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
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