# [FOM] A Query about Logical Content

Ken Gemes K.Gemes at bbk.ac.uk
Thu Sep 20 04:07:48 EDT 2007

```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

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