Incompleteness of Intuitionistic Propositional Logic with Respect to Proof-Theoretic Semantics

Piecha, Thomas; Schroeder‐Heister, Peter

doi:10.1007/s11225-018-9823-7

Cited by 16 publications

(13 citation statements)

References 9 publications

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confidence: 56%

Proof-Theoretic Semantics and Inquisitive Logic

Stafford

2021

J Philos Logic

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Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This article resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of the logic of inquisitive semantics, a uniform semantics for questions and assertions. The paper further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.

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confidence: 56%

Proof-Theoretic Semantics and Inquisitive Logic

Stafford

2021

J Philos Logic

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“…And the result mentioned above (that there is a counter example to every proof-theoretic validity) is sufficient to prove the correctness of the conjecture. This is in contrast with how proof-theoretic validity was treated in Piecha and Schroeder-Heister (2019) where the sets of atomic rules, not the proof-theoretic systems, are treated analogously with models. This proposal avoids the problem we have identified above and is not subject to Piecha and Schroeder-Heister (2019) general result about superintuitionistic validates.…”

Section: The Role Of the Atomic Formulas Explainedmentioning

confidence: 96%

“…Here we will discuss how the set of atomic rules used in the definition affects what is proof-theoretically valid, as has been shown by a growing body of formal work (Sandqvist 2009, Piecha, Campos Sanz, and Schroeder-Heister 2015, Goldfarb 2016, Piecha and Schroeder-Heister 2019, Stafford 2021).…”

Section: Atomic Rules and Proof-theoretic Systemsmentioning

confidence: 99%

“…This conjecture remained open for almost thirty years before negative results started to appear (Sandqvist 2009). Piecha and Schroeder-Heister (2019) show that Prawitz's conjecture is false for all of the most important definitions of proof-theoretic validity. They show that superintuitionistic formulas, which we shall understand here to mean formulas that are classically consistent but not intuitionistically derivable, are proof-theoretically valid.…”

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confidence: 99%

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‘Not at home, yet so completely at home’

Stafford¹

2023

Imperial Steam

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Several recent results bring into focus the superintuitionistic nature of most notions of proof-theoretic validity, but little work has been done evaluating the consequences of these results. Proof-theoretic validity claims to offer a formal explication of how inferences follow from the definitions of logic connectives (which are defined by their introduction rules). This paper explores whether the new results undermine this claim. It is argued that, while the formal results are worrying, superintuitionistic inferences are valid because the treatments of atomic formulas are insufficiently general, and a resolution to this issue is proposed.

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“…Tranchini, Pistone and Petrolo [23] show that the Russell-Prawitz translation of firstinto second-order logic preserves identity of proof with respect to a certain enriched system. Piecha and Schroeder-Heister [15] show that intuitionistic propositional logic is incomplete with respect to standard notions of prooftheoretic validity.…”

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confidence: 99%