Closed Structure

Abstract

According to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise.

Higher-Order Metaphysics and Propositional Attitudes

“…Since they can't appeal to such a translation, how might the higher-orderist justify the claim that their language is meaningful and not extremely vague?⁹ The best case I know of rests in part on two key ideas (for these points see Fritz et al, 2021, §5.2, Fritz, 2023. First, when we speak of the meaningfulness of higher-order languages, we mean such languages as endowed with a strong logic.…”

“…Moreover, natural language is the basis for our first exposure to the array of "ontological categories" that neo-Fregean pluralists claim there are. If higher-order languages are cut loose from anchors in natural language (or from some other 1⁰ A third idea emphasized in Fritz et al (2021) is that there are analogies between certain constants at different types. For instance, we can understand the universal quantifier at higher types by analogy to the universal quantifier at lower types.…”

“…On some relevant theories of propositional identity, this pattern would directly give us a counterexample to Atomic Congruence, so the hardliner would have to reject the intuitive judgment here. Different theories of propositional identity might allow us to escape this particular example, but most reasonable theories I'm aware of predict that there are some intuitive apparent counterexamples to Atomic Congruence (an exception is Fritz et al 2021). So in general hardline substitutionalists must not just take a hard line with [A]; they must (less plausibly) do so with these further examples as well.…”

Essence and Necessity

“…The rigid types employed here correspond to their extensional types, while the non-rigid types generated by clause (ii) below correspond to their intensional types. Fritz et al (2021) use an intensional relational type theory with a single extensional type for pluralities of propositions.…”

“…The appeal to "constituents" of propositions should be treated with caution, however, at least insofar as it presupposes a structured account of propositions, since such a theory is arguably inconsistent, due to the Russell-Myhill Antinomy (see note 50). For recent discussions of the implications of the Russell-Myhill Antinomy for theories of structured propositions, see Dorr (2016), Fritz et al (2021), Goodman (2017), Klement (2010a,b), and Uzquiano (2015).…”