Leon Horsten

The difficulties with formalizing the intensional notions necessity, knowability and omniscience, and rational belief are well-known. If these notions are formalized as predicates applying to (codes of) sentences, then from apparently weak and uncontroversial logical principles governing these notions, outright contradictions can be derived. Tense logic is one of the best understood and most extensively developed branches of intensional logic. In tense logic, the temporal notions future and past are formalized as sentential operators rather than as predicates. The question therefore arises whether the notions that are investigated in tense logic can be consistently formalized as predicates. In this paper it is shown that the answer to this question is negative. The logical treatment of the notions of future and past as predicates gives rise to paradoxes due the specific interplay between both notions. For this reason, the tense paradoxes that will be presented are not identical to the paradoxes referred to above

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics