Behavioral Algebraization of Logics

Abstract

Abstract. We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of many-sorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL towards providing a meaningful algebraic counterpart also to logics with a manysorted language, and possibly including non-truth-functional connectives. The proposed behavioral approach covers logics which are not algebraizable according to the standard approach, while also bringing a new algebraic perspective to logics which are algebraizable using the standard tools of AAL. Furthermore, we pave the way towards a robust behavioral theory of AAL, namely by providing a behavioral version of the Leibniz operator which allows us to generalize the traditional Leibniz hierarchy, as well as several well-known characterization results. A number of meaningful examples will be used to illustrate the novelties and advantages of the approach.Keywords: Abstract algebraic logic, many-sorted behavioral logic, non-truth-functionality. IntroductionThe general theory of abstract algebraic logic (AAL, from now on) was first introduced in [3] with the aim of extending the so-called Lindenbaum-Tarski method, as used for instance to establish the relationship between classical propositional logic and Boolean algebras, to the systematic study of the connection between a given logic and a suitable equational theory. This connection enables one to use the powerful tools of universal algebra to study the metalogical properties of the logic being algebraized, namely with respect to its axiomatizability, definability aspects, the deduction theorem, or interpolation properties [11,13]. Despite of its success, the scope of applicability of the standard tools of AAL is relatively limited. Logics with a many-sorted language, even if well behaved, are good examples of logics that fall out of their scope. It goes without saying that rich logics, with many-sorted languages, are essential to specify and reason about complex systems, as also argued and justified by the theory of combined logics [40]. However, even in the case of propositional based (single-sorted) logics, many interesting examples simply fall out of the scope of the standard tools of AAL. With the proliferation of logical systems, with applications ranging from computer science, to mathematics and philosophy, the examples of nonalgebraizable logics that, therefore, lack from a meaningful and insightful Studia Logica (2008) 0: 1-49

Behavioral algebraization of da Costa's C-systems

“…The idea was to associate to C 1 not a class of algebras over the signature of C 1 , but rather a class of algebras over a two-sorted signature of formulas and truth-values, where the valuation map is an operation between the two sorts. This work actually inspired a novel behavioral approach to AAL, as introduced in (Caleiro et al, 2007), whose main aim was extending the range of applicability of the usual tools of AAL towards providing a meaningful algebraic counterpart also to logics that, as C 1 , include non-truth-functional connectives and which are not algebraizable according to the traditional theory. The key idea of the behavioral approach to AAL is to substitute the role played by unsorted equational logic (Eqn) by a suitable many-sorted behavioral equational logic (BEqn).…”

“…The interested reader can find all the details in (Caleiro et al, 2007;Gonçalves, 2008). We can say, informally, that a propositional logic is behaviorally algebraizable if it is equivalent to the behavioral equational theory of a class of two-sorted algebras.…”

“…However, the work in (Caleiro et al, 2003) motivated a novel behavioral branch of AAL, as proposed and developed by Caleiro, Gonçalves and Martins in (Caleiro et al, 2007). Intuitively, while AAL is traditionally centered around the notion of congruence, the behavioral approach to AAL is centered around the weaker notion of behavioral equivalence.…”

“…This feature of behavioral logic plays a fundamental role in our development of behavioral AAL, since it allows us to cope also with non-truth-functionality. In a nutshell, the behavioral approach to AAL introduced in (Caleiro et al, 2007) extends the standard theory of AAL by simply using many-sorted behavioral logic in the role traditionally played by (unsorted) equational logic.…”

“…Using behavioral tools and techniques in AAL, it is possible to prove that the whole hierarchy of C-systems is behaviorally algebraizable, as shown in (Caleiro et al, 2007;Gonçalves, 2008). Furthermore, their connection with da Costa algebras, as well as with their bivaluation semantics, finally emerges naturally in this behavioral algebraic setting.…”