[FOM] Understanding Universal Quantification

[Hartley Slater]

Allen Hazen writes: "Another [division amongst positions] is over the 
question of whether the "understanding" of universal quantification 
presupposes some sort of delimitation of the domain quantified over". 
There is a technical result in this area which is not too well known, 
which shows the epsilon calculus is not as subject to limitation as 
the predicate calculus.  See, for instance, the last section of my 
entry 'Epsilon Calculi' in the Internet Encyclopedia of Philosophy. 
Briefly:

A convenient form of the epsilon calculus arises simply by modifying 
the predicate calculus truth trees, as found in, for instance, 
Richard Jeffrey's 'Formal Logic: Its Scope and Limits', (McGraw-Hill, 
New York, 1st Ed. 1967). Jeffrey has a rule of existential quantifier 
elimination, (Ex)Fx |- Fa, in which 'a' must be new, and a rule of 
universal quantifier elimination, (x)Fx |- Fb, in which 'b' must be 
old, unless no other individual terms are available. Clearly, upon 
adding epsilon terms to the language, the first of these rules can be 
changed to: (Ex)Fx |- FexFx, (where 'e' is epsilon) while also the 
two parts of the second rule can be replaced by the pair of rules: 
(x)Fx |- Fex-Fx, Fex-Fx |- Fb (where 'b' is invariably old) to 
produce an appropriate proof procedure.

But Jeffrey's rules only allow him 'limited upward correctness' 
(Jeffrey 1967, p167), since he has to say, with respect to his 
universal quantifier elimination rule, that the range of the 
quantification there be limited merely to the universe of discourse 
of the path below. This is because, if an initial sentence is false 
in a valuation so also must be one of its conclusions. But the first 
epsilon rule which replaces Jeffrey's rule ensures, instead, that 
there is 'total upwards correctness'. For if it is false that 
everything is F then, without any special interpretation of the 
quantifier, one of the given consequences of the universal statement 
is false, namely the immediate one, since Fex-Fx is in fact 
equivalent to (x)Fx. A similar improvement also arises with the 
existential quantifier elimination rule....

Not being able to specify the prime putative exception, 'ex-Fx', to 
the universal statement '(x)Fx' leaves Jeffrey believing that there 
must be a model for the quantifiers which restricts them to a certain 
domain, which means that they do not necessarily range over 
everything. But in the epsilon calculus the quantifiers do range over 
everything, and there is no need to specify their range.  This has 
consequences for 'The Domain Principle' found in Cantor (Hallett, M. 
'Cantorian Set Theory and Limitation of Size', Clarendon, Oxford, 
1984 p25), and defended to the last by Priest in his attempt to 
establish dialetheism ('Beyond the Limits of Thought' C.U.P. 1995). 
I have discussed these further matters in Ch7 of my 'Logic Reformed' 
(Peter Lang, Bern, 2002, ISBN 3-906768-57-0; US-ISBN 0-8204-5875-9).
-- 
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