Related papers: Smullyan's truth and provability
We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence $\alpha$ which extends a weak arithmetical theory…
In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. In particular, it is shown that mathematical models are introduced differently and used differently in different…
In this paper I propose the idea to establish a clear distinction between the foundations of truth and the foundations of meaning in Mathematics. I explore on the most basic example, the mathematical line, the possibility that the…
To develop rigorous knowledge about ML models -- and the systems in which they are embedded -- we need reliable measurements. But reliable measurement is fundamentally challenging, and touches on issues of reproducibility, scalability,…
We introduce a framework that can be used to model both mathematics and human reasoning about mathematics. This framework involves {stochastic mathematical systems} (SMSs), which are stochastic processes that generate pairs of questions and…
In this paper, we take a fresh look at three Popperian concepts: riskiness, falsifiability, and truthlikeness (or verisimilitude) of scientific hypotheses or theories. First, we make explicit the dimensions that underlie the notion of…
n this paper, we review and connect the three essential conditions needed by the collapse model to achieve a complete and exact formulation, namely the theoretical, the experimental, and the ontological ones. These features correspond to…
We propose a new definition of actual cause, using structural equations to model counterfactuals. We show that the definition yields a plausible and elegant account of causation that handles well examples which have caused problems for…
We give a simple rigourous treatment of the classical results of the abelian sandpile model. Although we treat results which are well-known in the physics literature, in many cases we did not find complete proofs in the literature. The…
We explore the theory of illfounded and cyclic proofs for the propositional modal $\mu$-calculus. A fine analysis of provability for classical and intuitionistic modal logic provides a novel bridge between finitary, cyclic and illfounded…
Knowing the truth is rarely enough -- we also seek out reasons why the fact is true. While much is known about how we explain contingent truths, we understand less about how we explain facts, such as those in mathematics, that are true as a…
We introduce SMProbLog, a generalization of the probabilistic logic programming language ProbLog. A ProbLog program defines a distribution over logic programs by specifying for each clause the probability that it belongs to a randomly…
Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform…
A knowledge system S describing a part of real world does in general not contain complete information. Reasoning with incomplete information is prone to errors since any belief derived from S may be false in the present state of the world.…
Research in information systems includes a wide range of approaches which make a contribution in terms of knowledge, understanding, or practical developments. The measure of any research is, ultimately, its validity: are its finding true,…
The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical…
Facts are subject to contingencies and can be true or false in different circumstances. One such contingency is time, wherein some facts mutate over a given period, e.g., the president of a country or the winner of a championship.…
We propose a new definition of actual causes, using structural equations to model counterfactuals.We show that the definitions yield a plausible and elegant account ofcausation that handles well examples which have caused problems forother…
We investigate abstract model theoretic properties which holds for models in which a truth or satisfaction predicate for a sublanguage of the signature is definable. We analyse in which cases those properties in fact ensure the definability…
Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial…