N.A. Vasiliev's "Imaginary" Experimental Modal Logic...

Evolutionary Thinking in Past Scientific Theories: A Logical Analysis by Antonino Drago, Dept. Phys. Sci., Univ. “Federico II”, Naples, Italy

Abstractions lead us to shape ideas, about which our minds argue by means of logic. An evolutionary thinking occurs when these ideas are not linked together by means of mechanistic deductions, but in a creative way. In this sense evolutionary thinking pushes us to shape a broader kind of logic. The phenomenon of a double negated statement whose corresponding positive statement is lacking of scientific evidence (=DNS) will be examined. It represents a failure of the double negation law; this law constitutes the borderline between classical logic and, broadly speaking, non-classical logic (in particular, intuitionistic logic). In fact, several scientific theories born in past times include in an essential way DNSs. In particular, quantum logic can be represented by means of DNSs inside intuitionistic logic. When DNSs pertain in an essential way to a theory, no more – as a comparative analysis upon the several instances shows – a deductive organization of the theory is possible; rather, the theory puts an universal problem by means of a DNS, then some double negated methodological principles (e.g.: “It is impossible a motion without an end”) follow in order to achieve a new scientific method, capable to solve the problem at issue. This arguing evolves through a cyclic pattern, according to the synthetic method as it was improved by L. Carnot. The crucial step in this pattern is an ad absurdum theorem (likely as in thermodynamics S. Carnot’s theorem is). This theorem reaches evidence for a possible conclusion, still enunciated by means of a DNS. Then by a move like Markoff principle this DNS is changed in a positive statement; it can now be put as a new hypothesis from which to develop a full deductive system. This move is illustrated at best in Lobachevsky’s – maybe first – presentation of a non-Euclidean geometry, but can be recognised also in S. Carnot’s thermodynamics, Avogadro’s atomic theory, Einstein’s founding special relativity. This pattern of arguing is examined by means of paraconsistent logic. In correspondence to the use by theoretical scientific research, of respectively paraconsistent logic, intuitionistic logic and classical logic about statements which are potentially principles for a theory, three kinds of principles are recognized; i.e., a guess, a methodological principle, an axiom-principle. These differences are expressed in a lucid way by Einstein again in his celebrated paper on special relativity: “We will raise the conjecture (the substance of which will be hereafter called the “[axiom-]principle of relativity”) to the state of a [methodological] postulate” LINK: In a previous paper I obtained a relevant result regarding paraconsistent logic. The founder of paraconsistent logic, N.A. Vasiliev, stated as a characteristic feature of his logic, three kinds of sentence, i.e., "S is A", "S is not A", "S is and is not A" ("indifferent judgment"). I was able to show that they hold true even when one substitutes "￢￢A" for "S" and "-->" for "is". One obtains respectively: "￢￢A-->A ", "￢￢A fails to -->A", "￢￢A-->A and ￢￢A fails to -->A".(substitute necessity [box] for --> everywhere) Let us remark that the three cases represent three different roles played (in) a sentence in an argument. i) ￢￢A-->A represents as an affirmative sentence, i.e. a sentence well-supported by scientific evidence; ii) ￢￢A fails to -->A represents a logical problem, i.e., it can represent a sentence still insufficiently supported by scientific evidence; iii) ￢￢A-->A and ￢￢A fails to -->A represents a sentence whose truth and falsity is not yet decided in scientific terms; this kind of sentence may be considered inside a theoretical framework as a guess, whose scientific qualification it is still yet to be decided. The last kind of sentence qualifies the characteristic sentence of paraconsistent logic as pertaining to a theory in construction. Antonino Drago   Imaginary(meaning sheet of assertions in imagination) Experimental Modal Logic: ￢￢A-->A(classical linear deduction) ￢￢A fails to -->A(non-classical/non-linear induction) ￢￢A-->A and ￢￢A fails to -->A(non-classical/non-linear abduction, hypothesis, theory)Antonino Drago on N.A. Vasiliev(my additions in italics)   Infinity__ Where all doubts are allowed… Let us consider Lobachevskii's geometry. By substituting "two straight lines meet" for A and "It is not true that two straight lines do not meet" for ￢￢A, i.e. Vasiliev's S, the three Vasiliev's above sentences describe respectively i) ￢￢A-->A, i.e. the hyperbolic secant lines, ii) ￢￢A fails to-->A, i.e. the hyperbolic ultra-parallel lines and, iii) ￢￢A-->A and ￢￢A fails to -->A, i.e. the parallel lines - which meet at a point which is located at infinity, i.e. where all doubts are allowed. This last meaning is presented by Lobachevskii himself in his most relevant writing; there, Lobachevskii refers to the meeting point at infinity by means of the following words: "In the uncertainty...", just the meaning of Vasiliev's third kind of sentence. That vindicates Vasiliev's reiterated claim, i.e. his logic represents just the logic of Lobachevskii's geometrical theory. "Handle two sorts of negations (logical and ontological)"; as paraconsistent logic does.   Conclusions The three main kinds of logic correspond to three characteristic ways of organizing a set of scientific data in a systematic way. Paraconsistent logic is a relevant logic since it represents the logic of the work of a scientist in his guessing new hypotheses for a given set of scientific data. I would add that the above exploration of the different roles played by the three kinds of logic has introduced us to a new kind of study, which can be called experimental logic; it is based upon evidence coming from the characteristic features of past scientific theories rather than the characteristic features of natural languages. Antonino  Drago Vasiliev affirmed, only ''positive'' sensations are possible, by which we can distinguish only contrary qualities. This is the basis of qualitatively different types of judgments - affirmative and negative. If one imagines a world in which not only positive but negative sensations are possible, then such a world will indeed require a different logic, and the introduction of supplementary qualitative judgments…