In his treatise Die Grundlagen der Arithmetik, Gottlob Frege tries to find a definition of number. First he rejects the idea that number could be a property of external objects. Then he comes with a suggestion that a numerical statement expresses a property of a concept, namely it indicates how many objects fall under the concept. Subsequently Frege rejects, or at least essentially modifies, also this definition, because in his view that a number cannot be a property – it should (...) be an object. In this article, I try to show that Frege’s first definition of number seems to be, despite his own opinion, much more promising than he supposed. I also argue that Frege’s argumentation against the empirical character of number is by no means convincing. (shrink)

Cantor's diagonal proof is sig­nificant both because the central method of proof used in it has been subsequently applied in a number of other proofs, and because it is considered to confirm the existence of infinite sets whose size fun­ damentally and by an order of magnitude exceeds the size of the "classical" infinite set represented by all natural numbers, while their size can theoretically exceed every conceivable limit. Although Can­tor's proof is generally accepted by the scientific community, some experts (...) are somewhat reserved about it. The aim of this paper is to present Cantor's proof in an accessible way, while pointing out its (hidden) assumptions and possible problematic points, and pointing out that some of its underlying assumptions are not indisputable mathematical truths, but rather postulated propositions that may or may not be accepted. (shrink)

Numerical identity is standardly considered to be a relation between things. This means that two things are identical if they are only one thing. It is not only Wittgenstein who finds this claim rather odd. Another possibility is to understand identity as a relation between names which denote the same thing; or as a relation between the senses of those names which are modes of presentation of the same thing. Or identity statements can be considered as expressions of the fact (...) that there is exactly one thing that has certain characteristics. These conceptions are, in fact, closely interlinked. There are some rather straightforward proposals on how to give these approaches a formal pattern in relative harmony with classical logic. What is important to emphasize and keep in mind is the specific character of identity and essential dissimilarities between identity and relations of objects in logic. (shrink)

Recent extensional logic considers a formula containing existential quantifier as an adequate way of rewriting a statement about the existence of something. Quine even claims that “to be is to be the value of a variable”. That means that if we consider a theory to be true, we have to treat all members of an appropriate universe as existing. This article intends to point out a difficulty in this conception: The primary task of an existential quantifier is only to claim (...) that something can be found in a universe, whatever the universe is. (shrink)

Instead of the existential quantifier Bernard Bolzano uses his notion of Gegenständlichkeit einer Vorstellung an sich . This approach makes it possible for him to solve the traditional problem of the predication of existence in accordance with an approach common today, namely to take existence to be primarily a property of a corresponding representation in itself, and not a characteristic of individuals. However, Bolzano considers a property of real existence to be attributable also to particular individuals, which is essentially different (...) from the usual practice. (shrink)

The notion of the extension of a concept has been used in logic for a long time. It is usually considered to be closely connected to the intuitive notion of a set and thus seems as though it should be embedded into set theory. However, there are significant differences between this “logical” concept of set and the notion of set (class) as defined via standard axiomatic systems of set theory; it may, therefore, be quite misleading to consider the two concepts (...) as being continuous with each other. When we look at the writings of Gottlob Frege and consider the development of his attitude to extensions, we can see what the differences consist in and which of the two notions is more apt to be used in foundations of logic. Frege himself eventually rejected sets entirely. (shrink)

The nominalist attitude in medieval logic supported a fully extensional conception of the sense of expressions. Many arguments against this approach were raised at that time. I would like to show in this article that there is a extensional conception of notions in current logic, namely in the interpretation of formal theories and the creation models of them and that, and how, traditional arguments against the extensional conception are relevant also for logic today.

Bernard Bolzano presents in his work Wissenschaftslehre a definition of derivability among sentences. Tarski publishes his well-known definition of logical consequence almost one hundred years later. This article intends to prove that the Bolzano´s definition is fully satisfactory compared to the Tarski´s one, even though no attention was drawn to Bolzano´s definition at that time.

Gottlob Frege rozlišuje mezi smyslem a významem výrazů, přičemž smyslem rozumí to, co je výrazem vyjadřováno, významem pak to, co je jím označováno. Toto rozlišení se týká jak singulárních, tak obecných výrazů, ovšem Fregovy rozbory týkající se významu obecných výrazů jsou všeobecně méně známy. Chci zdůraznit skutečnost, že Frege považuje za význam obecných výrazů pojem, který chápe jako tzv. nenasycenou funkci, nikoli jako množinu nebo zobrazení. To mu umožňuje zachovat některé podstatné intuice týkající se obecnin – můžeme např. mít obecný (...) pojem (znát nějakou vlastnost či relaci), aniž bychom věděli, jaké předměty pod něj spadají. Zároveň se snažím ukázat, že není jasné, nakolik Frege skutečně zastával tezi, která mu bývá obvykle přičítána, totiž že identita pojmů je dána jejich koextenzivitou. Gottlob Frege distinguished between sense and reference of expressions. 'Sense' according to him is that what the expression means, 'reference' is that what is designated by it. This distinction concerns both singular and general terms but Frege’s analyses focused on general terms are much less known. It is crucial, I argue, to appreciate the importance of the fact that the reference of a general term is, according to Frege, a concept – identified with an unsaturated function in the Fregean sense, i.e. not a set, and not a mapping. In this way Frege can keep some important intuitions about universals – for instance that we can have a general concept (know a property or a relation) even if we don’t know which objects fall under it. I also try to show that it is not clear whether Frege really defended the thesis usually attributed to him that the identity of concepts is guaranteed solely by their coextensivity. (shrink)