Re: Another way of expressing the difference between first order and second order languages?
- From: Chris Menzel <cmenzel@xxxxxxxxxxxxxxxxxxxx>
- Date: 18 Oct 2005 18:27:09 GMT
On 17 Oct 2005 17:29:49 -0700, andrewspencers@xxxxxxxxx
<andrewspencers@xxxxxxxxx> said:
> It seems to me that in a language with logical constants (including
> quantifiers and sentential connectives), variables, and nonlogical
> constants, the language is first order if it allows the quantifiers to
> apply only to variables or to nonlogical constants but not to both,
> and the language is second order if it allows the quantifiers to apply
> to both variables and to nonlogical constants. Is this a correct
> analysis?
No. The distinction between first- and second-order is *semantic*.
Second-order languages interpreted according to Henkin's so-called
"general models" are still expressively first-order. What makes a
semantics for a second-order language genuinely second-order is the
requirement that the n-place predicate quantifiers range over the entire
power set of the set of n-tuples over the domain of the first-order
quantifiers.
See Enderton's A Mathematical Introduction to Logic, ch. 4.
Chris Menzel
.
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