Tameness in generalized metric structures

We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves

Stationary logic of finitely determinate structures

AbstractIn this part we develop the theory of finitely determinate structures, that is, structures on which the dual quantifiers “stat” and “unreadable” have the same meaning. Among other genera

A simplified framework for first-order languages and its formalization in Mizar

A strictly formal, set-theoretical treatment of classical first-order logic is given. Since this is done with the goal of a concrete Mizar formalization of basic results (Lindenbaum lemma; Henkin, satisfiability, completeness and Lowenheim-Skolem theorems) in mind, it turns into a systematic pursue of simplification: we give up the notions of free occurrence, of derivation tree, and study what inference rules are strictly needed to prove the mentioned results. Afterwards, we discuss details of the actual Mizar implementation, and give general techniques developed therein.Comment: Ph.D. thesis, defended on January, 20th, 201

The allegory of isomorphism

Isomorphism has become a key concept for the analysis of representation in many contexts: perceptual experience, mental imagery, scientific theories, and visual artwork may all be described as standing in isomorphisms to their targets. Yet isomorphism is a technical term from mathematics—how are we to evaluate its use in fields such as philosophy, psychology, neuroscience, or physics? I suggest that we should understand appeals to isomorphism as allegorical; the upshot of this suggestion is that isomorphism claims always operate on two distinct levels of significance, with different standards of precision and evaluation. Recognizing these levels as distinct changes the landscape of debate for isomorphism-based accounts of representation: it both dissolves the well-known triviality objection to these accounts and undermines strong forms of structural realism

Cardinal Arithmetic: From Silver’s Theorem to Shelah’s PCF Theory

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2019-2020, Tutor: Joan Bagaria PigrauThe main goal of this master’s thesis is to give a detailed description of the major ZFC advances in cardinal arithmetic from Silver’s Theorem to Shelah’s pcf theory and his bound on 2אω. In our attempt to make this thesis as self-contained as possible, we have devoted the first chapter to review the most elementary concepts of set theory, which include all the classical results from the first period of developement of cardinal arithmetic, from 1870 to 1930, due to Cantor, Hausdorff, König, and Tarski

Reference and Reinterpretation

Reference is the relation held to obtain between an expression and what a speaker or thinker intends the expression to represent. Reference is a component of interpretation, the process of giving terms, sentences, and thoughts semantic content. An example of reference in a formal context involves the natural numbers, where each one can be taken to have a corresponding set-theoretic counterpart as its referent. In an informal context reference is exemplified by the relation between a name and the specific name-bearer when a speaker or thinker utters or has the name in mind. Recent debates over reference have concerned the mechanism of reference: How is it that we can refer? In informal contexts, externalists see the reference relation as explicable in terms of the salient causal relations involved in the naming of a thing, or a class of things, and the ensuing causal chains leading to a term’s use. Opponents of this view—internalists—see the reference relation as being conceptually direct, and they take the external approach to rely on untenable metaphysical assumptions about the world’s structure. Moreover, some internalists take the permutability—i.e. the consistent reinterpretation—of certain referential schemes to confound the externalist picture of reference. In this thesis I focus on the reference of theoretical terms in science, and I argue for an externalist treatment of natural kinds and other theoretical elements. Along the way I offer a defense of the externalist’s pre-theoretic metaphysical assumptions and emphasize their central role in the interpretation of scientific languages. The externalist approach acknowledges the necessary constraints on reference-fixing that account for the schemes we employ, and this, I argue, confounds the permutation strategy