Formalized Proofs of the Existence of God

Edward Nieznański, Agnieszka

Burakowska

Formalized Proofs of the Existence of

God

Collectanea Theologica 64/Fasciculus specialis, 109-122

Collectanea Theologica 64 (1994) fasc. specialis

FORMALIZED PROOFS OF THE EXISTENCE OF G O D1 Rev. Jan Salam ucha (1903-1944) was the first to formalize in 1934 A quinas’ argum ent ex m otu o f the existence of G od, inserted in Summa

contra Gentiles (1,13). This first in the history of theodicy formalized p ro o f

for the existence o f G od was published exactly in Collectanea Teologica 15 (1934), 53-92. To com m em orate the 50 th anniversary o f the execution of Rev. Jan Salamucha by Nazis and the 60th anniversary o f the publication o f the m entioned above form alization giving birth to the new m ethod of coming to G od, let us try to estimate m ore im portant achievements, tools and ways to the absolute o f the succusors o f Rev. Jan Salam ucha’s idea.

Now adays, years after the investigations around the form alization of theodicy we are ready to see the m ain sources of its complexity and also m istakes in the constant confusion o f two different ways o f thinking: logical and metaphysical. Since any qualifications and metaphysical relationships are in their nature m odal - existential a philosopher has no way out from that maze o f problems, even with the help o f a form alization if he confuses the very metaphysical modalities with logical ones or if he refuses to acknowledge the empirical basis to any kinds o f existential judgem ents.

Generally speaking, there are two po-Nible attitudes o f philosophers tow ards the treatm ent o f the problem o f the existence o f the absolute: either - according to some - it is a purely linguistic issue and its solution is a simple consequence o f terminological agreements or - according to the others - it is, precisely reverse, a factual problem and its solution does not comprise in only linguistic conventions. Detailed divergences are con­ siderable and, not rarely, also extreme.

0.1. PRO BLEM O F T H E A B SO L U T E A N D L IN G U IS T IC C O N V E N T IO N S

First o f all, some semantic conventions lead to contrasting solutions, when they themselves, a priori, set the logical value of existential sentences. To give an example, the so called semasiology which was practised by Stanisław Leśniewski in 1910-1915, claimed false all existential sentences. A sentence in its canonical form: subjectively-predicative - in accordance with the settlement o f semasiology - states only the possesion by the object

A the quality B. M eanwhile the „existence” is n ot a quality, so it cannot be possessed by any object. Therefore a sentence „The Absolute exists” is in this estimasion analytically false.

However, ju st an opposite metathesis to this proclaim ed by Leśniew­ ski about the logical valute o f existential sentences, was brougt forw ard by W illard Van O rm an Quine (1951)2, to whom the thesis Vx x exists is trivially true, because to be is to be the value o f a variable. Thus there is not any problem o f existence o f the absolute, but only the problem o f its nature.

0.2. T H E Q U E S T IO N O F TH E M IN IM U M O F A S S U M P T IO N S

Both o f these m utually exclusiwe options o f logicians are, as a rule, rejected by a philosopher for whom the issue o f the existence o f the absolute is a factual problem and the thesis o f the existence o f G od is neither trivially true nor trivially false and suggests itself with obviousness directly neither a priori n o r a posteriori, since it is not the consequence o f only linguistic conventions or a form ally recorded sentence from experience. Thus its acceptance can be m ade only on the way o f a p ro o f and not otherwise than within the theory to which this p ro o f belongs. But in the m atter o f proofs created in the scheme o f philosophy the tendentions o f a logician and metaphysician are also in general divergent. While a logican tends especially to the exposure o f the formal correctness o f deduction by formalizing it, a m etaphysician practically does not attach any im portance to form al considerations. A nd when, in turn, a m etaphysican concentrates his whole effort on gaining other people’s certitude m ultiplying unendingly axioms o f a theory, a logician, on the contrary, aims at the m inimalization o f the num ber o f original statem ents often having no regard to a sense or anybody’s willingness to accept them. Also, in the same way, there misled to pragm atic emptiness, i.e. to the suspension o f the judgem ent about the existence o f the absolute both uformalized argum ents for the thesis o f the existence o f G od which do not give any possibility o f the verification o f the logical corollary from premises to a conclusion, as well as taken in isolation, i.e. placed beyond any theory, formalized proofs o f the existence o f the absolute based on some axioms pulled from the context o f metaphysics. Hence, n o t easy to obtain, a comprom ise o f aims and a golden means in theodicy requires probably such a form alization which accom p­ lishes not one, but two m inima at the same time:

1) deductively inevitale m inimum of axioms w ithout which the thesis o f the existence o f the absolute could be proved;

2) pragm atically inevitable m inimum o f axioms w ithout which it is impossible to understand both simple and complex terms and to assent to oryginał and derivative propositions.

2 The number (n) in brackets occuring at a surname means here, as well as in further places, a bibliograhical abbreviation o f a gliven author’s work from the year n, according to the list o f bibliography placed at the end o f the article.

A t the same time the first minimum should be included in the second and not inversly.

1. Review of the accomplished formalizations of theodicy

All the attem ts made so far o f form alization o f chosen fragm ents of theodicy were undertaken either within the theory o f relations or under the basis o f the logic o f predicates.

1.1. F O R M A L I Z A T I O N S W IT H I N T H E T H E O R Y O F R E L A T IO N S

A nthony Kenny (1969) noticed th at num erous Thom ist argum ents come under the same form al scheme taken from the calculus o f relations. E. Nieznański (1980) constructed the generally-logical theory o f form al properties and extreme elements o f binar relations for the use o f the form alization o f theodicy.

1.1.1. The application o f the idea o f linear orders

However, in the initial phase o f the form alization o f the proofs for the existence o f G od the idea o f a chain predem inated indivisibly. J. Salamucha (1934), as well as J. Bocheński (1935), L. Koj (1954) and J. Bendiek (1956) treated the relationship o f metaphysical m ovem ent as a relation arranging lineary a set o f beings and assuming its finiteness, in the conclusion they were receiving a statem ent th at there is the element first and minimal at the same time, i.e. primum movens immobile. These authors also thougt that they were formalizing the argum ent ex m otu o f St. Thom as Aquinas enclosed in Summa contra Gentiles 1,13 and Summa Theologica I,q.2,a.3.

The idea o f a chain comes back later again, first it was suggested by Peter G each (1963) then in a certain version of a cosmological argum ent formalized by E. Nieznański (1992). This time however, not the chain of objects is a question, but a relation arranging lineary the family o f a set of objects. By assuming the sign A, to denote the set o f all beings actual in the m om ent t, one designates the set o f all „acual worlds” , i.e. the family of sets

{ A t } t e T (where „ T ” means time continuum ) and the relation arranging

lineary this family o f sets and then one assigns the conditions of unemptiness o f the product o f all actual worlds i.e. п 1етА,.

1.1.2. Relations partially arranging all beings

U nder the influence o f criticism o f the assum ption abo u t connectivity o f the relation o f m ovem ent as a thesis empirically false the criticism first carried out by F. Rivetti Barbo (1960) m any philosophers engaged in the form alization o f theodicy resigned from the idea o f a chain in support of relations partially arranging the set o f all real beings. A nd thus Korneliusz Policki (1975) presented for the first time a formalized argum ent ex m otu using the lemma o f K uratow ski-Zorn. In a similar way Z o rn’s lemma was

employed by Reinhard K leinknecht (1991) to the argum ent ex ratione causae efficientis.

The idea of a p artital arrangem ent was also used by K u rt Gödel, where he employed the relation o f inclusion in the family o f all subsets of the set of real beings. K u rt G ödel’s ontological p ro o f for the existence o f G od (from 10th. Feb., 1970) in its philosophical contents approxim ates mostly the ontological argum ent o f Leibniz from Monadology 41, 44 and 45, whereas in its form al layer it assimilates especially Ch. H artsh o rne’s (1941, 1961, 1962) formalized on the basis o f m odal logic S5 p ro o f o f St. Anselm for the existence o f a m aximal perfect being from Proslogion c. 2-3. In a simplified, not-m odal version it was presented by Essler (1991).

There are assum ed three axioms o f positive features: A x l. V F V G ( Fe G a FePs -► GePs)

„Oversets o f positive classes are positive” . Ax2. V F (-FfiPs ;=± -, FsPs)

„O nly a given class or its complement is positive” . Ax3. n P s ε Ps

„The product o f all positive classes is positive its e lf’.

The conception o f G od is identified by G ödel with summum bonum according to the definition:

Df. G t = n P s , i.e.

Vx [xeGt ^ V F (FePs ->xeF)]

In other words axioms A xl and Ax3 state tn at the family o f positive classes is a filter, while the axiom Ax2 additionally settles th at this family is a maximal filter (ultrafilter), generated by summum bonum G t. Three im portant theorem s result from the m entioned axioms:

T h l. V F [FePs -»■ -> (F = 0 )] „N o positive class is em pty” . The proof:

FePs, F = 0 1- 0fiPs, Ax 1, F Ç U (- UePs, Ax2 к -, (-U«Ps) н - (0sP s) i- contr. Th2. -, (G t = 0 , because Df, Ax3 and T h l.

,,Summum bonum exists’.

Th3. Vx Vy (xeGt -> x = y)

„A t m ost one being is summum bonum ”. The proof:

xcGt, yeGt / -, {x}ePs, Ax2 н -{x}aPs, D f н χε - {x} i— , (x = x) н contr. / н

{x}eP s, V F (FePs -*· ysF) к ye{x} ι- x = y

The theorem s Th2 and Th3 state exactly th at there is precisely one being which is summ um bonum . However, it is obvious th at only a formal principle, the idea with an ultrafilter is valuable in G ödel’s proof, whereas the metaphysical contents are so extremaly poor in it th at nothing interferes with giving in the above notation to the constant Ps the sense of a negative gualification and obtaining, as a re su lt, summum m alum - a one - element personification o f evil.

1.1.3. Multiplicative quasihalfstructures

Still another set theory form ulation besides the chain conception, partial arrangem ents and ultrafilters was the treatm ent o f metaphysical relationships as the so called m ultiplicative quasihalfstructures, i.e. the relation R, which for each couple o f elements x and y belonging to its field satisfies the condition: x = y v xRy v yRx v 3 z (zRx A zR y) The form ali­ zations o f F. Rivetti Barbo (1960, 1962, 1966, 1967), Ivo Thom as (I960) and W. K. Essler (1969) went exactly by this trail.

1.2. F O R M A L I Z A T I O N S O N T H E B A SIS O F T H E L O G IC O F P R E D I C A T E S

Fragm ents o f theodicy form alized within the calculus o f relations - it is to be emphasized - even if they avoided form al mistakes and made deduction - as to its logical form - verifiable, neglectes completely a pragm atic aspect shifting the whole weight o f the im plem entation o f its ontological assum ptions to classical m etaphysicians who do not occupy themselves with form alization. These, however, leave such expectations unnoticed refusing any value at all to formalizing endeavours, or only accusing them o f a semantic deform ation o f metaphysical judgem ents. One should rem ember th at the language o f classical philosophy is a system of sings for objects and qualifications and n ot for sets and elements. So even if we successfully use set theory constructs to m etatheoretical descriptions o f semantic models o f form alized theodicy, the very theory however should probably be created on the basis o f the logic o f predicates.

1.2.1. M odal formulations

In S5-modal calculus o f predicates o f an upper order are done (specially num erous) form alizations of the, so called, ontological argum ent and am ong them the m ost fam ous (mentioned above) form alizations of Ch. H arsthorne and K. Gödel. In as m uch as H artsh orne’s calculi are entangled in the equivocation o f logical and metaphysical modality, in G ödel’s p ro o f - as W. Essler showed (1991) - logical m odalities occur inessentially. The often raised objection o f paralogism 3 in relation to tertia

via o f St. Thom as A quinas, particulary on the account o f the thesis Quod possibile est non esse, quandogue non est, resembling an not-tautological

form ula in ordinary model systems < > p -> p, can be disproved by the application o f a special extention o f the system S5, in which the equivalence would be obligatory: D p < )p ^ p, i.e. when m odalities are used inessentially and the m odal calculus is generated by the semantic models of K ripke with identity as the equivalence relation o f the accessibility od possible worlds.

In S5-modal logic o f predicates of the first order A nthony Kenny (1969) and E. Nieznański (1991) formalized an argum ent ex possibili et

3 See K łósak, W poszukiw aniu Pierwszej Przyczyny, (Tow ards the First Cause), II, Warszawa 1957, p. 124.

necessario and also E. Nieznański (1991) form alized an argum ent ex ratione sufficienti.

1.2.2. Classical formulations

As it seems, the form alizations o f theodicy carried out in the classical logic o f predicates are least exposed to m aking semantic m istakes, since one avoids in them in a simple way - owing to the very nature o f a language - confusing a quality with m ultiplicity and a logical m odality with metaphysical one.

The argum ent ex m otu o f the existence o f G od unfolded by Leibniz in

Demonstratio Existentiae Dei ad M athematicam Certitudinem Exacta was

form alized twice by K rystyna Blachowicz (1982, 1992). These form aliza­ tions are based, however, on a special postulate o f Leibniz, which in K. Blachowicz’s n o tation is in the following shape:

Яу Vx (xPy ^ 0 (x))

(where ’xPy’ m eans ’x is a p art o f y’) and which immediatelly leads in an open way to Russell’s antinom y (when e.g. 0 (x) is a form ula xPx).

W ithin the classical logic o f predicates o f the first order also formalized: Józef M. Bocheński (1989) - all quinque viae o f St. Thom as, E. Nieznański (1977,1979,1980,1981,1982b)-classical argum ents ex ratione sufficienti, E. N ieznański (1980, 1982a, 1984) - argum ent ex ratione cause

efficientis, and H. G entahaler and P. Simons (1987) - the cosmological

argum ent o f B. Bolzano.

2. Argument ex ratione sufficienti

The consideration o f the existence and nature o f the absolute must originate in some model o f reality. Let us return, then, to (A t}teT - the family o f all „actual w orlds” .

2.1. T H E N O T I O N O F A B E IN G

A t the beginning, we assume the principle o f the conservation of existence establishing th at no actual world is an empty .set, which in the language o f the creating theory - in which individual variables x, y, z,... represent any uncontradictory individual objects, not only beings - we write in the axiom 4 „Som ething always exists” :

A l. V t Hx Axt (where „A xt” means as m uch as ,,x is present at the m om ent t”). The object, which is at any time actual, we call a real being (we read ,,Bx” : ,,x is a real being” ) according to the definition:

df.B: Bx T t Axt.

4 The axiom s used in deduction are designated by a capital letter ,,A ’-: A 1,A 2,...,A 10; whereas the axiom s which are not used as premises o f any proof, but are added because o f purely pragmatic reasons, are denoted by a small letter „a” : a l , a2, a3, a4.

2.2. T H E N O T I O N O F T H E R E A S O N O F E X IS T E N C E

The transm ission o f existence from some beings o f the actual world into other beings o f next worlds can be accomplished only through becoming or lasting, because either some things become other through a connection, disintegration, change o r they last as identical in time, as unchangeable. The m ost im portant existential connection is the s.c. reason o f existence. Rxy, i.e. x is a reason for the existence o f y if and only if y cannot exist w ithout x, when the existence (and the essence) of x is a necessary condition o f existence (and essence) o f y. A nd although the implication

a l. Rxy ;=± -, < > (By л —, Bx)

is universally im portant for all objects x and y, a reverse im plication, however, is not always satisfied, because the metaphysical m odality ,,y cannot exist w ithout x” is a connection (richer) stronger than the connection < > ( B y / \- , Bx)” . The m etaphysical sense o f the notion „im possibility o f existence w ithout” , i.e. „the reason o f existence” is not suitable for logical analysis by means o f simpler notion and because o f this we treat them as an original notion. First we state th at nothing can exist w ithout itself, so:

A2. Vx Rxx,

which m eans reflexivity o f the relation R, and we also assent - secondly - to the impossibility o f existence o f y without x, while y cannot exist w ithout z, and z w ithout x, so:

a2. Vx Vy Vz (Rxz л Rzy -> Rxy) which means then transitiveness o f the relation R.

Very likely, the m ost im portant feature o f the relation o f the reason of existence is the fact th at existence is hereditary because o f the converse of this relation, and nonexistence - because o f the very relation. Always, when у cannot exist w ithout x, the existence o f у guarentees the existence o f x, and the nonexistence o f x states also th e nonexistence o f y, so:

A3. Vx Vy (Rxy a By -*· Bx).

The reason o f existence is by no means a simple connection, on the contrary, it is a sum o f m any existential relations. First, the identity relation enters into its com position, which results directly from the axiom A2:

T l. V x V y (x = y -> Rxy).

2.3. B E C O M I N G A N D C H A N G E

Let us assume two abbreviations: „Sxy” for the predicate ,,x becomes y” and ,,Zx” for „x is changeable being” . Becoming is a process, which happens only in beings: one being becomes to other and both are real beings, so the relation o f becoming and its converse enter into the com position o f the relation o f the reason o f existence.

A4. Vx Vy [ÎSxy v Syx) ->· Rxy].

Becoming o f a being is also reverse to its identity at time, hence the axiom is obligatory:

A5. V x V y [Sxy -+ V t (Axt -*· -, Ayt)],

from which the thesis o f irreflexivity o f the relation o f becoming o f a being results. It means th at nothing becomes itself:

T2. Vx Vy (Sxy a By -> -, x = y), since:

Sxy, By, x = y, A5 h V t (Axt -»■ -, Axt), (p -* -, p) -> -, p н Vt -, Axt r- I— , H1 Axt, df. В I- Ht Ayt b Ht Axt I- contradiction

It is also easy to prove that beings which are becoming are not constant entities:

T3. Vx Vy (Sxy л By —» H t - , Axt л H t -, Ayt), since:

Sxy, By l· Rxy, A 3 1- Bx, df. В l· H t Axt, H t Ayt, A5 l·- V t (Axt -*· -> Ayt), V t (AVt -» -, Axt) l· H t -, A xt л Ht -, Ayt

Therefore each limit o f the relation o f becoming is called a changeable being according to the definition:

dt. Z: Zx Ну [By л (Sxy v Syx)]

W hen x is a constant being, i.e. when V t Axt, x is always one and the same x, so it is an unchangeable being:

T4. Vx (V t Axt -* -, Zx), since:

Vt Axt, Zx, df. Z l· Ba, Sxa v Sax, A5 н V t (Axt -+ -, Aat) k V t Axt -» -> V t -, A at l· V t -, A at, Ht A at b contradiction.

W hereas a reverse im plication is not valid, because one cannot a priori exclude the existence o f beings unchangeable in a time, arising ex nihilo or undergoing annihilation. Instead, it is obvious th a t there are changeable beings:

a3. Hx Zx.

One should distinguish contingency from changeability. The being x is contingent when it has at least one necessary condition o f its existence ab alio, so when H z (->z = x л Rzx). It is clear, at the same time, th at every changeable being is as well contingent, though a reverse connection does not have to happen:

T5. Vx [Zx -> Hz ( - ,z = x л Rzx)], since:

Zx, d f .Zb Ba, Sxa v Sax, A 4 Ь Rxa, Rax, A 3 Ь Bx, T21— , a = x I— , a = x л

A Rax b Hz (-, z = x л Rzx).

2.4. T H E N O T I O N A N D P R I N C I P L E O F A S U F F I C I E N T R E A S O N O F B E IN G

One special case o f the reason o f existence is sufficient reason of a being. The notion o f „a sufficient reason, which does not need any other reason” was introduced by G.W . Leibniz, who cut the left dom ain of the relation o f reason to the set o f beings which do not have the reason ab alio. By assuming the abbreviation „D xy” for the predicate ,,x is a sufficient reason o f existence o f y” we define:

d f. D: Dxy Rxy д -, Hz (-, z = x л Rzx).

Classical philosophy coped in three different though equivallent ways with the m ain metaphysical problem: why rather is there som ething than nothing? St. Thom as m ade use o f the rule: non est procedere in infinitum. Namely, if we take into account conected underrelations o f the relation R and we will call the maximal o f them series (even when they m ake cycles), a w hether а repress in infinitum is possible in them

w hat it means. J. Bendiek (1956) accurately noticed that: „das V erbot eines regressus in infinitum nicht a u f das Unendliche, sondern a u f die Anfang- losigkeit bezieht” (p.10). Thus St. T hom as’ impossibility o f regress in infinitum points out not its finiteness (as it was assumed in the first form alizations o f theodicy), b u t possessing a beginning. C onditioning reasons by reasons cannot be beginningless, even though it was infinite. St. T hom as’justification o f the impossibility o f beginningless series by the fact th at the nonexistense o f a beginning w ould involve the nonexistence o f the whole series, J. Bendiek estim ated as petitio principii, a vicious circle: the beginning o f a series is, because a beginningless series would be impossible, while a series cannot be beginningless, since it would not exist at all w ithout a beginnig. However one could not, in our opinion, im pute to such a m ature thinker as St. Thom as as primitive m istake as all th at vicious circle. If St. Thom as in his ways to the absolute was reality determining something m ore than only the obvious equivalence of the impossibility of beginninglessness with the necessity o f the beginning, he could only yield to a difficult to discover equivocation of the w ord „exist” : once in a sense of the existential quantifier and for the second time, as a predicate. F o r when speaking abo ut the beginninglessness o f a series we have in m ind the fact that -> 3 x Vz (Rzx -> z = x), when we, however, further say th at if there is not the beginning a, so there are not next elements o f the series, either since Vy (R ay л -i Ba —*■ —> By) - in accordance with the axion A3 - so there is m ade an equivocation because o f the ambiguity o f „existence” , and m oreover - in forbidden way - the notation „ a ” is attributed to a contradictory thing. In this situation we understandt finally that St. Thom as adopted axiomatically the theorem: V y (By - » 3 x [Rxy a

л Vz (Rzx -> z = x)}, though the added com m entary for the reinforcement o f the readiness to accept this axiom is not efficient and misses the aim.

The other approach to the attem pt o f the final solution o f the problem o f existence in the Thom ist axiom o f the impossibility o f the existence of series consisiting only of contingent beings: V y [By -> 3x (-.x = у л Rxy)]- However the com m entaries appearing sometimes which are to confirm the conviction abo u t the rightness o f this axiom because of the fact th at series o f contingent beings (as it is claimed then) are in themselves contingent beings - m ake fallatium com positionis5.

A particularly significant way o f solving the problem o f a genesis of existence is the application o f Leibniz’s principle o f a sufficient reason:

A6. Vy (By -> Hx Dxy)

Contingency, i.e. conditioning o f beings in existence ab alio, even though it reached infinity, does not account for the fact o f existence; the beginning­ lessness o f the series o f reasons does not give an explantion but only suspends it and pushes into infinity and begininnglessness, i.e. to nowhere.

We can notice straight away th at there are two, at the utm ost, possible attitudes tow ards the nature o f existence: either universal variabilism which

excludes the principle o f a sufficient reason is obligatory or - on the contrary - there is valid the principle o f a sufficient reason, which eliminates the possibility o f universal variabilism:

T6. Vy (By -» 3 x Dxy) - * n Vx (Bx -> Zx), since:

Vy (By -> 3 x Dxy), V x (Bx -» ZxL A l, af. В l· 3 x Dxa l·- D ba, df. D(- h R b a , V z(R zb -> z = b), А З к В Ы -^ Ь , df. Z I-Scb v Sbc, T2, A4l--iC = b, Rcb k c = b l · contradiction.

2.5. T H E N O T I O N O F T H E A B S O L U T E

Let ,,αχ” be the abbreviation for the predicate ,,x is the absolute” , df.a: ax ^ Bx л V z (Rzx ;=± z = x).

The absolute, therefore, is called the being which is itself the only and exclusive reason of its existence. Now we can prove the existence o f the absolute:

T7 since*

A 1, df. В I- 3 x Bx h Ba, À 6 1- 3 x Dxa l·- D ba, df. D Rba, - . 3 z ( ^ z = bA Rzb)l- V z(R zb -» z = b), A3,T1 hB b , V z ( R z b ^ z = b),df. a L a b l· 3 x a x . The thesis ab o u t uniqueness o f the absolute: Vx Vy (ax д ау - * x — y) is not, however, easy to prove or reject in the Thom ism , which - although it is philosophy - nevertheless knows the theology De Trinitate. On the basis o f the definition o f the absolute it is only obvious that Vx Vy (ах л ay a Rxy

-* x = y), which m eans th at if there were m ore absolutes than one they

could not be in any existential connections with one another.

The existence and uniqueness o f the absolute are, however, apparently protected by every possible m aterialism. It assumes th at m atter is the absolute, m atter which conditions the existence o f every being and, at the same time, is the being, which all the necessary conditions to its existence has only in itself: 3 ^ [ V y Rxy л V z (Rzx -> z = x)], which results directly from the - stronger than the principle of a sufficient reason - postulate of the final reason: 3 x [ Vy Rxy л V z (Rzx -> z = x)], on which materialism bases unconsciously. In this way, using an apparent nam e (onom atoid) „m atter” m aterialists - due to the confusion o f the category o f existence - find the final justification o f the fact o f existence, the final reason o f a being in the abstract idea of m atter undergone ad hoc reification.

2.6. I N T R IN S I C N E S S O F T H E A B S O L U T E

Since the essence and existence o f a being are completely fixed by the factors determ inating a se o r ab alio, then the being which would comprise all the reasons o f its existence in itself and only in itself, would not undergo any determ inations from the aoutside and under the influence o f its own reasons - according to the theorem T4 - it would not undergo any changes, which ju st m eans th at in its n ature it would be a constant being:

A7. V t Axt ^ Bx л V z (Rzx ^ z = x).

In this way we come to the conclusion th at the absolute and a constant being are one and the same being:

T8. Vx (αχ V t Axt), since df. a and A7.

F rom the theorem s T4 and T8 it is also results in a simple way th at the absolute is an unchangeable being:

T9. Vx (ax ► Zx).

A nd since we generally acknowledge th at every m aterial being (every physical body) is changeable being:

A8. Vx (Mx -> Zx),

where ,,M x” is the abbreviation o f the predicate ,,x is a m aterial being” , we m ust also assume th at the absolute in an im m aterial being:

T10. Vx (ax -» -i Mx), since T9 and A8.

Let us complete axiom atically our theory by the notion o f a proper p art as the next existential connection which makes up the relation o f the reason o f a being. Let the notation ,,x < y ” be the abbreviation o f the predicate ,,x is the p a rt of y” :

A9. Vx Vy ( x < y x = y), A10. V x V y [( x < v y < x ) ->■ Rxyl.

It is also obvious th at the connection o f a p art to a whole is a transitive relation (although it is not a chain):

a4. Vx Vy Vz ( x < y л y < z —*· x < z ).

On the basis o f the theory o f a p art we can now introduce the notion of a simple being („Px” m eans ,,x is a simple being”) and complete being („U x ” - „x is a complete being”):

df.P: Px B x л —i 3 y y < x , df.U: Ux ;=± В х л -i 3 y x < y .

Hence we receive two succeeding conclusions: that the absolute is a simple being and th at it is a complete being:

T i l . Vx (ax -» Px), since:

ax, Px, df. a, df. P F Bx, S t (Rzx z = x), Зу y < x а < х , A10, A9 I-a = x, a = x F contradiction.

T12. Vx (ax -> Ux), since:

ax, -.U x , df. a, af. U I- Bx, V z (Rzx z = x), З у х < у 1 - х < а , A10, A91- l· Rax l - a = x, ->a = x l· contradiction.

In this way we come to the conclusion th at there exists the absolute which is a simple, complete, constant, unchangeable, necessary, autonom ous and imm aterial being.

2.7. O M N IP O T E N C E O F T H E A B S O L U T E

Since it is even impossible to conceive such a situation, in which there exist absolute beings unrelated to one another in any existential way, we could assume th at V x V y (ax л ay -» Rxy), which - considerating df.a - would lead to the acceptance o f the uniqueness of the absolute and the conclusion th at Vx [ax -> Vy (By -> Rxy)] i.e. to thetheorem th at the absolute is also the First being. Adding the conception o f the causative reason (Cxy), by means o f axioms: V x Vy (Cxy -* x = y), V x V y V z (Cxy л Cyz -> Cxz), Vx Vy (Cxy -» Rxy) and V у (By л 3 t Ayt -> a x Cxy), we would have to notice that tne immaterial absolute cannot exert physical forcel on material beings, hence its impact on the real world is of

a special nature. Now th a t it has been credited an im m aterial character we could follow further the form alizations o f K u rt C hristian (1957) and Paul W eingartner (1974, 1979), assuming in the first place th at the absolute has a will: V x (ax -> 3 p W Lxp) - where ,,p” is a sentence variable, and the function „W L xp” m eans the same as ,,x wants p to be” - and, secondly, th at the absolute is om nipotent: V x [ax -» Y p (W Lxp -> p)], i.e. that everything which it w ants - is. Thereby philosophical ways would adjoin theological ways: because it turns out th a t the absolute - contrary to the rest of existence which is becoming - „is the one who is” and also is the one „to whom the heaven and earth are obedient” .

Bibliography

Bendiek, J. (1956), „Z u r logischen S truktur der Gottesbeweise” , Francis- kanische Studien 38: 1-38, 296-321.

Blachowicz, K. (1982), „Leibniz’s ’dem onstratio existentiae Dei ad m at­ hem aticam certitudinem exacta’. A Logical Analysis” , [in:] J. K opania (Ed.), Studies in Logic, G ram m ar and Rethoric II, Białystok: Papers of W arsaw University, 43-55.

Blachowicz, K. (1992), Leibniz, wczesne pojęcie substancji, (Leibniz, An Early Conception o f Substance) Białystok: Idea.

Bocheński, J. I. M. (1935), „C om pte rendu nr 936” , Bulletin Thom iste 12: 601-603.

Bocheński, J. I. M. (1989), „D ie fünf Wege” , Freiburger Zeitschrift für Philosophie und Theologie 36: 235-265.

Christian, C. (1957), „Eine N ote zum G ottesbegriff” , Religion-W issens­ chaft-K ultur 8: 227-228.

Essler, W. K. (1969), E inführung in die Logik, Stuttgart.

Essler, W. K. (1991), „G ödels Beweis” , [w:] F. Ricken (ed.), Klassische Gottesbeweise in der Sicht der gegenwärtin Logik und W issenschaftstgen- heorie, Stuttgart: K ohlham m er, 140-152.

G anthaler, H., Simons, P. (1987), „B ernard Bolzanos kosmologischer G ottesbeweis” , Philosophia N aturalis 24: 469-475.

Geach, P.T. (1963), „A quinas” , [w:] G .E .M. Anscom be, P. T. Geach Three Philosophers, Oxford: Basil Blackwell & M ott Ltd.

H artshorne, C. (1941), M an ’s Vision o f G od and the Logic o f Theism, New York.

H artshorne, C. (1961), „T he Logic o f the Ontological A rgum ent” , Journal o f Philosophy 58: 471-473.

H artshorne, C. (1962), The Logic o f Perfection, La Salle.

Kenny, A. (1969), The Five W ays. St. Thom as A quinas’ Proofs o f G o d ’s Existence, London.

Kleinknecht, R. (1991), „A usw ahlaxiom und causa prim a” , [in:] F. Ricken (ed.), Klassische Gottesbeweise in der Sicht der gegenwärtigen Logik und W issenschaftstaheorie, Stuttgart: K ohlham m er, 111-123.

Koj, L. (1954), Poglądy ks. Salamuchy na uściślenie filozofii, (Rev. Salam ucha’s Views on the Specification o f Philosophy) typescript of M.A. thesis, Catholic University o f Lublin.

Nieznański, E. (1977), „Logistyczny przyczynek do analizy pojęcia ’istoty, do której należy istnienie” ’ („A Logical C ontribution, to the Analysis o f the N otion o f ’Essence to which Existence Belongs’” ), Studia Philosophiae C hristianae 13: 139-156.

Nieznański, E. (1979), „Form alizacja tomistycznych podstaw dow odu na istnienie koniecznego bytu pierwszego” („A Form alization o f Thomistic F oundations o f a P ro o f for the Existence o f the Necessary First Being”), Studia Philosophiae Christianae 15: 163-180.

Nieznański, E. (1980), „Form alizacyjne próby ustalenia logiko-formal- nych podstaw stwierdzania pierwszych elementów rozważanych w tomis- tycznej teodycei” , („Form alized A ttem pts to Set Form a-logic Bases to State First Elements o f Relations Considered in Thom istic Theodicy”) [in:] E. Nieznański (ed.). W kierunku formalizacji tomistycznej teodycei, Miscellanea Logica, vol. 1, W arszawa: Academy o f Catholic Theology, 7-194.

Nieznański, E. (1981), „Ein form alisierter Beweis für die Existenz eines ersten notwendigen Seienden” , [in:] E. M orscher, O. N eum aier, G. Zecha (eds.) Philosophie als W issneschaft, Bad Reichenhall: Commes, 379-389. Nieznański, E. (1982a), „W poszukiw aniu pierwszej przyczyny z pom ocą logiki form alnej” („Tow ards the First Cause with Help o f Form al Logic”), Analecta Cracoviensia 14: 51-60.

Nieznański, E. (1982b), „Sform alizowany dow ód istnienia koniecznego bytu pierwszego” („Form alized P ro o f for the Existence of the Necessary First Being”), [in:] B. Bejze (Ed.), W kierunku Boga, W arszawa: Academy o f C atholic Theology, 405-409.

Nieznański, E. (1984), „Form alisierung des Gottesbeweises ’ex ratione causae efficientis’ , Salzburger Jahrbuch für Philosophie 27/29: 79-84. Nieznański, E. (1991), „G ründe, zureichende G ründe und Gottesbeweise ’ex contingentia m undi” ’, [in:] F. Ricken (ed.), Klassische Gottesbeweise in der Sicht der gegenwärtigen Logik und W issenschaftstheorie, Stuttgart: K ohlham m er, 124-139.

Nieznański, E. (1992), „M it o m itach” (A „M yth about M yths”), [in:] B., Bejze (Ed.), Studia z filozofii Boga, O filozoficznym poznaniu Boga dziś, W arszawa: Academy o f C atholic Theology, 250-261.

Policki, Κ. (1975), „W sprawie formalizacji dow odu ’ex m otu’ na istnienie Boga” („O n the Form alization o f the P ro o f ’ex m o tu’ for the Existence o f G o d ” ), Roczniki Filozoficzne 23: 19-30.

Quine, W.V.O. (1951), M athem atical Logic, Cam bridge, translated into Polish by L. Koj.

Rivetti-Barbo, F. (1960), „L a stru ttu ra logica della prim a via per provare l’esistenza di Dio. Applicazioni di logica simbolica e nessi di contenuti” , Rivista di Filosofia Neoscolastica 52: 241-320.

Rivetti-Barbô, F. (1962), „A ncora sulla prim a via per provare 1’esistenza di D io” , Rivista di Filosofia Neoscolastica 54: 596-616.

Rivetti-Barbô, F. (1966), Logica simbolica e prove dell’ esistenza di Dio. Rilievi metodologici, [in:] F. Selvaggi (ed.), De Deo in philosophia S. Thom ae et in hodierna philosophia, vol.2, Rom a, 359-363.

Rivetti-Barbô, F. (1967), „L a formalizzazione e la strutture propria delle ’vie’ diascesa a Dio. Considerazioni m etodologiche” , Rivista di Filosofia Neoscolastica 59: 161-177.

Salam ucha, J. (1934), „D ow ód na istnienie Boga. Analiza logiczna argum entacji św. Tom asza z A kw inu” („The P ro o f for the Existence of G od. Logical Analysis o f St. T hom as’ A rgum ent” ), Collectanea Theologi­ ca 15: 53-92.

Thom as, I. (1960), Rewiew o f the Rivetti (1960), Journal o f Symbolic Logic 25: 347-348.

W eingartner, P. (1974), „Religiöser Fatalism us und das Problem des Übels, [in:] E. Weinziel (ed.) D er M odernismus. Beiträge zu seiner Erfoschung, Festschrift für Th. Michels Graz: Styria, 369-409.

W eingartner, P. (1979), „K om m entar zu Thom as von Aquin, Summa Theologica 1 ,19,1, über den Willen G ottes” , [in:] E. Weinzierl (ed.), Kirche und Gesellschaft, Wien: Geyer, 167-187.