Abstract
In this paper, I ask whether we should see different logical systems as appropriate for different domains (or perhaps in different contexts) and whether this would amount to a form of logical pluralism. One, though not the only, route to this type of position, is via pluralism about truth. Given that truth is central to validity, the commitment the typical truth pluralist has to different notions of truth for different domains may suggest differences regarding validity in those different domains. Indeed, as we’ll see, the differences between the proposed multiple notions of truth are often of a type that is clearly significant in relation to logical features, such as whether or not a constructive notion of truth is at issue. I criticise domain-based logical pluralism. Having done so I introduce a context-based framework that operates with a context-relative notion of validity. I show that this context-based framework can be employed by the domain-specific logical pluralist, but that framework also allows for logical pluralism that does not involve several domains. Different contexts may demand rules of classical logic, where others only justify intuitionistic rules, even when the same domain (e.g. mathematics) is at issue.
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Notes
- 1.
Uses of logical systems can go beyond capturing logical consequence relations. For example, it might be useful to employ a paraconsistent logic to work with databases that may contain inconsistent data. But that is not to say that there are true contradictions, even though it can be on the record that p and on the record that not-p. See Keefe 2014, footnote 13.
- 2.
See Keefe 2000, chapter 5, and on pluralism about supervaluationist notions of consequence, see Keefe 2001. Hjortland 2012 offers the helpful term “intra-theoretic pluralism” for this kind of pluralism. A different relatively modest logical pluralism is one that maintains that there is no uniquely correct choice of logical constants and so we can construct different equally good logics by selecting different sets from the putative logical constants.
- 3.
- 4.
For example, Wright maintains that there is one concept but many properties and Lynch adopts a functionalism about truth whereby different concepts play the role.
- 5.
Pedersen (2014, p. 262) maintains that “alethic pluralism argues that any alethic pluralism that accepts both realist and anti-realist conceptions of truth “brings on a commitment to logical pluralism” and Lynch writes: ‘The alethic functionalist … is not required to endorse …logical pluralism. But it is likely that she will.’ (2009, p. 104).
- 6.
Pedersen 2014, p. 260.
- 7.
- 8.
- 9.
See Lynch 2008, p. 137.
- 10.
Lynch 2008, p. 139 responds to the objection that the first approach he considers—the logic of a mixed inference is the logic governing the weakest element of it—requires “the assumption that the logics in question can be ordered, in the sense that the stronger logics are extensions of the weaker logics” by maintaining, without further explanation, that this is not “an unreasonable constraint on those logics that apply only to specific domains of inquiry”. But he also considers the above objection of taking the logic to be the intersection of the logics governing the elements of the inference.
- 11.
Note that this denial of a unique property is compatible with accepting a unified concept of truth. On some conceptions of properties (e.g. concept nominalism), that would surely be enough for commitment to a single general property of truth, in which case the distinction between weak and strong pluralism would collapse.
- 12.
Along the way to showing this, he shows that his set of values with conjunction and disjunction forms a lattice with the top value taking 1 in every place and the bottom value taking 0 in every place.
- 13.
This illustration is an odd case of a sentence that doesn’t fall into any domain. The Law of Excluded Middle also fails for more typical sentences within some domain, as the first disjunct, and thus the second disjunct too, will typically take value ½ in some of the places and thus the disjunction will not take the top value (1 in all places). The only cases where it will take the top value is when the disjuncts have 1 or 0 in each place, which will be the rare cases of sentences simultaneously in all domains, for example, perhaps itself a disjunction with a disjunct from each domain.
- 14.
“In this section, I extend the algebraic account of validity to nonclassical domains, showing how the account can handle domains for which paracomplete, paraconsistent and intuitionistic logic seem most appropriate” (p. 13).
- 15.
Might it be represented instead by alternative Boolean algebra, so that the logic is still classical, even though bivalence does not hold? If bivalence is a logical feature of a domain (as Lynch argues, 2008), then surely only the two-valued Boolean algebra will be able to capture the domain.
- 16.
Domain-specific logical pluralism is not the kind of position Beall and Restall are interested in, for example. They would, I think, regard it as a form of relativism, since what is valid is relative to the domain/context. Beall and Restall’s logical pluralism turns on the different interpretations of case in the defining principle of logical consequence (GTT): “An argument is validx if and only if, in every casex in which the premises are true, so is the conclusion.” Pedersen (2014) seeks to adopt their framework by explaining how different notions of case could be appropriate to different domains. But (GTT) commands a universality in its biconditional, and limiting the notion of case to a single domain does not provide a “precisification” of “case” in the manner required for Beall and Restall’s framework.
- 17.
Field 2009, p. 344.
- 18.
- 19.
The arguments below do not require that the logic of mixed inferences is distinct from all of the logics of the pure domains—it could, rather, be the weakest, for example, intuitionistic logic if the domain-specific options are just classical logic and intuitionistic logic. The same questions arise as to the status of the stronger logics.
- 20.
Perhaps, then, we should deny that there is any mixed domain—the domain-specific logical pluralist could consider the logic governing mixed inferences (the intersection of the logics of the pure domains) without requiring a corresponding domain. Different logics may then result from different domains either by being logics of those different domains or resulting from the interaction of different domains in a more complex way. Whether there is a mixed domain or not, the argument regarding the universality of the logic of mixed domains still holds.
- 21.
Or, if SP is not strictly a domain, then the assessment via the most general logic still applies.
- 22.
Beall and Restall (2006), for example , identify the settled core features of logical consequence relations as necessity, normativity and formality: to qualify as one of the genuine consequence relations of their logical pluralism, a relation must have these features. Although details of Beall and Restall’s “settled core” are controversial, a requirement of formality is widely accepted and the domain-specific relations in question meet none of the candidate more detailed specifications of the formality criterion.
- 23.
See, for example, Shapiro 2014 for a view taking this line, focusing on a range of fruitful mathematical theories that use different logics (e.g. intuitionistic analysis).
- 24.
- 25.
Lewis 1982.
- 26.
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Keefe, R. (2018). Pluralisms: Logic, Truth and Domain-Specificity. In: Wyatt, J., Pedersen, N., Kellen, N. (eds) Pluralisms in Truth and Logic. Palgrave Innovations in Philosophy. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-98346-2_18
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