[FOM] Modal logic with scope-modifying operators

[Aatu Koskensilta]

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