XVI. Задача Льюиса Кэрролля и вопрос о догмате (к стр. 61).

The way out of doubt, indicated in the text on page 61, appears, from a formal logical point of view, to be a special case of the logical problem proposed by Lewis Carroll. For greater awareness of the step we take when we believe in the Truth, it is useful to consider the mental processes corresponding to the act of faith in abstracto, that is, to solve the above-mentioned logical problem in its general form. This task is formulated as follows: « q includes r; but p includes that q includes not-r; what should we conclude from this? – q implique r; mais p implique que q implique non-r; que faut-il en conclure?” 836. In ordinary language, although somewhat one-sidedly narrowing the task, we can convey it in the following expressions: “The truth of a proposition or concept r necessarily follows from the truth of another judgment or another concept q; but some third judgment or some third concept p is such that from its truth it necessarily follows that r cannot follow from q, as was said earlier, but the negation of r, not-r, necessarily follows; what can be concluded from such a set of premises? At first it may seem that we are talking about resolving some extraordinary and artificially created difficulty that has no vital significance. But this is far from the case. L. Carroll's task was not “composed”, but put forward by real need. But the most interesting thing is that the author of the problem himself, when solving it theoretically, fell into the same mistake that people usually fall into when solving it in practice. He argues this way: “If q includes r, then it is impossible for q to include not-r; this means that p includes the impossible and, therefore, is false” 837. But Carroll’s conclusion is erroneous, because it is possible that not p is false, but q is false, which simultaneously includes r and not-r, that is, two contradictory judgments or concepts. Thus, the conclusion from common sense does not provide a definite, strictly logical solution. On the contrary, the symbolic method of logistics allows us to obtain it through very elementary transformations. We will do them now, but first we will symbolically write down the conditions of the problem. Obviously, the first condition of the problem will be written: But the first inclusion (I), with the signs reversed, gives: and the second (II), after replacing the inclusion in the second part q Ɔ ¬ r with an equivalent sum ¬ q ∪ ¬ r, gives: Substituting in (II") instead of ¬ r what he included in (I") ¬ q we find: p" . Ɔ. ¬ q ∪ ¬ q (III), which gives the correct solution to Carroll's problem. We made the decision semi-symbolically. With just the same symbols it will be written like this: What is the meaning of the obtained solution (IV)? – That the truth of p entails the negation of q, i.e., in other words, that it is impossible to assert q, since, while, if, where p has the force. This, however, does not mean at all that p is absurd, as L. Carroll believed, nor does it mean that q, which includes a contradictory consequence, is absurd, as Couture believes it is possible to assert in the name of common sense. p Ɔ ¬ q - this solution satisfies both the first and second conditions of the problem, recognizing their truth and value, and common sense solutions do not satisfy either the first or second conditions, because they declare at least one of them to be a simple absurdity and, therefore, only a misunderstanding. Speaking figuratively, one can imagine that condition (I) is the testimony of one witness, and condition (II) is the testimony of another. The arbitrator - common sense - who intervened in this dispute, frivolously declares that either the testimony of the second witness, by virtue of his statement p, or the testimony of both, by virtue of the statement of both q, is nonsense, nonsense, absurdity. With these words “nonsense”, “nonsense”, “absurdity”, common sense does not mean that any of the disputants are lying or making mistakes - and then an actual verification of the testimony of both would be required. Not at all, he is simply saying that the words of at least one of them are meaningless and therefore do not deserve any factual verification, refuting themselves. Thus, common sense not only does not give the correct solution, but also does not give a solution at all, for it says: “either one or both are talking nonsense”; but, moreover, without giving a solution, he keeps those disputing from research, from actually verifying their statements, because there is no point in actually examining something that is already formally absurd. - Then, both witnesses, offended by this outcome of the case, turn to a more thorough judge - to logistics. This judge, having examined the case, pronounces a very definite sentence, namely: “p Ɔ ¬q”, i.e. That is, in other words, without offending either of the disputing parties with the reproach of the meaninglessness of the testimony and even recognizing the correctness of both, the judge asserts that neither one nor the other can talk about q at those times and under those conditions when p receives the force. If p, q is present, it is canceled; and in all other cases it is in force. The first party who approved conditions (I) is right; rights and the second, which approved conditions (II). But both of them must understand that the ordinary, everyday, ubiquitous q ceases to be such under special conditions, namely under the condition p. To clarify these abstract arguments about p, q and r, let’s replace the signs with concrete data, that is, we will solve the examples I invented ad hoc, due to the lack of ready-made ones. Example 1: “The sky is blue; at sunset the sky is red. What can we conclude from this? Let us denote: “sky” - q, “blue” - r, “at sunset”, i.e.: “when sunset”, “if sunset”, “sunset” - p and, finally, “red”, i.e. “not blue” - ⌐r. Symbolically, the conditions of our example will be presented as follows: q Ɔ r: ∩: p.Ɔ.q Ɔ ¬ r (IV), that is, it really turns out to be a special case of L. Carroll's problem. According to the verdict of common sense, it would turn out that either the expression p, “at sunset,” is meaningless, i.e., that sunset not only does not actually exist, but that purely logically it is something impossible and unacceptable, or, that q is absurd, i.e., that the very concept of “sky” is internally contradictory and that there cannot be any “sky.” The answer from logistics is given by (IV, V): i.e., that, although both “sky” and “sunset” are quite possible and, if conscientious observers show their reality, they actually exist, however, “at sunset” the observer does not deal with “sky”, does not observe “sky”, but with something else, not with the sky; for example, if you try to give a positive answer, the observer then sees the sun, although through the atmospheric layer, through the “sky”. The essential importance of this answer hardly requires proof. It is enough to recall at least the explanation of the “sky” by the dust theory of Ramsey and Tyndall in order to understand that here logistics introduces us into the medias res of the scientific work of a physicist 838. Example 2: “The rationalist says that the contradictions of Holy Scripture and dogma prove their non-divine origin; the mystic claims that in a state of spiritual enlightenment, these contradictions precisely prove the divinity of the Holy Scriptures and dogmas. The question is, what conclusion should be drawn from these statements? Let us again denote: “contradictions of the Holy Scripture and dogmas” - q, “non-divine origin” - r, “state of spiritual enlightenment” - r and, finally, “divinity”, i.e. “non-divinity” - ⌐r. Then, again, the conditions of this collision will be expressed by the formula: q Ɔ r: ∩: p . Ɔ. q Ɔ ¬ r (IV"), i.e. It is again discovered that our example fits L. Carroll's scheme. Therefore, solving the problem, as common sense would like, we would come to the conclusion that either p or q is meaningless, i.e. Either “spiritual enlightenment” is meaningless and impossible, or it is absurd to talk about “contradictions of the Holy Scriptures and dogmas.” In the first case, the statement of a mystic would be meaningless, and in the second, both a mystic and a rationalist. The logistics answer (IV, V) again gives: that is, both the rationalist and the mystic are right. Both the “contradictions of Holy Scripture and dogma” and “spiritual enlightenment” do not contain anything absurd and, therefore, if an honest rationalist and an honest mystic refer to them, then they actually exist. But what for ratio is a contradiction, and an undoubted contradiction, ceases to be a contradiction at the highest level of spiritual knowledge; is not perceived as a contradiction, it is synthesized, and then, in a state of spiritual enlightenment, there are no contradictions. Therefore, there is no point in training a rationalist to realize that there are no contradictions: they exist; yes, they are undeniable. But the rationalist must believe the mystic that these contradictions turn out to be the highest unity in the light of the Unsetting Sun, and then they precisely show that the Holy Scripture and dogmas are above carnal prudence and, therefore, could not be invented by man, i.e., divine. This is the same conclusion that we came to in this essay. Cout. , – Pr. d. M.,, p. 16, with reference to Carroll's article, is placed. in "Mind", 1905, Apr. and July, pp. 293, 400. Public exposition theory of the blueness of the sky, see: J. Tyndall, - The role of imagination in the development of natural sciences. Per. F. Pavlenkova, Vyatka, 1873 (“Speeches and Articles”, M., 1875). – His, – Dust and diseases (J. Tyndall, – Essays from the natural sciences, with a foreword and note by Helmholtz, translated from German by O. Bobylev, P. Gezehus, N. Egorova et al., St. Petersburg, 1876, pp. 1–62).