[FOM] A minor issue in modal logic
Richard Heck
rgheck at brown.edu
Sun Jul 4 17:20:57 EDT 2010
On 07/04/2010 11:40 AM, Michael Lee Finney wrote:
> It would seem to me that Np means that p is necessarily true without
> distinction of world, or that p is true in all possible worlds. In
> either case it would be true in the actual world because in one case
> the world was not distinguished and in the other case surely the
> actual world is possible. So you have Np => Np[a] no matter how you
> look at it.
>
>
In general, this is not true; that is, it is not true when we think of
necessity in very general terms.
In (propositional) modal logic, a model is a quadruple <W,R,V,@> where W
is a set of worlds, R is a relation of "accessibility" between the
worlds, V is a valuation function, which we can think of as a map from
worlds to sentence letters true in those worlds, and @ (a member of W)
is the actual world. A formula "N\phi", commonly written "\Box\phi", is
true at a world W in the model if \phi is true at all worlds that are
_accessible from_ w. One can therefore have "N\phi" be true at a world
without having \phi be true at all worlds; since this can happen at @,
"N\phi" can be true in the model without its being true at all worlds in
the model.
This is important, as it is not obvious that every reasonable notion of
necessity must validate, for example, "N\phi --> NN\phi". Why shouldn't
there be a proposition that is, as things stand, necessary but that, had
things been otherwise, might have been contingent? Why shouldn't there
be a proposition that isn't, as things stand, necessary but, had things
been otherwise, might have been necessary? Anyone who thinks that there
are contingent non-identities but who thinks (with Kripke) that all
identities are necessary holds this latter view.
Moreover, since the accessibility relation R need not be reflexive,
"N\phi" could be true at a world without \phi being true at that world.
Again, this can happen at @, so "N\phi" need not imply \phi, in general.
Of course, one might sensibly argue that any reasonable notion of
necessity must validate "N\phi --> \phi" and so argue that any modal
logic modelling any reasonable notion of necessity must have only models
in which the accessibility relation is reflexive. But that is a
substantive---i.e., not purely logical---claim.
Richard Heck
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