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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]

We show that monadic intuitionistic quantifiers admit the following temporal interpretation: "always in the future" (for $\forall $) and "sometime in the past" (for $\exists $). It is well known that Prior's intuitionistic modal logic ${\sf MIPC}$ axiomatizes the monadic fragment of the intuitionistic predicate logic, and that ${\sf MIPC}$ is translated fully and faithfully into the monadic fragment ${\sf MS4}$ of the predicate ${\sf S4}$ via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension ${\sf TS4}$ of ${\sf S4}$ and provide a full and faithful translation of ${\sf MIPC}$ into ${\sf TS4}$. We compare this new translation of ${\sf MIPC}$ with the Gödel translation by showing that both ${\sf TS4}$ and ${\sf MS4}$ can be translated fully and faithfully into a tense extension of ${\sf MS4}$ , which we denote by ${\sf MS4.t}$. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for ${\sf MS4.t}$ using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered. [ABSTRACT FROM AUTHOR]

Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be "sufficiently classical" in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. [ABSTRACT FROM AUTHOR]

I show that the logic $\textsf {TJK}^{d+}$ , one of the strongest logics currently known to support the naive theory of truth, is obtained from the Kripke semantics for constant domain intuitionistic logic by (i) dropping the requirement that the accessibility relation is reflexive and (ii) only allowing reflexive worlds to serve as counterexamples to logical consequence. In addition, I provide a simplified natural deduction system for $\textsf {TJK}^{d+}$ , in which a restricted form of conditional proof is used to establish conditionals. [ABSTRACT FROM AUTHOR]

Modal logic S5 has attracted significant attention and has led to several practical applications, owing to its simplified approach to dealing with nesting modal operators. Efficient implementations for evaluating satisfiability of S5 formulas commonly rely on Skolemisation to convert them into propositional logic formulas, essentially by introducing copies of propositional atoms for each set of interpretations (possible worlds). This approach is simple, but often results into large formulas that are too difficult to process, and therefore more parsimonious constructions are required. In this work, we propose to use Answer Set Programming for implementing such constructions, and in particular for identifying the propositional atoms that are relevant in every world by means of a reachability relation. The proposed encodings are designed to take advantage of other properties such as entailment relations of subformulas rooted by modal operators. An empirical assessment of the proposed encodings shows that the reachability relation is very effective and leads to comparable performance to a state-of-the-art S5 solver based on SAT, while entailment relations are possibly too expensive to reason about and may result in overhead. [ABSTRACT FROM AUTHOR]

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding 'strong negation'. Though all four values 'truth', 'falsity', 'neither' and 'both' are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for 'neither' or/and 'both' leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics. [ABSTRACT FROM AUTHOR]

Standard reasoning about Kripke semantics for modal logic is almost always based on a background framework of classical logic. Can proofs for familiar definability theorems be carried out using a nonclassical substructural logic as the metatheory? This article presents a semantics for positive substructural modal logic and studies the connection between frame conditions and formulas, via definability theorems. The novelty is that all the proofs are carried out with a noncontractive logic in the background. This sheds light on which modal principles are invariant under changes of metalogic, and provides (further) evidence for the general viability of nonclassical mathematics. [ABSTRACT FROM AUTHOR]

This article presents a unified logic that combines classical logic, intuitionistic logic and affine linear logic (restricting contraction but not weakening). We show that this unification can be achieved semantically, syntactically and in the computational interpretation of proofs. It extends our previous work in combining classical and intuitionistic logics. Compared to linear logic, classical fragments of proofs are better isolated from non-classical fragments. We define a phase semantics for this logic that naturally extends the Kripke semantics of intuitionistic logic. We present a sequent calculus with novel structural rules, which entail a more elaborate procedure for cut elimination. Computationally, this system allows affine-linear interpretations of proofs to be combined with classical interpretations, such as the λμ calculus. We show how cut elimination must respect the boundaries between classical and non-classical modes of proof that correspond to delimited control effects. [ABSTRACT FROM AUTHOR]

The sentences employed in semantic paradoxes display a wide range of semantic behaviours. However, the main theories of truth currently available either fail to provide a theory of paradox altogether, or can only account for some paradoxical phenomena by resorting to multiple interpretations of the language, as in (Kripke, 1975). In this article, I explore the wide range of semantic behaviours displayed by paradoxical sentences, and I develop a unified theory of truth and paradox , that is a theory of truth that also provides a unified account of paradoxical sentences. The theory I propose here yields a threefold classification of paradoxical sentences—liar-like sentences, truth-teller–like sentences, and revenge sentences. Unlike existing treatments of semantic paradox, the theory put forward in this article yields a way of interpreting all three kinds of paradoxical sentences, as well as unparadoxical sentences, within a single model. [ABSTRACT FROM AUTHOR]

Modal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics. [ABSTRACT FROM AUTHOR]

In two recent articles, Norihiro Kamide introduces unusual variants of Nelson's paraconsistent logic and its classical extension. Kamide's systems, IP and CP , are unusual insofar as double negations in these logics behave as intuitionistic and classical negations, respectively. In this article we present Hilbert-style axiomatizations of both IP and CP. The axiom system for IP is shown to be sound and complete with respect to a four-valued Kripke semantics, and the axiom system for CP is characterized by four-valued truth tables. Moreover, we note some properties of IP and CP , and emphasize that these logics are unusual also because they are contra-classical and inconsistent but nontrivial. We point out that Kamide's approach exemplifies a general method for obtaining contra-classical logics, and we briefly speculate about a linguistic application of Kamide's logics. [ABSTRACT FROM AUTHOR]

Proofs with logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe two case studies using the proof environment Beluga: First, we explain the mechanization of the weak normalization proof for the simply typed lambda-calculus; second, we outline how to mechanize the completeness proof of algorithmic equality for simply typed lambda-terms where we reason about logically equivalent terms. The development of these proofs in Beluga relies on three key ingredients: (1) we encode lambda-terms together with their typing rules, operational semantics, algorithmic and declarative equality using higher order abstract syntax (HOAS) thereby avoiding the need to manipulate and deal with binders, renaming and substitutions, (2) we take advantage of Beluga's support for representing derivations that depend on assumptions and first-class contexts to directly state inductive properties such as logical relations and inductive proofs, (3) we exploit Beluga's rich equational theory for simultaneous substitutions; as a consequence, users do not need to establish and subsequently use substitution properties, and proofs are not cluttered with references to them. We believe these examples demonstrate that Beluga provides the right level of abstractions and primitives to mechanize challenging proofs using HOAS encodings. It also may serve as a valuable benchmark for other proof environments. [ABSTRACT FROM AUTHOR]

In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic. [ABSTRACT FROM AUTHOR]

The article discusses fixed point axioms and related principles in Kripke-Platek set theory. It discusses fixed point axioms for set-bounded monotone Σ1 definable operators in the context of Kripke-Platek set theory KP. It analyzes their relationship to other principles such as maximal iterations, bounded proper injections, and Σ1 subset-bounded separation, and the results thereof.

Kripke recently suggested viewing the intuitionistic continuum as an expansion in time of a definite classical continuum. We prove the classical consistency of a three-sorted intuitionistic formal system IC, simultaneously extending Kleene’s intuitionistic analysis I and a negative copy C° of the classically correct part of I, with an “end of time” axiom ET asserting that no choice sequence can be guaranteed not to be pointwise equal to a definite (classical or lawlike) sequence. “Not every sequence is pointwise equal to a definite sequence” is independent of IC. The proofs are by Crealizability interpretations based on classical ω-models ${\cal M}$ = $\left( {\omega ,{\cal C}} \right)$ of C°. [ABSTRACT FROM AUTHOR]

We study the measure semantics for propositional modal logics, in which formulas are interpreted in the Lebesgue measure algebra${\cal M}$, or algebra of Borel subsets of the real interval [0,1] modulo sets of measure zero. It was shown in Lando (2012) and Fernández-Duque (2010) that the propositional modal logic S4 is complete for the Lebesgue measure algebra. The main result of the present paper is that every logic L aboveS4 is complete for some subalgebra of ${\cal M}$. Indeed, there is a single model over a subalgebra of ${\cal M}$ in which all nontheorems of L are refuted. This work builds on recent work by Bezhanishvili, Gabelaia, & Lucero-Bryan (2015) on the topological semantics for logics above S4. In Bezhanishvili et al., (2015), it is shown that there are logics above that are not the logic of any subalgebra of the interior algebra over the real line, ${\cal B}$(ℝ), but that every logic above is the logic of some subalgebra of the interior algebra over the rationals, ${\cal B}$(ℚ), and the interior algebra over Cantor space, ${\cal B}\left( {\cal C} \right)$. [ABSTRACT FROM PUBLISHER]

Kripke models, interpreted realistically, have difficulty making sense of the thesis that there might have existed things that do not in fact exist, since a Kripke model in which this thesis is true requires a model structure in which there are possible worlds with domains that contain things that do not exist. This paper argues that we can use Kripke models as representational devices that allow us to give a realistic interpretation of a modal language. The method of doing this is sketched, with the help of an analogy with a Galilean relativist theory of spatial properties and relations. [ABSTRACT FROM PUBLISHER]

In Kripke’s classic paper on truth it is argued that by adding a new semantic category different from truth and falsity it is possible to have a language with its own truth predicate. A substantial problem with this approach is that it lacks the expressive resources to characterize those sentences which fall under the new category. The main goal of this paper is to offer a refinement of Kripke’s approach in which this difficulty does not arise. We tackle this characterization problem by letting certain sentences belong to more than one semantic category. We also consider the prospect of generalizing this framework to deal with languages containing vague predicates. [ABSTRACT FROM AUTHOR]