Related papers: Smullyan's truth and provability
Argumentation problems are concerned with determining the acceptability of a set of arguments from their relational structure. When the available information is uncertain, probabilistic argumentation frameworks provide modelling tools to…
This paper revisits the foundations of mathematical proof through the lens of Aristotle's threefold conception of truth: sensory evidence, axiomatic definition, and syllogistic deduction. I argue that modern mathematics has too often…
Besides their use for efficient computation, quantum computers are a base for studying quantum systems that create valid physical theories using mathematics and physics. An essential part of the validation process for quantum mechanics is…
Reliability has long been treated as an engineering practice supported by testing, statistics and standards, yet its status as a scientific discipline remains unsettled. From a philosophical perspective, scientific truth is characterized by…
This paper studies the modal logical aspects of provability predicates and consistency statements for theories of arithmetic. First, we provide an overview of previous works on the correspondence between various derivability conditions for…
Topological models of empirical and formal inquiry are increasingly prevalent. They have emerged in such diverse fields as domain theory [1, 16], formal learning theory [18], epistemology and philosophy of science [10, 15, 8, 9, 2],…
Model explainability has become an important problem in machine learning (ML) due to the increased effect that algorithmic predictions have on humans. Explanations can help users understand not only why ML models make certain predictions,…
By Solovay's celebrated completeness result on formal provability we know that the provability logic $\mathrm GL$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable…
Learning correlations from data forms the foundation of today's machine learning (ML) and artificial intelligence research. While contemporary methods enable the automatic discovery of complex patterns, they are prone to failure when…
This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions.…
In this essay, I attempt to provide supporting evidence as well as some balance for the thesis on `Transforming socio-economics with a new epistemology' presented by Hollingworth and Mueller (2008). First, I review a personal highlight of…
The features of a logically sound approach to a theory of statistical reasoning are discussed. A particular approach that satisfies these criteria is reviewed. This is seen to involve selection of a model, model checking, elicitation of a…
Our approach is basically a coherence approach, but we avoid the well-known pitfalls of coherence theories of truth. Consistency is replaced by reliability, which expresses support and attack, and, in principle, every theory (or agent,…
In this paper, we propose standard statistical tools as a solution to commonly highlighted problems in the explainability literature. Indeed, leveraging statistical estimators allows for a proper definition of explanations, enabling…
This paper presents four theorems that connect continuity postulates in mathematical economics to solvability axioms in mathematical psychology, and ranks them under alternative supplementary assumptions. Theorem 1 connects notions of…
In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates $\Box$ and $\triangle$ that…
In this note I describe reliability standards for writing and reviewing mathematical papers; these standards are (in my opinion) vital for the progress of mathematics. I give examples of applying the described or other reliability…
Research on summarization has mainly been driven by empirical approaches, crafting systems to perform well on standard datasets with the notion of information Importance remaining latent. We argue that establishing theoretical models of…
Model uncertainty is a crucial issue in statistics, econometrics and machine learning, yet its definition remains ambiguous and is subject to various interpretations in the literature. So far, there has not been a universally accepted…
Satisfiability modulo theory (SMT) consists in testing the satisfiability of first-order formulas over linear integer or real arithmetic, or other theories. In this survey, we explain the combination of propositional satisfiability and…