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Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically…

We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…

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…

The superposition principle lies at the heart of many non-classical properties of quantum mechanics. Motivated by this, we introduce a rigorous resource theory framework for the quantification of superposition of a finite number of linear…

Rule-based reasoning is an essential part of human intelligence prominently formalized in artificial intelligence research via logic programs. Describing complex objects as the composition of elementary ones is a common strategy in computer…

This article supports the epistemological claim that sound human reasoning about ultimate knowledge is either foundational or circularly justified. In particular, questions which naturally arise in theology, philosophy, and related…

This paper examines the application of Tarski's Undefinability Theorem to first-order arithmetic. The generally accepted view is that for this case the Theorem establishes that arithmetic truth is not arithmetic. A careful examination of…

This paper proposes an alternative to standard first-order logic that seeks greater naturalness, generality, and semantic self-containment. The system removes the first-order restriction, avoids type hierarchies, and dispenses with external…

We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…

Empirical science needs to be based on facts and claims that can be reproduced. This calls for replicating the studies that proclaim the claims, but practice in most fields still fails to implement this idea. When such studies emerged in…

In this work, we develop a formal system of inductive logic. It uses an infinitary language that allows for countable conjunctions and disjunctions. It is based on a set of nine syntactic rules of inductive inference, and contains classical…

The introduction of explicit notions of rejection, or disbelief, into logics for knowledge representation can be justified in a number of ways. Motivations range from the need for versions of negation weaker than classical negation, to the…

We define notions of cautiousness and cautious belief to provide epistemic conditions for iterated admissibility in finite games. We show that iterated admissibility characterizes the behavioral implications of "cautious rationality and…

Within the program of finding axiomatizations for various parts of computability logic, it was proved earlier that the logic of interactive Turing reduction is exactly the implicative fragment of Heyting's intuitionistic calculus. That sort…

In settings from fact-checking to question answering, we frequently want to know whether a collection of evidence (premises) entails a hypothesis. Existing methods primarily focus on the end-to-end discriminative version of this task, but…

The tension between deduction and induction is perhaps the most fundamental issue in areas such as philosophy, cognition and artificial intelligence. In an influential paper, Valiant recognised that the challenge of learning should be…

Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common…

We focus on the persistence principle over weak interpretability logic. Our object of study is the logic obtained by adding the persistence principle to weak interpretability logic from several perspectives. Firstly, we prove that this…

We study a conservative extension of classical propositional logic distinguishing between four modes of statement: a proposition may be affirmed or denied, and it may be strong or classical. Proofs of strong propositions must be…

Decomposition methods are often used for producing counterfactual predictions in non-strategic settings. When the outcome of interest arises from a game-theoretic setting where agents are better off by deviating from their strategies after…