ABSTRACT. Improving a recent result of the author, it is shown here that, when a quasivariety K algebraizes a finitary sentential logic L, then L has a classical inconsistency lemma iff K is relatively filtral and its nontrivial members have only nontrivial subalgebras.  In the process, the theory of uniform congruence schemes, ideal varieties and filtrality is extended from varieties to quasivarieties, and it is proved that a quasivariety is relatively filtral iff it is relatively semisimple and has equationally definable principal relative congruences.  

ABSTRACT. This contribution addresses a general problem of abstract algebraic logic and its instantiation for the variety of semilattices, which is the underlying structure of a host of non-classical logics, in particular fuzzy logics. The general problem is how to associate a logic with a given variety. Since the general theory of abstract algebraic logic provides three procedures to associate a class of algebras with a given logic, our original problem gives rise to three different questions. We show that one of them cannot be answered affirmatively in general, while the other two always can. We determine and study the weakest solution. This leads to a natural definition of the notion of "the logic of a variety". We obtain some results on the poset of all logics of a variety generated by a primal algebra. In the second part we study the poset of the logics of the variety of semilattices, showing that it has a minimum and two maximals and is atomless, among other results.

ABSTRACT. We introduce the notion of the deductive system of the order of an equationally orderable quasivariety and we prove that for equationally orederable varieties this deductive system is congreuntial (fully selfextensional). We compare the result with previous results on deductive systems with the congruece property (selfextensional)and the deduction-detachment property or the property of conjunction.

ABSTRACT. We will consider MV-algebras as systems where A is a nonempty set of elements, 0 and 1 are distinct constant elements of A, + and · are binary operations on the elements of A, and - is a unary operation on elements of A. The class of all MV-algebras will be denoted be MV. It is known that the set Id(MV) of all identities fulfilled in the class MV determines a variety (i.e., nonempty class of algebras closed under subalgebras, homomorphic images and direct products) MV. Let Id(\tau) denote the set of all identities of type \tau. For a set S ⊆ Id(\tau) we denote by Cn(S) the deductive closure of S, i.e. Cn(S) is the smallest subset of Id(\tau) containing S such that: 1. x = x ∈ Cn(S) for every variable x; 2. p = q ∈ Cn(S) ⇒ q = p ∈ Cn(S); 3. p = q, q = r ∈ Cn(S) ⇒ p = r ∈ Cn(S); 4. Cn(S) isclosed under replacement; 5. Cn(S) is closed under substitution. If S = Cn(S) then S is called an equational theory. We will choose from the set Id(MV) a subset E. If the set E is a proper subset of the set Id(MV) and it is an equational theory, then its corresponding variety MV_E is bigger than MV with respect to inclusion. A natural question to describe a lattice of subvarieties of some bigger variety containing MV arises. A partial answer to this problem is presented in the paper. The research of subvarieties of the variety MV_E is a partial solution of the most general problem in this area: for a fixed language find all equational theories contained between the theory generated by the single identity x = x and the full theory determined by the single identity x = y. Logicians inquire about lattices of intermediate logics (for example between intuitionistic logic and classical logic), algebraists search for lattices of subvarieties. In our case the set E is related to the special structure of terms occurring in the identity. We consider a given type of algebras \tau : F → N, where F is a set of fundamental operation symbols and N is the set of non-negative integers. Let \Pi_F be the set of all partitions of F and let P ∈ \Pi_F. For any terms p and q of the type \tau, the identity p = q is P-compatible iff it is of the form x = x or both p and q are not variables and the outermost operation symbols in p and q belong to the same blok of the partition P. The notion of P-compatible identity was introduced by J. Płonka [7] and it is a generalization of an externally compatible identity introduced by W. Chromik in [2] and normal identity defined independently by J. Płonka [6] and I. I. Mel'nik [5]. An identity p = q of type \tau is externally compatible if it is of the form x = x or the most external fundamental operation symbols in p and q are identical (in other words, P-compatible identity is an externally compatible identity if P consists of singletons only). An identity p = q of type \tau is normal if it is of the form x = x or neither p nor q is a variable (i.e., P-compatible identity is a normal identity if P = {F}). For the variety V we will use the following notations: • P(V) - the set of all P-compatible identities satisfied in V, • Ex(V) - the set of all externally compatible identities satisfied in V, • N(V) - the set of all normal identities satisfied in V. • Id(V) - the set of all identities satisfied in V. The following inclusions are obvious: Ex(V) ⊆ P(V) ⊆ N(V) ⊆ Id(V). One can prove that P(V) is an equational theory. It is known that every equational theory corresponds to a variety of algebras. Let • V_P denotes the variety defined by P(V), • V_{Ex} denotes the variety defined by Ex(V), • V_N denotes the variety defined by N(V). It is a well known fact that the lattice of all equational theories extending the theory Id(V) is dually isomorphic to the lattice of all subvarieties of the variety V. Thus, for any partition P we have: V ⊆ V_N ⊆ V_P ⊆ V_{Ex}. Subvarieties of the variety MV have been studied by R. Grigolia, Y. Komori, A. Di Nola, and A. Lettieri. A. Lettieri and A. Di Nola gave equational bases for all MV. Y. Komori determined the lattice of subvarieties of the variety of MV. We will ask for the L(MV_{Ex}) lattice of subvarieties of the variety defined by the set Ex(MV). The full description of the lattice L(MV_{Ex}) is complicated and falls outside the scope of the talk. Of course, every subvariety of the class MV is also a proper subvariety of the variety determined the set Ex(MV). Beside giving a description of some chosen elements of the lattice L(MV_{Ex}) we will find all subdirectly irreducible algebras from the classes in the lattice L(MV_{Ex}) and we will give syntactical and semantical characterization of the class of algebras defined by P-compatible identities of MV-algebras. We use standard definitions from [1]. References [1] Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, Springer- Verlag, New York, 1981. [2] Chromik, W., `Externally compatible identities of algebras', Demonstratio Mathematica 23 (1990), 345--355. [3] Di Nola, A., and A. Lettieri, `Equational Characterization of All Varieties of MV-Algebras', Journal of Algebra 221 (1999), 463--474. [4] Yuichi Komori, `Super ŠLukasiewicz propositional logics', Nagoya Mathematical Journal 84 (1981), 119--133. [5] Mel'nik, I. I., `Nilpotent shifts of varieties', Mat. Zametki 14 (5) (1973) (in Russian). English translation: Math. Notes 14 (1973), 962--966. [6] Płonka, J., `On the subdirect product of some equational classes of algebras', Math. Nachr. 63 (1974), 303--305. [7] Płonka, J., `P-compatible identities and their applications to classical algebras', Math. Slovaca 40 (1) (1990), 21--30. [8] Tarski, A., `Equational logi and equational theories of algebras', in: H. A. Shmidt, K. Shütte, H. J. Thiele, (eds.), Contributions to Mathematical Logic, Nort Holland Publ. Co., Amsterdam, pp. 275--288.

ABSTRACT. We study the Stone-Priestley dual space SpecSubP of the lattice of subpolyhedra of a compact polyhedron P, with motivations coming from geometry, topology, ordered-algebra, and non-classical logic. From the perspective of algebraic logic, our contribution is a geometric investigation of lattices of prime theories in Łukasiewicz logic, possibly extended with real constants.

The main result we announce here is that SpecSubP has a concrete description in terms of a non-Hausdorff completion of the space P which holds great geometric interest.

If time allows, we discuss selected consequences of our main result, including compactness of the subspace of minimal primes of SpecSubP, and the fundamental property of SubP of being a co-Heyting algebra.

ABSTRACT. Fixing countable sets of propositional variables X and Y, we write Form(X) for the set of formulæ of Łukasiewicz (infinite-valued propositional) logic L, and Form(Y) for the set of formulæ of Intuitionistic (propositional) logic Int.

We prove that there exists a deductively closed theory Θ in Int, and a function T : Form(X) ⟶ Form(Y) satisfying T(⊤)=⊥, such that, for each α,β∈Form(X), the following holds:
     β ⊢ α in L     if, and only if,     Θ∪{T(α)} ⊢ T(β) in Int.
As a corollary, a formula α∈Form(X) is provable in L if, and only if, ¬T(α) is provable in Int, modulo the theory Θ.

In future work we plan to investigate the properties of T and Θ. Some obvious questions to be addressed include axiomatisability of Θ, and computability of T.

ABSTRACT. The aim of this paper is to give a complete classification of germinal MV-algebras. As an application, we will settle the fifth one of the eleven problems present by Mundici in [D. Mundici, Advanced Łukasiewicz calculus and MV-algebras, Trends in Logic, Studia Logica Library. Vol. 35, Springer, Berlin, 2011].

ABSTRACT. We prove that the extension of the known hypersequent calculus for standard first-order {G\"odel} logic with usual rules for second-order quantifiers is sound and (cut-free) complete for Henkin-style semantics for second-order {G\"odel} logic. The proof is semantic, and it is similar in nature to {Sch\"utte} and Tait's proof of Takeuti's conjecture.

ABSTRACT. We provide a methodology to introduce relational hypersequent calculi for a large class of many-valued logics, and a sufficient condition for their Co-NP completeness. Our results include the most important Co-NP fuzzy logics.

ABSTRACT. During the last decade a connection emerged between proof theory and algebra via which cut-elimination, one of the cornerstones of structural proof theory, can be proved by using completions, in particular, the MacNeille completion. This technique, which has been developed for a wide range of logics, including substructural ones, is far from trivial. Our modest aim in this paper is to establish what the technique of completions boils down to for "strong" logics such as full intuitionistic propositional logic, and what the connection is with other semantical proofs of cut-elimination.

ABSTRACT. One of the main topics of Abstract Algebraic Logic is the study of the Leibniz hierarchy, in which logics are classified by means of properties of the Leibniz operator which determine how nicely the Leibniz congruences and the truth predicates can be described in models of the logic. In this talk we will introduce and characterize logics, whose truth predicates are defined by means of equations with parameters. Then we will go through the consideration of weaker conditions on the truth predicates of a logic. This will give rise to a small hierarchy, in which logics are classified according to the way their truth predicates are defined; this new hierarchy can be thought of as an extension of the Leibniz hierarchy, since almost all the conditions we take into account turn out to be characterised by a property of the Leibniz operator.

ABSTRACT. In this paper we study the notion of an S-coherent family of S-compatibility operators, for a sentential logic S. This notion is tailored to be a common generalization of the well-known Leibniz and Suszko operators, which have been fundamental tools in recent developments of Abstract Algebraic Logic. The first main result we prove is a General Correspondence Theorem, which generalizes several results of this kind obtained for either the Leibniz operator (Blok and Pigozzi, 1986; Font and Jansana, 2001) or the Suszko operator (Czelakowski, 2003). We apply the general results to obtain several new characterizations of the main classes of logics in the Leibniz hierarchy in terms of the Leibniz operator or in terms of the Suszko operator.

ABSTRACT. In this paper, Church--Henkin simple type theories are constructed for finitary weakly implicative logics. The resulting type theory TT(L) over a given finitary weakly implicative logic L is the minimal (extensional, substitution-invariant) type theory closed under the rules of lambda-conversion and the intersubstitutivity of equals whose propositional fragment coincides with L and whose sound and complete Henkin semantics consists of Henkin-style general models over a generating class of L-algebras. The soundness and completeness theorem for TT(L) is obtained by a schematic adaptation of the proof for the ground theory TT0 which is a common fragment of all TT(L).