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A New Theory of Content II: Model Theory and Some Alternatives
Ken Gemes

Birbeck College

University of London

k.gemes@bbk.ac.uk

N.B. The pagination of this piece differs from that in its published form in Journal of Philosophical Logic, 26:449-476, 1997

1. Review of the Syntactical Specification of Content


Gemes (1994) offered two equivalent syntactic definitions of so-called basic (logical) content for a generic propositional language L~v&→, hereafter L. The prime motivation for the theory of content was to construct a notion of consequence that would not allow arbitrary disjunctions to count as part of the contents of their disjuncts; for instance, to disallow '(pvq)' from counting as part of the content of 'p'. Gemes (1994) was framed in terms of "basic content" rather than simply "content" because otherwise it would have yielded an account according to which disjunctions of content parts did not count as content parts. I have come to consider this result acceptable, perhaps even desirable.1 So for the remainder of this paper I will treat Gemes (1994) as if it gave an account of content rather than basic content.
Let '', 'ß' and 'ø' be variables for wffs, and, were specified, sets of wffs, of the propositional language L; '' stand for the classically defined relation of being a syntactical consequence; and dnf be a canonical Boolean disjunctive normal form of arbitrary (contingent) L wff  in the essential vocabulary of . For instance, presuming 'p', 'q' and 'r' are atomic wffs of L, ((pvq)&(rv~r))dnf is '(p&q)v(p&~q)v(~p&q)'.2 Then the following, using 'ß├c' to abbreviate ' is a (syntactically specified) content part of ß' and ‘ß├/c’ to abbreviate ' is not a (syntactically specified) content part of ß’, is a variant of one of Gemes (1994)'s definitions of content for wffs of L
(D1) ßc =df (i) and ß are contingent, (ii) ß├, and (iii) each disjunct of dnf is a sub-conjunction of some disjunct of ßdnf.3

In (D1), while '' ranges over single wffs, the variable 'ß' may be interpreted as ranging over both single wffs and sets of wffs. So (D1) may be taken as given both the contents of wffs and the content of theories, where theories are taken to be sets of wffs. From here on we shall generally exclude mention of the notion of (D1) and subsequent definitions giving the content of theories and concentrate on the notion of the content of wffs. For more on the content of theories see Gemes (1993).


Some examples of content relationships for wffs of L:
p├cp p├/c(pvq)

p├c(p&(rv~r)) p&(pvq)├/c(pvq)

(p&(rv~r))├cp (p&q)├/c(pvq)

(p&q)├cp (pq)├/c(p→q)

(p&q)├cq (p&q)├/(pq)

((p&q)v(r&s))├c(pvr) p&(qvr)├/cqv(p&r)



~p&(pvq)├cq q├/c(p→q)
As noted in Gemes (1994) this type of definition amounts to saying that  is a content part of ß, iff  and ß are contingent and  is the strongest consequence (up to logical equivalence) of ß constructible using just the essential atomic wffs occurring in .4 Furthermore such definitions readily admit of a mechanical decision procedure [Cf. Gemes (1994), p.606 and pp.610-611].

2. A Model-Theoretic Definition of Content for L.


We may easily create a model-theoretic analog of (D1). Such accounts are read of from (D1) since it makes use of disjunctive normal forms and such forms have obvious model-theoretic analogs.

Each disjunct of the dnf form of a wff  may be seen as corresponding to an "-relevantly specified model of ", or, more simply, an "-relevant model of ". For arbitrary L wff , an -relevant model of  is a model of  that specifies values for all and only those propositional variables whose truth values on some model or other are relevant to the truth value of .


For instance where  is '(pvq)', dnf is '(p&q)v(p&~q)&(~p&q)'. Here the first disjunct corresponds to that -relevant model of  which assigns 'p' the value true and 'q' the value true and makes no other assignments (to propositional variables). The second disjunct corresponds to that -relevant model of  that assigns 'p' the value true and 'q' the value false and makes no other assignments. The third disjunct corresponds to the -relevant model of  where 'p' is assigned the value false and 'q' true and makes no other assignments. Now suppose ß is '(pvq)&r'. Then ßdnf is '(p&q&r)v(p&~q&r)v(~p&q&r)'. The first disjunct of ßdnf corresponds to the ß-relevant model of ß which assigns 'p', 'q' and 'r' the value true and makes no other assignments. The second disjunct of ßdnf corresponds to the ß-relevant model of ß which assigns 'p' and 'r' the value true, and 'q' the value false and makes no other assignments. The third disjunct of ßdnf corresponds to the ß-relevant model of ß which assigns 'p' the value false and 'q' and 'r' the value true makes no other assignments. Thus we have the following table summarizing the various relevant models for our designated values for  and ß:
-relevant model of  ß-relevant models of ß

1. p:T q:T 1. p:T q:T r:T

2. p:T q:F 2. p:T q:F r:T

3. p:F q:T 3. p:F q:T r:T

Now given our specified values of  and ß,  is a content part of ß according to (D1). According to (D1), where  is a content part of ß each disjunct of dnf is a sub-conjunction of some disjunct of ßdnf. In other words, and as seen in the above table, where  is a content part of ß each -relevant model of  is a sub-model of some ß-relevant model of ß, or, equivalently, each -relevant model of  can be expanded to a ß-relevant model of ß.
Here already we have the makings of a model theoretic account of content. To make it more precise we need to give an exact account of what, for any wff  an -relevant model of  is. To do this we need to introduce the notion of a partial interpretation and some other notions.
A partial interpretation of L is a valuation function which assigns to at least one propositional variable of L one of the two truth values true (T) and false (F) and assigns to no propositional variable of L both T and F.
We can take a full interpretation of L to be a partial interpretation that assigns a truth value to every propositional variable of L. Truth under a full interpretation is defined in the usual way.5 Where wff  is true under full interpretation P we say P is a model of .
A propositional variable ß is relevant to wff  iff there is some model P of  such that there is some interpretation P' which differs from P in and only in the value P' assigns to ß and P' is not a model of .

An (full or partial) interpretation P' is an extension of partial interpretation P iff for any propositional variable ß, if P assigns T(F) to ß then so does P'. Where interpretation P' is an extension of interpretation P we may say that P is a sub-model of P'.


(D2.1) P is an -relevant model of wff  of L=df P is a partial interpretation of L such that P assigns values to each of and only those propositional variables relevant to  and is such that for any full interpretation P' which is an extension of P, P' is a model of .
Now, assuming the classical relation of semantical consequence (╞) and using 'ß╞c' to abbreviate ' is a (semantically specified) content part of ß' and ‘ß/cto abbreviate ' is a not a (semantically specified) content part of ß', we are at last in a position to give a model theoretic definition of content part for wffs of L:
(D2) ß╞c =df  and ß are contingent, ß╞, and every -relevant model of  has an extension which is a ß-relevant model of ß.
Alternatively, we might say that ß╞c, where  and ß are contingent, ß╞, and every -relevant model of  is a sub-model of some ß-relevant model of ß.
According to (D2), for instance, '(pvq)' is not a content part of 'p', since that '(pvq)'-relevant model of '(pvq)' which assigns 'p' the value F and 'q' the value T and makes no other assignments has no extension which is a 'p'-relevant model of 'p'.
According to (D2), for instance, 'p' is part of the content of '(p&q)'. There is only one 'p'-relevant model for 'p', namely that which assigns T to 'p' and makes no other assignments and that model has an extension to a '(p&q)'-relevant model of '(p&q)', namely that which assigns 'p' the value T and 'q' the value T and makes no other assignments.
More generally, (D2) yields the same results as (D1) since we have the correctness and completeness theorem:
Theorem 1: For any wffs  and ß of L, ß├c iff ß╞c.
Proof: Cf. definitions (D1) and (D2) and note that (i) each -relatively specified model of  uniquely characterizes and is uniquely characterized by a unique disjunct of dnf, (ii) each ß-relatively specified model of ß uniquely characterizes and is uniquely characterized by a unique disjunct of ßdnf and (iii) an -relatively specified model m of  has an extension which is a ß-relatively specified model m' of ß iff the disjunct of dnf which characterizes m is a sub-conjunction of the disjunct of ßdnf that characterizes m'.
The classical consequence relation is both reflexive and transitive. The content relation defined by (D2) and (D1) is clearly reflexive for any contingent wff , thus we have
Theorem 2: For any contingent wff  of L, ╞c.
Proof: Trivial.
Gemes (1994) made fairly heavy going of a proof of transitivity. (D2) makes for a transparent proof of transitivity which applies equally for definitions for more sophisticated languages than L.
Theorem 3: For any wffs , ß and ø of L, if ß╞c and ╞cø then ß╞cø
Proof: Assume ßc and cø. Then ß, , and ø are all contingent; ß and ø; every ø-relevant model of ø is a sub-model of some -relevant model of ; and every -relevant model of  is a sub-model of some a ß-relevant model of ß. So, ßø. Further, since the relation of being a sub-model is transitive, every ø-relevant model of ø is a sub-model of some ß-relevant model of ß. So ßcø.
The content relationship allows for substitution of classical equivalents. In other words
Theorem 4: For any wffs , ß, σ, and  of L, where ╡╞ and σ╡╞, ß╞c iff ╞cσ.
Proof: This follows from the fact that any two classically equivalent wffs  and ß share the same relevant propositional variables and hence, being equivalent, a partial interpretation P is an -relevant model of  iff P is a ß-relevant model of ß.

3. Content for a Generic Quantification Language Without Identity


In Gemes (1994) a syntactically based definition of content for wffs of a generic quantificational language without identity was briefly considered.
Consider the generic classical quantificational language L'. The vocabulary of L' is limited to an infinite stock of individual constants, 'a', 'a1', 'a2', ... ; an infinite stock of individual variables 'x', 'x1', 'x2, ... ; an infinite stock of predicate letters of varying degrees (one-placed, two-paced, etc.), 'F', 'G', 'H', 'F1, G1, H1, ... ; the sentential connectives '~', '&', 'v', '→' and '≡'; and the grouping indicators '(' and ')'. The wffs of L' are formed from the elements of L's vocabulary in the usual ways. The notions of derivable consequence (), semantical consequence (╞), contradiction, tautology, atomic wffs, basic wffs, free variables and closed wffs are defined as usual. Hereafter, unless directly specified, we shall use the term 'wffs' to refer to closed wffs. We shall use the Greek letters '', 'ß', and 'ø' and subscripted variants as meta-variables ranging alternatively over open wffs, closed wffs, quantifiers and individual constants of L', relying on specific indications and/or context to indicate the range in particular cases.
To define content for wffs of L we make use of the purely propositional fragment of L and the resources of propositional infinitary logics applied to that fragment of L. In particular, we allow for wffs of infinite length.
To form Dev() for arbitrary L' wff  (1) Put  into prenex normal form, (2) Where ø1.....øn A is the resultant wff, with ø1.....øn being a string of n, n0, quantifiers and A being a quantifier free wff, where n>0 eliminate the left-most quantifier øn, and (i) if øn is a universal quantifier replace A with an infinite conjunction such that each conjunct is the result of replacing in A all occurrences of the variable governed by øn by a given constant of L' and such that for each constant of L' there is one and only one such conjunct, or (ii) if øn is an existential quantifier replace A with an infinite disjunction such that each disjunct is the result of replacing in A all occurrences of the variable governed by øn by a given constant of L' and such that for each constant of L' there is one and only one such disjunct, (3) keep repeating step (2) until a quantifier free wff, being Dev(), results.
Let dnf, that is the canonical disjunctive normal form of , be the result of putting Dev() into a canonical Boolean disjunctive normal form in the essential vocabulary of Dev().
Then our old definition
(D1) ß├c =df (i)  and ß are contingent, (ii) ß├, and (iii) each disjunct of dnf is a sub-conjunction of some disjunct of ßdnf,
serves to define content for wffs of L'. (D1) applied to wffs of L' yields the result that 'Fa1' is a content part of '(x)Fx'. The dnf of 'Fa1' is 'Fa1' and the dnf of '(x)Fx' is 'Fa1&Fa2& ..... & Fan &.... &' and clearly each disjunct of the former - we take 'Fa1' to be a disjunction with a single disjunct - is a sub-conjunction of the
Some examples of content relationships for wffs of L':
(x)FxcFa1 (x)Fx/c(x)(FxvGx)

(x)(Fx&Gx)c(x)Fx (x)Fx/c(x)Fx

(x)(Fx&Gx)c(Fa&Gb) Fa/c(x)Fx
Since the (D1) definition of content applied to wffs of the quantificational language L' makes use of the classical notion of consequence applied to quantificational wffs it is not generally decidable. However the content relation, like the classical consequence relation, is decidable for the purely monadic fragment of L'.
Where  and ß are monadic wffs of L' we can use standard procedures to check whether  and ß are contingent and whether ß. If one of  or ß is not contingent or ß/ then  is not a content part of ß. Otherwise, to determine whether  is a content part of ß (1) form L'- from L' by deleting from L' all individual constants save those occurring in  or ß and then adding any two extra constants appearing in neither  or ß; then (2) form Dev(ß/L'-) and Dev(/L'-) according to the recipe given above for forming Dev(ß) and Dev() substituting in the recipe all reference to L' with reference to L'-; then (3) form dnf:L'- and ßdnf:L'- by putting Dev(ß/L-) and Dev(/L-), respectively, into a canonical Boolean disjunctive form in their essential vocabulary; then (4) check whether each disjunct of dnf:L'- is a sub-conjunction of some disjunct of ßdnf:L'-. If so then  is a content part of ß otherwise it is not. Note, since both dnf:L'- and ßdnf:L'- are both finite disjunctions of finite conjunctions of atomic wffs and negated atomics wffs step (4) is mechanically decidable.

This decision procedure for the monadic case suggests a method for defining content for all wffs of L' which does not make recourse to the use of infinitary logics. To see this imagine that we restrict dnf and ßdnf as defined above to finite subsets of atomic wffs occurring in dnf and ßdnf. More particularly, where ø is any arbitrary wff of L' and σ is any non-empty set of atomic wffs of L' including at least one atomic wff occurring in ødnf, let ødnf:σ be ødnf less all conjuncts containing atomic wffs not occurring in σ. For instance (x)~Fxdnf:{Fa,Fb} is '~Fa&~Fb'. Then we might define content for wffs of L' as follows


(D1.1) ß├c =df (i)  and ß are contingent, (ii) ß, and (iii) for any finite set σ of atomic wffs of L' including at least one atomic wff occurring in dnf, each disjunct of dnf:σ is a sub-conjunction of some disjunct of ßdnf:σ.
Since σ of (D1.1) is restricted to finite sets of atomic wffs in every case dnf:σ and ßdnf:σ are wffs of finite length.
(D1.1) applied to wffs of L' is equivalent to (D1) applied to wffs of L'.



  1. A Model-Theoretic Definition of Content for L'.

We may with a bit of work create a model-theoretic analog of (D1) for wffs of L'. To do this we need to construe the quantifiers of L' substitutionally and construct our model theory accordingly. In particular, an interpretation or model for L' will be an assignment to each of the (closed) atomic wffs of L' of one and only one of the truth values T and F. A universally quantified wff  will count as true in a given model of L' iff each substitution instance of  counts as true in that model, and an existentially quantified wff will count as true in a given model of L' iff some substitution instance of it counts as true in that model.


On such a reading each disjunct of dnf for arbitrary L' wff  will correspond an "-relevantly specified model of ", that is, a model of  that specifies values for all and only those atomic wffs whose truth values on some model or other are relevant to the truth value of 

.

Let us try to be a little more precise.


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 L both T and F.
We take a full interpretation of L to be a partial interpretation that assigns a truth value to every atomic wff of L'. Truth under a full interpretation is defined in the usual way with the appropriate substitution type clauses for quantificational wffs. Where wff  is true under full interpretation P we say P is a model of .
A (full or partial) interpretation P' is an extension of partial interpretation P iff for any atomic wff ß, if P assigns T(F)to ß then so does P'. Where P' is an extension of P and P is not an extension of P' we say P is a proper sub-interpretation of P'.
An atomic wff ß is relevant to wff  iff for some partial interpretation P, P assigns a value to ß and for every full interpretation P', if P' is an extension of P then P' is a model of , 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 .
(D2.2) P is an -relevant model of wff  of L' =df P is a partial interpretation of L' such that P is assigns values to each of and only those atomic wffs relevant to  and is such that for any full interpretation P' which is an extension of P, P' is a model of .
Given these preliminaries our old definition
(D2) ß╞c =df  and ß are contingent, ß╞, and every -relevant model of  has an extension which is a ß-relevant model of ß,
serves for wffs of L'.
According to (D2), for instance, 'Fa' is a content part of '(x)Fx'. The only 'Fa'-relevant model of 'Fa' is that which assigns 'Fa' the value T and makes no other assignments. By adding only the assignment T to every other wff of the form 'F' that model may be extended to an '(x)Fx'-relevant model of '(x)Fx'.

The proof of Theorem 1 above counts equally as a correctness and completeness proof for definitions (D1) and (D2) applied to wffs of L'. Thus we have


Theorem 5: For any wffs  and ß of L', ß├c iff ß╞c.
Similarly, the reflexivity, transitivity and substitution proofs of theorems 2, 3 and 4 work for definition (D2) applied to wffs of L'. Thus we have
Theorem 6: For any contingent wff  of L', ╞c.
Theorem 7: For any wffs  and ß of L' if ß╞c and ╞cø then ß╞cø
Theorem 8: For any wffs , ß, σ and ø of L', where  ╞σ and ß╞ø, ß╞c iff ø╞cσ.
Note, while in constructing our model theoretic definition of content part for wffs of L' we have construed the quantifiers of L' substitutionally, we may otherwise give them a fully objectual reading. That is to say, one can read, for instance, '(x)Fx' as stating that every object in the domain of quantification has property F, while using recourse to a substitutional reading in order to determine the content parts of '(x)Fx'. While this dual treatment may seem infelicitous I believe that a model theoretic definition of content which eschews the use of the substitutional interpretation can be constructed along the following lines suggested by Philip Kremer.
Let M be any standard objectual denumerable model of arbitrary wff  of L'. Where D is the domain of M and E is any finite sub-domain of M we construct (E) as follows: (1) form L'+ from L' by supplementing L' with a set I of new individual constants containing one new constant for each member of D not assigned by M to some individual constant of L', (2) form M+ from M by adding to M an assignment of each element of D not assigned by M to an individual constant of L' to a unique individual constant of I, (3) construct Dev() by replacing  with its non-quantificational equivalent in L+ using the recipe given in Section 3 above, substituting reference to L'+ for all references to L', (4) form Dev( (E)) from Dev() by removing from Dev() each atomic wff containing a constant which M+ assigns to an individual that is not a member of E - where every atomic wff occurring in Dev() contains some constant assigned by M+ to some individual that is not a member of E, Dev( (E)) is {/}, then (5) construct  (E) by putting Dev( (E)) in a canonical Boolean disjunctive normal form in the essential vocabulary of Dev( (E)) - where Dev( (E)) is {/} then  (E) is {/}.
We might say that where M is a model of wff  with domain D, and E is a sub-domain of D,  (E) tells us in a propositional form all the truth wholly about E contained in . We then have the preliminary definition:
(D2.2) ß╞c relative to model M of  and ß =df  and ß are contingent, ß╞, and for every finite sub-domain E of M's domain D, where  (E) is not {/}, ß(E)├c (E) by (D2).

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