Chaotic Logic -- Copyright Plenum Press 1994

Back to Chaotic Logic Table of Contents

Chapter Five


    Alfred Tarski, who pioneered the mathematical semantics of formal languages (Tarski, 1935), adamantly maintained the impossibility of a mathematical semantics of natural language. But nevertheless, his work spawned a minor intellectual industry in the analysis of natural languages using formal semantics. In this chapter I will add a new twist to this research programme -- I will give a mathematical analysis of language and meaning from a pattern-theoretic rather than formal-logical angle, with an emphasis on the fundamentally systemic nature of language.

    The idea that language has to do with pattern is not a new one. It was present in the structuralism of Ferdinand de Saussure. And, more pertinently, it played a central role in the controversial thought of Benjamin Lee Whorf. Although a great deal of attention has been paid to Whorf's linguistic relativity hypothesis (Whorf, 1956), very little has been written about the general philosophy of language underlying his work in comparative grammar and semantics. All of Whorf's thought was grounded in a conviction that language is, in some sense, made of pattern and structure:    

    Because of the systematic, configurative nature of higher mind, the "patternment" aspect of language always overrides and controls the "lexation" or name-giving aspect.... We are all mistaken in our common belief that any word has an "exact meaning.... [T]he higher mind deals in symbols that have no fixed reference to anything, but are like blank checks, to be filled in as required, that stand for "any value" of a given variable, like ... the x, y, z of algebra....

    We should not however make the mistake of thinking that words, even as used by the lower personal mind, represent the opposite pole fromthese variable symbols.... Even the lower mind has caught something of the algebraic nature of language; so that words are in between the variable symbols of pure patternment ... and true fixed quantities. The sentence "I went all the way down there just in order to see Jack" contains only one fixed concrete reference; namely, "Jack." The rest is pattern attached to nothing specifically....

According to Whorf, a language consists of patterns which interact according to certain rules, which can somehow take one another as arguments, and which only occasionally make direct "reference" to "real," external objects.

    In this chapter I will elaborate on this powerful insight, using the concepts developed in Chapters Two and Three. One result of this exploration will be a general model of language as a special kind of structured transformation system. Syntactic rules form a transformation system, and semantics determines the analogical structure of this system. This view of language will allow us to explore the relation between language and thought in a much clearer light than has previously been available. It will aid us in understanding how language relates with deduction, consciousness and belief, and how language aids in the development and maintenance of those constructions which we call self and external reality.


    Richard Montague was the first person to make a full-scale effort to prove Tarski wrong, by applying the abstract semantic ideas of mathematical logic to natural languages. Due to his pioneering papers, and the subsequent work of Partee (1975) and others, we can now analyze the semantics of particular sentences in terms of formal logic. This is no small accomplishment. In fact, at the present time, no other theory of semantics, mathematical or no, can boast as effective a track record.

    So, in this section, I will begin the discussion of language by reviewing the main points of Montague grammar -- the syntactical theory that underlies Montague semantics. Then I will move to the more general notion of syntactic system, which will lead toward a deeper understanding of linguistic dynamics.

5.1.1. Montague Syntax

    Natural languages are frequently ambiguous: one word or phrase can have more than one meaning. This creates problems for mathematical logic; therefore Montague chose to deal only with disambiguated languages. Within the context of the formal approach, this is not a restriction but rather a methodological choice: any formal language can be mapped into a corresponding disambiguated formal language, by one of a number of simple procedures.

    For instance, in the "language" of vector algebra, ixjxk is ambiguous, and to disambiguate it one must introduce parentheses, obtaining (ixj)xk, and ix(jxk). One way to disambiguate an English sentence is to draw an "analysis tree" for each interpretation of the sentence, and take these trees to be the elements of the disambiguated language. This is awkward, yes, but it is not a formal obstacle.

    So, according to Montague, a disambiguated language consists of:

    1) a set of syntactic operations, each of which maps sequences of syntactic expressions into single syntactic expressions,

    2) a set of syntactic categories, which contain all possible words,

    3) syntactic rules, telling which operations may be applied to words in which categories.

    The disambiguity of the language is ensured by further axioms stating, in effect, that each syntactic expression can be obtained in accordance with the syntactic rules in exactly one way.

    For instance, consider the operation F with three arguments, defined so that F(x,y,z) is the statement "x y z." Consider the categories "noun" and "transitive verb." There is a syntactic rule, in English, saying that this operation can generally be applied if x and z are nouns and y is a transitive verb. Thus yielding, for instance F(I, kill, you) = "I kill you". Montague's Axioms (*)

    Formally, in Montague's terminology, a disambiguated language is an ordered quintuple (A,{Fl},{Xd},S,d0), defined by the following axioms:

    1) {Fl,l in L} is a set of syntactic operations, where L is an index set. Each operation maps finite ordered sets of syntactic expressions into syntactic expressions.

    2) D is a collection of syntactic category names

    3) {Xd, d in D} is a set of sets of basic expressions, each associated with a category name. It is possible for the same basic expression to have two different category names and hence belong to two different Xd

    4) S is a set of syntactic rules, each rule having the interpretation "If x1 belongs to category d1, and ... and xn belongs to category dn, then Fl(x1,...,xn) must belong to category dn+1" for some l in L.

    5) d0 is a special category name, to be used for the set of basic expressions denoting truth values.

    6) A is the set of all expressions generated by freely applying compositions of elements of the set {Fl, l in L} to the set {x: x is in Xd for some d in D}.

    7) No basic expression can be an output of any syntactic operation

    8) No expression in A can be the output of two different syntactic operations

    9) No syntactic operation can produce the same output from two different input expressions (i.e. the Fl are one-to-one)

    The formalism is obscure and complex, but the ideas are not particularly subtle. The first six axioms define the basic set-up, and the last three axioms ensure disambiguity.

5.1.2. Syntactic Systems

    A Montague grammar is a transformation system, in the sense defined above -- the transformation rules are the "syntactic operations," and the initials are the "basic expressions." But it is a very special kind of transformation system. I will need to deal with a somewhat less restrictive transformation-system model of grammatical structure, which I call a syntactic system. The "syntactic system" contains the "disambiguated language" as a special case, but it also includes a variety of structures which the Montagovian analysis ignores.

    The first step toward syntactic systems is the Sausseurean structuralist observation that, syntactically, "John" and "Mike" are the same, as are "cat" and "rat." It is not the meaning of a word that matters, but only the way it relates to other words. Therefore, it is natural to define a word, for the purposes of syntax, as a relation between other words.

    More specifically, one may characterize a word as a fuzzy set of functions, each of which which takes in a sequence of syntactic expressions and puts out a singlesyntactic expression. And one may characterize a punctuation mark in the same way. The class of syntactic expressions need only be defined, at this point, as a subset of the set of ordered sets of words and punctuation marks. From here on I will omit reference to punctuation marks, and speak only of words; but this does not reflect any theoretical difficulty, only a desire to avoid tedious figures of speech.

    What makes a collection of functions a syntax is a collection of constraints. Constraints tell us which sorts of expressions may be put into which inputs of which words. Thus they embody explicit grammatical rules, as well as "grammatico-semantic" rules such as the rule which tells us that the subject of the word "to walk" must be an animate object.

    For instance, the word kiss is identified, among other functions, with a function fkiss that has three arguments -- a subject, an object and a modifier. fkiss(I,my wife,definitively) = I kiss my wife definitively.

    And the word wife is identified with, among other functions, a function fwife that has at least five arguments. How one lines them up is arbitrary -- one may write, for instance, fwife(x1,x2,x3,x4,x5,x6), where:

x1 is constrained to be the subject of a verb of which wife is the object,

x2 is constrained to be a verb phrase of which wife is the object,

x3 is constrained to be an adjectivial phrase modifying wife,

x4 is constrained to be a verb phrase of which wife is the subject,

x5 is constrained to be the object of a verb phrase of which wife is the subject.

Arguments that are not filled may simply be left blank. For instance, fwife( , ,my lovely, eats, too much pie) is "my lovely wife eats too much pie." And fwife(I,kiss,my, , ) is "I kiss my wife."

    fI is similar to fwife in its syntactic structure. And, more simply, fmy is identified with a function of two arguments, one of which is constrained to be a noun phrase, one of which is constrained to be an adverbial phrase.

    In these simple examples I have mentioned only strictly grammatical constraints. An example of a less grammatical constraint would be the restriction of the object of "kiss" to entities which are concrete ratherthan abstract. This is an example of a constraint which need not necessarily be fulfilled. People may use the word "kiss" metaphorically, as in "When Elwood threw his boss out the window, he kissed his job goodbye." But if something is concrete, it fulfills the constraint better than something that isn't animate. Thus a constraint is a fuzzy set -- it tells not only what is allowed, but what is better, more easily permissible, than what.

5.1.3. A Formalization (*)

    Let's get more precise. Given a set of "concrete" entities X (which may well be the empty set), a syntactic system over X may then be defined as:

    1) A collection H of subsets of X.

    2) A collection of constraints -- at least one for each set in H. Each constraint may be written in the form f(i,x1,x2,...), and defines a series of fuzzy sets Cj(i). Let d(j,i,x) denote the degree to which x belongs to Cj(i). Then the interpretation is that, in a situation in which infon i obtains, x can be "plugged into" the xj position in f with acceptability level d(j,i,x). C(f), the collection of Cj(i) corresponding to f, is the collection of constraints implicit in f.

    3) A collection W* = {W,W#W,W#W#W,...,W#n,...}, where A#B is defined as the set of all possible entities obtainable by applying the functions in A to the functions in B, and W#n denotes W#W#...#W iterated n times. These sets are called "possible syntactic expressions," and are the elements of the fuzzy sets Cj(i,f).

    Each element x of W* has a certain "admissibility" A(i,x) defined inductively as follows. The raw admissibility RA(x) of the abstract form "f(i,g1,g2,...)," where the gi are in W, is the sum over j of the quantities d(j,i,f,gj). And the raw admissibility of an abstract form "f(i,g1,g2,...)" where the gi are in W#(n-1), is the sum over j of the product d(j,i,f,gj) * RA(gj).

    Finally, each element of W* is potentially realized by a number of different abstract forms of this nature. The admissibility of an element E of W*, relative to a given situation s, is the maximum over all abstractforms x that yield E of the product RA(x) * di(s). This measures the extent to which the formation of the expression E is grammatical.

5.1.4. A Comparative Analysis

    Despite the different mathematical form, my definition of "syntactic system" is not tremendously different from the Montagovian definition of "disambiguated language." The syntactic system represents a generalization of, rather than a radical break from, Montague grammar. There are, however, several important distinctions that may be drawn between the two approaches.

    For one thing, loosely following Barwise and Perry (1985), Barwise (1989), and Devlin (1991), the definition of syntactic system incorporates an infon i at every juncture. Each real situation supports certain infons to certain degrees. Montague assumes that there is one big situation, in which everything applies; but his axioms could be "situated" without too much difficulty (by modifying 1, 4 and 6). And, correspondingly, my axioms could be de-situated by removing all references to infons.

    Next, the Montagovian approach describes only disambiguated languages, whereas the concept of syntactic system makes no such restriction. It could easily be restricted to disambiguated languages (by adding on a clause resembling condition 8 of the Montagovian definition -- conditions 7 and 9 are common sense and are automatically fulfilled). But there is no need. Real languages are of course ambiguous in many ways. Montague's possible worlds analysis of meaning requires disambiguity, but the semantical theory to be given below does not.

    Finally, the most substantial difference is that the definition of syntactic system defines a word as a set of syntactic operations, and assigns a set of grammatical rules to each word. The Montague approach, more manageably, assigns each word to a certain class and then sets up syntactic operations and grammatical rules to work via classes.

    One way to look at this difference is via algorithmic complexity. A random syntactic system would be useless -- no one could memorize the list of rules. Only a syntactic system that is truly systematic -- thatis simple in structure, that has patterns which simplify it a great deal -- can possibly be of any use. One way for a syntactic system to be simple in structure is for its constraints to fall into categories.

    In other words, suppose the class of all words can be divided into categories so that the constraints regarding a word can be predicted from knowledge of what category the word is in. Then the syntactic-system approach reduces to something very similar to the Montague approach (to a situated, ambiguous Montague grammar). But when dealing with syntactic systems in general, not only written and spoken language, it is unnecessarily restrictive to require that all rules operate by categories. It is better and more in line with the pattern-theoretic approach to speak about general syntactic systems, with the understanding that only syntactic systems of low algorithmic complexity are interesting.

5.1.5. What is Language?

    Basically, the definition of syntactic system says that each word (each fundamental unit) is a certain collection of functions, each of which takes in certain kinds of external entities and certain types of functions associated with other words. The kinds of entities and functions that a certain function takes in can depend upon the situation in which the associated word is used. There are certain rules by which words can be built up into more complex structures, and by which more complex structures can be built up into yet more complex structures -- each rule applies only to certain types of words or other structures, and the types of structures that it applies to may depend on the situation in which it is being used.     

    I will give a theory of meaning to go along with the general model of syntax outlined in the previous section. The basic idea of this theory is that the meaning of an entity is the fuzzy set of patterns related to its occurence.

    Using this characterization of meaning, I will define a semantic system as a set of entities which obtain much of their meanings from each other. In this context, it will become clear that the semantical structure of written and spoken language is not at all unique. Written and spoken language may be the most cohesive semantic system known to us. But subtle interdefinition, and the intricate interplay of form and content, can be found to various degrees in various domains.

     Language will then be defined as requiring a syntactic system, coupled with a semantic system in such a way that a property called continuous compositionality holds. Thus, I do not believe that one can give a thoroughly context-independent definition of language. The definition of syntactic system refers frequently to infons, which hold or do not hold in specific situations. At the very foundation of a language is the set of situations in which it evolved, and in which it is used.


    The next step after Montague grammar is Montague semantics. Also known as possible-worlds semantics, Montague semantics is just as forbiddingly formal as Montague grammar, perhaps more so. However, as I will show, it packs much more of a philosophical punch.

    First of all, Montague assumes that there is some set B of meanings. Then he assumes that, to each syntactic operation F, there corresponds a semantic operation G taking the same number of arguments and mapping n-tuples of meanings into meanings (rather than n-tuples of expressions into expressions). Finally, he assumes some function f mapping basic expressions into meanings. This setup determines, in an obvious way, the meaning of every element of A -- even those which areconstructed in violation of the syntactic rules. The existence of a correspondence between the F and the G is what Frege called the principle of compositionality.

    In order to specify B, Montague invokes the notion of possible worlds. This notion is used to build up a hierarchy of progressively more complex semantical definitions. First of all, assume that each basic expression contains a certain number of "variables", to be filled in according to context. Suppose that knowing what possible world one is in, at what time, does not necessarily tell one what values these variables must have, although it may perhaps give further information about the meaning of the expression. This does not contradict the very general axioms given above. Then, one may define a denotation of an expression as something to which the expression refers in a given possible world, at a specific time, assuming a certain assignment of values to its variables.

    And one may define the sense of an expression as that to which the expression refers regardless of the time and possible world. This is not a precise definition; however, one way to specify it is to define the sense of an expression as the function which assigns to each pair (time, possible world), the denotation which that expression assumes in that possible world at that time.

    Finally, one may define the Fregean meaning of an expression as the function which maps each triple (possible world, time, assignment of variable values) into the sense which that expression assumes under that assignment. Montague calls this simply the "meaning", but I wish to reserve this word for something different, so "Fregean meaning" it is.

    In this scheme, everything reduces to denotations, which may be divided into different "types" and then analyzed in some detail. For instance, one of the most important types of denotation is truth value. The truth value of an expression with respect to a given triple (possible world, time, assignment of variable values) is whatever truth-value it denotes in that model. Montague semantics does not, in itself, specify what sorts of expressions may denote truth values. It is possible to give a formal definition of the syntactic category "sentence," but not all sentences may take truth values. In English, it seems clear that statements, rather than imperatives or questions, may denote truth values; but this is an empirical observation, not a formal statement, and a great deal of work is required to formulate it mathematically.

    In general, a Fregean meaning of type T is a function mapping entities of the form (possible world, time, variable assignment) into a denotation of type T. Montague hypothesizes that each type corresponds to an element of D, so that Fregean meaning types and syntactic categories are matched in a one-to-one manner. Compositionality then requires that any rule holding for syntactic categories transfers over into a rule for Fregean meaning types. For instance, take a syntactic rule like F(x,y,z) = x y z. This maps vectors (noun, noun, transitive verb) into declarative sentences. Then F corresponds to a semantic function G which maps meanings of the type corresponding to nouns and verbs, into meanings involving truth-values as denotations.

5.2.1. Critique of Montague Semantics

    This thumbnail sketch is hardly an adequate portrayal of Montague semantics. The interested reader is urged to look up the original papers. However, I will not require any further development of possible-worlds semantics here. The reason is that I am highly skeptical of the whole project of possible-worlds semantics.

    I find it hard to accept that what "1+1=2" means is the same as what "2.718281828...i*3.1415926535... = -1" means. However, in the standard implementations of the possible worlds approach, these two assertions both denote truth in every possible world at every time, so they are semantically identical. It is true that each of these assertions can be derived from the other, given the standard mathematical axioms. But they still mean different things.

    Possible worlds semantics is formal in a very strange sense: it makes no reference to the actual empirical or psychological content of linguistic entities. Montague believed that the most important aspects of semantics could be developed in a purely formal way, and that considerations of content, being somehow more superficial, could be tacked on afterwards. Roughly speaking, he believed that content merely sets the values of the "parameters" provided by the underlying formal structure. But this is at best a debatable working hypothesis, at worst a dogma. The possible-worlds approach has not yet been shown to apply to any but the simplest sentences.

    It is remarkable that formal logic which ignores content can deal with semantically troublesome sentences like "John believes that Miss America is bald". But sentences like "Every man who loves a woman loses her"are still troublesome. And it is a long way from these formal puzzles to ordinary discourse, let alone to, say, a fragment of Octavio Paz's poetry (1984):


back flame


        you yourself the sun

        the moon

        turning for ever around you

pomegranate that bursts itself open each night

fixed star on the brow of the sky

and beat of the sea and the stilled light

open mind above the

        to and fro of the sea

The contemporary logicist approach can comprehend this fragment about as well as modern quantum physics can deal with the large-scale dynamics of the brain. There is a tremendous rift between theoretical applicability and practical application.

    I am not the only one to sense the fundamental impotence of possible worlds semantics. Many logicians and linguists share my frustration. In the absence of a constructive alternative, however, this frustration is not terribly productive.

    One possible alternative is situation semantics, a theory of meaning designed to transcend Montague semantics by making reference to information. However, the situation semanticists approach information in a very abstract way, starting from set theory. They define an abstract unit of information called an "infon," and attempt to delineate various axioms which infons must obey. While I admire situation semantics very much, I cannot agree with the abstract, set-theoretic approach to information. It seems clear that, just as physics models objects as elements of Euclidean space rather than general sets, a successful semantic theory must come equipped with a concrete, particular idea as to what information is. One way to do this is to take the algorithmic theory of information. This is the course that will be taken in the following section.


    Using the theory of pattern and algorithmic information, meaning can be defined without even mentioning syntax. Even entities that are not involvedin syntactic systems can have meanings. The meaning of an entity, I suggest, is simply the set of all patterns related to its occurence. For instance, the meaning of the concept cat is the set of all patterns, the occurence of which is somehow related to the occurence of a cat. Examples would be: the appearance of a dead bird, a litter box, a kitten, a barking dog, a strip dancer in a pussycat outfit, a cartoon cat on TV, a tiger, a tail,....

    There are certain technical difficulties in defining "related to" -- these will be dealt with shortly. But it is clear that some things are related to cat more strongly than others. Thus the meaning of cat is not an ordinary set but a fuzzy set. A meaning is a fuzzy set of patterns.

    In this view, the meaning of even a simple entity is a very complex construct. In fact, as is shown in the following section, meaning is in general uncomputable in the sense of Godel's Theorem. But this does not mean that we cannot approximate meanings, and work with these approximations just as we do other collections of patterns.

    This approach to meaning is very easily situated. The meaning of an entity in a given situation is the set of all patterns in that situation which are related to that entity. The meaning of W in situation s will be called the s-meaning of W. The degree to which a certain pattern belongs to the s-meaning of W depends on two things: how intensely the pattern is present in s, and how related the pattern is to W.

    These ideas are not difficult to formalize. Let MW,s(q) denote the degree to which q is an element of the s-meaning of W, relative to the situation s. Then one might, for instance, set

    MW,s(q) = IN[q;s] * corr[W,q]

where corr[W,q] denotes the statistical correlation between W and q, gauged by the standard "correlation coefficient," and IN[q;s] is the intensity of q as a pattern in s. The correlation must be taken over some past history of situations that are similar in type to s; and it may possibly be weighted to give preference to situation which are more strongly similar to s. The determination of similarity between situations, of course, is a function of the mind in which the meanings exist.

    Like all pattern-theoretic definitions, this characterization of meaning is unpleasantly messy. Thereare all sorts of loose ends and free parameters; things are not nearly so cut-and-dried as in, for example, the Montagovian possible-worlds approach. But unfortunately, this is the price which one must pay for being psychologically reasonable. Meaning exists only relative to a given mind, a given brain; and minds and brains are notorious for not adhering to conventional standards of mathematical nicety.

5.3.1. Meaning and Undecidability (*)

    It is clear that, according to the above definitions, determining the s-meaning of any entity W is an extremely difficult computational problem. In fact, if one considers sufficiently complex situations, the problem becomes so hard as to be undecidable, in the sense of Godel's Theorem.

    Godel showed that truth is not contained in any one formal system; and his results apply directly to the standard model-theoretic approach to semantics. But it is interesting that even a subjective, pragmatic approach to meaning cannot escape undecidability.

    Chaitin (1975, 1978, 1987) has given an incisive information-theoretic proof of Godel's Incompleteness Theorem. He has proved that, for any formal system G, there is some integer N so that

    1) for all n>N, there exists some binary sequence x so that the statement "H(x) > n" is undecidable in G (it can neither be proved true, nor proved false, in G).

    2) "H(x) > N" is not provably true (in G) for any x

    Where S is some subset of St(s), let us consider the statement "|S| > N" in this light.

    First, arrange the elements of S in a specified order (y1,...,yN), and set xS = L(y1)L(y2)...L(yN), where L maps patterns into binary sequences. Then, as |S| becomes arbitrarily large, so does H(xS). That is, for any N, there is some M so that when |S| >M, one has H(xS) > N. But for large enough N, the statement "H(xS) > N" is undecidable. Consequently, so is "|S| > M".      Finally, let MW,s;K denote the set of all Boolean q so that MW,s(q) > K. Then I have shown

     Theorem: For any formal system G, and any situation s of infinite structural complexity, there is some M so that the statement "|MW,s;K| > M" is undecidable in G.

    Godel showed that truth cannot be encapsulated in any formal system. According to this theorem, if semantics is defined in terms of information, complexity and pattern, Godel's proof applies equally well to meaning. This is philosophically interesting, becausethe informational approach to meaning makes no reference whatsoever to truth. But it is not surprising, not since Chaitin has already shown us that Godel's Theorem has as much to do with information as with truth.

5.3.2. Meaning and Possible Worlds

    Let us briefly return to Montague semantics. What does the present definition of s-meaning have to do with the Montagovian approach? Montague semantics speaks of denotations, senses and Fregean meanings. Where does meaning, as I have defined it, fit in?

    The present approach determines for each expression, given each situation s, a definite s-meaning. But each particular situation is, surely, a subset of the set of pairs (possible worlds, times). It may sometimes be useful to consider an entire possible world, from the beginning of time to the end, as one big situation; or perhaps to consider a "situation" as a class of events intersecting every possible world.

    The possible-worlds approach begins with denotations: the denotation of an expression is what it expresses regardless of the possible world and time. According to the informational approach, however, there is no reason to believe that denotations such as this exist. In each possible world over each interval of time, and more generally in each situation, each entity has a certain meaning. But since the meaning of an entity is defined relative to the structure of the situation it is used in, there is no reason to believe the meaning of any entity will be constant over all possible situations. Indeed, given most any entity, and most any pattern, one could cook up a situation in which that pattern was not relevant to that entity, and hence not a part of the meaning of that entity.

    This is related to a point made in Barwise (1989). Barwise argues, in effect, that the concept of what an expression expresses regardless of possible world and time is not meaningful, because the collection of all pairs (possible world, time) is not a set but a proper class. In order to make the possible-worlds approach set-theoretically meaningful, one must restrict consideration to some particular set of worlds.

    In reality, no entity experiences or envisions every mathematically possible world, nor even a reasonably large subset thereof. And it does mean something to talk about the meaning of W relative to some particular fuzzy set S of situations. Formally, where dS(x) denotes thedegree of membership of x in S, MW,S(q) may be defined as the sum over all x in S of MW,x(q)dS(x). If S is taken to be the collection of all situations in a given mind's memory, then one may omit the subscript S and simply write MW.

     This, finally, is what I mean by the "meaning" of a word or other entity W. In most practical cases, MW is actually not all that far off from the possible-worlds definition of meaning. Let's take the word "dog," for example. To an ordinary, intelligent, English-speaking person, the concept of "dog" is not that fuzzy: certain things are dogs (they belong to Mdog with degree 1) and most things aren't (they belong to Mdog with degree 0). Some things, like wolves or wolf-dog half-breeds, might belong to Mdog with intermediate degrees (say .25 or .75), but these are definitely the exception. In a vast majority of the situations in which the word "dog" is used, those things which are dogs, or various memories involving them, take part in patterns associated with the word "dog." In Montagovian terms, the elements of MW are very good candidates for the sense of the word "dog." They are, approximately, what "dog" refers to regardless of possible world and time. And for a simple expression like "dog," with no explicit variables, the sense is essentially (though not set-theoretically) the same as the Fregean meaning.

    In general, for more complex expressions which may have variables in them (say, "John eats ____'s pet ____"), MW may be computed either for the expression as an abstract formula, or for the expression given some particular assignment of variable values. The latter quantity will often be similar to the sense of the expression, given the same particular assignment of variable values. And the former quantity will often be similar to the Fregean meaning of the expression, since the Fregean meaning contains all senses for all possible worlds and times, and MJohn eats ____'s pet ____ contains, with some nonzero degree, all elements of MJohn eats x's pet y for every x and y.

    So, in many cases, the present situation-oriented, pattern-based definition of meaning coincides with the possible worlds definition (as well as with the situation-theoretic approach of Barwise or Devlin). This is because, to a large extent, the different approaches are getting at the same underlying intuition. However, it seems to me that the informational definition is psychologically a lot more sensible than the possible-worlds approach, and also a lot more sensible than the more abstract situation-theoretic analyses.


    Now, at long last, we are entering the final stretch of our quest to tie syntax and semantics together. Let me begin with Frege's "principle of compositionality." This axiom, if you recall, states that the meaning of a complex syntactic construct can be determined from a knowledge of: 1) the syntactic operations involved, and 2) the meanings of the simpler syntactic constructs of which the complex syntactic construct is formed.

    Mathematically, in the present formalism, compositionality says that for each syntactic operation F there is a "semantic operation" G so that

    MF(x,y)(q) = G(Mx(q),My(q)).

Clearly, this principle is not implied by the informational approach to meaning. But it is not forbidden either.

    For starters, let us consider a rule F(x,y,z) which takes in a noun x, a transitive verb y, and a noun z, and puts out the sentence xyz: F(Sandy, kisses, Andy) = Sandy kisses Andy. The question is, is there some G so that MSandy kisses Andy =

G(MSandy,Mkisses,MAndy), and, furthermore, MF(x,y,z) = G(Mx,My,Mz) for any x,y,z? In other words, is the meaning of the whole determined by the meaning of the "component parts"? Knowing the set of patterns related to "Sandy", "kisses" and "Andy", and the standard grammatical rules, can one predict the set of patterns related to "Sandy kisses Andy"?

    Or, to take an absurd example, what if English contained a rule F'(x,y,z), taking arguments x and z human beings, and y a transitive verb, defined so that

F'(x,y,z) =

    F(the father of x, the last transitive verb in the Standard High School Dictionary before y, the mother of z).

Montague's restrictions on semantic rules forbid this sort of construction, but the general definition of semantic system places no such restrictions. Then F'(Sandy, kisses, Andy) might equal, say, "Jack kings Jill". There would be no way to predict the meaning of "Jack kings Jill" from the meaning of "Sandy", "kisses" and "Andy". The point is that real written and spoken languages do not have crazy rules like this -- and the main reason they do not is compositionality.

    Finally, consider an example discussed in Barwise (1989), the opening sentence of Hoban's novel Riddley Walker:

On my naming day when I come 12 I gone front spear and kilt a wyld boar he parbly ben the las wyld pig on the Bundel downs any how there hadnt ben none for a long time befor him nor I aint looking to see none agen.

Barwise asks how a compositional account of meaning could possibly explain the meaning of the phrase "gone front spear" -- let alone the whole sentence. The same question could of course be asked in regard to much modern poetry and literature. The point is that we automatically assign meaning even to expressions that are formed in violation of the rules of grammar. If an expression is formed in violation of the rules of grammar, there is no way to compute its meaning by going from a function F to a function G as compositionality suggests.

    Barwise's "gone front spear" argument is fatal for strict Montague semantics. But it certainly does not imply that compositionality is totally absent from natural languages. I suggest that compositionality is a tool for estimating meanings, and a very powerful one. Without this tool, it would be hard to estimate the meaning of a sentence that one had never heard before. However, like all real tools, compositionality is not a complete solution for every problem.

    A language would be basically useless if it did not possess approximate compositionality for most words, most syntactic operations, F. Riddley Walker and Naked Lunch are more difficult to read than Huckleberry Finn and Catch-22, and this is precisely because when assigning meaning to the sentences of the former books, one must depend less on compositionality, and more on subtle structural clues internal to the semantics.

    One final note is in order. I have been talking about language in a very general way, but the examples I have given have been either Boolean logic or common English. These are good sources for examples, but they may also be misleading. In these cases, compositionality takes a particularly simple form: the deductive predecessors of an expression are also its components. For instance, where F(Sandy, kisses, Andy) = "Sandy kisses Andy", the arguments "Sandy," "kisses," and "Andy" are parts of the sentence "Sandy kisses Andy." Here,compositionality requires that the meaning of a whole predictable from the meaning of its parts.

    Fodor (1987), among others, thinks this is essential to the concept of compositionaliy. But on the other hand, nothing in the present theory of language requires that the relation between the output of a function and its arguments be a whole/part relationship. This point is particularly relevant in the context of the recent work of Tim van Gelder (1990), which suggests that certain neural network models of thought possess compositionality without displaying any sort of whole/part relationship between expressions and their deductive predecessors.

5.4.1. Formal Meaning and Dictionary Meaning

    What do all these abstract mathematical definitions of "meaning" have to do with meaning in the dictionary sense? When one looks up a word in a dictionary, one certainly does not find a huge fuzzy set of regularities spanning different situations: one finds a phrase, or a sentence, or a number of sentences!

    The answer to this most natural question is as follows. When one looks up a word like "high-falutin'" or "cosmological," one finds a sentence consisting hopefully of simpler words. Using compositionality (as well as of course all our knowledge of grammar and semantics), one construes the meaning of that sentence from the meanings of the simple words, and thus infers the meaning of the word in question. One never learns any word as well from the dictionary as from hearing it in practice, but for some words the dictionary can yield a good approximation.

    For other words, however, such as "the" or "a," the dictionary is totally useless for imparting meaning -- it can impart technical niceties to someone who already basically knows the meaning, but that's about it. And for words like "in" or "out" the dictionary almost as useless -- "in" refers you to "inside," which refers you back to "in," et cetera. The words that are possible to learn from the dictionary are those words which could reasonably be replaced in conversation by complex phrases, which could then be understood by appeal to compositionality.

5.4.2. Semantic Systems

    We have defined the syntactic system, identified the relation between syntax and semantics, and given a newtheory of meaning. What remains is to crystallize this theory of meaning into a definition of the semantic system.

    Intuitively, a semantic system V is a collection of entities whose meanings consist primarily of patterns involving other elements of the system. The systematicity of a collection of patterns is the extent to which that collection is a semantic system.

    More formally, let D(x/V) denote the percentage of the meaning of x which involves elements of V. This is an intuitively simple idea, but its rigorous definition is a little involved, and will be postponed to the end of the section.

A systematic collection of entities is characterized by a high average D(x/V) for x in V.

    Written and spoken languages are examples of collections with very high systematicity. The meaning of the word "dog" has a lot to do with physical entities. It also has to do with other linguistic entities: certain aspects of its occurence can be predicted from the fact that it is a noun, that it is animate, etc. But it is among the less dependent words; D(dog/English) is not all that large. On the other hand, the meaning of the word "the" has virtually nothing to do with nonlinguistic entities, and therefore "the" contributes a great deal to the systematicity of English. D(the/English) is certainly very large.

    The Sapir-Whorf hypothesis rests upon the assumption that languages are highly systematic. That is its starting-point. If the meanings of words and sentences had to do primarily with extra-linguistic phenomena, then how could language have the power Whorf ascribes to it? It is only after one realizes the extent to which linguistic entities depend on each other for their significance, that one can conceive of language as a coherent, acting entity.

    But written and spoken languages are almost certainly not the only systematic meaning systems. It seems that each sensory modality probably has its own semantic system. For instance, the set of patterns involving the visual entity "box" has a lot to do with other visual forms and not that much to do with anything else. And the same goes for most visual forms. Hence, intuitively, one might guess that the collection of visual forms is highly systematic. Formal Definition of D(x/V) (*)

    Finally, before moving on, let us deal with the problem of defining D(x/V). Although I will not be using this definition for any specific computations or theoretical developments, it is important to have a precise definition in mind when one speaks about a concept. Otherwise, one does not know what one is talking about.

    One approach to defining D(x/V) is as follows. First, for each q and each s, define the degree D(x/V;s,q) to which MW,s(q) involves V as the maximum, over all elements v of V, of the expression

    MW,s(v) * corr[v,q] * |St[v] St[q]|/ |St(q)|.

This product can never exceed 1; it is close to 1 only if v:

    1) is an element of the meaning of q in the situation s,

    2) is statistically correlated with q

    3) contains much of the same structure that q does

    Next, define D(x\V;s) to be the average of D(x/V;s,q) over all q. And, where S is a set of situations, define D(x/S) to be the average of D(x/V;s) over all s in S. Where S is taken to be all situations in the memory of a given mind, one may omit reference to it, and simply speak of D(x/V).

5.4.3 The Definition of a Linguistic System

    Now all the hard work is mercifully past. I am prepared to give a pattern-theoretic, "informational" definition of a linguistic system. First of all, let us state some minimal requirements. Whatever else it may be, every linguistic system must consist of

    1) a syntactic system, together with

    2) a collection of situations, so that this syntactic system, in this collection of situations, gives rise to

    3) a semantic system, in which the meanings of most expressions may be approximately determined by compositionality.

    This is quite a mouthful. But it is not quite enough to constitute an adequate definition of "linguistic system." To see what else is needed, let us recall the concept of structured transformation system, defined in Chapter Four. Now, a syntactic system is a transformation system -- this follows immediately from acomparison of the two definitions. But what about the "structured" part?

    Does semantics, combined with compositionality, have the capacity to induce a structure on the transformation system that is syntax? What is needed is that grammatically similar linguistic constructions (sentences) also tend to be structurally similar (where the complexity measure implicit in the phrase "structurally similar" is defined relative to the environment in which the sentences are used). But, if one knew that syntactically similar sentences tended to have similar meanings, this would follow as a consequence. One could form a sentence with meaning X by analogy to how one has formed sentences with meanings close to X.

    The principle of compositionality, under my loose interpretation, implies that for most syntactic operations F there is a "semantic operation" G so that MF(x,y) is close to G(Mx,My). But this does not imply that sentences formed by similar rules will tend to have similar meanings. I need an additional hypothesis: namely, that small changes in F correspond to small changes in G. It is not enough that each syntactic rule corresponds to a semantic rule -- this correspondence must be "stable with respect to small perturbations."

    This property may be called continuous compositionality. A little more formally, suppose that F(x,y) and F'(x,y) are close. Compositionality guarantees that there are G and G' so that:

    1) MF(x,y) is close to G(Mx,My), and

    2) MF'(x,y) is close to G'(Mx,My).

Continuity of compositionality requires that G and G' be close. But relations (1) and (2) render this "continuity requirement" equivalent to MF(x,y) and MF'(x,y) being close.

    So, all formalities aside, one may define a linguistic system as

    1) a syntactic system, together with

    2) a collection of situations,

    3) so that relative to these situations the expressions of the syntactic system form a semantic system

    4) which is related to the syntactic system according to continuous compositionality.

    From this definition, one has the immediate result that a linguistic system is a structured transformation system.

    Boolean logic, as analyzed in Chapter Four, is a specific example of a linguistic system; in fact it is a subset of natural languages. I have pointed out somerelations between analogical structure and deduction in the context of Boolean logic: these may now be understood as examples of the behavior of linguistic systems, and special cases of the complex dynamics of natural language.

5.3.4. Communication

    What is the purpose of language? The straightforward answer to this question is "communication." But what exactly does this elusive term denote? The so-called "mathematical theory of communication," founded by Claude Shannon, deals with the surprise value of a message relative to a given ensemble of messages. But although this is marvelous mathematics and engineering, it has little to do with meaning. The communication of patterns is different from the communication of statistical information.

    Let us consider the five "illocutionary categories" into which Searle (1983) claims all speech acts may be categorized:

     Assertives, which commit the speaker to the truth of an expression

    Directives, which attempt to get the speaker to do something. This category is inclusive of both commands and questions.

     Commissives, which commit the speaker to do something -- say to join the Navy, or to tell the truth in a court proceeding.

     Expressives, which express a psychological state on the part of the speaker

     Declaratives, which, by virtue of being uttered, bring about the content of the utterance. For instance, "I pronounce you man and wife."

    One could modify this list in various ways. For instance, what Searle calls "assertive" is sometimes called "declarative." And I am not sure about the boundary between assertives and expressives: it is not a crisp distinction. Many utterances combine both of these types in a complicated way -- for example, "My head hurts worse than yours." But these quibbles are irrelevant to what I want to do here.

    All of these categories have one obvious thing in common. They say that the speaker, by using a speech act, is trying to cause some infon to obtain. In the case of expressives and assertives, one is mainly trying to cause an infon (the content of one's statement) to obtain in the mind of the listener. In particular, among other things, one is telling the listener the situationin question/ speaker |-- this content. In the case of assertives, one may also be trying to cause the situation in question/ listener |-- this content to appear -- that is, one may be trying to convince the listener to agree with you. But at any rate, the most basic thing you are doing is trying to cause a record of what you think or feel to occur in her mind.

    In the case of directives, one is trying to cause the listener to respond either with an assertive statement of her own (in the case of a question) or with some other sort of action. One is trying to make a certain infon appear in one's present physical situation, or in some future situation.

    Finally, in the case of commissives and declaratives, things are even more direct. One is swearing oneself into the Navy, or declaring two people married. Within the network of beliefs that makes up one's subjective world, one is actually causing certain infons to obtain.

    So what communication really comes down to, is molding the world in a certain way. How does it differ from other means of molding the world, such as building something? Only, I suggest, in that it partakes of the deductive and analogical system associated with a given language. Rather than defining language as that which communicates, I propose to define communication as the process of doing something with language.

    In the context of the model of mind outlined in Chapter Three, the definition of language given above might be reformulated as follows: a linguistic system is a syntactic system coupled with a semantic system in such a way that the coupled system is useful for molding the world. After all, a syntactic system is useless for molding the world unless it is appropriately coupled with an analogical, associative-memory-based system. And a semantic system can serve in this role only if the property of continuous compositionality is present.     

    In Chapter Four I considered a very restrictive linguistic system -- Boolean logic. I showed in detail how the syntactic system of Boolean logic is useless in itself -- but extremely useful when appropriately coupled with a semantic, analogical network. With more general languages, many more issues are involved -- but the basic picture is the same. A linguistic system is a syntactic system coupled with a semantic system so as to make communication possible.


    I have theorized about general linguistic systems; but the only linguistic systems I have explicitly discussed are Boolean logic and written/spoken language. I will now briefly consider three other linguistic systems, which at least as essential to the functioning of mind. The treatment of these systems will be extremely sketchy, more of an indication of directions for development than a presentation of results. But it would be unthinkable to completely ignore three linguistic systems as essential as perception, motor control and social behavior.

5.5.1. Perception, Action and Language

    Let us begin with Nietzsche's analysis of the "inner experience" of an outer world as a construct of language and consciousness:

    The whole of "inner experience" rests upon the fact that a cause for an excitement of the nerve centers is sought and imagined -- and that only a cause thus discovered enters consciousness; this cause in no way corresponds to the real cause -- it is a groping on the basis of previous "inner experiences," i.e. of memory.... Our "outer world" as we project it every moment is indissolubly tied to ... old error.... "Inner experience" enters our consciousness only after it has found a language the individual understands. (p. 266)

In this view, experience enters consciousness only after it has found an appropriate language.

    Nietzsche also observed that language and perception are similar, both being based on making equal that which is not.

    First images.... Then words, applied to images. Finally concepts, possible only when there are words.... The tiny amount of emotion to which the "word" gives rise, as we contemplate similar images for which one word exists -- this weak emotion is the common element, the basis of the concept. That weak sensations are regarded as alike, sensed as being the same, is the fundamental fact.... Believing is the primal beginning even in every sense impression.... (p.275)

This penetrating observation implies that, in a sense, language is to the middle levels what systematic perception is to the below-conscious levels. Language is based on the identification of word-concepts, which is the recognition of common patterns among the outputs of lower-level, perceptual-motor processes. Perception, on the other hand, is based on the identification of common patterns among the outputs of "sensory organs" or else lower-level perceptual-motor processes. Both are systematic, with a grammar and a semantics; both are meaning-generating structured transformation systems.

    In humans, visual perception, at least, has a very complicated grammar. The visual cortex builds a scene out of the simple parts which it perceives, and it is this scene rather than the individual stimuli which it feeds up to consciousness. And the aural cortex does the same thing, in a less involved way: we listen to someone talking and hear words, but these words are pieced together according to a complex system of rules from particular blurry, superimposed sounds. These sensory-modality-dependent rules for building wholes out of parts are full-fledged, situation-dependent grammars.

    And there is no doubt that, in the sense defined above, visual and aural forms constitute very intricate semantic systems. Compositionality is slightly confusing: are the meanings of the raw sounds or visual stimuli experienced by low-level processes sufficient to determine the meanings of the complex combinations which the conscious mind experiences? Internally, from the point of view of the conscious perceiving mind, raw sounds and visual stimuli have meanings, in the sense of being algorithmically related to other things, only through these complex combinations. Therefore, from the phenomenological point of view, compositionality is only interesting above a certain level. Below that level, it is either obvious or meaningless: the meaning of the parts is the meaning of those wholes that the part contributes to; the parts have no independent significance.

    However, from the point of view of a real or hypothetical external observer, with access even to patterns below the level of consciousness, compositionality is interesting all the way down. It is perfectly sensible to ask whether the patterns associated with certain raw stimuli are sufficient to determine the patterns associated with something constructed out of them. And the answer to this question should be "yes" -- if, as proposed in Chapter Three, the perceptualhierarchy does indeed operate on the basis of pattern recognition.

    Similar arguments apply to motor control. Motions such as the wave of an arm, the kick of a leg, the fast walk, the jog, the shoulder shrug, the sigh -- none of these are indivisible units; all of them are formed by the systematic assemblage of more basic muscle movements. An excellent description of this process of systematic assemblage was given by Charles S. Peirce (1966):

    [M]ost persons have a difficulty in moving the two hands simultaneously and in opposite directions through two parallel circles nearly in the medial plane of the body. To learn to do this, it is necessary to attend, first, to the different actions in different parts of the motion, when suddenly a general conception of the action springs up and it becomes perfectly easy. We think the motion we are trying to do involves this action, and this and this. Then the general idea comes which unites all these actions, and thereupon the desire to perform the motion calls up the general idea. The same mental process is many times employed whenever we are learning to speak a language or are acquiring any kind of skills.

    As Peirce points out, learning a motion is a process much like learning a word or a grammatical form, or learning how to add, or learning to recognize a chair as a chair regardless of lighting and orientation. One combines different things into one -- learns to perceive them as one -- because they all serve a common purpose.

    But what Peirce does not point out is the systematicity of all these processes. There are certain tricks to learning complex motions, which may not be easy to formulate in words, but which everyone knows intuitively. Some people know more of these tricks than others, but almost all adults have the body of tricks down better than little children. When learning to throw something new -- say a football, or a frisbee, or a javelin -- one operates by putting together various accustomed motions. One combines familiar elementary motions in various ways, based on past experience of combining motions in similar situations, and then experiments with the results. What makes the process linguistic is the application of different combinatory rules in different situations, and the automatic,systematic assignment of meanings to the different combinations.

    So, in summary, I suggest that perceptual and motor systems are STS's and, more specifically, languages in the sense described above. Nietszche's perception of a similarity between sensorimotor processes and written/spoken language was right on target. This idea may be fleshed out by reference to the modern empirical literature on perception and control, but that is a major task which would take us too far afield.

5.5.2. The Language of Social Behavior

    What does it mean to say that a behavioral system is a language? Instead of "words," the fundamental units here are specific behaviors, specific acts. One communicates with acts: one acts in certain ways in order to cause certain infons to obtain in the minds of others or in physical reality.

    The system of behaviors used by a human being is inclusive of the system of speech acts used by that person, as well as of gestures, tones of voice and "body language." But it also includes less overtly communicative acts, such as walking out of a room, taking a job, getting married, cooking dinner, changing the TV channel, etc. This system is in fact so large that one might doubt whether it is really a cohesive system, in either the syntactic or semantic senses.

    But it is clear that we build up complex acts out of simpler ones; this is so obvious that it hardly requires comment. And there are of course rules for doing so. Thus there is a syntactic system of some sort to be found. The only question, then, is if this syntactic system coordinates with a semantic system in the proper way.

    I claim that it does. First of all, the structural definition of meaning is perfectly suitable for characterizing the meaning of an act. The meaning of an act is those regularities that are associated with it. In this context, it is not too hard to see that compositionality holds. For instance, the meaning of a woman kissing her husband, quitting her job, and writing a surrealist poem about her cat is approximately predictable from the meaning of a woman kissing her husband, the meaning of a woman quitting her job, and the meaning of a woman writing a surrealist poem about her cat. Or, less colorfully, the meaning of tapping one's feet while wrinkling one's brow during a lecture isapproximately predictable from the meaning of foot-tapping during a lecture, and the meaning of brow-wrinkling during a lecture.

    Compositionality is fairly simple to understand here, since the "syntactic" combination of acts tends to directly involve the component acts, or at least recognizable portions thereof. For instance, the meaning of carrying a gun on a New York City bus is easily predictable from, 1) the meaning of carrying a gun and, 2) the meaning of being on a New York City bus.

    Compositionality is not always the most useful way to compute meanings. For instance, the meaning of carrying a gun on an airplane is not so easily predictable from, 1) the meaning of carrying a gun and, 2) the meaning of being on an airplane. Carrying a gun on an airplane is highly correlated with hijacking; this is an added meaning that is not generally associated with the function F(x,y) = carrying x on y.

    Even in this example, some degree of compositionality may be present. The airport security check is part of the meaning of being on an airplane, so for a frequent airline passenger it may be part of the meaning-function G associated with F(x,y). But the degree Mbeing on airplane(security check) is fairly small, thus making the compositionality weak at best.

    The syntactic rules governing the formation of appropriate acts for different situations are extremely complex. It is not clear whether they are as complex as the syntactic rules of written and spoken language -- we know that the latter rules have been charted more thoroughly, and indeed are easier to chart, but that does not tell us much. Just as the rules of spoken language tell us how we should form verbal expressions in order to get across desired meanings, so do the rules of behavior tell us how we should form complex behaviors out of simpler components in order to get across desired meanings.

    The work of Erving Goffmann (1959, 1961), perhaps more than that of any other single investigator, went a long way toward elucidating the manner in which simple acts are built up into complex socio-cultural systems. In The Presentation of Self in Everyday Life, Goffman understood social interaction according to the dramaturgical metaphor. Each person, in each situation, has a certain impression which she wants to put across. She puts together a "performance" -- a complex combination of simple acts -- that she judges will best transmit this impression. One might say that the performance is the analogue in behavior of the"conversation" or "discourse" in speech -- it is the large-scale construction toward which basic units, and smaller combinations thereof, are combined. Goffman's ideas are particularly appropriate here because of their focus on situations. This is not the place to review them in detail, however -- the only point I want to make here is merely that performances are very complex.     We tend not to notice this complexity precisely because performances are so routine to us. But try to explain to a person of working-class background how to make a good impression at an interview for a white-collar job. Or try to explain to a person of upper-class background how to hang out in a ghetto bar for six hours without attracting attention. Experiments of this nature show us how much we take for granted, how complex and interconnected are the arrangements of simple acts that we use in our daily lives.

    Since the time of Goffmann's early work, a great number of social psychologists have investigated such phenomena, with often fascinating results. Callero (1991), thinking of the incredible complexity of social roles which this work has revealed, states that "a literal translation of a role into specific behavioral requirements for specific actors in specific situations is simply not possible." But I think this statement must be tempered. What is possible, what must be possible, is the expression of a role as a network of processes which implicitly computes a fairly continuous function mapping certain situations into certain fuzzy sets of behaviors. The network tells what a person playing that role is allowed to do in a given situation. But the mathematical function implicit in this network is far too complex to be displayed as a "list" of any manageable size. In this interpretation, Callero's statement is correct and insightful.