# [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
neither
"the unary predicate is a Q may be understood as the set {x|x is a Q}"
nor
"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

```