Jan Wolenski

This paper offers a logical analysis of scepticism. It is shown that dogmatism, academism and scepticism as characterized by Sextus Empiricus in Outlines of Pyrronism form a variety of views which can be ordered by an interpretation of the classical logical square. In particular, scepticism appears as a conjunction of the negations of dogmatism and academism. The next problem concerns the logic proper for scepticism. Logic based on a dual of the consequence operation is proposed as satisfying intuitive requirements associated with doubting. Finally, the attitude of the sceptic toward logic is discussed. In particular, it is argued that the principle of isosteny trivializes scepticism if it is applied to logic

This paper proposes a formal framework for the cognitive relation understood as an ordered pair with the cognitive subject and object of cognition as its members. The cognitive subject is represented as consisting of a language, conequence relation and a stock of accepted theories, and the object as a model of those theories. This language allows a simple formulation of the realism/anti-realism controversy. In particular, Tarski’s undefinability theorem gives a philosophical argument for realism in epistemology.

This paper investigates relations between truth and consistency. The basic intuition is that truth implies consistency, but the reverse dependence fails. However, this simple account leads to some troubles, due to some metalogical results, in particular the Gödel-Malcev completeness theorem. Thus, a more advanced analysis is required. This is done by employing the concept of ω-consistency and ω-inconsistency. Both concepts motivate that the concept of the standard truth should be introduced as well. The results are illustrated by an interpretation of the well-known logical square and its generalization.

Formal semantics is understood either as a formal analysis of semantical features of natural language or as model-theoretic semantics of formal(ized) languages. This paper focuses on the second understanding. The problem is how to identify the formal aspects of formal semantics, if we understand ‘formal’ as ‘independent of content’. This is done by showing that the form of semantical interpretation of a language L is given by its syntax and the parallelism of the signature of L and its interpretative structure SI. However, the content of interpretation, that is, the way of correlating of L-expressions with extralinguistic items depends of informal factors. The discussion shows that semantics is prior to syntax

There are two possible strategies for investigating questions on logic and mathematics. First, one can adopt the pattern recommended by the phenomenologists, which consists in looking for the actual essences of logic and mathematics in order to relate both fields. The second approach, adopted in this paper, starts with a historical review of the foundational standpoints. I will then try to extract on this base some insights on how logic and mathematics are mutually related. In particular, I am interested in the concept of logic suggested by the development of the foundational debate. Yet one more preliminary remark is here in order. I will take into account the classical foundational positions: logicism, intuitionism, and formalism, as well as their later modifications and changes.1 Of course, this, so to speak, Foundational Trinity, does not exhaust the complete map of the foundations and philosophy of mathematics. For instance, I entirely neglect views wich might be exemplified by Lakatos, Dieudonné or Hersh, who, roughly speaking, radically deny that the study of logic and its relation to mathematics provides any interesting problem. I do not feel competent myself to judge this matter from the point of view of working mathematicians. On the other hand, philosophers are traditionally interested in the nature of logic and its links with other fields. Consequently, I regard my problem rather as a way of understanding what logic is, than whether it is important for mathematics or not