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Abstract

Paradefinite (‘beyond the definite’) logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for introducing paradefinite logics. It consists of the four-valued matrices that expand the minimal matrix which is characteristic for first degree entailments: Dunn–Belnap matrix. We survey and study the expressive power and proof theory of the most important logics that can be developed in this framework.
... Indeed, the dichotomy in question reveals itself strikingly in the case of many-valued logics. 6 In contradistinction to paraconsistent logic, which is sometimes dubbed logic with truth-value gluts because a formula might be true and false simultaneously, paracomplete logic 7 is sometimes dubbed logic with truth-value gaps because a formula might be neither true nor false simultaneously. Let us confine ourselves to the case of three-valued logics for this approach to paracomplete logic seems to be the most popular in the literature: A formula of this kind is assigned the third truth-value which is not a designated one. ...
... Hence, the law of excluded middle and certain inference rules related to it fail (the italics are not ours): "A paracomplete logic is a logic, in which the principle of excluded middle, i.e., A ∨ ¬A is not a theorem of that 5 Notice that the first formal system, consciously conceived as a logic invalidating Duns Scotus law, was developed by Stanis law Jaśkowski in 1948 [23], while ideas that can be regarded as the basis of paracomplete logics were explored in the 1960s (for example, [50]), with formal investigations in [29]. 6 Note that each logic that is thoroughly discussed in this paper is bivalent. Manyvaluedness is needed to clarify the argument. ...
... Most fuzzy logics are paracomplete (and almost none of them is paraconsistent; see [7,15] for some rare examples of paraconsistent fuzzy logics). At last, let us mention paradefinite [6] or paranormal [9,43] logics, i.e., logics which are both paracomplete and paraconsistent. The most influential logic among them is Anderson-Belnap's FDE [5]. ...
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Jaśkowski's discussive (discursive) logic D2 is historically one of the first para-consistent logics, i.e., logics that 'tolerate' contradictions. Following Jaśkowski's idea to define his discussive logic by means of the modal logic S5 via special translation functions between discussive and modal languages and supporting at the same time the tradition of paracomplete logics being the counterpart of paraconsistent ones, we present a paracomplete discussive logic D p 2 .
... This means that BD ⊃,F (Σ ) is ¬-coherent with classical logic in the sense of [3]. There exist A, A ′ ∈ Form(Σ ) such that A, ¬A ⊭ A ′ . ...
... There exist A, A ′ ∈ Form(Σ ) such that A, ¬A ⊭ A ′ . Because BD ⊃,F (Σ ) is also ¬-coherent with classical logic, this means that BD ⊃,F (Σ ) is paraconsistent in the sense of [3]. ...
... There exist a Γ ⊆ Form(Σ ) and A, A ′ ∈ Form(Σ ) such that Γ , A ⊨ A ′ and Γ , ¬A ⊨ A ′ , but Γ ⊭ A ′ . Because BD ⊃,F (Σ ) is also ¬-coherent with classical logic, this means that BD ⊃,F (Σ ) is paracomplete in the sense of [3]. ...
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This paper concerns an expansion of first-order Belnap-Dunn logic, called BD^{\supset,F}_{\bot}, and an application of this logic in the area of relational database theory. The notion of a relational database, the notion of a query applicable to a relational database, and several notions of an answer to a query with respect to a relational database are considered from the perspective of this logic, taking into account that a database may be an inconsistent database and/or a database with null values. The chosen perspective enables among other things the definition of a notion of a consistent answer to a query with respect to a possibly inconsistent database without resort to database repairs. For each of the notions of an answer considered, being an answer to a query with respect to a database of the kind considered is decidable.
... This means that BD ⊃,F (Σ) is ¬-coherent with classical logic in the sense of [3]. ...
... There exist A, A ′ ∈ Form(Σ) such that not A, ¬A A ′ . Because BD ⊃,F (Σ) is also ¬-coherent with classical logic, this means that BD ⊃,F (Σ) is paraconsistent in the sense of [3]. ...
... There exist a Γ ⊆ Form(Σ) and A, A ′ ∈ Form(Σ) such that Γ, A A ′ and Γ, ¬A A ′ , but not Γ A ′ . Because BD ⊃,F (Σ) is also ¬-coherent with classical logic, this means that BD ⊃,F (Σ) is paracomplete in the sense of [3]. ...
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This paper concerns an expansion of first-order Belnap-Dunn logic, named BD,F\mathrm{BD}^{\supset,\mathsf{F}}, and an application of this logic in the area of relational database theory. The notion of a relational database, the notion of a query applicable to a relational database, and several notions of an answer to a query with respect to a relational database are considered from the perspective of this logic, taking into account that a database may be an inconsistent database or a database with null values. The chosen perspective enables among other things the definition of a notion of a consistent answer to a query with respect to a possibly inconsistent database without resort to database repairs. For each of the notions of an answer considered, being an answer to a query with respect to a database of the kind considered is decidable.
... (This is the equality between the first and third terms in the statement of Proposition 6.3 below.) − In showing this we use as a sort of intermediate link (actually also interesting in its own right) a relational characterization of this system previously given by Arieli and Avron [1]. § 2 below presents some basic definitions and remarks. ...
... We say (following Arieli and Avron [1]) that formula ψ(p 1 … p n ) (where p 1 … p n indicates the variables of the formula in their alphabetical ordering) characterizes n-ary function f if ∀a ∈ {T, F, B, N} n : ψ(a) is T-or-B iff f(a) is T-or-B. we have also f ∈ Pol{B}, so the disjunction ψ is not 'empty'. ...
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We define a concept of truthmaker function, and prove the functional completeness, w.r.t. truthmaker functions in this sense, of a set of four-valued functions corresponding to standard connectives of the system of relevance logic known as First-Degree Entailment or Belnap–Dunn logic.
... Due to involvement of four membership degrees in the characterization of the proposition, the ambiguous logic can be regarded as a family of four-valued logic. Many studies have been carried out with respect to fourvalued logic by the researchers (e.g., Dunn and Epstein [8], Belnap [9], Arieli and Avron [10], Font and Hájek [11], Béziau [12], and others). • Second, we examine the membership degrees of propositions using various kinds of connectives, including as negation, conjunction, and disjunction. ...
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