Logic and Philosophy of Logic

This paper examines the logic of conditional obligation, which originates from the works of Hansson, Lewis, and others. Some weakened forms of transitivity of the betterness relation are studied. These are quasi-transitivity, Suzumura consistency, acyclicity and the interval order condition. The first three do not change the logic. The axiomatic system is the same whether or not they are introduced. This holds true under a rule of interpretation in terms of maximality and strong maximality. The interval order condition gives rise (...) to a new axiom. Depending on the rule of interpretation, this one changes. With the rule of maximality, one obtains the principle known as disjunctive rationality. With the rule of strong maximality, one obtains the Spohn axiom (also known as the principle of rational monotony, or Lewis’ axiom CV). A completeness theorem further substantiates these observations. For interval order, this yields the finite model property and decidability of the calculus. (shrink)

Lewis (The Journal of Philosophy, 65(5), 113–126, 1968) attempts to provide an account of modal talk in terms of the resources of counterpart theory, a first-order theory that eschews transworld identity. First, a regimentation of natural language modal claims into sentences of a formal first-order modal language L is assumed. Second, a translation scheme from L-sentences to sentences of the language of the theory is provided. According to Hazen (The Journal of Philosophy, 76(6), 319–338, 1979) and Fara & Williamson (Mind, (...) 114(453), 1–30, 2005), the account cannot handle certain natural language modal claims involving a notion of actuality. The challenge has two parts. First, in order to handle such claims, the initial formal modal language that natural language modal claims are regimented into must extend L with something like an actuality operator. Second, certain ways that Lewis’ translation scheme for L might be extended to accommodate an actuality operator are unacceptable. Meyer (Mind, 122(485), 27–42, 2013) attempts to defend Lewis’ approach. First, Meyer holds that in order to handle such claims, the formal modal language L$$^*$$ ∗ that we initially regiment our natural language claims into need not contain an actuality operator. Instead, we can make do with other resources. Next, Meyer provides an alternative translation scheme from L$$^*$$ ∗ -sentences to sentences of an enriched language of counterpart theory. Unfortunately, Meyer’s approach fails to provide an appropriate counterpart theoretic account of natural language modal claims. In this paper, I demonstrate that failure. (shrink)

Traditional definitions of common ground in terms of iterative de re attitudes do not apply to conversations where at least one conversational participant is not acquainted with the other(s). I propose and compare two potential refinements of traditional definitions based on Abelard’s distinction between generality in sensu composito and in sensu diviso.

We present a logic of evidence that reduces agents’ epistemic idealisations by combining classical propositional logic with substructural modal logic for formulas in the scope of epistemic modalities. To this aim, we provide a neighborhood semantics of evidence, which provides a modal extension of Fine’s semantics for relevant propositional logic. Possible worlds semantics for classical propositional logic is then obtained by defining the set of possible worlds as a special subset of information states in Fine’s semantics. Finally, we prove that (...) evidence is a hyperintensional and non-prime notion in our logic, and provide a sound and complete axiomatisation of our evidence logic. (shrink)

The paper extends a framework of information-based semantics for intuitionistic logic with a paraconsistent negation and consistency-sensitive epistemic modalities. In this framework information states represent information collected from various sources and as such they can be inconsistent because they receive contradictory information either from a single inconsistent source or from various mutually incompatible sources. The modalities reflect only those sources that are consistent and trusted. For the paraconsistent logic generated by this framework, we present a Hilbert style system, we prove (...) completeness, the disjunction property, the finite model property and various other results. (shrink)

This paper develops De Morgan-Płonka sums, which generalise Płonka sums to contexts in which negation is not topically transparent but still respects De Morgan duality. We give a general theory of De Morgan-Płonka sums, on the model of the general theory of Płonka sums. Additionally, we describe free De Morgan-Płonka sums and apply our construction to give an algebraic proof of completeness for Kit Fine’s truthmaker semantics for Angell’s logic of analytic containment.

Assume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $, into the Esterle algebra of formal power series. (...) Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on X. (shrink)

Let M be a short extender mouse. We prove that if $E\in M$ and $M\models $ “E is a countably complete short extender whose support is a cardinal $\theta $ and $\mathcal {H}_\theta \subseteq \mathrm {Ult}(V,E)$ ”, then E is in the extender sequence $\mathbb {E}^M$ of M. We also prove other related facts, and use them to establish that if $\kappa $ is an uncountable cardinal of M and $\kappa ^{+M}$ exists in M then $(\mathcal {H}_{\kappa ^+})^M$ satisfies the (...) Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then $\mathbb {E}^M$ is definable over the universe of M from the parameter $X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$, and M satisfies “Every set is $\mathrm {OD}_{\{X\}}$ ”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “ $V=\mathrm {HOD}$ ”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters $u_n$. (shrink)

As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired by Serre’s algebraic twisting) to describe arbitrary fibres in cohomology sequences—yielding a useful “finiteness” result on cohomology sets.Applied to the special case of (...) differential fields and Kolchin’s constrained cohomology, we complete results from [3] by proving that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable. (shrink)

The Friedman–Stanley jump, extensively studied by descriptive set theorists, is a fundamental tool for gauging the complexity of Borel isomorphism relations. This paper focuses on a natural computable analog of this jump operator for equivalence relations on $\omega $, written ${\dotplus }$, recently introduced by Clemens, Coskey, and Krakoff. We offer a thorough analysis of the computable Friedman–Stanley jump and its connections with the hierarchy of countable equivalence relations under the computable reducibility $\leq _c$. In particular, we show that this (...) jump gives benchmark equivalence relations going up the hyperarithmetic hierarchy and we unveil the complicated highness hierarchy that arises from ${\dotplus }$. (shrink)

We continue with the study of the Katětov order on MAD families. We prove that Katětov maximal MAD families exist under $\mathfrak {b=c}$ and that there are no Katětov-top MAD families assuming $\mathfrak {s\leq b}.$ This improves previously known results from the literature. We also answer a problem form Arciga, Hrušák, and Martínez regarding Katětov maximal MAD families.

We obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial non-deterministic logical matrices. Our constructions preserve finite-valuedness in the context of multiple-conclusion logics, whereas, unsurprisingly, it may be lost in the context of single-conclusion logics. Besides illustrating our constructions over a wide range of examples, we also develop concrete applications of our semantic characterizations, namely regarding the semantics (...) of strengthening a given many-valued logic with additional axioms, the study of conditions under which a given logic may be seen as a combination of simpler syntactically defined fragments whose calculi can be obtained independently and put together to form a calculus for the whole logic, and also general conditions for decidability to be preserved by the combination mechanism. (shrink)

For which infinite cardinals $\kappa $ is there a partition of the real line ${\mathbb R}$ into precisely $\kappa $ Borel sets? Work of Lusin, Souslin, and Hausdorff shows that ${\mathbb R}$ can be partitioned into $\aleph _1$ Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of ${\mathbb R}$ into Borel sets can be fairly arbitrary. For example, given any $A \subseteq \omega $ with $0,1 \in A$, there is a forcing extension (...) in which ${A = \{ n :\, \text {there is a partition of } {{\mathbb R}} \text { into }\aleph _n\text { Borel sets}\}}$. We also look at the corresponding question for partitions of ${\mathbb R}$ into closed sets. We show that, like with partitions into Borel sets, the set of all uncountable $\kappa $ such that there is a partition of ${\mathbb R}$ into precisely $\kappa $ closed sets can be fairly arbitrary. (shrink)

This paper explores the hypothesis that proper names serve as anchors identifying the individuals in the possible or real world. This hypothesis is tested on Old Icelandic narratives. A prominent feature of Old Icelandic sagas is that the narrative matter is not quite new. A Saga is reliable iff it refers to the events relevant for its audience and accepted as true by the whole community. I argue that proper names must be regarded as references to the background knowledge of (...) the medieval audience. I follow two contemporary Sagas based on the fresh events and show that proper names in a Saga mind were arranged in form of a database, each type of a personal designation having its charachteristic text-building profile. The form of the name and its frequency in the Saga are linked with the rank of the character. The high number of people mentioned in the Sagas is explained by the fact that the Sagas aimed at describing the real world, where the names of the individuals and the events they were involved cross-identified each other. Every character had its rigid identifiers. The total amount of propria is always higher than the number of active figures. I classify the Saga characters into five ranks and discuss their information-structural profiles. (shrink)

Sengzhao (c.374–414) was a Chinese Neo-Daoist who converted to Mahāyāna Buddhism, and few people doubt his influence on Chinese Buddhist philosophy. In this article, provided his Neo-Daoism (xuanxue) and Madhyamaka Buddhism, I will present how Sengzhao featured a symbolic meaning of ‘void’ (śūnya) as rooted originally in Daoism. The Daoist contradictions, in particular between ‘being’ (you) and ‘nothing [non-being]’ (wu), are essential to the development of his doctrine of ‘no ultimate void’ (不真空論, Buzhenkonglun). To understand what Sengzhao meant by ‘void’, (...) which is in denial about the ultimate reality, I broach a notion of nihil (‘nothing’ but also ‘no value’) that bears on his discursive practice. In this light, I formulate a Daoist argument for contradictions and ECN (ex contradictione nihil—nothing follows from contradictions) from Laozi’s Daodejing. Furthermore, I elaborate on Sengzhao’s defence of ECN in his Buzhenkonglun. Reconstructing his negative approach to contradictions within the scope of the four-valued expressions (catuṣkoṭi) in the Madhyamaka tradition from Nāgārjuna, I consider a likely objection that a fifth value such as the ineffable may be inferred as void. Instead of subsuming the ineffable value under his discourse, I finally endorse Sengzhao’s purpose of linguistic and conventional approximation to the ultimate reality as silence. As such, I conclude the significance of void in Sengzhao’s denials via contradictions (ECN), i.e. an early philosophical peak of Chinese Buddhism from Daoism. (shrink)

The paper explores the potential of Byzantine diagrams in syllogistic logic. Byzantine diagrams are originated by Byzantine scholars in the early modern period to use as tools for teaching and studying Aristotelian logic. This paper presents pioneering work on employing Byzantine diagrams for checking syllogistic validity through reduction.

Recently, it has been proposed to understand a logic as containing not only a validity canon for inferences but also a validity canon for metainferences of any finite level. Then, it has been shown that it is possible to construct infinite hierarchies of "increasingly classical" logics—that is, logics that are classical at the level of inferences and of increasingly higher metainferences—all of which admit a transparent truth predicate. In this paper, we extend this line of investigation by taking a somehow (...) different route. We explore logics that are different from classical logic at the level of inferences, but recover some important aspects of classical logic at every metainferential level. We dub such systems meta-classical non-classical logics. We argue that the systems presented deserve to be regarded as logics in their own right and, moreover, are potentially useful for the non-classical logician. (shrink)

We present tableau proof systems for the annotated version of propositional justification logics, that is, justification logics which are formulated using annotated application operators. We show that the tableau systems are sound and complete with respect to Mkrtychev models, and some tableau systems are analytic and provide a decision procedure for the annotated justification logics. We further show Craig’s interpolation property and Beth’s definability theorem for some annotated justification logics.

In epistemic logic, the beliefs of an agent are modeled in a way very similar to knowledge, except that they are fallible. Thus, the pattern of an agent’s true beliefs is an interesting subject to study. In this paper, we conduct a systematic study on a novel modal logic with the bundled operator ⊡ϕ:=□ϕ∧ϕ as the only primitive modality, where ⊡ captures the notion of true belief. With the help of a novel notion of ⊡-bisimulation, we characterize the expressivity of (...) this new language on relational models, and offer various completeness results. We also discuss some interesting connections between our work and previous works on reflexive-insensitive logics and the so-called boxdot conjecture. Finally, we study two generalizations of the ⊡-operator: one is the iterated true belief operator □nϕ:=ϕ∧□ϕ∧⋯∧□nϕ, the other is the neighborhood semantics for the ⊡-operator. (shrink)

In 2008, Graff Fara presented relative-sameness semantics, a semantics for a first-order modal and temporal language with the explicit aim of being able to render true certain contingent/temporary identity claims (relative to certain contexts). Graff Fara achieves this aim by abandoning a straightforward analysis of de re modal/temporal claims in terms of identity. Instead, such a claim is analyzed in terms of her relative-sameness relations (which need not be the identity relation), with the relevant relative-sameness relations in play determined by (...) context and individual terms occurring in the claim. However, there is a significant problem for relative-sameness semantics concerning (i) iterated modality/temporality, and (ii) existence that renders it an inappropriate account of our modal and temporal talk. In this paper, I present that problem and a solution to it. (shrink)

Den här boken är en inledning till den s.k. modallogiken. Modallogiken studerar argument vars giltighet beror på modala ord såsom ”måste”, ”kan” och ”omöjlig”. Boken innehåller fem kapitel. Det första kapitlet är en kort inledning till modallogik. Kapitel 2 handlar om syntax. Det tar upp flera modallogiska språk; det beskriver hur dessa är uppbyggda och hur de förhåller sig till olika naturliga språk. Kapitel 3 handlar om semantik. Vad betyder olika symboliska tecken? Vad har olika satser för sanningsvillkor? Kapitel 4 (...) handlar om bevisteori. Det går bl.a. igenom hur man kan avgöra om ett argument är modallogiskt giltigt eller inte i olika modala system. Det nämner tre olika bevismetoder/härledningsmetoder: axiomatiska system, naturlig deduktion och semantiska tablåer. Kapitlet innehåller många exempel. Om man vill lära sig mer om logik, kan det vara bra att veta någonting om alla dessa olika metoder. Kapitel 5 innehåller viss allmän information om modallogiken och fler övningar. Boken tar upp information både om propositionell modallogik, som är en utvidgning av den s.k. satslogiken, och kvantifierad modallogik, som bygger vidare på den s.k. predikatlogiken. Grunderna till satslogiken beskrivs bl.a. i boken Satslogik: En Introduktion. Grunderna till predikatlogiken beskrivs bl.a. i boken Predikatlogik: En Introduktion. För att förstå materialet i den här boken bör läsaren först ha studerat dessa eller liknande böcker. Modallogik: En Introduktion innehåller en stor mängd övningar som kan hjälpa läsaren att öva upp sina logiska förmågor. (shrink)

The axiom of dependent choice ( $\mathsf {DC}$ ) and the axiom of countable choice ( ${\mathsf {AC}}_\omega $ ) are two weak forms of the axiom of choice that can be stated for a specific set: $\mathsf {DC} ( X )$ asserts that any total binary relation on X has an infinite chain, while ${\mathsf {AC}}_\omega ( X )$ asserts that any countable collection of nonempty subsets of X has a choice function. It is well-known that $\mathsf {DC} \Rightarrow (...) {\mathsf {AC}}_\omega $. We study for which sets and under which hypotheses $\mathsf {DC} ( X ) \Rightarrow {\mathsf {AC}}_\omega ( X )$, and then we show it is consistent with $\mathsf {ZF}$ that there is a set $A \subseteq \mathbb {R}$ for which $\mathsf {DC} ( A )$ holds, but ${\mathsf {AC}}_\omega ( A )$ fails. (shrink)

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 $$\varphi $$ φ and $$\psi $$ ψ into a true one whenever $$\varphi $$ φ and $$\psi $$ ψ 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 $$\textsf{LD}$$ LD, Logic of Descriptions with Suszko’s Axioms $$\textsf{LDS}$$ LDS, Logic of Equimeaning $$\textsf{LDE}$$ LDE ) with non-Fregean versions of certain well-known paraconsistent logics (Jaśkowski’s Discussive Logic $$\textsf{D}_2$$ D 2, Logic of Paradox $$\textsf{LP}$$ LP, Logics of Formal Inconsistency $$\textsf{LFI}{1}$$ LFI 1 and $$\textsf{LFI}{2}$$ LFI 2 ). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of $$\textsf{LP}$$ LP, $$\textsf{LFI}{1}$$ LFI 1, $$\textsf{LFI}{2}$$ LFI 2. Furthermore, we show that non-Fregean extensions of $$\textsf{LP}$$ LP, $$\textsf{LFI}{1}$$ LFI 1, $$\textsf{LFI}{2}$$ LFI 2, and $$\textsf{D}_2$$ 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. (shrink)

This expository paper presents an application, to the modal logic S4, of the valuation semantics technique proposed by Loparić for the basic normal modal logic K. In previous works we presented a valuation semantics for the minimal temporal logic Kt and several other systems modal and temporal logic. How to deal with S4, however, was left as an open problem—although we arrived at a working definition of \(A_1,\ldots,A_n\) -valuations, we were not able to prove an important lemma for correctness. In (...) this paper we solve this, presenting valuations for S4. (shrink)

\(\nabla \) -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. (shrink)

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. (shrink)

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, _LFI_s). 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, _LFU_s). 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. (shrink)

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. (shrink)

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.

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 \(\textsf{iP}^{-}\). (shrink)

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 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. (shrink)

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. (shrink)

We generalize two of our previous results on abelian definable groups in p-adically closed fields [12, 13] to the non-abelian case. First, we show that if G is a definable group that is not definably compact, then G has a one-dimensional definable subgroup which is not definably compact. This is a p-adic analogue of the Peterzil–Steinhorn theorem for o-minimal theories [16]. Second, we show that if G is a group definable over the standard model $\mathbb {Q}_p$, then $G^0 = G^{00}$. (...) As an application, definably amenable groups over $\mathbb {Q}_p$ are open subgroups of algebraic groups, up to finite factors. We also prove that $G^0 = G^{00}$ when G is a definable subgroup of a linear algebraic group, over any model. (shrink)

Drawing on recently published correspondence as well as on a survey of Polish and international philosophical activity published in 1937 and details concerning the publisher and bookseller Aleksander Mazzucato, I provide evidence that, contrary to some recent assertions (but in line with older bibliographical entries), Tarski's ‘Der Wahrheitsbegriff in den formalisierten Sprachen’ was not published in journal form until 1936, although preprints, lacking two corrections and a small addendum, were likely available in the late months of 1935.

This paper is about the concept of semantically closed languages. Roughly speaking, those are languages which can name their own sentences and apply to them semantic predicates, such as the truth or satisfaction predicates. Hence, they are “self-referential languages,” in the sense that they are capable of producing sentences about themselves or other sentences in the same language. In section one, we introduce the concept informally; in section two, we provide the formal definition of first-order semantically closed languages, which is (...) the Tarskian definition with some technical modifications. Then, we construct a semantic for this kind of language, and prove that the language is indeed semantically closed (according to Definition 1). Finally, we discuss whether the logic underlying the construction is classical, and future goals of this research. (shrink)

As is well known, any preorder R on a set U induces an Alexandrov topology on U. In some interesting cases related to data mining an Alexandrov topology can be transformed into different types of logico-algebraic models. In some cases, (pre)topological operators provided by Pointless Topology may define a topological space on U even if R is not a preorder. If this is the case, then we call R a crypto-preorder. The paper studies the conditions under which a relation R (...) is a crypto-preorder and how to transform a crypto-preorder into a proper preorder. (shrink)

Graph-based frames have been introduced as a logical framework which internalises an inherent boundary to knowability (referred to as ‘informational entropy’), due, e.g. to perceptual, evidential or linguistic limits. They also support the interpretation of lattice-based (modal) logics as hyper-constructive logics of evidential reasoning. Conceptually, the present paper proposes graph-based frames as a formal framework suitable for generalising Pawlak's rough set theory to a setting in which inherent limits to knowability exist and need to be considered. Technically, the present paper (...) establishes systematic connections between the first-order correspondents of Sahlqvist modal reduction principles on Kripke frames, and on the more general relational environments of graph-based and polarity-based frames. This work is part of a research line aiming at: (a) comparing and inter-relating the various (first-order) conditions corresponding to a given (modal) axiom in different relational semantics; (b) recognising when first-order sentences in the frame-correspondence languages of different relational structures encode the same ‘modal content’; (c) meaningfully transferring relational properties across different semantic contexts. The present paper develops these results for the graph-based semantics, polarity-based semantics, and all Sahlqvist modal reduction principles. As an application, we study well known modal axioms in rough set theory (such as those corresponding to seriality, reflexivity, and transitivity) on graph-based frames and show that, although these axioms correspond to different first-order conditions on graph-based frames, their intuitive meaning is retained. This allows us to introduce the notion of hyperconstructivist approximation spaces as the subclass of graph-based frames defined by the first-order conditions corresponding to the same modal axioms defining classical generalised approximation spaces, and to transfer the properties and the intuitive understanding of different approximation spaces to the more general framework of graph-based frames. The approach presented in this paper provides a base for systematically comparing and connecting various formal frameworks in rough set theory, and for the transfer of insights across different frameworks. (shrink)

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which (...) correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder. (shrink)