engleski

Evidence and self-evidence in the foundations of logic

We relate the question of the correctness of proofs and of a possible foundation of a logical system to a general, computational, concept of a formal system as a mechanical procedure, in the sense of a Turing machine, for producing provable formulas (Gödel 1934/64). By means of justification logic tools, the question about the evidence in a given system is united with the question of an abstract causal structure of a mechanical decision procedure. After introducing a translation procedure of the work of a Turing equivalent register machine into a suitable justification logic language, it is easy to show that, for each translated register program, the reason (evidence and cause, not necessarily by a register routine) could be proved for the program's halting/non-halting. The evidence of justification logic reasons exceeds the limits of a given formal axiomatic system (since not obeying the constrains of the incompleteness theorems, Gödel 1938). Further, justification logic (including its axiomatic description of reason operators) does not satisfy Gödel's (1938) constructivity requirements. Thus, as a foundational question and the question of the criteria of the correctness of reasoning, we discuss a possible ``meta-justification'' of the axioms about reasons by analyzing the work of a register (Turing) machine in causal terms in comparison with general self-evident structures of a human agent's reasoning. This includes a sort of abstract pragmatic considerations of the use of concepts by an abstract reasoner (attention to our own acts in using concepts, Gödel 1961).

correctness, proof, Turing machine, justification logic, causality

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