Applying logic to philosophical theology: a formaldeductive inference of affirming god’s existencefrom assuming the a-priori-ness of knowledgein the sigma formal axiomatic theory

Formal axiomatic theory of Sigma as a result of logical formalization of philosophical epistemology. Acquaintance with the main features of the application of logic to philosophical theology. The essence of the concept of "formal-axiological equivalence".

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Applying logic to philosophical theology: a formaldeductive inference of affirming god's existencefrom assuming the a-priori-ness of knowledgein the sigma formal axiomatic theory

V.O. Lobovikov

For the first time a precise definition is given to the Sigma formal axiomatic theory, which is a result of logical formalization ofphilosophical epistemology; and an interpretation of this formal theory is offered. Also, for the first time, a formal deductive proof is constructed in Sigma for a formula, which represents (in the offered interpretation) the statement of God's Existence under the condition that knowledge is a priori.

Keywords: formal axiomatic epistemology theory; two-valued algebra of formal axiology; formal-axiological equivalence; a-priori knowledge; existence of God.

Since Socrates, Plato, Aristotle, Stoics, Cicero, and especially since the very beginning of Christianity philosophy, the possibility or impossibility of logical proving God's existence has been a nontrivial problem of philosophical theology. Today the literature on this topic is immense. However, even in our days, the knotty problem remains unsolved as all the suggested options of solving it are controversial from some point of view.

Some respectable researchers (let us call them “pessimists”) believed that the logical proving of God's existence in theoretical philosophy was impossible on principle, for instance, Occam, Hume [1], and Kant [2] believed that any rational theoretic-philosophy proof of His existence was a mistake (illusion), consequently, a search for the logical proving of His existence was wasting resources and, hence, harmful; only faith in God was relevant and useful; reason was irrelevant and useless. At the very beginning of the 21st century, a theoretically interesting discourse of impossibility to demonstrate logically the existence of a Deity was developed at the level of contemporary symbolic logic and set theory by Bocharov and Yuraski- na [3]. Their attitude to the faith-reason-problem was an inverted and modernized one of Tertullian [4]. Their attitude was an inverted one (in relation to Tertullian) because they rejected resolutely his maxim “Credo quia absurdum est", but, being his worldview-opponents in relation to Faith, they agreed with Tertullian's statement that Reason and Faith were absolutely separated. As to the possibility of logical proving God's existence, Bocharov and Yuraskina were pessimists owing to their belief in the possibility of logical proving the impossibility of God's existence. In this concrete relation, Bocharov and Yuraskina, who have manifestly called themselves “convinced atheists” [3. P. 3], belong to the pessimists who think (together with Tertullian) that faith and reason are absolutely incompatible. According to some non-atheist-minded but sincerely religious representatives of the pessimists, looking for a perfect proof of God's existence is exposing the nonexistence of faith in His existence, i.e. exposing atheism.

However, some other eminent thinkers (let us call them “optimists”) believed that, on principle, the logical proving statement of God's existence within rational theoretic philosophy was possible and compatible with faith, namely: St. Anselm [5]; St. Thomas Aquinas [6-8]; Descartes [9-11]; Spinoza [12]; Leibniz [13, 14]; Gödel [15, 16]; Plantinga [17]; consequently, a search for the logical proving of His existence could be successful and useful; therefore, expending some limited resources for the search was worth undertaking. Equipped with the concrete logic tools available at their time, optimists attempted to invent (construct) a perfect logical proof of God's existence within some rational philosophical theology doctrine and thus to establish a perfect harmony of faith and reason. Before the 20th century, all the attempts were accomplished by means of natural language and traditional (not-mathematized) classical formal logic. In the 20th century, Gödel [15] made an original attempt to apply artificial language and modern symbolic logic machinery for creating a formal deductive proof of God's existence. This formal deductive proof initiated an interesting discussion [17]. Plantinga's applying modal logic to the problem is also worth mentioning here. A noteworthy critical analysis of Plantinga's modalizing the ontological argument is made in Gorbatova's publications and in her interesting dissertation [18]. A systematical critique of all the hitherto invented ontological arguments is given by Lewis [19] and Sobel [20]. A respectable survey of the theoretically significant literature on the theme is done by °ppy [21].

With respect to both sides: the pessimists and the optimists, in the present article I would like to take part in discussing the nontrivial problem. Below I will be writing within the paradigm of the optimists. However, in this article, the optimistic paradigm has undergone a significant modification, as the conceptual apparatus exploited for the deductive logical proving of God's existence differs much from the one used by the overwhelming majority of the optimists hitherto. I mean systematical exploiting (1) two-valued algebra of formal axiology [22-24] and (2) a formal axiomatic epistemology theory £ (Sigma) to be precisely defined for the first time below in the article. Hereafter the terms “proof” and “theorem” are used in the special meanings which have been defined in the 20th-century mathematical logic by the formalists (D. Hilbert et al). Gödel's famous proof of God's existence is a representative example of the systematical usage of the terms “proof” and “theorem” in the indicated formalistic meanings. In the present article, I shall use these terms in the formalistic meanings as well. Namely, by definition, a proof of a formula as a theorem in an axiomatic theory is such a finite succession of formulae of the theory, in which (succession) any formula belonging to the succession is (1) either an axiom of the theory or (2) obtained from previous formulae of the succession by an inference-rule of the theory.

Within the hitherto not considered formal axiomatic theory £, below a formal deductive proof of formula (Aa id [Dx]) as a theorem in £ is constructed for the first time. The article gives such an interpretation of the formal axiomatic epistemology theory £, in which the formula [Dx] of £ represents the famous theology tenet of God existence. According to the given interpretation, the formula Aa represents the assumption of a-priori-ness of knowledge. In the interpretation under discussion, “d” is “classical (material) implication”. Formally to prove that (Aa d [Dx]) is a theorem in £ and to attentively examine the proof, it is indispensable to have exact definitions of the terms involved into the discourse. Therefore, let us start with submitting precise definitions of the notions relevant to the case.

A Precise Definition of the Formal Axiomatic Epistemology

Theory £

Section 2 of this article is aimed at acquainting the reader with the rigorous formulation of the formal axiomatic epistemology theory £, which is a result of a further development (complementing substantially) the axiomatic epistemology system e originally submitted in [25, 26].

According to the definition, the logically formalized axiomatic epistemology system £ contains all symbols (of the alphabet), expressions, formulae, axioms, and inference-rules of the formal axiomatic epistemology theory e [25; 26] which is based on the classical propositional logic. But in £ several significant aspects are added to the formal theory e.

As a result of these additions, the alphabet of £'s object-language is defined as follows:

1) propositional letters q, p, d, ... are symbols belonging to the alphabet of £;

2) logic symbols --, d, o, &, v called “classical negation”, “material implication”, “equivalence”, “conjunction”, “not-excluding disjunction”, respectively, are symbols belonging to £'s alphabet;

3) technical symbols “(” and “)” belong to £'s alphabet;

4) axiological variables x, y, z, ... are symbols belonging to £'s alphabet;

5) symbols “g” and “b” called axiological constants belong to the alphabet of £;

6) axiological-value-functional symbols Akn, Bin, Cjn, Dmn, J, N, D, I, L, ... belong to the alphabet of £. The upper number index n informs that the indexed symbol is n-placed one. Nonbeing of the upper number index informs that the symbol is determined by one axiological variable. The value-functional symbols may have no lower number index. If lower number indexes are different, then the indexed functional symbols are different ones.

7) symbols “[” and “]”belong to the alphabet of £;

8) an unusual artificial symbol “=+=” called “formal-axiological equivalence” belongs to the alphabet of £;

9) a symbol belongs to the alphabet of object-language of £, if and only if this is so owing to the above-given items 1) - 7) of the present definition.

A finite succession of symbols is called an expression in the object-language of £, if and only if this succession contains such and only such symbols which belong to the above-defined alphabet of £'s object-language.

Now let us define precisely the general notion “term of £”:

1) the axiological variables x, y, z, ... (from the above-defined alphabet) are terms of £;

2) the axiological constants “g” and “b”, belonging to the alphabet of £, are terms of £;

3) If Fkn is an n-placed axiological-value-functional symbol, and ti, ... tn are terms (of £), then Fknti, ... tn is a term (compound one) of £ (here it is worth remarking that symbols ti, ... tn belong to the meta-language, as they stand for any term of £; the analogous remark may be made in relation to the symbol Fkn);

4) An expression in language of £ is a term of £, if and only if this is so owing to the above-given items 1) - 3) of the present definition.

Now let us agree that in the present article symbols a, ß, þ, ï, ... (belonging to meta-language) stand for any formulae of £. By means of this agreement the general notion “formulae of £” is defined precisely as follows.

1) All the above-mentioned propositional letters q, p, d, ... are formulae of £.

2) If a and ß are formulae of £, then all such expressions of the object- language of £, which possess logic forms --a, (a ç ß), (a -î- ß), (a & ß), (a v ß), are formulae of £ as well.

3) If ti and tk are terms of £, then (ti=+=tk) is a formula of £.

4) If ti is a term of £, then [ti]is a formula of £.

5) If a is a formula of £, then Ta is a formula of £ as well.

6) Successions of symbols (belonging to the alphabet of the object-language of £) are formulae of £, if and only if this is so owing to the above-given items 1) - 5) of the present definition.

The symbol T belonging to meta-language stands for any element of the set of modalities {?, K, A, E, S, T, F, P, Z, G, W, O, B, U, Y}. Symbol ? stands for the alethic modality “necessary”. Symbols K, A, E, S, T, P, Z, respectively, stand for modalities “agent knows that.”, “agent a-priori knows that.”, “agent a-poste- riori knows that.”, “under some conditions in some space-and-time a person (immediately or by means of some tools) sensually perceives (has sensual verification) that.”, “it is true that.”, “person believes that.”, “it is provable that.”, “there is an algorithm (a machine could be constructed) for deciding that.”.

Symbols G, W, O, B, U, Y, respectively, stand for modalities “it is (morally) good that.”, “it is (morally) wicked that.”, “it is obligatory that .”, “it is beautiful that .”, “it is useful that .”, “it is pleasant that .”. Meanings of the mentioned symbols are defined by the following schemes of own-axioms of epistemology system £ which axioms are added to the axioms of classical propositional logic. Schemes of axioms and inference-rules of the classical propositional logic are applicable to all formulae of £.

Axiom scheme AX-1: Aa ç (Dß ç ß).

Axiom scheme AX-2: Aa ç (D(a ç ß) ç (Da ç Dß)).

Axiom scheme AX-3: Aa î (Ka & (Da & ?--Sa & D(ß î Gß))).

Axiom scheme AX-4: Ea î (Ka & (--Da v --?--Sa v --D(ß î Gß))).

Axiom scheme AX-5:(Dß & DQß) ç ß.

Axiom scheme AX-6: (ti=+=tk) -î- (G[ti] -î- G[tk]).

Axiom scheme AX-7: (ti=+=g) ç G[ti].

Axiom scheme AX-8: (ti=+=b) ç W[ti].

Axiom scheme AX-9: Ga ç -Wa).

Axiom scheme AX-10: (Wa ç -Ga).

In AX-3 and AX-4, the symbol Q (belonging to the meta-language) stands for any element of the set = {?, K, T, F, P, Z, G, O, B, U, Y}. Let elements of be called “'perfection-modalities” or simply “perfections”.

The axiom-schemes AX-9 and AX-10 are not new in evaluation logic: one can find them in Ivin's famous monograph [27]. But axiom-schemes AX-5-AX-8 are perfectly new: they have not been published hitherto.

Defining Semantics of/for £

Meanings of the symbols belonging to the alphabet of the object-language of £ owing to items 1-3 of the above-given definition of the alphabet are defined by classical propositional logic.

Axiological variables x, y, z, ... range over (take their values from) such a set A, every element of which has: (1) one and only one axiological value from the set {good, bad}; (2) one and only one ontological value from the set {exists, not-exists}.

Axiological constants “g” and “b” mean, respectively, “good” and “bad”.

N-placed terms of £ are interpreted as n-ary algebraic operations (n-placed evaluation-functions) defined on the set A.

Speaking of evaluation-functions means speaking of the following mappings (in the proper mathematical meaning of the word “mapping”): {g, b} ^ {g, b}, if one speaks of the evaluation-functions determined by one evaluation-variable; {g, b} x {g, b} ^ {g, b}, where “x” stands for the Cartesian product of sets, if one speaks of the evaluation-functions determined by two evaluation-variables; {g, b}N ^ {g, b}, if one speaks of the evaluation-functions determined by N evaluation-variables, where N is a finite positive integer.

If ti is a term of £, then formula [ti] of £ means either true or false proposition “ti exists”. The proposition [ti] is true if and only if ti has the ontological value “exists”.

The formula (ti= + =tk) of £ is translated into natural language by the proposition “ti is formally-axiologfcally equivalent to tk”, whose proposition is true if and only if the terms ti and tk have identical axiological values from the set {good, bad} under any possible combination of axiological values of their axiological variables.

The one-placed term Dx is interpreted in this article as one-placed evaluationfunction “God of (what, whom) x in a monotheistic world-religion ”. This function is precisely defined by the following evaluation-table 1.

Table 1. One-placed evaluation-functions

In the above evaluation-table 1, the symbol Jx stands for the evaluationfunction “being (existence), life of (what, whom) x” Nx stands for the evaluationfunction “non-being (nonexistence), death of (what, whom) x” Dx stands for the evaluation-function “God of (what, whom) x in monotheistic world-religion”. Ix stands for the evaluation-function “deity of (what, whom) x in polytheistic local (pagan, heathen) religion”. Lx means the evaluation-function “daemon of x in polytheistic local (pagan, heathen) religion”. Ax - “'Anti-God (God's Enemy) of (what, whom) x in monotheistic world religion”.

Formal Proving (Aa ç [Dx]) in E

The proof of (Aa ç [Dx]) in E is the following succession of formulae.

1) (Dx= + =g) ç G[Dx] by substituting Dx for t in axiom-scheme AX-7.

2) (Dx= + =g) by the formal-axiological definition of God as absolute goodness.

3) G[Dx] from 1 and 2 by modus ponens.

4) Aa î (Ka & (Da & ?--Sa & D([Dx] î- G[Dx]))) by: substituting G for Q; and substituting [Dx] for ß in AX-3.

5) Aa ç (Ka & (Da & ?--Sa & D([Dx] î G[Dx]))) from 4 by the rule of elimination of î.

6) Aa assumption.

7) Ka & (Da & ?--Sa & D([Dx] î G[Dx])) from 5 and 6 by modus ponens.

8) D([Dx] î G[Dx]) from 7 by the rule of elimination of &.

9) [Dx] î G[Dx] from 8 by the rule of elimination of ?.

10) G[Dx] ç [Dx] from 9 by the rule of elimination of î.

11) [Dx] from 3 and 10 by modus ponens.

12) Aa |-- [Dx] by the above formula-succession 1--11.

13) |-- (Aa ç [Dx]) from 12 by the rule of introduction of ç.

Here you are.

1. Discussing the Theorem and Arriving to the Conclusion

Hume [1. P. 372-378] undertook a systematical critique of all possible a priori arguments demonstrating rational-philosophy statement of the existence of a Deity. In relation to his negating the a priori arguments in general, the result obtained above in the present article is a counter-example; at least some of the a priori arguments can be valid. In the above-defined interpretation of E, the theorem (Aa ç [Dx]) formally proved in E means that if agent a-priori knows that a, then God exists. Thus, in the present article existence of God is formally proved within the homogeneous system of a-priori knowledge exclusively. Talks of facts (=contingent truths) and empirical arguments are not involved into the discourse of God's being. This means that in the present article an abstraction from the empirical aspect of the problem under discussion is accepted and, consequently, the significance of the result obtained in this article is limited. Nevertheless, the above- submitted formal deductive inference is interesting theoretically and worth discussing among specialists with a view for further developing the analytic trend in philosophical theology investigations.

References

logical philosophical equivalence

1.Hume, D. (1996) Malye proizvedeniya [Small Works]. Translated from English. Moscow: KANON. pp. 297-426.

2.Kant, I. (1994) The Critique of Pure Reason. In: Adler, M. (ed.) Great Books of the Western World. Vol. 39. Chicago; Auckland; London; Madrid: Encyclopaedia Britannica.

3.Bocharov, V.A. & Yuraskina, T.I. (2003) Bozhestvennye atributy [Divine Attributes]. Mos¬cow: Moscow State University.

4.Tertullian, Q.S.F. (2015) On the Flesh of Christ. Savage, Minnesota: Lighthouse Christian Publishing.

5.St. Anselm of Canterbury. (1998) The Major Works. Oxford: Oxford University Press.

6.Aquinas St. Thomas. (1975) Summa contra Gentiles. Notre Dame: University of Notre Dame Press.

7.Aquinas St. Thomas. (1994) The Summa Theologica. Vol. 1. In: Adler, M. (ed.) Great Books of the Western World. Vol. 17. Chicago; Auckland; London; Madrid: Encyclopedia Britannica, Inc.

8.Aquinas St. Thomas. (1994) The Summa Theologica. Vol. II. In: Adler, M. (ed.) Great Books of the Western World. Vol. 18. Chicago; Auckland; London; Madrid: Encyclopedia Britannica, Inc.

9.Descartes, R. (1994a) Meditations on First Philosophy. In: Adler, M. (ed.) Great Books of the Western World. Vol. 28. Chicago; Auckland; London; Madrid: Encyclopaedia Britannica. pp. 295-329.

10.Descartes, R. (1994b) Objections against the Meditations, and Replies. In: Adler, M. (ed.) Great Books of the Western World. Vol. 28. Chicago; Auckland; London; Madrid: Encyclopaedia Britannica. pp. 330-519

11.Descartes, R. (1994c) Discourse on the Method of Rightly Conducting the Reason. In: Adler, M. (ed.) Great Books of the Western World. Vol. 28. Chicago; Auckland; London; Madrid: Encyclo¬paedia Britannica. pp. 265-291.

12.Spinoza, B. (1994) Ethics. In: Adler, M. (ed.) Great Books of the Western World. Vol. 28. Chicago; Auckland; London; Madrid: Encyclopaedia Britannica. pp. 589-697.

13.Leibniz, G.W.F. (1996) New Essays on Human Understanding. Translated from German by P. Remnant and J. Bennett. Cambridge; New York: Cambridge University Press.

14.Leibniz, G.W.F. (1952) Theodicy: Essays on the Goodness of God, the Freedom of Man, and the Origin of Evil. London: Routledge and Kegan Paul.

15.Gödel, K. (1995) Collected Works. Vol. 3. New York; Oxford: Oxford University Press.

16.Swiçtorzecka, K. (ed.) (2016) Gödel's Ontological Argument: History, Modifications, and Controversies. Warsaw: Wydawnictwo Naukowe Semper.

17.Plantinga, A. (1974) The Nature of Necessity. Oxford: Clarendon Press.

18.Gorbatova, Yu.V. (2012) Logiko-ontologicheskie osnovaniya sovremennoy analiticheskoy teologii (na materiale kontseptsii Alvina Plantingi) [Logical and ontological foundations of contempo¬rary analytical theology (instantiated by Alvin Plantinga conception)]. PhD dissertation. Moscow: HSE.

19.Lewis, D. (1970) Anselm and Actuality. Noûs. 4. pp. 175-88.

20.Sobel, J. (2004) Logic and Theism. New York: Cambridge University Press.

21.Oppy, G. (2019) Ontological Arguments. In: Zalta, N. (ed.) The Stanford Encyclopedia of Philosophy. [Online] Available from: https://plato.stanford.edu/archives/spr2019/entries/ontological- arguments/> (Acceaaed: 11th November 2019).

22.Lobovikov, V. (2015) The Trinity Triangle and the Homonymy of the Word “Is” in Natural Language (A Logically Consistent Discrete Mathematical Representation of the Trinity by Means of Algebra of Morality and Formal Ethics). Philosophy Study. 5(7). pp. 327-341. DOI: 10.17265/2159-5313/2015.07.001

23.Lobovikov, V. (2018) Vindicating Gödel's Uniting Logic, Metaphysics and Theology (God's omnipresence proved by computing compositions of evaluation-functions in two-valued algebra of metaphysics as formal axiology). Proceedings of the Round Table “Religion and Religious Studies in the Urals. Ekaterinburg: Delovaya Kniga. pp. 33-36.

24.Lobovikov, V. (2019) Analytical Theology: God's Omnipotence as a Formal-Axiological Law of the Two-Valued Algebra of Formal Ethics (Demonstrating the Law by “Computing” Relevant Evaluation-Functions). Vestnik Tomskogo gosudarstvennogo universiteta. Filosofiya. Sotsiologiya. Politologiya - Tomsk State University Journal of Philosophy, Sociology and Political Science. 1(47). pp. 87-93. DOI: 10.17223/1998863Õ/47/9

25.Lobovikov, V. (2018) Proofs of Logic Consistency of a Formal Axiomatic Epistemology Theory S, and Demonstrations of Improvability of the Formulae (Kq^ q) and (Dq ^ q) in It. Journal of Applied Mathematics and Computation. 2(10). pp. 483-495. DOI: 10.26855/jamc.2018.10.004

26.Lobovikov, V. (2019) A model \Sigma for the theory \Ksi. Formal Methods and Science in Philosophy III. Abstracts of the International conference. Dubrovnik, Croatia. April 11-13, 2019. Insti¬tute of Philosophy, Zagreb. pp. 19-20.

27.Ivin, A.A. (1970) Osnovaniya logiki otsenok [Foundations of Evaluation Logic]. Moscow: Moscow State University.

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