- Roman Murawski ‘John von Neumann and Hilbert's School of Foundations of Mathematics’
The aim of the paper is to describe the main achievements of John von Neumann in the foundations of mathematics and to indicate his connections with Hilbert's School. In particular we shall discuss von Neumann's contributions to the axiomatic set theory, his proof of the consistency of a fragment of the arithmetic of natural numbers and his discovery (independent of Godel) of the second incompleteness theorem.
- Mieczyslaw Omyla ‘Possible Worlds in the Language of Non-Fregean Logic’
The term “possible world” is used usually in the metalanguage of modal logic, and it is applied to the interpretation of modal connectives. Surprisingly, as it has been shown in Suszko [68a] certain versions of that notion can be dened in the language of non-Fregean logic exclusively, by means of sentential variables and logical constants. This is so, because some of the non-Fregean theories contain theories of modality, as shown in Suszko [71a]. Intuitively, possible worlds are maximal (with respect to an order of situations) and consistent situations, while the real world may be understand as a situation, which is a possible world and the fact.
Non-Fregean theories are theories based on the non-Fregean logic. Non-Fregean logic is the logical calculus created by Polish logician Roman Suszko in the sixties. The idea of that calculus was conceived under the influence of Wittgenstein's Tractatus. According to Wittgenstein, declarative sentences of any language describe situations. Different explanations of what is a situation may be found in Wittgenstein , Wolniewicz , Barwise and Perry , Wójcicki  and others.
- Katarzyna Doliwa ‘The Role of Language in the Philosophical System of Thomas Hobbes’
Linguistic consideration understood by Thomas Hobbes as an arranged system of signs making the basis of thinking creatures called speech appeared in his works relatively early. In the workbook of logic entitled Computatio sive logica which he had been preparing since 1655 (it was not until 1655 that it was published as the rst section of philosophy entitled De Corpore) Hobbes tackled issues of language, the topic which he would continue exploiting in his later works presenting his views on social philosophy.
The consequence characteristic for Hobbes's commitment to the issue of language and its meaning in his most important works allows us to state that “language” alone has a signicant role in his system. It is necessary to notice that what Hobbes had in mind was basically a language of science, that is to say, a humble, dry language consciously deprived of any glare of eloquence. Hobbes's ideas about other uses of language appeared accidentally in his works.
- Renata Jermolowicz ‘On the Project of a Universal Language in the Framework of the XVII Century Philosophy’
By the end of the seventeenth century Europe was fully in uenced by the philosophy of Renaissance which had aected nearly every field of life and aimed at the critical revision of the heritage left by the Middle Ages. The goal was to provide solid foundations for the new science based on reason and experiment. It was in that light that the issue of language had become the subject of discussion for almost every thinker of the period. According to Heinz, there were three main issues in the field of linguistics that had captured the interest of the seventeenth-century philosophy: universal language, universal grammar, and the origins of language. The attempt to find a universal pattern remained in accordance with the general direction of the new science that aimed at simplification and logical organization of things. It was no longer that dead languages such as Latin, Hebrew and Greek could provide sufficient material for linguistic research. Natural languages, which had remained in the shadow of the dead ones, appeared on the scene challenging philosophers to establish universal features of languages, the discovery of which would lead to the discovery of a universal language, also known as a philosophical language.
- Jan Wolenski ‘Logical consequence and the limits of first-order logic’
Logic as a formal mathematical theory is interesting in itself. It poses problems, like any other body of knowledge. Some of them, especially questions concerning axiomatization, consistency, completeness (in various senses of the term), decidability, etc. are more specific with respect to logic (and mathematics to some extent) than in the context of other theoretical systems. These questions are in principle independent of any application of logic to other branches of science. If logic is conceived in this manner, we speak about logica docens. On the other hand, logicians always claim that the main task of logic consists in governing intellectual activities. Thus, logica utens (logic in use) is deduction, indispensable device of mind, particularly in various reasonings. Leaving aside traditional prescriptions, usually considered as stemming from logic and helping us in processes of defining or classifying, the main aim of logical theory is to codify the rules of deductive proofs. These rules should be stated formally and effectively. In particular, checking whether a proof is correct or not should be subjected to mechanical or algorithmic procedures. However, the rules of deductive proofs must guarantee that they lead from true premises to true conclusions, that is, block deductive derivations of falsehoods from truth. We have here an important difference between deduction and induction. The label “correct deduction” is in fact pleonastic, unless it points out that a given deduction was more complex than the proof required, for instance, that is employs unnecessary premises or proceeds indirectly instead of directly. Yet too complicated deduction is still deduction, if any. Induction may be correct, despite starting with true premises and resulting with false conclusions, if its rules are preserved (of course, I am conscious that speaking about rules of induction is a delicate matter, but we think, for instance, that inductive reasoning is correct if it was performed as carefully as possible). Incorrect induction is still induction, although incorrect deduction is not a deduction.
- Witold Marciszewski ‘The Two Origins of Modern Logic 1879, 1936’
1 The first origin and ‘Universal Characteristic'
1.1 Merits and demerits of Aristotele's logic
1.2 Begrisschrift compared with the Leibnizian project
1.3 On a perspective in which there is no second origin
1.4 The right perspective
1.5 The idea of logic as embodied in Begriffschrift
2 On paving the way to the second origin
2.1 The problem of solvability attacked by modern logic
2.2 Hilbert's Entscheidungsproblem as the guiding idea
3 The impact of reformed logic on intelligence research
3.1 The confluence of ideas in 1936, Turing's position
3.2 Does Church-Turing Thesis apply to the physical world?
- 同上 ‘Do Post's Logics Belong to Alternative Logics?’
The University in Bialystok, Poland, in collaboration with the Bialystok Technical University, organizes the Workshop in the Centenary of Emil Post's Birth: Mutual Influences between Informatics and Logic, December 13/14, 1997. Post was born in 1897 in Augustow, a place in the vicinity of Bialystok, hence not only the time but also the place circumstances motivate commemorating him in this year and this city.
The Editors would enjoy publishing preliminary discussions which might assist working on papers to be read at the Symposium. The text below is to encourage such a discussion.
ここで気になるのは‘Post was born in 1897 in Augustow, a place in the vicinity of Bialystok,...’です。そこでこの‘Bialystok’を調べてみるとPolandの北東部にあるようです。Naziに手ひどくやられた街のようですが、写真を見るとよさそうなところです。なるほどね。昔TarskiがAmericaに渡ってPostに会った時、「America生まれのlogicianに初めて会えた」というような意味のことを言ってAmericaのlogicの後進性をそれとなく示唆するような発言をしたところ、Postが「私はPoland生まれです」と答えてTarskiがびっくりしたという話をどこかで読んだことがあります(これだとさらにAmericaがlogicに関して後進的であることが強調されてしまいますね)。PostのCollected Worksの解説だったか、CrossleyのReminiscences of logiciansだったかどこかで。そんな逸話があったのでPostの生まれ故郷にはちょっと興味がありました。この辺でPostは育ったのかなぁ。あるいはずっと小さい時にもうAmericaに移住してたんだったけ？ まぁいいか。