Ludwik Borkowski

A logical system based on rules and its application in teaching mathematical logic

As it is known, the contemporary formal logic has sufficient means to for? malize each proof, in particular each mathematical proof. But it is also known that the formalization of proofs by means of logical laws gives often very long and burdensome constructions even in the case of simple intuitive proofs. Thus some divergences arise between the theoretical and practical scope of logic in this domain. Hence there appears the tendency to introduce such logical means which would enable us to eliminate these divergences and to formalize proofs in a practically simple and didactically easy manner. To bring logic as near as possible to the practice of proofs was one of the aims of the authors of the logical systems based on rules.1 It seems however that the existing systems of logic based on rules are only to a small extent pra? ctically and didactically exploited. In the present article we shall describe a logical system based on rules2. For some years we were using it in the lectures of formal logic and in some branches of mathematics. The practice showed that students easily and quickly learn the rules of the system and ? without the help of the teachers ? perform

Reduction of arithmetic to logic based on the theory of types without the axiom of infinity and the typical ambiguity of arithmetical constants

The reduction of arithmetic of natural numbers, to logic based on the theory of types meets with two main difficulties: (1) the typical ambiguity of constants determining natural numbers ; (2) the necessity of assuming the axiom of infinity. In this article I show in what way arithmetic of natural numbers may be reduced ? without these defects ? to the system of logic based on the theory of types and extended in the following manner :

Deductive foundation and analytic propositions

1. The deductive foundation or the deductive proof is usually identified with the deductive inference. The deductive inference is defined as an inference whose conclusion follows logically from its premisses. The deductive proof defined in such a way is an indirect foundation of the truth of some proposi? tions (conclusions) by appealing to the truth of other propositions (premisses). The sequence of inferences which we can perform is always finite; hence there are always in our inferences ? if we avoid the fallacy of vicious circle ? some propositions which are not founded indirectly. It is obvious therefore that ? when using such a concept of a deductive proof ? we cannot say that some propositions are proved (or provable) deductively but only that they are proved (or provable) deductively by virtue of certain other propositions. The deductive science is defined as a science in which only deductive in? ference is performed. It is also almost commonly assumed that each deductive science is an axiomatic system. It follows from our previous remarks that we cannot say that a proposition of a given deductive science is proved (or pro? vable) deductively. We can only say that it is proved (or provable) deductively by virtue of the axioms. But the axioms cannot be founded deductively. Such a conception of the deductive proof as an indirect foundation is com? monly accepted in the methodology of sciences. As I know it has not been criticized till now.

Some theorems on the smallest sets closed under the classes of relations and their generalizations. I

The concept of a smallest set including an initial set and closed under the relations (in particular: operations) of a given class is of a considerable importance in the metho? dology of deductive sciences. The sets of significant expressions, the sets of theses of axiomatic systems, the sets of consequences of some sets of propositions are examples of the smallest sets including certain initial sets and closed under definite operations. In the methodology of deductive sciences we intuitively use various theorems on the smallest sets. The proofs of some theorems of such a kind are given in ? 1 of this article. These proofs are formalized (by means of the method presented in the book: J. Slupecki and L. Borkowski, Elements of Mathematical Logic and Set Theory). This formalization may be useful, on the one hand, in the applications of such theorems in teaching mathematical logic; on the other hand, it makes easy stating which of these theorems are valid for some generalizations of the concept of a smallest set in? cluding an initial set and closed under some relations. Such generalizations are also suggested by some examples used in the methodology of deductive sciences. If we say e.g. about expressions formed of the infinite number of symbols or if we say about the rule of infinite induction (co-rule), then we take into consideration operations which assign some expressions to infinite sequences of expressions or to infinite sets of ex? pressions. Some theorems in which we use a suitably generalized concept of a smallest set are given in ? 2 and ? 3 of this article.