Interpretations of modal logic. A contribution to phenomenological philosophy of science. (Interpretationen der Modallogik. Ein Beitrag zur phänomenologischen Wissenschaftstheorie.) (German) Zbl 0922.03007

Wiegand, Olav K.

Phaenomenologica. 145. Dordrecht: Kluwer Academic Publishers. x, 217 p. (1998).

The author’s aim is to point out interpretations of modal logic which are compatible with the phenomenological approach to mathematics. The book consists of three parts with ten chapters. In the first part (pp. 19-77) the author presents E. Husserl’s conception of a “mathesis universalis”. For Husserl, the mathesis universalis contains both, formal mathematics and formal (symbolic) logic. It has a hierarchical structure consisting of a pure logical grammar, a logic of consequences and a logic of truths. The author pays special attention to the differences between formal logic and formal mathematics which can be observed despite their extensional identity.
In the second part (pp. 81-143) the author presents what he calls “phenomenological semantics”, i.e. the phenomenological theory of modalization being a general analysis of intentions. The author distinguishes three levels of modalization, the level of protological passive synthesis, the level of protological active synthesis, and the level of (logical) predication.
The third part (pp. 147-194) combines the results of the preceding parts in a phenomenological criticism of modern modal logic, especially its interpretation as possible worlds semantics. The problems of applying this semantics to natural language are seen as anchor points of phenomenlogical criticism. The provability interpretation of modal logic is proposed as a genetic interpretation, notwithstanding the problems which Hilbert’s program and Husserl’s closely related idea of definite manifolds had with Gödel’s and Church’s results.

Reviewer: V.Peckhaus (Erlangen)

MSC:

03A05	Philosophical and critical aspects of logic and foundations
00A30	Philosophy of mathematics
03B45	Modal logic (including the logic of norms)
03F40	Gödel numberings and issues of incompleteness
03-02	Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations

Keywords:

mathesis universalis; formal logic; formal mathematics; axiomatics; definite manifolds; definition; phenomenological semantics; modalization; modal logic; provability; possible worlds semantics; interpretations of modal logic compatible with the phenomenological approach to mathematics; phenomenological criticism of modern modal logic; natural language