Re: what follows from denying an axiom

On Jan 17, 6:34 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
I never claimed Henkin's semantics was second-order anything.  

Oh, bull***.
You claimed it was an appropriate semantics for what you called
"second-order languages", which is itself a ridiculous locution.

 The only
thing I referred to as second-order were the languages for which it is a
semantics.

Well, I'm sorry, you DIDN'T do that, because THAT is NOT POSSIBLE.
The language by itself DOES NOT GET TO HAVE an order.
What its order is DEPENDS on what the quantifiers are SUPPOSED/
INTENDED
to be quantifying over.

 It is just a fact that languages with predicate quantifiers
are referred to as "second-order",

Exactly.
And if you choose to supply a first-order semantics then your
predicate quantifiers STOP BEING "predicate" quantifiers.
That would make about as much sense as insisting on a finite
universe for FIRST-order quantifiers. "But it would still be a first-
order language", you whine. NO, IT WOULDN'T. The fact that the
universe
was finite would mean that quantified statements would be EQUIVALENT
to finite conjunctions or disjunctions, which would make them
ELIMINABLE
IN PRINCIPLE, which would mean that every sentence WITH quantifiers
would become equivalent to one WITHOUT, so EVEN THE LANGUAGE
would no longer be clearly first-order.

.

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