Date: Oct 25, 2013 4:47 PM
Subject: Purport is not instantiation
In this link:
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:
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:
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:
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
There is an axiom for infintary matroids introduced here:
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
Purport is not instantiation.