The main issue to be discussed in this article is an exemplification of logical inference of statement of being from statement of value within a formal axiomatic theory of knowledge under the assumption of knowledge a-priori-ness. In this paper, the statement of being is exemplified by a law of thermodynamics; the statement of value is exemplified by a formal-axiological analog of the law of thermodynamics.

A short review of relevant literature: The nontrivial problem of logical deriving statements of value from statements of being (and statements of being from statements of value) has been raised originally in (Hume, 2000) and (Moore, 1903) with respect to philosophy of morals. In relation to philosophy of science, the discussion of fact/value dichotomy problem has produced an immense amount of literature; for instance, (Marchetti & Marchetti, 2017; Putnam, 2002; 2004; 2017; Lobovikov, 2020c). According to the positivism paradigm, being completely reduced to facts science has nothing to do with values (Carnap, 1931; Mach, 1914; 1960; 2006; Reichenbach, 1959; 1965; Schlick, 1974; 1979a; 1979b; Wittgenstein 1992), consequently, a proper axiological aspect of thermodynamics does not exist. However, in the relevant literature, there is a hypothesis (Lobovikov, 2012; 2017; 2019; 2020b) that, in its essence, metaphysics is nothing but an abstract formal axiology. If the unhabitual hypothesis is accepted, then metaphysics of nature (philosophical grounding physics) necessarily has a proper axiological aspect. Accepting this psychologically unexpected corollary from the extraordinary hypothesis under investigation (by the hypothetical-deductive method) makes a heavy problem (paradox) to be scrutinized carefully and solved below in the present paper. In (Lobovikov, 2020c), a rigorous formal proof (within a formal axiomatic theory Σ) is constructed for such a theorem-scheme , which means (in the precisely defined interpretation) that under the condition of knowledge a-priori-ness, a statement of formal-axiological equivalence of evaluation-functions is logically equivalent to logic equivalence of corresponding statements of being.

But, in (Lobovikov, 2020c), this philosophically significant theorem-scheme is not exemplified; its rigorous formal proof is constructed independently from its possible interpretations. Therefore, to support the above-mentioned unhabitual hypothesis of metaphysics of nature as its formal axiology, there is a theoretical necessity to exemplify the above-mentioned philosophically significant theorem-scheme by a concrete material taken from physics. For implementing the exemplification, it has been decided to utilize the concrete material of thermodynamics. Thus, the reason and significance of choosing the topic of this paper are clarified.

Due to such clarifying, the overall logical structure (somewhat complicated one) of the applied investigation becomes more evident. Namely, for obtaining and examining the main scientifically new result of this paper, it is necessary to have precise definitions of basic notions of two-valued algebraic system of metaphysics as formal axiology, which are already published, for instance, in (Lobovikov, 2012; 2019; 2020b). These precise definitions are contents of the following paragraph 2. Including these already published contents into the paragraph 2 of the present paper is indispensable; otherwise, the significantly new nontrivial scientific result (represented in the paragraphs 3 and 7 of this article) should be not understandable and not examinable. The set of exact definitions necessary and sufficient for perfect understanding and examining original contents of the paragraph 3 is submitted in the immediately following paragraph 2. The set of precise definitions necessary and sufficient for adequate understanding and examining original contents of the paragraphs 7 and 8 is given below in the paragraphs 2, 4, 5. As the significantly novel nontrivial result is obtained (in the paragraphs 3 and 7 of this article) within the framework of a qualitatively new paradigm, which scientists and philosophers are not used to, they have to have exact definitions of all the novel basic notions at their disposal before: 1) starting to read and understand formal deductive proofs and to scrutinize them carefully at syntax level; 2) interpreting the formally proved theorems and discussing the interpretations. Now let us move to submitting the system of basic definitions.

According to the contemporary view of algebra and logic, generally speaking, algebra may be based upon any set of objects having any nature. The habitual sets (of numbers, quantity relations, space forms, etc.) are implied by the well-known habitual concrete applications of algebra to the concrete (fixed) objects for solving the concrete (fixed) classes of problems of human life. For instance, originally, Boolean two-valued algebra of logic had broken the habitual paradigm of algebra as a mathematical apparatus for operating exclusively with numbers. Boolean algebra of logic is based upon the set of thoughts, which are either true of false ones. Numbers and thoughts have qualitatively different nature but it does not matter if one talks of abstract algebra in general. Consequently, from the universal algebra standpoint, one can create an algebraic system based on a set of any (even very unhabitual, extraordinary, odd) objects. Hence, in principle, nowadays it is possible rationally to talk of constructing and investigating even such an algebraic system which is based upon a set of objects having either proper ethical (moral) or proper metaphysical nature as well (Lobovikov, 2009; 2012; 2019; 2020b). Certainly, elements of the set which hypothetical algebra of metaphysics is to be based on are to be neither numbers of arithmetic, nor figures of geometry. According to the standpoint accepted in the present article, elements of the set which algebra of metaphysics is based on are objects of abstract axiology, which is a universal theory of abstract values. Obviously, the nature of objects which are elements of the set which algebra of metaphysics is based on is odd (extraordinary) one. Nevertheless, below in this paragraph, in spite of the oddity, relevant notions of algebra of metaphysics are to be introduced and defined precisely.

The odd (unhabitual) algebraic system mentioned in the title of this paragraph is based upon the set Δ. By definition, elements of Δ are such (and only such) either existing or not-existing objects, namely, things, processes, persons (individual or collective ones, it does not matter), which are either good, or bad ones from the standpoint of a valuator V, who is a person (individual or collective one, it does not matter), in relation to which all valuations are generated. Here the terms “good” and “bad” have abstract axiological meanings which are more universal in comparison to the particular ones exploited in ethics: n the present article, “good” means abstract positive value in general; “bad” means abstract negative value in general. Certainly, V is a variable: changing values of the variable V can result in changing valuations of concrete elements of Δ. However, if a value of the variable V is fixed, then valuations of concrete elements of Δ are quite definite.

Algebraic operations defined on the set Δ are abstract-valuation-functions (in particular, moral-value-ones). Abstract-valuation-variables of these functions take their values from the set {g, b}. Here the symbols “g” and “b” stand for the abstract positive values “good” and “bad”, respectively. The functions take their values from the same set. The symbols: “x” and “у” stand for axiological-forms of elements of Δ. Elementary axiological-forms deprived of their contents are independent abstract-valuation-arguments. Compound axiological-forms deprived of their contents are abstract-valuation-functions determined by these arguments.

In this article, talking of valuation-functions determined by (a finite integer of) valuation-arguments means talking of the following mappings (in the proper mathematical meaning of the word “mapping”): {g, b} → {g, b}, if one talks of the valuation-functions determined by one valuation-argument; {g, b} × {g, b} → {g, b}, where “×” stands for the Cartesian product of sets, if one talks of the valuation-functions determined by two valuation-arguments; {g, b}^N → {g, b}, if one talks of the valuation-functions determined by N valuation-arguments, where N is a finite positive integer. To exemplify the above-defined general notion, let us introduce and define precisely by tables the following evaluation-functions determined by one argument. This is not merely an exemplification as the below-introduced one-placed functions are to be exploited essentially for obtaining the main new nontrivial scientific result of this article.

Glossary for the below-submitted Table 1. B₁x, “being, existence of (what, whom) x”. N₁x, “nonbeing, nonexistence of (what, whom) x”. F₁x, “finite (what, who) x” or “finiteness of (what, whom) x”. I₁x, “infinite (what, who) x”, or “infiniteness of (what, whom) x”. T₁x, “physical time of (what, whom) x”. T₂x, “metaphysical time of (what, whom) x”. T₃x, “absolute time of (what, whom) x”. T₄x, “time (in general) of (what, whom) x”. M₁x, “matter, material, materialness of (what, whom) x”. M₂x, “movement, change, flow of (what, whom) x”. D₁x, “diminishing (what, whom) x”. The mentioned functions are defined by Table 1. (Attentively looking at this table, one can notice that in algebra of formal axiology, the functions T₂x and T₄x are mathematically identical. However, this psychologically odd fact does not make a real problem: although formal-axiological meanings of the symbols “T₂x” and “T₄x” (the evaluation-functions) do coincide, the ontological meanings of these symbols are not completely identical: they can be different, namely, in general, time can be not metaphysical but physical one.)

Glossary for the following Table 2. R₁x, “relativity (relativeness) of (what, whom) x”. O₁x, “order of (what, whom) x”, or “x’s order”, or “being ordered by (what, whom) x”. O₂x, “order for (what, whom) x”, or “ordered-ness of (what, whom) x”, or “x’s being ordered”. C₁x, “closed, isolated, protected (what, who) x”, or “closedness, isolated-ness, protected-ness of (what, whom) x”. S₁x, “sensation of (what, whom) x as an object, i.e. x’s being an object of sensation”. M₃x, “measurement of (what, whom) x as an object, i.e. x’s being an object of measurement”. P₁x, “possibility of (what, whom) x”. I₂x, “impossibility of (what, whom) x”. I₃x, “irreversibility of x”. R₂x, “reversibility of x”. V₁x, “x’s vector (direction)”, or “immanent direction (own vector) of (what, whom) x”. These functions are defined below by Table 2.

Now, let us move from the above-introduced evaluation-functions determined by one evaluation-argument to below-introduced evaluation-functions determined by two evaluation-arguments.

Glossary for Table 3, the symbol К²xy stands for the two-placed evaluation-function “a unity (one-ness) of x and y”, or “joint being of x and y”, or “x’s and y’s being together”. The symbol E²xy, “equalizing (identifying values of) x and y”, or “coincidence (identify) of x and y”. C²xy, “y’s being in (what, whom) x”. C₁²xy, “y’s being an immanent (inner) cause of (what, whom) x”. C₂²xy, “y’s being an external (transcendent) cause of/for x”. The mentioned evaluation-functions determined by two arguments are defined by Table 3.