Related papers: Reasoning by Analogy in Mathematical Practice
Analogical reasoning plays a critical role in human cognition, enabling us to understand new concepts by associating them with familiar ones. Previous research in the AI community has mainly focused on identifying and generating analogies…
Mathematical proof aims to deliver confident conclusions, but a very similar process of deduction can be used to make uncertain estimates that are open to revision. A key ingredient in such reasoning is the use of a "default" estimate of…
We re-examine the old question to what extent mathematics may be compared with a game. Mainly inspired by Hilbert and Wittgenstein, our answer is that mathematics is something like a rhododendron of language games, where the rules are…
The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years \cite{aA75,nB77,aA89,gP89,sH96}. In this note, we investigate some of these paradoxes and classify them, over minimal…
As David Berlinski writes (1997), the existence and nature of mathematics is a more compelling and far deeper problem than any of the problems raised by mathematics itself. Here we analyze the essence of mathematics making the main emphasis…
The logic LAE discussed in this paper is based on an approximate entailment relation. LAE generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision. To this end, properties are…
Inspired by empirical work in neuroscience for Bayesian approaches to brain function, we give a unified probabilistic account of various types of symbolic reasoning from data. We characterise them in terms of formal logic using the…
While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool…
This paper studies algebras arising as algebraic semantics for logics used to model reasoning with incomplete or inconsistent information. In particular we study, in a uniform way, varieties of bilattices equipped with additional…
A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the…
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
Across machine learning (ML) sub-disciplines researchers make mathematical assumptions to facilitate proof-writing. While such assumptions are necessary for providing mathematical guarantees for how algorithms behave, they also necessarily…
We study counting propositional logic as an extension of propositional logic with counting quantifiers. We prove that the complexity of the underlying decision problem perfectly matches the appropriate level of Wagner's counting hierarchy,…
In this paper, a new approximate syllogistic reasoning schema is described that expands some of the approaches expounded in the literature into two ways: (i) a number of different types of quantifiers (logical, absolute, proportional,…
Different types of reasoning impose different structural demands on representational systems, yet no systematic account of these demands exists across psychology, AI, and philosophy of mind. I propose a framework identifying four structural…
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us…
Current conceptions of expert problem solving depict physical/conceptual reasoning and formal mathematical reasoning as separate steps: a good problem solver first translates a physical Current conceptions of quantitative problem-solving…
Multiplicative linear logic is a very well studied formal system, and most such studies are concerned with the one-sided sequent calculus. In this paper we look in detail at existing translations between a deep inference system and the…
Abductive reasoning, reasoning for inferring explanations for observations, is often mentioned in scientific, design-related and artistic contexts, but its understanding varies across these domains. This paper reviews how abductive…
Mathematics is a critical part of much scientific research. Physics in particular weaves math extensively into its instruction beginning in high school. Despite much research on the learning of both physics and math, the problem of how to…