28 research outputs found
Categories of First-Order Quantifiers
One well known problem regarding quantifiers, in particular the 1storder
quantifiers, is connected with their syntactic categories and denotations.
The unsatisfactory efforts to establish the syntactic and ontological categories
of quantifiers in formalized first-order languages can be solved by means of the
so called principle of categorial compatibility formulated by Roman Suszko,
referring to some innovative ideas of Gottlob Frege and visible in syntactic
and semantic compatibility of language expressions. In the paper the principle
is introduced for categorial languages generated by the Ajdukiewicz’s classical
categorial grammar. The 1st-order quantifiers are typically ambiguous. Every
1st-order quantifier of the type k \u3e 0 is treated as a two-argument functorfunction
defined on the variable standing at this quantifier and its scope (the
sentential function with exactly k free variables, including the variable bound
by this quantifier); a binary function defined on denotations of its two arguments
is its denotation. Denotations of sentential functions, and hence also
quantifiers, are defined separately in Fregean and in situational semantics.
They belong to the ontological categories that correspond to the syntactic
categories of these sentential functions and the considered quantifiers. The
main result of the paper is a solution of the problem of categories of the
1st-order quantifiers based on the principle of categorial compatibility
Introduction. The School: Its Genesis, Development and Significance
The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School –
a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach towards philosophical studies and teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines.The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives
of the School (among others, Ajdukiewicz, Lesniewski, Łukasiewicz,Tarski), accompanying the names with short descriptions of their achievements.
The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name – Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period – the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of followers are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic
or computer science in Poland
Rejection in Łukasiewicz's and Słupecki's Sense
The idea of rejection originated by Aristotle. The notion of rejection
was introduced into formal logic by Łukasiewicz [20]. He applied it to
complete syntactic characterization of deductive systems using an axiomatic
method of rejection of propositions [22, 23]. The paper gives not only genesis,
but also development and generalization of the notion of rejection. It also
emphasizes the methodological approach to biaspectual axiomatic method of
characterization of deductive systems as acceptance (asserted) systems and
rejection (refutation) systems, introduced by Łukasiewicz and developed by
his student Słupecki, the pioneers of the method, which becomes relevant in
modern approaches to logic
Pathways of ROS homeostasis regulation in Mesembryanthemum crystallinum L. calli exhibiting differences in rhizogenesis
Knowledge, Vagueness and Logic
The aim of the paper is to outline an idea of solving the problem of the vagueness of concepts. The starting point is a definition of the concept of vague knowledge. One of the primary goals is a formal justification of the classical viewpoint on the controversy about the truth and object reference of expressions including vague terms. It is proved that grasping the vagueness in the language aspect is possible through the extension of classical logic to the logic of sentences which may contain vague terms. The theoretical framework of the conception refers to the theory of Pawlak's rough sets and is connected with Zadeh's fuzzy set theory as well as bag (or multiset) theory. In the considerations formal logic means and the concept system of set theory have been used. The paper can be regarded as an outline of the logical theory of vague concepts