Dear Ed, (01)
I'll try to make my answer very brief. (02)
1. We seem to try to catch more or less the same fourfold distinction, but
I think we are talking each other by because both of different preferred
terminology and different implicit ontological commitments. I will now try
to present my view in your terminology, but resttricted to discourses
about the spatiotemporal world. I think the fourfold distinction Q1-Q4 is
the result of combining two independent dichotomies: (03)
A: individuals vs. classes of individuals
B: type of class vs. subtype of class (04)
As I interpreted your distinctions the following holds true: (05)
Q1 = individuals of (the most specific?) subtype of class;
Q2 = individuals of type of class (such as those that make up dimensions
in the SI system?);
Q3 = subtype of class;
Q4 = type of class. (06)
2. For two reasons I consciously gave only an informal presentation of the
axioms. I am not sure what kind of logic you are willing to accept, and I
am not a logician. However, you seem to have misunderstood the axioms A3
and A4. A3 is meant to make additions such as '5 kg + 7 m' and '4 s + 8
A' impossible. A4 is meant to say, for instance, that one and the same
particle cannot have two determinate masses, two determinate volumes, two
determinate electric charges, etc. (07)
Ingvar J (08)
> ingvar_johansson wrote:
>> I agree that a fourfold distinction as yours below between Q1-Q4 is
>> necessary. However, to me you are using 'category' in an unusual sense,
>> and I think that in order to avoid confusion one should distinguish
>> between two senses of 'being an instance of', and then use different
>> terms
>> for them. On the one hand we have 'an instance' as a spatiotemporally
>> located individual entity, this I will call 'an instance'; on the other
>> hand we have 'an instance' as a determinate of a determinable (red is an
>> instance of color, 14 kg is an instance of mass, etc.), this I will call
>> 'a determinate'. In my opinion, VIM confuses these two senses.
>>
>>
> I didn't realize that I was using "category" in an unusual sense. I
> used the term to express the inverse of the subsumption relationship: A
> 'is a category of ' B iff B subsumes A. And in that I meant to imply
> that A is a unary relation (a class, the sense of "category" that I
> assume is "usual") in its own right, i.e. that A(x) is true if the
> individual x has the characteristics that define A and false otherwise.
> I would be happy to change all occurrences of 'is a category of' to 'is
> a subtype of'. I am less comfortable using 'is a kind of', because I
> think it has an ambiguous model.
>
> I never use "'an instance' as a spatiotemporally located individual
> entity". I prefer the term "individual" for things that appear in the
> domain of discourse (which is a potentially larger population than
> "spatiotemporally located individual entity"), in order to avoid
> precisely the confusion you identify. Further, I hope I only used
> "instance" in the relation "is an instance of", with the
> interpretation: x 'is an instance of' A iff (x is an individual in the
> domain of discourse) and A(x) is true. You may use "determinate of a
> determinable" for that relation if you please, but I doubt that it will
> be easily comprehended by much of our audience.
>
>> Furthermore, the fourfold distinction in question applies even to
>> non-quantified properties, and I think that to bring this out can help
>> to
>> clarify the issues at hand. Let me exemplify with the determinable
>> 'shape'
>> and one of its determinates, 'spherical'. A necessary requirement for a
>> quantification is that all the determinates of certain determinable can
>> be
>> and are linearly ordered, and this has not yet been done for the shape
>> determinable.
>>
>> Q1-spherical = the spherical instance of ball A at t, the spherical
>> instance of the planet B at t, etc. That is, Q1 = determinate property
>> instances.
>>
>> Q2-shape = the shape instance of ball A, the shape instance of the
>> planet
>> B, etc. That is, Q2 = determinABLE property instances.
>>
>>
> The examples are confusing, but the definition "Q2 = determinABLE
> property instances" is what I meant. I think the example of Q2-shape is
> "spherical" or "sphere", as distinct from "prismatic solid". So "ball A
> at t" is a 'determinate' that 'is an instance of' the 'determinable'
> "sphere". Thus the 'individual' "sphere" 'is a subtype of' Q1, and the
> same 'individual' "sphere" 'is an instance of' Q2.
>> Q3-spherical = the determinate universal 'spherical' if you are a
>> realist,
>> and the equivalence class of 'spherical property instances' if you are a
>> resemblance nominalist. That is, Q3 = determinate universal or
>> determinate
>> equivalence class.
>>
>> Q4-shape = the determinable universal 'shape' if you are a realist, and
>> the equivalence class of 'shape property instances' if you are a
>> resemblance nominalist. That is, Q4 = determinABLE universal or
>> determinABLE equivalence class.
>>
>> And here are some axioms:
>>
>> A1: Necessarily, if a Q1-instance (e.g., a spherical instance) then also
>> a
>> Q2-instance (shape instance).
>>
>>
> No. This is not an axiom I mean. What I want is:
>
> (forall (x D)(if (and (Q2 D) (D x))
> (Q1 x)
> ))
> Every instance of Q2 is a subtype of Q1.
>> A2: Necessarily, if a Q2-instance (e.g., a shape instance) then also an
>> instance of a determinate of Q2 (instance that is spherical, or square,
>> or
>> star-formed, etc.).
>>
> Again. I don't understand this axiom. What I mean is:
>
> (forall (x)(if (Q1 x)
> (exists (D) (and (Q2 D) (D x)))
> ))
> Every individual that is an instance of Q1 is also an instance of some
> determinable D that is an instance of Q2.
>> A3: Only quantified determinates of the same determinable can be added
>> and
>> subtraced (in a physically menaingful way).
>>
>>
> The VIM term is "comparable" ("added and subtracted" is perhaps too
> specialized a notion).
> But the axiom is stronger than that:
>
> (forall (x y)(iff (exists (D) (and (Q2 D) (D x) (D y)))
> (comparable x y) ))
> x and y are comparable if and only if there is some determinable D in Q2
> such that x and y are both instances of D.
>
>> A4 (for many but not all determinables): Two Q2-instances cannot
>> possibly
>> exist in the same spatiotemporal region.
>>
> I am going to assume I don't understand this, because what I understand
> it might be intended to mean is false.
>
> Assuming the relation (st-location x r) means x has a spatiotemporal
> existence in region r, then we probably do mean (although I am not sure):
>
> (forall (x)(if (Q1 x)
> (exists (r) (st-location x r)) ))
> Every instance of Q1 has some spatiotemporal location. Every measurable
> instance is physical.
>
> What I think Ingvar means is:
>
> (forall (x y rx ry)(if (and (exists D)(and (Q2 D) (D x) (D y)))
> (st-location x rx)
> (st-location y ry))
> (not (= rx ry)) ))
>
> If two 'determinates' (measurable instances) x and y are both instances
> of a common 'determinable', their spatiotemporal locations cannot be the
> same.
>
> But I don't think that is true, at least not of what I called Q2. For
> example, the width of a given box and the height of the same box are
> both instances of a Q2 (determinable) called "length". Do we really
> mean that those properties are not coincident in space-time? That seems
> to me to be an issue in philosphical geometry. ;-)
>
> -Ed
>
> --
> Edward J. Barkmeyer Email: edbark@xxxxxxxx
> National Institute of Standards & Technology
> Manufacturing Systems Integration Division
> 100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
> Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694
>
> "The opinions expressed above do not reflect consensus of NIST,
> and have not been reviewed by any Government authority."
>
> (09)
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