Behavioral algebraization of da Costa's <i>C</i>-systems

Caleiro, Carlos; Gonçalves, Ricardo

doi:10.3166/jancl.19.127-148

Cited by 6 publications

(7 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, using the tools of behavioral logic, it is possible to recover, in our behavioral approach, this class of structures. A full treatment of these questions about C 1 , which falls clearly beyond the scope of this paper, can be found in [22,9]. Furthermore, even the behavioral analysis of logics algebraizable in the standard sense seems to be useful.…”

Section: Discussionmentioning

confidence: 99%

Behavioral Algebraization of Logics

2009

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Discussionmentioning

confidence: 99%

Behavioral Algebraization of Logics

2009

Self Cite

View full text Add to dashboard Cite

show abstract

“…C 1 (ϕ • ) • Several semantics have been put forward in the literature for the logic C 1 . Namely, behavioral semantics [12,13], possible-translations semantics [9], and the so-called da Costa algebras investigated in [15,52] and generalizing the C 1 -algebras introduced by da Costa himself in [21]. Da Costa's proposal has the drawback (from an AAL point of view) of changing the underlying language of the class of algebras associated to C 1 .…”

Section: Hugo Albuquerque and Carlos Caleiromentioning

confidence: 99%

“…As a final note, observe that by considering axiom ϕ → ¬¬ϕ, (Ax16) one can define the logics Cilae and Ciloe 12 as the extensions of C 1 and Cilo by (Ax16), respectively. Although we will not study these logics in detail here, we shall nevertheless include them among the extensions of C 1 in Figure 1.…”

Section: Notice Thatmentioning

confidence: 99%

An Abstract Algebraic Logic Study of Da Costa’s Logic and Some of Its Paraconsistent Extensions

Albuquerque¹,

Caleiro²

2022

Bull. symb. log

Self Cite

View full text Add to dashboard Cite

Two famous negative results about da Costa's paraconsistent logic C1 (the failure of the Lindenbaum-Tarski process [44] and its non-algebraizability [39]) have placed C1 seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa's logic C1. On the one hand, we strengthen the negative results about C1 by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, C1 is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of C1 covered in the literature. We prove that for extensions S such as Cilo [26], every algebra in Alg * (S) contains a Boolean subalgebra; and for extensions S such as P 1 , P 2 or P 3 [17,53], every subdirectly irreducible algebra in Alg * (S) has cardinality at most 3. We also characterize the quasivariety Alg * (S) and the intrinsic variety V(S), with

show abstract

“…by Carlos Caleiro and Ricardo Gonçalves developed so-called Behavioral algebraization of da Costa's C-systems (cf. [2]). Proposition 1.…”

Section: Da Costa Algebrasmentioning

confidence: 99%

Potoses: Categorical Paraconsistent Universum for Paraconsistent Logic and Mathematics

Vasyukov¹

2017

View full text Add to dashboard Cite

It is well-known that the concept of da Costa algebra [3] reflects most of the logical properties of paraconsistent propositional calculi C n , 1 ≤ n ≤ ω introduced by N.C.A. da Costa. In [10] the construction of topos of functors from a small category to the category of sets was proposed which allows to yield the categorical semantics for da Costa's paraconsistent logic. Another categorical semantics for C n would be obtained by introducing the concept of potos -the categorical counterpart of da Costa algebra (the name "potos" is borrowed from W.Carnielli's story of the idea of such kind of categories)

show abstract

Behavioral algebraization of da Costa's C-systems

Cited by 6 publications

References 16 publications

Behavioral Algebraization of Logics

Behavioral Algebraization of Logics

An Abstract Algebraic Logic Study of Da Costa’s Logic and Some of Its Paraconsistent Extensions

Potoses: Categorical Paraconsistent Universum for Paraconsistent Logic and Mathematics

Contact Info

Product

Resources

About