FOM: subjunctive conditionals

[Robert Black]

In reply to Michael Detlefsen I am tempted to write pages on the exact
nature of and relations between material conditionals, English indicative
conditionals and English subjunctive conditionals.  However, I shall
restrict myself to some remarks bearing pretty directly on his claim that
we need to use subjunctive conditionals for a full account of Goedel's
theorems.

(Any FOMer who wants to consult the immense philosophical literature on the
general topic could start with the collection _Conditionals_ edited by
Frank Jackson - not to be confused with the *book* by Jackson with the same
title.)

First note that if our only reason for believing a material conditional to
be true is our belief in the falsehood of its antecedent (truth of its
consequent), it would be misleading to assert the conditional rather than
the stronger claim that the antecedent is false (consequent true).  So
although a material conditional says nothing about any connexion between
antecedent or consequent, we would only normally assert it if we knew of
some such connexion.  Being able to prove the consequent from the
antecedent is about as good a connexion as you can have.

Also note that English indicative conditionals are normally only used when
we are in some doubt about the truth of the antecedent, whereas subjunctive
conditionals make sense even when we are sure the antecedent is false.

Suppose now that in the context of the Hilbert program we have found a
finitary proof of the material conditional: if G is provable in PA then PA
is inconsistent.  We have however no finitary proof of the consistency of
PA, so if we stick to rigorous finitism we are unwilling to assert that
*since* PA is consistent G is not provable in PA.  We are quietly confident
that PA is in fact consistent, so we naturally express our discovery in the
subjunctive:  If G *were* provable in PA, PA *would be* inconsistent.  But
all we have actually proved is the material conditional.

Since it's still (in theory) epistemologically (though not, one hopes, in
reality) open that someone might provide a proof of G in PA, and since
combining that proof with ours would give a proof that PA is inconsistent,
of course we are happy to say that in that event we would no longer believe
PA consistent.  But ultimately the only mathematical content here is the
presence of a finitary proof of a material conditional.

On the usual semantics of subjunctive conditionals (somewhat simplified),
'If it were the case that A then it would be the case that C' comes out as
true iff C is true at all those possible worlds which are as similar as
they can be to the actual world except that A is true at them.  On this
account any subjunctive conditional with an impossible antecedent comes out
as vacuously true.  (Admittedly, 'If there were a largest prime p, then
p!+1 would be prime' is *assertible*, whereas 'If there were a largest
prime p, then there would be six regular solids' is not, but assertibility
‚ truth.  See the discussion of these examples in David Lewis -
_Counterfactuals_  pp.24-6.)  But on this semantics, whenever S is not
provable in PA, the fact that it's not provable is surely necessary, so
*any* subjunctive conditional 'If S were provable in PA then...' will come
out as true.  Hence it is completely natural that people have tried to make
sense of what Detlefsen says by providing a narrower semantics for the
subjunctive conditional; it would have to be one where the subjunctive
conditional 'If it were that A it would be that C' doesn't just come out as
automatically true when 'A & not-C' is necessarily false.  And the only way
I can see to do this is indeed to talk about the derivability of C from A
in some suitably restricted formal system.  The obvious candidate here, as
Vladimir Kanovei pointed out, is PA itself, and in that case Bill Tate's
example answers Detlefsen's question.  (It's a better example than
Detlefsen's own neg-Cons(PA), since Bill's S is true.)  One could try some
other restriction (relevance logic?), but if what I've said above about how
subjunctive conditionals are not really needed is true, what's the point?

[Incidentally, it's not that I think subjunctive conditionals have nothing
to do with f.o.m.  They seem to me essentially involved in the
constructivist account of 'every number has a successor' (i.e. whatever
number you *were* to construct you *would* be able to construct a successor
to it) and the intuitionistic conditional ( a proof of 'if A then C' is a
construction such that if you *were* to find a proof of A then you *would*
be able to apply it and produce a proof of C).  Just how these subjunctive
conditionals are to be understood is well worthy of discussion, and note
that for the second example we get the intuitionist justification of ex
falso sequitur quodlibet iff subjunctive conditionals with impossible
antecedents come out as true.]

Robert Black
Dept of Philosophy
University of Nottingham