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In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl's ideas of pure grammar, Leśniewski-Ajukiewicz's theory syntactic/semantic categories and in accordance with Frege's ontological canons, Bocheński's famous motto-syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic (extensional and intensional) categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers.Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science.

In the paper, the authors discuss two kinds of consequence operations characterized axiomatically. The first one are consequence operations of the type Cn + that, in the intuitive sense, are infallible operations, always leading from accepted (true) sentences of a deductive system to accepted (true) sentences of the deductive system (see Tarski in Monatshefte für Mathematik und Physik 37:361-404, 1930, Comptes Rendus des Séances De la Société des Sciences et des Lettres de Varsovie 23:22-29, 1930; Pogorzelski and S lupecki in Stud Logic 9:163-176, 1960, Stud Logic 10:77-95, 1960). The second kind are dual consequence operations of the type Cn − that can be regarded as antiinfallible operations leading from non-accepted (rejected, false) sentences of a deductive system to non-accepted (rejected, false) sentences of the system (see S lupecki in Funkcja Lukasiewicza, 33-40, 1959; Wybraniec-Skardowska in Teoria zdań odrzuconych, 5-131, Zeszyty Naukowe Wyższej Szko ly Inżynierskiej w Opolu, Seria Matematyka 4(81):35-61, 1983, Ann Pure Appl Logic 127:243-266, 2004, in On the notion and function of rejected propositions, 179-202, 2005). The operations of the types Cn + and Cn − can be ordinary finitistic consequence operations or unit consequence operations. A deductive system can be characterized in two ways by the following triple: by the triple: (+, −) < S, Cn + , Cn − > or by the triple: (−, +) < S, Cn − , Cn + >.We compare axiom systems for operations of the types Cn + and Cn − , give some methodological properties of deductive systems defined by means of these operations (e.g. consistency, completeness, decidability in Lukasiewicz's sense), as well as formulate different metatheorems concerning them.Mathematics Subject Classification (2010). Primary 03B22; Secondary 01A60, 03B99.Keywords. Axiom systems of theories of deductive systems, consequence operations, unit consequence operations, a rejection consequence operation, a dual consequence operation, asserted system, refutation system. 178U. Wybraniec-Skardowska and J. Waldmajer Log. Univers.