engleski

Proof and causation

In one of his notes Gödel briefly mentions a possible logical and philosophical program of a reduction of all ``logical (set-theoretical)'' axioms to the ``axioms of causality'', and, in philosophy, of all (Kantian) categories to the category of causality. From this standpoint, it seems natural to connect two Gödel's results: (1) a sketch of justification logic as a sort of a logic of proofs (Zilsel Lecture ; further developed only recently by S. Artemov and M. Fitting), and (2) Gödel's second-order modal ontological system GO, devised for his ontological proof of the existence of the most positive being (a note from 1970). On the ground of this we want to show that proofs can be formally conceived as a special (in a way, paradigmatic) case of causality, and causality as a central ontological concept. We show that Gödel's ontological system can be transformed into a sort of justification logic (a modification and extension of FOLP (Artemov - Yavorskaya 2011) in which the justification (proof) terms can be re-interpreted in a causal sense. To that end, we first analyze Mackie's INUS concept of causality, and relate it to historical concepts of causality in Aristotle's theory of proof (premises as causes of the conclusion), as well as to Leibniz' and Kant's concepts of causality (in connection with the principle of sufficient reason and the concept of consequence). After this conceptual-historical analysis, we apply the ontological concept of causality to the justificational transformation of GO, and transform it further into a causal ontological system CGO, where ``essence'' and ``necessary existence'' are reduced to causality, and in which it can be proved that every fact has a cause and that there is a first cause. We formally describe a first-order and a second-order version of CGO, and give an appropriate semantics with respect to which the soundness and completeness proofs for versions of CGO can be devised.

causality; justification; ontology; Gödel

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