[FOM] higher order logic, Slater

Hartley Slater slaterbh at cyllene.uwa.edu.au
Wed Oct 1 23:26:55 EDT 2003

Randall Holmes (FOM Digest Vol 10 Issue 1) has widened the discussion 
of numbers to higher order logic generally.   But one has, for a 
start, to be more careful than Holmes (and many others) to 
discriminate predicates.  He says

>In "higher-order logic", we allow quantification over _predicates_.
>So for example we might say
>(E0)(ES)((Ax)(~Sx = 0) & (Axy)(Sx = Sy -> x=y)),
>capturing the idea that there is a sequence with some of the properties
>of the sequence of natural numbers.
>The most common way to make sense of higher-order logic is to
>interpret it as a kind of set theory (actually type theory): the unary
>predicate P may be understood as the set {x | P(x)}

The 'S' Holmes illustrates is a mathematical function, and while that 
does bear comparison with a linguistic predicate, i.e. something like 
'is a Q', the latter is not a referring phrase, and in particular 
"the unary predicate is a Q may be understood as the set {x|x is a Q}"
"the unary predicate 'is a Q' may be understood as the set {x|x is a Q}"
makes sense.  Quantification over linguistic items like 'is a Q' has 
to be substitutional.

In line with his mention of 'properties', what Holmes is maybe 
wanting is the property of being a Q to be understood as the set {x|x 
is a Q}.  But, if so, what he is wanting to be understood as a set 
cannot be formalised in standard second-order predicate logic, since 
that incorporates no nominalising process by which to formulate the 
referring phrase to the property.  For more on nominalised 
predicates, see, for instance, Nino Cocchiarella's 'Logical 
investigations of predication theory and the problem of universals' 
Bibliopolis, Napoli 1986; Distributed in the U.S.A. and Canada by 
Humanities Press, Atlantic Highlands, N.J..  Without nominalised 
predicates there cannot be objectival quantification over what they 
refer to, of course, so the fact of the matter is that standard 
second order predicate logic cannot involve predication over 
properties, and these are mis-understood if it is thought there are 
elements in that logic which might replace them.  I have said a good 
deal more about these kinds of thing in 'Concept and Object in 
Frege', Minerva Vol 4, see http://www.ul.ie/%7Ephilos/vol4/frege.html.
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 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

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