Universe restriction in the logic of language

Pieter A. M. Seuren, Max Planck Institute for Psycholinguistics, Nijmegen

It is my pleasure to dedicate this study to myold friend and colleague Frans Zwarts, whoseinterest in the vagaries of negation in naturallanguage is almost as old as mine.

Abstract: Negation in language, far from being a simple representative of standard bivalentnegation, has its own idiosyncrasies. The present study is an attempt at subsuming many or mostof these under the rubric of universe (Un) restriction, meaning that natural language negationselects different complements under different conditions. Un restriction is manifest in threedifferent ways: static, dynamic and developmental. The focus of this study is on the static form ofUn restriction, leading to an investigation of the relation between unrestricted standard modernpredicate logic on the one hand and the classic Square of Opposition on the other. It is arguedfurther that the addition of a ‘strict’ existential quantifier ‘some but not all’ leads to increasedinsight into the matter.

1. Introduction

Although there can be no doubt about the universal mathematical validity of standardmodern predicate logic (SMPL), the status of uniqueness and untouchability it hasacquired over the past century is unwarranted and has proved an obstacle to deeperinsights into the richness of logical space and in particular its relation with language andcognition. Human cognition, with language in its wake, has made a far more ingenious useof the possibilities afforded by logical space than present day logicians, semanticists andpsychologists have given it credit for. The present study explores the way in whichcognition has found its way in logical space, with an emphasis on the one outstandingfeature of the resulting logic of natural language (LNL), its tendency to reduce SMPL’sinfinite universe (Un )of all metaphysically possible situations to functionally manageableproportions. Whereas SMPL operates in terms of one unchangeable and infinite universeof discourse, LNL operates within universes that are, for obvious functional reasons,restricted in a variety of ways. Although this principle was already emphasized at greatlength in the chapters 2 and 4 of De Morgan (1847), it is only now that the various aspectsof this basic fact are beginning to come to light, and as this is happening, it is becomingclear that the well known discrepancies between SMPL and natural logical intuitions are

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computer science); epistemic possible has a negative opposite in epistemic impossible butepistemic necessary lacks an epistemic negative counterpart (unnecessary is notepistemic). In all such cases, it is the contradictory of the element in the A position of theSquare structure (consisting of the proposition types <A, I, ¬I, ¬A>) these items fit intothat withstands lexicalization. This fact has been dubbed the unlexicalisability of the Ocorner. Also, for example, the quadruple <order, permit, forbid, *non order>, whichlogically corresponds to the traditional Square: order entails permit, forbid is the Unrestricted contradictory of permit and *non order of order, order and forbid are contrariesand, finally, permit and *non order are subcontraries. Typically, there is no lexical item forthe O corner item *non order in the logically expected sense of ‘either forbid or permit’.And analogously for <cause, allow, prevent, *non cause> in the logic of causality.

It was found recently (Jaspers 2012; Seuren & Jaspers, forthcoming; Seuren,forthcoming) that the regularity pervades the lexicon as a whole: in any lexical assemblywhose members can be arranged in the form of the classic Square, the O position remainsunlexicalised. To mention just one example out of a myriad, in Ancient Greek, which has aword Athenian, entailing Greek, and a word barbarian for ‘non Greek’, no word exists forthe O corner item ‘non Athenian’ covering both non Athenian Greeks and barbarians. Thisnon lexicalisability of items in the O position appears to be universal for ordinary, nontechnical language.

This regularity is reinforced when the Square is extended with two additional vertices,one for the strict existential quantifier ‘some but not all’ (the Y type) and one for itscontradictory (the ¬Y type, often called U). As shown below, this extends the Square tothe logical Hexagon, as proposed in Jacoby (1950, 1960), Sesmat (1951), Blanché (1953,1966). In the Hexagon it is not only the ¬A (=O) position but also the ¬Y (=U) position thatis unlexicalisable. Moreover, the (unlexicalised) negation of all, as in Not all childrenlaughed, takes one to ‘some but not all children laughed’, rather than to the expected‘either no or some but not all children laughed’.

In order to account for this class of observations, a cognitive mechanism is postulated(Seuren & Jaspers, forthcoming; Seuren, forthcoming) in virtue of which Un is stepwiseand recursively restricted in the course of early cognitive development—as a result ofincreased cognitive activity—so that the (default) negation selects different complementsaccording to the degree to which the logical system has been developed. Assume that theyoung child’s very first distinction, in the world of its experiences, will be between therebeing nothing and there being something. Subsequently, the child will discover that not allentities (‘somethings’) are equal, as some have this and others that property. Then, whencombining properties, the child discovers that, given entities with property R, some ornone of these also have property M, thus establishing the first or primary axis of a

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Both the Square and SMPL are predicated on two types of quantified propositions, typeA (All R is M) and type I (At least some R is M), where the variables R (restrictor predicate)and M (matrix predicate) range, in principle, over all first order predicates. In addition,both systems incorporate an external and an internal negation. The former negates entirepropositional types and is symbolized here as ¬ prefixed to the type name. The latternegates the propositional function in M position and is symbolized as * suffixed to thetype name. This gives eight type names, as shown in Table 1.

Table 1

type name linguistic expression formal languageA All R is M x(Rx,Mx)I At least some R is M x(Rx,Mx)¬A Not all R is M ¬[ x(Rx,Mx)]¬I No R is M ¬[ x(Rx,Mx)]A* All R is not M x(Rx,¬Mx)I* At least some R is not M x(Rx,¬Mx)

¬A* Not all R is not M ¬[ x(Rx,¬Mx)]¬I* No R is not M ¬[ x(Rx,¬Mx)]

The type names are merely a convenient shorthand; they are not meant to be a formallogical language. The logical language used throughout this study is, in principle, that ofthe theory of generalized quantifiers, where the quantifiers are taken to be binary higherorder predicates, as exemplified in the column “formal language” of Table 1. Any claimthat SMPL is superior to the Square in that the former is able to express multiple internalquantification, as in Some boys admire all popsingers, while the latter lacks that expressivepower, is thus false, as it is based on a confusion of the logical system itself on the onehand and the formal language used on the other. In principle, the same formal languagecan be used for both SMPL and the Square, or indeed any other variety of predicate logic.

In the language used, the quantifiers and are higher order predicates over pairs offirst order sets. Rx and Mx are first order set denoting predicates (propositionalfunctions). The quantifying predicates and are semantically defined as in (1) (thevariable x binds the x in Rx and Mx; [[P]] stands for the appropriate order extension of anypredicate P in the universe of entities Ent):

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Table 2

Un: 1 2 3 4———————————————————————————————–———A T F F T

I T T F F———————————————————————————————–———¬A F T T F

¬I F F T T

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I* F T T F

¬A* T T F F

¬I* T F F T———————————————————————————————–———

Conditions for T: [[R]] [[M]] [[R]] [[M]] [[R]] [[M]] [[R]]= Ø[[R]] [[M]] Ø [[R]] [[M]] Ø [[R]] [[M]]= Ø

__________________________________________________________

Each of the four valuations in Table 2 specifies a truth value (T or F) for each of the eightproposition types in SMPL, plus the conditions to be fulfilled for truth in each valuation.These truth conditions are simply derived from the satisfaction conditions specified forthe quantifying predicates and in (1) above. For valuation 4, this condition is theconjunction of [[R]] [[M]] (since A is true) and [[R]] [[M]] = Ø (since I is false), whichamounts to the simple condition that [[R]] = Ø. That this is so is easily shown: for anyarbitrary [[R]]and [[M]], if [[R]]= Ø, [[R]] [[M]]and [[R]] [[M]]= Ø; conversely, if [[R]] [[M]]and[[R]] [[M]]= Ø; then [[R]]= Ø. Note that the conditions for truth in the valuations 1,2 and 3all entail that [[R]] Ø.

This fact has not or hardly received any attention in the literature, yet it is remarkable inthat situations where [[R]] = Ø form the crucial difference between SMPL and the Square:the two systems are identical but for the fact that SMPL (in conformity with standard settheory) treats A type propositions as true when [[R]] = Ø, while the Square ignores suchsituations. This shows that leaving cases where [[R]] = Ø out of consideration is not anarbitrary decision but cuts into SMPL at a natural joint (a fact that does not hold forAristotelian Abelardian logic; see below). Dropping cases where [[R]]= Ø from Un creates awealth of new logical relations giving rise to the Square and giving the system a logicalpower unsurpassed by any other known logical system in terms of the eight proposition

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considered, under standard bivalence (in the present study).1 The basic way of defining asituation is by means of assigning truth values to the (types of) propositions in thelanguage fragment L considered. Thus, if L contains exactly three logically independentpropositions A, B and C, Un consists of eight (23) possible situations/valuations (see Table3). (If L consists of 25 logically independent propositions, the number of valuations is 225,or 33,554,432.) Each of the situations/ valuations 1–8 is an assignment of truth values toA, B and C. The logical compositions follow automatically, given the definitions of thepropositional operators. Thus, the VS /A/ of proposition A in Un as specified in Table 3 is{1,3,5,7}, /B/ = {1,2,5,6}, etc. (For more elaborate discussion see Seuren 2013, Ch. 8.)

Table 3Un: 1 2 3 4 5 6 7 8A T F T F T F T FB T T F F T T F FC T T T T F F F F

¬A F T F T F T F TA B T F F F T F F FA B T T T F T T T FA C T T T T T F T Fetc.

The diagrammatic representation of SMPL given in Figure 1 can thus be interpreted as aVS model for SMPL, with the four spaces corresponding to the four valuations of Table 2.

When the members of a pair of propositions considered are not logically independent,so that there exists a logical relation between them, some situations are inadmissible. For

example, if B entails C (B C), all situations in which B is valued true and C is valued false

are inadmissible, which eliminates the situations 5 and 6 from the Un of Table 3. Aproposition P is necessarily true in Un iff /P/ = Un; P is necessarily false in Un iff /P/ = Ø.Thus, if A is necessarily true, the valuations 2, 4, 6 and 8 are inadmissible and are removedfrom Un. If A is necessarily false, the valuations 1, 3, 5 and 7 are inadmissible and thusremoved.

Un Ø iff L contains at least one proposition P. If L contains precisely one proposition Pand P is contingent, there are two admissible situations, one in which P is true and one inwhich P is false, but if P is either necessarily true or necessarily false, there is only one

1 Both the notion and the term Valuation Space were introduced in Van Fraassen (1971),where, however, the notion was left underdeveloped.

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Yet, although this removes the defect of UEI, it also produces a logic that is different fromboth SMPL and the Square. The added condition that [[R]] Ø makes A false when [[R]]= Ø,so that space 4 is not left blank but reserved for ¬A, ¬A*, ¬I and ¬I*. This gives rise to afully consistent alternative predicate logic, which I have dubbed Aristotelian Abelardianpredicate logic (AAPL), as it reflects Aristotle’s original system, rediscovered andreconstructed by the medieval French philosopher Peter Abelard (1079–1142) (see Seuren2010, Ch. 5; 2013, Section 8.4.1). In AAPL (see Figure 2), the Conversions do not hold but

are replaced with the one way (strict) entailments A ¬I* and I ¬A*. Moreover,

subcontrariety disappears: in AAPL: I and I* are no longer subcontraries but are logicallyindependent, because, as pointedly observed by Abelard, if [[R]]= Ø, both I and I* are false.The construction of the classic Square from SMPL is thus brought about not by adding thecondition that [[R]] Ø to the semantics of but by restricting Un to those situationswhere [[R]] Ø.

Figure 2 The VS model for AAPL

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294

Figure 3 (a) VS model and (b) square representation of the Square

5. Adding the Y type

The 19th century Scottish philosopher Sir William Hamilton (1788–1856) proposed(Hamilton 1866) to interpret the natural language word some as ‘some but not all’, ratherthan as the standard ‘some perhaps all’ of SMPL and the Square. 2 In this he was followedby the Danish linguist Otto Jespersen (1860–1943) in Jespersen (1917, 1924). I call thismore restricted form of existential quantification strict existential quantification and,following a tradition established by the French philosopher Robert Blanché (1898–1975), Iuse the type name Y for the corresponding proposition type. The strict existential

quantifier, which I write as , is defined as follows:

(3) [[ ]] = {<[[R]],[[M]]> | [[R]] [[M]] Ø, [[R]] [[M]]}

2 Hamilton’s logic is characterized further by his theory of so called ‘quantification of thepredicate’, in virtue of which a sentence like All men are mortal should be analyzed as ‘all men aresome mortal’. This aspect of Hamilton’s logic is left out of account here, as it reflects a notion oflogical structure that is totally at odds with the approach followed here and does not seem tocontribute to a better insight into the logical aspects of language or cognition.

HamilogicipropoContr

Cois a p(1866

Blanc

fact,

quan

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Gi

valuefollow

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ilton’s propians, amounosition typeraries.ontrary to tpragmatic de6), Ginzberg

ché (1953, 1

the additio

tifier m

ning up newSeuren & Jape plus its tber of propfor ¬Y ¬Y

e 5

type na

Y

¬Y (=

Y*

¬Y* (

ven the sa

ed T in valuws that Yand vice vehe extensionL in Figure 4nly the Conmplete hexabe read off F

posal, whichnts to a triaes A, ¬I an

he now widerivate fromg (1913), J

1966), that

on of the Y

akes for a

w possibilitieaspers, fortthree negaosition typeY*).

ame

U)

= U)

tisfaction c

ation 2 andY*, since if

ersa. Conseqn of SMPL w4. Figure 5nversions aagon resultiFigure 4.

h was rejectadic predicnd Y, formi

despread asm the standespersen (1

deserves

Y type (plus

nontrivial

es for the sthcoming; Stive compoes from 8 to

linguistic ex

Some but

All R is M

Some but

All R is M

condition fo

d F in the vaf [[R]] [[M]]quently, ¬Ywith Y and ishows (a) tnd the coning from th

ted and eveate logic cong what is

ssumption tard ‘some p1917, 1924

s a place in

its negativ

enrichment

tudy of theeuren, forthositions aso 12. (NB: U

xpression

not all R is

OR No R is M

not all R is

OR No R is M

or as spe

aluations 1,]] Ø and [[R¬Y*.

ts negativethe dismantntradictoriee addition

en treatedonsisting ofknown as

hat the inteperhaps all’,4), Jacoby (

predicate lo

ve composit

t of both S

e natural loghcoming). Tspecified inU is often, f

f

M

M

not M

M

ecified in (3

, 3 and 4 ofR]] [[M]], t

compositiotled squares of the teof Y and its

with derisiof the threethe Hamilt

erpretation, I submit, f1950, 1960

ogic on a pa

tions) and t

SMPL and t

gic of languThe additionn Table 5 tollowing Bla

formal langu

(Rx,Mx)

¬[ (Rx,M

(Rx,¬Mx

¬[ (Rx,¬M

3) above, it

f Table 2 abhen [[R]]

ons results iof standarderms conces negative c

on by mainmutually cotonian Tria

‘some butfollowing Ha0), Sesmat

ar with an

the corresp

the Square

uage and con to Table 1thus increasanché 1953

uage

x)]

x)

Mx)]

t follows th

bove. MoreØ and

in the VS md SMPL, conrned, andcomposition

295

nstreamontraryngle of

not all’amilton(1951),

nd . In

ponding

e, while

ognition1 of theses the3, 1966,

hat Y is

over, itd [[R]]

model ofnsisting(b) thens, as it

296

Figure 4 The VS model for SMPL extended with Y and its negative compositions

Figure 5 The dismantled square for SMPL and the incomplete hexagon resulting fromthe incorporation of Y and its negative compositions

Figure 5 shows how the incorporation of Y and its negative compositions leads to aconsiderable enrichment of SMPL. An even greater enrichment results if the Square is thebeneficiary of the incorporation of Y and its negative compositions—that is, when space 4is disregarded. Whereas SMPL in its original form consists of only the Conversions, and theaddition of Y and its negative compositions to SMPL greatly increases the number of

instanegatHexawher‘logic

F

On<A,Y,of Opand 3thus

3 T(It hawith tlatterreacti1913:by for(Ginzbbut nfunctisay thtries t

ntiated logtive compogon as envre [[R]] = Ø iscal power’) a

Figure 6 VS

ne notes th,¬I>, the Tripposition <3. The additseen to ma

This was alres so far provthe Salomonr’s trip throuion by Louis: 256; translarmal logicianberg 1914) tnot all’ and tional advanthan (translatto make pro

ical relatiositions intovisaged by Js thus seena maximal i

S model an

hat the Logiangle of Su

<A,I,¬I,¬A> tion of the Yke good log

eady seen byved impossibn or Simon Gugh the UniCouturat (1

ation mine):ns in a way that no logicathat such a qages. In his stion mine) “[opositions ‘qu

ns, as showthe SquareJacoby, Sesn to lead sysncrease in t

nd correspon

ical Hexagoubcontrariesand thus SMY type plusgical sense.3

S. Ginzbergble to traceGinzberg whoited States1868–1914),“The problehat seems toal principle pquantifier, bshort final re[O]ne steps ouantitatively

wn in Figure makes thesmat and Bstematicallythe logical p

nding polyg

on containss <I,¬A,¬Y>MPL to theits negative

(1913), whothis author,o acted as Ain 1921.) Gia well knowm raised byo me to be sprohibits thebeing more pepartee (Couout of the fr’ precise. On

re 5–b, thee hexagon cBlanché (Figy to what ispower of the

onal repres

s the Hamilt, the classicextent that

e compositio

se argumentunless he is

Albert Einsteiinzberg’s littwn follower oMr Ginzbergsatisfactory aintroduction

precise thanuturat 1914),ramework ofne may well

e incorporacomplete, ygure 6–b). Ds probably (e system at

entation for

tonian Triac Aristoteliat it is validons to these

ts were sharps identical—in’s ad hoc stle article pof Russell, wg […] was soland decisive.n of a quant‘some perh, Couturat haf classical logelaborate al

ation of Yyielding theDisregarding(failing a thhand.

r the Hexag

ngle of Conn Boethianfor the spae logical sys

per than Hamwhich is dousecretary durovoked an

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297

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milton’s.ubtful—ring theacerbic

Couturatime agoretortedg ‘somecertain

better toent onegitimate

298

The Y type plus its negative compositions may also be incorporated into AAPL. Doing soleads to the VS model shown in Figure 7. A polygonal representation corresponding to theVS model may also be set up, but, due to the scarcity of equivalences, this provesunrewarding.

Figure 7 The VS model for AAPL extended with Y and its negative compositions

6. Conclusion and further perspectives

The question of what logical system is best taken to correspond to natural human logic isstill far removed from a final answer. But a few things are clear. First, the question is of anempirical nature. One empirical touchstone is formed by intuitive judgements ofnecessary consequence and of consistency through texts, just as intuitive judgementsregarding given categories of data are an empirical touchstone for many other humansciences, such as linguistics. Other such empirical touchstones may be found in

and consistent systems, but one cannot make the classical system more ‘perfect’ by introducinggreater ‘precision’, which is alien to its spirit.” It should be clear that this final reply by Couturatwas gratuitous and that, far from being alien to it, it is entirely in the spirit of the “classical system”to make it more perfect by introducing greater precision.

psychdevis

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Ththe rethe owhichtheirpervamind

7. Re

Blanc— (19

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hological exsed.hen, it seement study, s, interestinger aspectsmic (presupto be invets. And, imiction relatehis, it seemselation betwother. Anothh have beerepercussioasive. The nover the pa

eferences

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Paris: J. Vurat, Louis (

de Métap914). Répon

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es propositio

, Revue de M

d Logic. Vol.el and J. Ve

ve so far be

central featpic, but thecomplete: tboration anforms of Uif possibleow these tsistency in tlong overduognitive lingf phenomennguistic theense are mstudy of lano good.

, 89–130.ystématique

eur portée e

, 259–260.ence, Necess

ons particul

Métaphysiq

IV (Lectureitch. Edinbu

een carried

ture of LNLresults pre

there are nod elucidatioUn restrictio, by experithree formstexts.ue investigaguistic comna of logicalories evenomentous anguage and

e des concep

existentielle

sary and Pro

lières, Revu

ue et de Mo

s on Logic. Vurgh London

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. In theesentedo doubton. Theon alsoimentals of Un

ation ofplex onl scope,thoughand allof the

pts.

, Revue

obable.

e de

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