On Oct 8, 2010, at 6:48 PM, John Bottoms wrote:
> Can we say:
> FOL is a subset of HOL. (01)
Not really, at least under most definitions of the notion of a logic. This is a
bit better: Let L be a first-order language and let L' be the result of adding
second-order (and even higher-order, if you please) quantifiers to L. Then
every logical truth of L, under a standard first-order semantics, will be a
logical truth of L', under a standard second-order semantics. More generally,
for any set S of sentences of L and any particular sentence A of L, if A is a
logical consequence of S under a standard first-order semantics for L, then A
is a logical consequence of S under a standard second-order semantics for L'. (02)
Chris Menzel (03)
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