Scientific theories are often said to implicitly define their theoretical
terms: phlogiston theory implicitly defines "phlogiston", quantum mechanics
implicitly defines "spin". This is easily extended to non-scientific
theories: ectoplasm theory implicitly defines "ectoplasm", folk psychology implicitly defines "pain".
The first problem from the mathematical case applies here too: Since all
these theories make substantial claims about reality, their truth is not a
matter of stipulation. For example, no stipulation can make phlogiston
theory true. That's why, according to the standard Ramsey-Carnap-Lewis
account, what defines a term (or several terms) t occurring in a theory
T(t) is not really the stipulation of T(t) itself, but rather the
stipulation of something like its 'Carnap sentence' 
x T(x) 
T(t). All substantial claims in T(t) are here cancelled out by the
antecedent.
I vaguely believe that there are no implicit definitions. So I've decided
to write a couple of entries to defend this belief. The defence may well
lead me to give it up, though. Anyway, here is part 1.
Explicit definitions introduce a new expression by stipulating that it be
in some sense synonymous or semantically equivalent to an old expression.
For ordinary purposes this can be done without the use of semantic
vocabulary by stipulations of the form
This weekend, I've moved into my new
flat, which has both a bath room and a fridge, and also lot's of funny
records from the 1970s.
I often wonder to what extent different theories and approaches in
philosophy of language are conflicting theories about the same matter, or
rather different theories about different matters. For example, some
theories try to describe the cognitive processes involved in human speaking
and understanding; Others try to find systematic rules for how semantic
properties (like truth value or truth conditions) of complex expressions
are determined by semantic properties (like reference or intension) of
their components; Others try to spell out what mental and behavioural
conditions somebody must meet in order to understand an expression (or a
language); Others try to find physical relations that hold between
expression tokens and other things iff these other things are in some
intuitive sense the semantic values of the expression tokens; Others try to
discover social rules that govern linguistic behaviour; and so on. How are
all these projects related to each other?
I'm doing a visual memory test. On the table in front of me are twelve
green and fourteen red apples, and an empty basket. The lights go out, and
the instructor says to me:
"Put all the green apples into the basket". (1)
I try to do what he says. When the lights go on, you, the instructor's
assistant, are given a form on which you are to tick whether I've
correctly or incorrectly fulfilled the task. You see twelve green and two red apples in the basket. What do you tick?
Today I've been reading Hilbert. I must admit that I don't really
understand his view on the foundations of mathematics. It seems to me that
he always confuses truth with consistency. For example, he writes in his
"New Grounding":
If we can produce [a consistency proof of formalised mathematics], then
we can say that mathematical statements are in fact incontestable and
ultimate truths.
Obviously, Hilbert uses "true" in a very unusual way here: Both ZFC + the
Continuum Hypothesis and ZFC + its negation are consistent. Hence, on
Hilberts account, both CH and its negation are "incontestable and ultimate
truths".
A while ago, I
asked: "Could Frege's ontology be a Henkin model?". I now believe that
this question doesn't make sense: A standard model of second-order logic
is a (standard) Henkin model. I should have asked: "Could Frege's
ontology be a non-standard Henkin model?". Even this question is,
uh, questionable, because the late Frege would have certainly rejected both
a standard and a Henkin semantics, as both of these employ singular terms
to denote the semantic values of function expressions. So I should rather
have asked: "Are Frege's logical and semantical theses satisfiable in a
non-standard Henkin model?" But now, I guess, the answer is trivially Yes,
because nothing you can say in higher-order logic rules out a non-standard
Henkin interpretation. However, my question was not meant to be trivial.
I wanted to know whether Frege is comitted to there being more concepts
(values of second-order quantifiers) than objects (values of first-order
quantifiers), a claim that is true in standard models, but not in some
non-standard models of any (really?)* second-order theory. Unfortunately,
this question can't even be asked without violating Frege's semantical
theses. As he himself notes in a letter to Russell:
This is a problem that cropped up several times in my thesis on Lewis,
but which I never seriously discussed.
Lewis argues, or rather, stipulates, that all fundamental ("perfectly
natural") properties are intrinsic. I agree that fundamental extrinsic
properties would be strange. For if a thing x's being F depends on the
existence and the properties of other things, it seems that F-hood should
be reducible to intrinsic properties (and relations) of all the things
involved. Moreover, fundamental properties are supposed to be the basis for
intrinsic similarity between things, and they could hardly be if they were themselves extrinsic.
A problem from Kit Fine, "The Non-Identity of a Material Thing and Its Matter", Mind 112 (2003):
Suppose a certain piece of well made alloy coincides with a certain
badly made statue. Al makes an inventory of well made things. The only entry on his list is "that piece of alloy". Question: Does the entry on Al's list refer to
a badly made thing?
Kit Fine intuits that the answer is definitely "no", irrespective of the
context in which that question is asked. From which it seems to follow
that the piece of alloy and the statue are not identical. At least I think this is what he thinks would follow. Anyway,
here is an extension of the above story where "the entry in Al's list
refers to a badly made thing" appears to be true.
Strolling through the library, I just came across George Tourlakis' Lectures in Logic and Set
Theory. I wouldn't recommend it as a textbook for logic courses in
philosophy, unless you want to torture your students with a full
proof of Gödel's Second Incompleteness Theorem. But it's nice to have
that proof available somewhere. The second volume on set
theory (unfortunately only on ZFC) also looks useful, if only because there
are so few thoroughgoing introductions to set theory.