Posts on: Frege
Frege argued that number concept are, in the first place, second-order predicates. When we talk about numbers as objects, we use a logical device of "nominalization" that introduces object-level representations of higher-level properties. In Grundgesetze, he assumed that every first-order predicate can be nominalized: for every first-order predicate F, there is an associated object – the "extension" of F – such that F and G are associated with the same object iff ∀x(Fx ↔︎ Gx). The number N is then identified with the extension of 'having an extension with N elements'. Unfortunately, the assumption that every predicate has an extension turned out to be inconsistent, so the whole approach collapsed.
Suppose we want to follow Frege and distinguish an expression's
denotation from its sense. Suppose also we take the
denotation of a predicate to be its extension: the set of its instances. The following argument
appears to show that this leads to trouble.
- All humans are featherless bipeds, and all featherless bipeds are
human, but there could have been featherless bipeds that are not
human. In short, (Ax)(Hx <-> FBx) & <> (Ex)(~Hx & FBx)).
- By existential generalisation over the predicate positions, it
follows that (EX)(EY)((Ax)(Xx <-> Yx) & <> (Ex)(~Xx &
Yx)).
- If things in predicate position denote sets of individuals, this
can be read as: there is a set X and a set Y such that X and Y have
the same members and it is possible for something to be a member of Y
and not of X.
- But if X and Y have the same members, then they are identical; and then
nothing could belong to "one of them" without also belonging to "the
other".
- Hence things in predicate position do not denote sets of
individuals.
The argument is modeled on a brief passage (p.13) in Tim
Williamson's latest
paper on the Barcan Formula. Williamson there argues against the
plural interpretation of second-order quantifiers. On this
interpretation, the sentence in (2) can be read as "there are things
xx and things yy such that all xx's are yy's and all yy's are xx's and
it is possible for something to be one of the yy's but not of the
xx's". Williamson objects that if the xx's just are the yy's,
then it is not possible for something to belong to "the ones" without
also belonging to "the others".
When I prepared for my exam, I noticed something curious.
Richard Heck, in "The Julius Caesar Objection", claims that
In a letter to Russell, Frege explicitly considers adopting
Hume's Principle as an axiom, remarking only that the 'difficulties here'
are not the same as those plaguing Axiom V [p.274 in Language, Thought
and Logic].
The claim is repeated by Crispin Wright and Bob Hale in the introduction
to The Reason's Proper Study (p.11f., fn.21). The letter Heck,
Wright and Hale refer to is xxxvi/7 from July 1902.
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:
If nothing goes terribly wrong, I will finish the Frege paper tomorrow. Though I'm not sure if it's really the same Frege paper I mentioned previously.
Initially I just wanted to put together all the comments on
Fregean thoughts and Rieger's paradox that I had already posted to this
weblog. That looked like a cheap way to get a termpaper. For some reason
however the paper has now evolved into a discussion about the prospects and
dangers of developing a semantics that can be applied to its own
metalanguage.
Frege believes that predicate expressions have semantic values (Sinne and
Bedeutungen) which can't be denoted by singular terms. Hence "the
Bedeutung of 'is a horse'" does not denote the Bedeutung of 'is a horse'.
Before the discovery of Russell's paradox, the only reason he ever gave for
this view -- apart from claiming that it is a fundamental logical fact that
just has to be accepted -- is that otherwise the semantic values of a
sentence's constituents wouldn't "stick together". The more I think about
this reason, the less convincing I find it.
Frege uses second-order quantification in both his formal and informal
works. So far, I have always assumed that his second-order logic is
standard second-order logic. But couldn't it also be second-order logic
with Henkin semantics, which would in fact be a kind of first-order logic
(compact, complete and skolem-löwenheimish)? Unfortunately, I know far too
little about second-order logic to answer this question.
Are there any second-order statements that are satisfiable in standard
semantics, but not in Henkin semantics? (I guess there must be: Wouldn't
second-order logic with standard semantics have to be complete otherwise?
Not sure.) If so, do any of Frege's theorems belong to these?
Apple was very quick shipping the (free) replacement adapter.
I've decided to bring order into my thoughts about Fregean thoughts by
writing a little paper. If all goes well, I'll hand it in as the termpaper
required for my MA. Since my last entry on this topic, I've found out that
there is a lively discussion among Frege scholars about the structure of
thoughts. Some, in particular Dummett, argue that Frege is, or should be,
committed to this view:
In my last posting, I argued that to escape the cardinality problem
for thoughts Frege perhaps has to give up
1) For any things there is at least one concept under which all and only
those things fall.
Now (1) is clearly false if, as I think, all there is are objects --
that is, if it makes sense to quantify over absolutely everything. But if
not, as Frege thinks, denying (1) is not an option. A concept is a
function from things to truth values. Given that functions are not
themselves things, how could there fail to be such functions?
A while ago, I was discussing Adam Rieger's alleged paradox in Frege's
ontology (here, here, and here). I'm still confident that the Russellian
version of the paradox can be blocked. But on second thought, the
cardinality version of the paradox appears to be much more difficult. Here
it is again.
1) For any things there is at least one concept under which all and only
those things fall.
2) For each of these concepts, there exists the thought that Ben Lomond
falls under it.
3) All these thoughts are different.
4) All thoughts are objects.
From (1)-(3) it follows that there are more thoughts than objects (2^k
if k is the number of objects), contradicting (4).
Yesterday, I argued that Frege can escape Rieger's Paradox if it is allowed that the thought that Fb might equal the thought that Gb (briefly, [Fb]=[Gb]) even if F and G are not coextensive.
In particular, to escape the paradox there has to be a concept F, such that [Fb]=[Ob] even though O([Ob]) and 
F([Ob]). O, recall, is defined thus:
O(x) iff 
F(x=[Fb]
Fx)
I did not say how this F might look like. Here is a good candidate:
In the October issue of Analysis, Adam Rieger presents the following paradox in Frege's ontology.
For any object b and first order concept F, there is the thought [Fb] that b falls under F. Let Con and Obj be functions that yield the (referents of the) constituents of such thoughts: Con([Fb])=F, and Obj([Fb])=b. We stipulate moreover that 'b' shall denote the mountain Ben Lomond, and define O ("ordinary") as follows: