Context-dependence and equivocation
Dretske denies closure because he believes that the
zebra case is a counterexample to it: You know that the animal is
a zebra but you do not know that is it not a painted mule, even though
you know that being a zebra entails not being a painted mule. He
argues this in terms of what contrast-alternatives are respectively relevant
to the first two of those knowledge ascriptions. (In this handout
I shall follow Stine in understanding closure as an inference form rather
than as a thesis. Accordingly, I shall capitalize the "c" henceforth.)
Stine charges Dretske with "some logical sin akin
to equivocation" (p. 151):
If the relevant alternatives, which have after all to do with the truth or falsity of the premises and conclusion, cannot be held fixed, it is hard to see on what basis one can decide whether the argument form [(E) on p. 150] is valid or not. And if the set of relevant alternatives is one thing for the first premise and another for the conclusion, how do we determine what it is for the second premise, and how does this affect the truth of the second premise?Stine puts that gently, in terms of mere uncertainty, but it can be sharpened as by Heller on pp. 197-98 of his article:
John knows(-modulo-R) that the animal is a zebra.
John knows [the entailment].
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\ John knows(-modulo-R') that the animal is not a painted mule.
(Let's pass over Stine's question of how the relevant-alternative parameter
applies to the second premise, and to logical knowledge generally.)
But, Heller asks, how does this differ from offering
his (A), (B) and (C) (about the human and the boat) as a counterexample
to "conjunction" or &-Introduction? The latter move would be
absurd, because when we test an argument for validity, we perforce assume
that there is no equivocation as between premises and conclusion.
If the reference of one of the terms shifts between premises and conclusion,
the test is off--especially if it's admitted on all sides that the reason
the conclusion went false is precisely that a term's reference shifted.
The argument in question is not an instance of &-Introduction in the
first place, and so cannot counterexample it.
And exactly the same thing has happened in argument
(E) as represented above: The relevant-alternatives parameter has shifted
its reference from the ordinary set R to a much expanded set R'.
So there is no counterexample to Closure, for (E) is not an instance of
Closure in the first place.
(Something is suspicious here. If (E) is not
a counterexample to Closure, then it would seem Dretske can make the judgments
he does about the three knowledge ascriptions and now join the majority
of epistemologists and embrace Closure too. That would be a surprisingly
happy and easy outcome for him.)
Heller then does a clever thing. By switching
from the RA version of contextualism to a twiddled version of Nozick's
subjunctive-conditional version (ERA), he evades the equivocation charge.
Although the set of worlds relevant to evaluating one of the relevant conditionals
is a different set from the one relevant to evaluating the other, this
is not equivocation. Because, as he says (p. 200), the sets of worlds
are selected by a Stalnaker-type uniform "similarity" semantics for counterfactuals
that remains in place throughout. They are not referred to
at all. In ERA there is no analogue of a shifting parameter; nothing
changes its reference between premises and conclusion of a suitably translated
version of (E). (Rather, the work is begun by the counterfactuals'
antecedents specifying respective types of world, and then all the rest
is done by the uniform similarity relation.) He gives the simpler
example of Antecedent-Strengthening failure (pp. 199-200): Premise
is true and conclusion false, because a larger set of worlds is selected
by the conclusion (W)'s antecedent than by the premise (L)'s, but this
is not equivocation because the worlds are not themselves parametrically
referred to and nothing shifts its reference as between (L) and (W).
Well and good, but I think this is not really a
big advantage over RA. RA itself has what I think is a pretty good
reply to Stine's equivocation charge. To get the idea, let's stick
with Antecedent-Strengthening and now consider the subjunctives interpreted
according to a different semantics (a semantics more like Nozick's own
and also strangely similar, though not identical, to that presented in
my book Real
Conditionals (Oxford U.P., 2001)):
Understand "P > Q" as being true iff, in every P
world within a certain reference-class R, Q. "R" is again a parameter
referring to a set of worlds, so the spectre of equivocation appears again.
"R" is governed by pragmatic rules that determine, given an utterance context,
which worlds are in R in that context. The rule that is pertinent
here is the "Antecedent Requirement": that in any context, R must contain
at least one world in which the conditional's antecedent holds. (The
intuitive justification for that Requirement is that the whole function
of a conditional antecedent is to call attention to a particular possibility
and place it on the table for discussion.)
On this semantics, it is the Antecedent Requirement
that invalidates Antecedent-Strengthening. (L) is true because in
all the nearest worlds in which I strike the match, it lights. (W)
is false because in some of the nearest worlds in which I strike the match
and
it is wet, it does not light. (This assumes that any world in
which I strike the match and it is wet is farther away than is any of the
nearby worlds in which I strike the match and it does light. I don't
strike wet matches.) Similar reasoning would show why Heller's Nozickian
conditionals would behave as he says they do on our new parametric semantics
as well as on his Stalnaker-type semantics.
But on the parametric semantics, there is parameter
shift between premises and conclusion, so the equivocation charge applies,
as regards Antecedent-Strengthening as well as Closure.
Here is my reply, still in terms of Antecedent-Strengthening:
Yes, the premise has RL as its parameter while the conclusion
has RW, and the two are distinct. In this technical sense
the point is correct; the inference of (W) from (L) is, very strictly
speaking, not an instance of Antecedent-Strengthening. But that
very strict sense of "instance" is neither specified nor intended in logic
textbooks that present Antecedent-Strengthening as a valid form of inference.
What students and professional philosophers have always been told is that,
barring equivocation or overt indexicals, arguments of the sentential surface
form A > B / \ (A & C) > B are valid arguments, period.
And that is what is refuted by Heller's example. One can continue
to insist that Antecedent-Strengthening is valid for the strict sense of
"instance," but at the price of keeping us from telling easily and uncontroversially
when a set of ordinary English sentences is an "instance" of an argument
form. So I think it is better to use "instance" in the ordinary
loose and popular sense.
Parallel remarks apply to the RA approach to (E).
Yes, in a very strict sense of "instance," (E) is not an instance of Closure.
But in the looser and more colloquial and more useful sense, it still is,
and correspondingly, the zebra example interpreted in Dretske's way is
a counterexample to Closure.
And now we see what more exactly was suspicious
about Heller's equivocation objection to Dretske. Dretske could indeed
make the judgments he does about the three knowledge ascriptions and join
the majority of epistemologists and embrace Closure too, but only by denying
that (E) is an "instance" of Closure. In the strict sense, (E) is
not one. But in everyone else's looser and more intuitive sense,
it is, and Closure is still refuted if Dretske's judgments are right.
(Notice that inference-form names such as "Antecedent-Strengthening" and
"Closure" inherit the same strict/loose ambiguity from "instance."
Strict Closure remains intact, but Loose Closure, which is what everyone
means by "Closure," is counterexampled.)