engleski

The formal and the informal in the foundations of logic

According to J. M. Bocheński (Formale Logik, 2002), "formal logic" deals with formulas ("logical propositions", "laws of formal logic") whose instantiations give conditional propositions containing a consequent that necessarily follows from its antecedent. Strictly, formal logic does not include semiotics (semantics) nor general methodology (use of logical laws in science). According to Gödel's and Turing's results (Turing-reducibility of a formal system) and some Gödel's views (reflections on causality), it follows that the questions about the foundations of formal logical laws (the questions of their "formulation, critique, and systematization", Bocheński) lead beyond the dependency on a particular logical formalisms, and moreover, conjoin the concept of the "formal" with the content-related concept of causality. (1) According to Gödel, the universal concept of a formal system, independently of a given formalism, is given only by a clear and precise perception of the concept of a mechanical procedure, defined by a Turing machine, which can write down all the theorems of a given formal system (`Gibbs lecture', Collected Works, III, 308). By a formal description of the work of Turing machines by means of justification logic tools, we can confirm that evidence (justifications) about halting and non-halting exceeds the limits of a possible formal system (because of the violation of the second incompleteness theorem) and, in addition, that it is not constructive in a Gödelian sense (evidence obeys non-provable principles that are not definitions, nonenumerability of "all" justifications, 'Zilsel lecture', Collected Works, III). (2) The generating of evidence by a Turing machine is obviously an (idealized) mechanical, causal affair. On the ground of a formal presentation by means of a causal version of justification logic, we show that Turing machine is causally interconnected and dependent on a human mind as its intentional operator ("which infinitely surpasses the powers of any finite machine", 'Gibbs lecture', Collected Works, III, 310). In searching for logical forms (in themselves and as implemented in particular formalisms), we are thus led, along Gödelian lines, to the foundational concept of causality, although this concept in itself does not seem to be of a formal logical nature, but requires an objective content to which it refers. Aristotelian syllogism, which is Bocheński's historical paradigm of a logical form, has, according to Aristotle, a causal sense (e.g., premises are causes of the conclusion). On several examples, some of them historically preceding Gödel's results, it will be shown how the advances in the foundations of formal logic happen prior to and without the aid of an already established axiomatization, through the study of our use of logical forms in informal everyday or scientific discourse and the study of philosophical conceptions of logic and causality (Łukasiewicz, Tarski, Jaśkowski).

formal logic ; Turing machine ; causality

Dorađena inačica objavljena u knjizi "Formal and Informal Methods in Philosophy", M. Będkowski et al., Brill, 2020.