Abstract
This is a reply to the considerations advanced by Schroeder-Heister and Tranchini (Ekman’s paradox, Unpublished typescript) as prima facie problematic for the proof-theoretic criterion of paradoxicality, as originally presented in Tennant (Dialectica 36:265–296, 1982) and subsequently amended in Tennant (Analysis 55:199–207, 1995). Countering these considerations lends new importance to the parallelized forms of elimination rules in natural deduction.
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Notes
Note that the compactifying reduction procedure is not considered by Prawitz. This is why he was able to prove the normalization theorem for intuitionistic logic, despite basing it on the serial forms of elimination for \(\wedge \), \(\rightarrow \) and \(\forall \). Prawitz was considering as eligible reduction procedures only those provided for the logical operators’ introduction-elimination rule-pairs. These reduction procedures are designed to eliminate ‘maximal sentence occurrences’—ones standing as conclusions of introductions, and as major premises of the corresponding eliminations.
Space does not permit any further explanation here of the details of Schroeder-Heister’s and Tranchini’s positive proposal; the interested reader in search of such details is referred to their text. My concern here is to confront the problem they have raised, and to propose a more effective and simpler solution to it.
These are also known as ‘generalized’ elimination rules. They were first introduced in Schroeder-Heister (1984).
The sequent calculus in question is a single-conclusion one (succedents of sequents being at most singletons), whose sequent-antecedents are sets of sentences, not sequences of sentence-occurrences. Thus the Gentzenian structural rules of Contraction and Interchange (or Permutation) are irrelevant, because unnecessary; and modifications of such rules, in the manner of Zardini (2011), for example, are orthogonal to the discussion undertaken here.
A note on notation: we shall use \(\Pi \), \(\Sigma \), \(\Omega \) and \(\Xi \) for proofs, and use \(\Delta \) and \(\Gamma \) for sets of premises.
A similar proof, of \(\lnot ((A\!\rightarrow \!\lnot A)\wedge (\lnot A\!\rightarrow \! A))\), using parallelized elimination rules, but using the definition of \(\lnot \varphi \) as \(\varphi \rightarrow \bot \), was given by von Plato (2000), at p. 123, by way of solution of Ekman’s problem for the serial forms of elimination rules. Note that all applications of (\(\rightarrow \)E) in the proofs that we are giving here are applications of the parallelized form of that rule, even if some of those applications involve degenerate (i.e., single-sentence) major subproofs.
With an eye to the Paradox’s historical (i.e., Fregean) origins, we see also (at second order) that Naïve Abstraction is refutable:
Recall that a free logic is one that is not based on the standard presupposition that all singular terms denote. Put another way: free logic allows for non-denoting singular terms. If in addition the logic is not based on the standard presupposition that the universe is non-empty, then it is called a universally free logic. Such a logic was presented in Tennant (1978), Ch. 7. A completeness proof was also given there for the universally free logic of a first-order language with identity and the familiar primitive variable-binding operator \(\{x|\ldots x\ldots \}\) for forming (singular) set-abstraction terms.
Here I am indebted to Elia Zardini.
See Ramsey (1926), at pp. 352–3.
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Acknowledgments
I am grateful to Peter Schroeder-Heister and Luca Tranchini for generously providing a copy of Schroeder-Heister and Tranchini (Unpublished typescript) and thereby provoking this study. I thank the Editor, Elia Zardini, and two anonymous referees for their careful readings and helpful suggestions for improvements. Any defects that remain are my sole responsibility.
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Tennant, N. Normalizability, cut eliminability and paradox. Synthese 199 (Suppl 3), 597–616 (2021). https://doi.org/10.1007/s11229-016-1119-8
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DOI: https://doi.org/10.1007/s11229-016-1119-8