Сергей Антаков участвовал в XXIV Всемирном философском конгрессе в Пекине, где выступил с докладом «The Concept of Truth in Pragmatism and Its Logical Formalization».

Выступление на секции Конгресса и другие фотографии тех времён.

## The Concept of Truth in Pragmatism and Its Logical Formalization

The formalization of pragmatic concept of truth can be achieved so far only at the price of its reduction to the theory of validating the truth of assertive propositions. But the comprehension of the results leads to the “foundation” (justification) of the philosophy of pragmatism as a whole.

**1. Logical Definition of Truth**

To analyze the concept of truth let’s divide the logical and mathematical definitions of truth. The first one is a definition of *classical truth*, or *correspondence-truth*, going back to Parmenides, Plato and Aristotle.

In his “semantic theory of truth” A. Tarsky [*Tarsky A. *Der Wahrheitsbegriff in den formalisierten Sprachen. – Studia Philosophica, Bd. l. Lemberg, 1935] reproduces the definition given by Aristotle. Because of its features, Tarsky’s definition led to W. Quine’s “disquotation theory of truth”. And this theory can be considered as a repetition of A. Ayer’s thesis with some new arguments: “the concept of truth can be completely excluded from the science as a «pseudo-predicate»” [Ayer A*. **Language, Truth and Logic*. L., 1958.]. Disquotation theory and the like represent some kinds of deflationary theory of truth which is influential in analytic philosophy.

It seems that *logical* definition of truth finally leads to the theories which deflate and eliminate the truth. Here we assume scientific truth, empirical or factual. Quine and other analytics developed also a theory of logical truth as *tautology*.

**2. Formalization of Pragmatic Truth in Ch. S. Pierce’s Examples. Conclusion****Mathematical Definition of Truth**

The idea of mathematical definition can be found in Plato’s Myth of the Cave and his “astronomic task” [Plato. *Respublica*]. The truth of knowledge is a correspondence of the image (knowledge itself) to its pre-image (subject of knowledge). The difference of this definition from logical definitions does not immediately show up, but it starts manifesting itself when mathematics formalizes the concept of correspondence as a mapping or a function. “The image”, “the pre-image” and “the imaging” (“the mapping”) are the three strictly defined *mathematical* terms used to formulate the problems of mathematics, of mathematical natural sciences and their technical applications. In such (mathematical) understanding the truth is inseparable from the scientific method – the method of mathematical modeling.

Therefore, if the logicians who operate with logical definitions of truth oppose the concept of truth, then they oppose scientific knowledge and the corresponding broad practice, or they imply some special truth (or its equivalents, like correctness) that is detached from the natural-science intuition of truth which itself is formalized as a modeling procedure.

But the *scientific* truth understood in a mathematical way is not purely classical, because it takes on an important pragmatic or subjective moment: the non-uniqueness of the image-model of this pre-image (object) and the need to choose a model. A. Einstein and L. Infeld give a metaphor of a closed watch in order to express this essentially Kantian thought [Einstein, A., Infeld, L. *The evolution of physics*. New York, NY: Simon & Schuster. 1938].

The models turn out as best and worst, as fitting and not fitting the problems. The choice is consistent with the theoretical and practical goals of the person, that’s why it is a choice guided by values. One can draw a preliminary conclusion from here: the classical concept of truth is not alien to the pragmatic truth, which we are about to cover.

**3. Limits of the Correspondence Theory of Truth**

The correspondence theory of truth is not able to in many practically important cases, where empirical truth is implied and the pre-image is immanent to the image. For example, let us answer to the question “Is the chessboard white?”. If the answer “it is not” seems obvious, then we make the question more difficult showing a new chessboard where all cells are painted white except one (which is left black). What if there are two black cells left? Or three cells? Etc.

This question is only an illustrative example of a *typical* practical situation. There are identical questions solved in the problems of classification and pattern recognition in Artificial Intelligence. Or they arise when the court must decide if the suspect is guilty when his crime is neither fully proven nor refuted. In the latter case, one of two legal presumptions (guilt or innocence) is in effect. A summary of these presumptions is presented below.

A conclusion follows, consistent with the conclusion of the previous section: the boundary between the classical and pragmatic concepts of truth is not absolute.

**4. Pragmatic Definition of Truth by Pierce**

Pierce starts his well-known article [Peirce, C. S. (1902). Truth and Falsity and Error. In J. M. Baldwin (Ed.), *Dictionary of Philosophy and Psychology, Vol. II* (pp. 718-720). London: Macmillan and Co.] with a definition, in which he introduces an explanatory mathematical metaphor: “Truth is a character which attaches to an abstract proposition, such as a person might utter. It essentially depends upon that proposition’s not professing to be exactly true. But we hope that in the progress of science its error will indefinitely diminish, just as the error of 3.14159, the value given for π, will indefinitely diminish as the calculation is carried to more and more places of decimals. What we call π is an ideal limit to which no numerical expression can be perfectly true <…>. Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth”.

In another Pierce’s example a seller told to the buyer that a horse is sound and free from vice. Later the buyer found out that the horse is dyed with undesirable real color. He complains of the deception but the seller objects saying that he could not tell every fact about a horse, but the facts he revealed were true.

**5. Logical Presumptions**

Let us turn to the logical means, which will allow us to consider both examples in a uniform way and, moreover, describe the method of formalizing the pragmatic truth.

*Logical presumption* is practical rule (criterion) determining the truth of propositions. This sets up a correspondence between the rule and the truth where truth is not classical (but pragmatic). It is closely connected with classical truth and depends on classical truth as on its basis, but the pragmatic truth violates the laws of classical logic: the law of contradiction, the law of excluded middle. Though, it is quite justified in context of radical, but fruitful criticism of these laws, unfolded 100 years ago by J. Lukasiewicz [Lukasiewicz, Jan.* O zasadzie sprzecznosci u Arystotelesa. Studium krytyczne* [*On the Principle of Contradiction in Aristotle. A Critical Study*], Krakow: Polska Akademia Umieijetnosci, 1910] and N. A. Vasiliev [Vasiliev, N.A., *Imaginary (non-Aristotelian) Logic*, in *Logique et Analyse*, 46 (2003), n. 182, pp. 127-163].

The definition of presumptions depends on the known definitions of conjunction and disjunction, the logical operators expressed in languages by “and” and “or” words. Presumptions are applied to propositions (sentences) whose truth is to be evaluated when the classical theory of truth does not help. To apply the presumptions, the subject (the subject of the proposition), particularly the process, is speculatively divided into spatial or temporal parts called *aspects*. The original proposition becomes a conjunction or disjunction of propositions that tell the same things about aspects which the original proposition claimed about the subject as a whole. Thus, one can obtain two identical or two different truth-bound estimates of the original proposition. The first will be justified by a conjunctive (strong) presumption, and the second by a disjunctive (weak) presumption. The choice of presumption is made by deciding person in the interests of achieving their own goals (which in turn don’t need any formalization).

At the same time, the evaluation of propositions about aspects comes from the classical understanding of truth, and it is possible in case of proper separation of aspects of the subject. But then these classical assessments are integrated by the rule of strong or weak presumption, resulting in a pragmatic truth that may or may not coincide with the classical truth.

**6. Formalization of Pragmatic Truth in Ch. S. Pierce’s Examples. Conclusion**

In the horse example the proposition “horse is flawless (acceptable)” is a subject for evaluation of truthfulness. This proposition seems true to the seller and false to the buyer. What is the ideal image for a horse that we should consider if these ideal images are different for the buyer and seller? Let’s formalize this using the presumptions. The seller keeps the “health” aspect of horse in mind when claiming horse’s acceptability. The fact that this horse is healthy is a classical truth. Denying the same proposition, the buyer has the aspect of “natural color” in his mind. It is classically false that the horse is not dyed.

The conjunction “the horse is healthy and the horse has natural color” representing the original proposition “the horse is flawless” is false, therefore the latter proposition is false and the buyer is right, *as if* he had chosen the strong presumption. But the disjunction of the same two parts is true, and the same proposition “the horse is flawless” is true. That’s why the seller is right who implicitly selected the weak presumption. He has hidden the aspect of the object which was important for the buyer, and this is where the buyer found lie.

We could consider Pierce’s example about the value of π in similar way, but this would require some basic knowledge of mathematical analysis. Further analysis of the two examples shows their structural union and comes to a conclusion that the relationship between classical and pragmatic truths in this case is similar to the relationship between definitions of classical (exact) and pragmatic (approximate) equality: the content of the first one (truth) is bigger than of the second, thus the scope of the second is wider than the scope of the first. This is the formal basis for W. James’s thesis that pragmatic truth subordinates the classical truth. In the strict sense the latter appears to be some kind of the first one.

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