Partial Model Theory -- Ultraproducts and Compactness

TL;DR

This paper extends traditional model theory to partial structures by proving a compactness theorem using ultraproducts and quasi-truth, bridging philosophical notions with formal mathematical properties.

Contribution

It introduces a partial model theory that generalizes classical model theory to include partial structures and quasi-truth, with a proof of compactness using ultraproducts.

Findings

Proves compactness theorem for partial structures

Develops a formal framework for quasi-truth

Extends model theory to partial structures

Abstract

In the present paper we prove the compactness theorem with respect to partial structures and quasi-truth, using the technique of ultraproducts. Partial structures and quasi-truth are two notions developed within the partial structures approach, which is a philosophical conception that emerged in the context of contemporary philosophy of science. Nevertheless, the notions developed within this conception, in particular the two mentioned, have a model-theoretic content that has not been explored so far, so that this paper is part of a project where we intend to analyze their formal properties by means of the development of a partial model theory, which is an extension of traditional model theory to partial structures.