universe restriction in the logic of languageuniverse restriction in the logic of language pieter a....
TRANSCRIPT
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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285
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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):
(1)
ThatM is r
Githe q
for Aexpre
for I
The Cin SMneithcontivaluasix restand
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s ‘the class onull intersec
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MPL. Beyondhey can benecessarily
ossible valufirst two ro
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of Rs is inclction betwese equivalecalled duals
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iff [[R]]ence I ¬A
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; [[R]] [[M]]
; [[R]] [[M*.
nd I are logently of eacfalse). Th
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287
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288
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
A* F F T T
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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289
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290
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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291
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Un)
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292
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
When the situations where [[R]] = Ø are removed from Un and neither A nor I is eithernecessarily true or necessarily false, the Conversions are unaffected and, in addition:
A
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293
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re from
trarietytact forll leavealse outes, with
e classicgure 3–rariety,odel fora (or byt issue,
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
open(seeY typnumbused
Table
Gi
valuefollow
,Th
SMPLof onincomcan b
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
who wrote (Clved a long t.” Ginzberg rifier meaninaps all’, hasad nothing bgic the momlternative leg
297
and itsLogicalg caseseory of
gon
ntrariesSquare
aces 1,2stems is
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
Thpresehere,furthdynaneedresulrestri
Ththe rethe owhichtheirpervamind
7. Re
Blanc— (19
Coutu
— (19De M
Ginzb
— (19
Hami
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
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there are nod elucidatioUn restrictio, by experithree formstexts.ue investigaguistic comna of logicalories evenomentous anguage and
e des concep
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