On 7/4/2013 11:42 PM, apoorv wrote: > My understanding is very imprecise, but the compactness/ completeness theorems > ' A theory is consistent iff it has a model'. - > The model itself is a theory in some other language, > So, should it be, ' a theory is consistent iff it has a Consistent model'. > Which really means that semantic consistency is somewhat of a regress.
This is not quite accurate.
Although one may argue as to how to view certain modern presentations, the transition from syllogistic logic to modern compositional forms of logic had been accomplished by Frege. In Frege's writings, he refers to intensional logicians and how his views disagree with those positions.
Frege's language seems to reflect the Aristotelian arrangement from "Posterior Analytics" where the definition of an object (its "essence") had been fundamentally associated with "substance".
However, the abstractness of mathematical objects creates a problem in this regard. Frege, himself, viewed his theories of arithmetic as grounded on a "logical object" taken to be substantive. He eventually retracted those views.
Having said that, I agree with your analysis and do not think that there is any real advantage to the received paradigm which holds to that.