Inscrutability

Inscrutability of reference is the thesis that we can scramble the referents of singular terms and extensions of predicates freely so long we preserve the distributions of truth values across sentences without thereby affected the adequacy of the theories as theories usable for interpreting object language sentences.

 We can see right away that the crucial issue has to do with the claim that preserving the distributions of truth values across sentences suffices for preserving adequacy for interpretation.  Since we know being true is not ipso facto adequate, it is a mystery why anyone would accept this.  Even if we required that the scrambling be compatible with maintaining the lawlike status of the theorems there is no reason to think it would preserve interpretability.  And even if it preserved lawlikeness there is no reason to think that such a theory would be equally confirmable -- in the first instance -- from the standpoint of the racial interpreter.

 Having said all of that, let us take a quick look at some of the issues that came up in discussion in regard to the use of the permutation function and what you can do with the theories generated from it.

 First, the standard story: Let P(x) be a permutation function.  For  every object x, it maps it to just one other, and every object has some object mapped onto it, and not every object is mapped onto itself.  if P(x) = y we say that y is the image of x under the permutation.  For a set or extension {x, y, z, ...} we say that the image of the set or extension is the set consisting of the images of each member of the first set, {P(x), P(y), P(z),...}.

 A reference scheme is a set of reference and satisfaction axioms for a truth theory.  Let our starting scheme be R.  Take some sample reference and satisfaction axioms.  (I'll use square brackets as Quinean corner quotes, forming a description of a expression in terms of the concatenation of the enclosed elements.)

 R

 1.      Ref('KL')=Kirk Ludwig

2.       For any function f, f satisfies 'phil(x)' iff f('x') is a philosopher.

3.       For any name N, function f, f satisfies [phil(N)] iff ref(N) is a philosopher.

Sample canonical T-theorem.

 4.       'Phil(KL)' is true iff Kirk Ludwig is a philosopher.

Now we  construct a new reference scheme R' using P.  On R', every proper name in the object language is assigned as a reference the image of its reference on R.  If 'N' is the original name used in the metalanguage to give the referent of a given object language name, let 'N*' be  the metalanguage name used on scheme R' to assign to the object language name its new referent.  Every predicate is said to be satisfied by an object just in the case a predicate is true of it whose extension is the image of the extension of the predicate used to give the satisfaction conditions on R.  If 'F' is the original predicate, we let 'F*' represent the predicate we use on the new scheme whose extension is the image of F.  Then for our corresponding axioms, on R', we have

R’

1’. Ref('KL')=Kirk Ludwig*

2’. For any function f, f satisfies 'phil(x)' iff f('x') is a philosopher*.

3’. For any name N, function f, f satisfies [phil(N)] iff ref(N) is a philosopher*.

Canonical T-theorem:

4’. 'Phil(KL)' is true iff Kirk Ludwig* is a philosopher*.

Given the construction, we know that

5. Kirk Ludwig is a philosopher iff Kirk Ludwig* is a philosopher*,

and so we can be sure that we have preserved the truth value of this, and, as it turns out, every other sentences in the object language.

I complained that this was not interpretive, and in particular that the T-theorems of the new theory would NOT be interpretive if those of the old one were (and vice versa for that matter).

Miguel suggested that if the only goal was to provide a theory that met Convention T (or the appropriate analog for a natural language) then we could mess with the reference scheme and still get a a theory that worked in that sense.

Miguel suggested (I believe) the following.  Let us illustrate with just the axioms we’ve given as examples.  1' is drawn from R', and 2 from R, and 3'' is the new axiom and 4’’ the new T-theorem.  Let PI(x) be the inverse of P(x), i.e., if P(x) = y then PI(y)  = x.

R''

1'. Ref('KL')=Kirk Ludwig*

2. For any function f, f satisfies 'phil(x)' iff f('x') is a philosopher.

3''. For any name N, function f, f satisfies [phil(N)] iff PI(ref(N)) is a philosopher.

Canonical T-theorem:

4''. 'Phil(KL)' is true iff PI(Kirk Ludwig*) is a philosopher.

Given that PI(Kirk Ludwig*) = Kirk Ludwig, and assuming a direct reference theory of proper names, 4'' is equivalent to our original 4.

So this is an illustration of a theory that doesn’t meet Convention A but does meet Convention T.  Does it show inscrutability?  No, for in giving the goal as meeting Contention T and admitting it requires reproducing the original (whether or not we know a change of truth value has occurred) means that we are tacitly accepting that meaning (in the relevant dimension) varies with extensions of contained terms.

I suggested we might get something going with our original R' to prove interpretive T-sentences.  But I don’t think it quite works now.  Here is the idea.  We don’t achieve the right result with the canonical theorems, using the same canonical proof procedure.  But if all we want is that the theory satisfy Convention T we might try to add some additional axioms, for example, like 5, which would enable us to get from 4' back to 4, after all.  Well that works if we add 5 to get back to 4 from 4', but that’s just for one of the many cases we want to treat.  If we had to add one for each T-theorem the theory would not be finitely axiomatizable.  With the permutation function in hand we could always use the inverse to produce a predicate  that would have the extension of the orginal, but extension is not guaranteed to preserve meaning.  We’d have to be able to recover  predicates the same in  meaning.  Is there a way?  I suppose if you add enough there is.  For example, we could add the original theory to the new one and then there would be a route from the new T-theorems to the old!  But that would not be very interesting.