Let me begin by distinguishing three kinds of expressions and noting ways in which they have been confused. Later I shall make some moves that tend rather to merge them again, though without confusion.
The three kinds are predicates, general terms, and class names. The predicate may be pictured as a sentence with gaps left in it where a singular term could be inserted to com plete the sentence. This is what C. S. Peirce called a rheme.
I am thinking at first only of one-place predicates, but the gaps may still be many, corresponding to recurrences of some one singular term. Or, with Frege, we may regard the predicate not as composed of signs at all, but rather as a
way of forming a sentence around a singular term. This also was Wittgenstein's conception.
The general term, on the other hand, is a sign or a con tinuous string of signs. It may be a verb or verb phrase, a noun or noun phrase, an adj ective or adjective phrase ; these
I read this paper in South Africa in October 1980 and at Boston University in December. Early portions hint somewhat of "Clauses and classes," which I presented to the Societe Frarn;aise de Logique in 1978. That paper was subsequently circulated in the bulletin of the society. The last part of the present paper is adapted, by permission of the American Mathematical Society, from "Predicate functors re visited," which is to appear in the Journal of Symbolic Logic copyright © by the Association of Symbolic Logic.
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Predicates, Terms, and Classes 165
distinctions are immaterial to logic. If we think of a predicate as a sentence with gaps, then a general term is that special sort of predicate where the gap comes at one end. I am still limiting my attention to the monadic.
Finally a class name is not a general term, anyway not until further notice, and not a predicate. It is a singular term, simple or complex, designating a single abstract ob ject, a class. The corresponding general term denotes any number of objects, each member of the class.
The schematic letter 'F' in the 'Fx' of symbolic logic is quite properly called a predicate letter, for 'Fx' stands for any open sentence in 'x', however numerous and scattered the occurrences of 'x' in it may be. On the other hand, the letters 'S', 'M', and 'P', as used in schematizing syllogisms in traditional textbooks, are schematic letters for general terms. The place holders for class names, finally, are genuine bindable variables whose values are classes. They may be general variables such as 'x' and 'y', or they may be distinc
tive ones such as 'a' and '(3' in Principia Mathematica.
A major part of the traditional exercises in syllogisms con sisted in preparing each sentence by recasting it in one of the four categorical forms 'All S are P', 'No S are P', 'Some S are
P', and 'Some S are not P', known as A, E, I, and 0 : 'A' for 'all', we might say, 'E' for 'exclusion', 'I' for 'intersection', and 'O' for 'overflow'. The job of recasting can be visualized in two stages. First the given sentence had to be paraphrased in such a way as to say explicitly that everything or some thing satisfying such and such a condition, 'Fx' let us say, sat isfies also or fails to satisfy such and such a further condition, say 'Gx'. The remaining step consisted in effect in maneuver ing the 'Fx' and 'Gx' into the forms 'x is an S' and 'x is a P',
with nicely segregated general terms in place of 'S' and 'P'.
This step consisted, thus, in devising general terms coex tensive with given predicates. It was the easier step of the two, for there is a uniform grammatical construction to the purpose : the relative clause. We can immediately convert any sentence about an obj ect a into the form 'a is something which' followed by a contorted rendering of the original sen tence with pronouns where 'a' had been.
166 Theories and Things
in the 'such that' idiom, which is simpler grammatically : 'a
is an x such that Fx'. The bound variable prevents ambiguity of cross-reference where clauses are nested. The relative clause, whether in the 'such that' form or the 'which' form, is not a class name but merely a complex general term.
Throughout the history of modern logic there has been a tendency to confuse the general term with the abstract sin gular. The schematic letters for general terms, in syllogistic logic and elsewhere, were thus commonly viewed as class variables, and the 'such that' clauses were viewed as class names. At the same time there was a laudable reluctance to obj ectify classes at the elementary level of logic ; one saw the wisdom of relegating them to where they were really needed, in ulterior parts of mathematics and elsewhere. In the neo-classical logic, consequently, schematic letters for general terms were avoided, and so was the relative clause, the 'such that' construction. Logicians made do with predi cate letters, always with variables or singular constants ap pended as arguments. Such is the standard schematism of quantification theory or the predicate calculus, and it has regularly been adhered to even in the monadic case, where Boolean algebra can cover the same ground more simply and graphically. It was not appreciated that the letters of Boolean algebra can be received innocently as standing schematically for general terms, and that their Boolean com
pounds can be seen as standing not for class names but for compound general terms.
Scruples against premature reification of classes or prop erties were probably what led Frege to stress the ungesattigt
character of predicates, or what he called functions : they need to be filled in with arguments. I applaud the scruples and I agree about the predicates. What I deplore is his failure to see that general terms can be schematized without reify ing classes or properties. This failure was due to the dim ness, back then, of the distinction between schematic let ters and quantifiable variables. Notice that even his predi cate letters creep hesitantly into quantifiers on occasion. He was still feeling his way.
Once we appreciate the ontological innocence of the 'such that' idiom, we can admit it with equanimity to the language
Predicates, Terms, and Classes 167
of elementary logic. Now what about a compact notation for it ? One might think we ought to keep conspicuously clear of the set-theoretic notation '{x : Fx }', in view of the mel ancholy history of confusion between general terms and class names. However, I shall now propose quite the con trary : that we write ' { x : Fx }' for our innocent relative clause, and 'e' correspondingly for the innocent copula 'is an'
that is the inverse of 'such that', and that we then simply deny that we are referring to classes. I find that this course is suited to a philosophically attractive line on classes.
Here is a myth of genesis of the notion of classes. It need not be true, though it seems fairly plausible. In the beginning there were general terms, including relative clauses, for which I am boldly proposing the notation '{x : Fx}'. Prodded then by certain analogies, on which I have speculated else where, 1 between general terms and singular terms, people
began to let the general terms do double duty as singular terms. Thus they posited a single abstract object for each general terms to designate. They called it a property, but we may slim properties down to classes for the well-known benefits of extensionality.
Quantifying over classes began thus in a confusion of gen eral with singular, but it proved to be a happy accident, en riching science in vastly important ways that I shall not pause over.2 And then, in the fullness of time, people whom we can name found that not every general term could have its class, on pain of paradox. The relative or 'such that' clauses, written as term abstracts '{x : Fx} ', could continue without restriction in the capacity of general terms, but some of them could not be allowed to double as class names, while the rest of them still could. Where to draw the line is the question what set theory to adopt. Wherever drawn, the distinction is easily expressed :
(1) ( 3 y) (y = {x : Fx} )
tells us that there is such a class. Mathematics transcends logic at just this point.
1. Roots of Reference, pp. 84-88, 97-106. 2. See E ssay 1 above, §II.
168 Theories and Things
What we have here is the theory of virtual and real classes as of my Set Theory and Its Logic, but seen no longer in terms of virtual classes as simulated classes. The abstracts are seen now as unpretentious relative clauses, some of which may also be class names and some not.
Logic, then, in the narrow sense represented by quantifi cation theory, can make free with the abstraction notation ' {x : Fx} ', but with no thought of substituting such an ab stract for a variable in instantiating a quantification. Such a move would require a premise of the form ( 1 ) , which would belong to a higher level of mathematics, namely, set theory.
Recoiling from that higher level, let us see how neat an elementary logic can be based on '{x : Fx} ' in the innocent sense of term abstraction. This will be the only variable binding operator. In addition I shall assume a copula ; not 'e'
,
however, not the singular 'is an', but one of the categorical copulas, as our syllogizing forefathers called them. I shall adopt the universal negative copula, the E of the mediaevals, exclusion. 'S excl P' will mean that the S's exclude the P's ; no S are P. We must bear in mind that 'excl' is not a singular verb 'excludes', joining two class names, but a copula, 'no are', or 'exclude'. The plural verb 'exclude' is indeed a copula, from a logical point of view, and is equivalent to 'no are', whereas the singular verb 'excludes' is a dyadic general term predicable of pairs of classes. When the further step is made of positing classes as designata of the general terms, the distinction lapses.
Term abstraction and the categorical copula 'excl' suffice for expressing the truth functions and quantification. We can define as follows.
'p
I
q' ,_ p' ' 3 {x : Fx ) ' for '{x : p} excl {x : q}', for 'p1
p', for ' - ( {x : Fx} excl {x : Fx} ) '.As usual, 'p' and 'q' are schematic letters for sentences lack ing any free variables relevant to the context. The first defi nition thus exploits vacuous abstraction and delivers the truth function 'not both', Sheffer's stroke function. The last
Predicates, Terms, and Classes 169
definition gives existential quantification, ' ( 3 x) Fx'. Every one knows how to proceed from these acquisitions to the rest of the truth functions and universal quantification. So we see that the needs of the predicate calculus are met by just the relative clause and the exclusion copula, without use of the singular copula 'e', 'is an'. The embedded predicate let
ters 'F', 'G', and so on, do remain in the schematism, with variables always attached, just as in the standard schema tism of the predicate calculus. On the other hand, the sche matic sentence letters are a mere convenience, here as in the standard predicate calculus ; we could always use 'Fx' or
'Gy' or the like instead of 'p'. Our notation thus comprises just term abstraction and exclusion and the usual schematism of predicate letters adjoined to variables. The exclusion copula occurs only between abstracts.
The reason for distinguishing between general terms and other predicates was that the predicate was not always a segregated and continuous string of signs. This meant keep ing predicate letters attached to their arguments. With term abstracts at our disposal, however, this contrast has less point ; the predicate can always be gathered up into a gen eral term by abstraction. We could begin to think of our predicate letter 'F' as a term letter after all, if it were only a question of monadic predicates. But it is not ; not now. The schematism of predicate letters that is called for in this en capsulation of the predicate calculus is the usual full array, including 'Fxy' and the rest.
However, this line of thought opens up an interesting al ternative course of development that is oriented utterly to general terms, monadic and polyadic. Turning to this new course, we reassess all predicate letters as term letters. When 'F' was a predicate letter, the combination 'Fx' was merely a composite symbol standing for any open sentence containing 'x'. Now that 'F' stands for a general term, on the other hand, the juxtaposing of 'F' and 'x' must be under stood as a logical operation of predication, a binary opera tion upon a general term and a variable. I would now write it with a copula as 'x e F' were it not that I want to preserve
170 Theories and Things
ternary operation of predication, operating on a dyadic gen eral term and two variables. Correspondingly for 'Fxyz' and beyond.
Numerical indices will now be wanted on the term letters to indicate the degree of each, that is, the number of places. This is because the attached variables are destined to dis appear, as we shall see, so that we can no longer count them to determine degree.
Also other supplementary devices will be introduced. The benefit they will confer is the full analysis and elimination of the relative clause, or abstract, and its variables. Predica tion will disappear as well.
What I am leading up to is what I have called predicate functor logic. I published on it in 1960 and again in 1971.3
My reason for reopening it now is that the logic of term ab straction and exclusion which we have just been seeing affords easy new access to a predicate-functor version.
The purpose of the relative clause was to integrate what a sentence says about an object. Its instrument is the bound variable, which marks and collects scattered references to the object. In predicate-functor logic this work is accom plished rather by a few fixed functors that operate on gen eral terms to produce new general terms. These functors have the effect of variously permuting or fusing or supple menting the argument places. Four will suffice. There are
major and minor inversion, explained thus : (Inv pn) X2 . . . XnX1 = Fnx1 . . . Xn,
(inv Fn) X2X1X3 • • • Xn = Fnx1 • • • Xn.
There is a functor that I call padding :
( Pad Fn) XoX1 . . . Xn = Fnx1 . . . Xn.
Finally there is reflection, the self or reflexive functor : (Ref Fn) x2 . . . Xn = F"X2X2 . . . Xn.
3. "Variables explained away,'' reprinted in Selected Logic Papers ;
"Algebraic logic and predicate functors,'' reprinted with revisions in the 1976 edition of Ways of Paradox. In its elimination of variables the plan is reminiscent of the combinatory logic of Schonfinkel and Curry, but unlike theirs it stays within the bounds of predicate logic.
Predicates, Terms, and Classes 171 In iteration these four functors suffice to homogenize any two predications-that is, to endow them with matching strings of arguments-and to leave the arguments in any desired order, devoid of repetitions. For example, the heterogeneous predications 'F5wzwxy' and 'G4vxyz' are verifiably equivalent to these homogeneous ones :
( Pad Ref Inv inv F5) vwxyz, (inv Pad G4) vwxyz,
These four functors accomplish the recombinatory work of variables. One further functor suffices for the rest of the burden of the predicate calculus. It is a two-place functor that I call the divergence functor. Applied to two nadic gen eral terms, it produces a term 'Fn 1 1 Gn' of degree n - 1 . In
particular then 'F1 1 1 GP is a term of degree 0, that is, a sen tence. Its interpretation is to be 'F1 excl GP ; 'No F are G'. 'F2 1 1 G2' is to be interpreted as the monadic general term :
{ y : { x : F2xy} ex cl { x : G2xy } } .
For example, where 'F2xy' and 'G2xy' mean ' x reads y ' and
'x understands y', ' (F2 1 1 G2) y' means that y is understood by none who read it. The general term 'F2 II G2' amounts to the words 'understood by no readers thereof'. In general,
(2) (F" I I G") x2 • • • X,, =. {x1 : Fnx, . . . Xn} excl {x1 : Gnx, . . . Xn } = (x1 ) (F"x1 . . . x,, I G"x1 . . . x,, ) .
It can now be quickly shown that these five functors, ap plied in iteration to term letters, are adequate to the whole of the predicate calculus. Abstraction, bindable variables, and predication all go by the board. Functors and schematic term letters remain.
For, consider our last version of the predicate calculus, in terms of term abstraction and the exclusion copula. Given any closed sentence schema S in that notation, we can trans late it into terms of our five functors as follows. Choose any innermost occurrence in S of the exclusion copula ; that is, any occurrence that is flanked by abstracts devoid of the copula. It is flanked thus :
(3) {x : F . . . } excl {x : G . . . }
172 Theories and Things
our four combinatory functors to bear, we homogenize the
'F .. .' and 'G . . .', giving the variable 'x' initial position in
each. Thus (3) goes over into something of this sort :
(4)
where 'f' and '!l' stand for complex general terms built from
'F' and 'G' by the combinatory functors. But ( 4) reduces by
(2) to the single predication ' (r 11 !l) Y1 . . . Yn'· The variable 'x'
and its abstracts have disappeared. Then we proceed simi larly with another innermost occurrence of the exclusion copula. As we continue this procedure, exclusion copulas that were not innermost become innermost and give way to single predications ; and variables and abstracts continue to disap pear. In the end S reduces to a single predication, '®z1 . . . z/.
But S had no free variables ; all its variables were bound by abstracts, and all are now gone. So k = 0 ; we are left with merely '®', which is some zero-place term schema, some sen tence schema, built up of term letters by the four combina tory functors and the divergence functor.
The five functors that have thus proved adequate to the predicate calculus can in fact be reduced to four. George Myro showed me in 1971 that the two inversion functors can be supplanted by a single functor of permutation, ex plained thus :4
4. This can be seen with the help of Wo,ys of Paradox, 1976, p. 298. The 'p' of that page is 'Perm'. The cropping functor there used is definable as the complement of our present 'F 1 1 F', complement being definable thus : '-G' for 'Pad G 1 1 Pad G'.
21
Responses
Some of my reading elicits responses in rebuttal or further explanation. Some strikes a responsive chord. The ensuing fragments comprise responses of all three sorts.
R E S P O N D I N G TO S A UL K R I P K E 1
A rigid designator is one that "designates the same obj ect in all possible worlds," or, as Kripke presently corrects him self, "in any possible world where the obj ect in question
does exist." He reassures us regarding his talk of possible worlds : it is not science fiction, but only a vivid way of phrasing our old familiar contrary-to-fact conditionals. Let us recall then that some of us have deemed our contrary-to fact conditionals themselves wanting in clarity. It is partly in response to this discomfort that the current literature on possible worlds has emerged. It is amusing to imagine that some of us same philosophers may be so bewildered by this further concept that we come to welcome the old familiar contrary-to-fact conditionals as a clarification, and are con tent at last to acquiesce in them.