Sunday, 19 May 2013

Der logische Aufbau der Welt

The title of Rudolf Carnap's 1928 book, Der logische Aufbau der Welt, is normally translated as "The Logical Structure of the World", although apparently a more accurate rendition would be "The Logical Construction of the World".

In working on how to make sense of the claim/significance of Leibniz Equivalence from spacetime theories (roughly: isomorphic spacetime models represent the same possible worlds), I've been trying to work out a version of a propositional view of possible worlds. This has some similarities which Carnap's theory of "state descriptions" (and with Wittgenstein's "picture theory" which influenced Carnap).

The propositional diagram account of possible worlds can be put like this:
$w$ is a possible world if and only if
$w = \hat{\Phi}_{\mathcal{A}}[\vec{R}]$,
where $\mathcal{A}$ is a model, and $\vec{R}$ is a sequence of relations-in-intension.
Here, given a model $\mathcal{A}$ (say, $(A, \vec{S})$ with domain $A$), then $\Phi_{\mathcal{A}}(\vec{X})$ is a formula of pure second-order logic (perhaps infinitary: it has cardinality $max(\omega, |A|^{+})$). $\Phi_{\mathcal{A}}(\vec{X})$ defines the isomorphism type of $\mathcal{A}$. The variables $\vec{X}$ are free second-order variables. I call $\Phi_{\mathcal{A}}(\vec{X})$ the diagram formula for the model $\mathcal{A}$. It correspoinds very closely to what model theorists call the elementary diagram of a model. And then $\hat{\Phi}_{\mathcal{A}}$ is the corresponding (second-order) propositional function, and $\hat{\Phi}_{\mathcal{A}}[\vec{R}]$ is then the result of ``saturating'' $\hat{\Phi}_{\mathcal{A}}$ with the relations $\vec{R}$.

On this "Propositional Diagram" conception of possible worlds, a world $w$ has the form
$w = \hat{\Phi}_{\mathcal{A}}[\vec{R}]$
One might now think of $\hat{\Phi}_{\mathcal{A}}$ as the abstract structure of the world $w$, and think of the sequence $\vec{R}$ as expressing the intensional content of $w$.

This is a kind of form/content distinction. Another way of putting this is to try and define the "representation" relation that holds between models and worlds. Let $\mathcal{A}$ be a model. Let $w$ be a world. Let $\vec{R}$ be a sequence of relations-in-intension with signature matching $\mathcal{A}$. Then:
$\mathcal{A}$ represents $w$ relative to $\vec{R}$ iff $w = \hat{\Phi}_{\mathcal{A}}[\vec{R}]$.

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