Tuesday, 12 March 2013

Possible Worlds as Saturated Propositional Functions

In the previous post, I suggested identifying the abstract structure $\hat{\mathcal{A}}$ of a given model,
$\mathcal{A} = (A, \vec{R})$ 
with a certain kind of categorical second-order propositional function, $\hat{\Phi}$.
Roughly, the procedure is this:
1. Consider a model $\mathcal{A} = (A, \vec{R})$.
2. Introduce a (possibly infinitary) language $L(\mathcal{A})$, with a constant $c_a$ for each $a \in A$, and define the categorical axiomatization $\psi$ of $\mathcal{A}$.
3. First-order ramsify away all constants and let replace all predicate symbols by second-order variables in $\psi$. Let $\Phi_{\mathcal{A}}(\vec{X})$ be the corresponding categorical second-order formula.
4. Then, consider the corresponding second-order propositional function $\hat{\Phi}_{\mathcal{A}}$. 
I propose that this entity, $\hat{\Phi}_{\mathcal{A}}$ is the abstract structure of the model $\mathcal{A}$. So, we just define:
$\hat{\mathcal{A}} := \hat{\Phi}_{\mathcal{A}}$.
There are certainly "class/set issues" that arise with this approach (as noted by Sam Roberts in the comments to the previous post.) However, setting this aside, and assuming a suitably typed/sorted approach, I believe that we can get a notion of abstract structure such that
Leibniz Abstraction
$\hat{\mathcal{A}} = \hat{\mathcal{B}} \Leftrightarrow \mathcal{A} \cong \mathcal{B}$
The main alternative to the propositional function approach is to try an isomorphism-class approach, and define:
$\hat{\mathcal{A}} := \{ \mathcal{B} \mid \mathcal{B} \cong \mathcal{A}\}$.
again somehow making adjustments for the "class/set" issues that arise. But I'm somewhat unhappy with this approach.

Both approaches eliminate any distinguished domain. So, what then are "nodes"? The closest thing one gets to a domain of "nodes" on the propositional function approach is the sequence of existentially bound first-order variables in the linguistic description of the structure. Skolemizing these variables leads to a sequence of constants, and these constants can then play the role of parametrized "nodes". Perhaps one can then make sense of "occupying" a node -- this will be a semantic notion, somewhat like realizing a type in model theory. On the isomorphism-class approach, one has no domain either.

A partial motivation for the propositional function approach is to get a theory of possible worlds which automatically implements
Leibniz Equivalence
Isomorphic spacetime models represent the same possible world. 
If one thinks (as I do) that this is a genuine metaphysical constraint on what worlds are like, then how might one given an account of possible worlds in which Leibniz Equivalence is automatically implemented?

Consider a possible world $w$ in which
there are precisely three spatial points $p_1, p_2, p_3$ such that $p_2$ is properly between $p_1$ and $p_3$, and no other combinations hold.
Then I can describe this world $w$ categorically by the following formula:
\exists p_1 \exists p_2 \exists p_3(p_1 \neq p_2 \wedge p_1 \neq p_3 \wedge p_2 \neq p_3 \wedge Sp_1 \wedge Sp_2 \wedge Sp_3 \\
\wedge Bp_1p_2p_3 \wedge \dots \wedge \forall x(x = p_1 \vee x = p_2 \vee x = p_3))\end{eqnarray*}$
 So, we may think of this formula as expressing an genuinely interpreted proposition. But we can also disinterpret the predicate symbols in the formula by replacing them with second-order variables, and give a second-order formula,
\exists p_1 \exists p_2 \exists p_3(p_1 \neq p_2 \wedge p_1 \neq p_3 \wedge p_2 \neq p_3 \wedge Xp_1 \wedge Xp_2 \wedge Xp_3 \\
\wedge Yp_1p_2p_3 \wedge \dots \wedge \forall x(x = p_1 \vee x = p_2 \vee x = p_3))\end{eqnarray*}$
And we can then abbreviate this formula as $\Phi(X,Y)$. We can then consider the corresponding propositional function, denoted $\hat{\Phi}$. This is then the abstract structure of the possible world $w$. As before, $\hat{\Phi}$ is the abstract structure $\hat{\mathcal{A}}$, for some representative model $\mathcal{A}$. If we take $S$ to be the property of being a spatial point, and $B$ to be the relation of betweenness, then we get:
$w = \hat{\mathcal{A}}[S, B]$
The argument properties/relations here are properties/relations amongst "concreta".

So, in other words, a possible world is the image of certain properties and relations under a certain propositional function. If you like, a possible world is a saturated propositional function. This then suggests the following explication:
$w$ is a possible world iff $w = \hat{\mathcal{A}}[\vec{R}]$, for some model $\mathcal{A}$ and some sequence $\vec{R}$ of concretum properties/relations. 
So long as we are careful in defining "represents", this will then automatically implement Leibniz Equivalence.


  1. It would seem to me that in order to second-order characterize a structure A you have to be able (among other things) to characterize its cardinality kappa. But not all cardinals are so characterizable (there are only countably many second-order propositional functions). [Enderton's entry on the SEP refers to an article by Garland on this topic, there might be more.]

  2. Aldo, thanks.

    The language is infinitary, so for a model A with cardinality kappa, we have kappa many constants, and kappa many variables; so we may have highly infinitary boolean compounds and quantifier prefixes. I was not sure for a while that the categoricity claim I assume here is right, but speaking to several others, it seems to be ok.

    The number of propositional functions involved is going to be uncountable, because the languages are infinitary. So, for set-sized models, I think the propositional functions will form a proper class.

    (Sam Roberts made some good comments about these "set/class" issues in the comments below.)



    1. Ah, I missed the bit about the language being infinitary. But it won't be a run-of-the-mill infinitary language, essentially each structure $\mathcal{A}$ of cardinality $\kappa$ will be described by a formula in $\mathcal{L}_{\kappa^+,\kappa^+}$. What is then the residual advantage over taking $\mathcal{A}$ itself as a representative of its own isomorphism type?

  3. Hi Aldo,

    Yes, the formula is like a huge conjunction of the elementary diagram.

    The idea is to get an explication of possible worlds out of this. I take one of these propositional functions, and apply it to a sequence of relations (Fregean intensions), giving a proposition that ....blah-blah.

    But I'm not sure how that would work on the isomorphism-type view of abstract structure? Possible worlds aren't themselves structures or models. But perhaps there's a way of identifying a model for a given possible world, and then considering the isomorphism type. But it seems that this will conflate distinct worlds. I think it will eliminate the "quiddities" of the particular relations.

    So, a world w1 where

    there are exactly two things x,y and such that Ax, Ay, Rxx, Rxy, -Ryy and -Ryx

    where A and R are definite relations amongst concreta, will be conflated with a world w2 where,

    there are exactly two things x,y and such that Bx, By, Sxx, Sxy, -Syy and -Six

    where B and S are relations amongst concreta, but different from A and R.

    This is because they will give rise to the same isomorphism-type.



  4. Thanks for this clarification, Jeff. I suppose a true structuralist would not care much about the "quiddities", but what do I know?