[FOM] A Query about Logical Content

[Ken Gemes]

Dear List Members,

Some time ago I published some articles about the notion of logical content.  The basic idea is that (where A and B are atomic wffs)  AvB should not count as part of the logical content of A. Note, if AvB counts as part of the logical content of A then on the evidence of B one would have to say that part of A is true, and hence A is partially true. There are many other reasons for not counting AvB as part of the logical content of A. 

The basic idea was to say X is part of the logical content of Y iff Y logically entails X and every relevant model of X can be extended to a relevant model of Y.

A relevant model of X is a partial valuation function V which assigns values to all and only the atomic wffs relevant to X and is such that every full extension of  V is a model of X. An atomic wff Z is relevant to X iff there are assignments A and A’ of truth values to atomic wffs such that A and A’ differ only in their assignment to Z and one and only one of assignments A and A’ makes X true.

So  (where A and B are atomic wffs)  AvB is not part of the content of A because that relevant model that assigns A the value F and B the value T cannot be extended to a relevant model of A. But (where A and B are atomic wffs) A is part of the content of A&B since A&B logically entails A and the sole relevant model of A, namely that which assigns A the value T and makes no other assignments, can be extended to a relevant model of A&B.

To handle quantificational wffs I simply treated the quantifiers substitutionally. So a relevant model for (x)Fx (where Fx is an atomic predication) is one that assigns T to Fa, Fb, Fc, etc.  So Fa is part of the logical content of (x)Fx  since (x)Fx entails Fa and  the only relevant model of Fa, namely that which assigns T to Fa and makes no other assignments, can clearly be extended to a relevant model of (x)Fx . But FavGa is not part of the logical content of (x)Fx since that relevant model that assigns Fa the value F and Ga the value T cannot be extended to a (relevant) model of (x)Fx.

For a quantificational languages with identity the notion of relevant wff has to be defined with more complications.  Let L be a standard quantificational language with an identity operator then 

A partial interpretation of L is a valuation function which assigns to at least one atomic wff of L one of the two truth values true (T) and false (F) and assigns to no atomic wff of L both T and F and fulfils the following conditions regarding identity statements
(i) No identity statement ┌a=a┐ is assigned the value F.
(ii) For any identity statement ┌a=ß┐, if ┌a=ß┐ is assigned 
       a value then ┌ß=a┐ is assigned the same value.
(iii) For any pair of identity statements ┌a=ß┐ and ┌ß=ø┐ if ┌a=ß┐ and ┌ß=ø┐ are assigned T then so is ┌a=ø┐.
(iv) If an atomic wff S is assigned a truth value V and ß is a constant that occurs in S, then for any individual constant ø, if ┌ß=ø┐ is assigned the value T, then the  
     value V is assigned to any atomic wff S' which differs from S only in that one or more occurrence of the individual constant ß is replaced by ø.

An atomic wff ß is relevant to wff a iff for some partial interpretation P, every full interpretation P' that is an extension of P is a model of a and each such extension of P assigns the same value to ß, and there is no proper sub-interpretation P'' of P, such that for every full interpretation of P''' that is an extension of P'', P''' is a model of a.

Given these definitions the old definition of content, that X is part of the content of Y iff Y logically entails X and every relevant model of X can be extended to a relevant model of Y, holds. Again we here treat the quantifiers substitutionally.

QUESTION:  Does anyone have any idea about how to give an elegant wholly objectual account of this notion of logical content for a quantificational language with an
                    identity operator?

If you need further specification of the problem please feel free to e-mail me at k.gemes at bbk.ac.uk or look at my piece on logical content at http://www.bbk.ac.uk/phil/staff/academics/gemes-work/NewTheoryContentII.doc

Yours,

Ken

Ken Gemes

Reader in Philosophy
School of Philosophy
Birkbeck College
University of London
Malet Street
London, WC1E 7HX

Reader in Philosophy
Philosophy
School of Humanities
University of Southampton
Southampton, S017 1BJ

web page: http://www.bbk.ac.uk/phil/staff/academics/gemes/