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Categorical abstract algebraic logic : Wójcicki's conjecture and Malinowski's theorem

Voutsadakis G.

Reports on Mathematical Logic

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2017

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Vol. 52

69--82

EN

During the Autumn School on Strongly Finite Sentential Calculi held in Międzygórze in 1977, Wójcicki conjectured that a propositional logic has a strongly adequate matrix semantics consisting of matrices with a singleton designated filter, which we call a Rasiowa semantics since it is possessed by all implicative logics of Rasiowa, if and only if it satisfies a simple technical condition that we name the Wójcicki condition. Malinowski proved the conjecture in 1978. We revisit Malinowski's Theorem in the setting of logics formalized as π-institutions.

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Categorical abstract algebraic logic weakly referential π-institutions

Voutsadakis G.

Reports on Mathematical Logic

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2016

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Vol. 51

91--103

EN

Wójcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wójcicki asserts that a logic has a referential semantics if and only if it is selfextensional. A second theorem of Wójcicki asserts that a logic has a weakly referential semantics if and only if it is weakly self- extensional. We formulate and prove an analog of this theorem in the categorical setting. We show that a π-institution has a weakly referential semantics if and only if it is weakly self-extensional.

3

Secrecy logic : protoalgebraic S - secrecy logics

Voutsadakis G.

Reports on Mathematical Logic

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2012

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Vol. 47

3-28

EN

In recent work the notion of a secrecy logic S over a given deductive system S was introduced. Secrecy logics capture the essential features of structures that are used in performing secrecy-preserving reasoning in practical applications. More precisely, they model knowledge bases that consist of information, part of which is considered known to the user and part of which is to remain secret from the user. S-secrecy structures serve as the models of secrecy logics. Several of the universal algebraic and model theoretic properties of the class of S-secrecy structures of a given S-secrecy logic have already been studied. In this paper, our goal is to show how techniques from the theory of abstract algebraic logic may be used to analyze the structure of a secrecy logic and draw conclusions about its algebraic character. In particular, the notion of a protoalgebraic S-secrecy logic is introduced and several characterizing properties are provided. The relationship between protoalgebraic S-secrecy logics and the protoalgebraicity of their underlying deductive systems is also investigated.

4

CAAL : Categorical Abstract Algebraic Logic : coordinatization is algebraization

Voutsadakis G.

Reports on Mathematical Logic

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2012

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Vol. 47

125-145

EN

The methods of categorical abstract algebraic logic are employed to show that the classical process of the coordinatization of abstract (affine plane) geometry can be viewed under the light of the algebraization of logical systems. This link offers, on the one hand, a new perspective to the coordinatization of geometry and, on the other, enriches abstract algebraic logic by bringing under its wings a very well-known geometric process, not known hitherto to be related or amenable to its methods and techniques. The algebraization takes the form of a deductive equivalence between two institutions, one corresponding to affine plane geometry and the other to Hall ternary rings.

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Categorical Abstract Algebraic Logic: Syntactically Algebraizable pi-Institutions

Voutsadakis G.

Reports on Mathematical Logic

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2009

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Vol. 44

105-151

EN

This paper has a two-fold purpose. On the one hand, it introduces the concept of a syntactically N-algebraizable pi-institution, which generalizes in the context of categorical abstract algebraic logic the notion of an algebraizable logic of Blok and Pigozzi. On the other hand, it has the purpose of comparing this important notion with the weaker ones of an N-protoalgebraic and of a syntactically N-equivalential pi-institution and with the stronger one of a regularly N-algebraizable pi-institution. N-protoalgebraic pi-institutions and syntactically N-equivalential pi-institutions were previously introduced by the author and abstract in the categorical framework the protoalgebraic logics of Blok and Pigozzi and the equivalential logics of Prucnal and Wroński and of Czelakowski. Regularly N-algebraizable pi-institutions are introduced in the present paper taking after work of Czelakowski and of Blok and Pigozzi in the sentential logic framework. On the way to defining syntactically N-algebraizable pi-institutions, the important notion of an equational pi-institution associated with a given quasivariety of N-algebraic systems is also introduced. It is based on the notion of an N-quasivariety imported recently from the theory of Universal Algebra to the categorical level by the author.

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Categorical Abstract Algebraic Logic: Strong Version of a Protoalgebraic pi-Institution

Voutsadakis G.

Reports on Mathematical Logic

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2007

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No. 42

19-46

EN

An analog of the strong version of a protoalge- braic logic, introduced by Font and Jansana, is presented for N- protoalgebraic -institutions. Some properties of this strong ver- sion of an N-protoalgebraic -institution are explored and they are related to the explicit denability of N-Leibniz theory sys- tems. N-Leibniz theory systems were introduced in previous work by the author, also taking after the corresponding theory of Font and Jansana in the sentential framework.

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Categorical Abstract Algebraic Logic: Full Models, Frege Systems and Metalogical Properties

Voutsadakis G.

Reports on Mathematical Logic

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2006

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Nr 41,spec. iss.

31-60

EN

Font and Jansana studied the full models of sentential logics under the presence of a variety of metalogical properties. Their theory of full models was adapted, in recent work by the author, to cover the case of institutional logics. In the present work, the study of metalogical properties is carried out in the -institution framework and the way they aect full models of -institutions is investigated.