Comment On Eklund And Kolak

Abstract

I appreciate greatly the occasion that Eklund and Kolak (referred to in the sequel as 'the authors') have provided me to reflect on the nature of independence-friendly (IF) logic. Naturally I also welcome their sup port of my position. However, the issues they raise cut even deeper than the authors bring out. In order to help our readers to put their paper in perspective, I therefore propose to offer a few comments on it. The most general comment concerns the framework the authors are using. They consider different logics the received first-order logic IF first-order logic, the E{ fragment of second-order logic, the whole second order logic, and so on, as different 'systems' of logic, each with their own 'commitments'. This way of talking and thinking clearly presupposes some idea of a system of logic as an independent unit of inquiry, some thing like a structure of formulas held together by inferential relations are to be captured by suitable axioms and inference rules. Now G?del-type incompleteness results show that one cannot conceive of mathematical theories as such formal systems. Even arithmetical truth can only be char acterized model-theoretically, not formally. However, the necessity of the same shift away from a syntactical and deductive viewpoint to a model theoretical (semantical) one in the case of systems of logic has not reached the consciousness of philosophers. The culprit here is historically speaking G?del's completeness proof for ordinary first-order logic which had the disastrous effect of leading philosophers and philosophical logicians to believe that logical theories (branches of logic) can be thought of as formal systems, that is, thought of in axiomatic and deductive terms. One of the most important impacts of IF logic on the philosophy of logic is to destroy this belief for good. In spite of its crystal clear semantical meaning, IF first order logic is not axiomatizable, that is to say, representable as a formal system. It can only be characterized semantically (model-theoretically). And comparisons with other kinds of logic can a fortiori be carried out only in model-theoretical terms.