Abstract
Anti-exceptionalism about logic is the thesis that logic is not special. In this paper, I consider, and reject, a challenge to this thesis. According to this challenge, there are basic logical principles, and part of what makes such principles basic is that they are epistemically exceptional. Thus, according to this challenge, the existence of basic logical principles provides reason to reject anti-exceptionalism about logic. I argue that this challenge fails, and that the exceptionalist positions motivated by it are thus unfounded. I make this case by disambiguating two senses of ‘basic’ and showing that, once this disambiguation is taken into account, the best reason we have for thinking that there are basic principles actually implies that those principles do not require a special epistemology. Consequently, the existence of basic logical principles provides reason to accept, rather than reject, anti-exceptionalism concerning the epistemology of logic. I conclude by explaining how an abductivist, anti-exceptionalist approach to the epistemology of logic can accommodate the notion of basic logical principles.
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Notes
I take the term ‘systematically basic’ from Carlson (2015, Sect.1.3), and I use it in largely the same sense here. The only significant difference is that here I am concerned with the basic status of individual logical principles, as opposed to that of logical theory as a whole.
I should also note that the explicatory emphasis here is on ‘principle’, not ‘logical’. There are, of course, difficult theoretical issues concerning the “bounds” of logic; concerning whether, e.g., second order logic is really logic properly so-called. I’ll skirt this issue by focusing on principles that are, I think, clearly logical.
If you don’t find this so compelling, consider instead as an example an extremely complicated valid principle; a principle whose validity you cannot simply determine “on sight.”
This is roughly the sense in which the term is used in Wright (2002). While it may be a proof, a one-line proof consisting solely of a statement of an axiom of the sort described in Williamson (forthcoming) is not a cogent inference in this sense.
Elsewhere, Dummett calls this a “proof-theoretic justification of the first grade” (Dummett, 1991, p. 188).
Of course, I might also employ resources in my argument that are not themselves logical principles. I might apply a tableaux procedure in which I perform various operations on the formulae of the principle (e.g. I operate on a disjunction by splitting into into two branches, etc.). But here, again, I’ve reduced the question whether the principle is valid to the question of whether those operations track validity. Or, I might employ resources that are not simply syntactic; I might avail myself of a theory of truth-in-an-interpretation and prove metatheoretically that the succedent is true in any interpretation in which the antecedent is true. Here I’ve reduced the question of the justification of that principle to the justification of the metatheoretic resources employed in this semantic proof.
The reasoning here, typically left implicit by advocates of the reduction picture, is essentially that of Agrippa’s trilemma, famous from the regress problem in epistemology. According to this reasoning, our beliefs must be supported either by infinite sequences of reasons, sequences of reasons which are ultimately circular, or sequences of reasons that terminate in foundational beliefs. The argument for thinking that there must be basic logical principles ultimately relies on the thought that the overall structure of logical principles cannot be circular or infinitely descending. Thanks to an anonymous reviewer for helping me to clarify this point.
This is a central idea in Frege’s logicist project, and I think its influence has extended well beyond the bounds of logicism. Frege’s idea was that all proof-chains will eventually terminate in unprovable “primitive truths”, and we can determine the status of the theorem proved (i.e. whether it is analytic, synthetic a priori, or synthetic a posteriori) by determining the statuses of those primitive truths. Arithmetic was to turn out to be analytic, on this view, because the primitive truths from which all theorems of arithmetic were to be proven are analytic (i.e. are definitions and logical laws) (Frege, 1950, Sects. 3–4).
Caveat: you might suppose, inspired by the work of Prawitz (2006), that modus ponens can be reduced, in the system we are considering, to a principle of conditionalization. That may be so, but then of course the issue we are considering here will simply arise for that latter principle. So I will ignore this possibility for now.
Dag Prawitz puts the point nicely: “Tarski’s definition...involves exactly [this] kind of circularity...Whether e.g. a sentence \(\exists {x}\lnot {}P(x)\) follows logically from a sentence \(\lnot \forall {x}P(x)\) depends according to this definition on whether \(\exists {x}\lnot {}P(x)\) is true in each model (D, S) in which \(\lnot \forall {x}P(x)\) is true. And this again is the same as to ask whether there is an element e in D that does not belong to S whenever it is not the case that every e in D belongs to S, i.e. we are essentially back to the question whether \(\exists {x}\lnot {}P(x)\) follows from \(\lnot \forall {x}P(x)\)” (Prawitz, 1974, p. 68).
This also agrees with Dummett’s characterization of basic principles of a logical system (Dummett, 1991).
There’s an additional question we should ask, though due to limited space I won’t address it in any detail here. This question is: Are we in any sense working in a system; in an all-encompassing scheme within which all proving takes place? One reason for thinking that we are is that the central goal of studies in the foundations of logic and mathematics is to systematize or rationally reconstruct these disciplines. Such a project, naturally associated with Frege, aims to show that our successful logical and mathematical practice can be rationally reconstructed so as to constitute a system in the desired sense, and it is via this rational reconstruction that we can best understand how and why our practices are successful. I will have a bit more to say about this idea at the end of this paper.
It is important to clarify that I am not claiming here that the systematically basic principles of a given logic must be complete for that logic in the usual metatheoretic sense. This is because I am not assuming that systematically basic principles must be finitely axiomatizable.
With this word choice I aim to call to mind Tarski’s criterion of material adequacy. Essentially the idea here is that a set of principles is adequate when it produces all the results that we want it to. In the case of systematically basic principles, what we want is that they underwrite (or maybe better, comprise) the entire logical system. As Frege put it, the basic laws of his system were to contain the whole of arithmetic, as a seed contains a plant.
We’ll have a bit more to say on this issue at the end of this paper.
Here I share Goodman’s general attitude. Concerning a related point, he claims that “[a] philosophic problem is a call to provide an adequate explanation in terms of an acceptable basis.” And, “[t]he disproportionate emphasis put on the problem has resulted in a gross exaggeration of the consequences of the failure to solve it. Lack of a general theory of goodness does not turn vice into virtue; and lack of a general theory of significance does not turn empty verbiage into illuminating discourse” (Goodman, 1983, p. 31). Similarly, lacking an account of the epistemology of logic does not deprive us of justification for accepting logical principles. That said, there is a possibility of a skeptical worry creeping in here. If, despite our best efforts, we find ourselves unable to explain how logical principles can be justified, one possible explanation for this state of affairs is that logical principles cannot be justified after all. As should become clear by the end of this paper, I don’t think this degree of skepticism is warranted in this case. Thanks to an anonymous referee for this journal for pressing me to clarify this point.
More accurately, Schechter’s view is that reasoning in accord with basic logical principles is a byproduct of a trait that our ancestors were selected for, but this subtlety does not make a difference to my overall point here.
Williamson is using ‘mathematical’ here in a broad sense that also encompasses logic.
Where, as usual, \(\phi (x/a)\) is the result of uniformly substituting a for all free occurrences of x in \(\phi \), and where a does not occur in \(A\).
Gödel’s first incompleteness theorem may be relevant to this point. If we assume that (1) there are only finitely many (independent) epistemically basic principles, (2) the epistemically basic principles are consistent, and (3) our overall system is sufficiently strong, then Gödel’s theorem implies that the epistemically basic principles are not adequate with respect to our system.
The point raised in this paragraph bears some similarity to Sellars’s famous attack on the “myth of the given” (Sellars, 1956). But to ward off undue confusion, I will point out an important difference. According to Sellars, foundationalism’s “basic beliefs”, including importantly perceptual beliefs, must be directly accessible in some way. But, because of this, these beliefs must be nonceptualized. But if this is so, they will be incapable of entered into inferential relations, and will thus be isolated from the non-basic beliefs which they are supposed to support. If this is right, the basic beliefs are grossly inadequate. By contrast, the inadequacy I am urging for the epistemically basic principles is not so gross. My claim is not that they are incapable of interacting with other principles—they can be employed to prove the validity plenty of other principles—but only that they do not suffice to prove all other principles that we accept.
It is insufficient to derive the validity of even fairly simple and uncontroversial logical principles such as modus tollens.
This assumption engenders no loss of generality. For, if P is not systematically basic, the issue we are about to discuss will nevertheless arise for whichever principle(s) P is ultimately based on, and these cannot be in or based on principles in C.
To be clear, this sort of account might not be adequate for other explanatory and justificatory aims. For example, if the task is to explain why we find some logical principles to be obvious, a core/periphery approach might be appropriate. Similarly, if our task is to bolster the justification of basic logical principles as much as possible, we might accomplish this by finding multiple, independent sources of justification for basic logical principles. Perhaps they get some justification simply for being systematically basic, and additional justification due do some other special property that some of them enjoy. Thanks to an anonymous referee for this journal for raising this interesting issue. See also Shapiro (2009) for some important additional issues related to, but distinct from, those considered here.
See Williamson (2016) for more discussion of this idea. Note, too, that while Russell’s proposed methodology is broadly abductivist, he also holds that logical principles can be justified a priori. Once again, this shows that the exceptionalist/anti-exceptionalist distinction cuts across the naturalist/a priorist distinction, contrary to the claims of Hjortland (2017).
With this formulation, I aim to call to mind Goodman’s point that general principles of inference and specific inferences are both justified by being brought into “equilibrium” with one another (Goodman, 1983). But this general idea leaves a lot of work to be done in specifying the details. Most notably, an important question remains: what exactly are the inferences to be systematized? At the very least, I think, they include inferences contained in our most successful deductive practice, that of mathematical proof. But they may also include vernacular inferences whose apparent validity we would like to account for. See Martin and Hjortland (2021) for detailed and helpful discussion of this issue.
See Carlson (2015) for further discussion of the structure/source distinction and its relationship to the justification of logical systems.
References
Boghossian, P. (2000). Knowledge of logic. In P. Boghossian & C. Peacocke (Eds.), New essays on the a priori (pp. 229–254). Oxford University Press.
BonJour, L. (1998). In defense of pure reason: A rationalist account of a priori justification. Cambridge University Press.
Carlson, M. (2015). Logic and the structure of the web of belief. Journal for the History of Analytical Philosophy, 3(5), 1–29.
Corcoran, J., & Hamid, I. S. (2016). Schema. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy, Fall (2016th ed.). Metaphysics Research Lab: Stanford University.
Dummett, M. (1978). The justification of deduction. In Truth and other enigmas (pp. 290–318). Harvard University Press.
Dummett, M. (1991). The logical basis of metaphysics. Harvard University Press.
Frege, G. (1950). The foundations of arithmetic. Blackwell.
Frege, G. (1979). Posthumous writings. University Of Chicago Press.
Frege, G. (2013). Basic laws of arithmetic. Oxford University Press.
Goodman, N. (1983). Fact, fiction, and forecast (4th ed.). Harvard University Press.
Hale, B. (2002). Basic logical knowledge. Royal Institute of Philosophy Supplement, 51, 279–304.
Hjortland, O. T. (2017). Anti-exceptionalism about logic. Philosophical Studies, 174(3), 631–658. https://doi.org/10.1007/s11098-016-0701-8
Irvine, A. D. (1989). Epistemic logicism & Russell’s regressive method. Philosophical Studies, 55(3), 303–327. https://doi.org/10.1007/BF00355328
Maddy, P. (2002). A naturalistic look at logic. Proceedings and Addresses of the American Philosophical Association, 76(2), 61–90.
Maddy, P. (2007). Second philosophy: A naturalistic method. Oxford University Press.
Martin, B., & Hjortland, O. (2021). Logical predictivism. Journal of Philosophical Logic, 50(2), 285–318. https://doi.org/10.1007/s10992-020-09566-5
Parsons, C. (2008). Mathematical thought and its objects. Cambridge University Press.
Peacocke, C. (1998). Implicit conceptions, understanding and rationality. Philosophical Issues, 9, 43–88.
Prawitz, D. (1974). On the idea of a general proof theory. Synthese, 27(1/2), 63–77.
Prawitz, D. (2006). Natural deduction: A proof-theoretical survey. Dover Publications.
Russell, B. (1973). The regressive method of discovering the premises of mathematics. In D. Lackey (Ed.), Essays in analysis (pp. 272–283). George Braziller Inc.
Russell, G. (2020). Logic isn’t normative. Inquiry: An Interdisciplinary Journal of Philosophy, 63(3–4), 371–388. https://doi.org/10.1080/0020174x.2017.1372305
Schechter, J. (2010). The reliability challenge and the epistemology of logic. Philosophical Perspectives, 24(1), 437–464. https://doi.org/10.1111/j.1520-8583.2010.00199.x
Schechter, J. (2013). Could evolution explain our reliability about logic? In T. S. Gendler, & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 4, p. 214). Oxford University Press.
Schechter, J. (2018). Is there a reliability challenge for logic? Philosophical Issues, 28(1), 325–347. https://doi.org/10.1111/phis.12128
Sellars, W. S. (1956). Empiricism and the philosophy of mind. Minnesota Studies in the Philosophy of Science, 1, 253–329.
Shapiro, S. (2009). We hold these truths to be self-evident: But what do we mean by that? The Review of Symbolic Logic, 2(01), 175–207.
Sher, G. (2013). The foundational problem of logic. Bulletin of Symbolic Logic, 19(2), 145–198. https://doi.org/10.2178/bsl.1902010
Williamson, T. (2016). Abductive philosophy. Philosophical Forum, 47(3–4), 263–280. https://doi.org/10.1111/phil.12122
Williamson, T. (2017). Semantic paradoxes and abductive methodology. In B. Armour-Garb (Ed.), Reflections on the Liar (pp. 325–346). Oxford University Press.
Williamson, T. (forthcoming). Absolute provability and safe knowledge of axioms. In L. Horsten, & P. Welch (Eds.), The limits of mathematical knowledge. Oxford University Press.
Wright, C. (2002). (Anti-)sceptics simple and subtle: G.E. Moore and John McDowell. Philosophy and Phenomenological Research. https://doi.org/10.1111/j.1933-1592.2002.tb00205.x
Wright, C. (2004). Warrant for nothing (and foundations for free?). Proceedings of the Aristotelian Society, Supplementary Volumes, 78, 167–245.
Zermelo, E. (1908). A new proof of the possibility of a well-ordering. In J. van Heijenoort (Ed.), From Frege to Gödel: A sourcebook in mathematical logic (pp. 1879–1931). Harvard University Press.
Acknowledgements
This paper had a lengthy gestation period, and far more people helped me wrestle with the ideas in it than I can adequately acknowledge in this space. I presented earlier versions of the material in this paper at the Eastern APA, the Society for Exact Philosophy, and at Smith College, Wabash College, and the University of New Mexico, and I am grateful to audiences on all of these occasions for helpful questions and comments. Moreover, I am grateful to several anonymous referees for this journal whose careful and astute suggestions made this paper much stronger. Finally, I owe special debts of gratitude to Joshua Schechter, Gary Ebbs, and David McCarty for their patient and helpful comments on earlier versions of this paper and the ideas it contains.
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This paper is dedicated to the memory of David McCarty, who taught me a great deal about logic and its philosophy.
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Carlson, M. Anti-exceptionalism and the justification of basic logical principles. Synthese 200, 214 (2022). https://doi.org/10.1007/s11229-022-03715-w
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DOI: https://doi.org/10.1007/s11229-022-03715-w