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Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula A of the one-variable fragment is a theorem of LQ (QL) iff its translation is a theorem of L5 (L.5). The proof is model-theoretic. In one direction, semantics based on the Mares-Goldblatt [15] semantics for quantified L are transformed into ternary (plus two binary) relational semantics for S5-like extensions of L (for a general presentation, see Seki [26-27]). In the other direction, a valuation is given for the full first-order relevant logic based on L into a model for a suitable S5 extension of L. I also discuss this work's relation to finding a complete axiomatization of the constant domain, non-general frame ternary relational semantics for which RQ is incomplete [11]. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We take Carnap's problem to be to what extent standard consequence relations in various formal languages fix the meaning of their logical vocabulary, alone or together with additional constraints on the form of the semantics. This paper studies Carnap's problem for basic modal logic. Setting the stage, we show that neighborhood semantics is the most general form of compositional possible worlds semantics, and proceed to ask which standard modal logics (if any) constrain the box operator to be interpreted as in relational Kripke semantics. Except when restricted to finite domains, no modal logic characterizes exactly the Kripkean interpretations of $\Box $. Moreover, we show that, in contrast with the case of first-order logic, the obvious requirement of permutation invariance is not adequate in the modal case. After pointing out some known facts about modal logics that nevertheless force the Kripkean interpretation, we focus on another feature often taken to embody the gist of modal logic: locality. We show that invariance under point-generated subframes (properly defined) does single out the Kripkean interpretations, but only among topological interpretations, not in general. Finally, we define a notion of bisimulation invariance—another aspect of locality—that, together with a reasonable closure condition, gives the desired general result. Along the way, we propose a new perspective on normal neighborhood frames as filter frames, consisting of a set of worlds equipped with an accessibility relation, and a free filter at every world. [ABSTRACT FROM AUTHOR]

In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result where negation is included is also discussed). It is proved that BG satisfies the variable sharing property and is Halldén complete, while failing to satisfy the disjunction property. Sound and complete subscripted tableaux are presented for BG. An axiomatization (in the positive language extended with fusion) is presented, which is sound with respect to BG and, with respect to a related ternary relational semantics, complete; the problem of identifying a complete axiomatization for BG itself is left open. Connections between BG and the quasi-relevance logic KR are also discussed. [ABSTRACT FROM AUTHOR]

The joint logic of problems and propositions QHC introduced by S. A. Melikhov, as well as intuitionistic modal logic QH4, is studied. An immersion of these logics into classical first-order predicate logic is considered. An analog of the Löwenheim–Skolem theorem on the existence of countable elementary submodels for QHC and QH4 is established. [ABSTRACT FROM AUTHOR]

We explore two ways of formalizing Kreisel's addendum to the Brouwer–Heyting–Kolmogorov interpretation. To do this, we compare Artemov's justification logic with simply typed |$\lambda $| calculus. First, we provide a completeness result for Kripke style semantics of the implicational fragment of the intuitionistic logic of proofs. Then we introduce a map from justification terms into |$\lambda $| terms, which can be viewed as a method of extracting the computational content of the justification terms. Then we examine the interpretation of Kreisel's addendum in justification logic along with the image of the resulting justification terms under our map. [ABSTRACT FROM AUTHOR]

In this paper, we introduce provability multilattice logic P M L n and multilattice arithmetic M P A n which extends first-order multilattice logic with equality by multilattice versions of Peano axioms. We show that P M L n has the provability interpretation with respect to M P A n and prove the arithmetic completeness theorem for it. We formulate P M L n in the form of a nested sequent calculus and show that cut is admissible in it. We introduce the notion of a provability multilattice and develop algebraic semantics for P M L n on its basis, by the method of Lindenbaum-Tarski algebras we prove the algebraic completeness theorem. We present Kripke semantics for P M L n and establish the Kripke completeness theorem via syntactical and semantic embeddings from P M L n into G L and vice versa. Last but not least, the decidability of P M L n is shown. [ABSTRACT FROM AUTHOR]

The present paper establishes systematic connections among the first-order correspondents of Sahlqvist modal reduction principles in various relational semantic settings, including crisp and many-valued Kripke frames, and crisp and many-valued polarity-based frames (aka enriched formal contexts). Building on unified correspondence theory, we aim at introducing a theoretical environment which makes it possible to: (a) compare and inter-relate the various frame correspondents (in different relational settings) of any given Sahlqvist modal reduction principle; (b) recognize when first-order sentences in the frame-correspondence languages of different types of relational structures encode the same "modal content"; (c) meaningfully transfer and represent well known relational properties such as reflexivity, transitivity, symmetry, seriality, confluence, density, across different semantic contexts. These results can be understood as a first step in a research program aimed at making correspondence theory not just (methodologically) unified, but also (effectively) parametric. [ABSTRACT FROM AUTHOR]

This paper introduces the logics of super-strict implications, where a super-strict implication is a strengthening of C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the modal cube. it is shown that all logics of super-strict implications are connexive logics in that they validate Aristotle's Theses and (weak) Boethius's Theses. A proof-theoretic characterisation of logics of super-strict implications is given by means of G3-style labelled calculi, and it is proved that the structural rules of inference are admissible in these calculi. It is also shown that validity in the S5-based logic of super-strict implications is equivalent to validity in G. Priest's negation-as-cancellation-based logic. Hence, we also give a cut-free calculus for Priest's logic. [ABSTRACT FROM AUTHOR]

We introduce fuzzy relational modalities of necessity and possibility designed to handle propositions that can be truth-valueless. We identify several important types of these modalities, corresponding to families of connectives and quantifiers of fuzzy partial logic (Bochvar, Sobociński, Kleene) and show some of their properties. We also present a faithful syntactic translation of the resulting fuzzy partial modal logic to first-order fuzzy partial logic. [ABSTRACT FROM AUTHOR]

We define Kripke semantics for propositional intuitionistic logic with Suszko's identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko's propositional iden- tity connective. [ABSTRACT FROM AUTHOR]

The modal characterization theorem by J. van Benthem characterizes classical modal logic as the bisimulation invariant fragment of first-order logic. In this paper, we prove a similar characterization theorem for intuitionistic modal logic. For this purpose we introduce the notion of modal asimulation as an analogue of bisimulations. The paper treats four different fragments of first-order logic induced by their respective versions of Kripke-style semantics for modal intuitionistic logic. It is shown further that this characterization can be easily carried over to arbitrary first-order definable subclasses of classical first-order models. [ABSTRACT FROM AUTHOR]

The standard first-order reading of modality does not bind individual variables, i.e. if x is free in F(x), then x remains free in F(x). Accordingly, if stands for 'provable in arithmetic,' ∀x F(x) states that F(n) is provable for any given value of n=0,1,2,...; this corresponds to a de re reading of modality. The other, de dicto meaning of F(x), suggesting that F(x) is derivable as a formula with a free variable x, is not directly represented by a modality, though, semantically, it could be approximated by compound constructions, e.g. ∀xF(x). We introduce the first-order logic FOS4* in which modalities can bind individual variables and, in particular, can directly represent both de re and de dicto modalities. FOS4* extends first-order S4 and is the natural forgetful projection of the first-order logic of proofs FOLP. The same method of introducing binding modalities obviously works for other modal logics as well. [ABSTRACT FROM AUTHOR]

We consider intermediate predicate logics defined by fixed well-ordered (or dually well-ordered) linear Kripke frames with constant domains where the order-type of the well-order is strictly smaller than ωω. We show that two such logics of different order-type are separated by a first-order sentence using only one monadic predicate symbol. Previous results by Minari, Takano and Ono, as well as the second author, obtained the same separation but relied on the use of predicate symbols of unbounded arity. [ABSTRACT FROM AUTHOR]

We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin–Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF. The result in its most basic form states that the category FC (S et , S et) of finite-coproduct-preserving endofunctors of S et is equivalent to the presheaf category [ UF , S et ]. Using this result, and some of its evident generalisations, we re-find in a natural manner the important model-theoretic realisation relation between n -types and n -tuples of model elements; and draw connections with Makkai and Lurie's work on conceptual completeness for first-order logic via ultracategories. As a further application of our main result, we use it to describe a first-order analogue of Jónsson and Tarski's canonical extension. Canonical extension is an algebraic formulation of the link between Lindenbaum–Tarski and Kripke semantics for intuitionistic and modal logic, and extending it to first-order logic has precedent in the topos of types construction studied by Joyal, Reyes, Makkai, Pitts, Coumans and others. Here, we study the closely related, but distinct, construction of the locally connected classifying topos of a first-order theory. The existence of this is known from work of Funk, but the description is inexplicit; ours, by contrast, is quite concrete. [ABSTRACT FROM AUTHOR]

We present a categorical formalization of a variant of first-order logic. Unlike other texts on this topic, the goal of this paper is to give a very transparent and self-contained account without requiring more background than basic logic and set theory. Our focus is to show how the semantics of first-order formulas can be derived from their usual deduction rules. For understanding the core ideas, it is not necessary to investigate the internal term structure of atomic formulas, thus we abstract atomic formulas to (syntactically opaque) relations; in this sense, our variant of first-order logic is "relational". While the derived semantics is based on categorical principles (even the duality that arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other), it is nevertheless "constructive" in that it describes explicit computations of the truth values of formulas. We demonstrate this by modeling the categorical semantics in the RISCAL (RISC Algorithm Language) system which allows us to validate the core propositions by automatically checking them in finite models. [ABSTRACT FROM AUTHOR]