Description
Consider the following two facts about classical first-order logic (FOL): : first, due to the Löwenheim-Skolem theorem, a countable first-order theory with an infinite model cannot determine it uniquely up to isomorphism (thus, it allows for models of different infinite cardinalities, i.e., non-standard models) and, secondly, as proved in [Carnap 1943], the formal rules and axioms for the propositional operators and quantifiers do not uniquely determine their meanings (thus, there are models of FOL in which the propositional operators and quantifiers have the normal meanings and models in which they have a non-normal meaning). Each of these two facts shows that FOL is non-categorical, i.e., it allows non-standard models and non-normal interpretations. Nevertheless, since the operators and quantifiers may have a normal meaning in all non-isomorphic models that are due to the Löwenheim-Skolem theorem, while they have different meanings in the non-isomorphic models associated with Carnap’s results, the two notions of categoricity should not be confused with one another. But more importantly, what is the relation between these two distinct notions of categoricity? If the formal natural deduction rules of FOL are taken to be open-ended (i.e., they continue to hold even if the language expands), then it is argued that they uniquely determine the meanings of the logical terms that they introduce and, consequently, the universal quantifier should be taken as ranging over absolutely everything, rather than over a subset of the universal set. This would then seem to imply, against the Löwenheim-Skolem theorem, that all models of a countable first-order theory with an infinite model have the same cardinality – that of the universal set. This project proposes an investigation of the class of non-standard models of classical logic in relation to the idea of open-ended logical principles.
Objectives
The present project has three specific objectives:1. To prove that existence of the non-normal interpretations discovered by Carnap is independent of the non-standard models that follow from the Löwenheim-Skolem theorem.2. To prove that the open-ended formal rules of deduction for the propositional operators and for the first-order quantifiers do not uniquely determine their standard meanings. The first step is to show that these rules eliminate at best the non-standard general models, introduced by [Antonelli 2013] and adapted by [Bonnay and Westerståhl 2016], provided that everything is nameable, but they do not eliminate the non-normal interpretations of the propositional operators and of the quantifiers that Carnap pointed out.3. To analyze the connection between open-endedness, the assumption of nameability, and the Löwenheim-Skolem theorem. This will also require an analysis of the non-standard models of Peano Arithmetic.
Philosophical Motivation
The analysis of this problem, besides its logical relevance, has two fundamental philosophical motivations:i) If the formal natural deduction rules of first-order logic are taken to be open-ended and we take every object to be nameable, it seems that the universal quantifier should be taken as ranging over absolutely everything, rather than over a subset of the universal set. This implies, against the Löwenheim-Skolem theorem, that all models of a countable FOT with an infinite model have the same cardinality – that of the universal set. A philosophical consequence of this idea is that the ontological discourse, which is widely open to fit everything there is (and thus requires unrestricted quantification), is not, at least from this point of view, impossible. My analysis will confront this consequence with the idea that ontological discourse is relative to a conceptual framework.ii) The open-ended requirement does not make the logical rules language-specific, but rather general, universally applicable, and this would also fulfil Aristotle’s aim of providing universally applicable logical principles for correct reasoning. Likewise, my analysis will provide an analysis of this idea in connection with with the idea that logical principles are always work in a certain conceptual framework.