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

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

Narration in judiciary fact-finding : a probabilistic explication

Legal probabilism is the view that juridical fact-finding should be modeled using Bayesian methods. One of the alternatives to it is the narration view, according to which instead we should conceptualize the process in terms of competing narrations of what (allegedly) happened. The goal of this paper is to develop a reconciliatory account, on which the narration view is construed from the Bayesian perspective within the framework of formal Bayesian epistemology

Stanislaw Lesniewski: rethinking the philosophy of mathematics

Near the end of the nineteenth century, a part of mathematical research was focused on unification: the goal was to find 'one sort of thing' that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterprise hit a rough patch with Frege's failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. In parallel to this development was the work of Stanislaw Lesniewski (1886-1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Lesniewski attempted to find a nominalistically acceptable replacement for set theory in the foundations of mathematics. His candidate was Mereology - a theory which, instead of sets and elements, spoke of wholes and parts. The goal of this paper will be to present Mereology in this context, to evaluate the feasibility of Lesniewski's project and to briefly comment on its contemporary relevance

A Note on Identity and Higher Order Quantification

It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics (where the variables range over all subsets of the domain) in which the identity relation is not definable. The point is that the definability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed. This paper is a follow-up to (Urbaniak 2006), where it has been proven that no actual axiomatization of Leśniewski’s Ontology determines the standard semantics for the epsilon connective

Different Arguments, Same Problems. Modal Ambiguity and Tricky Substitutions

I illustrate with three classical examples the mistakes arising from using a modal operator admitting multiple interpretations in the same argument; the flaws arise especially easily if no attention is paid to the range of propositional variables. Premisses taken separately might seem convincing and a substitution for a propositional variable in a modal context might seem legitimate. But there is no single interpretation of the modal operators involved under which all the premisses are plausible and the substitution successful

Capturing dynamic conceptual frames

The main focus of this paper is to develop an adaptive formal apparatus capable of capturing (certain types of) reasoning conducted within the framework of the so-called dynamic conceptual frames. I first explain one of the most recent theories of concepts developed by cognitivists, in which a crucial part is played by the notion of a dynamic frame. Next, I describe how a dynamic frame may be captured by a finite set of first-order formulas and how a formalized adaptive framework for reasoning within a dynamic frame can be developed

Probabilistic legal decision standards still fail

Various probabilistic explications of the phrase the court's decision regarding a fact, given the evidence, is justified have been proposed. In this paper I evaluate them against two conceptual challenges: the difficulty about conjunction and the gatecrasher paradox. I argue that despite arguments to the opposite, all proposed models fail to solve these two problems