# [FOM] Modal logic with scope-modifying operators

tulenhei@mappi.helsinki.fi tulenhei at mappi.helsinki.fi
Thu Dec 29 18:26:04 EST 2005

```Aatu Koskensilta <aatu.koskensilta at xortec.fi> wrote:

> 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.

The logic proposed by Koskensilta can be put in a larger perspective as
follows.

In modal logic it is natural to distinguish between two types of dependency
relation between states that 'interpret' modal operators. One might call these
(a) *logical* dependence, and (b) *contextual* dependence.

(a) An operator O is logically dependent on all those operators whose semantic
values ('interpretations') are available when choosing a semantic value for O.
This is naturally explicated in terms of the notion of Skolem function
(suitably extended): an operator depends on exactly those operators whose
interpretations are allowed as arguments of the Skolem function corresponding
to O. Usually Skolem functions are defined for existential quantifiers, but
the notion makes obvious sense for disjunctions and diamonds (existential
modal operators) as well.

In usual modal logic any diamond, or any disjunction, is dependent on
precisely all syntactically preceding boxes (universal modal operators), and
conjunctions.

In 'independence-friendly' modal logic further possibilities are allowed in
addition to this default option: a syntactic device is introduced, which
allows to indicate that some operator is dependent on fewer syntactically
superordinate operators, instead of being dependent on all of them. In other
words, that the corresponding Skolem function has fewer arguments.

(b) The semantics of modal logic is relational: a state s' interpreting an
operator O is always required to satisfy a condition of the form R(s,s'),
where R is a relation specified by the semantics of O, and s is a state
introduced earlier in the evaluation process. In fact in usual modal logic
such a state s is always introduced by the unique modal operator that
syntactically precedes O, or if none exists, then s is the initial state of
evaluation.

If I have understood Koskensilta's proposal correctly, in his logic one has
more freedom in indicating relative to which state s the interpretation of an
operator is chosen: it may be the one introduced by the immediately preceding
operator, but likewise, eventually, any earlier state introduced in the
evaluation.

Hence the scope modification Koskensilta is speaking of would be a
modification in the relation of contextual dependency. The other possible type
of modification concerns logical dependencies and would be implemented by an
independence-friendly modal logic.

Allowing to express arbitrary contextual dependencies is a trademark of so-
called hybrid logics, studied by Patrick Blackburn, Balder ten Cate and many
others. However, hybrid logics seem not to have been studied in a general
mathematical framework of the kind sketched by Koskensilta.

The two types of dependency, (a) and (b), are clearly conceptually distinct,
and in fact can be shown not to be interdefinable.

To the best of my knowledge, no one has up to now systematically studied
independence-friendly modal first-order logic; Hintikka however has several
suggestions related, in particular, to IF epistemic first-order logic, e.g. in
his book Principles of Mathematics Revisited (1996).
```