Modal logics and varieties of modal algebras: The beth properties, interpolation, and amalgamation

2008

LLP

Abstract. The provability logic GL was in the field of interest of A.V. Kuznetsov, who had also formulated its intuitionistic analog-the intuitionistic provability logic-and investigated these two logics and their extensions.In the present paper, different versions of interpolation and of the Beth property in normal extensions of the provability logic GL are considered. It is proved that in a large class of extensions of GL (including all finite slice logics over GL) almost all versions of interpolation and of the Beth property are equivalent. It follows that in finite slice logics over GL the three versions CIP, IPD and IPR of the interpolation property are equivalent. Also they are equivalent to the Beth properties B1, PB1 and PB2.

“…(1) and (2) are proved in [11], (3) in [12] and (4) in [16]. ⊣ It follows that B1 and PB1 are also equivalent to the super-amalgamation property of the corresponding variety.…”

Section: Interpolation and Amalgamationmentioning

confidence: 88%

Interpolation and implicit definability in extensions of the provability logic

2008

LLP

“…As it was shown in [14], in modal logics the Property B1 is equivalent to the Craig Interpolation Property that is satisfied rather seldom [8]. The properties IPD and B2 are independent in N E(K) [13]. At the same time, all logics in N E(K4) possess B2 [15].…”

mentioning

confidence: 81%

“…If L is locally tabular, FGFI(V (L))) contains only finite algebras, and this class is finite, whenever L is tabular. The following statements are proved in [13].…”

mentioning

confidence: 99%

See 1 more Smart Citation

Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics

1999

Banach Center Publ.

We consider the families L of propositional superintuitionistic logics (s.i.l.) and N E(K) of normal modal logics (n.m.l.). It is well known that there is a duality between L and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also N E(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig's interpolation property (CIP), the disjunction property (DP), the Beth property (BP), Hallden-completeness (HP) etc. have suitable properties of varieties as their images, and many natural algebraic properties are in accordance with natural properties of logics. For example, a s.i.l. L has CIP iff its associated variety V (L) has the amalgamation property (AP); L is Hallden-complete iff V (L) is generated by a subdirectly irreducible Heyting algebra. For any n.m.l. L, the amalgamation property of V (L) is equivalent to a weaker version of the interpolation property for L, and the superamalgamation property is equivalent to CIP; L is Hallden-complete iff V (L) satisfies a strong version of the joint embedding property.Well-known relational Kripke semantics for the intuitionistic and modal logics seems to be a more natural interpretation than the algebraic one. The categories of Kripke frames may, in a sense, be considered as subcategories of varieties of Heyting or modal algebras. We discuss the question to what extent one may reduce problems on properties of logics to consideration of their semantic models. Preliminaries.A language of propositional modal logic contains the connective → and the propositional constant ⊥, and an unary modal operation 2. Other connectives

“…The interplay between interpolation and amalgamation in varieties of algebras was investigated in many works (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14]). In this paper we come up with a weak version of the amalgamation property and find its algebraic counterpart.…”

Section: Interpolation and Amalgamationmentioning

confidence: 99%

A weak form of interpolation in equational logic

2008

Algebra Logic

Self Cite

The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety.In the present paper, varieties of so-called bounded algebras are investigated. This name derives from the notion of bounded lattices having a greatest element and a least element ⊥. Various classes of bounded algebras are tied with logic. For example, classical logic is interpreted via Boolean algebras, modal logic via modal algebras, and intuitionistic logic via Heyting algebras. In a two-valued Boolean algebra, the constant is identified with a truth value "true," and the constant ⊥ with a truth value "false." In many logical calculi, such as classical, intuitionistic, etc., the inconsistency of a theory is equivalent to the absurdity ⊥ being derivable in that theory. In this event the theory becomes trivial, i.e., all formulas are derivable in it. Under the passage to algebraic semantics, investigations of the calculi reduce to studying suitable classes of algebras, and often varieties of algebras, i.e., classes that are defined by some sets of identities. In varieties of algebras associated with logics, inconsistency is frequently associated with derivability of the identity ⊥ = . The sentence "contradiction implies everything" has an algebraic interpretation as the statement ⊥ = ⇒ ∀x∀y (x = y). The algebras satisfying this condition are said to be bounded.The interpolation theorem, proved in [1] for classical predicate logics, has different equivalent formulations which become nonequivalent in intuitionistic and modal logics. Here we consider a weak interpolation property for varieties of bounded algebras. Some results concerning weak interpolation in modal logics were obtained in [2]. These results may be naturally re-worded in the language of varieties of modal algebras. In the present paper we show that part of the results in [2] can well be extended to arbitrary varieties of bounded algebras.