negation in logic and in natural language

Negation in Logic and in Natural LanguageAuthor(s): Jaakko HintikkaSource: Linguistics and Philosophy, Vol. 25, No. 5/6 (Dec., 2002), pp. 585-600Published by: SpringerStable URL: http://www.jstor.org/stable/25001865 .

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JAAKKO HINTIKKA

NEGATION IN LOGIC AND IN NATURAL LANGUAGE

ABSTRACT. In game-theoretical semantics, perfectly classical rules yield a strong negation that violates tertium non datur when informational independence is allowed. Con

tradictory negation can be introduced only by a metalogical stipulation, not by game rules.

Accordingly, it may occur (without further stipulations) only sentence-initially. The res

ulting logic (extended independence-friendly logic) explains several regularities in natural

languages, e.g., why contradictory negation is a barrier to anaphase. In natural language, contradictory negation sometimes occurs nevertheless witin the scope of a quantifier. Such sentences require a secondary interpretation resembling the so-called substitutional inter

pretation of quantifiers. This interpretation is sometimes impossible, and it means a step beyond the normal first-order semantics, not an alternative to it.

If you review the discussions of negation in the last couple of decades in

journals like Linguistics and Philosophy, you will find a curious difference between the treatments of negation in logic and in linguistics. In logic, negation is usually taken to be a simple, unproblematic notion, perhaps a

mere reversal of the concepts of truth and falsity. An extreme form of this view is found in Wittgenstein's Tractatus (1921). An important part of the

argument there is an extension of the picture idea from elementary pro positions to their truth-functions. These truth-functions include negation. For Wittgenstein, the negation of a truth-function qua picture is not only also a picture. It is the same picture, but taken into the opposite polarity or, as Wittgenstein sometimes expresses it, with the opposite Sinn (sense), as shown in Hintikka and Hintikka (1986).

In contrast, linguists have found negation in natural language a highly complex subject full of puzzling phenomena. In an able recent book (Horn 1989), it takes over six hundred pages to survey "the natural history of

negation". Logicians' course-grained treatment of negation means that lin

guistics have so far not been in a position to expect much help from them. Even when linguists have found generalizations that according to their current methodological jargon "account for" certain phenomena, those

generalizations themselves often cry out for a deeper explanation. This entire contrast demands an explanation. It is obvious that the

source of the complexities of negation in natural languages is the fact

LA Linguistics and Philosophy 25: 585-600, 2002. 9 ? 2002 Kluwer Academic Publishers. Printed in the Netherlands.



586 JAAKKO HINTIKKA

that the underlying structure of our Sprachlogik have not been located by logicians. One of the thrusts of this paper is to propose an explanation.

As an example of puzzling natural language phenomena related to

negation, one can mention the fact that negation is normally a barrier to

anaphora. It is acceptable to say

(1) Some of the students passed the examination. They must have studied hard.

but not

(2) Not all the students failed the examination. They must have studied hard.

This is the case in spite of the fact that, logically speaking, (1) and (2) should be equivalent. If someone actually uttered (2), we might be able to

figure out that he or she actually means the same as (1), but we would still think of (2) as being somehow deviant.

This cannot be accounted for by pointing out that in (2) They does not have a syntactical antecedent. For the acceptability of an anaphoric pronoun is usually a matter of its having a semantical antecedent, not a

syntactical one. Another group of puzzling phenomena - in this case somewhat less

sharp ones - manifest a tendency in natural languages to avoid a negation that semantically speaking would be governed by a quantifier. If I say to

you

(3) Everything that glitters is not gold.

you will spontaneously understand it as having the logical form

(4) -(V?x) (x glitters D x is gold)

not the form

(5) (Vx) (x glitters D --(x is gold)).

This violates the (admittedly weak) preference in ordinary language for

left-to-right order as an indication of logical priority ("scope"). If (5) were what I meant, it would have to say

(6) Nothing that glitters is gold.



NEGATION IN LOGIC AND IN NATURAL LANGUAGE 587

Here we can in fact see a generalization that in a sense "accounts for" the relevant data. It says roughly that if a negation occurs, semantically speaking, within the scope of a universal quantifier in English, one should use one of the negative quantifier words no, nothing, nobody, nowhere, etc. Similar regularities hold in other languages.

This explains also the preferred reading (4) of (3). For if - were in the

scope of (Vx), one would have to utter (6) and not (3). So where is the puzzle? What has happened is that one puzzle has been

replaced by another. The new puzzle is: Why should there exist in natural

languages a separate negative quantifier? I realize that questions like this are currently not popular among linguists, possibly because they ask for a functional explanation. As an epistemologist, I must, nevertheless, in sist on the tremendous theoretical interest of such questions - if they can be answered. In the present case, no reasonable explanation seems to be

forthcoming from current logical theory. Another question - or what ought to be a puzzle - is the phenomenon

known as neg-raising. It is illustrated by the possible, and possibly preferential, reading of sentences ostensibly of the form

(7) x does not believe that S

as having the same form as

(8) x believes that not S.

Still another question originates from traditional discussion in old fashioned logic and philosophy and not only from the facts of ordinary use. It is the traditional distinction between contradictory and contrary negation. In spite of having been part and parcel of old-fashioned logic, it has not received a natural rational reconstruction in terms of modem

logic. Here the development of game-theoretical semantics (GTS) and of

independence-friendly (IF) logic has put a new complexion on things. (See here Hintikka and Sandu (1997) and Hintikka (1996).) One of the first

questions that arises in the GTS framework is whether negation should be treated by means of a separate game rule or whether it is merely a matter of some kind of omnipresent symmetry in propositions. To this question, a game-theoretical analysis of English quantifiers yields an answer. (See here Hintikka and Kulas 1985.) We must postulate a game rule for nega tion. By so doing we can capture certain regularities of the semantics of

English which otherwise do not seem to be amenable to reasonable ex

planation. These regularities concern the principles that regulate the order



588 JAAKKO HINTIKKA

of applications of the different game rules. Such ordering principles are characteristic of GTS. They do not all reduce to structural conditions such as the rules of Chomsky's Government and Binding (GB) theory, for they

may depend in some cases on the lexical items involved. For instance, the game rule for any has priority over the rule for negation. This gives sentences like the following their normal meaning:

(9) George does not listen to anybody.

The natural game rule for negation stipulates that the two players of a semantical game sometimes called the verifier and the falsifier exchange roles. This is an explicit, well-defined rule. It has some general interest in that rules of this kind have not been studied (to the best of my knowledge) in the general theory of games. Also, this rule implements in a natural way philosophers' idea of negation as merely exchanging the rules of truth and falsehood.

Here differences between different quantifier words are significant. Some quantifier words have a priority over ("wider scope than") negations, some others do not.

A major novelty comes into play when we move from received first order logic to IF first-order logic. This step is not only possible and

natural, but unavoidable. It concerns the relations of (substantial) depend ence between quantifiers and, hence, between the variables bound to them. These relations are represented in GTS by relations of informational de

pendence and independence (in the sense of general game theory) between different moves in a semantical game. In the received first-order logic we cannot represent all possible patterns of dependence and independence between quantifiers (and, hence, between variables). Such a representation is obtained by introducing a notation (Qlx/Q2y) which exempts a quanti fier (Qlx) from its dependence on another one (Q2y) in whose semantical

scope it would otherwise be. In principle, we do not even need any new notation. The same result could be obtained by liberalizing the use of

scope-indicating parentheses (,). (See here Hintikka 1997.) All the other semantical rules, in the first place those governing the

different logical constants, will remain unchanged. In this sense, IF first order logic is not a "nonclassical" logic. It is hyperclassical logic in that in it the applicability of the classical semantical rules (rules for semantical

games) is extended more than before. This includes the rule for negation. Unsurprisingly, most of the usual laws of traditional first-order logic

hold in IF first-order logic. These laws include all the usual distribu tion laws, DeMorgan's laws, interdefinability of the two quantifiers, the law of double negation, and so on. Likewise, most of the metatheorems




characteristic of the traditional first-order logic remain valid, including compactness, the downwards L6wenheim-Skolem theorem, and the sep aration theorem (even in a strengthened form). But one surprising thing happens: the law of excluded middle does not any longer hold in general. It holds only in the fragment of IF first-order logic consisting of the re ceived ("ordinary") first-order logic. Formally, this is the slash-free part of IF first-order logic. IF logic is thus reached not by changing anything in the "classical" (read: old-fashioned) first-order logic, but by liberalizing it in the spirit of the original enterprise.

A failure of tertium non datur perhaps should not be surprising to a

game theorist. In IF logic, the truth of a sentence S is defined as the existence of a winning strategy in the correlated two-person game G(S) for one of the players, sometimes called the verifier. The falsity of S is

accordingly defined as the existence of a winning strategy for his/her/its

opponent called the falsifier. The law of excluded middle as applied to S then becomes the thesis that G(S) is determinate. And, as any aficionado of game theory knows, determinacy is an exception rather than the rule in

game theory in general. A couple of explanatory comments are in order on this point. First, IF

theory is even on the propositional level not the same as intuitionist logic. Among other discrepancies, several of the laws that intuitionists reject are valid in IF first-order logic.

Second, the failure of the law of excluded middle is a purely structural, combinatorial matter. Whether or not (S v -S) is true depends on whether there exists a winning strategy for either player in G(S). And this depends simply on the structure of the model ("possible world") on which G(S) is

played, i.e., has nothing to do with the limitations of human knowledge or of the knowledge of any particular human being. This point is easy to miss because game-theoretical semantics makes liberal use of informational notions and locutions, including the idea of informational independence that gave rise to IF logic. The game-theoretical terminology is nevertheless

only one possible way of speaking of patterns of substantial dependence and independence which quantifier structures express. Informational no tions are part of the metatheory, not of what an IF first-order language expresses.

One truly remarkable thing here is that this failure of tertium non datur results from completely classical ("hyperclassical") assumptions with the

only possible exception of the admission of, as it were, ad hoc informa tional dependence. And this admission is not a "nonclassical" idea, but a

necessary correction to "classical" (i.e., received) first-order logic needed to enable it to fulfill its job description. It follows that we cannot consider



590 JAAKKO HINTIKKA

a failure of the law of the excluded third as a deviation from "classical

logic", whatever such a creature otherwise is or may be. As we might put the same point otherwise, there cannot exist in any language obeying the laws of our first-order logic any negation which would be more classical than the IF logic negation -, for ~

already obeys the perfectly classical semantical game rules for negation. In still other words, in any language

whose logic includes first-order logic and allows informational independ ence there is inevitably present a negation behaving like -. This behavior does not include obeying the law of excluded middle in general. These

languages obviously include natural languages like English, for informa tional independence is arguably expressible in them. Hence in the logic of a natural language, too, the logically fundamental negation is a dual one which does not obey the law of excluded middle. Of course, its presence can be tacit or, so to speak, a virtual one.

But this conclusion appears to be blatantly paradoxical, not to say ab surd. For the negation that is used in natural languages like English is

obviously intended to be the contradictory negation and to be "classical" in the sense of obeying the law of excluded middle. How can this be

compatible with logical primacy of the dual (strong) negation? In order to answer this question, let us examine what happens if we

try simply to add a contradictory negation -- to an IF first-order language. Surely there cannot be any laws against doing so, logical, legal, moral, or divine - I can hear you think. Indeed, we can in fact add to IF first order logic simply by stipulating that - is true iff S is not true. But it

follows from what has been found that this is all that can be said of the semantics of -. In an IF logic, a sentence S says, in effect, "There exists a winning strategy for the verifier in G(S)". This is understood as soon as the game rules G(S) are given. But there are no game rules for - and hence no such games as G(-,S). The reason is that the game rules must of course be the "classical" ones that we already have in ordinary first order logic, but these rules lead to the dual negation - rather than to the

contradictory negation -. This negation -- does not govern any moves in a game. Its meaning does not refer to what happens during a semantical

game. The meaning of -,S is parasitic on the games that involve other

logical constants, and all it can serve to express is that the verifier does not have a winning strategy in the game G(S). In this definition, the designated not must be understood as being a contradictory one. Here contradictory negation in an object language can only be defined in a metalanguage that

already possesses contradictory negation. On the formal level, this is reflected by the fact that strictly speaking

the contradictory negation can only have a meaning when it occurs pre




fixed to an entire sentence. It does not make sense inside a quantificational sentence. It cannot be prefixed to an open formula.

This explains the apparently paradoxical fact that we can have in our first-order logic an apparently (and really) classical contradictory negation present, even though the "classical" semantical rules (in a GTS context) for negation define a different negation, viz. the dual negation -. The ex

planation is that the contradictory negation - is not present in an extended IF first-order language in the same sense as the other logical constants. It does not serve to guide any semantical games. It does not operate by means of semantical rules in the same sense as -, but by means of a metalogical truth definition.

We might thus say that in a natural language like English we have to

try to express strong negation which can occur anywhere in a sentence but which does not get expressed in the language by means of a contradictory negation which occurs there, but only sentence-initially. No wonder the behavior of negation in natural language is messy.

There is an interesting exception to this restriction of contradictory negation to a sentence-initial position. In many applications it can - and,

maybe, must - be assumed that each atomic sentence is either true or false. If so, contradictory negation can occur prefixed to atomic formulas even

though they are open. Indeed, in such a position the difference between -- and - then disappears.

This presence - at least implicit presence - of two different notions of

negation in natural language serves to explain the puzzles pointed out in the beginning of this paper. It is the logical source of the complexities of

negation in ordinary discourse mentioned earlier. For one thing, the distinction between the contradictory negation

- and the strong (dual) negation - can be thought of as a rational reconstruction of the time-honored (or perhaps time-abused) contrast between contradict ories and contraries. I do not claim that the match is perfect. The numerous dead logicians and rhetoricians who have proposed or defended the con trast might very well have had a multitude of different ideas in mind. What is nevertheless interesting is that the distinction between -, and - is quite general. It is present on the most fundamental level of our language, viz. in first-order logic without any intensional notions.

One of the most striking consequences of what has been found is that in natural languages having contradictory negation there is inevitably some semantical pressure not to use negation inside a sentence (perhaps more

accurately, within the scope of a quantifier), that is, other than sentence

initially. Of course, sentence-initial here refers to the logical form of the sentence in question rather than its syntactical form. However, since left



592 JAAKKO HINTIKKA

to-right order is one of the ordering principles of English, this pressure is to some extent manifested also on the syntactical level.

We also obtain a neat explanation of why natural language negation is a barrier to anaphora. This explanation turns on the way anaphora works in natural language as described in a GTS framework. This way is ex amined in Hintikka and Kulas (1985). In a nutshell, pronouns pick out certain individuals during a play of a semantical game. The reason why these individuals cannot be ordinary references independent of semantical

games is that they are selected by the players in the course of the play, not

absolutely. For instance, suppose I read the following pair of sentences

(10) A couple was sitting on a bench. Suddenly he stood up.

In a sense, I know perfectly well who the he is. It is the male member of the

couple mentioned in the first sentence. But this couple is only an "arbitrary couple", that is, only a value of an existential quantifier. It is not a definite

pair of entities whose members could be referred to in a normal way. But in any semantical game with (10) an actual couple will have to be picked out. And then he can simply refer to the male member of that pair.

But, according to what has been found out, a sentence of the form -S is not interpreted by reference to any play or set of possible plays of a semantical game. It is never the initial sentence of a game of verification and falsification in GTS. It is merely a statement that there does not exist a winning strategy for the verifier in the game G(S), that is, in the game played on S rather than -'S.

Hence there are no individuals picked out in G(-,S) and hence nothing to serve as values of anaphoric pronouns. Such pronouns therefore become

uninterpretable and the entire sentence containing them unacceptable. This

explains why negation is a barrier to anaphora in natural languages. At the same time, it illustrates the fact that the explicit negation present in natural

languages is indeed the contradictory negation. Another feature of natural languages like English that now becomes

understandable is the presence of two syntactically different kinds of

negation. On the one hand, one can negate the verb ("He isn't dumb") and, on the other hand, one can negate the quantifier term (e.g. "Not

many people are that dumb"). What has been found has several implic ations concerning these two kinds of negation. First, the possibility of verb negation is now reconcilable with the fact that in natural languages like English explicit negation, being the contradictory negation, can occur

only sentence-initially. The answer lies in the exception to this sentence

initiality noted above. In ordinary discourse, it is often, perhaps even

typically, possible to assume that the law of excluded middle applies to




atomic sentences - or what corresponds to them in the relevant use of natural language. Then verb negation is interpretable, and because of this

typical interpretability verb negation is syntactically acceptable. But the actual semantical interpretability of an instance of verb negation depends on whether in the logical form of the sentence in question the negation really governs only an atomic formula. For instance, the sentence

(11) Everybody has a different hobby.

is naturally taken to mean

(12) (Vx)(Vy)(3z/Vy)(3u/Vx) (x f y D ((z is a hobby of x's but not y's) & (u is a hobby of y's but not x's)))

But the negated sentence

(13) Everybody does not have a different hobby.

does not have a natural interpretation. One regularity which now receives an explanation concerns the admiss

ibility of quantifier negation. It is permissible for some quantifier phrases of English, but not for all. For instance, we can say

(14) not every not a

but not

(15) not any not some

What makes the difference? An answer follows from what has been found. What is common to quantifier words figuring in (14) is that the ordering principles governing them do not give them priority over negation. Hence

negation is not in (14) within the semantical scope of the quantifier. In

contrast, the quantifier words occurring in (15) do have priority over nega tion. Hence if (15) were acceptable, negation in sentences involving them

would occur within the semantical scope of a quantifier, which was found to be discouraged in natural languages. This explains the unacceptability of (15).

For reasons of space, I will not try to present extensive evidence for the relevant ordering principles here. A few simple sample sentences should



594 JAAKKO HINTIKKA

convince the reader of this feature of the ordering principles governing the relevant quantifier words. At the same time, we can see why the regularity in question is not a perfect one. Ordering principles can be overruled and a prima facie contradictory negation occurring in the scope of a quanti fier can sometimes be given a nonstandard interpretation that restores its

meaningfulness. Now we are in a position to understand better some of the puzz

ling ordinary-language regularities mentioned earlier and even to reach some fundamental new insights into the meaning of natural-language expressions.

Take, first, putative ordinary-language sentences of the form

(16) everything + neg + formula

One's first idea might be to express them as

(17) (Vx) -

S[x]

In some cases (12) is the force of a natural-language expression of the form

(11). But this can scarcely be the only possible interpretation of (11) nor the normal one because it is not in keeping with the general characteristics of natural-language semantics. For natural-language negation is obviously in the first place the contradictory one ("not true") rather than the dual one

("definitely false"). Hence one might be tempted to express (14) by

(18) (Vx)-S[x]

But (18) is not well formed, because - cannot be prefixed to an open formula. So it is only to be expected that sentences of the form (16) do not occur in natural language. Or if a sentence like (16) is nevertheless used in natural language, its force will be

(19) --(Vx)S[x]

This explains one of the puzzling phenomena noted earlier. This phe nomenon is that ordinary-language sentences of the form (16) are often understood as having the force of (19). Indeed this explains why (3) is

spontaneously read as having the force (4). This peculiarity of the every + neg construction is not due to an ordering

principle giving negation priority over every. For if the negation in question is a term negation, it does not jump to the beginning of the logical form:

(20) Everybody is sometimes illogical.




(21) Everyone refrained from smoking.

Similar exceptions are not found with real ordering principles, e.g.

(22) John refrained from ridiculing anybody.

where the negation is brought in by the word refrain. Another regularity that receives an explanation here is that sentences

whose force apparently is (17) are typically expressed in English by means of negative quantifiers like no, nobody, nothing, ... Once again, in natural

language we spontaneously avoid having explicit negation to occur within the scope of a quantifier.

But this prompts the question of the meaning of the negative quantifier no. According to what has been said, the force of a sentence of the form

(23) nothing + formula

should be (18). But (18) is ill formed. So what does (23) mean? The answer to this question is that the negative quantifier in English

has a meaning that does not reduce to that of any expression of ordinary first-order logic, of IF first-order logic, or even of IF first-order logic exten ded by a sentence-initial contradictory negation. This meaning falls within the purview of extended IF first-order logic, however. Let us introduce a

symbol (Nx) for the negative universal quantifier. What then is meant by

(24) (Nx)S[x]

if it is not equivalent with (17)? An answer is obviously not hard to find. For (16) can be taken to mean

that of each member of the domain of individuals, designated by the con stant b, it is the case that -,S[b]. This is not the same meaning as (17), for (17) entails that for each such b it is the case that S[b]. Thus negative quantifiers operate by means of a kind of substitutional interpretation.

Thus we have reached two highly interesting conclusions. First, we have found that the precise meaning of the English negative quantifier words "no", "nothing", "nobody", etc. is not expressible in ordinary or unextended IF first-order logic. No wonder, therefore, that there exists in English a separate quantifier to express this meaning. (More generally speaking, we have here an especially poignant reminder of the fact that the semantics of the quantifier system of natural languages is in several

respects quite different from, but more complex than, the semantics of the

quantifiers of first-order logic.)



596 JAAKKO HINTIKKA

Quantifiers like (Nx) can be studied systematically in the same way as extended IF first-order logic. Indeed, here is a useful chore for logicians to do for the purpose of throwing light on natural language phenomena.

However, there is a second more general observation that can be made here. The way of interpreting (Nx) that I diagnosed can be used - or per haps, should I say, can be tried - generally. We can try to interpret so as to

give a meaning to occurring within the scope of a quantifier. We can try to understand (Vx) - in the same way as (Nx).

An even wider perspective opens up here. What was found is that we can make sense of the initially forbidden symbol combination (18) by re-interpreting the universal quantifier (Vx). This re-interpretation is in a sense substitutional. It is not in terms of a game of the form G((Vx)-S[x]), for there does not exist any such game. Instead, it is in terms of the to

tality of games G(S[b]) for all the different substitution-values b of the variable x. Such an interpretation is a viable one, and it can be extended to corresponding existential sentences of the apparent form (3x)-S[x]. Furthermore, the same re-interpretation can be applied in the substitution instances S[b], if needed. Thus by repeated applications of the substitution value idea we can reach a coherent interpretation of any sentence of an

ordinary first-order language. It is obviously a prima facie implementation of what is commonly called the substitutional interpretation of first-order sentences.

This substitutional interpretation calls for a number of comments. Per

haps the most important one is that the substitutional interpretation cannot be extended to the whole of IF first-order logic. The stepwise procedure just explained requires that quantifiers are transitively ordered by their

dependence relations. It is not a universally applicable interpretation of

natural-language quantificational sentences. With such qualifications, however, the substitutional interpretation is

in principle always available. I have no doubt that it is part and parcel of how the speaker of a natural language understands quantifiers. The

negative quantifiers thus have the great interest of providing an example of a natural-language expression whose semantics forces us to recognize explicitly this omnipresent second line of interpretation.

These observations throw some light also on what has been called in the literature the substitutional interpretation of quantifiers. For one thing, even though the defenders of such an interpretation are usually laboring under serious misconceptions, what has been found out here shows that there is a grain of truth - albeit a pretty small one - in the substitutional idea. In some cases, though not all, a substitutional truth definition is pos sible. Moreover, there are some natural-language quantifier expressions




whose meanings have to be explained substitutionally. At the same time it is seen that such quantifier expressions are not the most fundamental ones, but rather parasitic on others.

This in turn illustrates the pitfalls of relying on the intuitions we hear so much about in philosophy and in linguistics these days. Even when

they are in some sense correct, they do not come with instructions for their use. An earnest scholar may contemplate the meaning of negative quantifiers, their whole meaning, and nothing but their meaning, and by so

doing brainwash himself or herself into believing that all quantifiers have to be defined substitutionally.

It is to be noted, however, that the semantics I have ascribed to (Nx) is not literally the same as the substitutional interpretation in the received sense (if any). The crucial special feature is not the use of substitution

values, but to give an interpretation that is not in terms of semantical

games. Or, rather, that it is not in terms of one single semantical game at the time. I will nevertheless continue to call this interpretation substitutional. In the special case of the received first-order logic, the substitutional inter

pretation coincides with the game-theoretical one. However, in the case of

irreducibly IF sentences, the interpretational difference matters. In the case of universally quantified sentences, what the substitutional interpretation does is to interpret universality (the meaning of the universal quantifier) as mere exceptionlessness.

Perhaps the most remarkable observation here is that the substitutional

interpretation of quantifiers is not just an alternative, but an equivalent way of looking at the semantics of quantifiers. It does not apply in all cases, but, in those cases in which it does, it can be a genuine alternative to the normal

("game-theoretical") interpretation in the sense of yielding a different force to the sentences in question.

To return to the interpretation of natural-language sentences of the form

(16) and (23), one virtue of the analysis presented in this paper is that it

predicts its own exceptions. In many particular examples, perhaps in most

examples linguists are likely to contemplate, the difference between ~ and - does not make any difference. Hence competent speakers' Sprachfiihl may be tolerant enough in such cases to allow one of them the same

privileges as to the other. Other predictable exceptions come about in other ways. For instance,

the context shows that the point of uttering a sentence of the form (16) is to stress the exceptionlessness of the negative generalization, in other

words, to stress that not only are there some negative cases, all the cases are negative. Then every + negation can assume the "substitution instance"



598 JAAKKO HINTIKKA

sense and become acceptable ad hoc and still have the sense of (Nx). Here is an example from ordinary usage:

(25) He was scanning X-rays on the walls.

"This doesn't fit with his victimology", he said. "Not at all".

"Everything about this case doesn't fit", I replied. "Except that once again, a saw was used....."

Patricia Cornwall, Unnatural Exposure, Warner Books, Lon

don, 1997, pp. 60-61.

Notice how the intended meaning of "Everything ... doesn't fit" is made clear only by the context. For once, here an exception literally helps to

prove the rule. On the basis of the logical laws of IF logic, we can also explain the phe

nomenon of neg-raising. Horn (1989, p. 309) is right in one respect. Just because neg-raising reflects such underlying logical laws it is "manifested across distinct, but systematically related, classes of predicates in genet ically and typologically diverse families of languages." But he is wrong in another respect, viz. in thinking that neg-raising is a "fundamental"

phenomenon. It is in fact explainable on the basis of the nature of IF logic. One can introduce the explanation by asking how the semantical force

of a sentence of the following form can be explained in ordinary discourse:

(26) BaS

where Ba is the logical counterpart of a believes that. It is clear that (26) has a well-defined content, in that a might perfectly well believe the con

tradictory of the proposition that S. But how can this content be expressed in natural language? The first candidate might be the literal translation of

(26), that is, a sentence of the form

(27) a believes that + neg + S.

But this will not do, the reason being that Ba is a tacit universal quantifier (quantifier ranging over possible scenarios). Hence in (27) contradictory negation occurs in the scope of a quantifier, which is being avoided in natural language. Tacitly re-interpreting neg as having the force of the dual

negation - does not help, for it would amount to changing the content of the proposition that is believed by a. In such circumstances, the nearest

ordinary-language expression for (26) is in fact

(28) a doesn't believe that S.




This makes it understandable why (28) should easily be taken to have the force of (26), and thus helps to explain neg-raising.

This is not the whole story, however. There is another way of looking at the behavior of negation in the context of propositional attitudes. When we

describe, for instance, the content of a person's knowledge or belief, the

logical notions used can be expected to be independent of the epistemic operator as possible. This requirement of independence does not apply to

quantifiers, for in different possible scenarios compatible with a person's attitude there may exist different individuals for the quantifier to range over. But negation is the same in all scenarios, and therefore it must be

informationally independent of the propositional attitude operator. And if

so, the order of the operator and the negation will not matter. Thus on the

neg-raising reading, the logical form of (28) will be

(29) (Ba/-)S

which is logically equivalent with

(30) Ba(-/Ba)S.

Now (30) is the logical form of

(31) a believes that it is not the case that S.

This explains how (28) and (31) can mean the same in English. These observations show that the role of informational independence

and of negation in epistemic and doxastic logics has to be studied more

carefully than has happened so far. There is a different important new perspective on negation gained by

IF logic. Here I can only sketch it briefly. The usual propositional con nectives - conjunction, disjunction, and contradictory negation - have

geometrical counterparts in logical space, corresponding as they do to

intersection, union, and complement, respectively. This correspondence is utilized in Venn diagrams, and it is also exemplified by the geometry of state spaces in physics. But what is the analogous geometrical counterpart to the strong negation ~? It does not seem to have any natural geometrical interpretation. This fact might even be used to argue for the primacy of the

contradictory negation -. These first impressions are misleading, however. It turns out (see

Hintikka, forthcoming) that if the state space in question is a Hilbert

space and if we consider subspaces specified by sentences with mutually dependent quantifiers, then the strong negation corresponds to orthocom

plementation. This suggests the general possibility of considering strong



600 JAAKKO HINTIKKA

negation in general as the logical counterpart to orthogonality. Two pro positions S1 and S2 are then orthogonal if and only if it is the case that

(SI D S2). This opens up all sorts of interesting possibilities. One of them is the

following: when we have at our disposal the concept of orthogonality we can define the concept of dimension. A model has at most n dimensions if and only if there cannot be in that model more than n pairwise mutu

ally orthogonal propositions. Thus the basic features of the concepts of

orthogonality and dimension can be dealt with on a purely logical level. Once again, IF logic opens highly interesting new perspectives on the

theory of negation. It does not exhaust the potentialities of IF logic to throw

light on different aspects of negation. One result worth mentioning here is that when only mutually dependent quantifiers are present in a sentence S, its strong negation -S has the same logical form as S except for negations prefixed to atomic sentences and identities.

REFERENCES

Chomsky, N.: 1981, Lectures on Government and Binding, Foris, Dordrecht.

Hintikka, J.: 1996, Principles of Mathematics Revisited, Cambridge University Press.

Hintikka, J.: 1997, 'No Scope for Scope?', Philosophy and Linguistics 20, 515-544.

Hintikka, J.: forthcoming, 'Quantum Logic as a Fragment of a Independence-Friendly Logic'.

Hintikka, J. and J. Kulas: 1985, Anaphora and Definite Descriptions: Two Applications of Game-Theoretical Semantics, D. Reidel, Dordrecht.

Hintikka, J. and G. Sandu: 1997, 'Game-theoretical Semantics', in Johan van Benthem and Alice ter Meulen (eds.), Handbook of Logic and Language, Elsevier, Amsterdam,

pp. 361-410.

Hintikka, M. and J. Hintikka: 1986, Investigating Wittgenstein, Basil Blackwell, Oxford.

Horn, L.: 1989, The Natural History of Negation, The University of Chicago Press.

Klima, E. S.: 1964, 'Negation in English', in Jerry A. Fodor and Jerrold J. Katz (eds.), The Structure of Language, Prentice Hall, Englewood Cliffs, NJ, pp. 246-324.

Wittgenstein, L.: 1921, Tractatus Logico-Philosophicus, Kegan Paul, London.

Jaakko Hintikka Boston University Department of Philosophy 745 Commonwealth Avenue Boston, Massachusets 02215 E-mail: [email protected]



negation in logic and in natural language

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