[FOM] The influence of Leibniz on Russell
Allen Hazen
allenph at unimelb.edu.au
Sun May 13 21:49:23 EDT 2007
John McCarthy asked:
>"In Russell's book on Leibniz, he remarks on Leibniz's determined
>rejection of predicates with more than one argument and considers it
to be the major inadequacy of Leibniz's ideas. Does anyone
understand why Leibniz did that?"
Rough and approximate, but I think it's something like this. Leibniz
didn't want to be a REALIST in the scholastic sense: didn't want to
believe in universals (as things "separated" from their instances).
So he wanted to be able to explain (away) apparent references to
universals in term of the particular, concrete, things which are
their instances: to say "Red is a color" means "Red things are
colored things" (roughly!).
And this sort of explaining away is harder in the case of relations:
features of the two individuals related to each other may go part
way, but they don't seem to make the LINK between the two.
The late David Lewis's philosophy had points of comparison with
Leibniz's. In his "Parts of Classes" (Blackwell, 1986) he argues
that (at least in the actual world) spatio-temporal relations are the
only irreducible "external" relations between distinct objects. (An
external relation is one that does not supervene on the monadic
characteristics of the relata-- "resembles" is the paradigm
NON-external relation.) "Is thinking about"? No, because this
reduces to the thinker's mental image (part of the intrinsic, or
monadic, character of the thinker) and its quasi-semantic relation to
the thought-of, where the quasi-semantic relations can be explained
in terms of resemblance and causal history. "Causes"? No, because
as Hume argued causation is a matter of the spatio-temporal
regularities in the occurrence of similar events. Leibniz would, I
think, have wanted to explain away relations in roughly this sort of
way. He went beyond Lewis in thinking that spatio-temporal relations
could be explained away in a similar manner, as derivative from the
monadic features of things that a non-Leibnizian would say are caused
by (and so reflect) the fact that one object passes close by another.
Note that Leibniz did NOT have First-Order Logic in mind as a
canonical notation. I think it is wrong to say that he was against
using predicates of arity > 1. He was happy to say that two objects
were related because they POSSESSED accidents (=monadic properties)
that MIRRORED each other. There are two dyadic predicates here, but
neither expresses an "external" relation between distinct objects:
one is a relation between an object and its accidents, the other can
be more complicated than simple resemblance, but is, like
resemblance, an "internal" relation. This general pattern for
explaining relations COULD have inspired (I have no evidence that,
historically, it did) the proof that the theory of two equivalence
relations is a reduction class for the Entscheidungsproblem. (I'm
not sure where this proof comes from originally; I found it in notes
by M. Rabin and D. Scott.) Let one equivalence relation relate each
object to its accidents (and to nothing else). Let the other have
the objects as unit cells, and otherwise relate "mirroring" accidents
of distinct objects. In this way an arbitrary model of the theory of
a symmetric irreflexive relation can be embedded in a model of the
theory of two equivalences.
Allen Hazen
Philosophy Department
University of Melbourne
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