Abstract
Logical Pluralists maintain that there is more than one genuine/true logical consequence relation. This paper seeks to understand what the position could amount to and some of the challenges faced by its formulation and defence. I consider in detail Beall and Restall’s Logical Pluralism—which seeks to accommodate radically different logics by stressing the way that they each fit a general form, the Generalised Tarski Thesis (GTT)—arguing against the claim that different instances of GTT are admissible precisifications of logical consequence. I then consider what it is to endorse a logic within a pluralist framework and criticise the options Beall and Restall entertain. A case study involving many-valued logics is examined. I next turn to issues of the applications of different logics and questions of which logic a pluralist should use in particular contexts. A dilemma regarding the applicability of admissible logics is tackled and it is argued that application is a red herring in relation to both understanding and defending a plausible form of logical pluralism. In the final section, I consider other ways to be and not to be a logical pluralist by examining analogous positions in debates over religious pluralism: this, I maintain, illustrates further limitations and challenges for a very general logical pluralism. Certain less wide-ranging pluralist positions are more plausible in both cases, I suggest, but assessment of those positions needs to be undertaken on a case-by-case basis.
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Notes
Field (2009a, pp. 346–347) argues that this idea of “inter-theoretic sameness of meaning of connectives” is not so straightforward, as he illustrates by considering the translations of negation between different competing logics.
Beall and Restall (2006), p. 91 briefly addresses the possibility of non-transitive logics (e.g. Tennant’s) which can’t fit the form of GTT; see also Read (2006) for discussion of the limitation. Bueno and Shalkowski (2009a) advocate a form of pluralism employing a modalist view whereby the necessity involved in logical consequence isn’t to be spelled out quantificationally, so GTT is no longer central.
Similarly, of course, predicate calculus does not recognise the validity of, say, arguments of the form “necessarily p, therefore p” and this problem generalises for any logic that does not recognise all candidate logical connectives (e.g. temporal or epistemic operators). It may be easier to bite the bullet on such cases than it is for the quantificational arguments discussed in the text: “necessarily p, therefore p” could be declared not definitely valid, so only valid in some senses and on some precisifications.
Tarski’s definition of logical consequence takes GTT to involve models that differ in their assignments to non-logical constants. He expressed scepticism about the possibility of providing grounds on which to draw a sharp boundary between logical constants and other expressions, though he did assume that the universal quantifier was definitely among the constants (1936, p. 418). He acknowledges the possibility that “we shall be compelled to regard such concepts as ‘logical consequence’... as relative concepts which must, on each occasion, be related to a definite ... division of terms into logical and extra-logical”, taking this to reflect the fluctuation in the common usage of the concept of consequence (p. 420). Although this allows for several acceptable logics, it is not pluralism as advocated by Beall and Restall, as it is relativist in ways they reject and does not allow divergence between logics with the same logical constants (e.g. classical and intuitionistic logic).
Beall and Restall (p. 92) consider and reject a closely related view, prompted by an envisaged objector proposing that validity requires truth-preservation in all cases of all kinds. (The view is not the same, however: arguments that are not valid according to some logics but valid according to others count as invalid on that view and borderline valid on the supervaluationist one.) Bueno and Shalkowski (2009a, p. 300) argue that Beall and Restall’s necessity constraint drives them to this view, and that once we consider the full range of alternative logics, this would result in logical nihilism whereby nothing is valid.
This paragraph appears to ignore any types of necessity—perhaps conceptual necessity—that can’t be effectively treated in this way as restricted necessity in this way. Any potential types of necessity that don’t fit the form of (GLT) will be of no use to Beall and Restall here, though. Considering other possible types—perhaps a two-dimensional treatment of conceptual necessity—would take us well beyond the scope of this paper. The point at present is that, at the very least, Beall and Restall’s treatment of necessity (which also ignores these kinds of case) does not provide an analogy that helps illuminate their logical pluralism.
Beall and Restall sometimes write as if the pluralist’s different consequence relations correspond to different senses of “consequence” and several commentators have taken their position as such (see, e.g., Beall and Restall 2001, p. 1, Field 2009a and Hjortland 2013). I cannot here explore different, more subtle types of ambiguity, but we can say, at the very least, that drawing an analogy with other standard cases of ambiguity does not help in understanding the logical pluralist position in question.
Note that on pp. 82–83, the definition of “strongly endorsing” a consequence relation requires it to be “an instance” of GTT, whereas weak endorsement of a consequence relation requires it to be an “admissible instance” of GTT. I take it that the admissibility of the instance is also required for strong endorsement as the authors are clear that strong endorsement entails weak endorsement.
What if, for example, you are not sure whether or not there are non-classical values or whether the classical models exhaust the possibilities? You might naturally be described as an agnostic not a pluralist (though for Beall and Restall, you count as weakly endorsing all the options and strongly endorsing none of them). By characterising their pluralism in terms of the logics that the advocate endorses, they leave no room for the debate between someone—surely a monist—who maintains there is a single correct logic, though they’re not sure which and a pluralist who thinks there are several correct logics, but is uncertain about which pass the relevant tests.
Beall and Restall’s constraints of formality, normativity and necessity don’t help here either: again, they are too easy to pass, as would be shown by replicating the arguments offered to show that their chosen logics satisfy them (e.g. p. 55, p. 69).
Bueno and Shalkowski (2009a) also focus on uses of logic of this kind, where, again, what we should say is that stronger logics are useful for capturing something other than a consequence relation. The modalist element of their theory—that necessity is taken as primitive and not to be understood in terms of quantification over cases—does not justify their claim that in different domains, different judgements of necessity are warranted. A database may represent a contradiction (a student both has and has not returned some library books) and it can be useful to have a system whereby we can discriminate among the things that we can take this to commit us to (perhaps that they owe a 3 fine, but not that they owe a 3000 fine, as would also be true if we took any arbitrary conclusion to follow from the contradiction). But that’s not to say that there’s a sense of possibility in which it is possible that the student both has and hasn’t returned the same books. It’s possible for both “he has returned the books” and “he has not returned the books” to be recorded, but there’s no contradiction there, and it isn’t a domain in which he has both returned and not returned the books. Similarly, the situation in which Bueno and Shalkowski judge a constructivist view to be warranted is in “intensional contexts”: it allows us to model “I am not certain that the Yankees will not win the World Series” as not-not-p, where the inference to p (the Yankees will win) does not go through. But, again, this doesn’t give a sense in which p fails to follow from not-not-p, just a situation where a different, but related inference doesn’t hold. Modelling the above claim as not-not-p does not make it a genuine instance of that form.
On pp. 101–102, Beall and Restall consider the interpretation of logical consequence whereby an argument is logically valid iff it is logically valid according to one of the accepted logics (i.e. according to one admissible interpretation of “case”). They call this “a perfectly singular sense of ‘logically valid”—though it isn’t clear what this means, given that each sense is surely singular—but deny that this supports monism if relations also count as logical consequence relations. I am arguing that they shouldn’t so count given the availability of this “univocal” sense. Elsewhere, Beall and Restall address a related argument from Graham Priest which considers some inference that follows according to one logic but not according to another and asks whether we are entitled to accept the conclusion (p. 93). But they just take up a part of Priest’s argument that pushes it further than necessary, namely to the end-point that “we should ... apply the notion of validity appropriate to the smallest class that s is in [i.e. {s}]”. They respond that “all that Priest’s argument shows is that the more we know about s, the more we know about what is true in s” (p. 93). Whereas they may be right that we will only be able to use Priest’s limit case of quantification over the actual situation alone if we know enough about what’s actually true, this doesn’t address the point as explained above, according to which we should use the strongest consequence relation of those endorsed—where we can take ourselves to know the truth of the premises and know the validity of the argument independently of knowing the truth of the conclusion. See also Read (2006) on Priest’s argument.
See e.g. Field (2009b) and Milne (2009) for some discussion of degree-theoretic treatments of the normative requirements of a valid argument, where a valid argument dictates, not just that you should (fully) believe the conclusion if you fully believe the premises, but also the degree to which you should believe the conclusion when you merely partially believe the premises.
Could we accommodate a pluralism that allows both classical logic and intuitionistic logic in a similar way? Hjortland doesn’t want to rule out a semantics that would accommodate both of those logics. Maybe this would help establish that pluralist claim and maybe some other story could be told that allows both to count as true consequence relations, but the devil would be in the detail.
E.g. Varzi 2007 rejects my 2001 pluralist thesis by defending a particular definition of “validity” within the framework in question.
References
Beall, J. C., & Restall, G. (2000). Logical pluralism. Australasian Journal of Philosophy, 78, 475–93.
Beall, J. C., & Restall, G. (2001). Defending logical pluralism. In Logical Consequence: Rival Approaches. In J. Woods, & B. Brown (Eds.), Proceedings of the 1999 conference of the society of exact philosophy (pp. 1–22). Stanmore: Hermes.
Beall, J. C., & Restall, G. (2006). Logical pluralism. Oxford: Clarendon Press.
Bueno, O., & Shalkowski, S. A. (2009a). Modalism and logical pluralism. Mind, 118, 295–321.
Field, H. (2009a). Pluralism in logic. Review of Symbolic Logic, 2, 342–359.
Field, H. (2009b). What is the normative role of logic? Proceedings of the Aristotelian Society, Supplementary Volume, 83, 251–268.
Fine, K. (1975). Vagueness, truth and logic. Synthese, 30, 265–300.
Hjortland, O.T. (2013). Logical pluralism, meaning-variance and verbal disputes. Australasian Journal of Philosophy, 91, 355–373.
Keefe, R. (2000). Theories of vagueness. Cambridge: Cambridge University Press.
Keefe, R. (2001). Supervaluationism and validity. Philosophical Topics, 28, 93–105.
Lewis, D. (1993). Many, but almost one. In J. Bacon, K. Campbell, & L. Reinhardt (Eds.), Ontology, causality and mind: Essays in honour of D.M. Armstrong. Cambridge: Cambridge University Press.
Milne, P. (2009). What is the normative role of logic? Proceedings of the Aristotelian Society, Supplementary Volume, 83, 269–98.
Paseau, A. (2007). Review of logical pluralism. Mind, 116, 391–396.
Read, S. (2006). Monism: The one true logic. In D. DeVidi, & T. Kenyon (Eds.), The logical approach to philosophy. Berlin: Springer.
Restall, G. (2002). Carnap’s tolerance, meaning and logical pluralism. Journal of Philosophy, 99, 426–43.
Smiley, T. (1998). Conceptions of consequence. In E. Craig (Ed.), Routledge encyclopedia of philosophy. London: Routledge.
Tarski, A. (1936). On the concept of logical consequence. In A. Tarski (Ed.), Logic, semantics, metamathematics (2nd ed.). Indianapolis: Hackett Publishing.
Varzi, A. (2007). Supervaluationism and its logics. Mind, 116, 633–75.
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Many thanks to two referees for very helpful comments.
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Keefe, R. What logical pluralism cannot be. Synthese 191, 1375–1390 (2014). https://doi.org/10.1007/s11229-013-0333-x
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DOI: https://doi.org/10.1007/s11229-013-0333-x