[FOM] natural language and the F of M

[Hartley Slater]

Robert Williams writes (FOM Digest Vol 4, Issue 16):

>  >Harvey Slater glosses: `(nx)Fx' as `there are exactly n Fs'
>  >
>  >This seems grammatical for count predicates e.g.
>  >
>  >              (nx) horse (x)
>  >             there are exactly n horses
>  >
>  >But is it even grammatical in the case of mass expressions? e.g.
>  >
>  >              (nx) water(x)
>  >(???)    there are exactly n waters.
>  >

I do not have any problem with the latter form of speech.  Those 
kinds of expression are simply false.  As a result they would not 
come up very often in adult life, maybe, since the separation between 
mass and count nouns is so basic, but in a teaching situation in 
early life one could surely easily say 'there are not two snows, 
Alfred, there are two patches of snow', or 'there are not zero snows, 
Alfred, it's merely that there is no snow'.

The latter point is significant in connection with Cripsin Wright's 
attempt to save Fregean/Neo-Fregean definitions of the number series 
- using a definition of zero, and Hume's Principle - from becoming 
historical curiosities, along with von Neumann ordinals, and the like 
(see my own posting in FOM Digest Vol 4, Issue 16), once the 
count/mass distinction is fully incorporated into Logic.  In 'The 
Reason's Proper Study' (Oxford 2001, p315), as mentioned before (FOM 
Digest Vol 4, Issue 12), Wright tries to be more constructive than 
Dummett, Tiles, and Hale over the issue of mass terms.  Perhaps, 
having set up Hume's Principle for sortals, there is a possibility of 
including Nx:Fx for non-sortals this way: if Nx:(Fx.Px)=Nx(Fx.Qx) for 
all appropriate sortals 'P' and 'Q', couldn't that be Nx:Fx, if 'F' 
is non-sortal? In the one case that matters that might seem to work, 
since Nx:(Fx.Px) = 0, if 'P' is sortal, and ~(Ex)Fx.

Not so, since, as before, there might be no snow (~(Ex)Sx), but not 
zero snows ((0x)Sx; Nx:Sx = 0).  That means that ~(Ex)Fx only entails 
Nx:Fx=0 if 'F' is sortal, so there is no general equivalence; and to 
show that Nx:~(x=x)=0 from ~(Ex)~(x=x), for instance, as a result, 
would require showing that the specific predicate in question was 
sortal.  But Wright has himself provided a clear proof that 'x=x' is 
not sortal: 'white' is non-sortal, 'man' is sortal, and so 'white 
man' is sortal, since the relation 'non-sortal with sortal makes 
sortal' holds generally.  But if 'x=x' were sortal that would mean 
that 'white and identical with itself' was sortal, whereas it is 
equivalent to 'white', which is, of course, non-sortal.

How is zero to be defined, if not simply through non-existence?  The 
definitions of the exact numerical quantifiers from 1 upwards are 
unproblematic, so if 'm' ranges from1 upwards, 'F' is count iff 
(Em)(mx)Fx v [~(Ex)Fx.M(Em)(mx)Fx].  It is the second disjunct which 
defines '(0x)Fx', giving a simplified definition of count terms via 
'(En)(nx)Fx', where 'n' varies from 0 upwards.

ps, I am honoured to be thought to be 'Harvey' (Sandy Hodges also 
made the same mistake).  Whatever the resemblance, however, my 
preferred name is 'Hartley'.

-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html