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The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis. Recently, a number of nonstandard versions of Goedel's system T have been introduced ([2,9,12]), and it was shown in [26] that the…
Intuitionistic logic extended with decidable propositional atoms combines classical properties in its propositional part and intuitionistic properties for derivable formulas not containing propositional symbols. Sequent calculus is used as…
Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable…
This paper presents a simple notion of proof net for multiplicative linear logic with units. Cut elimination is direct and strongly normalising, in contrast to previous approaches which resorted to moving jumps (attachments) of par units…
It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by double-negation, while the…
This paper introduces a new algebra, the crossing algebra, that is applied to count the number of components for arborescent knots, links, tangles or states (of a state polynomial expansion such as the Kauffman bracket). This algebra is…
Deep and shallow embeddings of non-classical logics in classical higher-order logic have been explored, implemented, and used in various reasoning tools in recent years. This paper presents a method for the simultaneous deployment of deep…
Linear implication can represent state transitions, but real transition systems operate under temporal, stochastic or probabilistic constraints that are not directly representable in ordinary linear logic. We propose a general modal…
In this paper we investigate the complexity-theoretical aspects of cyclic and non-wellfounded proofs in the context of parsimonious logic, a variant of linear logic where the exponential modality ! is interpreted as a constructor for…
Commonsense reasoning is an appealing topic in natural language processing (NLP) as it plays a fundamental role in supporting the human-like actions of NLP systems. With large-scale language models as the backbone, unsupervised pre-training…
This chapter presents probability logic as a rationality framework for human reasoning under uncertainty. Selected formal-normative aspects of probability logic are discussed in the light of experimental evidence. Specifically, probability…
We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave…
Human reasoning can often be understood as an interplay between two systems: the intuitive and associative ("System 1") and the deliberative and logical ("System 2"). Neural sequence models -- which have been increasingly successful at…
Proof equivalence in a logic is the problem of deciding whether two proofs are equivalent modulo a set of permutation of rules that reflects the commutative conversions of its cut-elimination procedure. As such, it is related to the…
We prove undecidability and pinpoint the place in the arithmetical hierarchy for commutative action logic, that is, the equational theory of commutative residuated Kleene lattices (action lattices), and infinitary commutative action logic,…
A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms…
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…
The paper introduces a propositional linguistic logic that serves as the basis for automated uncertain reasoning with linguistic information. First, we build a linguistic logic system with truth value domain based on a linear symmetrical…
We propose a novel framework seamlessly providing key properties of both neural nets (learning) and symbolic logic (knowledge and reasoning). Every neuron has a meaning as a component of a formula in a weighted real-valued logic, yielding a…
Cathoristic logic is a multi-modal logic where negation is replaced by a novel operator allowing the expression of incompatible sentences. We present the syntax and semantics of the logic including complete proof rules, and establish a…