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[ontolog-forum] Is modal logic first-order?

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Till Mossakowski <till@xxxxxxxxxxxxxxxxxxxxxxxx>
Date: Thu, 22 Feb 2007 23:41:11 +0100
Message-id: <45DE1C07.5030404@xxxxxxxxxxxxxxxxxxxxxxxx>
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)

Till    (03)

Till Mossakowski    Office:      Phone +49-421-218-64226
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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