## 1. LNC as Indemonstrable




The twin foundations of Aristotle's logic are the law of
non-contradiction (LNC) (also known as the law of contradiction, LC) and the
law of excluded middle (LEM). In *Metaphysics* Book G,
LNC--"the most certain of all principles"--is
defined as follows:



>  It is impossible that the same thing can at the same time
> both belong and not belong to the same object and in the same respect,
> and all other specifications that might be made, let them be added to
> meet local objections (1005b19-23).




It will be noted that this statement of the LNC is an explicitly modal
claim about the incompatibility of opposed properties applying to the
same object (with the appropriate provisos). Since Lukasiewicz
(1910), this *ontological* version of the principle has been
recognized as distinct from, and for Aristotle arguably prior to,
the *logical* formulation ("The opinion that opposite
assertions are not simultaneously true is the firmest of
all"--Met. 1011b13-14) and the *psychological*
formulation ("It is impossible for anyone to believe that the
same thing is and is not, as some consider Heraclitus
said"--Met. 1005b23-25) offered elsewhere in Book G;
we return to Heraclitus below. Wedin (2004a), who argues for the
primacy of the ontological version (see also Meyer 2008, Other
Internet Resources), formalizes it as
!*([?]*x*)(*Fx* [?]
!*Fx*). These three formulations of LNC differ in important
respects, in particular as to whether the law is explicitly modal in
character, whether it applies to propositions or to properties and
objects, and whether it requires the invocation of a metalinguistic
truth predicate. (See also the entry 
 Aristotle on non-contradiction.) 




For Aristotle, the status of LNC as a first, indemonstrable principle
is obvious. Those who mulishly demand a proof of LNC clearly
"lack education": since "a demonstration of
everything is impossible", resulting in infinite regress. At
least *some* principles must be taken as primitive
*axiomata* rather than derived from other
propositions--and what principle more merits this status than
LNC? (1006a6-12). In first philosophy, as in mathematics, an axiom is
both indemonstrable and indispensable; without LNC, Aristotle argues, "a is
*F*" and "a is not *F*" are
indistinguishable and no argumentation is possible. While Sophists and
"even many physicists" may claim that it *is*
possible for the same thing to be and not to be at the same time and
in the same respect, such a position self-destructs "if only our
opponent says something", since as soon as he opens his mouth to
make an assertion, *any* assertion, he must accept LNC. But
what if he does *not* open his mouth? Against such an
individual "it is ridiculous to seek an argument" for he
is no more than a vegetable (1006a1-15).




The celebrated Arab commentator Avicenna (ibn Sina,
980-1037) confronts the LNC skeptic with a more severe outcome than
Aristotle's vegetable reduction: "As for the obstinate, he must
be plunged into fire, since fire and non-fire are identical. Let him
be beaten, since suffering and not suffering are the same. Let him be
deprived of food and drink, since eating and drinking are identical to
abstaining" (*Metaphysics* I.8, 53.13-15).


The role of LNC as *the* basic, indemonstrable "first
principle" is affirmed by Leibniz, for whom LNC is taken as
interdefinable with the Law of Identity that states that everything is
identical to itself: "Nothing should be taken as first
principles but experiences and the axiom of identity or (what is the
same thing) contradiction, which is primitive, since otherwise there
would be no difference between truth and falsehood, and all
investigation would cease at once, if to say yes or no were a matter
of indifference" (Leibniz 1696/Langley 1916: 13-14). For
Leibniz, everybody--even "barbarians"--must
tacitly assume LNC as part of innate knowledge implicitly called upon
at every moment, thus demonstrating the insufficiency of Locke's
empiricism (ibid.,
 77).[1]




In accounting for the incompatibility of truth and falsity, LNC lies
at the heart of Aristotle's theory of opposition, governing both
contradictories and contraries.
 (See traditional square of opposition.)
 Contradictory opposites ("She is sitting"/"She is
not sitting") are mutually exhaustive as well as mutually
inconsistent; one member of the pair must be true and the other
false, assuming with Aristotle that singular statements with 
vacuous subjects are always false. As it was put by the medievals,
contradictory opposites divide
the true and the false between them; for Aristotle, this is the
primary form of
 opposition.[2]
 Contrary opposites ("He is happy"/"He is
sad") are mutually inconsistent but not necessarily exhaustive;
they may be simultaneously false, though not simultaneously true. LNC
applies to both forms of opposition in that neither contradictories
nor contraries may belong to the same object at the same time and in
the same respect (*Metaphysics* 1011b17-19). What distinguishes
the two forms of opposition is a second indemonstrable principle, the
law of excluded middle (LEM): "Of any one subject, one thing
must be either asserted or denied" (*Metaphysics*
1011b24). Both laws pertain to contradictories, as in a paired
affirmation ("*S* is *P*") and denial
("*S* isn't *P*"): the negation is true
whenever the affirmation is false, and the affirmation is true when
the negation is false. Thus, a corresponding affirmation and negation
cannot both be *true*, by LNC, but neither can they both be
*false*, by LEM. But while LNC applies both to contradictory
and contrary oppositions, LEM holds only for contradictories:
"Nothing can exist between two contradictories, but something
may exist between contraries" (*Metaphysics* 1055b2): a
dog cannot be both black and white, but it may be neither.




As Aristotle explains in the *Categories,* the opposition
between contradictories--"statements opposed to each other
as affirmation and negation"--is defined in two
ways. First, unlike contrariety, contradiction is restricted to
statements or propositions; terms are never related as
contradictories. Second, "in this case, and in this case only,
it is necessary for the one to be true and the other false"
(13b2-3). 




Opposition between terms cannot be contradictory in
nature, both because only statements (subject-predicate combinations)
can be true or false (*Categories* 13b3-12) and because any two
terms may simultaneously fail to apply to a given
 subject.[3]
 But two statements may be members of either a contradictory or a
contrary opposition. Such statements may be simultaneously false,
although (as with contradictories) they may not be simultaneously
true. The most striking aspect of the exposition for a modern reader
lies in Aristotle's selection of illustrative material. Rather than
choosing an uncontroversial example involving *mediate*
contraries, those allowing an unexcluded middle (e.g. "This dog
is white"/"This dog is black"; "Socrates is
good"/"Socrates is bad"), Aristotle offers a pair of
sentences containing *immediate* contraries, "Socrates is
sick"/"Socrates is well". These propositions may
both be false, even though every person is either ill or well:
"For if Socrates exists, one will be true and the other false,
but if he does not exist, both will be false; for neither
'Socrates is sick' nor 'Socrates is well' will
be true, if Socrates does not exist at all" (13b17-19). But
given a corresponding affirmation and negation, one will always be
true and the other false; the negation "Socrates is not
sick" is true whether the snub-nosed philosopher is healthy or
non-existent: "for if he does not exist, 'he is
sick' is false but 'he is not sick' true"
(13b26-35).




Members of a canonical pair of contradictories are formally identical
except for the negative particle:



> An affirmation is a statement affirming something of
> something, a negation is a statement denying something of something...It
> is clear that for every affirmation there is an opposite negation, and
> for every negation there is an opposite affirmation...Let us call an
> affirmation and a negation which are opposite a *contradiction*
> (*De Interpretatione* 17a25-35).




But this criterion, satisfied simply enough in the case of singular
expressions, must be recast in the case of quantified expressions,
both those which "signify universally" ("every
cat", "no cat") and those which do not ("some
cat", "not every cat").




For such cases, Aristotle shifts from a formal to a semantically based
criterion of opposition (17b16-25). Members of
an *A*/*O* pair ("Every man is
white"/"Not every man is white")
or *I*/*E* pair ("Some man is
white"/"No man is white") are contradictories
because in any state of affairs one member of each pair must be true
and the other false. Members of an *A*/*E*
pair--"Every man is just"/"No man is
just"--constitute contraries, since these cannot both be
true simultaneously but can both be false. The contradictories of
these contraries ("Not every man is just"/"Some man
is just") can be simultaneously true with reference to the same
subject (17b23-25). This last opposition of *I*
and *O* statements, later to be dubbed *subcontraries*
because they appear below the contraries on the traditional square, is
a peculiar opposition indeed; Aristotle elsewhere (*Prior
Analytics* 63b21-30) sees *I* and *O* as
"only verbally opposed", given the consistency of
the *I* statement, e.g. "Some Greeks are bald",
with the corresponding *O* statement, "Some Greeks aren't
bald" (or "Not all Greeks are bald", which doesn't
necessarily amount to the same thing, given existential import;
see traditional square of opposition). 




The same relations obtain for modal propositions, for propositions
involving binary connectives like "and" and
"or", for quantificational adverbs, and for a range of
other operators that can be mapped onto the square in analogous ways
utilizing the same notions of contradictory and contrary opposition
and with unilateral entailment definable on duals (see Horn
1989). Thus for example we have the modal square below, based
on *De Interpretatione* 21b10ff. and *Prior Analytics*
32a18-28, where the box and diamond symbols denote necessity and
possibility, respectively. As with universal affirmatives and
universal negatives, necessity and impossibility constitute
contraries: "A priest must marry" and "A priest
can't marry" can both be (and, on the Episcopalian reading, are)
false but cannot both be true. "A priest can marry" and
"A priest can (if he wants) not marry" are subcontraries;
these can be simultaneously true but not simultaneously false. And
necessity, as in "A priest must marry", unilaterally
entails its possibility dual counterpart, "A priest may
marry".



> (1) **Modal Square**  
>   
> 
> ![modal square](modal.jpg)




In the twelfth century, Peter of Spain (1972: 7) offers a
particularly elegant formulation in his *Tractatus*; it will be
seen that these apply to the modal propositions in (1) as well as to
the quantificational statements in the original square:


* The law of contradictories is such that if one
contradictory is true the other is false and vice versa, for nothing
can be simultaneously true and false.
	+ Each contradictory is equivalent to (entails and is entailed by) the
	negation of the other.
* The law of contraries is such that if one is true the other is false
but not vice versa.
	+ Each contrary statement entails the negation of the other but
	not vice versa. [E.g. "I am happy" unilaterally entails
	"I am not unhappy"; "It is necessary that
	Ph" unilaterally entails "It is not impossible that
	Ph".]
* The law of subcontraries is such that if one is false the other is
true but not vice versa.



By these definitions, the three central species of opposition--contradiction, contrariety, and subcontrariety--are mutually inconsistent.


## 2. LEM and LNC




The law of excluded middle, LEM, is another of Aristotle's first
principles, if perhaps not *as* first a principle as LNC. Just
as Heraclitus's anti-LNC position, "that everything is and is
not, seems to make everything true", so too Anaxagoras's
anti-LEM stance, "that an intermediate exists between two
contradictories, makes everything false" (*Metaphysics*
1012a25-29). Of any two contradictories p and !p, LNC entails
that at most one be true while LEM entails that at least one be
true. A logic validates LEM if *p v !p* is a theorem in that
logic. LEM thus imposes a constraint on logical syntax and is distinct
from the Principle of Bivalence, the purely semantic property
dictating that any given proposition is either true or false. The
latter principle is rejected in some multivalued and supervaluationist
logics that validate LEM,
a point to which we return in SS6 (see also 
 many-valued logics,
 Sorites Paradox,
 truth values). Despite the logical
distinction between these two principles, in practice they are often conflated.




For Aristotle, the status of LEM and bivalence comes down to the problem of future contingents. In a passage that has launched a thousand treatises,
Aristotle (*De Interpretatione*, Chapter 9) addresses the
difficulties posed by apparently contradictory contingent statements
about future events, e.g. (2a,b).



> 
> (2a) There will be a sea-battle tomorrow.  
> 
> (2b) There will not be a sea-battle tomorrow.




Clearly, (2a) and (2b) cannot both be true; LNC applies to future
contingents as straightforwardly as to any other pair of
contradictories. But what of LEM? Here is where the difficulties
begin, culminating in the passage with which Aristotle concludes and
(apparently) summarizes his account:



>  It is necessary for there to be or not to be a sea-battle
> tomorrow; but it not necessary for a sea-battle to take place
> tomorrow, nor for one not to take place--though it is necessary
> for one to take place or not to take place. So, since statements are
> true according to how the actual things are, it is clear that wherever
> these are such as to allow of contraries as chance has it, the same
> necessarily holds for the contradictories also. This happens with
> things that are not always so or are not always not so. With these it
> is necessary for one or the other of the contradictories to be true or
> false--not, however, this one or that one, but as chance has it;
> or for one to be true *rather* than the other, yet not
> *already* true or false. Clearly, then it is not necessary
> that of every affirmation and opposite negation one should be true and
> the other false. For what holds for things that are does not hold for
> things that are not but may possibly be or not be; with these it is as
> we have said. (*De Interpretatione* 19a30-b4)




Unfortunately, given the systematic ambiguity and textual variations
in the Greek text, the difficulty of telling when Aristotle is
speaking with his own voice or characterizing an opponent's argument,
and the lack of formal devices for the essential scopal distinctions
at issue, it has never been clear exactly just what has been said here
and in the chapter more generally. Some, including Boethius and
Lukasiewicz, have seen in this text an argument for rejecting LEM for
future contingent statements, which are therefore to be assigned a
non-classical value (e.g. "Indeterminate") or no
truth-value at
 all.[4]
 Their reasoning is based in part on the premise that the alternative
position seems to require the acceptance of determinism. Others,
however, read Aristotle as rejecting not simple bivalence for future
contingents but rather determinacy itself. This interpretive
tradition, endorsed by al-Farabi, Saint Thomas, and
Ockham, is crystallized in this passage from Abelard's
*Dialectica* (210-22) cited by Kneale and Kneale (1962:
214):



> 
> No proposition *de contingenti futuro* can be determinately
> true or determinately false..., but this is not to say that
> no such proposition can be true or false. On the contrary, any such
> proposition is true if the outcome is to be true as it states, even
> though this is unknown to us.




Even if we accept the view that Aristotle is uncomfortable with
assigning truth (or falsity) to (2a) and (2b), their disjunction
in (3a) is clearly seen as true, and indeed as necessarily true.
But the modal operator must be taken to apply to the disjunction as a
whole as in (3b) and not to each disjunct as in (3c).



> 
> (3a) Either there will be or there will not be a sea-battle tomorrow.
>   
> 
> (3b) # (Ph
>  [?]
>  !Ph)
>   
> 
> (3c) # Ph
>  [?]
>  # !Ph
> 




For Aristotle, LNC is understood primarily not as the principle that
no proposition can be true simultaneously with its negation, but as a
prima facie rejection of the possibility that any predicate
*F* could both hold and not hold of a given subject (at the
same time, and in the same respect). A full rendering of the version
of LNC appearing at *Metaphysics*
1006b33-34--"It is not possible to truly say at the
same time of a thing that it is a man and that it is not a
man"--would require a representation involving operators
for modality and truth and allowing quantification over
 times.[5]
 In the same way, LEM is not actually the principle that every
statement is either true or has a true negation, but the law that for
any predicate *F* and any entity *x*, *x* either
is *F* or isn't *F*. 




But these conceptualizations of LNC and LEM must be generalized, since
the principle that it is impossible for a to be *F* and not to
be *F* will not apply to statements of arbitrary complexity. We
can translate the Aristotelian language, with some loss of
faithfulness, into the standard modern propositional versions in (4) and (5) 
respectively, ignoring the understood modal and temporal
modifications:



> 
> (4) LNC: !(Ph [?] !Ph)   
> 
> (5) LEM: Ph
>  [?]
>  !Ph
> 


 Taking LNC and LEM together, we obtain the result that exactly one
proposition of the pair {Ph, !Ph} holds, where !
represents contradictory negation. 


## 3. Contradictory Negation in Term and Propositional Logic




Not every natural language negation is a contradictory operator, or
even a logical operator. A statement may be rejected as false, as
unwarranted, or as inappropriate--misleading, badly pronounced,
wrongly focused, likely to induce unwanted implicatures or
presuppositions, overly or insufficiently formal in register. Only in
the first of these cases, as a toggle between truth and falsity, is it
clear that contradictory negation is involved (Horn 1989, Smiley
1993). Sainsbury (2004) takes truth-functional contradictory negation
to be a special case of a generalized *option negation* as a
deselection operator: If there are two mutually exhaustive and
exclusive options A and B, to select A is to deselect B. But the
relevant options may involve not truth, but some other aspect of
utterance form or meaning as in the standard examples
of *metalinguistic negation* (Horn 1989; see the entry on
 negation). In such cases, a speaker
uses negation metalinguistically or echoically to object to a previous
utterance on any grounds whatsoever, including its phonetic or
grammatical form, register, or associated presuppositions or
implicatures: "That's not a car, it's a Volkswagen",
"Cancer selection is not a but the major force in the emergence
of complex animal life", "He's not your old man, he's your
father", "We didn't call the POlice, we called the
poLICE". In such cases, the relevant target for deselection is
what the right thing is to say in a particular context, where
"truth is not sufficient for being right, and may not even be
necessary" (Sainsbury 2004: 87). Thus the apparent LNC violation
(if it's a Volkswagen, it both is and isn't a car) is not a real
one. 




Given that not every apparent sentential negation is contradictory, is
every contradictory negation sentential? Within propositional logic,
contradictory negation is a self-annihilating operator:
!(!Ph) is equivalent to Ph. This is explicitly
recognized in the proto-Fregean Stoic logic of Alexander of
Aphrodisias: "'Not: not: it is day' differs from
'it is day' only in manner of speech" (Mates 1953:
126). The Stoics' *apophatikon* directly prefigures the
iterating and self-cancelling propositional negation of Frege and
Russell. As Frege puts it (1919: 130), "Wrapping up a thought in
double negation does not alter its truth value." The
corresponding linguistic principle is expressed in the grammarians'
bromide, "Duplex negatio affirmat."




Not all systems of propositional logic accept a biconditional law of
double negation (LDN), !(!Ph) [?] Ph. In particular,
LDN, along with LEM, is not valid for the Intuitionists, who reject
!(!Ph) - Ph while accepting its converse, Ph
- !(!Ph). But the very possibility of applying
negation to a negated statement presupposes the analysis of
contradictory negation as an iterative operator (one capable of
applying to its own output), or as a function whose range is
identical to (or a subset of) its domain. Within the categorical term-based logic of
Aristotle and his Peripatetic successors, every
statement--whether singular or general--is of
subject-predicate form. Contradictory negation is not a one-place
operator taking propositions into propositions, but rather a *mode
of predication*, a way of combining subjects with predicates: a
given predicate can be either affirmed or denied of a given
subject. Unlike the *apophatikon* or propositional negation
connective introduced by the Stoics and formalized in Fregean and
Russellian logic, Aristotelian predicate denial, while toggling truth
and falsity and yielding the semantics of contradictory opposition,
does not apply to its own output and hence does not syntactically
iterate. In this respect, predicate denial both anticipates the form
of negation in Montague Grammar (see the entry on
 Montague semantics) and provides a more plausible
representation of contradictory negation in natural language, whether
Ancient Greek or English, where reflexes of the iterating one-place
connective of the Stoics and Fregeans ("Not: not: the sun is
shining") are hard to find outside of artificial constructs like
the "it is not the case" construction (Horn 1989,
SS7.2). In a given natural language, contradictory negation may be
expressed as a particle associated with a copula or a verb, as an
inflected auxiliary verb, as a verb of negation, or as a negative
suffix or prefix. 




In addition, there is a widespread pragmatically motivated tendency
for a formal contradictory negation to be strengthened to a semantic
or virtual contrary through such processes as litotes ("I don't
like prunes" conveying that I dislike prunes) and so-called
neg(ative) raising ("I don't think that Ph" conveying
"I think that !Ph"). Similarly, the prefixal
negation in such adjectives as "unhappy" or
"unfair" is understood as a contrary rather than
contradictory (not-Adj) of its base. These phenomena have been much
discussed by rhetoricians, logicians, and linguists (see the
entry on negation and Horn 1989: Chap. 5).




In addition to predicate denial, in which a predicate *F* is
denied of a subject *a*, Aristotelian logic allows for
narrow-scope *predicate term negation*, in which a negative
predicate *not-F* is affirmed of *a*. The relations of
predicate denial and predicate term negation to a simple affirmative
proposition (and to each other) can be schematized on a generalized
square of opposition for singular (non-quantified) expressions (*De
Interpretatione* 19b19-30, *Prior Analytics* Chapter
46):



> (6) **Negation Square**  
>   
> 
> ![negation square](negation-square.jpg)




If Socrates doesn't exist, "Socrates is wise" (*A*)
and its contrary "Socrates is not-wise" (*E*) are
both automatically false (since nothing--positive or
negative--can be truly affirmed of a non-existent subject), while
their respective contradictories "Socrates is not wise"
(*O*) and "Socrates is not not-wise" (*I*)
are both true. Similarly, for any object *x*, either *x*
is red or *x* is not red--but *x* may be neither
red nor not-red; if, for instance, *x* is a unicorn or a prime
number.




While Russell (1905) echoed (without acknowledgment) Aristotle's
ambiguist analysis of negation as either contradictory
("external") or contrary ("internal"), by
virtue of the two logical forms assigned to "The king of France
is not bald" (see descriptions),
such propositionalized accounts are bought at a cost of naturalness,
as singular sentences of subject-predicate grammatical form are
assigned the logical form of an existentially quantified conjunction
and as names are transmuted into predicates. 




The difference between denying *P* of *S* and
affirming *not-P* of *S* is realized in Ancient Greek as
a scopal distinction reflected in word order: *S P [not is]
(Socrates healthy not-is)* vs. *S [not P] is (Socrates
not-healthy is)*. As indicated in (6), for Aristotle only
sentences can be in contradictory opposition. *P*
and *not-P* both yield falsity when predicated of a
non-existent subject but one or the other of the two terms is
truthfully predicable of any existent subject in the relevant
domain. *P* and *not-P* are "logical
contraries" that exclude a true middle, an existent entity which
is neither *P* nor *not-P*. But naturally occurring
cases of prefixal adjectives, those marked by *a*(*n*)- in
Greek, may involve an unexcluded middle, as do polar contraries or
antonym pairs. Modern grammatical discourse departs from Aristotle in
allowing for contradictory terms: middle-allowing contrary adjectives
(*white*/*black*,
*happy*/*unhappy*) are distinguished from middle-excluding contradictory adjectives
(*transitive*/*intransitive*, *alive*/*dead*). 




Jespersen (1917: 144) describes the logical status of negatively
prefixed adjectives in English:



> 
> 
> The modification in sense brought about by the addition of the
> prefix [*un-*] is generally that of a simple negative:
> *unworthy* = 'not worthy', etc. The two terms
> [P, unP] are thus contradictory terms. But very often the prefix
> produces a "contrary" term...: *unjust* (and
> *injustice*) generally imply the opposite of *just* (*justice*);
> *unwise* means more than *not wise* and approaches
> *foolish*, *unhappy* is not far from miserable, etc.
> 
> 
> 




Like Aristotle, Jespersen predicts that the negation of true
contraries like *unhappy*, *unjust*, or *unwise* will be
semantically distinct from their positive bases. Thus, *not
unhappy* fails to reduce to *happy* by virtue of allowing an
unexcluded middle: one can be neither happy nor unhappy but just blaah,
in the same way that something can be neither black nor white but one
of the fifty-plus shades of gray. At the same time, even those
adjectives that are semantic contradictories, e.g. *impossible*,
may be coerced under negation into virtual contraries. While
technically any action or event must be either possible or impossible,
to assess something as *not impossible* is often to portray its
occurrence as a more remote possibility than to assess it as
*possible* simpliciter, as reflected in attestations of
"It's possible, or at least not impossible". Similar
instances of virtual contrariety are readily attested with negated verb
phrases ("I don't NOT love the dog" [?] I love the
dog) or predicate nominals ("We're not NOT friends"
*[?]* We're friends); see Horn 2017.




Drawing on an epistemic theory of vagueness, Krifka (2007) argues
that prefixal negation always yields semantic contradictories. On this view,
*unhappy* is literally just 'not happy', with the
characteristic stronger understanding derived pragmatically. The
incomplete cancellation of the two negators in *not unhappy* is
taken to be a purely pragmatic phenomenon, conflating this case with
that of *not impossible*. But the classical theory has its
advantages. On that approach, *un-Adj* antonyms (like their
morphologically simplex classmates, *sad* or *bad*) are
lexical items that can constitute contraries vis-a-vis the
corresponding positive. By virtue of their lexical status, they are
candidates to undergo further semantic drift, unlike *not Adj*
sequences (or *non-Adj* forms), as evidenced in the semantic
and phonological opacity of *infamous* or
*impious*. Note too that many *un*- and *in*-
adjectives (*unkempt*, *inchoate*, *incorrigible*) lack
corresponding simple bases. Furthermore, the prefix *non-*
yields staunch contradictories (typically with objective and/or
technical senses) often contrasting minimally with *un-Adj* or
*iN-Adj* contraries favoring gradable and evaluative
contexts:


* *non-American* vs. *un-American; non-professional* vs.
 *unprofessional*
* *non-Christian* vs. *un-Christian; 
non-rational* vs. *irrational*
* *non-moral* vs. *immoral; 
non-realistic* vs. *unrealistic*
* *non-natural* vs. *unnatural; 
non-scientific* vs. *unscientific*




Even more problematically for a unified treatment, the treatment of
all negatively prefixed adjectives as semantic contradictories would
seem naturally to extend from *unhappy* or *unwise* to
simplex antonymic pairs like *happy*/*sad* or
*wise*/*foolish*, where the evidence for semantic
contrariety appears to be incontrovertible. While Krifka (2007: 174)
supports the analysis of e.g. *happy* and *unhappy* as
"literally contradictories that receive their interpretations as
contraries only via pragmatic strengthening", Horn (2017) argues
that a traditional (neo-Aristotelian) approach invoking parallel but distinct semantic
and pragmatic strengthening processes is on sounder empirical
footing.


## 4. Gaps and Gluts: LNC and Its Discontents




In addition to the future contingent statements discussed in SS2,
vacuous subjects like those in (7a,b) have sometimes been taken to
yield a violation of LEM through the emergence of a truth-value
gap.



> 
> (7a) {The present king of France/King Louis} is bald.  
> 
> (7b) {The present king of France/King Louis} isn't bald.
> 




While Aristotle would see a republican France as rendering (7a) false
and (7b) automatically true, Frege (1892) and Strawson (1950) reject
the notion that either of these sentences can be used to make a true
or false assertion. Instead, both statements presuppose the existence
of a referent for the singular term; if the presupposition fails, so
does the possibility of classical truth assignment. Note, however,
that such analyses present a challenge to LEM only if (7b) is taken as
the true contradictory of (7a), an assumption not universally
shared. Russell, for example, allows for one reading of (7b) on which
it is, like (7a), false in the absence of a referent or denotatum for
the subject term; on that reading, on which the description has
primary occurrence, the two sentences are not contradictories. In this
way, Russell (1905: 485) seeks to guide the French monarch out of the
apparent trap without recourse to wigs or truth value gaps: 



> 
> 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.




In those systems that do embrace truth value gaps (Strawson, arguably
Frege) or non-classically-valued systems (Lukasiewicz, Bochvar,
Kleene), some sentences or statements are not assigned a (classical)
truth value; in Strawson's famous dictum, the question of the truth
value of "The king of France is wise", in a world in which
France is a republic, simply fails to arise. The negative form of such
vacuous statements, e.g. "The king of France is not wise",
is similarly neither true nor false. This amounts to a rejection of
LEM, as noted by Russell 1905. In addition to vacuous singular
expressions, gap-based analyses have been proposed for future
contingents (following one reading of Aristotle's exposition of the
sea-battle; cf. SS2 above) and category mistakes (e.g. "The
number 7 likes/doesn't like to dance").




While LNC has traditionally remained more sacrosanct, reflecting its
position as the primus inter pares of the indemonstrables,
transgressing this final taboo has become increasingly alluring in
recent years. The move here involves embracing not gaps but truth
value *gluts*, cases in which a given sentence and its negation
are taken to be both true, or alternatively cases in which a sentence
may be assigned more than one (classical) truth value, i.e. both True
and False. Parsons (1990) observes that the two non-classical theories
are provably logically equivalent, as gluts arise within one class of
theories precisely where gaps do in the other; others, however, have
argued that gaps (as in Intuitionist non-bivalent logics) are easier
to swallow than gluts (see papers in the Priest et al. 2004 collection
for further debate). Dialetheists reject the charge of incoherence by
noting that to accept some contradictions is not to accept them all;
in particular, they seek to defuse the threat of logical armageddon or
"explosion" posed by Ex Contradictione Quodlibet, the
inference in (8):




> 
> (8) *p*, !*p*  
> 
>     \_\_\_\_\_  
> 
>     [?]*q*
> 




Far from reduced to the silence of a vegetable, as Aristotle ordained,
the proponents of true contradictions, including self-avowed
dialetheists following the lead of Sylvan (ne Routley) and
Priest have been eloquent.




Is the status of Aristotle's "first principle" as obvious
as he believed? Adherents of the dialetheist view that there are true
contradictories (Priest 1987, 1998, 2002; see also the entries on
 dialetheism and
 paraconsistent logic) would
 answer firmly in the 
 negative.[6]
 In the Western tradition, the countenancing
of true contradictions is typically--although not
exclusively--motivated on the basis of such classic logical
paradoxes as "This sentence is not true" and its
analogues; such a statement is evidently true if and only if it is
not true. The Liar, or really the family of Liar's paradoxes
(see Liar Paradox), and Russell's
paradox, its set-theoretic analogue (a set that is not a member of
itself both is and is not a member of
itself--see the entry on 
 Russell's Paradox), would
no longer constitute immediate threats to logical coherence in the
absence of LNC (see papers in Priest et al. for extended discussion).
If we are indeed prepared to jettison LNC, we can regard "This
sentence is not true" (or "This statement is false")
as simultaneously true and not true without deriving the resultant
absurdity taken by Aristotle and his heirs to result from such a
contradiction. As Smiley (1993: 19) has remarked, "Dialetheism
stands to the classical idea of negation like special relativity to
Newtonian mechanics: they agree in the familiar areas but diverge at
the margins (notably the paradoxes)."




Related to the classic paradoxes of logic and set theory is the
Paradox of the Stone. One begins by granting the basic dilemma, as an
evident instance of LEM: either God is omnipotent or God is not
omnipotent. With omnipotence, He can do anything, and in particular He
can create a stone, call it *s*, that is so heavy even He
cannot lift it. But then there is something He cannot do, viz. (ex
hypothesi) lift *s*. But this is a violation of LNC: God can
lift *s* and God cannot lift *s*. This paradox, and the
potential challenge it offers to either LNC or the possibility of
omnipotence, has been recognized since Aquinas, who opted for
retaining the Aristotelian law by understanding omnipotence as the
capacity to do only what is not logically impossible. (Others,
including Augustine and Maimonides, have noted that in any case God is
"unable" to do what is inconsistent with His nature,
e.g. commit sin.) For Descartes, on the other hand, an omnipotent God
is by definition capable of *any* task, even those yielding
contradictions. Mavrodes (1963), Kenny, and others have sided with
St. Thomas in taking omnipotence to extend only to those powers it is
possible to possess; Frankfurt (1964), on the other hand, essentially
adopts the Cartesian line: Yes, of course God can indeed construct a
stone such that He cannot lift it--and what's more, He can lift
it! (See also Savage 1967 for a related solution.)



As we have seen, the target of Aristotle's psychological (doxastic)
version of LNC was Heraclitus: "It is impossible for anyone to
believe that the same thing is and is not, as some consider Heraclitus
said--for it is not necessary that the things one says one also
believes" (Met. 1005b23-26). But as Aristotle acknowledges here
(even if he is less politic elsewhere), there is considerable uncertainty
over exactly what Heraclitus said and what he believed. Heraclitus
could not have literally rejected LNC, as he is often accused of (or
praised for) doing, as his writings preceded the statement of that
principle in *Metaphysics* G by well over a century. But
the question remains: do his words, as represented in the extant
fragments, anticipate the Dialetheists and other rejectionists? Yes
and no. To be sure, Heraclitus was proud to wear the mantle of
"paradoxographer" (Barnes 1982: 80) and enjoyed nothing
more than to *epater les bourgeois* of his day. But the
key fragments supporting his proclamation of the Unity of Opposites
can be taken in more than one way. He points out that sea water is
salutary (if you're a fish) and unhealthy (if you're a human), just as
garbage is preferable to gold (for a donkey) but then again it isn't
(for a person). And given the inevitability of flux (as Heraclitus
memorably illustrated by his river into which one cannot step twice),
what is true (today) is also false (tomorrow). But these astute
observations do not so much refute the LNC as much as demonstrate
(through what Barnes calls the Fallacy of the Dropped Qualification)
the need for Aristotle's crucial codicil: sea water, for example,
cannot be both healthful and unhealthful for the same experiencer at
the same time and in the same respect. (In a similar way, "It is
raining" may of course be deemed true at this moment in Seattle
and false in Palo Alto; we need admit no contradiction here, whether
we deal with the issue by endorsing unarticulated constituents or in
some other manner; cf. Recanati 2002 inter alia.) Ultimately, whether
one follows Kirk (Heraclitus [1954]) in charging Aristotle with
misrepresenting Heraclitus as an LNC-denier or sides with Barnes
(1982) and Wedin (2004b) in sustaining Aristotle's accusation, it is
hard to see in what respect the evidence presented by Heraclitus,
however subtle a guide he may be for our travels on that path on which
up and down is one and the same, threatens the viability of LNC. (See
also Heraclitus.)




Within the modern philosophical canon, Hegel has often been seen as
the echt LNC-skeptic, well before his reputed deathbed lament,
"Only one man ever understood me, and he didn't understand
me." Hegel saw himself as picking up where Heraclitus left
off--"There is no proposition of Heraclitus which I have
not adopted in my logic" (Barnes 1982: 57)--and indeed the
Heraclitean view of a world shaped by the unity of opposites through
strife and resolution does seem to foreshadow Hegelian dialectic. In
fact, however, an unresolved contradiction was a sign of error for
Hegel. The contradiction between thesis and antithesis results in the
dialectical resolution or superseding of the contradiction between
opposites as a higher-level synthesis through the process
of *Aufhebung* (from *aufheben,* a verb simultaneously
interpretable as 'preserve, cancel, lift up'). Rather than repudiating
LNC, Hegel's dialectic rests upon it. In Marxist theory, too,
contradictories do not simply cancel out but are dynamically resolved
(*aufgehoben*) at a higher level in a way that both preserves
and supersedes the contradiction, motivating the historical
dialectic. (See Horn 1989: SS1.3.2.)




For Freud, there is a realm in which LNC is not so much superseded but
dissolved. On the primary, infantile level, reflected in dreams and
neuroses, there is no *not*: "'No' seems not
to exist as far as dreams are concerned. Anything in a dream can mean
its contrary" (Freud 1910: 155). When the analysand insists of a
dream character "It's not my mother", the analyst
knowingly translates, "So it *is* his mother!"
Freud sought to ground this pre-logical, LNC-free (and negation-free)
realm not just in the primal realm of the dreamer's unconscious but
also in the phenomenon of *Gegensinn*, words
(especially *Urworte*, primal words) with two opposed meanings
purported attested widely in ancient and modern languages. The
empirical basis for this latter claim, however, has been widely
discredited; see Benveniste 1956.




Given Aristotle's observation (*Metaphysics* 1006a2) that
"even some physicists" deny LNC and affirm that it is
indeed possible for the same thing to be and not to be at the same
time and in the same respect, he might not have been surprised to
learn that quantum mechanics has brought such challenges once more
into play. Thus, we have Schrodinger's celebrated imaginary cat,
placed (within the context of a thought experiment) inside a sealed
box along with radioactive material and a vial of poison gas that will
be released if and only if that material decays. Given quantum
uncertainty, an atom potentially inhabits both states--decayed
and non---simultaneously, seeming to render the cat (in the
absence of an observer outside the system) both alive and dead. But
most physicists would argue that while quantum mechanics may challenge
some aspects of classical logic it does not threaten LNC. If we could
in fact observe a cat, or a particle, as A and not-A at the same time,
then there would be a violation of the Law of Non-Contradiction; the
mere potential for an entity to be in either of two mutually
inconsistent states does not in itself violate the LNC.  




As we have seen, Aristotle himself anticipated many of the challenges
that have since been raised against LNC. One more such challenge is
posed by the ubiquity of doxastic inconsistency. Take the desires of
Oedipus, for example. In seeking Jocasta as his mate, did he wish to
marry his mother? Certainly he did on the de re reading: Oedipus's
mother (Jocasta) is such that he wanted to marry her, although he
would not have assented to the claim that he wanted to marry his
mother. In a sense, then, "Oedipus wanted to marry his
mother" is true (de re) and false (de dicto), but no violation
of LNC is incurred, since these represent different propositions, the
semantic distinction neutralized within the sentential form. But what
of the de dicto reading itself: it is *really* false? After
all, as a young boy Oedipus can be assumed (by some) to have exhibited
the eponymous complex, according to which the falsity of the (de
dicto) proposition that he wanted to marry his mother on
a *conscious* level belies the truth of this proposition on
an *unconscious* level. But this does not entail that he both
wished and did not wish to "marry" his mother at the same
time *and in the same respect*. Whether it concerns the
unacknowledged incestuous conflict of the Theban king, the indecision
of Zerlina's "Vorrei e non vorrei" response to Don
Giovanni's invitation, or the unspecified ambivalence of the
respondent in Strawson's exchange (1952: 7)



> 
> --Were you pleased?
>   
> 
> --Well, I was and I wasn't.




we have ample opportunity to reflect
on the foresight of Aristotle's rider: "a is
*F*" and "a is not *F*"
cannot both hold *in the same sense, at the same time, and in the
same respect*. 


## 5. LNC and the Buddhist Tetralemma




Beyond the Western canon, the brunt of the battle over LNC has been
largely borne by the Buddhists, particularly in the exposition by
Nagarjuna of the *catuskoti* or tetralemma (c. 200
A.D.; cf. Bochenski 1961: Part VI, Raju 1954, Garfield 1995, Tillemans
1999, Garfield & Priest 2002), also known as the four-cornered or
fourfold negation. Consider the following four possible truth
outcomes for any statement and its (apparent) contradictory:



> 
> 
> 
> |  |  |
> | --- | --- |
> | (9)   (i) | *S* is *P* |
> | (ii) | *S* is not *P* |
> | (iii) | *S* is both *P* and not-*P* |
> | (iv) | *S* is neither *P* nor not-*P* |
> 
> 
> 




For instances of the positive tetralemma, on Nagarjuna's
account, all four statement types can or must be accepted:



> 
>  Everything is real and not real.  
> 
>  Both real and not real.  
> 
>  Neither real nor not real.  
> 
>  That is Lord Buddha's teaching.  
> 
>     --*Mula-madhyamaka-karika*
> 18:8, quoted in Garfield (1995: 102)
>  




Such cases arise only when we are beyond the realm to which ordinary
logic applies, when "the sphere of thought has ceased." On
the other hand, more use is made of the negative tetralemma, in which
all four of the statements in (9) can or must be rejected, and thus
one cannot assert either Ph, !Ph, both Ph and !Ph, or
neither Ph nor !Ph. Is this tantamount, as it appears, to the
renunciation of LEM and LNC, the countenancing of both gaps and gluts,
and thus--in Aristotle's view--the overthrow of all bounds
of rational argument?




It should first be noted that the axiomatic status of LNC and LEM is
as well-established within the logical traditions of India as it is
for the Greeks and their
 epigones.[7]
 Garfield (1995) and Tillemans (1999) convincingly refute
the claim that Nagarjuna was simply an 
"irrationalist".[8]
 In the first place, if Nagarjuna simply rejected LNC, there
would be no possibility of *reductio* arguments, which hinge on
the establishment of untenable contradictions, yet such arguments are
standardly employed in his logic. In fact, he explicitly
prohibits *virodha* (contradiction). Crucially, it is only in
the realm of the Absolute or Transcendent, where we are contemplating
the nature of the ultimate, that contradictions are embraced; in the
realm of ordinary reality, LNC operates and classical logic
holds. (Recall Freud's dichotomy between the LNC-observant conscious
mind and the LNC-free unconscious.) In this sense, the logic of
Nagarjuna and of the Buddhist tradition more generally can
be seen not as inconsistent but paraconsistent. Indeed, just as
Aristotle ridiculed LNC-skeptical sophists as no better than
vegetables (see SS1), the Buddhists dismissed the arch-skeptic
Sanjaya and his followers, who refused to commit themselves to a
definite position on any issue, as
"eel-wrigglers" *(amaravikkhepa)*. Sanjaya
himself was notorious for his periodic lapses into the extended
silence Aristotle described as the last refuge of the LNC-skeptic (see
Raju 1954). 




One aspect of the apparent paradox is precisely parallel to that
arising with some of the potential counterexamples to the LNC arising
in Western thought. In various Buddhist and Jainist systems of
thought, the apparent endorsement of *Fa* &
!*Fa* (or, in propositional terms, Ph [?] !Ph)
is upon closer examination qualified in precisely the way foreseen by
the codicils in Aristotle's statement of the law: From a certain
viewpoint, Ph (e.g. Nirvana exists); from a certain viewpoint,
!Ph (e.g. Nirvana does not exist). (Compare the observation of
Jainists two millennia ago that "*S* is *P*"
and "*S* is not
*P*" can both be true from different standpoints; cf. Raju
1954: 698-701; Balcerowicz 2003.)




To further explore the status of truth-value gluts, in which both
classical values are simultaneously assigned to a given proposition
(e.g. "*x* is real"), let us consider the analogous
cases involving gaps. Recall, for example, the case of future
contingents as in (2a,b) above: we need not maintain that "Iraq
will become a secular democracy" is neither true nor false when
uttered today, but only that neither this statement nor its
contradictory "Iraq will not become a secular democracy" is
*assertable* today in the absence of foreknowledge. Similarly
for past unknowables, such as (to adapt an example from Quine) the
proposition that the number of blades of grass on the Old Campus lawn
during the 2005 Yale commencement exercises was odd. This is again
more plausibly viewed as unassertable than as truth-valueless, even
though its truth-value will never be known. To take a third example,
we can argue, with Grice (1989: 80ff.), that a negation outside the
scope of a conditional is generally intended as a refusal (or
hesitation) to assert "if *p* then *q*"
rather than as the contradictory negation of a conditional, whose
truth value is determined in accord with the standard material
equivalence:



> 
> (10) !(*p*-*q*) [?] (*p* &
> !*q*)




Thus, in denying your conditional "If you give her penicillin,
she will get better", I am allowing for the possibility that
giving her penicillin might have no effect on her, but I am not
predicting that you will administer the penicillin and she will fail
to recover. Nor does denying the apothegm (typically though
inaccurately attributed to Dostoyevsky or Nietzsche) that if God is
dead everything is permitted commit one to the conjoined proposition
that God is dead and something is forbidden. As Dummett (1973: 328-30)
puts the point, we must distinguish negation outside the scope of a
Fregean assertion operator, not
 ([?]*p*),
 from the assertion of a negative proposition,
 [?](not *p*).
 The former interpretation "might be taken to be a means of
expressing an unwillingness to assert" *p*, in particular
when *p* is a conditional:



> 
> 
> 
> |  |  |
> | --- | --- |
> | (11)   *X*: | If it rains, the match will be canceled. |
> | *Y*: | That's not so. (*or,* I don't think that's the case.) |
> 
> 
> 



*Y*'s contribution here does not constitute a negation of
*X*'s content; rather, we can paraphrase *Y* as
conveying (11'a) or (11'b):



> 
> (11'a) If it rains, the match won't necessarily be canceled.  
> 
> 
> (11'b) It may [*epistemic*] happen that it rains and yet
> the match is not canceled.




Dummett observes, "We have no negation of the conditional of
natural language, that is, no negation of its sense: we have only a
form for expressing refusal to assent to its assertion."




Similarly with disjunction. Consider the exchange in (12) preceding
the 2000 election, updated from an example of Grice:



> 
> 
> 
> |  |  |
> | --- | --- |
> | (12)   *X*: | Bush or Gore will be elected. |
> | *Y*: | That's not so: Bush or Gore or Nader will be elected. |
> 
> 
> 



*Y*'s rejoinder cannot be a contradictory of the content of
*X*'s claim, since the (de jure) election of Bush rendered both
*X*'s and *Y*'s statements true. Rather, *Y*
objects on the grounds that *X* is not in an epistemic position
to assert the binary disjunction.




Unassertability can be read as the key to the apparent paradox of the
*catuskoti* as well. The venerable text in
*Majjhima-nikaya* 72, relating the teachings of the
historical Buddha, offers a precursor for Nagarjuna's
doctrine of the negative tetralemma. Gotama is responding to a monk's
question concerning the doctrine of rebirth (quoted in Robinson 1967:
54):



> 
>  Gotama, where is the monk reborn whose mind is thus freed?  
> 
>     Vaccha, it is not true to say that he is
>  reborn.  
> 
>  Then, Gotama, he is not reborn.  
> 
>     Vaccha, it is not true to say that he is not
>  reborn.  
> 
>  Then, Gotama, he is both reborn and not reborn.  
> 
>     Vaccha, it is not true to say that he is both
>  reborn and not reborn.  
> 
>  Then, Gotama, he is neither reborn nor not reborn.  
> 
>     Vaccha, it is not true to say that he is neither
>  reborn nor not reborn.  
> 
> 




Note the form of the translation here, or similarly that of the
standard rendering of the negative *catuskoti* that "it
profits not" to assert Ph, to assert !Ph, to assert
both Ph and !Ph, or to assert neither Ph nor !Ph:
the relevant negation can be taken to operate over an implicit modal,
in particular an epistemic or assertability operator. If so, neither
LEM nor LNC is directly at stake in the tetralemma: you can have your
Aristotle and Buddha too.




We tend to recalibrate apparent violations of LNC as conforming to a
version of the law that incorporates the Aristotelian qualifications:
a sincere defense of "*p* and not-*p*"
plausibly involves a change in the context of evaluation or a shift in
viewpoint, or alternatively a suppression of modal or epistemic
operators. This practice can be seen as an instance of a general
methodological principle associated with
 Davidson
 and Quine that has come to be called the *principle of
charity* (or, alternately, the *principle of rational
accommodation*): when it is unclear how to interpret another's
argument, interpret it in a way that makes the most sense. At the same
time, this procedure evokes the standard Gricean mode of explanation
 (Grice 1989; see the entry on
  implicature):
granted the operation of the Cooperative Principle and, more broadly,
the shared premise of rationality, we reinterpret apparent violations of
valid principles or maxims so as to conserve the assumption that one's
interlocutor is a rational and cooperative agent. And as Aristotle would
remind us, no principle is more worthy of conservation than the Law of
Non-Contradiction.


## 6. Vagueness and borderline contradictions




We have seen that two of the more significant threats to the
unchallengeable status of LNC tend to dissolve under closer scrutiny.
Heraclitus-type contradictions *(Sea water is healthy and sea water
is not healthy)* are rendered LNC-compatible by the Aristotelian
rider designed for that purpose ("a is *F*" and
"a is not *F*" cannot both hold *in the same
sense, at the same time, and in the same respect*) once the
"dropped qualification" or contextual specification is
restored (SS4). Buddhist-type contradictions *(Nirvana exists
and Nirvana does not exist)* can be understood as modalized,
embedded under an epistemic modal or assertability operator.




We are left with two significant challenges for LNC: (i) the case of
the Liar and related paradoxes of self-reference as touched on in
SS3 and work cited therein (see the entries on Liar Paradox,
Russell's Paradox, Dialetheism, Paraconsistent Logic) and (ii)
the case of vagueness and its implications for borderline truth,
discussed in this section. As we have seen, the admission of
truth-value gaps (exceptions to LEM) is mirrored by the admission of
truth-value gluts (exceptions to LNC). As intellectual heirs of the
Hegelian tradition, Marxists have willing to accept paradox, and
indeed the pre-Revolutionary theorist Plekanov (1909) suggests that
Sorites (in the form of the bald man; see sorites entry) can be
resolved by rejecting LNC. Hyde (1997) traces the history of
vagueness-based arguments for paraconsistency from Plekhanov through
Jaskowski, McGill, and Parry in the 1940s to the dialetheist
logics of today. Pointing out the parallels between treatments of
vagueness via truth-value gap theories (including the supervaluation
theory of van Fraassen 1969) and their truth-value glut counterparts,
Hyde laments the unwarranted neglect of the latter option: "The
thought that an adequate response might require the recognition of
cases of overdetermination and and truth-value gluts has few
supporters" (Hyde 1997: 641). In the two decades since the
publication of his important paper, that neglect has been largely
rectified, especially with the accretion of empirical support for
glut-oriented approaches endorsing the acceptability of a range of
true contradictions of the form *a is P and a is not P,* in
particular when P is a vague predicate like *tall*, *vague*,
or *red*.



Supervaluation theory utilizes the concept of admissible sharpening or
precisification of vague predication (see the entry on
 vagueness). I may be unable to assert
 truthfully of a given color
chip in the red-orange range that it is red, while also being unable
to assert that it is not red. Once the notion of red is sharpened or
precisified to a particular interval of wave-lengths, I am in a
position to assert either the positive proposition "a is
red" or its negation "a is not red", but which one
depends on the details of the sharpening. A sentence *S* is
supertrue iff it is true on every sharpening and superfalse if it is
false on every sharpening; if is true on some sharpenings and false on
others, it is neither supertrue nor superfalse, thus corresponding to
a truth-value gap. One salutary result of this approach is that LEM
can be retained: *P or not P* is supertrue since it is true on
every sharpening. Hyde (1997) draws on the dual status of gaps and
gluts to define a paraconsistent "subvaluation" theory:
*S* is subtrue iff is true on at least one sharpening, subfalse
iff it is false on at least one sharpening, and neither subtrue nor
subfalse if it is true on some sharpenings and false on others. Given
the status of supervaluation and subvaluation as duals, each is equally
defensible and applicable to the semantics of vagueness. 
Crucially, it is the notion of supertruth, or subtruth as the case may
be, that tracks our natural language intuitions of truth, and similarly
for falsity. In this sense, the logic of supervaluation is
paracomplete (allowing for violations of LEM) and the logic of
subvaluation paraconsistent (allowing for violations of LNC). In the
tradition of paraconsistent logics, Hyde stresses the need for
"quarantining the gluts" to avoid the explosive
consequences of *ex contradictione quodlibet* (see SS4).




Where supervaluation theory allows *a is not
tall* or *a is not bald* to be neither (super)true nor
(super)false if *a* represents a borderline instance of tallness,
subvaluation theory would characterize such an assessment as both
(sub)true and (sub)false. But since tautologies are preserved if
validity is defined in terms of truth and not subtruth (or
supertruth)--an assumption that some might challenge--*a
is tall and a is not tall* can never be true, even when *a is
tall* and *a is not tall* are both true:



> 
> A vague
> sentence '*A*' and its negation might both be true
> (since they each have some true disambiguation--i.e. the sentence
> '*A*' itself as some true and some false
> disambiguation--but their conjunction can never be.
> '*A & !A*' has only false disambiguations;
> LNC is retained. (Hyde 1997: 654)




For Akiba (1999), however, paraconsistent subvaluation theories,
like paracomplete supervaluation theories, should be recast in a modal
light, by supplying an epistemic possibility operator *Pos* in the former
case (reminiscent of the modalized accounts of the tetralemma explored in
SS5) and its dual, an abstract modal concept Akiba defines as
*Def* (corresponding to necessary truth) in the latter. 
Thus, while *P & !P* remains a full-fledged
contradiction, its modalized counterpart *Pos(P) &
Pos(!P)* is coherent, and while *P v !P* is a valid
instance of LEM, *Def(P) v Def(!P)* is not:
"Super- and subvaluation systems can be considered not really
alternatives to classical logic but in fact just modal extensions of
it." Akiba's key departure from Hyde's approach is in
rejecting the move to identify truth *simpliciter* with subtruth
(or supertruth). 




Confronted with the task of applying semantically vague predicates
like *tall* and *red* to borderline cases, the intuitions
of respondents in psycholinguistic studies often tend to become
uncertain. Subjects produce varying responses for the crucial
data, and the framers of the studies have produces varying explanations
for the data. In one of the earlier studies, Bonini et al. point to the
asymmetries in judgments of truth and falsity for borderline cases of
*a is red/tall* and ultimately reject truth-gap and truth-glut
accounts in favor of an LEM- and LNC-compatible epistemic theory of
"vagueness as ignorance". That is, the subject *S*
"mentally represents vague predicates in the same way as other
predicates with sharp true/false boundaries of whose location
*S* is uncertain" (Bonini et al. 1999: 387). 




Kyburg (2000) offers a pragmatic explanation for why *a is
tall* might be judged neither true nor false in a case of
borderline tallness, or why *a is tall and a is not tall* might
be judged true in the same circumstance. Without conceding gaps
or gluts, she points to a disconnect between truth-value intuitions and
assertability intuitions for a cooperative speaker aiming to be
maximally informative. Of course, in some cases, the felicity of *a
is tall and a isn't tall* might simply reflect the awareness
that different cut-off points apply depending on the relevant
comparative class of entities (children vs. adults, male adults vs.
male professional basketball players, etc.)--the instances
foreseen by the Aristotelian rider. But in other cases, such a
statement would indeed convey that a possesses a borderline degree of
tallness, while simply affirming or denying the vague
predicate--*a is tall* or *a is not
tall--*would communicate too high or too low a degree. Just as
speakers might a sentence might refrain from uttering, or classifying
as true, a perfectly and "sharply" true sentence whose use
would mislead the hearer--*Snow is white or snow is
purple--*so too the assertion of a conjunction of opposites in
borderline cases--*a is tall and he's not
tall--*might be preferred, despite its contradictory nature,
to the misleading assertion of either *a is tall or a is not
tall* in the same circumstance.




For Sorensen (2001), an individual's acceptance of the truth
of certain false claims, including those necessarily false claims of
the form *a is tall and a is not tall,* reflect not the
inadequacy of our semantic and logical analyses but the tendency for
speakers to be wrong, confused, or ignorant about the truth conditions
of given sentences or about their own beliefs, especially those
involving vagueness, similarly to the way in which perceptions can
mislead in the case of visual illusions.




The vagueness-as-ignorance theory of Bonini et al. (1999), the modal
theory of Akiba (1999), the assertability theory of Kyburg (2000), and
the error theory of Sorensen (2001) are all semantically classical in
their preservation of LNC (along with LEM and bivalence). In essence,
each of these approaches to the assessment of vague predications
reflects a wariness about jettisoning the classical account of truth
and contradiction in favor of an approach endorsing either
supervaluations (and gaps) or subvaluations (and gluts). Despite the
difficulties posed by vagueness, these theorists would thus adopt the
philosophical corollary to the familiar legal adage: "Hard cases
make bad laws."




In two recent papers the reconciliatory approach is rejected in
favor of defenses of paraconsistency, although not precisely along the
lines laid out by Hyde. On Hyde's account, *a is tall* and
*a* *is not tall* can both be true when *a* is
borderline tall (say, 5'11'') while the full or reduced conjunction of
these statements, *a is tall and (a is) not tall,* will be
false, given the standard supervaluationist approach. Alxatib &
Pelletier, however, find the opposite judgments among a majority of the
subjects of their study, who accept the conjunction while rejecting the
individual ascriptions of truth. In addition to reinforcing the view of
gappy and glutty theories as mirror images, Alxatib & Pelletier
present their results as evidence against the neo-classical
vagueness-as-ignorance position of Bonini et al. (1999), a position
which also has difficulty accounting for the paucity of subjects
choosing the "can't tell" option in the surveys. Like
the assertability-based pragmatic theory of borderline contradictions
proposed by Kyburg, Alxatib & Pelletier (2011: 321) offer a Gricean
explanation for subjects' endorsement of the truth of apparent
contradictory statements--*a is tall and not tall* when a
is just short of 6 feet tall--which yields (in just this case) a
subvaluationary interpretation on which the speaker is taken to convey
the borderline status of *a*'s height. Their central
finding (2011: 298-9) is that theories admitting truth-value gaps like
that of Hyde (1997) or the paraconsistent logic of Priest 2006 fare
better in accounting for the full range of interesting data presented
in their experiments--and those of Ripley 2011. 



Like Alxatib & Pelletier, Ripley (2011: SS2) conducts his own
survey of reactions to apparent contradictions involving vague
predications (e.g., "The circle is both is and isn't near
the square"; "The circle neither is nor isn't near
the square"). He leverages the results of his study to argue
against error (or epistemic) theories on which borderline
contradictions are always false even when believed true; he also
refutes fuzzy theories of vagueness, which fail to predict why
subjects are so willing to accept the truth of borderline
contradictions rather than assigning them an intermediate
acceptability ranking. Ripley also finds pragmatic or contextualist
theories not insufficiently fleshed out to yield the correct
predictions. He adopts a dialetheist account that enables him to line
up with a majority of his subjects in finding both "*a*
is *F* and *a* is not *F*" and "*a* is
neither *F* nor not *F*" true for borderline
applications of *F*. Supporting this view, he notes (2011:
186-7) that there may be variation with respect to the "cultural
aversion to contradictions"; there is evidence tending to
suggest that Asian subjects are less likely than Westerners
(specifically, as it happens, Canadians) to object to contradictions
like "It's raining and it's not raining." But as Ripley
concedes, more research is needed on this point.


## 7. Contradiction in everyday life




The logical incoherence of contradictions is the ground both for
indecision (as with Zerlina's ambivalent *Vorrei e non
vorrei* in our epigraph) and for the pragmatic exploitation of
apparent contradictions for communicative ends. One locus is the
oxymoron, a phrasal contradiction recognized for millennia as a figure
of speech. The dramatic effect of yoking contradictories is as old as
the classical *festina lente* ('make haste slowly',
adopted as a motto by Augustine and the Medicis. The German
counterpart, deployed by Goethe and others, adds rhyme: *Eile mit
Weile*.




The use of actual or apparent contradiction to generate meanings via
reconstrual may convey advice to seek the middle way, reminiscent of
the function of borderline contradictions surveyed in SS6, or may,
in the case of an apparently incoherent phrasal structure, serve as a
signal that the modifier constrains the denotation of the head. Thus
*bittersweet* is not taken as self-contradictory property, while
a *tragicomedy* is a work or production that spans the two
normally opposed genres--recall Walt Whitman's "I am
large; I contain multitudes". Classic oxymorons include *cruel
kindness,* *living death,* and *true lies;* jocular
ones are regularly proposed, from *military intelligence* and
*congressional ethics* to *airplane food* and *open
secrets*. A particularly transparent form is offered by those
promoting *undesign design* or the *non-concept
concept.*




The oxymoron, and in particular the catalogue of oxymorons, may
signal the breakdown of logical coherence, given the explosive effect
of *ex contradictione quodlibet*. In *Romeo and Juliet,*
we ascend from the former's risible inventory of battling
binaries to the latter's ardent valediction on her balcony:



> 
> 
> Why, then, O brawling love! O loving hate!
>   
> 
> O anything, of nothing first create!
>   
> 
> O heavy lightness! Serious vanity!
>   
> 
> Misshapen chaos of well-seeming forms!
>   
> 
> Feather of lead, bright smoke, cold fire, sick health!  
> 
>    --*Romeo and Juliet*, I.i
> 
> 
> 
> Good night, good night. Parting is such sweet sorrow
>   
> 
> That I should say good night till it be morrow.  
> 
>    --*Romeo and Juliet*, II.ii
> 
> 
> 




Expanding from the phrasal to the sentential level, we find
self-contradictory propositions, whether in full or elliptical form,
functioning in forums from high culture to social media to generate a
variety of implicatures. In her recent study of the phenomenology of
ambivalence, Razinsky takes the affirmation of conjoined
contradictories "a is *v* and a is not *v*"
for a given value judgment *v,* or the affirmation of "a
is *v* and a is *v *" where *v,
v * are contraries (e.g. *a is good and a is
bad*), to be "central to the logic of value and of
value judgment" (2017: 229). Some apparent instances of
ambivalence (and of any underlying violation of LNC) can be explained
away. Citing Mill, Razinsky (2017: 201) accepts that there are
instances in which *a is just and a is unjust* reflects only an
apparent opposition, to be understood essentially as *a is just in
some respects and a is unjust in other respects*. Such cases
illustrate the prescience of the Aristotelian rider: "a is
*F*" and "a is not *F*" cannot both
hold *in the same sense, at the same time, and in the same
respect*. Similarly, the imposition of the edict *a*
*ought to F and a ought not to F* results in difficulty for the
unfortunate *a*, to be sure, but not necessarily in an LNC
violation, given the availability of distinct sources of obligation.
But other cases involve true ambivalence: "Meaningful
contradictions are moored in the possibility of conceptual
ambivalence" (Razinsky 2017: 44). The analysis of such cases must
preserve the dynamic tension in affirming that a is *v* and a is
*not-v* and at the same time acknowledge the speaker's
awareness of the violation of LNC, not an easy task: "Do people
believe contradictions when it comes to value? I do not think there is
a good answer to give here" (Razinsky 2017: 228)--they
believe them and do not believe them.




Perhaps as a reflection of this tension, the A-not-A meme has widely
proliferated in popular entertainment, advertisements, and social
media. "Sorry not sorry" has become a standard device for
a sarcastic or insincere apology (also known as a "non-apology
apology"), while #SorryNotSorry is a trending Twitter hashtag as
well as a Demi Lovato song title (see Carey 2014 in Other Internet
Resources). A related meme is "I'm not saying, I'm just
saying", a now notorious "get out of jail free"
discourse ploy showing up in hundreds of google hits, as the subject
of an urbandictionary entry and several online commentaries by
linguists, and as the title of a 2012 flash fiction paperback by
Matthew Salesses. The category of "Christmas-not-Christmas
movies" makes its perennial appearance on the web during holiday
season, and the #worknotwork has been adopted as the house meme of
Samsonite luggage.
 #GuiltyNotGuilty
is another popular hashtag as well as a theme for Gucci ads.




A relatively new means for expressing ambivalence in colloquial U.S.
and Australian English is the "Yeah no" (or, less
frequently, "No yeah") response, which has been analyzed as
conveying a variety of possible discourse functions (see Burridge &
Florey 2002, Lee-Goldman 2011, and Liberman 2008 in Other Internet
Resources).




The conjunction of contradictory predicates of the form *not A
and/but not NOT A* is used to indicate the borderline status of an
entity with respect to a category and its complement:
 


* **They not brown but they not NOT brown**.  

[Saturday Night Live (faux commercial 
Levi's Wokes),
 Nov. 30, 2017]
* Karl Mitze, Geoff Manyin, Nick Montopoli, and Zachariah Matteson
redefine what it means to be a string quartet. Described as
"**not classical...but not not
classical**", bowed and fretted Invoke continues to
successfully dodge even the most valiant attempts at genre
classification.   

[youtube.com]
* **Not punk enough to be punk but not not punk enough to be not
punk**.   

 [skyblock forums]
* Strayed and I are heading out for a stroll in Portland, Oregon, in
the kind of weather for which that city is famous: **not raining,
but not not-raining**.   

[Schulz 2014, Other Internet Resources]
* Mozzarella sticks are one of my favorite appetizers...The thing
is, are they healthy? No. Are they even in the middle, like **not
healthy but not not healthy**?
No.  

 [Creme de la Crumb recipe column]




While these appear fully contradictory, that diagnosis relies on the
status of *S is not not P* as an instance of !(!(S is
P)) within a logic employing a robust Law of Double Negation. Arguably,
however, in each of these attestations, the inner
negation--*not brown, not punk, not healthy,* and even
*not-raining--*can be viewed as a coerced or virtual
contrary, in which case the two negations do not fully cancel out.
Thus, just as *not impossible* fails to reduce to *possible,
not not classical* is distinct from *classical,* and no
actual violation of LNC is ultimately condoned. But, as on
Grice's (1989) analysis of irony and tautology, it is the
recognition of the apparent contradiction that the speaker exploits to
generate the higher-order coherence of the message.




A less, or perhaps differently, mediated case of coherence in
apparent contradictories involves prosody. In this case, the
hearer/reader seems to have to reconcile LNC with the joint predication
of *A* and *not A* rather than the more complex *not A
and not not A*. But on closer examination, the task is less
daunting than it appears, as it's really a matter of affirming
*A* while rejecting ***A****,* or of
metalinguistically rejecting *A* in favor of
***A***:


* After playing a staggering 40,000 minutes (in just 11
seasons) and carrying four straight Finals teams, LeBron [James] might
be battling the long-term effects of a historically ridiculous two-way
burden. **He's still great, but he's not
GREAT**.   

[Simmons 2014, Other Internet Resources]
* I'm short. ***Not SHORT, but short***.
 I'm 5'8'' (on a good day) with a 30'' inseam.  

 [Total Motorcycle Forum posting, May 31, 2007]
* "Rita", he whispered. "I'm
tired." And the way he said the word, she understood.
**He wasn't tired, he was *tired****,*
a condition of the mind as well as the body. A state of the soul.  

[Lisa Gardner, 2008, *Say Goodbye*, p. 151]
* But - I can tell you this, and maybe I'm stating the
obvious here but - Vegas is a DESERT.
In other words, it's hot. No, **it's not hot...
it's HOT**. REALLY HOT. People drop like flies during the
day from \*walking\* too much.  

 [Skate Log Forum posting, 
Nov. 5, 2007]




The prosodic focus in such cases serves to invoke a scalar contrast
in degrees or qualities of A-ness; the result is analogous to instances
in which a descriptor is contrasted with a clone or contrastive focus
reduplication instance of the same descriptor, yielding variants of the
true borderline contradictions discussed in SS6:


* Average American man is 5'9''. **You're tall, but not TALL
tall**... if that makes sense.  

 [https://www.girlsaskguys.com/other/q1740369-is-6-1-1-85m-tall-for-a-guy]
* (**So he's dead but not *dead*-dead**. Get
it?)  

 [https://tinyurl.com/y8656rqy,
 of Jon Snow on Game of Thrones]




Finally, it is worth noting the role contradiction plays as a lively
source of ironic humor, especially for the highbrow reader. In one
 New Yorker cartoon 
(Benjamin Schwartz, March 30, 2015), a veterinarian comes into the
waiting room, places a comforting arm on the worried man's
shoulder, and breaks the news: "About your cat, Mr.
Schrodinger--I have good news and bad news." (See
SS4 above for more on the poor cat's indeterminate
prognosis.) In
 another
(J. B. Handelsman, March 9, 1987), a mutton-chopped publisher, seated
at his Victorian office desk and riffling through an immense
manuscript, explains to the dejected bearded author, "I wish you
would make up your mind, Mr. Dickens. Was it the best of times or was
it the worst of times? It could scarcely have both." .





We end by borrowing the last words of Samuel Beckett's *The
Unnamable* (1954)--"I can't go on, I'll go
on"--as my valedictory, since in fact I can't go
on.