Abstract
Carnap’s result about classical proof-theories not ruling out non-normal valuations of propositional logic formulae has seen renewed philosophical interest in recent years. In this note I contribute some considerations which may be helpful in its philosophical assessment. I suggest a vantage point from which to see the way in which classical proof-theories do, at least to a considerable extent, encode the meanings of the connectives (not by determining a range of admissible valuations, but in their own way), and I demonstrate a kind of converse to Carnap’s result.
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Notes
Since Raatikainen’s (2008) reminder of the result, Murzi and Hjortland (2009) have considered how ‘intuitionists like Dummett and Prawitz’ (p. 480) could respond to this issue, arguing that it does not vitiate their programme. They have also raised but left open the issue of how ‘classical inferentialists’ (p. 480) might respond. Detailed technical work by Peregrin (2010) has explored ‘the power of various inferential frameworks as measured by that of explicitly semantic ones’ (Peregrin 2010, p. 255). See also Smith and Incurvati (2010), Hjortland (2014) and Bonnay and Westerståhl (2016).
For instance, Raatikainen's gloss of the result is:
It can be shown that no ordinary formalization of logic, and not the standard rules of inference (of the natural deduction) in particular, is sufficient to ‘fully formalize’ all the essential logical properties of logical constants. That is, they do not exclude the possibility of interpreting logical constants in any other than the ordinary way. (Raatikainen 2008, p. 283)
The working-out will be easiest in tree systems, followed by natural deduction systems (since they have separate rules for each connective). In an axiom system with few axioms the working-out will be possible in principle but often very involved.
References
Bonnay D, Westerståhl D (2016) Compositionality solves Carnap’s problem. Erkenntnis 81(4):721–739
Carnap R (1943) Formalization of logic. Harvard University Press, Cambridge
Hjortland OT (2014) speech acts, categoricity, and the meanings of logical connectives. Notre Dame J Form Log 55(4):445–467
Murzi J, Hjortland OT (2009) Inferentialism and the categoricity problem: reply to Raatikainen. Analysis 69(3):480–488
Peregrin J (2010) Inferentializing semantics. J Philos Log 39(3):255–274
Raatikainen P (2008) On rules of inference and the meanings of logical constants. Analysis 68(300):282–287
Smith P, Incurvati L (2010) Rejection and valuations. Analysis 70(1):3–10
Acknowledgements
Thanks to N. J. J. Smith for encouragement when this result was obtained in the course of work conducted under his supervision in 2010.
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Haze, T. A Note on Carnap’s Result and the Connectives. Axiomathes 29, 285–288 (2019). https://doi.org/10.1007/s10516-018-9393-3
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DOI: https://doi.org/10.1007/s10516-018-9393-3