Abstract
We examine the proof-theoretic verificationist justification procedure proposed by Dummett (1991). After some scrutiny, two distinct interpretations with respect to bases are advanced: the independent and the dependent interpretation. We argue that both are unacceptable as a semantics for propositional intuitionistic logic.
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Notes
The formulation of an introduction rule for \(\bot \) is not necessary for our purposes.
Definitions are going to be numbered x.y, where x indicates the section and y indicates the position inside the section. The numbering system is intended to make a parallelism between the various definitions and characterizations. For instance, Characterization 3.1, Definitions 6.1 and 7.1, where \(y=1\), all deal with the notion of canonical argument.
When discussing this example, Dummett (1991, p. 263) doesn’t follow his own definition. He claims that both the premiss A and the final conclusion
are in the main stem. However, since the sentence 
(which depends on the hypotheses 
) occurs in the path from A to the conclusion, A is not, after all, in the main stem. This causes no further difficulties for understanding his definitions.We do not consider instances as in Dummett’s original formulation because they are only relevant for predicate logic.
According to the substitutional point of view, hypothetical arguments, i. e. arguments with open assumptions, should be explained in terms of closed arguments by transforming canonical closed arguments for the open assumptions into canonical closed arguments for the conclusion. For a more detailed discussion of the substitutional point of view concerning open arguments, see (Schroeder-Heister 2012, Sect. 2.2).
As a limiting case, we have canonical arguments for atomic sentences by an empty series of introduction rules.
Among the transformations that Dummett had envisaged, we think that reduction steps of roundabouts would be included.
This quotation is extracted from a later chapter, after Dummett had already presented his verificationist justification procedure.
This remark refers to the dependent interpretation.
Our proof is essentially the same given by Dummett (1991, p. 263).
Here, we assume that the introduction rules comply with a complexity condition, as formulated by Dummett (1991, p. 258). The introduction rules of \({\textsf {NJ}}^{}\) are all examples of such rules.
The proof depends on the restriction to basic rules without discharge. In particular, it depends on the absence of discharges among the rules in \(\varPi _{6}\). We thank an anonymous referee for pointing this out.
This result is due to Goldfarb (2015).
The case with
in the main stem is trivial.
are in the main stem. However, since the sentence 
(which depends on the hypotheses 
) occurs in the path from A to the conclusion, A is not, after all, in the main stem. This causes no further difficulties for understanding his definitions.
in the main stem is trivial.References
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Acknowledgments
Luiz Carlos Pereira made valuable suggestions on an early draft of the paper. We also thank the anonymous referees for their comments which certainly improved the paper (any faults it still contains are our responsibility). The work was supported by CNPq grant PDE 202174/2014-0, Wagner de Campos Sanz, and DAAD grant 91562976, Hermógenes Oliveira.
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de Campos Sanz, W., Oliveira, H. On Dummett’s verificationist justification procedure. Synthese 193, 2539–2559 (2016). https://doi.org/10.1007/s11229-015-0865-3
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DOI: https://doi.org/10.1007/s11229-015-0865-3