On Extensions of Intermediate Logics by Strong Negation

Kracht, Marcus

doi:10.1023/a:1004222213212

Cited by 59 publications

(49 citation statements)

References 11 publications

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“…First of all, note that it is well known that (A | Γ )/ θ A is a Heyting algebra [43,24]. From the definition of N -lattice we can conclude that [∼ (…”

Section: Theorem 23 If a Is A Heyting Algebra Then A Is A N -Latticementioning

confidence: 92%

“…Thus, in order to obtain a constructive logic with this property, IPL was extended with an unary connective for strong negation satisfying the desired property. We closely follow the notation of Kracht [24] and denote by N the constructive logic with strong negation. It is well known that N is algebraizable according to the standard notion and that its equivalent algebraic semantics is the class of so-called N -lattices [43,38] (also known as Nelson algebras [41] or quasi-pseudo-Boolean algebras [33]).…”

Section: Constructive Logic With Strong Negationmentioning

confidence: 99%

“…It is well known that N is algebraizable according to the standard notion and that its equivalent algebraic semantics is the class of so-called N -lattices [43,38] (also known as Nelson algebras [41] or quasi-pseudo-Boolean algebras [33]). The variety of N -lattices has been extensively studied [33,43,38,24]. One important result is the characterization of N -lattices through Heyting algebras.…”

Section: Constructive Logic With Strong Negationmentioning

confidence: 99%

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Behavioral Algebraization of Logics

2009

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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

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“…First of all, note that it is well known that (A | Γ )/ θ A is a Heyting algebra [43,24]. From the definition of N -lattice we can conclude that [∼ (…”

Section: Theorem 23 If a Is A Heyting Algebra Then A Is A N -Latticementioning

confidence: 92%

Section: Constructive Logic With Strong Negationmentioning

confidence: 99%

Section: Constructive Logic With Strong Negationmentioning

confidence: 99%

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Behavioral Algebraization of Logics

2009

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“…By strong completeness, we have Π |= ϕ ⇔ Π ϕ, for all Π, ϕ. Further properties of N 5 are studied in [19]. By adding to N 5 the axiom schema ¬¬α → α we obtain a 3-valued logic, called classical logic with strong negation, denoted by N 3 .…”

Section: Logical Backgroundmentioning

confidence: 99%

Simplifying Logic Programs Under Answer Set Semantics

Pearce

2004

Logic Programming

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Abstract. Now that answer set programming has emerged as a practical tool for knowledge representation and declarative problem solving there has recently been a revival of interest in transformation rules that allow for programs to be simplified and perhaps even reduced to programs of 'lower' complexity. Although it has been known for some that there is a maximal monotonic logic, denoted by N5, with the property that its valid (equivalence preserving) inference rules provide valid transformations of programs under answer set semantics, with few exceptions this fact has not really been exploited in the literature. The paper studies some new transformation rules using N5-inference to simplify extended disjunctive logic programs known to be strongly equivalent to programs with nested expressions.

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“…A constructive logic N − , which is a variant of logics with strong negation, was proposed by Almukdad and Nelson [1], who showed a relationship between constructive logics N − and N + , and Cleave's three-valued logic [3]. Logics such as N − , N and N + have been studied by many logicians and computer scientists (e. g. [9,11,12,14,15]). …”

Section: Introductionmentioning

confidence: 99%