# [FOM] Leibnizian principles

Charlie silver_1 at mindspring.com
Fri Nov 14 14:39:06 EST 2014

```	   Leibniz says different things at different times, but I think the following is well-accepted:  Every object in every world (including ours) is “compossible” with every other in that world.  Every object in any world is distinct from all others in that world, but “reflects” all of the other properties of all other objects in that world. I think it’s a mystery what things Leibniz counts as properties — i.e., one-place relations. Re. Indiscernibles below, every object in any world is distinct in terms of its properties (I don’t know what they are, whether they’re first-order, or anything else about them; as mentioned it’s a mystery).  So, I think the biconditional is correct, as it expresses both of L’s laws. The second principle (“dual”) doesn’t make sense to me in a couple of ways.  The idea of two distinct objects having all their first-order properties, but not all their second-order ones is an interesting thought, though how it would apply to Leibniz I don’t know.  Objects for Leibniz are different from properties, so I don’t understand how they could be “on the same footing”.

On Nov 13, 2014, at 1:32 AM, Alex Blum <Alex.Blum at biu.ac.il> wrote:

> A points position on a line is an essential property of a point despite it being relational, hence it must be counted a property.
> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of F.A. Muller
> Sent: Wednesday, November 12, 2014 9:18 AM
> To: <fom at cs.nyu.edu>
> Subject: [FOM] Leibnizian principles
>
>> Message: 1
>> Date: Tue, 11 Nov 2014 18:39:33 +1300
>> From: W.Taylor at math.canterbury.ac.nz
>> To: W.Taylor at math.canterbury.ac.nz
>>
>> If one doesn't wish to go to a hierarchy of predicate variables, and
>> is prepared (largely ignoring 1st/2nd-order distinctions) to treat
>> objects and properties on the same footing, then Leibnitz' law has an
>> interesting "dual" form, which might be useful.
>>
>> IDENTITY OF INDISCERNABLES:    (Leibnitz)
>>
>> [all x,y]  x = y  <=>  [all P] Px <=> Py
>
> If true, a line has only one point, because all points share their properties.
>
>>
>> IDENTITY OF INDISCRIMINABLES:      (dual)
>>
>> [all P,Q]  P = Q  <=>  [all x] Px <=> Qx
>
> If true, "having a unique successor" is identical to "having a unique prime decomposition" (when x is a natural number variable).
>
> --> F.A. Muller
>
> Sent from my iPhone
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```