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)

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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)

V.O. Lobovikov

Annotation

The paper submits a significantly new method for analytical theology - mathematical representing and solving knotty problems of theology by means of constructing and investigating their discrete mathematical models at the level of artificial language of algebraic system of formal ethics. For the first time God's omnipotence is formulated by the artificial language and demonstrated as a formal-axiological law by “computing” relevant evaluation- functions.

Keywords: algebra-of-formal-ethics, formal-axiological-equivalence, formal-axiological- contradiction, formal-axiological-law, omnipotence-of-God .

We proceed thus to the Third Article: It seems that God is not omnipotent. ... Further, sin is an act of some kind. But God cannot sin, nor deny Himself, as it is said II Tim. 2. 13. Therefore He is not omnipotent. ... On the contrary, it is said: No word shall be impossible with God (Luke I. 37). I answer that,All confess that God is omnipotent; but it seems difficult to explain in what His omnipotence consists.

Thomas Aquinas. Summa Theologica (Part I. Question 25. Article 3)

The knotty paradoxical problem of omnipotence came to us from ancient times: Augustine [1]; Pseudo-Dionysius [2]; Aquinas [3]; Descartes [4]; Leibniz [5]. Today there are plenty of profound theological and philosophical writings devoted to this problem [6-33]. Therefore, publishing a new article on this topic can be justified only if it submits a really new nontrivial attitude missed by the others.

I think that this is just the case. The above-mentioned authors have concentrated almost all their attention on proper logic semantics of the natural language used in discussing the omnipotence problem. As a rule, theologians and philosophers have discussed statements of being or non-being (or possibility or impossibility) of the omnipotence. Statements of (moral) value of the omnipotence are presumed but they do not undergo a systematical formal-axiological analysis using discrete mathematics. Therefore, the submitted paper is targeted at rational filling in this blank in the literature on the topic. To make the text understandable first of all it is indispensable to introduce, precisely to define, and to instantiate the minimal set of basic definitions necessary and sufficient for proving strictly that God's omnipotence is a formal-ethical law of the algebraic system of formal ethics. Therefore, let us introduce the novel conceptual apparatus (new terms) systematically to be used below for obtaining the result.

A Two-Valued Algebraic System of Formal Ethics (A Set of Basic Definitions Necessary-And-Sufficient for Proving Strictly That God's Omnipotence Is a Formal-Ethical Law of the Algebraic System)

In this part of the paper, I make the reader aware of the basic definitions of algebra of formal ethics which are already published, for instance, in [34-42]. Starting with this already published set of basic definitions is necessary for understanding the substantially novel nontrivial result submitted in this article. The paper's novel statement of provability of the formal-ethical law of omnipotence of God by computing relevant moral-evaluation-functions is still not published elsewhere.

Two-valued algebra of formal ethics is based upon the set A of either moral acts or moral agents (persons). By definition, moral acts are such and only such operations, which are either good, or bad ones in the moral meaning of the words “good” and “bad”. In general, any elements of A (and, in particular, any moral agents) are such and only such entities which are either good, or bad ones in the moral meaning of the words “good” and “bad”. The set A is homogenized by accepting such an identity-abstraction according to which an agent (person) is identified with the compound moral action uniting all moral acts of that person in his conduct as a whole. Thus, a moral agent is nothing but his complex moral act (called his moral life as a whole) made up by all his moral acts.

Algebraic operations defined on the set A are moral-evaluation-functions (or moral-value-ones). Moral-evaluation-variables of these functions take their values from the set {g, b}. Here the symbols “g” and “b” stand for the moral values “good” and “bad”, respectively. The functions take their values from the same set. The symbols: “x” and “у”stand for moral-forms of acts (or agents). Elementary moral-act-or-agent-forms deprived of their contents are independent moral-evaluation-variables. Compound moral-act-or-agent-forms deprived of their contents are moral-evaluation-functions determined by these variables.

Let symbol X stand for the moral evaluator, i.e. the person (individual or collective one - it does not matter), in relation to whom all evaluations are generated. In the moral-evaluation-relativity theory, X is a variable: changing values of the variable X can result in changing moral evaluations of concrete acts and agents. However, if a value of the variable X is fixed, then moral evaluations of concrete acts and agents are definite.

Speaking of moral-value functions in this paper, I mean the following mappings (in the proper mathematical meaning of the word “mapping”): {g, b} {g, b}, if one speaks of the moral-value functions determined by one moral-value variable; {g, b}x{g, b} {g, b}, where “x” stands for the Cartesian multiplication of sets, if one speaks of the moral-value functions determined by two moral-value variables; {g, b}N {g, b}, if one speaks of the moral-value functions determined by N moral-value variables, where N is a finite positive integer.

Below let us consider some concrete examples of mathematically elementary moral-value-functions immediately related to the theme of this article. Let us start with the functions determined by one moral-value variable.

The glossary for the below-submitted moral-value-table 1 : Let the symbol Bx stand for the moral-value function “being (existence), life of (what, whom) x” Nx stands for the moral-value function “non-being (nonexistence), death of (what, whom) x” Gx stands for the moral-value function “God of (what, whom) x in monotheistic world religion ”. Іаstands for the moral-value-function “god1of (what, whom) x in polytheistic local (barbaric) religion”. Dx means the moral-value- function “daemon of x in polytheistic local religion”. Ax - “'Anti-God (God's Enemy) of (what, whom) x in monotheistic world religion”. Zx means the moral-value function “self-destruction, self-extermination, suicide of (what, whom) x” Px means the moral-value function “self-preservation, self-conservation, selfprotection, self-defense of (what, whom) x”. The introduced functions are defined by the following Table 1. (This tabular definition may be found in [41; 42].)

Table 1

The glossary for the below-submitted moral-value-table 2: Let the symbol L2xy stand for the moral-value-function “necessity of y for x”. (The lower numerical index 2 informs that the indexed capital letter stands for a moral-value-function determined by two arguments.) The symbol M2xy stands for the moral-value-function 'possibility of y for x”. I2xy stands for the moral-value-function “impossibility of y for x”. F2xy - “x's freedom from y”, i.e. “nonbeing of necessity of y for x”. These functions are defined by Table 2. (The tabular definition may be found in [34; 35; 37-39; 43; 44].)

formal ethic axiological

Table 2. The Binary Functions

Definition 1 (offormal-ethical-equivalence-relation): in two-valued algebraic system of formal ethics, moral-evaluation-functions (moral-forms of human activity) Q and W are formally-ethically equivalent (this is represented by the symbol “Y = + = Q”), if and only if they acquire identical moral values (from the set (g (good), b (bad)}) under any possible combination of moral values of their moral-evaluation-variables.

Definition 2 (offormal-ethical law): in two-valued algebra of formal ethics, a moral-evaluation-function (moral activity form) is called formally-ethically (or invariantly) good one (or a law of algebra of formal ethics), if and only if it acquires the moral value g (good) under any possible combination of moral values of its variables.

Definition 3: (of formal-ethical contradiction): in two-valued algebra of formal ethics, a moral-evaluation-function is called formally-ethically (or invariantly) bad one, if and only if it acquires the moral value b (bad) under any possible combination of moral values of its variables.

Now the set of basic definitions necessary and sufficient for constructing the above-promised proof (of God's omnipotence as the formal-ethical law of the algebraic system) is already presented. Therefore, let us start proving.

Proving the Formal-Ethical Law of God's Omnipotence by Computing Moral-Evaluation-Functions and Systematic Using the Above-Given Definitions

As (according to the table 1) it is true that Gx = + = g, only Rows 1 and 2 of Table 2 are relevant to the main theme of the article. Attentively looking at these rows one can see that the following four equations are true.

1) Lg = + = b.

2) Ig = + = b.

3) F2gy = + = g.

4) Mg = + = g.

From the above four equations it is easy to obtain the below four ones, respectively, by substituting Gx for g (as according to the table 1 it is true that Gx = + = g).

5) L2Gxy = + = b: necessity of some (any) y for God (of x) is a formal- axiological contradiction.

6) I2Gxy = + = b: impossibility of some (any) y for God (of x) is a formal- axiological contradiction.

7) F2Gxy=+=g: the formal-axiological law of God's freedom: any yis not- necessary for God (of x).

8) M2Gxy = + = g: the formal-axiological law of God's omnipotence: any y (i.e. everything) is possible for God (of x).

Thus, constructing the proof is finished. Here you are. From the purely mathematical technical viewpoint the submitted proof is surprisingly elementary, but I think that from the conceptual metaphysical viewpoint it is quite nontrivial. In any way, it is worth recognizing that accepting all the above-given nontrivial definitions necessarily results in accepting God's omnipotence as the formal-ethical law.

His omnipotence is not the only law of algebra offormal ethics important for mathematical theology as a system of the laws. According to the following equation also God's existence is the formal-ethical law.

9) BGx = + = g.

Consequently, the existence and omnipotence of God are quite compatible. Moreover, according to the below equation, they are equivalent.

10) BGx = + = M2Gxy.

However, it is quite natural to expect that normal human creatures equipped with commonsense, formal logic, and empirical knowledge of facts can generate a lot of alleged objections against the above-submitted seemingly paradoxical equations 1-10. Also, it is quite natural that, in fact, many of such objections are already generated, presented, and discussed in the literature, for instance, in works of Englebretsen [7], Frankfurt [8], Mavrodes [22], Rosenkrantz and Hoffman [25; 26], Ramsey [27], Savage [29], Schlesinger [31] et al.

Nevertheless, I think that often the so-called refutations of God's omnipotence are not proper refutations but illusions of the ones naturally produced by the ambiguity and homonymy of the words “is”, “means”, “implies”, “entails”, “equivalence”, “compatibility”, “inconsistency”, etc. in the natural language [42]. I think so because often the refutations invented (artificially constructed on purpose) and submitted by the sophisticated critics contain a well-camouflaged and hence not- recognized violation of the principle of logic autonomy of moral-values (evaluations) and facts. According to this principle, it is strictly forbidden to make up a formal logic inference from purely evaluative a priori statements to purely factual a posteriori ones and conversely. Generally speaking, it is not logical to go from empirical “what is” to “what is good” and from “what is good” to empirical “what is”. In general, the gap is unbridgeable by means of formal logic-inference rules. Forbidden attempts to bridge up this gap by logic-inferences generate various paradoxes which could be dissolved by systematical using the logic-autonomy principle. In accordance with this principle the above equations 1-10 only seem paradoxical from the empirical viewpoint because they are not logic connections of empirical statements about facts but a priori formal-ethical statements about formal-axiological relations between moral-value-functions.

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