*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 aHere, 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 thepossible worldif 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.

*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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