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.

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.

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.

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