Related papers: Lax Modal Lambda Calculi
Unlike in classical modal logic, in non-classical modal logics the box and diamond operators frequently fail to be interdefinable. Instead, these logics impose some compatibility conditions which tie the box and diamond operators together…
This paper introduces a natural deduction calculus for intuitionistic logic of belief $\mathsf{IEL}^{-}$ which is easily turned into a modal $\lambda$-calculus giving a computational semantics for deductions in $\mathsf{IEL}^{-}$. By using…
We extend the {\lambda}-calculus with constructs suitable for relational and functional-logic programming: non-deterministic choice, fresh variable introduction, and unification of expressions. In order to be able to unify…
Large Language Models (LLMs) are increasingly being used in education, yet their correctness alone does not capture the quality, reliability, or pedagogical validity of their problem-solving behavior, especially in mathematics, where…
A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic. The proofs of soundness and completeness are constructive and…
In this paper we investigate the Curry-Howard correspondence for constructive modal logic in light of the gap between the proof equivalences enforced by the lambda calculi from the literature and by the recently defined winning strategies…
We introduce the $L_!^S$-calculus, a linear lambda-calculus extended with scalar multiplication and term addition, that acts as a proof language for intuitionistic linear logic (ILL). These algebraic operations enable the direct expression…
Remarkable progress has been made on automated reasoning with natural text, by using Language Models (LMs) and methods such as Chain-of-Thought and Selection-Inference. These techniques search for proofs in the forward direction from axioms…
Term modal logics (TML) are modal logics with unboundedly many modalities, with quantification over modal indices, so that we can have formulas of the form $\exists y. \forall x. (\Box_x P(x,y) \supset\Diamond_y P(y,x))$. Like First order…
In this paper, we present a typed lambda calculus ${\bf SILL}(\lambda)_{\Sigma}$, a type-theoretic version of intuitionistic linear logic with subexponentials, that is, we have many resource comonadic modalities with some interconnections…
In this article we show that hybrid type-logical grammars are a fragment of first-order linear logic. This embedding result has several important consequences: it not only provides a simple new proof theory for the calculus, thereby…
A variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. In the present article, we extend…
We study the correspondence theory of intuitionistic modal logic in modal Fairtlough-Mendler semantics (modal FM semantics) \cite{FaMe97}, which is the intuitionistic modal version of possibility semantics \cite{Ho16}. We identify the…
Nonmonotonic logics are usually characterized by the presence of some notion of 'conditional' that fails monotonicity. Research on nonmonotonic logics is therefore largely concerned with the defeasibility of argument forms and the…
In this paper, we introduce a general family of sequent-style calculi over the modal language and its fragments to capture the essence of all constructively acceptable systems. Calling these calculi \emph{constructive}, we show that any…
We propose a categorial grammar based on classical multiplicative linear logic. This can be seen as an extension of abstract categorial grammars (ACG) and is at least as expressive. However, constituents of {\it linear logic grammars (LLG)}…
We prove that the constructive and intuitionistic variants of the modal logic $\mathsf{KB}$ coincide. This result contrasts with a recent result by Das and Marin, who showed that the constructive and intuitionistic variants of $\mathsf{K}$…
Differentiable logics are a family of quantitative logics originated in the machine learning literature. Because of their origin, differentiable logics often come equipped with analytic properties that guarantee that they are…
In this paper we show that the intuitionistic monotone modal logic $\mathsf{iM}$ has the uniform Lyndon interpolation property (ULIP). The logic $\mathsf{iM}$ is a non-normal modal logic on an intuitionistic basis, and the property ULIP is…
Large language models (LLMs) and theorem provers (TPs) can be effectively combined for verifiable natural language inference (NLI). However, existing approaches rely on a fixed logical formalism, a feature that limits robustness and…