. A rule \(r\) corresponds to a property \(E\)
iff the rule \(r\) is valid on a compatibility frame just in case the
frame satisfies \(E\). Greg Restall (2000) observed that double
negation elimination corresponds to a property of both \(C\)
**and** the relation of possible expansion of information
states \(\leq\), the other negation principles have been shown to
correspond to properties only of the compatibility relation \(C\), see
Dunn 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 existential
quantification, respectively, in the metalanguage:

 
\[ \begin{align}
A \vdash \osim \osim A & & \forall x \forall y (xCy \Rightarrow yCx) \\
 A \vdash B, \; A \vdash \osim B &/ A \vdash \osim C &\forall x \forall y (xCy \Rightarrow xCx) \\
 A \vdash B, \; A \vdash \osim B &/ A \vdash C &\forall x (xCx), \forall x \forall y (xCy \Rightarrow yCx)\\
 \osim \osim A \vdash A & & \forall x \exists y(xCy \amp \forall z(yCz \Rightarrow z \leq x)) 
\end{align} \]


The following first-order property of \(C\) alone also corresponds to
double 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 negations
can be equilibrated, for example, by inserting the sequent schema
\(\osim \osim \osim A \vdash \osim A\). This schema corresponds to the
following first-order condition on \(C\) (as calculated with the help
of the SQEMA algorithm for computing first-order correspondences in
modal logic due to Georgiev, Tinchev, and Vakarelov (see
 Other Internet Resources):
 
\[ \forall x \forall y (xCy \Rightarrow \exists z (xCz \amp \forall u (zCu \Rightarrow uCy))).\]




![[a diagram of 7 points, 6 connected with lines in a hexagon and one below connected by a line to the hexagon. The points are labeled in clockwise order: 'weak De Morgan \sim\sim\sim A \vdash \sim A', 'De Morgan \sim \sim A \vdash A', 'ortho', 'intuitionistic A \vdash B, A \vdash \sim B/A \vdash C', 'minimal A \vdash B, A \vdash \sim B/A \vdash \sim C', 'quasi-minimal A \vdash \sim B/B \vdash \sim A', and the solo point connected to the last is labeled 'preminimal A \vdash B/\sim B \vdash \sim A'.]](figure4-kite.png)

Figure 4




Negation as unnecessity gives rise to a dual lopsided kite of
negations that can be combined with the lopsided kite into a
"united" kite of negations, see Shramko 2005, Dunn and
Zhou 2005, and
 Section 2.4.
 An even richer inclusion diagram of negations can be found in Ripley
2009.



Whilst satisfying the contraposition rule \(A \vdash B \slashrel \osim
B \vdash \osim A\) is a basic property of negation as a normal
impossibility operator, there exist unary connectives that are
referred to as negations, although they do **not**
satisfy contraposition. Prominent examples of logics with a
non-contraposable negation in addition to the already mentioned logics
**K3**, **L\(\_3\)**, and
**LP**, are Nelson's logics **N3**,
**N4**, and **N4**\(^{\bot}\) of
constructive logic with so-called strong negation (see Nelson 1949;
Gurevich 1977; Almukdad and Nelson 1984; Wansing 1993, 2001; Dunn
2000; Odintsov 2008). These logics contain intuitionistic implication
as a primitive connective. Nelson (1959), however, also considers a
variant of **N3** with a contraposable strong negation.
In this system **S**, the contraction axiom



\[ (A \rightarrow (A \rightarrow B))\rightarrow (A \rightarrow B)\]


is replaced by

 
\[ (A \rightarrow (A \rightarrow (A \rightarrow B))) \rightarrow (A \rightarrow (A \rightarrow B)).\]


This replacement avoids a collapse into classical logic. Strong
negation is called "strong" because it captures a notion
of negation as definite falsity and because in the system
**N3** the strong negation of a formula entails its
intuitionistic negation. The conjunction, disjunction, and strong
negation fragment of **N4** coincides with the logic of
first-degree entailment **FDE**, also known as Dunn and
Belnap's useful four-valued logic (Belnap 1977a,b; Dunn 1976;
Omori and Wansing 2017). Interestingly, contraposition as stated above
holds for **FDE**, whereas it fails in
**FDE** for multiple-premise sequents (see Problem 7,
Section 8.10, p. 162 in Priest 2008).



The system **FDE** is a well-known system of relevance
logic (see the entry on
 relevance logic)
 and it shares with other relevance logics the property of being a
paraconsistent logic, see the entry on
 paraconsistent logic.
 Paraconsistent logics fail to satisfy the unrestricted *ex
contradictione* rule, which is usually presented in a
multiple-antecedent framework by the sequent: 
\[ A, \osim A \vdash B.\]




Logics that do not satisfy the negative *ex contradictione*
rule are paraconsistent in a stricter sense. Double negation
elimination and classical contraposition fail to be valid in
intuitionistic logic (see the entry on
 intuitionistic 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 or
unneccessity? According to Berto (2014), the meaning of negation is
grounded in the notion of compatibility, together with its opposite
concept of incompatibility. Moreover, Berto takes compatibility and
incompatibility to be symmetric relations and, as a result, holds that
nothing can be justifiably called a negation if it does not satisfy
the contraposition rule and the double negation introduction rule.
According to Berto and Restall (2019), "because
incompat*ibility* is modal, negation is a modal operator as
well." This view is contentious, in light of non-contraposable
negations in systems such as **L3**,
**K3**, **LP**, and **N4**, and
it has been argued by De and Omori 2018 that there is more to negation
than modality.


### 2.3 Interactions with negation



As already remarked, the classification of unary connectives as
negations may depend on the presence or absence of other logical
operations. If the propositional language to which a negation
operation is added contains only conjunction and disjunction (and
atomic formulas), a natural starting point is to assume that one is
dealing with a so-called distributive lattice logic (cf. Dunn and Zhou
2005). A distributive lattice logic is a single-antecedent and
single-conclusion proof system in the language with only conjunction
\(\wedge\) and disjunction \(\vee\). In addition to reflexivity and
transitivity of the derivability relation \(\vdash\), the following
inferential schemata are assumed:


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



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



\[ \begin{align}
(\osim A \vee \osim B) &\vdash \osim (A \wedge B)\\
 \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}\]


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

 
\[ \begin{array}{c}
\begin{array}{cc}
\begin{array}{c}
(\osim A \wedge \osim B) \vdash (\osim A \wedge \osim B) \\
\hline (\osim A \wedge \osim B) \vdash \osim A \\
\hline A \vdash \osim (\osim A \wedge \osim B) 
\end{array} & \begin{array}{c}
(\osim A \wedge \osim B) \vdash (\osim A \wedge \osim B) \\
\hline (\osim A \wedge \osim B) \vdash \osim B \\
\hline B \vdash \osim (\osim A \wedge \osim B) 
\end{array} 
\end{array} \\
 \hline \begin{array}{c}
(A \vee B) \vdash \osim (\osim A \wedge \osim B) \\
\hline (\osim A \wedge \osim B) \vdash \osim (A \vee B)\\
 
\end{array} 
\end{array} \]


The proof system for preminimal negation in the extended language
given by the rules for distributive lattice logic together with
contraposition is incomplete, and a complete proof system is obtained
if \((\osim A \wedge \osim B) \vdash \osim (A \vee B)\) 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 the
presence of double negation elimination. The following derivation
makes use of both double negation elimination and classical
contraposition (cf. Restall 2000):

 
\[ \begin{array}{c}
\begin{array}{cc}
\begin{array}{c}
\osim A \vdash \osim A \\
\hline \osim A \vdash (\osim A \vee \osim B) \\
\hline \osim (\osim A \vee \osim B) \vdash A 
\end{array} & \begin{array}{c}
\osim B \vdash \osim B \\
\hline \osim B \vdash (\osim A \vee \osim B) \\
\hline \osim (\osim A \vee \osim B) \vdash B 
\end{array} 
\end{array} \\
 \hline \begin{array}{c}
\osim (\osim A \vee \osim B) \vdash (A \wedge B) \\
\hline \osim (A \wedge B) \vdash (\osim A \vee \osim B) 
\end{array} 
\end{array} \]


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, negative *ex contradictione* can be
stated as \((A \wedge \osim A) \vdash \osim B\) and unrestricted
*ex contraditione* as \((A \wedge \osim A) \vdash B\). It is
also natural to assume a constantly true formula \(\top\) and a
constantly untrue formula \(\bot\), so that the following sequents are
valid: \(A\vdash \top\), \(\bot\vdash A\), and \(\top \vdash \osim
\bot\). (With negation as unnecessity, \(\neg \top \vdash \bot\) is
valid.) Whereas these sequents are indeed valid on any compatibility
frame, 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,
unrestricted *ex contradictione* can be stated as \((A \wedge
\osim A) \vdash \bot\), and in this form it is characterized by the
reflexivity 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 as
unnecessity in a language with \(\top\), \(\bot\), \(\wedge\), and
\(\vee\) has been developed in Lahav, Marcos, and Zohar 2017, using
methods of the theory of basic sequent systems from Lahav and Avron
2013 in order to obtain various cut-admissibility results. Moreover,
Lahav and his co-authors also consider the addition of so-called
adjustment operators and the definability of classical negation.



Another interesting classification of negation operators arises if it
is assumed that the language under consideration contains a primitive
implication connective, \(\rightarrow\), that is not defined by
putting \((A \rightarrow B) \coloneq (\osim A \vee B)\), or a
primitive 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 negated
implications is conveyed by the equivalence \(\osim (A \rightarrow B)
\leftrightarrow (A \wedge \osim B)\). Dually, the classical reading of
negated co-implications is expressed by \(\osim (A \coimp B)
\leftrightarrow (\osim A \vee B)\). Co-implication is the dual of
implication, insofar as it stands to disjunction as implication stands
to 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 implication
and co-implication are primitive and not defined as in classical logic
(and some other logics), further readings of negated implications and
co-implications are given by the following equivalences:



(24)
\(\begin{align}\osim (A \rightarrow B) &\leftrightarrow (A
\coimp B),\\ \osim (A \coimp B) &\leftrightarrow (A \rightarrow
B),\\ \osim (A \rightarrow B) &\leftrightarrow (\osim B \coimp
\osim A),\\ \osim (A \coimp B) &\leftrightarrow (\osim B
\rightarrow \osim A). \end{align}\)


In the literature, however, one may also find a less-standard reading
of negated implications (and consequently also a corresponding
non-standard understanding of negated co-implications). This unusual
reading of negated implications is usually tracked back to Aristotle
and Boethius, and is referred to as the *connexive* version of
(negated) implications (cf. Wansing 2005, McCall 2012 and the
 connexive logic
 entry). Equivalences characteristic of connexive implication and
co-implication are:



(25)
\(\osim (A \rightarrow B) \leftrightarrow (A \rightarrow \osim
B)\), \(\osim (A \coimp B) \leftrightarrow (\osim A \coimp B)\).


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



The constructive implication of positive intuitionistic logic is
present in the systems with subminimal negation from Colacito, De
Jongh, and Vargas 2017. They also treat negation as a kind of modal
operator by considering frames \((W, R, N)\), where \((W, R)\) is a
relational frame, \(R\) is a partial order on \(W\), and \(N\) is a
unary 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 a
valuation function that is persistent for atomic formulas. Moreover,
the following condition is imposed on the function \(N\):



\[\tag{\*} \mbox{for every } w\in W\!: w\in N(X) \mbox{ iff } w \in N(X \cap \{u \mid wRu\}). \]


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

 
\[\begin{array}{lcl}
M, w \models \neg A & \mbox{ iff } & w \in N{\llbracket A\rrbracket} \\
 M, w \models (A \rightarrow B) & \mbox{ iff } & \mbox{for every } u\in W\!: \\
 & &\; wRu \mbox{ implies } (M,u \not \models A \mbox{ or } M,u \models B). 
\end{array} \]


The basic subminimal logic **N** characterized by the
class of all frames can be presented as the extension of the standard
axiomatization of positive intuitionistic logic by the congruence
axiom scheme \( (A \leftrightarrow B ) \rightarrow (\neg A
\leftrightarrow \neg B)\). The validity of that axiom (its truth at
every state of every model) is guaranteed by condition (\*), but it is
not distinctive of negation. The following additional schematic axioms
are considered in Colacito, De Jongh, and Vargas 2017:


* \((A \rightarrow \neg A) \rightarrow \neg A\),
* \((A \rightarrow B) \rightarrow (\neg B \rightarrow \neg
A)\),
* \((A \wedge \neg A) \rightarrow \neg B \),
* \(\neg \neg A \rightarrow A \),
* \(\neg (A \wedge B) \rightarrow (\neg A \vee \neg B) \),



and it is shown that the logic obtained by adding the contraposition
axiom \((A \rightarrow B) \rightarrow (\neg B \rightarrow \neg A)\) is
characterized by the class of all frames that satisfy for all upwards
closed sets \(X, X'\): if \(X \subseteq X'\) then \(N(X') \subseteq
N(X)\). Moreover, it is shown that the negative *ex
contradictione* 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. Although
classical and intuitionistic logic as well as the familiar systems of
modal logic comprise only one negation operation, there are also very
naturally arising logical systems with more than just one negation,
and the motivation for taking into account multiple negations not only
comes from natural language semantics but also from the field of
knowledge representation, see, for instance, Wagner 1994.



A well-known example of a logic with two negation operations is
Heyting-Brouwer logic, also known as bi-intuitionistic logic, see
Rauszer 1980, Gore 2000. In addition to intuitionistic
negation, bi-intuitionistic logic contains a so-called co-negation
that is in a sense dual to intuitionistic negation. In
bi-intuitionistic logic \(\top\) is definable as \((p \rightarrow p)\)
and \(\bot\) as \((p \coimp p)\) for some atomic formula \(p\). The
intuitionistic negation \(\osim A\) of \(A\) is then definable as \((A
\rightarrow \bot\)) and the co-negation \(- A\) of \(A\) as \((\top
\coimp A)\). Whereas intuitionistic negation is a forward-looking
impossibility operation with respect to the information 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, called
**2Int**, with a different notion of co-implication and
hence co-negation has been developed in Wansing 2016a. In particular,
in the semantics of **2Int** a distinction is drawn
between a support of truth and a support of falsity relation between
states and formulas, and a state supports the truth of \(-A\) iff it
supports the falsity of \(A\).



Other examples of logics with more than just one negation are provided
by logics with Galois negations. Moreover, in so-called
**trilattice logics** (cf. Shramko and Wansing 2011) a
distinction is drawn between a truth negation \(\osim \_t\) and a
falsity negation \(\osim \_f\). Whereas truth negation is interpreted
by a unary algebraic operation that inverts a truth order on a set of
generalized truth values (see the entry on
 truth values),
 falsity negation is interpreted by an operation inverting a falsity
order on generalized truth values. Furthermore, there is an
information negation \(\osim \_i\) understood as an information order
inversion. The three negations satisfy not only contraposition, but
they are also "period two" (involutive), i.e., they
satisfy the double negation law in both directions. Obviously, in such
a setting various double and triple negation laws may be considered,
see also Kamide and Wansing 2012. An in-depth investigation of a
hierarchy of double negation principles can be found in Kamide
2013.



After examining the interplay between negation, implication, and
co-implication, recent work on negation as a modal operator in display
calculi, a generalization of Gentzen's seuqent calculi, can be
considered. There the above notions of negation as impossibility and
unnecessity can be captured by structural sequent rules, see the
supplement
 Substructural 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 quite
different ideas of expressing semantic opposition. A meta-level
conception of negation, for example, is the so-called **negation
as failure** that has been developed in logic programming. The
seminal paper Clark 1978 suggests the higher-level negation as failure
rule (in a slightly irritating notation): \(\vdash \osim \vdash p
\mbox{ infer } \vdash \osim p\). The idea is that \(\osim p\) may be
inferred if the exhaustive search for a proof of the atomic statement
\(p\) failed.



In Hintikka's (1973) game-theoretical semantics, negation is
modeled by a role-switch between two players in a semantic game (cf.
the entry on
 logic and games).
 A geometrical intuition of **negation as inversion** can
be 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 a
> symbolism is only inconvenient because we are not trained to perceive
> complicated symmetry about a horizontal axis, and if we adopted it, we
> should be rid of the redundant "not-not", for the result
> of negating the sentence "\(p\)" twice would be simply the
> sentence "\(p\)" 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 Warglien
2003, see also Shramko and Wansing 2011 for negation as
order-inversion in a logic of generalized truth values.



In order to extend Dummett's verificationism (cf., e.g., Dummett
1996) from mathematical to empirical discourse, a notion of
"empirical negation" has been suggested (see De 2011,
2013). 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 be
denoted as \(\neg\) (if not stated otherwise):



Negation as the Routley star
The notion of Routley star negation is more general than the
notion of empirical negation. The Routley star is a unary function
\(^\*\) on possible worlds that delegates the semantic evaluation of a
negated 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 that
the negation of \(A\) expresses that \(A\) implies (or allows to
derive) something absurd or even something
"unwanted".
Negation as contradictoriness
The idea of negation as contradictoriness is to explicate negation
by understanding \(\neg A\) as the contradictory of \(A\), where the
relationship of contradiction may be defined in terms of certain
logical laws, such as the Law of Excluded Middle and the Law of
Non-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 is
related to the view that a proof of \(\neg A\) is a direct
falsification of \(A\).
Negation as cancellation
Negation as cancellation develops the idea that the content of the
proposition expressed by \(\neg A\) erases or annihilates the content
of the proposition expressed by \(A\).
Negation by iteration
The idea of negation by iteration is to obtain a negation by a
double 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 recent
references, Dalla Chiara, Giuntini, Leporini, and Sergioli 2018, Paoli
2019, and the entry on
 sentence connectives in formal logic.
 This clearly reminds one of the *duplex negatio negat* in
negative concord, see
 Section 1.8.2.
Perfect negation
Perfect negation is a rather restrictive notion of negation that
has been developed by Avron (1999, 2002) in terms of proof-theoretical
as well as semantic conditions.

### 2.5 Negation, rejection, and denial



As already remarked, negation has been analyzed, for example, as a
truth-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 act
of denial is contentious. There is, as a kind of orthodox view, a
thesis defended by Frege (1919) and Geach (1965), namely that denying
\(A\) is the same as asserting \(A\)'s negation. This view implies
what Ripley (2011b, 623) calls **the denial
equivalence**:



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



(Note that Parsons (1984) refers to the claim that denying \(A\) is
always the same as asserting \(\osim A\) as the "Equivalence
Thesis.") There is, however, no clear syntactic restriction on
speech acts of denial, as denials can be realized not only by
assertions of negated sentences but, for example, also by means of
irony. Moreover, whereas negated sentences can be embedded into
compound sentences, speech acts cannot be constituents of other speech
acts. Therefore, if it is held that to deny **is** to
assert a negation, the idea is that acts of denial can be analyzed as
assertions of (propositions expressed by) negated sentences. It may,
for instance, be held that it is revealing to understand denials of
\(A\) as assertions of \((c) A\), for some contrary-forming negation
operator \((c)\).



But there is also a position called "rejectivism" defended
by, for example, Price (1983, 1990), Smiley (1996), and Rumfitt
(2000). Lloyd Humberstone (2000a: 331) characterizes rejectivism as
follows:



> 
> 
> Whether assent ("acceptance") and dissent
> ("rejection") are thought of as speech acts or as
> propositional attitudes, the idea of *rejectivism* is that a
> grasp of the distinction between them is prior to our understanding of
> negation as a sentence, this operator then being explicable as
> applying to \(A\) to yield something assent to which is tantamount to
> dissent from \(A\).
> 
> 
> 



At issue is the conceptual priority of the notions of assertion and
denial over the concept of negation. But if the notion of denial is
conceptually prior to the concept of negation, one may wonder why
negation is needed at all and how Frege's argument that an
account of negation in terms of denial does not make sense of embedded
negations can be met.



As Ripley (2011b) remarks, rejectivists are typically inferentialists,
i.e., they hold that the meaning of the logical operations can be
explicated in terms of meaning-conveying rules. If inferentialism is
developed in terms of rules for asserting **and** rules
for denying compound formulas (as, for example, in Price 1983, 1990;
Rumfitt 2000) according to Ripley (2011b), the above questions can be
answered by explaining that negation is a switch between warranted
assertability conditions and warranted deniability conditions. This
role of negation is similar to the role strong negation in
Nelson's logics plays in turning support of truth conditions
into support of falsity condition, and *vice versa*. Price
(1990, 225) argues that



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



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



> 
> 
> opens the way for an attempt to construe the meaning of negation as
> deriving 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 for
the negation, conjunction, disjunction fragment of
**CPL**, since in classical propositional logic the
Bendall normal form (bnf) of a conjunction of atomic formulas \((p
\wedge \osim q)\) is \(\osim (p \rightarrow q)\). Moreover, the
translation of \((\osim p \rightarrow q)\) into bnf is \((p \vee q)\),
and the translation of \(\osim \osim p\) is \(p\). These pairs of
formulas are not logically equivalent in intuitionistic propositional
logic (**IPL**). The translation of \((\osim p
\rightarrow \osim q)\) into bnf is \((q \rightarrow p)\), which is not
logically equivalent to \((\osim p \rightarrow \osim q)\) in
**N3** and **N4**. An in-depth investigation
of sequent calculi using formulas prefixed by an assertion sign [+] or
a rejection sign [-], understood as non-embeddable force
indicators, can be found in Bendall 1978, Humberstone 2000a.



Humberstone (2000a, 368) challenges rejectivism, by asking the
rejectivists to "show how the claim for the conceptual priority
of rejection over negation is any more plausible than the
corresponding claim for the conceptual priority of alterjection over
disjunction--or indeed, ambi-assertion over conjunction,"
where alterjection (ambi-assertion) is the supposedly primitive speech
act the linguistic embodiment of which is disjunction
(conjunction).



Another way of repulsing the denial equivalence is to argue that
denial and rejection are conceptually independent from the concept of
negation. Such an independence is defended, for example in van der
Sandt and Maier 2003
 (Other Internet Resources)
 and Priest 2006, Chapter 6. As mentioned in the
 supplement, section 3,
 according to Priest negation is for the main part a
contradictory-forming operator. Nevertheless, Priest believes that
there exist "dialetheia", sentences \(A\) and \(\osim A\)
that are both true. In the
 dialetheism
 entry, Berto, Priest, and Weber explain that



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



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