We present our work on a new modal logic of optimality, OPT, whose semantics are modeled in terms of optimal paths through reward-weighted transition systems. We prove some basic properties of OPT, including its status as a normal modal logic, as well as its relation to some of the standard modal axioms. We end with a discussion of applications to AI and future research directions and extensions.

Large language models (LLMs) have demonstrated impressive capabilities in natural language understanding and generation, but they exhibit problems with logical consistency in the output they generate. How can we harness LLMs’ broad-coverage parametric knowledge in formal reasoning despite their inconsistency? We present a method for directly integrating an LLM into the interpretation function of the formal semantics for a paraconsistent logic. We provide experimental evidence for the feasibility of the method by evaluating the function using datasets created from several short-form factuality benchmarks. Unlike prior work, our method offers a theoretical framework for neurosymbolic reasoning that leverages an LLM’s knowledge while preserving the underlying logic’s soundness and completeness properties.

Abstract This paper argues for two claims, one conservative and the other one less so. First is the thesis that the range of linguistic features that can be considered authentically semantic is far wider in breadth than is typically acknowledged. In particular, linguistic phenomena that are entirely orthogonal to veridical considerations—including ethical matters—are sufficiently robust to support e.g. theories of meaning and inference. Second, we aim to argue that philosophy’s prioritization of truth obscures the existence of an important prior concern for care or the reduction of harm. Specifically, we offer a topic-theoretic interpretation of the bounds consequence reading of sequents put forward by Restall and Ripley. We argue that this reading allows us to provide a rigorous semantics using the newly introduced categories of care and harm. An important objection may be that philosophy requires the centrality and priority of truth above all else insofar as truth-like features are necessary to ground theories of meaning or reasoning. Anticipating such an objection, we will argue that care not only is an authentically semantic feature but also that care is sufficient to ground such theories.

Abstract This paper introduces a sequent calculus, $\textbf{M}_{\textbf{S}}$, the minimal structural logic, which includes all structural rules while excluding operational ones. Despite its limited calculus, $\textbf{M}_{\textbf{S}}$ unexpectedly shares a property with intuitionistic logic and modal logics between $\textsf{S1}$ and $\textsf{S5}$: it lacks sound and complete finitely-valued (deterministic) semantics. Mirroring Gödel’s and Dugundji’s findings, we demonstrate that $\textbf{M}_{\textbf{S}}$ does possess a natural finitely-valued non-deterministic semantics. In fact, we show that $\textbf{M}_{\textbf{S}}$ is sound and complete with respect to any semantics belonging to a natural class of maximally permissive non-deterministic matrices. We close by examining the case of subsystems of $\textbf{M}_{\textbf{S}}$, including the “structural kernels” of the strict-tolerant and tolerant-strict logics $\textbf{ST}$ and $\textbf{TS}$, and strengthen this result to also preclude finitely-valued deterministic semantics with respect to variable designated value frameworks.

Abstract In this paper, the notion of structural completeness is explored in the context of a generalized class of superintuitionistic logics that also involve systems that are not closed under uniform substitution. We just require that each logic must be closed under $D$ -substitutions assigning to atomic formulas only $\vee$ -free formulas. For these systems, we introduce four different notions of structural completeness and study how they are related. We focus on superintuitionistic inquisitive logics that validate a schema called Split and have the disjunction property. In these logics, disjunction can be interpreted in the sense of inquisitive semantics as a question-forming operator. It is shown that a logic is structurally complete with respect to $D$ -substitutions if and only if it validates Split. Various consequences of this result are explored. For example, it is shown that every superintuitionistic inquisitive logic can be characterized by a Kripke model built from $D$ -substitutions. We also formulate an algebraic counterpart of this result that says that the Lindenbaum–Tarski algebra ${\mathscr{H}}$ of any inquisitive logic can be embedded into the Heyting algebra formed from left ideals of endomorphisms on ${\mathscr{H}}$ . Additionally, we resolve a conjecture concerning superintuitionistic inquisitive logics due to Miglioli et al. and show that a false conjecture about superintuitionistic logics due to Minari and Wroński becomes true in the broader space of regular generalized superintuitionistic logics.

Abstract Nelson’s logic of constructible falsity $\textsf{N}$ and Rauszer’s Heyting–Brouwer logic $\textsf{HB}$ are well-known cases of extensions of intuitionistic logic $\textsf{Int}$ enriched with novel connectives. Wansing has suggested that Gödel’s provability interpretation of $\textsf{Int}$ can be extended to these systems by pairing the category of formal proofs with a distinct category of formal refutations. In this paper, we extend the framework of Artemov’s justification logic to provide explicit analyses of $\textsf{N}$ and $\textsf{HB}$ (and the dual-intuitionistic logic $\textsf{DualInt}$) that respect a distinction between proofs and refutations. The application distinguishes the categories by reinterpreting the agents of multiple-agent justification logic as devices that operate exclusively on one or the other category. The analyses reveal that differences between $\textsf{N}$ and $\textsf{HB}$ can be reduced to competing interaction principles characterizing the coordination between proofs and refutations. We conclude by reappraising some of the unusual features of $\textsf{HB}$ in light of the explicit analysis of $\textsf{HB}$.

Abstract A wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes a non-classical truth-value that is ‘contaminating’ in the sense that whenever the value is assigned to a formula $\varphi $, any complex formula in which $\varphi $ appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behaviour is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper.

In this article, we consider variations of Nuel Belnap's ‘artificial reasoner’. In particular, we examine cases in which the artificial reasoner is faulty, e.g. situations in which the reasoner is unable to calculate the value of a formula due to an inability to retrieve the values of its atoms. In the first half of the article, we consider two ways of modelling such circumstances and prove the deductive systems arising from these two types of models to be equivalent to Graham Priest's first-degree entailment with an ‘emptiness’ value (FDE ϕ ) and Richard Angell's analytic containment (AC), making computational interpretations of these systems possible. The Belnap-type semantics for AC bring FDE ϕ and AC in line with other containment logics in their neighborhood. The second half of the article examines formal questions, such as whether AC admits an analysis along the lines of that given to the related system of William Parry's system of analytic implication (PAI), as suggested by Kurt Gödel and confirmed by Kit Fine. Furthermore, a natural means of extending these systems to languages with an intensional implication connective is investigated.