Examination of the Axiomatic Foundations i

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N o t r e Dam e Jou rn al of Fo rm al LogicVolume IX, Number 4, October 1968

X

EXAMINATION OF THE AXIOMATIC FOUNDATIONS

OF A THEORY OF CHANGE. I

LAURENT LAROUCHE

In 1934, Jan Salamucha published an ar tic le in which he was making alogical analysis of Thomas Aquinas' argument in his proof "E x Motu" forthe existence of God, [ l] , [2]. Here was an attem pt to apply mathematicallogic to a cr it ical study of a cla ssical problem in philosophy. As ProfessorBolesiaw Sobociiski pointed out in a biographical note , Salamucha's mono-graph was intended to show the important role that mathematical logiccould play in studying complicated logical reason ing in the realm ofphilosophy, [3] . Some twenty years lat er, inspired by Salamucha's ideas,Johannes Bendiek published a historical study of the logical techniques used

in the classical proofs for the existence of God, [4]. In recent yearsF r a n c e s c a Rivetti- Barb has published some extensive ar ticles applyingthe tools of modern logic to Thomas Aquinas' proof "Ex Motu" for theexistence of God, [5], [6]. It is our intention to evaluate the above- men-tioned works with regard to our own study at the end of the presen tmonograph.*

Our purpose here is to work out the axiomatization of the foundationsof the theory of change which serves as the starting point and basis for theThomistic proof "Ex Motu" for the existence of God. It is well known thatAquinas drew the elements of his theory of change from the Aristotelian

system. We could thus con sider our study as dealing also with the theoryof change in the G reek Philosopher. We shall develop our subject in th reeparts followed by some appendices. The first part will present a formal-ization of the theory of change in question . In the second par t we shallcompare our formalization with some Thomistic and Aristot elian texts.The third part will deal with the proofs of the propositions which willappear in the formalization and with the proof of the consistency of the

*The present monograph constitutes the first part of the author's doctoral dis-sertation, Untersuchung zur axiomatischenGrundlegung der Bewegungslehre,writtenunder the direction of Professor Hans H ermes and accepted by the WestfalichenWilhelmuniversitat zu M unster in partial fulfillment of the requirements of thedegree of D r. re r. nat. in M athematicalLogic, June, 1964.

Received May 20, 1964


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372 LAURENT LAROUCHE

axiomatic system introduced in part one. Also in that last part we shallestablish the independence of a certain set of axioms.

No one would expect to find already in Aquinas' or Aristotle's writingseverything needed for a formalization. Although we were unable to findclear answers to certain questions relevant to our study in the texts of thetwo philosophers here concerned, it has been our aim to remain faithful totheir philosophy. It is up to specia lis ts in scholastic philosophy to decidein what measure we have succeeded in this.

1 . Introduction. The reasons for our choice of the word ' 'change"instead of one of the words "movement, motion" to translate the Latin word"mo tus " is that we want to avoid eventual misunderstanding. In one 'sdaily English usage, when speaking of movement or motion, one thinks firstand almost exclusively of local movement or motion. The word ' 'mo tus"has a much broader meaning in the Thornistic texts. It comprises all kindsof change. By the notion "mo tus", we shall mean a process in which oneand the same subject in a temporal succession of phases exhibits differentproper ties. Following Aristotle, Thomas Aquinas takes as the start ing-point for his proof the empirical reality of change in the world. In ourdaily life we experience a manifold alteration and transformation in theinner world as well as in the outer world. We observe the alteration oflandscape, the growth and the decay in vegetal and animal life, the constanttransformation of human society, the pr ogress in the art s and sciences, andso on. It is these empirically ascertainable facts which Aquinas takes asbasis for his proof "Ex Motu".

St. Thomas' works are written in Latin, a natural language. For astudy of the type which concerns us here, we need a so-cal led artific ial orformalized language. The preci se structure , explicitly given, of such alanguage is what we shall need. Let us explain briefly what is meant by:1) a formalized language; 2) a calculus; 3) an axiomatic system; 4) con-sistency; and 5) independence.

1) A formalized language consists of a system of symbols together with

a set of rul es for their use. Such a language must be so specially devisedas to enable us to express in it certain propositions of the theory underconsideration. To this end, some definite symbols are selected to expressthe primitive notions (basic concepts) of the theory.

2) By a calculus we mean first the determination of the rule s of forma-tion governing the construction of expressions . These rules effectivelydetermine which finite linear sequences of the given symbols constituteexpressions of the formalized language. To a calculus also belongs asystem of rules (relations on expressions) according to which certainlogical connections between expressions can effectively be determined so

that one expression can be said to be derivable from a set of expressions.3) The formalization of a theory consists in the selection of some

expressions of the theory in such a way that all the remaining propositionsof the theory could be derived from the selected ones, called the axioms ofthe theory. The theory in question is then said to be represented by anaxiom system.


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A THEORY OF CHANGE 373

4) Co nsistency is an obvious requ isite of any axiom system . An axiomsystem is said to be inco nsistent whenever an expr ession together with it s

negation ar e derivable within the axiom system, o therwise c on siste n t Thecon sistency of a given axiom system could be established by constru ctin ga model of the axiom system .

5) An expression is said to be independent in an axiom system if itcannot be derived in the axiom system o An axiom system itself is calledindependent if every axiom of the axiom system S is indepen dent in theaxiom system S

obtained from S by the rem oval of o The independence of

a given axiom system can be shown by con struc tin g m od els. The re qu ire -ment of independence is often only a matter of elegance.

In formalizing a theory we need first to state the con cept s per tain ing

to the th eory. Th ese belong to two grou ps: the prim itive notions (unde-fined con cepts) and the defined notions. The con nec tion s between theconcepts ar e ascert ain ed through axiom s and pro po sitions. Axioms ar eput forward without proofs, while the prop ositio ns must be derived in theaxiom system.

2. Symbolic Language. Th ere is no single way to d ete rm ine asymbolic language. It depends on the p roblem to be dealt with and on theobjective in view. We shall now stat e clearly the symbolic language to beused hereafter.

1. We will use a two- sorted pred icate calcu lus. In the usual symboliclanguages it is custom ary to distinguish var iab les of different kin ds. Wedivide them into types in such a way th at var iable s of one of th ese types arecla ssified as being of the lowest type. We then subdivide the class of thelowest type vari ab les into two so r t s which we will denote by " " and " b " .Th e rea son for using h er e a two- sorted symbolic language is tha t we needto have at our dispo sal two sepa rat e cl asse s of ind ividuals. The varia blesof sort will rep resen t the "m om enta neo us subjects" and those of sort bwill represent the "pro pe r t ies" .

2. The following logical signs will be used throughout this paper.Sign Usage Meaning

p It is not the case that p. ("p" is apropositional variable)

- p- *q If p t h e n q P q p a n d qv P v q P or q (the non exclusive "o r ")

P^~>q P if and only if qV Vx For all*, (" " is a variable for

expressions)3 3x T h e r e is an x s u c h t h a t .

= DfD i = Df >2 "

D i" means the same thing as "D 2"("Di" denotes the definiendum and

"D 2" the definiens.)


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3. In or der to eliminate a grea t accumu lation of paren th eses, weestab lish the following ru le for om itting pa re n th eses. To the conn ective

signs, we assign the following or de r of precedence "", ">", " v " ," A " ? " "I " . Of th ese, each has great er precedence or great er scope thanany listed to its right 0 It is also out of practical expediency that we agreeto omit all universal quantifiers at the left of an axiom, a proposition and adefinition, whenever the quan tifier h as the whole expression as it s operan d(scope).

4. We give now a syno psis of the individual vari ab les of pa rt icu lartypes along with a list of the pr im itive notions with the ir respec tive type.The meaning of these notions will of course be explained later.

Type Letters

Momentaneous sub ject s.. .x,y,z, . . . , Xi,yi, . . .b Proper t ies , ,y,

Types of the primitive notions

~ is of type ( )< is of type ( )A is of type ( b)F is of type ( b)M is of type ( b)B is of type ( b)-3 is of type (b b)

3. Domain of momentan eous subjects. Following Aristot le, Aquinasma kes variou s statem en ts about change, the most importan t statemen tbeing: ' f anything changes, it is changed by someth ing e ls e ". It is, in th isstudy, our pr incipa l aim to form alize the theory of change to such an extentas to provide a proof of the above statement,.

To make change intelligible, we introduce first that which undergoesthe change, the "su bje c t". Such subjects could be a "t a bl e ", a "block ofmarble" , an "a p p le ", and so on. Th ese subjects are not taken as abstr actentities, but r at h er as con cret e beings with all th eir specific and individualch ar ac te ri sti c s. When speaking t herefo re of a subject, we do not refer toan arbitrary member of a particular species, but to this definite being, withits quantity, qualities, location, e t c , uniquely deter m ined .

To the ch ar ac te rist ics of a con crete subject belon gs the sequence ofthe successive mom ent s of its existen ce . If we con sider a pr ec ise point oftime at which a subject exists, we speak then of a "momentaneous subject"whereby th is par tic ula r point of time is included. Beyond that it would bepossible to r efer to the c la ss of the mom entan eous subjects which are"genidentical" (the term "genidentical" was first introduced by K o Lewinin [7].), i.e . one and the same subject at all th e mo men ts of its tem po ralexistenceo It is important to point out h er e that in our study, we shallalways consider a subject as one whole in a similar way as for examplephysicists speak of a closed system of en ergy.


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A TH EORY OF CHANG E 375

Latin l e t t e r s x, y, z, . . . . will be used to denote individual variables

for momentaneous subjects.

There obviously are certain relations between momentaneous subjects.

The simultaneity of two momentaneous subjects is such a relation which weintroduce now as our first primitive notion.

Pn 3.1. x ~ y: x an d y are simultaneous , i.e. the momentaneous sub-jects represented by the variables x an d y coincide in time.

The relation " ~ " is evidently an equivalence relation, i.e. it is

reflexive, sym m etric and tra n sitive. Th is fact we state in our first axioms.

A3.1.1 x~x

A3.1.2 x ~ y - >y ~ x

A3.1.3 x ~ y y ~ z >x ~ z

In considering one and the same subject at two different po int s of t ime ,

we are dealing with two non identical m om entaneous subjects. They are of

course related since they belong to the same concrete subject. We i n t r o -duce here our secon d prim itive notion which we denote by " < " .

Pn 3.2. x < y: x an d y are genidentical an d x is earlier in time than y.

What pr op ert ies does th is relation have? It is not difficult to convinceourse lves that the re la t ion " < " is irreflexive, asymmetric , t ransi t ive anddense. If one holds that there is a smallest interval of t ime in the physicalworld, a modification of the axiom 3.2.4 below would be n ec essar y. How-ever, it would not impeach our resu l t s in any essential way. The nextaxioms will formulate these formal properties.

A3.2.1 lx


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376 LAUREN T LAROUCHE

call it the relation of "genid en tit y". (T his relation, as it is used h ere , was

first introdu ced by H. H erm es in [8]).

D 3 . 2 Gxy = D f x y v y < x

(Gxy means that x an d y are genidentical.)

The next two additional axioms exp re s s how momentaneous subjectsar e relat ed . When each of two mom entaneou s subjects x x an d x 2 stands inthe relation " < " with one and the sam e th ird mom entaneou s subject x s,

e i ther x a n d # 2 ar e iden tical or they stand in the relation " < " and thu s aregenidentical.

A3 O 5 ^ I < 3 ' A X 2 < 3 ' - *G# 1#2

A3.6 x < 3 x < y 2 - +Gyiy 2

Using D 3.2, it can be shown th at th e re lat ion of geniden tity is anequivalence rela tio n . An addition al pro position is introdu ced for abbr evia-tion purposes .

S3.1 x < y - >Gxy

53.2.1 Gxx53.2.2 Gxy - >Gyx53.2.3 Gxy A Gyz - +Gxz

Since the relat ion of geniden tity is an equivalence relat ion , the field of" G " could be divided into m utu ally exclusive classes that would satisfy th etwo following con dition s: a) G holds for each p air of individuals in any oneof these c lasses ; and b) if an individual in one of th ese c l a s se s bears therelation " G " to anoth er individual, then this second individual belongs to

th e same c l a s s as the first ind ividual. We could th us define th e notion ofsubject as being the c l a s s of geniden tical m om entan eous subjects.

The following five pro positions ar e tr ivial. The reaso n for introducingthem is that they will allow us to shorten cer tain proofs .

53.3 Gxy x ~ y >x = y

53.4 x ~y A i = 3; *iGxy53.5 iGxy A Gyz - *iGxzS 3

O6 X < y A x 2 < y A X X ~ x 2 *Xi = #2

S 3 7 x < y 1 A x < y 2 A y x ~y 2 - *y x = y 2

4. D omain of pr op er tie s. P rim itive notion "A" (actual). After deal-ing with the be a re r of chan ge, i. e. that which und ergoes the chan ge, we turnto what we agree to call "p r o p e r t ie s". A pro pert y is that which am om entan eous subject h as or could have. D epending on whether am om entan eous subject ha s or could have a ce rt ain pro per ty, we shall speak

respect ively of an "a c t u al " or "po ssible " pro perty of th is mom entaneoussubject. Let u s clarify what is meant here by giving an exam ple. A manwho at this very mom ent is sitting, ha s the actu al prop erty of the sittingpo sition. He could as well be standing up or lying down; because of this thestanding up or lying down positions are not actual but possible propert ies ofthis momentaneous man


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Greek letters a, , 9 . . . will be used to denote the individualvariables for properties.

In the scholastic philosophy there are hardly any notions that have suchan important role as the pair "a c t " and "pot en cy". It plays a decisiverole in explaining such problem s as coming to be and duplication ofspec ies. Th is pair of notions have been used with various shades ofmeaning. A formalized language requ ires however an exact determinat ionof notions. For th is we shall in the future speak of the pa ir "in actu" and"in poten tia", respectively "ac tu al " and "pot en tia l".

Before giving the definition of change, it is necessary to elucidate thenotion "in ac tu". We already explained what was meant by an actualproperty (determination). The primitive notion "in a ct u" will express thata momentaneous subject possesses a determination as an actual determ ina-tiono

Pn4.1 Ax : x is actual with regard to a.

F or Aquinas as well as for Aristotle no concrete subject could existwithout some determination. A subject always exists as an individual, i.e.with one or more of the ch arac te rist ics referred to above. The next axiomstates this fact.

A4.1 laxa

5. Pr imit ive notion " F " (capable of). Defined notions " P " and "V"(potential and change). The notions of subject and property provide us withthe basis for the definition of change. We want, however, to introduce anadditional primitive notion, namely "capacity", and a defined notion "inpo ten t i a " (be ing- in- potency).

It would be a mistake to consider the notion "in pot en tia" as implyinga mere denial of being. It is our intention to prevent such a misunder stand-ingo In the scholastic philosophy the notion "in po tentia" seems to be takenas a prim itive notion. We have chosen instead to introduce first the

primitive notion "cap ac it y" and to lead back to it the notion "in pot en tia ".When we say that a c erta in property is a possible determinat ion of amomentaneous subject, the justification for our saying so must reside insome relat ionship. It is this relation ship between momentaneous subjectan d property that we want to express by the notion "capacity".

Pn .l Fxa: x is capable of a*

Undoubtedly a momentaneous subject which possesses a particularproperty, is capable of th is propert y. Moreover, we seem to be justified instating that a subject which at a given point in time is capable of a certain

determina t ion , was also at ea rl ie r moments of its existence capable of th isdetermina t ion . We so obtain the following two axioms.

A5.1 kxa - * Fxa

A5.2 x x < x 2 x2a > FxiCi

The notion "in po ten tia" can now be defined.


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D 5.1 Pxa =of Fxa Axa

P # could read: x is potential in , i.e. x is capable of having the property , but at the particular moment of its existence, x does not possess thispr ope rt y. A few prop osition s could now be d erived.

55.1 G # i # 2 kxiC Px 2a * lx x ~x 255.2 kxa Px - = 055.3 P# 3j3(A*|3 A a = )55.4 j i < x 2 A#i x 2a * P^i

In everyday language, S5.1 says that it is not possible for two genident icalmomentaneous subjects to exist simultaneously when one is ac tua l and theother poten tial with regar d to the same pro per ty. The propo sition S5.2stat es t hat for one and the same mom entaneous subject to be actu al andpotential is only possible with regar d to different de ter m inat ion s.

We have now all the requ ired assum pt ion s for the explicit definition ofchange. As it was men tioned before, by change we mean a p ro ce ss in theco urse of which one and the same subject shows at different poin ts in timedistinct p ro per t ie s. One and the same subject is observed at differentmoments of its existen ce . Suppose that at the lat er moment th is subjectshows a determ ination which it did not have at the ea rli er mo men t. We thensay that th is subject has undergone a ch ange. Th ree th ings ar e to bedistinguished: the initial state (before the change), the final state (after thechange), and th at which un dergoes the change, the subject. An apple whichfirst was gree n , has becom e red; the apple has undergone a change, yet itremains the same apple.

A change could also be d escribe d as a p ro ce ss of the tr an sition frombe ing- in- potency to being- in- act with regard to a certain determ ination.Let us con sider the example of the formation of a statue from a block ofmarble. In it s initial state the m arble block con tains m erely the capacityto become a stat ue . In its final stat e, the m arble block has actually takenthe form of a statu e. The tra n sition from the possible determ ination -

marble statue - to the actual determin ation - marble statue - con stitutesthe change of the mar ble block. The marble is that which un dergoes thechange, the subject. It is worth n oting here that one could say that a subjectis in a state of change with regard to a certain determination only when thatparticular change could be conceived as one continuous p ro ce ss of act u-alizat ion . Th is would not be the case when dealing with instant aneo uschanges. Were we to form alize the definition of change as one con tinuouspro ce ss of actu alization , we would need a new pr im itive concep t. We wantto avoid using such a notion which, we think, would unn ecessar ily com plicateour formalization o We shall show in an appendix how such a form alization

could be obtained,,In our pr esen t formalization we con sider the essenc e of a change to be

th e en d- resu lt, i.e. t he acqu isition by one and the same subject of a newdetermination.

D 5O2 Vyy 2ot ^ / J ' K ^ ky 2a A Vy(y 1*y


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VyiyzQ. re ad s: The subject to which belong the mom entaneou s subjectsrep resen ted by the variables y and y 2) has undergone a change with regard

to the determination a during the time interval determined by y 1 and y 2i andthis change came to an end at the point in time belonging to y 2 .

It should be understood that in our definition of change we have avoidedany reference either to an active or to a passive component. We want tocon sider here change as neu tral. The following propositions could now bederived.

55.5 \ y y 2a - Gy y 255.6 yy 0y 2tt

A V yi ^ - *Gy oy55.7 Vy 0y 2 a A y o


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Pn .l Mxya: x changes y to a> i.e. the subject represented by x moves the

subject represented by y to a, namely in the time interval determined by x

and y.

It is important to remark that the possibility or im po ssibility of thegenidentity of th e subjects rep rese n te d by x an d y remains an openquestion . Th is prim itive n otion 6.1 will be henceforth r eferr ed to as the

relation ship ' 'm over - m oved '\ We so introduce explicitly the active

component (mover) along with the passive component (moved).

It would seem obvious tha t for two subject s to stand in the rela tio n

' 'mo ver- m oved" they mus t exist sim ultaneously. Th is requirem ent is

stated in the next two axioms.

A6.1 Nlxya --> ?x x (xi ~ y X < x )A6.2 Mxya - * ly 0 (x ~ y 0 A y Q < y)

Th ese two axiom s togeth er with the fact that the relatio n " < " is dense

allow the inference that a subject which exists at two distinct moments,exists also in the tim e inte rval determ ined by th em . We can now derive

some addi tional propo sit ion s.

S ol Y xya - x ~ y56.2 fJixya y 0 < y < y y 0 ~ x -> 2x 1 {x 1 ~ 3 i i A^ < x 1 )

56.3 Mxya X < x 1 < x 2 A X 2 ~ y - * 3 y x ( X - y 1 A y i < y)

With regar d to the relation "M ", we want to specify the moment at whichth e change is com pleted. F or th is, we use two a xiom s.

A . 3 Mxya > kya

A6.4 V xya * x ~ y 0^ y x< y > Aî

(x ~ y o< y Ax 2ya - Ôxx 2

We now obtain the following t h r e e p r o p o s i t i o n s th e meaning of which i s

c l e a r .

S6p4 Mxya - > 3;y 0 (x ~ y 0 VÔ^Q:)S6.5 h xy 2 a X ~ y x< y 2 - * Vyiyza

S6 e6 Mxya kxa Gxy

Up to now no an swer h as been given to the question as to whether oneand the same subject with regard to one and the same determ ination could

be at th e same tim e m over and moved in a chan ge. The prop osition S6.6

gives the an swer for the case where th e m over is actu ally in po ssession of

th e determination,,A third requ irem en t for the nu m erica l unity of a change is the


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A THEORY O F CHANGE 381

numer ica l unity of the time interval during which th e change takes place.We introduce it with the following axiom.

A606 fsAxya x lxilyibc 1 ~y 1 Ay o^y 1


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(par ticipat ion) in the comm and of the E nglish language, we could right ly saythat the first has a great er share than the second, and the latte r a great er

participation than the th ird. The following primitive notion will serve toestablish the comparison referred to above.

Pn7.1 Bxya : x has at most as large a share in a as y.

It is evident that this relation "B" is reflexive and tra n sitive. Let usthen form ulate th is in two axiom s.

A7.1 BxxaA7 O 2 Bxya A Byza > Bxza

The relation " B " is also a smaller- equal- than relationship. To simplify

our use of the preceding relation, we introduce now two additional concepts.

D 7 .1 \Nxya =of Bxya A iByxaD7 O 2 \xya = D/ Bxya A Byxa

The relat ion "W"i s irreflexive, asymm etr ic and tra n sitive as the followingpropositions formulate it.

S7.1 VIxxaS7 O2 Wxya - > 1 WyxaS7.3 W xya A W yza - Wx

Let us consider now a momentaneous subject x which actuallypossesses a certain determination a and a mom entaneous subject y which iscapable of possessing a but does not as yet possess the determination a.We should be justified to say th at y has a smaller participation in a than x.We formulate this in an axiom.

A7 O 3 Pxa Ay a - * Wxya

If any two momentaneous subjects happen both to be actua lly in possessionof a ce rt ain det erm ina tion , we find our selves justified to say that they haveequal share in .

A7 O 4 Axa A Ay a > \xya

In the hypothesis that a subject rep resen ted by x brings a subject y into thedetermination a during the time interval determined by x and y, and th iswithout any ou tside influence, one would seem justified to formulate thefollowing axiom which sta te s t hat the momen taneous subject x has at leastas large a part icipation in the determ ination a as any momentaneoussubject z which actually possesses the determination .

A7.5 b xya A Aza * Bzxa

We then obtain the proposition :

S7 4 VAxya A Pza - * Yizxa

We should re ca ll he re that, in setting up the domain of mom entaneoussubjects, we em phasized th at each m omen taneous subject has to be con-sidered as a whole. With th is in mind, we can see that the proposition S7.4


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A THEORY OF CHANG E 383

im plies the imp ossibility for a m om ent ane ous subject to be m over withrespect to a determination a and at the same t im e to be also in a state of

potency with regard to th is same determ inat ion. It is importan t to re m ar kt h a t we have now reached the nucleus of our study o Whenever a mo m en -taneous subject is m over with re gar d to a det erm in at ion a, the followingtwo clai m s hold: first, th ere exists a relationship between this momen-taneous subject and the dete rm inat ion a; secondly, the momentaneoussubject ha s at le ast as large a pa rt icip at ion in a as any momentaneoussubject which is ac tua lly in possessio n of . A fundamental question mustbe settled h er e. Whenever a mo me nt an eou s subject stan ds as m over withregard to a determ ination , is it requ ired that it be itself in actualpo ssession of the det erm ina tion a? If this were the case, then it would of

co ur se be impo ssible for t hi s m om ent an eou s subject to be at the same t im ealso in a stat e of potency with resp ec t to the det erm in at ion in questionsince being- in- act and being- in- poten cy with regard to the one and samedetermination at the same time would be contradictory.

S7.5 b xya iPxa

I t is now po ssible to give a satisfact ory an swer t o the question wheth era momentaneous subject could be at the same time mover and moved.

57 06 Mxya- *~\Gxy

Th is pro po sition can be gene raliz ed t o cover a tim e int erval as follows:

57 07 Mx y a *xi


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384 L A U R E N T L A R O U C H E

R E F E R E N C E S

[1] Salamucha, J. , "Dow d 'ex motu ' na istnienie Boga, analiza logiczna argumen-tacji sw. Tomasza z Akwinu," in Collectanea Theologica, XV Lw w (1934), 53 ff.

[2] Salamucha, J. , "The Proof 'ex motu ' for the Existence of God: Logical Analysisof St. Thomas' Arguments," in New Scholasticism, XXXII (1958), pp. 334- 372,English translation of [1].

[3] Sobocinski, B., "Jan Salamucha (1903- 1944): A Biographical N o t e , " in NewScholasticism, XXXII (1958), pp. 327- 333.

[4] Bendiek, J. , "Zur Logischen Struktur der G ottesbeweise," in FranziskanischeStudien, XXXVIII (1956), pp. 1-25.

[5] Rivetti- Barb , F ., "La Struttura Logica Delia Prim a Via Pe r Pro vare L sis-tenza Di Dio, " in Rivista di Filosofia N eo- scolastica (1960), pp. 241-320.

[6] Rivetti- Barb , F ., "Ancora Sulla Prim a Via Pe r P rovare L sisten za Di D io, "in Rivista di Filosofia N eo- scolastica (1962), pp. 596- 616.

[7] Lewin, K., Der Begriff der Genese in Physik, Biologie und Entwicklungs-geschichte, Springer, Berlin (1922).

[8] H ermes, H., Eine Axiomatisierung der allgemeinen Mechanik, Leipzig (1938).

(T o be continued).

Laurentian UniversitySudbury, Ontario, Canada

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Examination of the Axiomatic Foundations i