Logics, Misc

A number of general theories of physics provide a model for the fundamental rules that govern our universe, becoming a structural framework to which the new discoveries must conform. The theory of relativity is such a general theory. The theory of relativity is a complex theoretical framework that facilitates the understanding of the universal laws of physics. It is based on the curved space-time continuum fabric abstract concept, and it is well suited for interpreting cosmic events. More so, a general (...) theory based on abstract concepts and imagination facilitates the emergence of countless new extravagant theories. A new simplified theory of the natural world is necessary, a simple theory that provides a verifiable framework on which new discoveries can be integrated. The paper describes a view of the abstract time/space concept, and also a very simple model of our ever-changing universe. Views of physicists, mathematicians, chemists, engineers and of course philosophers have to be all in harmony with such a theory. We leave in a beautiful, uniform, and logical world. We live in a world where everything probable is possible. Contact email: [email protected] . (shrink)

This paper deals with higher-order vagueness in Williamson's 'logic of clarity'. Its aim is to prove that for 'fixed margin models' (W,d,α ,[ ]) the notion of higher-order vagueness collapses to second-order vagueness. First, it is shown that fixed margin models can be reformulated in terms of similarity structures (W,~). The relation ~ is assumed to be reflexive and symmetric, but not necessarily transitive. Then, it is shown that the structures (W,~) come along with naturally defined maps h and s (...) that define a Galois connection on the power set PW of W. These maps can be used to define two distinct boundary operators bd and BD on W. The main theorem of the paper states that higher-order vagueness with respect to bd collapses to second-order vagueness. This does not hold for BD, the iterations of which behave in quite an erratic way. In contrast, the operator bd defines a variety of tolerance principles that do not fall prey to the sorites paradox and, moreover, do not always satisfy the principles of positive and negative introspection. (shrink)

The logic of qualia is different than classical logic. We take the first steps in defining it and applying it to the Hard Problem.

Logical analysis of the applicability of nominals (which are introduced by hybrid logic) in the formal descriptions of the world (within modern knowledge representation and semantics-based systems) is very important because nominals, as second sorts of propositional symbols, can support logical identification of the described world at specific [temporal and/or spacial] states. This paper will focus on answering the philosophical-logical question of ‘how a fundamental world description in description logic (DL) and a nominal can be related to each other?’. Based (...) on my assumption that nominals can support more adequate identification of the world in DL, this paper will deal with the concept of ‘world identification’. Accordingly, based on a logical-terminological analysis of nominals, the paper will analyse hybridised fundamental world descriptions. The research will finally reach the idea that we can have a hybrid description logic based on the analysed concepts. (shrink)

Description Logics (DLs) are a family of formal knowledge representation formalisms and the most well-known formalisms in semantics-based systems. The central focus of this research is on logical-terminological characterisation/analysis of possibilistic and probabilistic descriptions of events in DLs. Based on a logical characterisation of the concept of `being', this paper conceptualises events within DLs world descriptions. Accordingly, it deals with the concepts of `possibility of events' and `probability of events'. The main goal of this research is to investigate how possible (...) and probable notations can, terminologically and logically, be interpreted and analysed within DLs world descriptions. In other words, the research logically-terminologically investigates how possibilistic and probabilistic descriptions of events can be expressed in DLs. (shrink)

Description Logics (DLs) are a family of well-known terminological knowledge representation formalisms in modern semantics-based systems. This research focuses on analysing how our developed Occurrence Logic (OccL) can conceptually and logically support the development of a description logic. OccL is integrated into the alternative theory of natural language syntax in `Deviational Syntactic Structures' under the label `EFA(X)3' (or the third version of Epi-Formal Analysis in Syntax, EFA(X), which is a radical linguistic theory). From the logical point of view, OccL is (...) a formal logic that mainly deals with the occurrences of symbols as well as with their priorities within linguistic descriptions, i.e. natural language syntax, semantics and phonology. In this article---based on our OccL-based definitions of the concepts of `strong implication' and `occurrence value' as well as of the logical concept `identical occurrence constructor (IDOC)' that is the most fundamental logical concept in our formalism---we will model Occurrence Description Logic ($\mathcal{ODL}$). Accordingly, we will formally-logically analyse `occurrence(s) of symbol(s)' within descriptions of the world in $\mathcal{ODL}$. In addition, we will analyse and assess the logical concepts of `occurrence' and `occurrence priority' in $\mathcal{ODL}$. This research can make a strong logical background for our future research in the development of a Modal Occurrence Description Logic. (shrink)

Inquisitive first-order logic, InqBQ, is a system which extends classical first-order logic with formulas expressing questions. From a mathematical point of view, formulas in this logic express properties of sets of relational structures. This paper makes two contributions to the study of this logic. First, we describe an Ehrenfeucht–Fraïssé game for InqBQ and show that it characterizes the distinguishing power of the logic. Second, we use the game to study cardinality quantifiers in the inquisitive setting. That is, we study what (...) statements and questions can be expressed in InqBQ about the number of individuals satisfying a given predicate. As special cases, we show that several variants of the question how many individuals satisfy α(x) are not expressible in InqBQ, both in the general case and in restriction to finite models. (shrink)

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical statements from (...) syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence, and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation. (shrink)

We present an abstract model of rationality that focuses on structural properties of attitudes. Rationality requires coherence between your attitudes, such as your beliefs, values, and intentions. We define three 'logical' conditions on attitudes: consistency, completeness, and closedness. They parallel the familiar logical conditions on beliefs, but contrast with standard rationality conditions like preference transitivity. We establish a formal correspondence between our logical conditions and standard rationality conditions. Addressing John Broome's programme 'rationality through reasoning', we formally characterize how you can (...) (not) become more logical by reasoning. Our analysis connects rationality with logic, and enables logical talk about multi-attitude psychology. (shrink)

At its core this book is concerned with logic and computation with respect to the mathematical characterization of sentient biophysical structure and its behavior. -/- Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. -/- These theories are based upon a new explanation of experience in (...) nature, the construction of senses, and motile behavior. This new approach is developed from first principles to enable a rigorous and systematic explanation of the variety of associated intelligent behaviors. -/- Alongside this development is a further account that focuses upon the nature of our work. It discusses the existential aspects of scientific inquiry, its epistemology and logic. It seeks to clarify the nature of the mathematical characterization and computation of natural behaviors, dealing with questions in the foundations of logic. It explores methodological issues related to reduction and the refinement of ideas from intuition to formal logical structure. -/- In support of this inquiry we work toward the development of a calculus for biophysical construction and its dynamics. If successful this mechanics mathematically characterizes sensory and motile behavior. -/- Upon this foundation we propose a model of apprehension and explore how its products are processed by the organism. Finally, we develop a probabilistic theory that enables us to reason about inaccessible factors in group behavior. -/- The mechanics we propose suggests the design and physical realization of a new model of computation; one in which structure and the concurrency of action are a first-order consideration. -/- We identify opportunities for experimental verification of the theory and we suggest a proof of our results in practice by the identification of this mechanism, allowing the construction of machines that experience. (shrink)

A low linear order with no computable copy constructed by C. Jockusch and R. Soare has Hausdorff rank equal to $2$. In this regard, the question arises, how simple can be a low linear order with no computable copy from the point of view of the linear order type? The main result of this work is an example of a low strong $\eta $ -representation with no computable copy that is the simplest possible example.

In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study (...) of certain principles of self-referential truth. (shrink)

We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from classical modal logic.

We study the first-order consequences of Ramsey’s Theorem for k-colourings of n-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm {RCA}^*_0$. Using the Chong–Mourad coding lemma, we show that in a model of $\mathrm {RCA}^*_0$ that does not satisfy $\Sigma ^0_1$ induction, $\mathrm {RT}^n_k$ is equivalent to its relativization to any proper $\Sigma ^0_1$ -definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe. We give (...) a complete axiomatization of the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^n_k$ for $n \ge 3$. We show that they form a non-finitely axiomatizable subtheory of $\mathrm {PA}$ whose $\Pi _3$ fragment coincides with $\mathrm {B} \Sigma _1 + \exp $ and whose $\Pi _{\ell +3}$ fragment for $\ell \ge 1$ lies between $\mathrm {I} \Sigma _\ell \Rightarrow \mathrm {B} \Sigma _{\ell +1}$ and $\mathrm {B} \Sigma _{\ell +1}$. We also give a complete axiomatization of the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^2_k + \neg \mathrm {I} \Sigma _1$. In general, we show that the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^2_k$ form a subtheory of $\mathrm {I} \Sigma _2$ whose $\Pi _3$ fragment coincides with $\mathrm {B} \Sigma _1 + \exp $ and whose $\Pi _4$ fragment is strictly weaker than $\mathrm {B} \Sigma _2$ but not contained in $\mathrm {I} \Sigma _1$. Additionally, we consider a principle $\Delta ^0_2$ - $\mathrm {RT}^2_2$ which is defined like $\mathrm {RT}^2_2$ but with both the $2$ -colourings and the solutions allowed to be $\Delta ^0_2$ -sets rather than just sets. We show that the behaviour of $\Delta ^0_2$ - $\mathrm {RT}^2_2$ over $\mathrm {RCA}_0 + \mathrm {B}\Sigma ^0_2$ is in many ways analogous to that of $\mathrm {RT}^2_2$ over $\mathrm {RCA}^*_0$, and that $\mathrm {RCA}_0 + \mathrm {B} \Sigma ^0_2 + \Delta ^0_2$ - $\mathrm {RT}^2_2$ is $\Pi _4$ - but not $\Pi _5$ -conservative over $\mathrm {B} \Sigma _2$. However, the statement we use to witness failure of $\Pi _5$ -conservativity is not provable in $\mathrm {RCA}_0 +\mathrm {RT}^2_2$. (shrink)

We show that the replacement rule of the sequent calculi ${\bf G3[mic]}^= $ in [8] can be replaced by the simpler rule in which one of the principal formulae is not repeated in the premiss.

In this paper, we define a class of dialogue games for Johansson’s minimal logic and prove that it corresponds to the validity of minimal logic. Many authors have stated similar results for intuitionistic and classical logic either with or without actually proving the correspondence. Rahman, Clerbout and Keiff [17] have already specified dialogues for minimal logic; however, they transformed it into Fitch-style natural deduction only. We propose a different specification for minimal logic with the proof of correspondence between the existence (...) of winning strategies for the Proponent in this class of games and the sequent calculus for minimal logic. (shrink)

The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that result. Utilizing (...) recent developments in structural proof theory, we develop a G3-style sequent calculus for Quarc and briefly demonstrate its structural properties. We put these properties to use immediately to construct direct proofs of the meta-theoretical properties of the system. We then incorporate an abstract (and, as we shall see, logical) predicate into the system in a way that preserves all the structural properties. This allows us to identify a system of Quarc which is deductively equivalent to PC, and also yields a constructive method of demonstrating the Craig interpolation theorem (which speaks in favor of the aforementioned predicate being logical). We further generalize this extension to develop a bivalent system of Quarc with defining clauses that still maintains all the desirable properties of a good proof system. (shrink)

This paper gives a semantics for schematic logic, proving soundness and completeness. The argument for soundness is carried out in ontologically innocent fashion, relying only on the existence of formulae which are actually written down in the course of a derivation in the logic. This makes the logic available to a nominalist, even a nominalist who does not wish to rely on modal notions, and who accepts the possibility that the universe may in fact be finite.

We introduce Arbitrary Public Announcement Logic with Memory (APALM), obtained by adding to the models a ‘memory’ of the initial states, representing the information before any communication took place (“the prior”), and adding to the syntax operators that can access this memory. We show that APALM is recursively axiomatizable (in contrast to the original Arbitrary Public Announcement Logic, for which the corresponding question is still open). We present a complete recursive axiomatization, that includes a natural finitary rule, and study this (...) logic’s expressivity and the appropriate notion of bisimulation. We then examine Group Announcement Logic with Memory (GALM), the extension of APALM obtained by adding to its syntax group announcement operators, and provide a complete finitary axiomatization (again in contrast to the original Group Announcement Logic, for which the only known axiomatization is infinitary). We also show that, in the memory-enhanced context, there is a natural reduction of the so-called coalition announcement modality to group announcements (in contrast to the memory-free case, where this natural translation was shown to be invalid). (shrink)

This paper introduces the logic _Q__L__E__T_ _F_, a quantified extension of the logic of evidence and truth _L__E__T_ _F_, together with a corresponding sound and complete first-order non-deterministic valuation semantics. _L__E__T_ _F_ is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (_FDE_) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘_A_ entails that _A_ behaves classically, ∙_A_ follows from _A_’s violating some classically valid inferences. The semantics of _Q__L__E__T_ (...) _F_ combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of _FDE_, _K3_, and _LP_, we show how these tools, which we call here the method of _anti-extensions + valuations_, can be naturally applied to a number of non-classical logics. (shrink)

After observing that the truth conditions of connectives of non–classical logics are generally defined in terms of formulas of first–order logic, we introduce ‘protologics’, a class of logics whose connectives are defined by arbitrary first–order formulas. Then, we introduce atomic and molecular logics, which are two subclasses of protologics that generalize our gaggle logics and which behave particularly well from a theoretical point of view. We also study and introduce a notion of equi-expressivity between two logics based on different classes (...) of models. We prove that, according to that notion, every pure predicate logic with \ variables and constants is as expressive as a predicate atomic logic, some sort of atomic logic. Then, we prove that the class of protologics is equally expressive as the class of molecular logics. That formally supports our claim that atomic and molecular logics are somehow ‘universal’. Finally, we identify a subclass of molecular logics that we call predicate molecular logics and which constitutes its representative core: every molecular logic is as expressive as a predicate molecular logic. (shrink)

This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift (...) to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems. (shrink)

The aim in this paper is to define an Abductive Question-Answer System for the minimal logic of formal inconsistency \. As a proof-theoretical basis we employ the Socratic proofs method. The system produces abductive hypotheses; these are answers to abductive questions concerning derivability of formulas from sets of formulas. We integrated the generation of and the evaluation of hypotheses via constraints of consistency and significance being imposed on the system rules.

We study the logic of so-called lexicographic or priority merge for multi-agent plausibility models. We start with a systematic comparison between the logical behavior of priority merge and the more standard notion of pooling through intersection, used to define, for instance, distributed knowledge. We then provide a sound and complete axiomatization of the logic of priority merge, as well as a proof theory in labeled sequents that admits cut. We finally study Moorean phenomena and define a dynamic resolution operator for (...) priority merge for which we also provide a complete set of reduction axioms. (shrink)

In recent years, some theorists have argued that the clogics are not only defined by their inferences, but also by their metainferences. In this sense, logics that coincide in their inferences, but not in their metainferences were considered to be different. In this vein, some metainferential logics have been developed, as logics with metainferences of any level, built as hierarchies over known logics, such as \, and \. What is distinctive of these metainferential logics is that they are mixed, i.e. (...) the standard for the premises and the conclusion is not necessarily the same. However, so far, all of these systems have been presented following a semantical standpoint, in terms of valuations based on the Strong Kleene truth-tables. In this article, we provide sound and complete sequent-calculi for the valid inferences and the invalid inferences of the logics \ and \, and introduce an algorithm that allows obtaining sound and complete sequent-calculi for the global validities and the global invalidities of any metainferential logic of any level. (shrink)

The aim of this article is to study the notion of derivability and its semantic counterpart in the context of non-transitive and non-reflexive substructural logics. For this purpose we focus on the study cases of the logics _S__T_ and _T__S_. In this respect, we show that this notion doesn’t coincide, in general, with a nowadays broadly used semantic approach towards metainferential validity: the notion of local validity. Following this, and building on some previous work by Humberstone, we prove that in (...) these systems derivability can be characterized in terms of a notion we call absolute global validity. However, arriving at these results doesn’t lead us to disregard local validity. First, because we discuss the conditions under which local, and also global validity, can be expected to coincide with derivability. Secondly, because we show how taking into account certain families of valuations can be useful to describe derivability for different calculi used to present _S__T_ and _T__S_. (shrink)

An extension of the propositional probability logic \ given in Ognjanović et al. that allows mixing of propositional formulas and probabilistic formulas is introduced. We describe the corresponding class of models, and we show that the problem of deciding satisfiability is in NP. We provide infinitary axiomatization for the logic and we prove that the axiomatization is sound and strongly complete.

An extension of the propositional probability logic \ given in Ognjanović et al. that allows mixing of propositional formulas and probabilistic formulas is introduced. We describe the corresponding class of models, and we show that the problem of deciding satisfiability is in NP. We provide infinitary axiomatization for the logic and we prove that the axiomatization is sound and strongly complete.

As shown in Fan, there are some similarities/resemblances between contingency and accident. Given this, one may naturally ask if we can unify the two operators to manifest all of their similarities/resemblances. In this article, instead of looking at the interactions between the two operators like in Fan, we turn our attention to the resemblances between the two operators. We extend the unification method in Fan to the current setting. The main results include some model-theoretical ones, such as expressivity, frame definability, (...) bisimulation, and some axiomatization ones. (shrink)

Many analyses of notion of _metainferences_ in the non-transitive logic ST have tackled the question of whether ST can be identified with classical logic. In this paper, we argue that the primary analyses are overly restrictive of the notion of metainference. We offer a more elegant and tractable semantics for the strict-tolerant hierarchy based on the three-valued function for the LP material conditional. This semantics can be shown to easily handle the introduction of _mixed_ inferences, _i.e._, inferences involving objects belonging (...) to more than one (meta)inferential level and solves several other limitations of the ST hierarchies introduced by Barrio, Pailos, and Szmuc. (shrink)

We introduce a game for Gödel logic where the players’ interaction stepwise reduces claims about the relative order of truth degrees of complex formulas to atomic truth comparison claims. Using the concept of disjunctive game states this semantic game is lifted to a provability game, where winning strategies correspond to proofs in a sequents-of-relations calculus.

In this paper, we shall give another proof of the faithfulness of Blass translation of the propositional fragment \ of Leśniewski’s ontology in the modal logic \ by means of Hintikka formula. And we extend the result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logic. As a result of observing the proofs we shall give general theorems on the faithfulness of B-translation with respect to normal modal logics complete to certain sets of well-known accessibility relations (...) with a restriction that transitivity and symmetry are not set at the same time. As an application of the theorems, for example, B-translation is faithful for the provability logic \ ), that is, \ \ \ \supset \Box \phi \). The faithfulness also holds for normal modal logics, e.g., \, \, \, \. We shall conclude this paper with the section of some open problems and conjectures. (shrink)

The “official” history of connexive logic was written in 2012 by Storrs McCall who argued that connexive logic was founded by ancient logicians like Aristotle, Chrysippus, and Boethius; that it was further developed by medieval logicians like Abelard, Kilwardby, and Paul of Venice; and that it was rediscovered in the 19th and twentieth century by Lewis Carroll, Hugh MacColl, Frank P. Ramsey, and Everett J. Nelson. From 1960 onwards, connexive logic was finally transformed into non-classical calculi which partly concur with (...) systems of relevance logic and paraconsistent logic. In this paper it will be argued that McCall’s historical analysis is fundamentally mistaken since it doesn’t take into account two versions of connexivism. While “humble” connexivism maintains that connexive properties only apply to “normal” antecedents, “hardcore” connexivism insists that they also hold for “abnormal” propositions. It is shown that the overwhelming majority of the forerunners of connexive logic were only “humble” connexivists. Their ideas concerning connexive implication don’t give rise, however, to anything like a non-classical logic. (shrink)

Logic and African Philosophy: Seminal Essays on African Systems of Thought is an edited work by Jonathan Chimakonam who is affiliated to the Philosophy Department at the University of Pretoria, Sou...

_T__S_ is a logic that has no valid inferences. But, could there be a logic without valid metainferences? We will introduce _T__S_ _ω_, a logic without metainferential validities. Notwithstanding, _T__S_ _ω_ is not as empty—i.e., uninformative—as it gets, because it has many antivalidities. We will later introduce the two-standard logic [_T__S_ _ω_, _S__T_ _ω_ ], a logic without validities and antivalidities. Nevertheless, [_T__S_ _ω_, _S__T_ _ω_ ] is still informative, because it has many contingencies. The three-standard logic [ \(\mathbf {TS}_{\omega (...) }, \mathbf {ST}_{\omega }, [{\overline {\emptyset }}{\emptyset }, {\emptyset } {\overline {\emptyset }}]\) ] that we will further introduce, has no validities, no antivalidities and also no contingencies whatsoever. We will also present two more validity-empty logics. The first one has no supersatisfiabilities, unsatisfabilities and antivalidities ∗. The second one has no invalidities nor non-valid-nor-invalid (meta)inferences. All these considerations justify thinking of logics as, at least, three-standard entities, corresponding, respectively, to what someone who takes that logic as correct, accepts, rejects and suspends judgement about, just because those things are, respectively, validities, antivalidities and contingencies of that logic. Finally, we will present some consequences of this setting for the monism/pluralism/nihilism debate, and show how nihilism and monism, on one hand, and nihilism and pluralism, on the other hand, may reconcile—at least according to how Gillian Russell understands nihilism, and provide some general reasons for adopting a multi-standard approach to logics. (shrink)

We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al., Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to (...) a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of \ :1941–1958, 2010) and in a sequent-of-relations calculus, Automated reasoning with analytic tableaux and related methods, Springer, Berlin, 1999, pp. 36–51) respectively. (shrink)

In a recent paper, Barrio, Pailos and Szmuc (BPS) show that there are logics that have exactly the validities of classical logic up to arbitrarily high levels of inference. They suggest that a logic therefore must be identified by its valid inferences at every inferential level. However, Scambler shows that there are logics with all the validities of classical logic at every inferential level, but with no antivalidities at any inferential level. Scambler concludes that in order to identify a logic, (...) we at least need to look at the validities and the antivalidities of every inferential level. In this paper, I argue that this is still not enough to identify a logic. I apply BPS’s techniques in a super/sub-valuationist setting to construct a logic that has exactly the validities and antivalidities of classical logic at every inferential level. I argue that the resulting logic is nevertheless distinct from classical logic. (shrink)

The main aim of this paper is to introduce first-order versions of logics of evidence and truth, together with corresponding sound and complete Kripke semantics with variable and constant domains. According to the intuitive interpretation proposed here, these logics intend to represent possibly inconsistent and incomplete information bases over time. The paper also discusses the connections between Belnap-Dunn’s and da Costa’s approaches to paraconsistency, and argues that the logics of evidence and truth combine them in a very natural way.