Date: Oct 25, 2013 4:47 PM
Author: fom
Subject: Purport is not instantiation

In this link:

http://en.wikipedia.org/wiki/Original_proof_of_G%C3%B6del%27s_completeness_theorem#Extension_to_first-order_predicate_calculus_with_equality

it is made clear that the completeness of

first-order logic with identity is, in fact,

completeness up to equivalence.

The formulas used describe an equivalence relation.

In this link:

http://en.wikipedia.org/wiki/Incidence_algebra#Examples

one must scroll down a little bit. What one

will find is that partition lattices are described

by incidence algebras.

Note that partition lattices are precisely the lattices

of equivalence relations.

Now, ask yourself what "singular term" is supposed

to represent. Next, ask yourself how Tarski's semantics

differs from Goedel's completeness theorem.

The syntactic purport of singular reference can never

be the same as the instantiation of a referent.

Tarski's semantics only makes reference to a single

element of a partiton lattice -- the node in which

the individual elements of the domain are to be found

partitioned into singletons.

The purport of identity in the logic makes reference

to any such partition.

Now, in the link:

https://groups.google.com/forum/#!original/sci.logic/Zc2egCgC9Nw/Q9u7tmEqrJ4J

I prove that the terms of a language for a consistent

theory have the structure of a matroid.

Matroids have their origins in the consideration of

independent sets of vectors in linear algebra.

In the link:

http://rutcor.rutgers.edu/pub/rrr/reports2002/35_2002.pdf

one can find a discussion of incidence algebras and matrix

algebra. So, it should not be surprising that the language

terms of a consistent theory have the structure of a

matroid.

There is an axiom for infintary matroids introduced here:

http://arxiv.org/pdf/1003.3919v3.pdf

Well, one can make up one's own mind. But there are reasons

why I look at the foundations of mathematics in the way that

I do.

Purport is not instantiation.