[FOM] Wittgenstein?

Tue Apr 22 23:08:49 EDT 2003

Charles Silver suggests we fudge the use-mention distinction (FOM 
Digest Vol 4 Issue 23):

>Philosophers seem to think they are making profound
>points by blathering on about such stuff.  I see no foundational
>significance here and do not believe the current fussing is in any
>way fruitful.

But not only is the contingent-necessary distinction obscured if ''If 
p then 'p' is true' is conflated with 'if p then it is true that p', 
as I explained in the posting Silver objects to (FOM Digest Vol 4 
Issue 22), more confusion arises that way in connection with 
understanding Goedel's Theorems, and specifically with understanding 
their relevance to the question of what distinguishes us from Turing 
machines, as I explained previously in FOM Digest Vol 3 Issue 18. 
These are profound points, even if 'fussing' is certainly needed to 
appreciate them.  Fussing has a better name: it is 'taking care'.

Silver's suggestion, of course, illustrates Wittgenstein's general 
thesis very well: 'Mathematical Logic has completely deformed the 
thinking of mathematicians and philosophers, it has done so by 
erecting an entirely superficial interpretation on the sentences of 
ordinary speech'.  The historical aspects of Wittgenstein's thought 
are not my speciality (on that, see, for instance, Mathieu Marion's, 
'Wittgenstein, Finitism and the Foundations of Mathematics', 
Clarendon, Oxford, 1998, which was reviewed by Juliet Floyd in 
'Philosophia Mathematica' 10.1, Feb 2002, pp67-87).  But I have been 
prepared to defend, independently, this overriding attitude of 
Wittgenstein's in my recent book, and specifically I have provided a 
re-assessment of Goedel's Theorems in the above way: check chapters 3 
and 4 of 'Logic Reformed', as before available via
http://www.peterlang.com/all/index.cfm?vSuche=vSuche&vDom=3&vRub=3060

Readers of FOM have also seen how greater attention to the mass-count 
distinction affects the relevance to the foundations of Mathematics 
of von Neumann ordinals and the like (FOM Digest Vol 4 Issue 16), and 
Fregean and Neo-Fregean attempts to generate the number series from 
Logic using a definition of zero (FOM Digest Vol 4, Issue 17).   I 
will now show how Hale and Shapiro's attempts to produce a 
Neo-Fregean account of the reals are also defeated through attention 
to other aspects of natural language predicates - clearing up a 
number of other confusions, and the Paradox of Predication on the way:

The confusions start with a confusion between facts and properties, 
which are commonly distinguished in natural languages, but which have 
come to be conflated in lambda abstraction and set abstraction 
languages.  Isn't the property, in the Russellian Paradox case, 
[lambda]x(x is not a member of x), which is a constant?  No: the 
predicate is 'is not a member of itself', so it is that, and not 'x 
is not a member of x' that expresses the property, which, therefore, 
is not constant, but functional.  It is not a property but a fact if 
x is a member of its complement.  Another case might help seeing 
this, since it is even more obviously unnatural to conflate actions 
with facts.  Abstracting 'John' from 'John shaves John' does not 
produce the action John does, but simply the form of a fact about 
him.  The action of shaving oneself is not the fact that one shaves 
oneself: the fact that one shaves oneself instead speaks of ones 
performance of the action.  Certainly those people with the property 
in question are those people for whom the same form of fact holds, 
i.e. (x)(x shaves himself iff x shaves x).  But that does not mean 
that each of those people has the same property, i.e. that the 
property in question is not functional.  Indeed, the equivalence 
itself is what gives the form of the function in this case.

The difficulty with seeing this compounds if one is attached to the 
'Px' symbolisation of elementary sentences in Fregean Logic, for if 
'P' stands for a property, and one abstracts 'x' from 'Px', then what 
is left but the property?  If we wrote instead 'x has P' for the form 
of an elementary predication it would be more clear that abstracting 
the subject leaves the possession of the property instead.  In the 
symbolism 'Px' the 'P' is doing double duty for an expression 
involving a finite form of the verb, like 'shaves', along with a 
referring phrase to the property alone, like 'shaving', which would 
allow objectival second-order quantification.  But, of course, on the 
latter reading of 'Px', 'x, the property P' is not a coherent 
sentence. I have pointed out, in Chapter 8 of 'Logic Reformed', how 
distinguishing predicates from their nominalisations relieves Frege 
of his problem with the concept Horse.

That, of course, also relieves us of one difficulty with the paradox 
of predication: '(EP)(x=P.~Px)' is ungrammatical.  But if one 
introduces, for instance, a nominaliser '*', so that '*P' is 
referential while 'P' remains descriptive, that would still allow a 
related expression, '(EP)(x=*P.~Px)', to be well formed.  Then, 
however, we would have the question, as before, whether this 
predicate expressed a single, unambiguous property.  Indeed, if 
(EP)(x=*P.~Px) iff Qx, then there would be a contradiction when x=*Q. 

How is it that '(EP)(a=*P.~Pa)', and '(EP)(b=*P.~Pb)', do not say a 
and b have the same property?  It is well known that 'Sa.Tb' does not 
entail '(ER)(Ra.Rb)': we could readily construct counter-examples 
like: Sa.Tb.(R)(Ra -> ~Rb), ~Sb. ~Ta ...  Nevertheless, 'Sa', of 
course, entails '(EP)Pa', and 'Tb' entails '(EP)Pb', and so it might 
easily seem that any such a and b must share a property: the property 
of simply having a property.  Certainly we can say 'a has a property' 
and 'b has a property', and so produce two sentences with the same 
grammatical predicate, and we can even say 'a and b each has a 
property'.  But as the latter expression indicates, the property 
referred to which a has might still be different from the property 
referred to which b has.  The epsilon reduction of the second-order 
quantifiers above also shows this since '(EP)Pa' is '[ePPa]a', and 
'(EP)Pb' is '[ePPb]b' (where the square brackets are merely for ease 
of reading).  The epsilon terms contain traces of the subject, 
allowing differentiation of the two properties, and showing  that 
'(EP)Pa.(EP)Pb' does not entail '(ER)(Ra.Rb)'.  The point can 
obviously be replicated with '(EP)(a=*P.~Pa)' and '(EP)(b=*P.~Pb)': 
although these may seem to say the same thing about a as about b, the 
sameness is only in the linguistic expression, which is 
systematically ambiguous.  The grammatical predicate here might be 
represented as a lambda expression '[lambda]x(EP)(x=*P.~Px)', but, as 
before, that only gives the form of a fact, not a property. 

Now in Stewart Shapiro's 'Frege meets Dedekind; a Neologicist 
Treatment of Real Analysis' (Notre Dame Journal of Formal Logic, 
41.1, 2002, p343) we find a definition of a property Q which 
supposedly is going to generate the least upper bound of a set of 
reals; it compares with Bob Hale's definition of H, in 'Reals by 
Abstraction', ('Philosophia Mathematica' 8.2, 2000, p112): Hx iff 
(EF)(phi(#F).Fx).  Here, '#F' is a cut in the rationals, and 'F' a 
property of the rationals related to it by means of the abstraction 
principle 'Cut':  #F = #G iff (x)(Fx iff Gx).  The question these 
Neo-Fregeans have not asked is whether there is just one property of 
the supposed kind H, i.e. whether 'H' is univocal.  For (EF)Fa, and 
(EF)Fb predicate different things of a and b, as we have seen. The 
predicate 'has a property of the Phi sort' contains an indefinite 
article, which might be the linguistic antecedent to anaphoric uses 
of the definite article in discriminations between 'the property of 
the Phi sort which a has' [eF(phi(#F).Fa)] and 'the property of the 
Phi sort which b has' [eF(phi(#F).Fb)].  So while the same syntactic 
item is involved it is context sensitive with respect to its 
referent.  Specifically, the indefinite article is sensitive to the 
subject provided in expressions like 'has a property of the Phi 
sort', as before.  An epsilon reduction of the second-order 
quantifiers also shows this context sensitivity: thus the given 
lambda term above can also be written '[lambda]x(x=*Q.~Qx),', where 
Q=eP(x=*P.~Px).

Take care.

-- 
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