[FOM] Mathematical explanation

Richard Heck rgheck at brown.edu
Mon Oct 31 21:17:02 EST 2005

Murphy wrote:

> Can you tell me what proposition the utterance "3+4=7" expresses when
> it is, say, being uttered by a mathematician and is expressing a
> necessary proposition?

It expresses the proposition that 3+4=7. And, as many people have
argued, Searle's example does nothing to show that /that/ proposition
might have been false.

Suppose I say instead that it expresses the proposition that, if there
are 3 Fs and 4 Gs, and no F is a G, then there are 7 F-or-Gs. I think
that proposition is both true and necessary. Is there an argument to the

> I also wrote:
> The point, as I read it, is that utterances of "3+4=7" can be true or
> false while its literal meaning remains the same....
> To which Richard responded:
> ...I've argued above that the example fails to make that point. But
> let's assume I'm wrong. Then the  question at issue is what, even if
> Searle were correct, would follow about the necessity of *the
> //proposition expressed* by "3+4=7" on various occasions of utterance.
> I am conceding, for the moment, that this sentence may express
> different propositions on different occasions. [...]
> Me:
> It is not merely a matter of making explicit unarticulated semantic
> constituents in order to transform an incomplete proposition into a
> complete one. Take the non-mathematical example again, "The ink is
> blue."  Firstly, what is incomplete about the proposition it seems to
> express, that the ink is blue? But perhaps, in order to be more
> precise, we say, "The ink is blue on the page." Is this any more
> complete?  I don't know, but let's assume it is, and that now we have
> a complete proposition--that the ink is blue on the page. 

There are many open questions here, such as:
(i) Is there really context dependence in such cases as Travis's "The
ink is blue"? I think, myself, that many of these examples are doubtful.
Some of them rest upon simple ignorance of basic syntax. Some of them I
find simply unconvincing. Some of them can be explained as implicatures.
And any example that, like this one, involves color terms I find
especially dubious. Since the semantics of color terms is so utterly
unclear, it's hardly surprising that we don't know what to say about
such cases. But there's been much recent work, and things are improving.
(ii) If there is context dependence in such cases, can it be traced to
syntactic constituents of the sentence? In many cases, there are
plausible and independently motivated analyses that would deliver a
"yes". Whether all cases of true context-dependence can be so resolved
is an open question. There's research strategy there that is being pursued.

> But then, is the ink blue on the page under natural light, or
> ultraviolet light, or etc.?

This slip is very revealing: The ink might /look/ different colors under
different lighting conditions, but surely you do not really think that
changing the lighting conditions changes the ink's color. Nor, as we
shall now see, do you think that changing contexts changes whether 3+4=7.

The question at issue is whether it is necessary that 3+4=7. To show
that it is not, one has to show that 3 plus 4 might not have been 7. My
point was that it does no good to show that "3+4=7" might express
different propositions on different occasions. It does no more good to
show---conceding yet again the "radical contextualist" point, for the
sake of argument---that "we have to appeal to a series of assumptions
[about] context to determine the truth value of the proposition". Part
of the problem here is that radical contextualists insist upon using the
term "proposition" in a way that is very non-standard and so liable to
confuse matters in so far as they fail to understand the differences
between their own usage and standard usage. But the points can all be
made without using the notion of proposition.

Suppose we are in a context in which "That ink is blue" can truly be
uttered. (It's the color on the page that matters, etc., etc.) Suppose
someone instead says "That ink might have been black". Maybe that's true
and maybe it's false. But the fact that there are *other* circumstances
in which the sentence "That ink is black" could truly have been uttered
(it's the appearance in the bottle that matters, etc, etc) is
irrelevant. But that's *exactly* how Searle's example is alleged to bear
upon the necessity of the proposition that 3+4=7. Hence the example is
equally irrelevant to the question of necessity.

> ...I doubt that anything of interest in the philosophy of language was
> definitively settled during the 60's and 70's.

The preceding just reworks old points of Kaplan's very slightly. Perhaps
we should show his work greater respect.

I know this has already gone too far into philosophy of language, but
let me add one point. I think one can generate near-fatal worries for
Searle's view in a similar fashion. Suppose Bob says, "That ink is
blue". Suppose context shifts and he now says "That ink is not blue".
Has Bob contradicted himself? Have his beliefs changed? Not necessarily,
and certainly that's not how Searle wants us to take the example. It's
the context that's shifted, not Bob's beliefs.

So we need to be able to record what is constant in Bob's cognitive
state, and we need to be able to record the fact that he is not
contradicting himself. If Searle denies us the resources to do so---and
he seems to be doing so, if he thinks the same "complete" proposition is
expressed in both contexts by "That ink is blue"---then so much the
worse for Searle. If he does not---if he agrees that the sentence is
used to express different beliefs on the two occasions---then I do not
see why we should not say that the sentence expresses different
"propositions" on these occasions, in which case radical contextualism
has been abandoned. (Note that it's utterly irrelevant whether there is
some context-independent way we theorists can say what proposition Bob
believes. Failure to appreciate this point is pretty common in the

Richard Heck

Richard G Heck, Jr
Professor of Philosophy
Brown University
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