3.1 Introductory observations
NATURAL SET THEORY, or NST, is a hypothesis, open to experimental testing. It is a novel hypothesis which, therefore, has not, so far, been subjected to any systematic experimental scrutiny and is based exclusively on what seems to be plausible at an intuitive or pretheoretical level. As a consequence, the present chapter can only be of an exploratory nature. Yet the underlying thought (a) that humans have at their disposal a natural, cognition-based, set theory and a predicate and propositional logic derived from it, (b) that this set theory and this logic are likely to differ in important respects from standard set theory and standard logic, and (c) that these differences may go a long way towards explaining the differences between natural set-theoretic and logical intuitions on the one hand and the corresponding standard systems on the other is, though novel, robust and, it would seem, hard to counter.
NST describes how humans deal cognitively with plural objects. It appears that NST lives by default or prototype and is thus open to correction or overriding as a result of more precise thinking shaped, in particular, by institutional education in more sophisticated societies. NST, like its corresponding logic, is thus open to bootstrapping into higher levels of precision and generality. This implies a gradient cline between two extremes of cognitive achievement, which we nameNATURAL SET THEORYandCONSTRUCTED SET THEORY, respectively. However, since too little is known, as yet, to define such a cline in a sufficiently precise format, I follow, for the moment, the easier course of distinguishing different levels of (un)naturalness, in the expectation that these distinctions, when more fully elaborated, will unfold into a system of gradual distinctions. For now, I consider natural set theory to be manifest in two forms: asBASIC-NATURAL SET THEORY(BNST), which stays within the limits imposed by all defaults, and as STRICT-NATURAL SET THEORY (SNST), which overrides one or more defaults. CONSTRUCTED SET THEORYoverrides all defaults.
NST instantiates the general hypothesis that there are hidden reserves in the mathematical, or, more generally, the formal powers of human cognition.
Inquisitive and formally creative humans are capable of cranking up the current level of mathematical or, more generally, formal performance to a higher degree of explicitness and formal precision by thinking through the consequences of their formal operations. Prime examples in the context of the Western world are, of course, Aristotle, Euclid, Al Hua´rizmi—who gave his name to the term algorithm—and the many other creators of the modern formal sciences known from history. When such efforts lead to new insights, which are then widely accepted in a given society and integrated into the educational system, we may consider this a measure of the degree of civiliza-tion of the society in quesciviliza-tion.
We thus assume a gradient in the analytical powers of humans ranging from the ‘rawest’ or most basic level of unsophistication to the most advanced levels of abstract thinking as found in the centres of science. The degree of achievement is taken to depend on the cultural, educational and other con-ditions that trigger the use and development of available cognitive reserves at the right age interval.
Some extremely interesting and challenging work has been done over the past few decades regarding the arithmetical capacities of infants, schoolchil-dren, and members of illiterate cultures.1Pica et al. (2004) investigated the arithmetical ability of speakers of Munduruku´, an Amazonian language spoken by some7000, mostly illiterate, Brazilian Indians. The language has no numerals beyond five (the word for ‘five’ being the equivalent of ‘hand’, as in many South-East Asian languages), reflecting the fact that the speakers were unable to count beyond five and had great difficulty doing simple arithmetical sums. Yet after some training, they quickly extended their count-ing and computcount-ing abilities both cognitively and lexically, creatcount-ing expres-sions like ‘two hands’, even though their achievements never matched those of humans born into culturally more developed societies.2
Butterworth states (1999: 7):
What makes human numerical ability unique is the development and transmission of cultural tools for extending the capability of the Number Module. These tools include aids to counting, such as number words, finger-counting, and tallying; and also the
1 For example, Ginsburg et al. (1984), Dehaene (1997), Butterworth (1999), Pica et al. (2004).
2 Pierre Pica, p.c. The fact that these speakers created new expressions to name numbers from existing ones strongly suggests, if not proves, that it was not the availability of the lexical items that enabled them to ‘think’ further along the number line but that it was in the first place their cognitive development that required the new lexical expressions, which were then readily composed and which probably helped them along in a secondary sense, in that the very availability of the expressions enhanced performance.
accumulated inventions of mathematicians down the centuries—from numerals to calculating procedures, from counting-boards to theorems and their proofs.
Though Butterworth is no doubt right as regards the role of culture in the development of human subjects’ arithmetical powers, one must fear that he errs on the issue of uniqueness. Far from being unique to the human numerical ability, the power to extend a naturally given ability through cultural development appears to be much more general. Although the exact boundaries of this phenomenon are unknown, it does seem that, along with numeracy and, apparently, also the reading ability, ‘significant evolutionary precursors [. . . ] may exist for other currently understudied cultural domains of human competence such as geometry, algebra, music and art’ (Dehaene 2005: 150–1; see also Dehaene et al. 2006). It thus does not seem too hazardous to surmise that logic and its underlying set theory are also worthy candidates to be considered. This is what is investigated in the present chapter.
The method followed here, however, differs from what is found in the studies mentioned above, which are experimental, performance-driven, and sometimes supported by neurophysiological evidence. Here, a hypothesis is proposed and offered for empirical testing, whereby the data are allowed to be largely intuitive. This method consists in observing how ‘ordinary’ speakers, whose degree of literacy and cultural sophistication is considered ‘normal’ in Western society and who distinguish themselves mainly by NOT being aca-demically trained, interpret and use logical expressions—in particular the logical constants—in their daily speech. An attempt is then made to reduce the (well-known) differences with standard modern logic, which is based on highly constructed mathematical set theory, to naturally given restrictions in the human way of cognitively dealing with plural objects.
This difference in method is, to a large extent, motivated by the object of research. In arithmetic or geometry, the technical aspects are relatively simple and straightforward. As regards arithmetic, it doesn’t take much theory to define a subject’s restricted arithmetical ability when this subject cannot count or carry out arithmetical computations beyond five. Against the background of our superior knowledge of arithmetic, such restrictions are quickly for-mulated. In fact, we have just done so. It is also easy to understand that early school training will trigger the growth and development of dormant arith-metical powers in the brains and minds of young children who grow up in a culturally developed environment. In fact, a moderately developed arithmeti-cal ability is simply taken for granted in our society, so much so that we are surprised to find out that individuals who grow up in a society without any scholastic tradition are not even able to count beyond the very lowest
numbers and, therefore, lack words for higher numbers. We also have a relatively good picture of what it has taken successive Chinese, Indian, Arabic, and Western civilizations to come to the highly sophisticated science of arithmetic that we have today. From the point of view of arithmetical theory, therefore, no great obstacles present themselves in the study of basic-natural arithmetic.
As regards logic, however, the situation is different. The very task of formulating restrictions on the logical powers of unsophisticated humans is the opposite of trivial. It requires considerable technical and theoretical effort and insight, and standard logical lore fails to provide the tools for doing so.
The foundations and basic notions of logic are far less clear and well under-stood than those of arithmetic, and the relation with natural cognition and language is still as problematic as ever. It is hardly surprising, therefore, that before one can pass on to any experimental work, one is forced to start with an identification and formulation of the restrictions involved.
The much greater conceptual difficulty of logic compared to arithmetic is borne out by the fact that what looks like the most firmly embedded logical intuitions of the human race appear to support a logic that depends for its application on complete situational knowledge, so that more developed forms of logic were required for use in situations where knowledge is not complete—a development that might explain the well-known discrepancies between natural logical intuitions on the one hand and the concepts and terms of the first, largely Aristotelian, ‘official’ logic on the other. Such discrepancies do not occur in the case of arithmetic.
One may also look at our hypothesized basic-natural logic in the light of the psychological theory of PROTOTYPES (see also Sections 8.6 and 8.8 in Volume I). In general, prototypes seem to be characterized by the fact that they maximize common features and thus avoid extremes or limiting cases.
Thus, it is proposed in Section3.2.2 below that the first principle of BNST consists in not taking into account the so-called extreme values—that is, the null set (), singletons (sets consisting of precisely one element) and the totality of objects (OBJ). Perhaps one may, therefore, just as well call BNPC by the name of prototypical logic. The problem is, however, that so little is known about the conditions that make for prototypes. Frequency won’t do as a criterion, as is shown in Section8.8 in Volume I. But what will do is simply to a large extent still a mystery, to do with hard-to-define notions such as
‘normal’ or ‘obvious’. Standard modern logic, for its part, can then be seen as the result of the exploration of the extreme cases: when these are taken into account, the basic-natural notions have to be sharpened. When one looks at the question from this angle, it becomes clear why one will have to distinguish
degrees of ‘naturalness’. This is precisely because the transition from intuitive prototypes to well-defined concepts is gradual.
Anthropological linguists and language typologists, driven as they are by a strong ecologistic motivation, will probably object that it is far from certain that thereISa naturally given, innate ‘basis’ as part of the genetic endowment of the human race and hence a basic-natural set theory or logic. It is uncertain, they will say, whether all natural languages possess (equivalents of) logical constants, in particularALL,SOME,NOT,AND, andOR, and, if they do, whether their meanings are the same in all natural languages.3All I can say to that is that there must be some genetically fixed cognitive substrate for dealing with plurality and for the making of inferences involving plural objects—that is, a predicate logic—or else the entire international machinery of bringing education and other forms of development to underdeveloped parts of the world would be built on quicksand. But, since our knowledge about these matters is as underdeveloped as those parts of the world we treat to our aid programmes, there is little else I can do now but make the famous inductive leap and embrace the simplest and most general hypothesis that, until proven otherwise, the naturally given lexical meanings of the logical operators under scrutiny are universally fixed.4
3.2 Some set-theoretic principles of natural cognition 3.2.1 A re´sume´ of standard set theory
It is important, at the outset, to emphasize the distinction between set-theoretic functions that map n–tuples of sets (a) onto sets and (b) onto truth values. The former are usually just called set-theoretic functions; the latter are called relations, denoted by predicates, since predicates typically
3 Steve Levinson tells me that there are languages which use the same word for ‘many’ and ‘all’ and also languages which use the indefinite article or the numeral meaning ‘one’ for ‘some’, referring to discussions in Wierzbicka (1996: 74–6, 193–7). What this evidence means remains to be seen. Modern Greek, for example, has the one word polı´ for both ‘very’ and ‘too’, making a phrase like polı´ mega´los ambiguous between ‘very big’ and ‘too big’ (though mega´los by itself can also mean ‘too big’, just as English late also has the meaning ‘too late’). Yet this does not mean at all that Greek speakers cannot or do not distinguish between the concepts ‘very’ and ‘too’. They clearly do and, when pressed, they use parapolı´ for ‘too’, even though parapolı´ still means ‘very much’ or ‘a whole lot’, but it seems to get closer to ‘too’ than simple polı´.
4 The programme thus outlined in effect amounts to an attempt at replacing current Gricean explanations for the disparity between logic and language in terms of generalized conversational implicatures with an explanation based on natural set theory and the cognitive faculty of forming mental propositions. If the objections raised in Section1.3.3.2 of Volume I against attempted pragmatic explanations along Gricean lines have any validity, this seems a worthwhile exercise.
denote functions from n–tuples of objects (or sets of objects) to truth values.
COMPLEMENT,INTERSECTION,UNION, andSUBTRACTIONare set-theoretic functions, butINCLUSION, for example, is a relation: for any given sets A and B, the binary relation ofINCLUSION, as in A B, is either true or false. By contrast, the set functions A (complement), A \ B (intersection), A [ B (union), or A–B (subtraction) do not have a truth value. Given the proper number of arbitrary sets (one for complement; more than one for intersection and union; exactly two for subtraction) they denote a new set defined by the Boolean functions complement, intersection, union, and subtraction, respectively. Complement is unique in that it involves the nonarbitrary set OBJ (the totality of all objects) as part of its definition. It can be described as a special case of subtraction, with OBJ–A as output for any set A. In logic, a further use of the term complement is to denote a relation (‘be in complement with’) yielding truth for two sets A and B just in case A[ B ¼ OBJ and A \ B ¼ .
The set-theoretic relations have a twofold use in the reduction of logic to set theory. First, they correspond to (meta)logical relations expressed in terms of valuation space (VS) analysis. For example, the inclusion relation translates into a possible metalogical statement that, say, the set of situations /P/ is included in the set of situations /Q/—that is, P entails Q (P‘ () Q), which is true or false depending on the meanings of P and Q, including the mean-ings of any truth-functional propositional operators they may contain. Sec-ondly, some set-theoretic relations correspond to quantifiers. For example,ALL
F is G translates, in principle, as saying that the set denoted by F is included in the set denoted by G.
By contrast, the set-theoretic functions correspond to the propositional logical constants of the object language LLas realized in any particular natural language. Just as the set-theoretic functions take sets and deliver sets, the propositional functions take valuation spaces and deliver valuation spaces.
For example, the operator AND in an L-proposition of the form P AND Q delivers /PANDQ/—that is, the set of those situations that make PANDQ true, corresponding to /P/\ /Q/, the intersection of /P/ and /Q/. Figure 3.1 shows how set-theoretic relations and functions are interpreted onto metalogical relations, object-language quantifiers, and propositional operators.
The counterpart in metalogic of the set-theoretic relation MUTUAL PARTIAL INTERSECTION(henceforth M-PARTIAL INTERSECTION, symbolized as A OOB: the two sets A and B partially intersect each other and do not severally or jointly equal eitherU or , as in Figure 3.3b) is logico-semantic independence, which plays no part in the machinery of logic: when /P/ and /Q/ M-partially
intersect, the L-propositions P and Q are logically (and semantically) inde-pendent in that the actual situation sitactcan be in /P/ but not in /Q/, in /Q/
but not in /P/, in both /P/ and /Q/, or in neither /P/ nor /Q/. That is, no entailment relation whatsoever holds between logically independent P and Q.
The relations complement, identity and full union lack a quantificational counterpart in predicate calculus, perhaps because they lack logical interest or perhaps because the makers of modern logic did not see far enough beyond natural language. Finally, the set-theoretic function subtraction appears to lack a single-morpheme propositional counterpart in most languages, but is expressed as and/but not in English: the valuation space of an L-proposition of the form PBUT NOTQ is the VS of P minus the VS of Q, or /P/ – /Q/.
Before a start is made with specifying the restrictions we intend to impose on standard set theory so as to slim it down to natural set theory, taken to reflect the way human cognition deals with sets, we will, for the sake of convenience, summarize the standard definitions of the functions and rela-tions of mathematical set theory. The standard Boolean funcrela-tions of comple-ment, union, intersection and subtraction are defined as follows (x ranges over elements inOBJ):
FIGURE3.1 The reduction of metalogical relations and object-language operators to set-theoretic relations and functions
(3.1) Standard Boolean functions on sets A and B:
a. A–
=def the set Z such that for all x2 Z, x =2 Aandforallx=2 Z,x2 A b. A[ B =def the set Z such that for all and onlyx 2 Z, x 2 A or x 2 B c. A\ B =def the set Z such that for all and only x2 Z, x 2 A and x 2 B d. A–B =def the set Z such that for all and only x2 Z, x 2 A and x =2 B The standard relations are defined in (3.2) (x ranges over elements in OBJ;
FULL UNIONhas been added for good measure):
(3.2) Standard relations between sets A and B:
a. COMPLEMENT 8x((x 2 A ! x =2 A–
)∧ (x =2 A ! x 2 A– )) b. IDENTITY 8x((x 2 A ! x 2 B) ∧ (x 2 B ! x 2 A)) c. MUTUAL EXCLUSION 8x(x 2 A ! x =2 B)
d. M-PARTIAL INTERSECTION ∃x(x 2 A ∧ x 2 B) ∧ ∃x(x 2 A ∧ x =2 B) ∧
∃x(x =2 A ∧ x 2 B) ∧ ∃x(x =2 A ∧ x =2 B) e. INCLUSIONof A in B 8x(x 2 A ! x 2 B)
f. FULL UNION 8x(x 2 A ∨ x 2 B)
These relations can also be defined in terms of the standard Boolean functions:
(3.3) Standard relations between sets A and B:
a. COMPLEMENT iff A[ B = OBJ and A \ B = Ø
b. IDENTITY iff A = B
c. MUTUAL EXCLUSION iff A\ B = Ø d. M-PARTIAL
INTERSEC-TION
iff A\ B 6¼ Ø; A \ B– 6¼ Ø; B \ A– 6¼ Ø;
A[ B 6¼ OBJ;
e. INCLUSIONof A in B iff A[ B 6¼ Ø f. FULL UNION: iff A[ B = OBJ 3.2.2 The restrictions imposed by NST
So much for standard set theory and standard logic. The question is now:
what restrictions are to be imposed on this system so that the discrepancies between logic and language are correctly predicted? To begin with, let it be assumed that NST entails that the mind, though naturally capable of proces-sing and operating with sets or ‘plural objects’, does not naturally represent a set as either the null set () or the universe of all objects (OBJ). It also seems
what restrictions are to be imposed on this system so that the discrepancies between logic and language are correctly predicted? To begin with, let it be assumed that NST entails that the mind, though naturally capable of proces-sing and operating with sets or ‘plural objects’, does not naturally represent a set as either the null set () or the universe of all objects (OBJ). It also seems