GYULA KLIMA

On Being and Essence in St. Thomas Aquinas s Metaphysics

and Philosophy of Science

In this paper I would like to present the outlines of a formal

reconstruction of St. Thomas Aquinas s concepts of being and essence as

they function in his metaphysics and philosophy of science. This will

necessitate the introduction of some formalism, however, I try to keep

certain balance between formal and informal presentation so that we can

steer our way safely between the Scylla of empty concepts and the

Charybdis of blind intuition.

Now, as we all know well, "esse duobus modis dicitur. Uno modo

secundum quod significat veritatem propositionis, secundum quod est

copula ... Alio modo dicitur esse, quod pertinct ad naturam rei, secundum

quod dividitur secundum decem genera." (3SN.d.6.q.2.a.2.)1 So the

copulative est signifies the truth of a proposition.

But what is it that makes a proposition true? Well, it is the actual

existence of an individualized form, or nature, signified by the predicate

term in the individual supposited for, i.e. referred to by the subject term

at the time of the predication.2

For what individualizes a form in the first instance is the individual

of which it is a form. Such a form is what St. Thomas speaks of as

"forma in supposito singulari existens per quod individuatur". (STl.q.13.

1

[Since severely restricted space did not allow me to indulge in detailed textual

analysis, let me ask the reader to "take my notes seriously",i.e. to read my paper with

an eye on the texts referred to below, secundum illud vulgo dictum: melius est esse

iimum quam caecum.]Cf. 1SN 19.5.1.adl.; 1SN 33.1.1.adl.; 2SN 34.1.1.; 2SN 37.1.2.adl.&ad3.; De Fnte 1.; De

Pot 7.2.adl.; De Malo I.l.adl9.; Quodl 9.2.2(3).; in Meta 4.1.; in Meta 5.9.; in Meta 6.2.;

in Meta 6.4.; in Meta 11.8.; ST1 3.4.ad2.; ST1 16.3.ad2.; ST1 48.2.ad2.; ST1-2 36.1; ScG

1.12.; ScG 1.58.; ScG 3.9.; cf. also Cajetan: Comm. in de Ente, c.l. in princ. in:

Opuscula Omnia, (Lugduni, 1577); Alamannus: Summa Philosophiae, Tom. 1.Sect. II.5.1.

(Paris, 1888); R.W.Schmidt: The Domain of Logic according to Saint Thomas Aquinas

(The Hague, Martinus Nijhoff, 1966) Part II. ch.4. & Part III. ch.VIII.

Cf. "Sicut cum dico, Socrates est homo, veritas huius enuntiationis causatur ex

compositione formae humanae ad materiam individualem, per quam Socrates est hie

homo." in Meta 9.11. vide totum locum; cf. also Schmidt, op. cit. pp.212-214 & 224-226.

Being and Essence in Tliomas Aquinas 211

a.9.)3 But the other individuating condition is time: for even if an in

dividual can have numerically the same form at different times, still, the

form once emitted cannot recur numerically the same, "quia quod omnino

in nihilum decidit idem numero resumi nonpotest". (4SN.d.22.q.l. a.l.)

So predicates signify individualized forms, which are numerically

different in different individuals (except for the case of divinity, of

course ) and may be different in the same individual at different times.

But the same predicate in the same individual at the same time cannot

signify different forms. So we can speak of the significate of a predicate

P in an individual u at time t, which, therefore, can be denoted as a

value of a function for these arguments like this: Sgt(P)(u)(t).

Now it is the existence of such an individualized form that accounts

for the truth of a predication (namely, of predicating P of u at t): "ex

hoc enim quod aliquid in rerum natura est sequitur veritas vel falsitas in

propositione, quam intellectus significat per hoc verbum est prout est

verbalis copula." (in Meta 5.9.)6

But so we can say that a proposition of the form S est P is true

at time t according to a given supposition, or acception of its subject

term, if the significate of P in the individual supposited for by S at time

Cf. "Non enim oportet si hoc est homo, et illud homo, quod eadem sit numerohumanitas utriusque, sicut in duobus albis non est eadem albedo numero" 2SN 17.1.1.;

cf. e.g. ST1 85.1. & 2.ad2.; cf. also Alamannus, op.cit. q.2. aa.1-3. Note that from the

point of view of this reconstruction it makes no difference whether we speak of

Socrates s humanity or of the humanity individualized by Socrates s matter, i.e. by the

materia signata that makes Socrates this individual. Indeed, these are one and the same

form, the forma totius of Socrates. Cf. also the references of the next note.

4Cf. In Phys. 5.6.; Quodl. 4.3.2.; Quodl. 11.6.; ScG 4.80 & 81; 4SN 44.1.1.; Comp.

Theol. 1.154.

Cf. ST1 39.2 & 3.; 1SN 9.1.2.; De Pot. 9.6. By the way, this approach offers a very

good criterion of truth for relative identity statements, like "a is the same F asb",

as

opposed to absolute identity statements like "a is identical with b". The former holds

iff the significate of F in the suppositum of a is identical with the significate of

F in the suppositum of b . So "Filius est idem Deus cum Patre" is true, for Deus

signifies the same nature in the suppositum of Filius and in the suppositum of

Pater . But "Filius est idem cum Patre" is false, since the Son and the Father are

distinct supposita of this nature, while this sentence states the identity of these

supposita.

1

Cf. 1SN 19.5.1.; 1SN 33.1.1.adl: "esse quod significat veritatem compositionis in

propositionibus ... fundatur in esse rei"; Schmidt, op. cit. pp.232-237.

212 Klima

t, at time t exists. It is the actual existence of this significate that

founds the truth of this proposition.7

But from this it does not follow that this form is the significate of

this proposition. For the proposition involves also the copula, which

signifies composition, which need not have a direct counterpart in reality.

For the copula is "significans compositionem cuiuslibet enuntiationis quam

anima facit, unde hoc esse non est aliquid in rerum natura, sed tantum in

actu animae componentis ct dividends". (Quodl.9.2.2(3)).

But so we can say that what is signified by a propositional

composition is a sort of ens rationis signified by the copula, which is in

the second sense, if and only if the form signified by the predicate in

the suppositum of the subject is in the first sense.9

(In the case, of

course, when the predicate is such that it signifies some real form, not a

privation, negation or relation of reason. In these latter cases also the

significate of the predicate would be an ens rationis. See n.17.)

Now to give this idea a formal expression consider the following.

First, let us suppose that everything that can be signified by any means

is either actual or not actual at a given time t.10 Let us suppose further

that everything which is actual is either a mere ens rationis or also an

ens reale.11 The significate of a predicate P in an individual u at time t

is an element of one of these domains: Sgt(P)(u)(t) W(t), where W(t) is

the set of all signifiable things which are either actual or not actual at

Cf. Schmidt, op.cit. pp.224-228., cf. also II. Weidcmann: TheIx>gic

of Iking in

Thomas Aquinas",in: S. Knuuttila - J. Hintikka: The Logic of Being (Dordrecht,

Holland, 1986).

^This point is brought out nicely in Wcidemann, op.cit. sect. IV; cf. also Schmidt,

op.cit. pp.238-239.

I would tentatively identify the significate of a proposition as the enuntiabile

expressed by the proposition, expressly called by St. Thomas a res rationis in 1SN

41.1.5. I say: "tentatively",because of St. Thomas s tendency to use the term

enuntiabile as a synonym for enuntiatio (although "cmphasi/.ing the objective meaning

of enunciation" Schmidt, op.cit. p.223. n.84.). For St. Thomas s use of the term see 3SN

24.1.1b.; 1SN 38.1.3.; De Ver. 2.13.ad7.; 1.6.; 14.8.; 2.7.; 1.5.; 14.12.; Quodl. 4.9.2.; ST1

14.14.; ST1 14.15.ad3.; ST1 16.7.; ST3 1.2.ad2. For a clear expression of the view that

an enuntiabile is the significate of a proposition see e.g. Logica Modernorum, vol.11. -

part two (ed.: L.M. de Rijk, Assen, 1967) pp.208-213. See also: Peter of Spain: Traaatus

(ed.: L.M. de Rijk, Assen, 1972.) pp.205-207. Cf. also G. Nuchelmans: Theories of the

Proposition- Ancient and Medie\>al Conceptions of the Bearers of Truth and Falsity

(Amsterdam-London, 1973.) pp. 165-194.

10Cf. e.g. De Princ. c.l. For a medieval-style resolution of the problems involved in

referring to and quantifying over nonexistents see my "Existence, Quantification and

the Mediaeval Theory of Ampliation", Doxa 9(1987) pp.83-112.

11Cf. 2SN 34.1.1.

Being and Essence in Tfwmas Aquinas 213

time t, so that the set A(t) / = Ra(t)/ is a part of W(t), and Re(t) is a

part of Ra(t).

Now this significate may be construed as the value of a function

for the argument t. But so the function itself can be got from this by

lambda-abstraction as follows:

At(Sgt(P)(u)(t))

But this, again, can be regarded as a value of a function for the

argument u. So, again, we get the function itself by applying lambda-

abstraction to u too - and let me call the result the signification ofP:u

Sg(P)=

Au(At(Sgt(P)(u)(t)))

Now let us suppose further that what the copula, the sign of

composition composes are this function and its consecutive arguments,

supplied by the subject term and the time of the predication. So we can

write:

Sgt(S est2 P)(t)(Sp)=

Sgt(est2)(Sg(P))(Sp(S)(t))(t),

where Sp(S)(t)e{u: Sgt(S)(u)(t)EA(t)}, if this set is not empty,

otherwise Sp(S)(t)= 0,

13 and Sgt(est2)(V)(u)(t) eW(t), where

V(u)(t)eW(t) and Sgt(est2)(V)(u)(t)eA(t) iff V(u)(t)eA(t).

That is, what is signified by a proposition (at time t according to a

given supposition, or acception of its subject term) is what is signified

by the copula when it composes the nature signified by the predicate

(according to its absolute consideration 14) with the suppositum of its

subject (at time t) at time t: for "compositio enuntiabilis significat

aliquod esse rei" (STl.q.l4.a.l4.ad2.). It is this composition of the intellect

For the use of lambda-abstraction see A. Church, Introduction to Mathematical

Logic (Princeton, 1956.), pp.15-23. Concerning the close parallelism between functional

abstraction, on the one hand, and the Aristotelian conception of abstraction, on the

other, see P.T. Geach, "Form and Existence", in: God and the Soul (London, 1969) and

my "St. Thomas Aquinas on the Meaning of Words", Magyar Filozofiai Szemle 3-4(1984)

pp.298-313. (in Hungarian with English abstract). As for the terminology used, of

course, the terms"signification"

and"significate"

are not to be regarded as strictly

corresponding to St. Thomas s use of significatio and significatum. What I call

"signification" is most frequently referred to by St. Thomas as forma significata (per

pracdicatum), and what I call"significate"

is St. Thomas s forma \<cl natura individuata .

Cf. e.g. ScG 4.49.; ST1 39.4.adl.; ST 16.2.; 1SN 25.1.4.; 3SN 7.1.1.ad5.; 1SN 4.1.2. &c.

As can be seen, is the semantic value of empty terms. If we add the

condition that for any predicate non Huberts vim ampliandi Sgt(P)(0)(t)?A(t), andthat an A proposition is true (at time t) iff its predicate is true of every suppositumof its subject (i.e., |Omne S est

2 P|t= T iff for every ue{u: for some Sp,

Sp(S)(t)=

u}, Sgt(P)(u)(t)eA(t)), then all relations of the Square of Opposition and all

syllogistical forms are saved. Cf. My "Modernorum Logica Modernorum", in: Festschrift

for Imre Ruzsa (ed.: L. Polos, Budapest, 1988).

Cf. De Ente c.4. in fine. Cf. etiam Cajetanum ad hunc locum.

214 Klima

which answers the composition that is found in the thing.15 And just as

from the real composition in the thing there results an esse reale, so

from the composition of the intellect results an esse rationis answering

the esse reale which ultimately founds the truth of the proposition.16

But so we can say that the proposition is true if and only if what it

signifies exists in the second sense, i.e. if it is an ens rationis and that

this is so if and only if the nature or form signified by the predicate in

the suppositum of the subject exists in the first sense, i.e. if it is an ens

reale (provided that the predicate is such that it docs not signify

negation, privation or rationale relation) at the time of the predication:17

|S est2 P|t,Sp= T iff Sgt(S est2 P)(t)(Sp)eRa(t)

iffSgt(P)(Sp(S)(t))(t)Re(t)

But what about the case when the copulative est is used absolutely,

without the addition of a predicate term, when it answers the question an

est!

Well, we may say that despite appearances this case is not so

different from the former: for just as in the former case the copula

signified the existence of what is signified by a predicate in a

suppositum, so it signifies in this case the existence of the suppositum-

the absence of the predicate term means that it is not some determinate

mode of existence that is attributed to the suppositum, but existence

simplidter. So in our reconstruction we may suppose that in this case

what holds the place of the signification of the missing predicate term is

an "identical" operation, i.e. a function which, somewhat loosely speaking,

sends its argument into itself:

Sgt(S est2)(t)(Sp)

=Sgt(est2)(I)(Sp(S)(t))(t),

where I(u)(t)= u.

15Cf. Schmidt, op.cit. pp.224-226. See also the texts referred to by him.

16 Tertio modo dicitur esse quod significat vcritatcm compositionis in propositi-

onibus, secundum quod est dicitur copula: et secundum hoc est in intellectu

componente et dividente quantum ad sui complcmentum; sed fundatur in esse rei, quodest actus essentiae." (1SN 33.1.1.adl.) Cf. Scmidt, op.cit. pp.2 15-222.

17I think when the predicate signifies an ens rationis, then (and only then) we can

identify the significate of the predicate with that of the copula: if Sgt(P)(u)(t) e

Ra(t)-Re(t), then Sgt(est2)(Sg(P))(u)(t)=

Sgt(P)(u)(t). In this way the esse of the

significate of the predicate will consist indeed in actu animae components et

dividends. However, against this identification cf.: Cajetan: Commentaria in

Praedicamenta Aristotelis (Romae, 1939) pp.210-212.

Being and Essence in Tlwmas Aquinas 215

From this we can derive that

|S est2 |t,Sp

= T iff Sp(S)(t)Ra(t),

i.e., that the sentence S est2 is true, iff an S is a rationale being.18

Now this reconstruction may perhaps gain further confirmation from

the fact that if we take into consideration St. Thomas s claim that a

substantive name, as opposed to an adjective name, can be taken for its

suppositum even in predicate position,19

then from this reconstruction it

follows that deletion of a substantive predicate term does not affect the

signification of a proposition, thereby doing justice to the intuition

behind the"ellipsis"

theories of the copula.20 For if Sg(P)

=I, i.e., if the

predicate P is taken to signify not a form in a thing distinct from the

thing, but the thing itself, then from the above equivalences it follows

that

Sgt(S est2 P)(t)(Sp)

=Sgt(S est2)(t)(Sp).

So we can say that est in the second sense, whether it is used as

a copula, i.e. as tertium adjacens, or absolutely, as secundum adjacens,

10Of course, through |S est

2 |t,Sp= T iff Sgt(S est

2)(t)(Sp)eRa(t). For, in general,for any proposition p: |p|t,Sp

= T iff Sgt(p)(t)(Sp)eA(t). Cf. Mohan Matthen: "Greek

Ontology and the Is of Truth", Phronesis 2(1983) pp.1 13-135. And from this: |p|t= T

iff for some Sp: Sgt(p)(t)(Sp)e A(t), secundum regulam: indefinite! aequipollet

particular!.

19Cf. 3SN 5.3.3. I think I should briefly comment on St. Thomas s remark in this text

that in this case the predication is a praedicatio per identitatem as opposed to a

praedicatio per informationem sive denominationem, the latter being a magis propriapraedicatio for praedicata tenentur formaliter. (Cf. e.g. in Meta 9.11.; ST3 16.7.ad4.; ST113.12.; 85.5.ad3.) Now this reconstruction, as it stands, of course, favors the "inherence

theory"as opposed to the

"identity theory"of predication (cf. e.g. L.M. de Rijk s

"Introduction" (pp.37^48.) to his edition of Abelard s Dialectica (Assen, 1956); D.P.

Henry: Medieval Logic and Metaphysics (London, 1972) pp.55-56; P.T. Geach:

"Nominalism", in: God and the Soul (London, 1969)) in that it assigns the predicate the

semantic function of signifying inherent forms through its abstract signification.

However, this does not preclude the fact that even in this reconstruction a"proper

predication" is always equivalent to an identity statement: S est2

P o S = P;

if

Sgt(=)(u,)(u2)(t)eA(t) iff Ul

= u2 6A(t), whence Sgt(S

=P)(t)(Sp)

=Sgt(

=)(Sp(S)(t))

(Sp(P)(t))(t). Now since supposition of a term is defined through the actual inherenceof the form signified by the term, this reconstruction expressly shows that a

predication is true iff its terms supposit for the same thing, i.e. iff the forms

signified by its terms inhere in their common suppositum. Furthermore, if we take the

predicate (in St. Thomas s, but not e.g. Ockham s and his followers, view, improperly)

to stand immediately for its suppositum, i.e., if we identify its significates with its

supposita, then we can identify also the significate of a predication with that of an

identity statement. So in this case taking the copula in the sense of identity will notaffect even the sense of a proposition, thereby doing full justice also to the identity

theory of the copula.

Cf. e.g. R.M. Dancy: "Aristotle and Existence", in: Hintikka - Knuuttila: The LogicofBeing (Dordrecht, 1986); A. Kenny, The Five Ways (Ixjndon, 1969) pp. 91-95.

216 Klima

predicates existence in the second sense, i.e. existence proper to rationale

beings.21

But this kind of existence is founded on existence in the first

sense, proper to real beings, which is signified by est in the first

sense:

ISest^Sp = TiffSgt(est 1 )(Sp(S)(t)eA(t),

where Sgt(est 1)(u)(t)A(t) iff ueRe(t).

But so

jScstJt.Sp= TiffSp(S)(t)eRe(t)

while

|Sest2 |t,Sp= TiffSp(S)(t)Ra(t)

and so, since Re(l)Ra(l), therefore S cslj implies S est2 but not

conversely. (Whether the converse implication holds or not depends on

the meaning of S: if S denotes real beings, then, of course, also the

converse implication holds.)

So the verb est primarily signifies actual, real existence (as its

"focal meaning").22 But this signification is extended also to rationale

beings, which exist in a secondary, derivalive sense, owing ihis derivalive

exislence lo ihe real existence of real beings, whether these be subsistenl

individuals, or real forms inhering iherein.-

For ihis real exislence is allribuled lo somelhing in a Iwofold

manner: "Uno modo, sicul ei quod proprie el vere habel essc, vel esl ...

Omnia vero quae non per se subsislunl, sed in alio, vel cum alio, sive

sinl accidenlia, sive formae subslanliales, aul quaelibet paries, non habenl

esse ul ipsa vere sinl, sed allribuitur eis esse alio modo, idesl ul quo

aliquid esl. Sicul albedo dicitur esse, non quia ipsa in se subsistat, sed

quia ea aliquid habcl esse album." (Quodl.9.2.)

Now, since any individual substance can have only one esse

substantiate,,

24 therefore we can say lhat a form is subslanlial to an

individual if and only if Ihe esse of Ihis form (ul quo aliquid est) is

idenlical with the esse of ihe individual (quod est). Bui so, furlhcr, a

21This is why (pace Schmidt, p.235) est means the same in Caecitas est and

Aliquid est caecum ,or even in Deus est when this is an answer to the question An

Deus est? Cf. texts referred to in n.l.

22I have borrowed the term from G.E.L. Owen through Weidemann, op.cit. p.190.

23Cf. inMeta4.1.

24"Impossible

est enim quod unum aliquid haheat duo esse substantialia" 3SN 6.2.2.

Being and Essence in Tliomas Aquinas 217

predicate P is essential, or substantial to an individual u, if and only if it

signifies such a form in u:

from which, besides, it follows that if P is essential to u, then it is

necessary that if u exists, then it is P.25

Now, as the significate of a substantial predicate is a substantial

form, St. Thomas s thesis of the unity of substantial form can be

expressed in this framework as follows:26

If G and S are substantial predicates of u, then

Sgt(S)(u)(t)=

Sgt(G)(u)(t).

On the other hand, since a species term signifies the quiddity, or

essence of a thing, therefore the thesis of the real distinction of essence

and existence in the creatures may be expressed as follows:27

If S is a species term, then

Sgt(S)(u)(t) * Sgt(estl)(u)(t).

Now this is how esse and essentia are found in the individuals; and

it is by abstracting from these that we arrive at the cognition of

universals, a class of rationale beings.28

But so, if the significate of a predicate P in an individual u at

time t is a natura individuata of P-ness, then what we get by this

abstraction is the nature of P as considered absolutely, without any

individuating conditions:29

Nat(P) =AuAt(Sgt(P)(u)(t)) /= Sg(P)/

"Haec autem natura habet duplex esse, unum in singularibus, aliud in

anima, et secundum utrumque consequuntur dictam naturam accidentia..."

Namely, for example: "ratio specie! accidit naturae humanae secundum

illud esse quod habet in intellectu ... Et quamvis haec natura intellecta

25Cf. Porphyry s definition of accident; also my paper referred to in n.10.

26Cf. e.g. De Ente c.3.

27Cf. e.g. De Ente c.5.

28The "basic texts" for St. Thomas s theory of abstraction are the following: in De

Anima 3.8-12; ST1 13.12.adl.; ST1 85.1.; in De Trin. 3.5.3. cf. Also Alamannus, op.cit.,

torn. l.sect.l.q.2.aa. 1-3; Schmidt, pp.177-202.

99Concerning the connection between abstraction and reduplicative constructions

extensively used by St. Thomas in this connection (cf. e.g. De Ente c.4 & comm.

Cajetani ad idem), see J. Lear: "Aristotle s Philosophy of Mathematics", The

Philosophical Review, April 1982, pp.161-192. Concerning 13th century treatment of

reduplicative propositions in general, see my "Libellus pro Sapiente- a Criticism of

Allan Back s Argument against St. Thomas Aquinas Theory of the Incarnation", TheNew Scholasticism, 58(1984) pp.207-219.

218 Klima

habet rationem univcrsalis secundum quod comparatur ad res quae sunt

extra animam, quia est una similitudo omnium, tamen secundum quod habet

esse in hoc intellectu vel in illo est species quaedam particularis". (De

Ente 4.)

Now from these and related remarks we may form the following

picture: by concretion, the inverse operation of abstraction as presented

above, we can go, as it were, in two directions: either ad extra, and then

we arrive at the real individualized natures of individuals, or ad animam,

and then we arrive at the universal intentions of particular minds.30 But

these intentions are universal only insofar as they are got by abstraction,

carried out by these particular minds, from representations of individuals,

namely from phantasms.

Now from these mental representations (universal intentions and

phantasms), in a similar manner as we could construct the significatcs of

propositions ad extra, we can construct their significates apud mentem? 1

Correctness of belief, then, consists in the adaequatio of these two kinds

of significates. But this correctness is based on the evident truth of first

principles, which, in turn, owe their evidence to induction, based on

correct, essential abstraction; for "scnsus est quodammodo et ipsius

universalis. Cognoscit enim Calliam, non solum inquantum est Callias, sed

etiam inquantum est hie homo, et similiter Sortem, inquantum est hie

homo. Et inde est, quod tali acceptione in sensu praeexistente, anima

intellectiva potest considerare hominem in utroque. ... Sic enim, scilicet

per viam inductionis sensus facit univcrsale in anima, inquantuam

considerantur omnia singularia." (An. Post.2.20)

However, limited space does not allow me to present here the

relevant reconstructions. All I hope to have shown in this paper is that

by the semantic approach presented here St. Thomas s thoughts

concerning essence and existence can be given such a strict formulation

that meets even present day standards of exactitude, whereby these

thoughts can be treated as something very substantial, and highly

relevant even to our modern ways of doing philosophy.

30Cf. Schmidt, op.cit. pp.98-130 & 212-215.

Concerning how this approach can be developed into a fully-fledged formal

semantics see my "Understanding Matters from a Logical Angle",in: Gyula Klima: Ars

Artium: Essays in Philosophical Semantics, Mediae\>al and Modem, Budapest, 1988.

32Cf. in Meta 1.1.; in Anal.Post. 2.20.; Schmidt, op.cit. pp.270-302.

Being and Essence in Tliomas Aquinas 219

Appendix

In this Appendix I supply a brief, exact description of the formal theory

outlined in the body of the paper, lest any technically obscure point

should remain.

The language of the theory, the language of categorical proposi

tions, is defined as follows: L:=<C,Pr,F>, where C: = {est 1 ,est2,-,Q,= }

(where - is the sign for negation and Q is Omne or Quoddam or their

equivalents), Pr is a set of predicate parameters (S, P, etc.), and F, the

set of formulae, or sentences is defined by the following clauses:

l\l If S,PePr, then S est2 P, Q S est2 P

,S estj ,

S est2 ,

S - P eF.

/ii/ IfpeF, then -p F.

A model for this language is defined as follows: M:= < W,T, < ,A,Ra,

Re, 0,Sg>,where W(t) is a nonempty set, T is a set of time-points

ordered by <, A(t) is a part of W(t), Ra(t)=A(t), and Re(t) is a part of

Ra(t), where t is an element of T. Intuitively, W(t) is the set of all

signifiable things which are either actual or not at time t, A(t) is the set

of things which are actual at time t, Ra(t) is the set of rationate beings

and Re(t) is the set of real beings at time t. 0, the zero-entity, the

semantic value of empty terms falls outside the whole universe of

discourse W!, i.e. 0gW!, where W!:= UteT:W(t), i.e., W! is the union

of all W(t) s. Sg, the signification function is defined for any expression

(primitive as well as complex) as follows:

Sg(exp) := A Cl (... Aen(Sgt(exp)(ei ) ... (en)) ... ),

where Sgt(exp)(e 1 )... (en) is the significate of any expression in respect

of any entities e:

... en whatever (including elements of W! U {0}, T,

elements of L and functions defined on these).

Note that I use the lambda-operator and repeated pairs of

parentheses after functional expressions according to the following

equivalences:

If f and g are functions, then

f(x)(y)=g(y) iff f(x)

=g iff Ay(f(x)(y))

=g.

Now Sgt(exp)(e!) ... (en ) is defined by the following clauses (if not

otherwise indicated, it is supposed throughout that teT and u\V!):

/i/ Sgt(P)(u)(t)W(t)

/ii/ Sgt(P)( )(t) A(t), provided P is non-ampliative

220 Klima

/m/ Sgt(est2)(V)(u)(t)eW(t), where V(u)(t) W(t), and

Sgt(est2)(V)(u)(t)eA(t) iff V(u)(t)A(t) (Intuitively, V is the

place-holder of the signification function of the predicate term,

while u is the place-holder of the suppositum of the subject

term. Cf. /viii/ below.)

/iv/ Sgt(est 1)(u)(t) W(t), and Sgt(est 1 )(u)(t) e A(t) iff u eRe(t)

/v/ Sgt(=)(u 1 )(u2)(t)W(t) and Sgl(

=)( Ul )(u2)(t)eA(t) iff

Ul= u2 A(t)

/vi/ Sgt(-)(u) W(t) and Sgt(-)(u) eA(t) iff u *A(t)

/vii/ Sgt(0)(N*)(E(V))(t)eW(t), and Sgt(Q)(N*)(E(V))(t)eA(t) iff

for O ueN*, E(V)(u)(t)eA(t), where Q is the English

equivalent of Q, E(V)(u)(t)\V(t), N(u)(t)W(t), and N* := {u:

N(u)(t)eA(t)}, if {u: N(u)(t)eA(t)} is not empty, otherwise

N*: = {0}. (Intuitively, N and E(V) are the place-holders of the

signification functions of the NP and VP of the quantified

statement, E is the placeholder of the signification of the

copula, while N* is the place-holder of the range of values of

the NP, i.e., of the subject of the quantified statement. Cf.

/iii/ above, and /xii/ below.)

These were the clauses for the primitive expressions of L. Nowhere follow the clauses for the complex expressions of L:

/viii/ Sgt(S est2 P)(t)(Sp)=Sgt(est2)(Sg(P))(Sp(S)(t))(t), where Sp(S)(t)

is an element of {u: Sgt(S)(u)(t)eA(t)}, if this set is not

empty, otherwise Sp(S)(t)=

/ix/ Sgt(S est2)(t)(Sp)=Sgt(est2)(I)(Sp(S)(t))(t), where I(u)(t)

= u

/x/ Sgt(S est1)(t)(Sp)=Sgt(est1)(Sp(S)(t))(t)

/xi/ Sgt(S =P)(t)(Sp)=Sgt(=)(Sp(S)(t))(Sp(P)(t))(t)

/xii/ Sgt(Q S est2 P)(t)(Sp)=Sgt(Q)(Sp(S)(t)*)(Sg(est2)(Sg(P)))(t),where Sp(S)(t)*:

= {ueW! U{0}:for some Sp, u = Sp(S)(t)} and

Sg(est2)(Sg(P))= Au( At(Sgt(est2)(Sg(P))(u)(t))).

/xiii/ Sgt(-p)(t)(Sp)=Sgt(-)(Sgt(p)(t)(Sp)).Now the definition of truth for any formula p, at time t, according

to a given supposition, or acception of its terms is the following:

|p|t,Sp= TiffSgt(P)(t)(Sp)eA(t);

wherefrom the definition of truth at time t is as follows:

| p 1

1 = T iff for some Sp, | p | t,Sp= T.

Now, if we define: Quoddam S non est2P >df.

- Omne S est2P

and Nullum S est2P df. - Quoddam S est2 P

,then all the relations

Being and Essence in TJiomas Aquinas 221

required by the Square of Opposition among the four categoricals are

provably valid in this system.

For possible extensions of this approach see my paper referred to in

note 31.

Institute of Philosophy of the

Hungarian Academy of Sciences