The semantic view of theories and higher-order languages

Dialectica, 2005

Based on an idea of Ajdukiewiu, a method of equifunctionality is developed to provide a formal explication of the notion of sameness of use relative to some system of rules. Given this, a set-theoretic explication of Lauener's context dependent conception of synonymy is introduced by looking at languages of propositional logic, and compared both with Ajdukiewicz's original conception and with Carnap's explication of synonymy based on his method of extension and intenrion ** Wolfson College, Oxford, GB.

2011

Abstract In this paper we discuss two approaches to the axiomatization of scientific theories in the context of the so called semantic approach, according to which (roughly) a theory can be seen as a class of models. The two approaches are associated respectively to Suppes' and to da Costa and Chuaqui's works.

The question of the semantic interpretation of higher-order logics has long been a matter of contention. Even though second-order quantification is quite natural, entangled interpretations have famously caused philosophers of logic such as Quine to reject second-order logic completely. In this paper I take a liberal attitude, open to maximizing the scope of logic, but careful to avoid conflation with other disciplines – and to avoid epistemological confusion. Higher-order logic (HOL) is perfectly acceptable, but one should be careful as to which semantics deserves to be called " standard ".

2004

In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, bi-interpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We provide some examples to separate notions across categories. In contrast, we show that, in some cases, notions in different categories do coincide. E.g., we can, under such-and-such conditions, infer synonymity of two theories from their being equivalent in the sense of a coarser equivalence relation. We illustrate that the categories offer an appropriate framework for conceptual analysis of notions. For example, we provide a ‘coordinate free’ explication of the notion of axiom scheme. Also we give a closer analysis of the object-language/ meta-language distinction. Our basic category can be enriched with a form of 2-structure. We use this 2-structure to characterize a salient subclass of...

In the paper we consider the classical logicism program restricted to first-order logic. The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don't impose restrictions on the size of their domains, can be reduced to pure logic. Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others. It is interesting to note that the initial formulation of the problem of reduction of mathematics to logic is principally insoluble. As we know all theorems of logic are true in the models with any number of elements. At the same time, many mathematical theories impose restrictions on size of their models. For example, all models of arithmetic have an infinite number of elements. If arithmetic was reducible to logic, it would had finite models, including an one-element model. But this is impossible in view of the axiom 0 ̸ = x ′ .

In this paper we present a unified approach to three topics: structures, formal languages and models. We begin by presenting a general theory of set theoretical structures. Formal languages and models are both structures inside this framework. We also present the link between languages and models given through a set theoretical predicate in the style of Suppes. The association of languages with structures satisfying certain conditions (given by set theoretical predicates) allow us to present an interesting application of the resulting framework: one is now able to characterize rigorously some classes of models. The classes of models so characterized play the role of scientific theories according to a version of the semantic approach to scientific theories. This is a first step in making explicit some of the underlying assumptions of the semantic approach to theories. In the end we give the example of how particle mechanics may be viewed as a theory according to that approach inside our framework.

Synthese, 2014

First, I discuss the older "theory-centered" and the more recent semantic conception of scientific theories. I argue that these two perspectives are nothing more than terminological variants of one another. I then offer a new theory-centered view of scientific theories. I argue that this new view captures the insights had by each of these earlier views, that it's closer to how scientists think about their own theories, and that it better accommodates the phenomenon of inconsistent scientific theories. Keywords Semantic conception of scientific theories • Inconsistent theories • Formal languages • Scientific theories as models • Bridge principles

Journal of Philosophical Logic, 2021

Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power and meta-theoretic properties when comparing first-order and second-order logic.

Proceedings of the 14th Meeting on the Mathematics of Language (MoL 2015), 2015

We discuss the model theory of two popular approaches to lexical semantics and their relation to transcendental logic.

2010

Abstract. A theory is stable up to ∆ if any ∆-type over a model has a few extensions up to complete types. I prove that a theory has no the independence property iff it is stable up to some ∆, where each ϕ(x; ¯y) ∈ ∆ has no the independence property. Definability of one-types over a model of a stable up to ∆ theory is investigated. 1.