arXiv subjects

Paaras Padhiar

Publications and source records attributed to Paaras Padhiar.

5 recordsLinked to original sources

arXiv2026

Justification Logic of the Lambda Calculus

The simply typed λ-calculus is a model of computation where typed terms correspond to proofs of intuitionistic propositional logic (IPL) via the Curry-Howard correspondence. Justification logic is an operational modal logic in which the standard box modality is replaced by an explicit proof term, allowing the logic itself to reason directly about proofs of its formulas. Standard justification logics reason about proofs of IPL by embedding Hilbert-style axiomatic proofs of IPL as proof terms of the logic. We instead introduce a justification logic of the λ-calculus, in which the proof terms of the modality are exactly the typed λ-terms themselves: a modal logic that reasons about computation and proof simultaneously, as both notions coincide under the Curry-Howard interpretation. First, we provide an axiomatisation of this logic. We then propose a natural deduction system and a Curry-Howard interpretation, where a formal connection between the term calculus and the proof terms of the logic is provided. To complete the picture, we give a Gentzen-style sequent calculus for which we prove cut-elimination, and consequently obtain a normalisation result by working in the negative fragment.

cs.LO↗

arXiv2026

Intuitionistic Justification Logic, Semantically

Justification logics are explicit versions of modal logic. In the classical setting, this means boxes are refined with explicit proof terms and interact with each other through proof operations. This exercise was extended to intuitionistic modal logic with native diamonds. In this setting, diamonds are refined to satisfier terms and come equipped with additional operations. Justification logic enjoys a connection to its corresponding modal logic through a realisation theorem. In the classical setting, this is achieved through either proof-theoretic or semantic methodology. So far, intuitionistic justification logic with satisfiers has only been presented syntactically with a proof-theoretic realisation theorem. We present two classes of semantics for intuitionistic justification logic with soundness and completeness results: basic modular models, which extend possible world semantics for intuitionistic propositional logic; modular models which contain Kripke-style machinery to promote "backwards compatibility" to modal logic. Using modular models, we present a realisation theorem to establish a connection between intuitionistic justification logic and its corresponding intuitionistic modal logic.

cs.LO↗

arXiv2026

The proof theory and semantics of second-order (intuitionistic) tense logic

We develop a second-order extension of intuitionistic modal logic, allowing quantification over propositions, both syntactically and semantically. A key feature of second-order logic is its capacity to define positive connectives from the negative fragment. Duly we are able to recover the diamond (and its associated theory) using only boxes, as long as we include both forward and backward modalities (`tense' modalities). We propose axiomatic, proof theoretic and model theoretic definitions of `second-order intuitionistic tense logic', and ultimately prove that they all coincide. In particular we establish completeness of a labelled sequent calculus via a proof search argument, yielding at the same time a cut-admissibility result. Our methodology also applies to the classical version of second-order tense logic, which we develop in tandem with the intuitionistic case.

cs.LO↗

arXiv2025

Justification Logic for Intuitionistic Modal Logic (Extended Technical Report)

Justification logics are an explication of modal logic; boxes are replaced with proof terms formally through realisation theorems. This can be achieved syntactically using a cut-free proof system e.g. using sequent, hypersequent or nested sequent calculi. In constructive modal logic, boxes and diamonds are decoupled and not De Morgan dual. Kuznets, Marin and Straßburger provide a justification counterpart to constructive modal logic CK and some extensions by making diamonds explicit by introducing new terms called satisfiers. We continue the line of work to provide a justification counterpart to Fischer Servi's intuitionistic modal logic IK and its extensions with the t and 4 axioms. We: extend the syntax of proof terms to accommodate the additional axioms of intuitionistic modal logic; provide an axiomatisation of these justification logics; provide a syntactic realisation procedure using a cut-free nested sequent system for intuitionistic modal logic introduced by Straßburger.

cs.LO↗

arXiv2024

Nested Sequents for Quasi-transitive Modal Logics

Previous works by Goré, Postniece and Tiu have provided sound and cut-free complete proof systems for modal logics extended with path axioms using the formalism of nested sequent. Our aim is to provide (i) a constructive cut-elimination procedure and (ii) alternative modular formulations for these systems. We present our methodology to achieve these two goals on a subclass of path axioms, namely quasi-transitivity axioms.

cs.LO↗