[FOM] Possible Worlds

[Allen Patterson Hazen]

It has been clear for a very long time-- at least since I was a student in
the 1970s-- that simple possible-worlds treatments do badly at accounting
for EPISTEMIC modality of the sort in question.  (This despite the fact
that, not long after the beginning of modern possible-worlds semantics in
the 1960s, Hintikka described a possible-worlds semantics for epistemic
and doxastic notions in his influential book "Knowledge and Belief."  I
think he was explicit in saying that the theory was an idealization, not
applicable to mathematical ignorance without refinement.)

(((Vocabulary: "Epistemic" logic is often construed broadly, as covering
knowledge and a variety of related notions.  When knowledge is being
distinguished from (mere) belief, "doxastic" refers to the belief
operators.)))

It is, I think, much harder to model MATHEMATICAL ignorance in a
possible-worlds framework than it is to give an account of the sense in
which, e.g., water MIGHT (have turned out) not (to) be H20.  Kripke
("Naming and Necessity") pointed out that the latter is an epistemic
"might" (and not the "metaphysical" possibility he was primarily concerned
with), but it seems possible to define THIS kind of epistemic possibility:
roughly
      It is metaphysically possible for their to be someone
       whose evidence for a statement of the same logical
       form as "water is H20" is similar to our evidence
       (before modern chemistry) for "water is H20" even
       though their statement is false.
This won't work for mathematical statements if the "evidence" is allowed
to include knowledge of the truth of axioms from which the statement
follows logically.

Many approaches to this problem have been suggested; none to my knowledge
have been worked out in adequate detail:

   (1) Early workers in RELEVANT (or relevance) logics often gave, as a
motivating hope, the possibility (so to speak) of developing an
epistemic logic based on them which would tolerate ignorance of
mathematical and logical truths.  (Since the Routley-Meyer semantics for
relevant logics makes use of "worlds" -- they preferred to call them
"set-ups" in their early papers -- in which logical truths fail, this
suggestion could be seen as connecting to the idea of partial worlds,
though their set-ups could be inconsistent as well as incomplete.)

   (2) Other suggestions about the use of partial or inconsistent
structures instead of full, classical, models as possible worlds have been
made.  One of Hintikka's compatriots (sorry, I've forgotten the exact
reference and will try to give it in a supplementary post if no one else
beats me to it) published a paper in an early (1970s) volume of the
"Journal of Philosophical Logic" proposing that "surface" models be used.
This is a notion derived from Hintikka's work on Distributive Normal
Forms: it would allow "possible worlds" in which sentences
inconsistent by First Order Logic are true, provided that the proof of
their inconsistency exceeds some measure of quantificational complexity.
  (Sorry to give such a vague account of it, but this letter is long
enough as it is!  If you are not familiar with Hintikka's work and are
wondering whether you would be interested enough for it to be worth
following up, think "Herbrand expansions, complexity of.")

    (3) David Lewis, in conversation, in probably the late 1970s or 1980s,
said he was attracted by an idea he attributed to Robert Stalnaker: that
what is really being claimed as possible when someone says "P might be
false" (P a mathematical statement) is the falsity of the SENTENCE: that
there is a possible world in which we so use language that the words which
actually express the (for sake of example, assume true) conjecture express
some falsehood.

    I think the problem is an open one, and perhaps-- given the use of
epistemic logics in AI-- an important one.  Note that approaches (2) and
(3) above, and probably even (1) (though Meyer and Routley
themselves were not Platonists) are consistent with quite extreme forms of
Platonism.  I would expect a satisfactory solution to be so
consistent.

---

Allen Hazen
Philosophy Department
University of Melbourne