8,856 research outputs found
Potential infinity, abstraction principles and arithmetic (Leniewski Style)
This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties
Perturbations of isometries between Banach spaces
We prove a very general theorem concerning the estimation of the expression
for different kinds of maps
satisfying some general perurbated isometry condition. It can be seen as a
quantitative generalization of the classical Mazur-Ulam theorem. The estimates
improve the existing ones for bi-Lipschitz maps. As a consequence we also
obtain a very simple proof of the result of Gevirtz which answers the
Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone
theorem which improves the results of Jarosz and more recent results of
Dutrieux and Kalton
Reconciling Bayesian Epistemology and Narration-based Approaches to Judiciary Fact-finding
Legal probabilism (LP) claims the degrees of conviction in juridical
fact-finding are to be modeled exactly the way degrees of beliefs are modeled
in standard bayesian epistemology. Classical legal probabilism (CLP) adds that
the conviction is justified if the credence in guilt given the evidence is
above an appropriate guilt probability threshold. The views are challenged on
various counts, especially by the proponents of the so-called narrative
approach, on which the fact-finders' decision is the result of a dynamic
interplay between competing narratives of what happened. I develop a way a
bayesian epistemologist can make sense of the narrative approach. I do so by
formulating a probabilistic framework for evaluating competing narrations in
terms of formal explications of the informal evaluation criteria used in the
narrative approach.Comment: In Proceedings TARK 2017, arXiv:1707.0825
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