Abstract
The Russell–Myhill theorem threatens a familiar structured conception of propositions according to which two sentences express the same proposition only if they share the same syntactic structure and their corresponding syntactic constituents share the same semantic value. Given the role of the principle of universal instantiation in the derivation of the theorem in simple type theory, one may hope to rehabilitate the core of the structured view of propositions in ramified type theory, where the principle is systematically restricted. We suggest otherwise. The ramified core of the structured theory of propositions remains inconsistent in ramified type theory augmented with axioms of reducibility. This is significant because reducibility has been thought to be perfectly consistent with the ramified approach to the intensional antinomies. Nor is the addition of reducibility to ramified type theory sufficient to restore other intensional puzzles such as Prior’s paradox or Kripke’s puzzle about time and thought.
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Notes
It is not uncommon to construe the vicious circle principle as a constraint on explicit comprehension principles, which provide a partial specification of the range of the variables of each type. In the propositional case, comprehension follows from a suitable formulation of existential generalization for quantification into sentence position. In the more general case, comprehension is derivable from existential generalization in the presence of a device of lambda abstraction governed by extensional beta equivalence.
Uzquiano (2021) discusses the link between the two results in more detail.
Thanks to an anonymous referee for the question of what is the role of sets in Kripke’s puzzle about time and thought.
In this respect, our presentation comes closer to Hazen and Davoren (2000). Other versions of ramified type theory are developed in Myhill (1979), Hazen (1983), and Hodes (2015). While these versions of ramified type theory differ from the present one in other respects, these differences are generally unimportant for present purposes.
This is one slight departure from Church (1976), who defines the order of a variable (rather than that of the r-type to which it belongs.)
Thanks to an anonymous referee for the suggestion.
(Church 1976, p. 750) takes for granted a standard axiomatization of quantificational logic to which the comprehension axioms are supposed to be added.
Instances of the first schema become instances of the second, more general schema when formulas are construed as 0-place propositional functions.
This is not to deny there may be other motivations for ramification. Hodes (2015), for example, does considers an independent motivation grounded on a substitutional interpretation of quantification into certain grammatical types.
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Acknowledgements
I presented earlier versions of this paper at a conference on higher-order metaphysics at the University of Birmingham and at the conference on Paradox, Context, and Generality at the University of Salzburg in summer of 2019. I greatly benefitted from the feedback I received from the two audiences. Thanks as well to Paul Egré, Peter Fritz, Michael Glanzberg, Jeremy Goodman, Nick Jones, Julien Murzi, Agustín Rayo, Lorenzo Rossi, James Studd. Special thanks to Amirhossein Kiani, Øystein Linnebo, and an anonymous referee for detailed comments on an earlier version of the paper.
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Uzquiano, G. Ramified structure. Philos Stud 180, 1651–1674 (2023). https://doi.org/10.1007/s11098-022-01780-y
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DOI: https://doi.org/10.1007/s11098-022-01780-y