Abstract
Intra-theoretical logical pluralism is a form of meaning-invariant pluralism about logic, articulated recently by Hjortland (Australas J Philos 91(2):355–373, 2013). This version of pluralism relies on it being possible to define several distinct notions of provability relative to the same logical calculus. The present paper picks up and explores this theme: How can a single logical calculus express several different consequence relations? The main hypothesis articulated here is that the divide between the internal and external consequence relations in Gentzen systems generates a form of intra-theoretical logical pluralism.
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20 December 2019
In the original publication of the article, in Definition 4, the sixth line which reads as
Notes
This is in contradistinction with what we may call historical pluralism, i.e., a range of views expressed or suggested by philosophers like Tarski (2002), Carnap (1959) or Quine (1970), which are quite naturally interpreted as requiring meaning variance in order to obtain a plurality of consequence relations.
The contexts are, roughly, the collections of parametrised formula occurrences in a rule application, i.e., those occurrences that are neither used nor produced by an application of the rule. As per usual, Roman capitals from the beginning of the alphabet range over formulae, while those from the end of the alphabet range over sets of formulae and, in the schematic formulation of the rules, over contexts (which need not be sets).
To keep things simple, we will look at sequent calculi simpliciter; the observations to follow generalise to other types of Gentzen systems.
For more on the relation between the various internal crs that can be associated with a calculus, see Avron (1991).
The observations to follow are formulated without disambiguating the symbol \(\vdash ^{i}_{ \mathrm {LK}}\) according to whether it is the single- or the multiple-conclusion relation that is under discussion because, by and large, the differences do not matter. Besides, such disambiguation is trivial.
Weakening on the right is restricted in the single-conclusion case, so it is only the multiple-conclusion relation that is fully monotonic on the right.
This is not to say that classical logic itself has some special dignity, for the same kind of cr can be encountered in the case of other logics, such as, for instance, intuitionist logic. It’s just that it is the most familiar cr that fits the bill.
Logicalcrs are usually required to be substitution invariant. I have bracketed this feature in the definition because it will play no role subsequently.
The parentheses around the sequents with empty antecedents are added in order to facilitate readability.
The reader can find a considered defence of them in Mares and Paoli (2014).
Of course, this distinction would have to be generated by something other than context-differences in the formulation of the rules. Otherwise, there would no progress over Restall’s version of pluralism.
It is unclear to me whether Hjortland actually made this connection. But this is the easiest way to reconcile the views he expressed in Hjortland (2013, 2014). Incidentally, it’s worth mentioning that the initial impetus for considering external crs was precisely an attempt to defuse the ‘meta-Quinean objection’, cf. Paoli (2014) and also Dicher and Paoli (forthcoming a).
Whether this really is the case is a contentious matter. But I have no reasons to go into this matter here. An argument against \(\mathrm {ST}\)’s alleged classicality, recently proposed in Dicher and Paoli (forthcoming b), will be briefly mentioned below.
Notice, however, that as long as the truth rules are not in the picture, Cut is perfectly safe, cf. Ripley (2012).
A more precise name would be metainferences of level one, cf. Barrio et al. (2018).
This is in contradistinction to the definition of metainferential validity in Cobreros et al. (2013), on which a metainference is valid iff if every valuation satisfies the premises, the conclusion is likewise satisfied. This ‘global’ notion of metainferential validity is too weak to be of much interest: any metainference whose premises are not logically valid will be valid.
Even model-theoretically, we have it that the connectives of these two logics are given via the same truth functions.
Depending on how one construes sameness of process, this putative objection may be used against internal/external pluralism even in the guise discussed in the previous section.
By abusive use of the power of baptism, these calculi had names based on ‘\(\mathrm {LK}\)’ in Dicher and Paoli (forthcoming b).
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Acknowledgements
I am grateful to the audiences at the ALOPHIS Research Seminar (University of Cagliari), The Institute for Research in the Humanities (University of Bucharest), Triennial Conference of the Italian Society for Logic and Philosophy of Science (University of Bologna), ENFA 7: Encontro Nacional de Filosofia Analitica, 7th edition (University of Lisbon) where I have presented previous versions of this paper. I owe special thanks to Francesco Paoli for many helpful discussions on the issues broached here as well as for encouraging me to actually write this paper. At different stages, this work was supported by Regione Autonoma Sardegna within the Project CRP-78705 (L.R. 7/2007) “Metaphor and argumentation” and by the FCT – Fundação para a Ciência e a Tecnologia, Portugal, through the grant SFRH/BPD/116125/2016.
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Appendix
Appendix
1.1 The n-sided calculus for \(\mathrm {K3}\) and \(\mathrm {LP}\)
The structural rules of this calculus are:
The official language of this calculus contains only conjunction, disjunction and negation. I present here only the rules for negation and conjunction (the rules for disjunction are dual to these last).
1.2 The structural rules of \(\mathrm {LK}\)
1.3 The \(\mathrm {PK}\) family of calculi
These calculi have set-based sequents, rendering explicit Contraction and Exchange rules unnecessary. They all contain Id and W. \(\mathrm {PK}\) consists these alongside Cut and the rules below that do not have the subscript i appended to their labels. \(\mathrm {PK}^{-}\) is PK minus Cut. \(\mathrm {PK^{-}_{INV}}\) is obtained by adding to \(\mathrm {PK}^{-}\) the inverses of the operational rules—those rules depicted below that have i subscripted to their label.Footnote 21
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Dicher, B. Variations on intra-theoretical logical pluralism: internal versus external consequence. Philos Stud 177, 667–686 (2020). https://doi.org/10.1007/s11098-018-1199-z
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DOI: https://doi.org/10.1007/s11098-018-1199-z