Meaning Relations among Predicates

Meaning Relations among Predicates

Bas C. van Fraassen

Noûs, Vol. 1, No. 2. (May, 1967), pp. 161-179.

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Meaning Relations among Predicates'

BAS C . VAN FRAASSEN YALE UNIVERSITY

The point of departure of this paper is a traditional distinction between two kinds of meaning relations, which we shall call relations of content and relations of intent. The subject of relations of intent has so far remained relatively unexplored. We intend to argue briefly that this subject forms a legitimate and in all likeli- hood fruitful field of semantic inquiry, to explore the means whereby relations of intent may be formally represented, and to describe and investigate a class of artificial languages (semi-interpreted languages) in which statements concerning relations of intent may be formulated.

1. RELATIONS AMONG PREDICATES

Traditionally, logicians have distinguished between two sorts of relations among predicates: relations of extension and relations of meaning. Extension has proved to be a relatively unproblematic subject; for further reference, we list here the three main relations of extension which may obtain between two predicates F and G:

a. F extensionally oovrlaps G if and only if there is something of which both F and G are true.

b. F is extensionally included in G if and only if all things of which F is true are such that G is also true of them.

c. F is extensionally identical with G if and only if each is extensionally included in the other.

l The theory of meaning relations presented here was developed in connection with a formalization of event language, as part of my Ph.D. dissertation (University of Pittsburgh, 1966) which was written under the direc- tion of Dr. Adolf Griinbaum. I should also like to acknowledge my debt to Dr. N. D. Belnap Jr., University of Pittsburgh, and Dr. Karel Lambert, West Virginia University.

These are the relations which are generally represented by means of Venn diagrams. This pictorial representation is no longer feasible when we also consider relational predicates, and demand that all possible extensional relations among them be taken into account. But contemporary quantification theory is demonstrably adequate for this.

The subject of meaning relations is not nearly so transparent or unproblematic even today. At least some of the problems in this area have been occasioned by a failure to maintain the distinction between two kinds of meaning relations, both of which appear in the traditional accounts. I shall call them relations of content (or: contensive relations) and relations of intent (or: intensive relations). I shall begin by explaining this distinction.

The content of a predicate is the set of predicates which are, by the definition of F, true of a thing of which F is true. Thus G belongs to the content of F ( G is contensively included in F) if the conventional definiens of F is a conjunction of G and some other predicates. My content is what Keynes called the conventional in- t e n ~ i o n . ~It also seems to fit fairly well what Bradley, Bosanquet, and Lewis meant by intension, Kant by Inhalt and the Port-Royal Logic by comprehension.3

To explain the notion of intent, it may be easier to begin with a listing of traditional characterizations. The intent of a predicate has been characterized as:

"the whole range of objects real or imaginary to which the name can be correctly applied, the only limitation being that of logical con~eivability."~ "the classil3cation of all possible or consistently thinkable things to which the term would be correctly appli~able."~ "the range, which comprehends those possible kinds of ob-jects for which the predicate hold^."^

J. N. Keynes, Studies and Exercises in Formal Logic (London, 1894), p. 24.

"F. H. Bradley, The Principles of Logic (London, 1922) 1, 168; B. Bosanquet, Logic (Oxford, 1911), 1, 44; I. Kant, Sammtliche Werke, ed. G. Hartenstein (Leipzig, 1868), Bd. 8, p. 93; C. I. Lewis, "The modes of meaning," in Linsky (ed.) Semantics and the Philosophy of Language (Urbana, 1952), p. 238; Arnauld and Nicole, The Port-Royal Logic, trans. T. S. Baynes (Edinburgh, 1850), p. 49.

*Keynes, p. 30 Lewis, p. 238 'R. Carnap, Maaning and Necessity, 2nd ed. (Chicago, 1958), p. 239

MEANING RELATIONS AMONG PREDICATES 163

These authors did not use the term "intent"; Keynes uses "subjec- tive extension," Lewis "comprehension" (an unfortunate usage, given the Port-Royal tradition), and Carnap "intension." Bradley and Bosanquet assimilated intent to extension (or vice versa); pre- vious writers seem not to have distinguished these two notions.

Of the two notions content and intent, the former is the clearer; appeal to conventional definitions is not problematic in the way in which appeal to possible entities is.? But it would not do to limit a discussion of meaning relations to contensive relations. For this would amount to relinquishing any inquiry into meaning relations among predicates not definable in terms of each other. Thus each of the following sentences would seem to be true (speaking naively) in virtue of the meanings of the relevant predicates:

( i ) Whatever is red, is coloured. (ii) Whatever is red (all over), is not green (all over). (iii) Nothing is warmer than itself.

But an appeal to conventional definitions will not establish the truth of these propositions.s For example, to be coloured is not to be red-and-something else. Nor is being red the same as being coloured-and-something else (unless this something else is being red itself). So ( i ) is not true in virtue of contensive meaning relations between "is red" and "is coloured." For this reason it is not fruitful to ban the notion of intensive meaning relation from the class of acceptable semantic concepts. The remainder of this paper is devoted to an explication of this notion.

2. INTENSIVE MEANING RELATIONS

There is, in my opinion, no strong reason to attempt to expli- cate the notion of the intent of a predicate. Since this notion only makes its appearance in connection with the description of a certain kind of meaning relations, it is these relations themselves to which we must address ourselves. The remarks quoted above from Keynes, Lewis, and Carnap, suggest that the following statements were regarded as definitions:

d. F intensively overlaps G if and only if G is true of some possible object of which F is true

'For a critical account of the notion of 'possible entities', cf. Quine, Methods of Logic (New York, 1959), pp. 201-202.

For supporting arguments, see J. G. Kemeny, "Analyticity vs. Fuzzi-ness", Synthese, XV ( 1963), pp. 57-80.

e. F is intensively included in G if and only if G is true of any possible entity of which F is true

For us, these do not qualify as definitions, because discourse about possible entities is just as much in need of explication as discourse about meaning relations.

But although d. and e. cannot be taken as definitions, they are not without value. For they furnish us with two important criteria of adequacy for our explication. Traditional logic texts do en- gage in discourse about possible entities, and this discourse is subject to certain principles. The most important of these is that what is actual, is possible. From this it is concluded that if something is the case about some actual entities, then it must a fortiori be the case about some possible entities. This has an immediate conse-quence for the relation of intensive overlap:

Criterion I. If F extensionally overlaps G, then they also overlap intensively.

A corollary is this: if F is intensively included in G, then the extension of F must be part of the extension of G. In general, the intensive relations 'force' the extensive relations of inclusion, while the extensive relations 'force' the intensive relations of o ~ e r l a p . ~

Secondly, it is very important to note that the phrases "some possible entity" and "all possible entities" have the form "some A" and "all A". These expressions have a certain logic, exhaustively studied by traditional logic, and faithfully observed also in discourse about possibles. This suggests that quantification theoly which makes possible an adequate representation of all extensional relations, is formally also a candidate for the representation of all intensive relations. I propose:

Criterion II . There is a complete formal analogy between the structure of relations of intent and the structure of relations of exten-sion.

This principle has an interesting history. At the turn of the century there was considerable bias against any consideration of intension in logic. Frege and Russell were exceptions of course; Peano and his student Padoa explicitly adopted the extensional point of view. So did Venn, who credits much of de Morgan's and Boole's success

'Cf. C. I. Lewis & C. H. Langford, Symbolic Logic (Dover, New York 19559) p. 49.


to the disregard of intension. Both Venn and Schroder included substantial arguments in their treatises on logic for preferring the extensional to the intensional point of view.lO Their arguments, like those of similarly inclined logicians in our own day, consist largely in listing the multifarious diificulties which according to them would be introduced by taking intension into account. More important here seems to me to be the early logical work of E. Husserl, in which he argued that the standard logic of class relations can also be interpreted as a theory of relations of intent. Venn was of the opinion that Husserl's theory is not truly a logic of intension, but this appears to be because Venn thought exclusively of content while Husserl thought of intent.ll

Although Husserl's attempt no longer has much technical in- terest, it appears to be the clearest early formulation of the principle that if a relation of intent obtains, then the analogous relation of extension also obtains. For this reason I propose that this be called Husserl's thesis. This thesis is explicitly accepted by Lewis and Langford.12 They point out that the converse must of course be rejected: if a given relation of extension obtains, it does not follow that the analogous relation of intent also holds. From this they conclude that an adequate logic would be one which deals with both kinds of relation. Such a logic would need to represent the relations of intent and those of extension in different ways.

The Boole-Schrijder algebra is inadequate to such complete representation of the logic of terms, though it is completely applicable to the relations of extension, and likewise com-pletely applicable to the relations of intension.13

We turn now to the task of providing an acceptable explication of intensive meaning relations which will satisfy our two criteria of adequacy.

3. INTERPRETATION OF INTENSIVE RELATIONS

When a concept is to be given an interpretation, the question necessarily arises: interpretation in terms of what? There would

A. Padoa, La Logique DBductive (Paris 1912) pp. 24-25, and J. Venn, Symbolic Logic (London 1894, Ch. 19) , both provide interesting historical dis- cussions of this matter.

Venn, loc. cit.; M. Farber, Foundations of Phenomnology (New York 1962) pp. 76-79.

Op. cit. p. 49 l3 Op. cit. p. 69. That "intension" refers here to what we call "intent"

may reasonably be inferred from the penultimate paragraph of p. 66.

seem to be no one answer to all questions of this form, no pana- cea for bestowing meaningfulness and philosophical respectability on a new theoretical term. There are of course slogans: we may ask for operational procedures, verifiability, reduction to the familiar. But such slogans give little guidance in a specific case. For example, if statement a. in section 1. reduces extensional overlap to the familiar, does statement d. in section 2. do so for intensive overlap? I t has the same pattern, and uses no new, artificial or indeed unfamiliar terminology. To the contrary, only after reflec- tion on what d. might mean cuts through its guise of familiarity do we feel doubts that it has any literal significance at all.

Similarily, we might have phrased a. as follows:

a". F extensionally overlaps G if the set which is the extension of F overlaps the set which is the extension of G.

We might then similarly rephrase d. in terms of properties, inten- sions or attributes. And we might argue, for example, that property is a more familiar concept than set, the latter being of relatively recent (nineteenth century) vintage. And that sets are abstract entities not further identifiable; true, they satisfy the axioms of at least one of the proposed set theories (we hope), but we can also state axioms satisfied by properties. Quine has advanced serious doubts regarding the existence of properties, but Russell had no less embarrassing puzzles regarding sets. Most of us know how to play the 'set' language game, but neither do we have much difficulty playing the 'property' language game.

This line of argument seems to me to be equally inappropri- ate. Not that I wish to deny the above assertion that the status of sets is not much clearer than that of properties-rather, I doubt entirely the relevance of this approach. Sets are not needed to interpret relations of extension (aQ is not needed, given a,) , and neither do I think that properties are needed to interpret relations of intent.

The concept of synonymy, understood either as identity of content or identity of intent is a concept belonging to semantics, just as are, for example, the concepts of denoting and identity of denotation. When we ask for an interpretation of a semantic concept, we may merely be requesting a definition in terms of other semantic concepts. In this sense, the question will not apply to the simpler, more basic concept of the discipline. We cannot take ex- ception to the claim of semantics to have its own technical con-


cepts. But this does not rule out the continued demand for an interpretation of these concepts: in that case, however, answers which go outside semantics proper must be acceptable. Such a demand for interpretation can only be construed as a request that we show the concepts in question to have a signscant role.

There is in fact an obvious field which may provide such a grounding for semantic concepts, namely, pragmatics. And I pro-pose accordingly that we construe the demand for an interpretation of semantic concepts as answerable by the exhibition of a clear pragmatic counterpart. For example, the semantic statement "The tenn W denotes the object 0 has as pragmatic counterpart "The person X uses the term W to refer to the object 0 (at time t)." The exhibition of a pragmatic counterpart does not define the semantic concept. It is perfectly legitimate to use the semantic concept of denotation, even if no term is used to refer to the same thing by all persons or even by any person at all times. The use of the semantic concept signals a certain level of abstraction, which involves disregarding the vagaries of contextual factors, idiosyn- cratic usage, changes in usage with time, and so But had the concept of denotation no clear pragmatic counterpart, its use would have no clear relevance to the study of language at all.

So now our task is to provide similar pragmatic counterparts to relations of intent. For brevity, we shall look only at identity of intent, which we regard as one plausible explication of synonymy. The pragmatic counterpart to "F is intensively identical with G" is "I? and G are synonymous for X (at time t)." But what is meant by the latter? Notwithstanding the well-known puzzles and doubts which have been raised regarding synonymy, it seems to me that the literature contains sufficient grounds for regarding "I? and G are synonymous for X" as in no way more problematic than "X uses W to refer to 0.We may briefly recount the relevant points here.

An early, but very thorough discussion, is Carnap's "Meaning and Synonymy in Natural Languages".16 In particular he holds that proper interrogation of the person X will provide factual evidence for or against the assertion that two terms are synonymous for X. Thus he describes how a linguist might determine the intention of "Pferd" for a German Karl.

14Cf . Church's review o f Nielsen's disparagement of such abstraction, T h e Journal o f Symbolic Logic X X V I I (1962), p. 118.

"Phil . Studies, VI (1955), pp. 33-47. Reprinted as Appendix D. o f Meaning and Necessity.

The linguist could simply describe for Karl cases, which he knows to be possible, and leave it open whether there is anything satisfying those descriptions or not. He may, for example, describe a unicorn (in German) by something correspond- ing to the English formulation "a thing similar to a horse, but having only one horn in the middle of the forehead". Or he may point toward the thing and then describe the intended modification in words . . . or, finally, he might just point to a picture representing a unicorn. Then he asks Karl if he is will- ing to apply 'Pfercl' to a thing of this kind.l6

Carnap describes this procedure as that of a linguist who is deter- mining the 'intension' of "Pferd" for Karl at that time-'intension' in the sense of "range, which comprehends those possible kinds of objects for which the predicate holds." Thus he is clearly referring to what we call intent. In a note on this paper, Chisholm argued that Carnap's account is an oversimplification, and Carnap agrees with this.l7

We may object more specifically both to the procedure of the linguist described (e.g. what experimental safeguards is he using with respect to this sort of questioning?) and to the interpretation in terms of a 'range of possibles' which Carnap puts on the results of this procedure. Neither objection would apply, as far as I know, to Arne Naess' study Interpretation and Preciseness, which satisfies the standards of empirical investigation in psychology and sociology, and does not state its conclusion in Carnap's objec- tionable terms.ls The hypotheses tested by Naess are of the form "A and B are synonymous for X at t", and the testing is by means of questionnaires. Questions are of the form "Can you imagine circumstances in which you would use A but not B?"

To have empirical procedures for the confirmation of asser-tions of synonymy-for-X, is perhaps already enough to welcome these assertions as legitimate. But for the philosopher the questions remains what exactly these procedures establish. Does the person for whom A and B are synonymous rely on an insight into the essences of things? Or is he the unwitting slave of definitional con-

laMeaning and Necessity, p. 238 R. Chisholm, "A note on Carnap's meaning analysis", Phil. Studies, VI

(1955), pp. 87-89; R. Carnap, "On some concepts of pragmatics", Phil. Studies VI (1955), pp. 89-91, reprinted as Appendix E. of Meaning and Necessity.

Is A. Naess, Interpretation and Preciseness: A Contribution to the theory of communication ( Oslo, 1953).


ventions established by the social contract of an earlier generation? With respect to this kind of question, the correct answer appears to be that of G. Maxwell and W. S e l l a r ~ . ~ ~ Maxwell points out that, for example, Quine's article "Two Dogmas of Empiricism" is pred- icated on the assumption that statement of the form '- means -' (for John, for the seventeenth century English, . . .) are statements "ostensibly confirmable or disconfirmable by data concern-ing actual linguistic practices". But, he argues, this ignores the pre- scriptive force of such statements. We may spell out the implied analysis in some detail.

It is to be noted that a questionnaire on linguistic usage might ask two kinds of questions. It might ask the subject about his actual linguistic behaviour so far, eg.:

( i ) Have you ever ( to the best of your knowledge) come across anything which you call an A but olf which you said that it was not a B?

It would be very surprising to find such an entry on any questionnaire concerning synonymy. One expects the second kind of question: questions dealing with linguistic commitment:

(ii) Would you call something an A ifyou knew it to be a B?

Faced with this kind of question, the subject may express a long- standing commitment ("Of course not!"), or a new intention ("I have in the past, but now I think that is wrong."). Or he may have to make a decision, because prior to the questionnaire, the question may never have arisen. The reasons for this commitment, intention, or decision can be of various sorts. It may be that he believes the usage in question to be "incorrect", "not standard, "not the Queen's English, and that he wishes to conform to norms which (he believes to) exist in his society. Or he may be concerned with the emotive effect on others. Or again his grounds may be largely inductive: he may adhere to a theory so strongly that he wishes to replace at least part of his prescientific framework with this theory.

The difference between questions ( i ) and (ii) is that the former inquires about factual belief, and the latter about commitment or intention. It is certainly not a difference of retrodiction

Is G. Maxwell, "The Necessary and the Contingent", ~Minncsota Studies in the Philosophy of Science, III (1962), pp. 398404; W. Sellars, Science, Perception and Reality (New York 1963), Chs. 10, 11.

and prediction, as my examples might suggest at first sight. I would, for example, answer no to the question whether I shall ever say of something which I know to have a heart, that it does not have kidneys-at least if the answer is meant to be a prediction. On the other hand, I can easily envisage circumstances that would make this prediction false, for the reason for my answer is not a linguistic commitment. The difference is approximately that between the repentant but pessimistic drunk who says "I don't think that I will stay on the wagon", and the unrepentant, optimistic drunk who utters those same words, in a slightly different tone of voice.

Having established that relations of intent do have clear pragmatic counterparts, we may now use them in semantics. But now it may well be thought that this is a hollow victory. For these semantic concepts, as opposed to those of denotation and extension, have no application in the semantics of logic and mathematics. The artificial languages constructed in formal semantics, and which suffice for the semantic study of mathematics and logic (in- cluding intuitionist and modal logics) are adequately discussed in terms of denotation and extension. Nor does there appear to be any need here for any other kind of language.

We have no intention of disputing this. But logic and mathematics are not the only subject of philosophical inquiry. In specific, many attempts have been made to apply formal methods to the empirical sciences. And there we find just the sort of state-ments whose truth or falsity derives from relations of intent: wit- ness our example "Nothing is warmer than itself."20 Recalling Car- nap, the reader may dismiss these examples as follows: such statements are meaning postulates, and would appear among the axioms of a complete formalization. I rather doubt that this analysis provides us with a very faithful model of scientific the0ries.2~ In particular, it is striking to note that the best way to distinguish 'meaning postulates' from 'ordinary' postulates is by the fact that the former are the ones which the physicists do not state in their mono- graphs and texts. Rather than speculate any further on possible applications, however, we shall now turn to the task of providing intensive meaning relations with a formal representation.

2 0 Cf. Maxwell and Sellars, op. cit. I have attempted to provide an alternative, using the machinery of

sections 4 and 5 below, in my forthcoming "On the Extension of Beth's Semantics of Physical Theories".


4.REPRESENTATION OF INTENSIVE RELATIONS

Relations of extension are pictorially represented by means of Venn diagrams. But in what, exactly, does the use of a Venn diagram consist? It consists in the choice of a set S (the set of points inside the square) and a function f (the function which assi,gns a part f ( F ) of S to each predicate F ) . The parts, or subsets, of S, and the function f, together represent the relations of extension among the predicates, provided only:

(i) f ( F ) and f (G) overlap if and only if there is something which is both F and G;

(ii) f ( F ) is included in f ( G ) if and only if all things which are F are also G.22

According to our criterion 11, the intensive meaning relations should be capable of exactly the same kind of representation. That is, we should be able to choose some set T and function g by which we can likewise represent the intensive relations. The map- ping of the predicates by g into the subsets of T needs to satisfy:

(iii) g ( F ) and g (G) overlap if and only if F intensively overlaps G;

(iv) g ( F ) is included in g ( G ) if and only if F is intensively included in G.

There is certainly no need to speak of the members of g ( F ) as being all the possible entities of which F is true. On the contrary, we may choose T to be a set of chairs, or the set of points in a certain spatial region. In particular, we may use the ordinary Venn diagrams.

The other criterion at which we arrived specaes that the relations of intent and the relations of extension 'force' each other into a certain pattern. What this means exactly is the following: it is always possible to choose S and f such that S is part of T, and f ( F ) is part of g ( F ) for every predicate F. For this is ex-actly equivalent to: f ( F ) must be included in f (G) if g ( F ) is part of g (G) , and g ( F ) must overlap g (G) if f ( F ) overlaps f (G) . As long as we are dealing with monadic predicates, we can still easily

22 We refer to Venn diagrams in which all regions are speciKed as 'occupied' or 'not-occupied'; if this is not so, some relations of extension are left unrepresented.

draw this on paper: we get a Venn diagram within a Venn dia-gram. We can also apply the traditional terminology of 'possibles' provided we 'bracket' its metaphysical commitments. We would then say: the set S represents the existents, which form part of the possibles (represented by T ) . And the set f ( F ) represents the set of real things of which F is true, which of course is part of the set of possible things of which F is true.

When a set is used as T is above, for the representation of intensive meaning relations, I propose to call it a logical space. In my opinion, this is fully in accordance with the use of that term by Wit tgen~tein .~~We may show this by considering two examples of logical spaces: the space of locations on my desk, and the 'space' in which objects are located by virtue of their colour (colour spec-trum). In the first place my predicates might include "is at least one foot from the back of the desk" and "is no more than half a foot from the back of my desk". These predicates are intensively disjoint: no object can be such that both these predicates are true of it. (More colourfully: no possible object falls under both predi-cates.) In the second case my predicates may include "is scarlet" and "is r e d ; of these the first is intensively included in the second. (Any possible object which is scarlet, is ( a possible object which is) red.) In the first case, the set of points on the desk top can be T, and g assigns to the predicates the regions suggested by the English meaning of the expressions in question. In the second case, we represent the colour spectrum, say by a straight line segment; the set of points on this segment is T, and the parts of this line segment assigned by g to the predicates must be such that g(scar-let) is part of g(red).

When we construct an artacial language, we may introduce such a logical space if we wish to have object-language counter-parts to assertions such as "I? is intensively included in G." (In the sense that A may be the object-language counterpart of "A is valid" in a modal system.) Such a construction will therefore spec-ify three things:

28 Similar concepts are explored by R. Kauppi ("Ueber den Begriff des Merkmalraumes", in Proc. 11th International Congress of Philosophy, Amster-dam 1953, Pt. V ) and H. Weyl ("The Ghost of Modality", in Philosophical Essays in Memory of Edmund Husse~l,ed. M . Farber (Cambridge, Mass. 1940)). I conjecture that the historical origin of Wittgenstein's notion of logical space is the use of vector spaces in physics ("phase space", "configuration space").


A. The syntax of the language: its vocabulary and grammar. B. The logical space of the languages set-and an interpretation

function. C. The models of the language.

When the definition of an artificial language comprises only A. and C., as is usually the case, it is often called an 'uninterpreted language'. When it comprises B. as well, I propose we call it a semi-interpreted hnguage. In the next two sections I shall describe two kinds of semi-interpreted languages, and describe some of their properties.

5. INTENSIVE RELATIONS AND S5

We shall begin by describing a very simple case, which never- theless will show an important connection between intensive meaning relations and modal logic. I t seems most natural to classify semi-interpreted languages by their syntax. The first syntax we describe, SQ, thus gives rise to the class C(SQ) of semi-interpreted languages. The syntax SQ is as follows:

Vocabuhry: One singular term b Various monadic predicates: F, G, . . . . Logical signs: -, &, +, (,).

Grammar: The atomic well-fonned formulas (wffs) consist of a predicate followed by a singular term: Fb, Gb, . . . If A and B are d s , then so are -AJ (A & B), (A+ B)

All the languages in the class C(S") have this syntax. The logical space must in each case be some nonempty set H, and the interpretation function f must assign to each predicate F some subset f ( F ) of H. Now we t u n to the notions of model, truth, and val-idity. We define these for an arbitrary member L of C(SQ ), whose logical space and interpretation function we denote as H and f.

Models: A model for L is an ordered couple < loc,X >,where X is some entity and loc a function defined only for X, such that loc(X) E H. We call loc(X) the location of X in H.

It will be clear from the following truth-definition that the term b is used as the name of this individual X.

Truth: Every sentence A of L has a truth-set h(A) in H, as follows: h(Fa) = f ( F ) h (A & B) =h(A) A h(B) h ( -A) =H -h(A) h(A +B) = TH if h(A) h(B)

A , the null set, otherwise The sentence A is true in the model < loc,X > for L if and

only if loc(X) e h(A).

We may paraphrase this as follows: the truth-set of a sentence A(b) is the set of positions which the individual denoted by b may have in the logical space to make A(b) true.

Validity: The sentence A is valid in L if and only if it is true in every model for L. The sentence A is universally valid (in C ( S e ) ) if and only if it is valid in every language L belonging to C ( S" ) .

It is very easy to see that all the theorems of classical propositional logic (with primitives & and N ) are universally valid sentences in this class of languages. But what sort of arrow is our + 2 The answer is that it is the S5 arrow, in the following precise sense: the set of universally valid sentences of C(S") is exactly the set of theorems of the Lewis modal system S5.24

Moreover, our truth-definition for A + B is a very natural one. Each of our sentences is about the individual denoted by the term b, so we can write A + B also as A(b) +B(b) . Doing so, we see that the sets h(A) and h (B) are exactly those which are being used to represent the relations of intent between the predicates A(- -) and B(- -). Our truth-definition now amounts to:

A +B is true if and only if A(- -) is intensively included in B(- -)

or, in the older terminology,

A +B is true if and only if all possible things which are A, are also B.

a4 To prove this, it suffices to notice that a logical space, interpretation function, and model together generate a normal S5, matrix in the sense of H-N. Castaiieda, The Journal of Symbolic Logic X X I X (1964), pp. 191-192, and that conversely every such matrix gives rise to a logical space, interpretation function, and model for the syntax S".


or again, if we think of the model as specifying the actual state of affairs with respect to the individual being talked about,

A +B is true if and only if whatever is A, is B, under all possible conditions.

This discussion does not throw new light on the nature of S5, which has long been known to admit of such interpretations. But it places the relation of intensive inclusion in a familiar frame of discussion.

The languages of the class C (S" ) provide a means for codify- ing the intensive meaning relations among monadic predicates. But we clearly must also consider the question how we could represent all possible intensive meaning relations among an arbitrary set of predicates of various degrees. The problem is exactly analogous (due to criterion 11) to the problem: Venn diagrams picture extensional relations among monadic predicates, but how could we represent all possible extensional relations among an arbitrary set of predicates of various degrees? And the answer to this is of course provided by quantification theory, using quantifiers over the domain of discourse. Hence we can expect to solve our problem through the use of quantifiers ranging over logical space. And this is indeed already suggested by those who talked about relations of intent through the use of such phrases as "for all possible things".

Let us write the universal quantifier over the logical space as "(/x)", and read it "for all possible x". This reading is of course purely heuristic. The kinds of things which make up the member- ship of logical space is essentially arbitrary: they may be chairs, points, vectors, cabbages, kings, or bits of sealing wax. This quantifier, which is to provide the explicans for discourse about possi- b l e ~ has already appeared in the work of many logicians; for example, Stahl, Lejewski, Prior, Rescher, and Leonard. Of these, only Leonard has given an interpretation which is not vague and not metaphy~ical .~~Leonard's semantics for this quantifier involves the use of non-referring singular terms, which makes it technically a bit more complicated than our 0wn.~6 It is instructive to note that

26 H. S. Leonard, "Essences, Attributes, and Predicates", Proceedings of the American Philosophical Association, XXXVII (Oct. 1964), pp. 25-51.

26 This is not an objection to Leonard's ingenious approach. The main reason I prefer the use of logical spaces is an intended application: see my forthcoming paper mentioned in footnote 21.

all these logicians agree that (/x) has at least the properties of the ordinary quantifier and either state or conjecture that the properties of (/x) are exactly the properties of (x). We shall see below that this is demonstrably so for the present interpretation.

Let us take a concrete example: a language which has predicates F,G, . . . ,of various degrees, variables x,y, . . . , the quantifier (/x), and so on. The sentence (/x)(Fx>Gx) has as in-tended interpretation:

The predicate F is intensively included in the predicate G. I t is read as:

All possible x which are F, are also G.

Given that the language has a logical space H and an interpretation function f, it is true in any given model provided:

f ( F ) c f ( G )

But a model will have a domain of discourse, each member of which also has a location in the logical space H. How can we express the statement that, say, all the members of the domain of discourse belong to the extension of F? For this purpose we introduce a predicate E!, to be read as "exists". We write the statement as usual as (x) (Fx > Gx), but this is short for:

(/x) (E!x > .Fx > Gx)

So that in general (x) . . . means (/x)(E!x . . .). This state-ment has as intended meaning:

The predicate F is extensionally included in the predicate G.

I t is read as:

All actual x which are F, are also G.

And it is true in any given model provided

where f(E!) is exactly the set of elements of H which are locations of members of the domain of d i s c o ~ r s e . ~ ~

Before we make this precise, and extend it to polyadic predicates, we must discuss a possible objection. This objection has its origin in the idea (or slogan) that the quantiiier of a language is

For simplicity we assume that each member of the domain has a distinct location in the logical space.

MEANING RELATIONS AMONG P~DICATES 177

the 'carrier of its ontological commitment'. Our quantifier (/x) ranges over the logical space H; ordinary quantifiers are defined in terms of (/x) and the predicate El. But that means that we do not really have a quantifier ranging over the domain of discourse at all. Does it not follow then that we are open to the charge -leveled by Quine against Carnap - that extension has been en-tirely eliminated from our language? In a statement quoted by Car-nap, Quine asserts:

Every language system, in so far as it uses quantifiers, assumes one or another realm of entities which it talks about. The determination of this realm is not contingent upon varying metalinguistic usage of the term 'designation'. . . , since the entities are simply the values of the variables of quantifica-tion . . . . The question of what there is frosm the point of view of a given language . . . is the question of the range of values of its variables28

Thus it appears that in Quine's view, a semi-interpreted language with ( /x) can be used only to talk about the elements of its logical space.

This is patently false, since it is possible to formulate all the usual assertions about the domain of discourse in such a language.29 And this does hinge on what 'designation' means here. Let us sup- pose that the value assigned to a variable x is the element h of H. Then if h is the location of some element d of the domain of discourse, then x designates d. And otherwise, x designates nothing at all; it certainly does not designate its value 11. The sentence (/x) (E!x> Fx), which we abbreviate as (x)Fx, means that whatever can be designated by x belongs to the extension of F. Upon proper understanding of the term "designates", this is entirely com- patible with the fact that the variables range over H, and do not range over the set of designata.

Furthermore, caution must be exercised when some tradi-tional reading of the formulas is used. For example, it would be pure confusion to say that (/x)Fx means that every element of H belongs to the intent of F. Certainly, that statement is true if and only if every element of H belongs to the set f (F) . But the set

IsMeaning and Necessity, p. 196 And it is not possible to formulate any assertions about the elements

of the logical space except logical and numerical ones (such as that it contains at least one element, that it contains at least two elements, and so on).

f ( F ) is only a set being used to represent the intensive meaning relations between F and other predicates.

We can now turn to a precise description of this kind of semi- interpreted language; then we shall be able to specify exactly the sense in which ( /x) has exactly the same formal properties as the standard quantifier (x). The class of languages in question is that which arises from the syntax S; so we shall call this class C(S) . The syntax S is as follows:

Vocabulary: Variables: x, y, z, . . . xl, x2, . . . . Predicates of various degrees: Fn, Gn, . . . Logical signs -, &, ( /X ), ( ,), =,E!

Grammar: The atomic wffs are of the form Fnxl . . . x,, E!x, and X1 =X2

If A, B are wffs then so are (-A), (A & B), (/x)A Statements are wffs in which no variable occurs free.

The logical space H ( of an arbitrary member L of C ( S) ) we again specify to be an arbitrary non-empty set. The interpretation function f is defined for every predicate, and if Fn is an n-ary predicate, then f ( F n ) is a set of n-tuples of elements of H.30

A model for L is a couple < loc,D>, of which D is any set, possibly empty, and loc is a function with domain D and range H. For simplicity's sake, we specify lac to be a one-one function, so that it assigns distinct locations in H to distinct members of the domain.

To define truth in a model, we must refer to assignment func- tions: an assignment function d is defined for every variable x and assigns it an element d (x) of H. We write "df =.d" for "df is like d everywhere except perhaps at x". Then we define "The wff A of L is true in the model M =<loc,D> relative to the assignment d , briefly "MT (d, A)", as follows:

MT (d, E!x) iff d (x ) = loc ( k ) for some element k of D; MT (d, XI=~ 2 )iff d(x1) =d(x2); MT (d, Fnxl . . . xu) iff [d(xl), . . . , d(xn)] is a member of

f (Fn) ; MT (d, -A) iff not MT (d, A); MT (d, (A & B ) ) 8 MT (d, A) and MT (d, B) ; MT (d, ( /x)A) iff MT (df, A) for every d l= .d

Note that we have listed El among the logical signs instead of with the predicates; this simplifies the presentation somewhat.


If A is a statement, then we may say that A is true in M pro-vided MT(d,A) for every d; otherwise A is false in M. Further- more, A is a valid statement of L provided it is true in every model for L, and A is a universally valid statement (of C ( S ) ) if and only if it is a valid statement of every language L in C(S) .

The reader will have no difficulty seeing that the usual quantification and identity theory remains valid for any member L of C(S) . Moreover, we have the following completeness theorem:

The universally valid statements of C (S) are exactly the theorems of standard quantification-cum-identity theory.S1

This is the precise form of the assertion that ( /x) has the same formal properties as (x). Note that any given member L of C(S) will in general have valid statements which are not quantifica-tional theorems. These residual valid statements reflect the meaning relations among the predicates represented by means of H and f. They are essentially what Carnap called the "meaning postulates". This term is misleading here, however, because the meaning structure may be so complex that no formal postulate system can capture all these residual valid statements (the theory of the subject may be essentially incomplete, by Godel's theorem). In this sense our method of logical spaces is essentially stronger than Carnap's method of meaning postulates.

"For proof, see section II.C.5 of my dissertation (footnote 1). The reasoning is essentially similar to that which we applied to C(S*) and S5 above.

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Meaning Relations among PredicatesBas C. van FraassenNoûs, Vol. 1, No. 2. (May, 1967), pp. 161-179.Stable URL:


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14 Review: [Untitled]Reviewed Work(s):

Language as Existent by Harry A. NielsenAlonzo ChurchThe Journal of Symbolic Logic, Vol. 27, No. 1. (Mar., 1962), p. 118.Stable URL:

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