In the dark old days of early logic, there was only syntax. People
introduced formal languages and laid down axioms and inference rules,
but there was nothing to justify these except a claim to
"self-evidence". Of course, the languages were assumed to be meaningful,
but there was no systematic theory of meaning, so the axioms and rules
could not be justified by the meaning of the logical constants.
All this changed with the development of model theory. Now one could
give a precise semantics for logical languages. The intuitive idea of
entailment as necessary truth-preservation could be formalized. One
could check that some proposed system of axioms and rules was sound, and
one could confirm – what had been impossible before – that it was
complete, so that any further, non-redundant axiom or rule would break
the system's soundness.
How much can you say about the world in purely logical terms? In
first-order logic with identity, one can construct formulas like
'(Ex)(Ey)~(x=y)'. But arguably, this doesn't yet mean anything. As we
learned in intro logic, formulas of first-order logic have no fixed
interpretation; they mean something only once we provide a domain of
quantification and an assignment of values to predicate and function
symbols. As it happens, '(Ex)(Ey)~(x=y)' doesn't contain any
non-logical predicate and function symbols, so to make it mean
anything we just need to specify a domain of quantification. For
example, if the domain is the class of Western black rhinos, then the
formula says that there are at least two Western black rhinos.
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:
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?
In §27 of Meaning and Necessity, Carnap announces that all
mathematical concepts can be defined without the use of any class
expressions. The basic idea is to use Frege's system, but to replace all
occurrences of class variables with higher order variables. In particular,
the cardinal number of a property F is defined as the second order property
of being equinumerous to F (definition 27-4). "Thus, for example, '2' is a
predicator of second level" (p.117).