During today's NIST-Ontolog-NCOR Mini-Series conference, there was
a short discussion about whether modal logic with the usual
Kripke semantics is first-order or not. (01)
Just to clarify the point:
- in general, modal logic (let it be propositional or first-order modal
logic) with Kripke semantics is of second-order nature.
- more exotic modal logics may be truly second-order, like provability
logic obtained from Loeb's formula
Box (Box p -> p) -> Box p
The second-order nature comes in here, because the above formula
implicitly is universally quantified over p, but p is interpreted as a
set of worlds - i.e. we have a second-order quantification.
- For the usual systems like S4 or S5 etc. (S5 is used for DOLCE), this
second-order quantification.luckily is equivalent to a first-order
sentence. For example, the S4 axiom
Box p -> Box Box p
holds iff the accessibility relation is transitive, which is easily
expressed in first-order logic.
Hence, the second-order quantification can be eliminated and these
logics are of first-order nature. (02)
Greetings,
Till (03)
--
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DFKI Lab Bremen Cartesium Fax +49-421-218-9864226
Robert-Hooke-Str. 5 Enrique-Schmidt-Str. 5 till@xxxxxx
D-28359 Bremen Room 2.051 http://www.tzi.de/~till (04)
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supervisory board: Prof. Hans A. Aukes (chair)
Amtsgericht Kaiserslautern, HRB 2313 (05)
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