Tuesday, June 08, 2010

logic and ontology

This article helps to summarize a number of the positions debated last night.
Logic and Ontology
Some of the highlights:

We might wonder why we should think that quantifiers are of great importance for making ontological commitments explicit. After all, if I accept the apparently trivial mathematical fact that there is a number between 6 and 8, does this already commit me to an answer to the ontological question whether there are numbers out there, as part of reality? The above strategy tries to make explicit that and why it in fact does commit me to such an answer. This is so since natural language quantifiers are fully captured by their formal analogues in canonical notation, and the latter make ontological commitments obvious because of their semantics. Such formal quantifiers are given what is called an ‘objectual semantics’. This is to say that a particular quantified statement ‘∃xFx’ is true just in case there is an object in the domain of quantification that, when assigned as the value of the variable ‘x’, satisfies the open formula ‘Fx’. This makes obvious that the truth of a quantified statement is ontologically relevant, and in fact ideally suited to make ontological commitment explicit, since we need entities to assign as the values of the variables. Thus (L1) is tied to (O1). The philosopher most closely associated with this way of determining ontological commitment, and with the meta-ontological view on which it is based, is Quine, in particular his (Quine 1948).

Logically valid inferences are those that are guaranteed to be valid by their form. And above we spelled this out as follows: an inference is valid by its form if as long as we fix the meaning of certain special expressions, the logical constants, we can ignore the meaning of the other expressions in the statements involved in the inference, and we are always guaranteed the the inference is valid, no matter what the meaning of the other expressions is, as long as the whole is meaningful. A logical truth can be understood as a statement whose truth is guaranteed as long as the meanings of the logical constants are fixed, no matter what the meaning of the other expressions is. Alternatively, a logical truth is one that is a logical consequence from no assumptions, i.e. an empty set of premises.

Whatever one says about the possibility of proving the existence of an object purely with conceptual truths, many philosophers have maintained that at least logic has to be neutral about what there is. One of the reasons for this insistence is the idea that logic is topic neutral, or purely general. The logical truths are the ones that hold no matter what the representations are about, and thus they hold in any domain. In particular, they hold in an empty domain, one where there is nothing at all. And if that is true then logical truths can't imply that anything exists. But that argument might be turned around by a believer in logical objects, objects whose existence is implied by logic alone. If it is granted that logical truths have to hold in any domain, then any domain has to contain the logical objects. Thus for a believer in logical objects there can be no empty domain.

What the philosophers aim to ask, according to Carnap, is not a question internal to the framework, but external to it. They aim to ask whether the framework correctly corresponds to reality, whether or not there really are numbers. However, the words used in the question ‘Are there numbers?’ only have meaning within the framework of talk about numbers, and thus if they are meaningful at all they form an internal question, with a trivial answer. The external questions that the metaphysician tries to ask are meaningless. Ontology, the philosophical discipline that tries to answer hard questions about what there really is is based on a mistake. The question it tries to answer are meaningless questions, and this enterprise should be abandoned. The words ‘Are there numbers?’ thus can be used in two ways: as an internal question, in which case the answer is trivially ‘yes’, but this has nothing to do with metaphysics or ontology, or as an external question, which is the one the philosophers are trying to ask, but which is meaningless.

If there is an explanation of this similarity [the similarity between the subject-predicate structure of though, and the object-property structure of reality] to be given it seems it could go in one of two ways: either the structure of thought explains the structure of reality, or the other way round.