# [FOM] Modal logic with scope-modifying operators

Aatu Koskensilta aatu.koskensilta at xortec.fi
Thu Dec 29 04:26:33 EST 2005

```On Dec 28, 2005, at 7:58 PM, Thomas Forster wrote:

> [The logic mentioned in the subject] sounds very like Hintikka's
> "Independence-friendly" logic.....

Indeed it does, if for no other reason than the underlying idea being
allowing the scope of a modal operator to be non-contiguous. There are
independence friendly modal propositional logics, studied by Tero
Tulenheimo in his dissertation. These are genuinely stronger than
ordinary propositional modal logic, unlike the extension I outlined
which reduces to ordinary modal logic in the propositional case. The
semantics (or, rather, several alternative semantics) are (obviously)
based on different ideas. It's been a while since I read Tulenheimo's
dissertation and can't recall the details, but I have a hazy
recollection that at least one semantics is based on the idea that we
can express e.g. that a propositional holds regardless of which of the
several paths trough the worlds (the possible set of paths determined
by the formula) we choose when evaluating the truth of the formula. I
don't know if Tulenheimo or anyone else has done any work in
independence friendly modal predicate logic and how such a logic would
relate to "my" extension. My entirely groundless gut feeling is that
such logics would be weaker.

I haven't done any work to determine what sort of classes of frames
etc. are definable in the extension. I gave up quite soon after
realizing that with the most liberal semantics (every first order
structure of the given signature is accessible from every first order
structure of the signature) the extension is so ludicurously expressive
as to be rather uninteresting. I can't think of any mathematical
structure not definable (apart from examples concoted just for that
purpose).

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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