[FOM] category mistakes

[Hartley Slater]

Dean Buckner writes (FOM Digest, Vol 4, Issue 31) writes:

>Rendering what is grammatically (i.e.logically) a sentence by an
>expression that is grammatically a referring phrase, namely rendering "7 is
>a prime number" as
>
>  is_a_prime_number(7)
>
>Prof. Hartley Slater has (as I understand) discussed the error underlying
>this move at great length on FOM..

The example Dean provides suggests it is the whole of 
'is_a_prime_number(7)' which is the referring phrase in question, 
rather than just (one way of reading) the part before the '(7)'.  The 
standard predicate logic symbolism for elementary sentences, 'Px', 
abbreviates them, and in so doing assimilates them to strings of 
referring terms which could be read 'the property P, x'.  On that 
interpretation supposedly predicative expressions like 'is a prime 
number' become referential.  It is on account of this that people 
(like Russell) have seen problems about the unity of the proposition 
(i.e. the generation of Bradley's regress), and it is also, I 
believe, the origin of Frege's difficulty with the concept Horse.

The abbreviation conflates predicates involving finite forms of the 
verb, like 'shaves', or 'is a shaver', which are descriptive, with 
forms that involve gerunds/present participles, such as  'shaving', 
'being a shaver', which are referential.  There is no difficulty with 
the unity of 'John shaves' like there is with 'John, shaving', and 
once one recognises that 'is a horse' is not referential there is 
naturally no difficulty about the concept it refers to - the 
associated concept is referred to by a different expression, 'being a 
horse'.  The grammatical category mistake materialises most commonly, 
and perhaps most pointedly, when one tries to read a second-order 
quantification, like '(EP)Px' as 'there is a property P which x has' 
.  For to get a comprehensible reading in English one has to insert 
the 'has', turning the 'P' in 'Px' from what it was in the '(EP)', 
namely a referential form, into what it should be, namely a 
descriptive form like 'has P'.  Another informal device sometimes 
used is such readings is to say 'there is a property P such that x 
Ps' - the added 's' giving something like the move to a finite verb 
form.

These points are quite standard in one sense, although they may not 
be too well known amongst mathematical logicians.  A predicate 
nominaliser was introduced in the classic history of Logic 'The 
Development of Logic' by W & M Kneale, Clarendon, Oxford, 1962, 
p601f, and William Kneale later made use of the corresponding 
sentence nominaliser in  'Propositions and Truth in Natural 
Languages', Mind 81, 1972, pp225-43, see also Susan Haack's 
'Philosophy of Logics', C.U.P. 1978, p150f.   It was essentially 
Kneale's analysis of the Liar which I gave in response to Hodges (FOM 
Digest Vol 4, Issue 10), although I made it more sophisticated, and I 
think more accurate, by also using propositional epsilon terms.  For 
other details see my 'Prior's Analytic Revised' Analysis 61.1, 2001, 
pp86-90.  People who think of 'logic' as just the mathematics 
surrounding the predicate calculus will maybe have difficulty 
adjusting to these extensions of the object language - and also to a 
notion of truth which allows that language to be semantically closed 
without contradiction, unlike in Tarskian semantics, and so to be 
unproblematically its own meta-language.  But 'Natural Language 
rules: OK' , even if appropriate abbreviations and symbolisations 
need to be constructed to reveal all the mathematcal mechanisms which 
underlie our (very subtle and complicated) natural speech.
-- 
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