[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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