Abstract
Alethic pluralism holds that there are many truth properties. The view has been challenged to make sense of the notion of logical validity, understood as necessary truth preservation, when inferences involving different areas of discourse are concerned. I argue that the solution proposed by Edwards to solve the analogous problem of mixed compounds can straightforwardly be adapted to give alethic pluralists also a viable account of validity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Here I use ‘property’ without any realist connotation. On this reading, also a nominalist could be an alethic pluralist.
See Wright (1992) for a first version.
Supporters of strong versions of pluralism are in the minority, and perhaps limited to two authors: Aaron Cotnoir and (more controversially) Crispin Wright. In general, strong pluralism should probably be better characterized as a position holding that there is no generic substantial truth property. In a sense, it is deflationism for what concerns the generic truth property.
The principal supporter of moderate pluralism is Michael Lynch, see for instance Lynch (2009).
Given the nature of the problem of mixed inferences, the present treatment will focus on propositional logic, putting, for the sake of simplicity, mostly aside the possible further complications of an extension to first order logic.
Since the topic is a logical issue, I take, for the sake of simplicity, sentences as truth bearers.
The existence of a pre-theoretical notion of validity is assumed. In this sense I speak of the nature of validity, although the expression might be, strictly speaking, inappropriate for an abstract entity.
The modification is the addition of the qualification ‘general truth’ to the property that is preserved. If the generic property preserved were not a truth property, then the account would be revisionary, violating the first constraint, while if it were not a general property, then it could straightforwardly be combined with strong pluralism.
See for instance Pedersen (2010) for a discussion of the problem.
Given that what is at stake here is an explanation of the nature of validity, not just extensional correlations, the biconditional involved in such equivalences should not be interpreted as a mere material biconditional. I use ‘\(\Leftrightarrow \)’ as a symbol for the material biconditional (of the metalanguage).
The polish logician Roman Suszko proved that any (Tarskian) logic has an adequate bivalent semantics, and took this as evidence for the thesis that any logic is logically two valued. Accordingly, many-valuedness is only apparent. For an application of Suszko’s result to alethic pluralism see Marrano and Strollo (to appear).
See Pedersen (2006) for the details.
Things are more complex, given that theories of truth are generally independent from a nominalism/realism debate about properties (Edwards 2013) but for the sake of simplicity I leave such complications aside. Note, however, that it is not clear how a plural understanding of second order quantifiers could help in a nominalist setting.
See Cotnoir (2013) for other criticisms of Pedersen’s proposal.
The recent proposal by Yu (Forthcoming) is arguably open to objections similar to those raised against Beall and Pedersen’s strategies.
See Tappolet (2000) for a more extensive discussion of the problem.
Edwards’ treatment is focused on conjunctions, but the proposal can be extended in a natural way to cover also other compounds. For analogous considerations, close to the spirit of Edwards’ solution, see also Wright (2013, p. 135).
The exploration of the nature of such a property is left as a topic for a different work.
Indeed, we would be in the case sketched above, according to which every atomic proposition would always have two true properties and truth-for-logic would be had by any proposition (true compounds would only have the truth property of truth for logic).
See Cotnoir (2009) for an objection in this direction.
To be precise, I should specify also the area of discourse, stating something like: (p is in area 1 and p is true1 or p is in area 2 and p is true2...) and (q is in area 1 and q is true1 or q is in area 2 and q is true2...).
Edwards (2009), footnote 3.
A feature of the semantics that might appear troublesome is that, for instance, a sentence like ‘Napoleon lost at Waterloo but he won at Austerlitz’ should be considered a case of truth in the logical area instead of being a simple example of an historical truth, as, intuitively, it should be. However, the point of identifying the truth property of true compounds with truth in the logical area is just that of taking the logical composition of connectives and the order of determination embodied by their logical behaviour seriously. Thus, saying that ‘Napoleon lost at Waterloo but he won at Austerlitz’ is true-for-logic is nothing but a way to say that its truth amounts, in a rather deflationary fashion, to the mere logical combination of two historical truths. Moreover, ‘Napoleon lost at Waterloo but he won at Austerlitz’ and ‘\(2+2=4\) and \(2+5=7\)’, although both true-for-logic, would still be grounded in two different kinds of truths. See Edwards (2008) and Wright (2013, p. 135).
Note that if anyone considered the proposal also implausible, she should attack not the conditional claim defended in this paper, but the original papers addressing the correctness of such a semantics.
Abusing of the language, for the sake of simplicity, I use the label ‘Deduction Theorem’ for the semantic counterpart of the theorem.
Assuming that there is no logical sentential constant for truth (T). In this case, however, it would not be hard to consider it a sentence true-for-logic.
If we move at first order, identity claims are possible exceptions. However, since identity is usually interpreted as a logical constant, also identity claims should belong to the logical area, so that true identities can be taken to instantiate truth for logical discourse on their own.
If one found this approach objectionable, thinking, i.e., that many claims in science are identity claims but not logical claims, at least two alternative options could be explored. An option is that of considering identity claims as cases of mixed atomics. Two strategies would then be available. First, the relevant subject-specific truth property could be determined by the predicate of the atomic sentence. This way, true identity claims could involve non logical notions and, nonetheless, be true-for-logic. Second, we could adopt a treatment à la Cotnoir (2009), attributing more than one subject-specific truth property at the same claim. Identity claims, then, would be also true-for-logic, and the strategy could be applied again.
If, instead, one insisted that identity claims are neither mixed nor logical claims, then the simplest move would be that of denying that identity claims are logical truth. In this case, one might recover the necessity of identity claims by interpreting it in terms of the relevant specific truth property. For instance, ‘Hesperus is Hesperus’ should be a necessary truth of physics but not a logical truth.
To enlighten the point, Michael Lynch proposes an analogy with an expression like ‘the colour of the sky at noon’. Lynch (2009, p. 61).
Once a monist reformulates a valid inference in terms of a true conditional, she is forced to say that its truth consists in the monist truth property, not in the property of having the consequent true anytime its antecedent is true. Otherwise different true compounds, true for different reasons, would exhibit different truth properties and monism would be lost. See Edwards (2008, 2009).
As explained above, this formulation is available to the strong pluralist and does not force a commitment to a generic truth property.
It is worth stressing that this reliance on traditional tools (namely logics validating the Deduction Theorem) is of a different kind from the one Cotnoir makes, and for which his account can be criticised. In traditional cases of valid inferences, in fact, I do offer a truth-theoretic account, while Cotnoir defends a revisionary one, just claiming that this move is not problematic because truth preservation is replaced with something we are familiar with. In non standard cases Cotnoir makes the same move, while I suggest that, even if I could not treat them, a revisionary account of validity would be less worrisome, given that the logical systems are already deviant. Thus, my proposal could be accused, at most, of having a limited range, not of being a revisionary account in familiar clothes.
The fact that certain logics are recalcitrant does not make them completely hopeless. For instance, the lack of a Deduction Theorem in certain logics could be imputable to the expressive limitations of that logic, namely to the absence of a suitable conditional adequately reflecting the underlying structure of the relation of logical consequence. Alternatively, one might try to reformulate the account at the meta-level, exploiting the conditional of the meta-theory.
Remarkably, the detour through the Deduction Theorem can be seen as a standard way to defend the characterization of logical validity as truth preservation. A criticism of the Theorem, in fact, can easily lead to question this very characterization. See Field (2015).
One might argue that the failure of the Theorem in certain logics signals a conceptual gap between the two sides of the equivalence, undermining the whole proposal. However, lacking a detailed argument, one could insist that such a failure is, instead, a sign that some deviant feature is embodied in those logics, making the possible gap irrelevant in this context. In any case, one might also think that the gap is limited to those logics.
References
Beall, J. C. (2000). On mixed inferences and pluralism about truth predicates. Philosophical Quarterly, 50, 380–382.
Cotnoir, A. J. (2009). Generic truth and mixed conjunctions: Some alternatives. Analysis, 69(3), 473–479.
Cotnoir, A. J. (2013). Validity for strong pluralists. Philosophy and Phenomenological Research, 86(3), 563–579.
Edwards, D. (2008). How to solve the problem of mixed conjunctions. Analysis, 68, 143–149.
Edwards, D. (2009). Truth-conditions and the nature of truth: re-solving mixed conjunctions. Analysis, 69(4), 684–688.
Edwards, D. (2013). Truth as a substantive property. Australasian Journal of Philosophy, 91(2), 279–294.
Field, H. (2015). What is logical validity? In Colin R. Caret & Ole T. Hjortland (Eds.), Foundations of logical consequence. Oxford: Oxford University Press.
Lynch, M. (2009). Truth as one and many. Oxford: Oxford University Press.
Marrano, R., & Strollo, A. (to appear). A renewed challenge for strong alethic pluralism: mixed inferences and Suszko’s reduction.
Pedersen, N. J. (2006). What can the problem of mixed inferences teach us about alethic pluralism? Monist, 89, 103–117.
Pedersen, N. J. (2010). Stabilizing alethic pluralism. Philosophical Quarterly, 60(238), 92–108.
Sainsbury, M. (1996). Crispin wright: Truth and objectivity. Philosophy and Phenomenological Research, 56, 899–904.
Tappolet, C. (1997). Mixed inferences: a problem for pluralism about truth predicates. Analysis, 57, 209–210.
Tappolet, C. (2000). Truth pluralism and many-valued logics: A reply to Beall. Philosophical Quarterly, 50, 382–385.
Williamson, T. (1994). A critical study of ’truth and objectivity. International Journal of Philosophical Studies, 30, 130–144.
Wright, C. (1992). Truth and objectivity. Cambridge, MA: Harvard University Press.
Wright, C. (2013). A plurality of pluralisms? In N. J. L. L. Pedersen & C. D. Wright (Eds.), Truth and pluralism: Current debates (pp. 123–153). New York: Oxford University Press.
Yu, A. (Forthcoming). Logic for alethic pluralists. Journal of Philosophy.
Acknowledgements
I would like to acknowledge the Scuola Normale Superiore, Massimo Mugnai and Giorgio Lando for support during the time this paper was written. A special thanks goes to Cogito-Research Center in Philosophy for the insightful discussions and the exciting events on alethic pluralism in which I have been kindly involved. I must also thank Andy Yu and the anonymous referees for very helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Strollo, A. A simple notion of validity for alethic pluralism. Synthese 195, 1529–1546 (2018). https://doi.org/10.1007/s11229-016-1280-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-016-1280-0