Abstract
Structural realists contend that the properties and relations in the world are more fundamental than the individuals. However, the standard model theory used to analyse the structure of logical theories can make it difficult to see how such an idea could be coherent or workable: for in that theory, properties and relations are constructed as sets of (tuples of) individuals. In this paper, I look at three ways in which structuralists might hope for an alternative: by appealing to predicate-functor logic, Tractarian geometry, or cylindric algebras. I argue that the first two are problematic, but that the third is promising.
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Notes
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Strictly speaking, I am changing the example: Diehl distinguishes between the variable-binding and existence-asserting functions of the existential quantifier, and takes the operator \(\mathsf {G}\) to be used by the religious community only in place of the latter function (so they must also use a lambda operator to bind variables). I leave out this complication for ease of exposition.
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With the result that all lines are parallel or perpendicular.
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Recall that two sets are isomorphic iff they’re equinumerous.
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This isn’t quite the way Turner does things: he characterises M by noting that the cardinality of M is the same as the number of hypersurfaces in the associated Tractarian geometry, where a hypersurface is, roughly, all the surfaces of almost-maximal dimension that involve a given diagonal point. But characterising D by using diagonal points seems to be simpler than doing so using hypersurfaces.
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That said, the approach I outline below has an important missing piece: in order this idea to be more fully developed, one would need a proper characterisation of the universe of sets relative to which one is working (and over which the universal quantifier in the universal property can be taken to range). Perhaps the simplest way to do this would be to supplement the account given here with a preferred axiomatisation of the category of sets, e.g. Lawvere’s Elementary Theory of the Category of Sets [20]. I’m grateful to Hajnal Andréka and István Németi for raising this point to me.
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Recall that two Tarski-models \(\fancyscript{M}\) and \(\fancyscript{M}'\), in signatures \(\varSigma \) and \(\varSigma '\), are definitionally equivalent if (i) \(M = M'\); (ii) for every m-ary \(P \in \varSigma \) there is some m-place \(\varSigma '\)-formula \(\phi \) such that \(P^\fancyscript{M} = \phi ^{\fancyscript{M}'}\); and (iii) for every n-ary \(R \in \varSigma '\) there is some n-place \(\varSigma \)-formula \(\psi \) such that \(R^{\fancyscript{M}'} = \psi ^{\fancyscript{M}}.\)
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The discussion below draws on the foreword of [14].
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Acknowledgements
I am grateful to Judit Madarász and Gergely Székely for the opportunity to contribute to this volume, and to Hajnal Andréka and István Németi for their comments on a previous draft; and I am grateful to all four for their generous hospitality during a visit to their group. I would also like to thank Erik Curiel and Jeremy Butterfield for a detailed discussion of this paper.
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Dewar, N. (2021). Freeing Structural Realism from Model Theory. In: Madarász, J., Székely, G. (eds) Hajnal Andréka and István Németi on Unity of Science. Outstanding Contributions to Logic, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-030-64187-0_15
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