Studia Logica最新文献

Even though the strong relationship between proof-theoretic and model-theoretic notions in one’s logical theory can be shown by soundness and completeness proofs, whether we can define the model-theoretic notions by means of the inferences in a proof system is not at all trivial. For instance, provable inferences in a proof system of classical logic in the logical framework SET-FMLA do not determine its intended models as shown by Carnap (Formalization of logic, Harvard University Press, Cambridge, 1943), i.e., there are non-Boolean models that satisfy its provable inferences. In the literature, this is known as the Categoricity problem or Carnap’s problem. In this paper, we will discuss the Categoricity problem (or Carnap’s problem) for three-valued logics K3 and LP. We will provide three different restrictions on admissible models that will deliver us categoricity results, some of which draw from the solutions provided for the Categoricity problem for classical logic in Belnap and Massey (Stud Log 49(1):67–82, 1990) and Bonnay and Westerståhl (Erkenntis 81(4):721–739, 2016). We will then argue that two of those solutions are philosophically well-motivated: (1) restricting the admissible models where negation is interpreted as a Strong Kleene truth-function, and (2) restricting the admissible models where a complex formula is assigned the third value when its immediate subformulas are assigned the third value.

In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic raises some technical and conceptual issues. We relate Sandqvist’s (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness and completeness arguments, thereby demonstrating the naturality of Sandqvist’s constructions. This naturality includes Sandqvist’s treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist’s semantics can also be viewed as the natural disjunction in a category of sheaves.

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. In McKinsey and Tarski’s setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation ( sqsubseteq ) defined by (a sqsubseteq b) if a is contained in the topological closure of b, for a, b subsets of some topological space. A specialization poset is a partially ordered set endowed with a further coarser preorder relation ( sqsubseteq ). We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Specialization semilattices have the amalgamation property. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate contexts, even far removed from topology.

In this paper, we use the concept of very true operator to pre-semi-Nelson algebras and investigate the properties of very true pre-semi-Nelson algebras. We study the very true N-deductive systems and use them to establish the uniform structure on very true pre-semi-Nelson algebras. We obtain some properties of this topology. Finally, the corresponding logic very true semi-intuitionistic logic with strong negation is constructed and algebraizable of this logic is proved based on very true semi-Nelson algebras.

The aim of this paper is to define a partial Propositional Type Theory. Our system is partial in a double sense: the hierarchy of (propositional) types contains partial functions and some expressions of the language, including formulas, may be undefined. The specific interpretation we give to the undefined value is that of Kleene’s strong logic of indeterminacy. We present a semantics for the new system and prove that every element of any domain of the hierarchy has a name in the object language. Finally, we provide a proof system and a (constructive) proof of completeness.

In this paper we study the variety (textsf{WL}) of bounded distributive lattices endowed with an implication, called weak Lewis distributive lattices. This variety corresponds to the algebraic semantics of the ({vee ,wedge ,Rightarrow ,bot ,top })-fragment of the arithmetical base preservativity logic (mathsf {iP^{-}}). The variety (textsf{WL}) properly contains the variety of bounded distributive lattices with strict implication, also known as weak Heyting algebras. We introduce the notion of WL-frame and we prove a representation theorem for WL-lattices by means of WL-frames. We extended this representation to a topological duality by means of Priestley spaces endowed with a special neighbourhood relation between points and closed upsets of the space. These results are applied in order to give a representation and a topological duality for the variety of weak Heyting–Lewis algebras, i.e., for the algebraic semantics of the arithmetical base preservativity logic (textsf{iP}^{-}).

As Classical Propositional Logic finds its algebraic counterpart in Boolean algebras, the logic of Quantum Mechanics, as outlined within G. Birkhoff and J. von Neumann’s approach to Quantum Theory (Birkhoff and von Neumann in Ann Math 37:823–843, 1936) [see also (Husimi in I Proc Phys-Math Soc Japan 19:766–789, 1937)] finds its algebraic alter ego in orthomodular lattices. However, this logic does not incorporate time dimension although it is apparent that the propositions occurring in the logic of Quantum Mechanics are depending on time. The aim of the present paper is to show that tense operators can be introduced in every logic based on a complete lattice, in particular in the logic of quantum mechanics based on a complete orthomodular lattice. If the time set is given together with a preference relation, we introduce tense operators in a purely algebraic way. We derive several important properties of such operators, in particular we show that they form dynamic pairs and, altogether, a dynamic algebra. We investigate connections of these operators with logical connectives conjunction and implication derived from Sasaki projections in an orthomodular lattice. Then we solve the converse problem, namely to find for given time set and given tense operators a time preference relation in order that the resulting time frame induces the given operators. We show that the given operators can be obtained as restrictions of operators induced by a suitable extended time frame.

A non-Fregean framework aims to provide a formal tool for reasoning about semantic denotations of sentences and their interactions. Extending a logic to its non-Fregean version involves introducing a new connective (equiv ) that allows to separate denotations of sentences from their logical values. Intuitively, (equiv ) combines two sentences (varphi ) and (psi ) into a true one whenever (varphi ) and (psi ) have the same semantic correlates, describe the same situations, or have the same content or meaning. The paper aims to compare non-Fregean paraconsistent Grzegorczyk’s logics (Logic of Descriptions (textsf{LD}), Logic of Descriptions with Suszko’s Axioms (textsf{LDS}), Logic of Equimeaning (textsf{LDE})) with non-Fregean versions of certain well-known paraconsistent logics (Jaśkowski’s Discussive Logic (textsf{D}_2), Logic of Paradox (textsf{LP}), Logics of Formal Inconsistency (textsf{LFI}{1}) and (textsf{LFI}{2})). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of (textsf{LP}), (textsf{LFI}{1}), (textsf{LFI}{2}). Furthermore, we show that non-Fregean extensions of (textsf{LP}), (textsf{LFI}{1}), (textsf{LFI}{2}), and (textsf{D}_2) are more expressive than their original counterparts. Our results highlight that the non-Fregean connective (equiv ) can serve as a tool for expressing various properties of the ontology underlying the logics under consideration.

(nabla )-algebra is a natural generalization of Heyting algebra, unifying many algebraic structures including bounded lattices, Heyting algebras, temporal Heyting algebras and the algebraic presentation of the dynamic topological systems. In a series of two papers, we will systematically study the algebro-topological properties of different varieties of (nabla )-algebras. In the present paper, we start with investigating the structure of these varieties by characterizing their subdirectly irreducible and simple elements. Then, we prove the closure of these varieties under the Dedekind-MacNeille completion and provide the canonical construction and the Kripke representation for (nabla )-algebras by which we establish the amalgamation property for some varieties of (nabla )-algebras. In the sequel of the present paper, we will complete the study by covering the logics of these varieties and their corresponding Priestley-Esakia and spectral duality theories.

Belnap–Dunn’s relevance logic, (textsf{BD}), was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. (textsf{BD}) is a four-valued logic which is both paraconsistent and paracomplete. On the other hand, De and Omori, while investigating what classical negation amounts to in a paracomplete and paraconsistent four-valued setting, proposed the expansion (textsf{BD2}) of the four valued Belnap–Dunn logic by a classical negation. In this paper, we introduce a four-valued expansion of BD called ({textsf{BD}^copyright }), obtained by adding an unary connective ({copyright }, )which is a consistency operator (in the sense of the Logics of Formal Inconsistency, LFIs). In addition, this operator is the unique one with the following features: it extends to (textsf{BD}) the consistency operator of LFI1, a well-known three-valued LFI, still satisfying axiom ciw (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, LFUs). Moreover, ({textsf{BD}^copyright }) is maximal w.r.t. LFI1, and it is proved to be equivalent to BD2, up to signature. After presenting a natural Hilbert-style characterization of ({textsf{BD}^copyright }) obtained by means of twist-structures semantics, we propose a first-order version of ({textsf{BD}^copyright }) called ({textsf{QBD}^copyright }), with semantics based on an appropriate notion of four-valued Tarskian-like structures called (textbf{4})-structures. We show that in ({textsf{QBD}^copyright }), the existential and universal quantifiers are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the classical negation. Finally, a Hilbert-style calculus for ({textsf{QBD}^copyright }) is presented, proving the corresponding soundness and completeness theorems.