Introducing doppelgangers

Yesterday, I mentioned that one might reinterpret the quantifier symbols in a language so as to introduce doppelgangers: extra quasi-entities that just don't exist, but get to be talked about using standard quantifier inference rules. Here I want to give a bit more detail, and then offer a curious application to the philosophy of mind: an account of how a materialist could use a doppelganged reinterpretation of language to talk like a hard-core dualist. The application shows that it is philosophically crucial that we have a way to distinguish between real quantifiers and mere quasi-quantifiers (in the terminology of the previous post) if we want to distinguish between materialism and dualism.

Onward! Let L be a first-order language with identity. A model M for L will be a pair (D,P), where D is a non-empty set ("domain") and P is a set of subsets ("properties") of D. A doppel-interpretation of L is a pair (I,s) where I is a function from the names and predicates other than identity to D and P respectively and s is a function from names to the set {0,1} ("signature"). The signature function s tells us which name is attached to an ordinary object (0) and which to a doppelganger (1).

Now a substitution vector for a doppel-interpretation (I,s) will be a partial function v from the names and variables of L to D such that v(a)=I(a) whenever a is a name. A signature vector is a partial function f from the names and variables of L to D such that f(a)=s(a) whenever a is a name. If x is a variable and u is in D, then I will write v(x/u) for the substitution vector that agrees with v except that v(x/u)(x)=u. I.e., v(x/u) takes v and adds or changes the substitution of u for x. Likewise, if f is a signature vector, then s(x/n) agrees with s except that s(x/n)(x)=n for n in {0,1}.

We can now define the notion of a substitution and signature vector pair (v,f) doppel-satisfying a formula F in L under a doppel-interpretation (I,s). We begin with the normal Tarskian inductive stuff for truth-functional connectives:

• (v,f) doppel-satisfies F or G iff (v,f) doppel-satisfies F or (v,f) doppel-satisfies G
• (v,f) doppel-satisfies F and G iff (v,f) doppel-satisfies F and (v,f) doppel-satisfies G
• (v,f) doppel-satisfies ~F iff (v,f) doesn't doppel-satisfies F.

Now we need our quasi-quantifiers:

• (v,f) doppel-satisfies ∃xF iff for some u in D, (v(x/u),f(x/0)) doppel-satisfies F or for some u in D, (v(x/u),f(x/1)) doppel-satisfies F
• (v,f) doppel-satisfies ∀xF iff for every u in D, (v(x/u),f(x/0)) doppel-satisfies F and for every u in D, (v(x/u),f(x/1)) doppel-satisfies F.

Then we need atomic formulae:

• (v,f) doppel-satisfies h=k (where h and k are variables-or-names) iff v(h)=v(k) and f(h)=f(k)
• (v,f) doppel-satisfies Q(h₁,...,h_n) where Q is other than identity iff (I(h₁),...,I(h_n))∈I(Q).

And finally we can define doppel-truth: a sentence S of L is doppel-true provided that it is doppel-satisfied by every substitution and signature vector pair.

It is easy to see that if s_m is the usual sentence that asserts that there are m objects, and if D has n objects, then s_m is doppel-true if and only if m=2n. It is also easy to see that all the first order rules for quantifiers are valid for our quasi-quantifiers ∃ and ∀.

Now on to our fake dualism. We need a more complex doppelganging. Specifically, we need to divide our stock of predicates into the mental and non-mental predicates. Start with a materialist's first order language L that includes mental predicates (the materialist may think they are in some sense reducible). Add a predicate SoulOf(x,y) (we won't need to specify whether it's mental or not) which will count for us as neither mental nor non-mental. I will assume for simplicity that the mental predicates are all unary (e.g., "thinks that the sky is blue")—things get more complicated otherwise, but one can still produce the fake dualism. Now, instead of doppelganging all the objects, we only doppelgang the minded objects. Thus, our models will be triples (D,D_m,P) where D_m is a subset of D (the minded objects, in the intended interpretation). We say that a substitution and signature vector pair (v,f) is licit if and only if f(a)=1 implies v(a)∈D_m ("only members of D_m have doppelgangers"), and in all our definitions we only work with licit pairs. Moreover, our reinterpretations do not need to give any extension to the predicate SoulOf(x,y): that's handled in the semantics.

Finally, we modify doppel-satisfaction for quantifiers and predicates:

• (v,f) doppel-satisfies ∃xF iff for some u in D, (v(x/u),f(x/0)) doppel-satisfies F or for some u in D_m, (v(x/u),f(x/1)) doppel-satisfies F
• (v,f) doppel-satisfies ∀xF iff for every u in D, (v(x/u),f(x/0)) doppel-satisfies F and for every u in D_m, (v(x/u),f(x/1)) doppel-satisfies F.
• (v,f) doppel-satisfies h=k (where h and k are variables-or-names) iff v(h)=v(k) and f(h)=f(k)
• (v,f) doppel-satisfies Q(h₁,...,h_n) where Q is non-mental and other than identity iff (I(h₁),...,I(h_n))∈I(Q) and f(h₁)=...=f(h_n)=0
• (v,f) doppel-satisfies Q(h) where Q is mental iff I(h)∈I(Q) and f(h)=1
• (v,f) doppel-satisfies SoulOf(h,k) iff v(h)=v(k), f(h)=1 and f(k)=0.

And then we say we have doppel-truth of a sentence when every licit pair is satisfied.

The intended materialist doppel-interpretation (I,f) consists of the usual materialist interpretation I of the names and predicates other than Soul(x) and that gives to Soul(x) the extension of all the minded objects, sets D_m to be the set of minded objects, plus has a signature f such that f(n)=1 where n is a name of a minded object and otherwise f(n)=0.

Now let's speak an informal version of our doppelganged materialist language. Let M be any mental predicate and Q any non-mental one. Say that a soul is any x such that ∃y(SoulOf(x,y)). Suppose "Jill" is the name of a minded object. The following sentences will be doppel-true:

• Some objects have souls.
• Every object that has a soul is a non-soul.
• Only souls satisfy M.
• No soul satisfies Q.
• Jill is a soul.
• Jill does not satisfy Q.

The doppel-language is strongly dualist. Only the souls have mental properties predicated of them and only the non-souls have non-mental properties (or stand in non-mental relations). A materialist community could stipulate that henceforth their language bears this kind of doppel-interpretation. They could then talk like dualists. But they wouldn't be dualists.. Hence the doppelganged ∃ and ∀ aren't really quantifiers.

Assume materialism. If there are n material objects, including m minded objects, then in the doppelganged language it will be true to say something like "There are n+m objects." For every one of the minded objects has a doppelganger.