Abstract
Logical nihilism is the view that the relation of logical consequence is empty: there are counterexamples to any putative logical law. In this paper, I argue that the nihilist threat is illusory. The nihilistic arguments do not work. Moreover, the entire project is based on a misguided interpretation of the generality of logic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
I use the turnstile, ‘\(\vdash \)’, to denote consequence. Later, I shall use the colon, ‘:’, to denote inferences and rules of inference. The Roman capitals are all metavariables, standing for (formulae expressing) sentences or for collections of (formulae expressing) sentences of a (formal) language. They are used to express schematic consequence claims as well as inference rules. I write specific (in a sense to be clarified later) consequence claims and inferences using sentential variables, p, q, r, etc.
I shall also assume that the logical vocabulary undergoes no changes of meaning as the space of interpretations is enlarged. This is a substantive assumption, particularly in a model-theoretic framework. See also infra, p. 5. (Thanks to an anonymous referee for pointing this out.)
What ‘features in’ means is open to speculation. In the simplest case, which is enough for my purposes, it simply means ‘is’.
An anonymous referee takes exception to this claim, pointing out that (first-order) FDE can represent actual mathematical proofs, as these are carried on in, as it were, FDE plus the (para-logical) assumptions of consistency (no gluts) and completeness (no gaps) of the respective domain. This I happily grant. A better way of expressing my claim would be to say that the strictly logical resources of FDE are insufficient for a full codification of actual mathematical proofs; for that to be possible, extraneous assumptions must be in place. This is an important nuance. If a logic is not strong enough to provide sufficient resources for reasoning about a certain domain, then one can find solutions for this—in the present context, Beall’s shrieking and shrugging are the obvious examples (Beall 2018). Nevertheless, this does not reflect back (positively) on the expressive resources of the logic itself. (Although I shall not get into this here, the issue of combining mathematics with subclassical and in particular relevant logics is more subtle than the discussion above may lead one to believe—see, for discussion, Mares (2012).)
Russell seems to be in good company when she sees paucity as (almost) as bad as absolute lack. For instance, Estrada-González (2011, p. 117), ostensibly engaging in an exegesis of the anti-non-necessitarian argument in Priest (2006), states the Priestian thesis that ‘a [general, domain independent] logic exists’ as follows: ‘There is at least one collection of inferences holding in all situations and this collection is large enough’. And later on p. 118: ‘even though if the collection of valid inferences were not empty, if it consisted of, say, only one or just [a] few inferences, it would be vacuous in practice to call “logic” such a small number of valid inferences’. While I cannot find any textual evidence that Priest is actually concerned with the size of the set of valid inferences, it does seem that Estrada-Gonzales himself is sympathetic to the view.
This is controversial but see Dicher and Paoli (2018) for arguments in favour of the claim. I shall have more to say about this once I have sketched more details of the picture—see footnote 12.
A rule is admissible in a sequent calculus iff its addition to the calculus does not increase the stock of provable sequents. The rule instantiated by (2) is, in fact, derivable (given the premiss, there is a derivation of the conclusion) in many logics, as long as Cut it present.
Blok and Jónsson call this relation similarity. Their notion of equivalence is more complex, being designed to deal with substitution invariance in the very abstract way required by the arbitrariness of carrier sets of \(\vdash _{1}\) and \(\vdash _{2}\). For my present purposes, this simplified definition suffices.
Now I can add more details to this picture, as promised in footnote 8. (In this footnote I use ‘consequence relation’ as shorthand for ‘logical consequence relation’.) On the metainferential account of logic inspired by the Blok-Jónsson generalisation of Tarski’s definition of consequence, all metainferential consequence relations are Tarskian. Moreover, the aforementioned recovery of consequence on formulae yields Tarskian consequence relations even for substructural logics. So every logic, qua consequence relation, is structural and, contra the received view, the failure of these rules does not amount to the failure of the homonymous properties. Substructural logics are still logics, but they are not substructural consequence relations. For some this may be too steep a price to pay, cf. Barrio et al. (2019a, b). The same authors put forward a different, if equally metainferential, conception of logic, on which a logic is identified as the transfinite union of all its (meta)inferential levels. However, this is a quarrel between metainferentialists and it is of little help to the (Russellian) nihilist. Either metainferentialist position makes room for true logicality, albeit differently. (My thanks to an anonymous referee for pressing me to clarify these issues.)
Russell sketches a Lakatosian way of resisting the nihilist threat by way of lemma incorporation, i.e., tweaking the statement of our logical (meta)theorems to rule out potential counterexamples from their range. Ultimately, however, this gives up on the generality of logic. So it either isn’t a remedy or, if it is, then it undercuts the problem. I do not object to the procedure: a cursory survey of elementary textbooks will show that logicians have been Lakatosians all the time. I do, however, object to embracing Lakatosianism because of nihilism. To do that is to blow in the yoghurt before getting burned by hot soup.
Could it be that this is all that the nihilist is claiming? I believe that this is not so. But if it is, then we would benefit more by analysing nihilism from the perspective of modern marketing, rather than that of the philosophy of logic.
As Quine’s influence waned, second-order logic (re)gained more and more friends (see, e.g. Shapiro 1991).
I assume that natural language instances of quantification over ‘properties’ are best explained via the standard semantics of second-order logic. This is contentious because the grammar of natural language quantification over properties is obscure. When one says, e.g., ‘for every property’ it looks as though a certain reification of the ‘property’ happens, which suggests that the quantifier is still first-order. That may support Henkin semantics rather than to the standard one. Be that as it may, in this context the matter is ‘academic’. It is plain that the quandary described in this footnote supports my main claim just as much as the scenario in the main text. (I am indebted to Bruno Jacinto for raising this issue.)
A caveat: this is not a change of topic, although prima facie, when one complains of infringements to the generality of logic one does not complain about what’s left out by a given logical theory. Instead, one complains about how that theory deals with phenomena it can accommodate. However, for the sake of the argument, here we join the nihilist and take an intra-linguistic and rather exhaustive view of generality, whereby its content is determined by natural language.
A bit of care is required here. As an anonymous referee points out, neither idealisation nor abstraction lead to generality. However, I am not invoking idealisation and abstraction as means of attaining generality. Rather, I am saying that, on the modelling half of the coin, they are parameters that must be considered before generality.
References
Anderson, A. R., & Belnap, N. D. (1975). Entailment: The logic of relevance and necessity (Vol. 1). Princeton: Princeton University Press.
Allo, P. (2017). A constructionist philosophy of logic. Minds and Machines, 27, 545–564.
Barrio, E. A., Pailos, F., & Szmuc, D. (2018). Substructural logics, pluralism and collapse. Synthese (Online First: 1–17). https://doi.org/10.1007/s11229-018-01963-3.
Barrio, E. A., Pailos, F., & Szmuc, D. (2019a). (Meta)Inferential levels of entailment beyond the Tarskian paradigm. Synthese (Online First: 1–25). https://doi.org/10.1007/s11229-019-02411-6.
Barrio, E.A., Pailos, F., & Szmuc, D. (2019b). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic (Online First: 1–28). https://doi.org/10.1007/s10992-019-09513-z.
Barrio, E. A., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.
Beall, Jc. (2017a). There is no logical negation: True, false, both, and neither. Australasian Journal of Logic, 14(1).
Beall, Jc. (2017b). Christ—A contradiction: A defense of contradictory christology (Unpublished manuscript).
Beall, Jc. (2018). The simple argument for subclassical logic. Philosophical Issues, 28, 30–54.
Beall, Jc, & Restall, G. (2006). Logical pluralism. Oxford: Oxford University Press.
Bimbó, K. (2015). Proof theory: Sequent calculi and related formalisms. Boca Raton, FL: CRC Press.
Blok, W. J., & Jónsson, B. (2006). Equivalence of consequence operations. Studia Logica, 83(1–3), 91–110.
Brouwer, L. E. J. (1981). Brouwer’s Cambridge lectures on intuitionism. In D. van Dalen (Ed.). Cambridge: Cambridge University Press.
Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122(488), 841–866.
Cook, R. T. (2010). Let a thousand flowers bloom: A tour of logical pluralism. Philosophy Compass, 5, 492–504.
Cotnoir, A. (2019). Logical nihilism. In N. Kellen, N. J. Pedersen, & J. Wyatt (Eds.), Pluralisms in truth and logic. Cham: Palgrave Macmillan.
Dicher, B. (2018). Variations on intra-theoretical logical pluralism: Internal versus external consequence. Philosophical Studies (Online First: 1–20). https://doi.org/10.1007/s11098-018-1199-z.
Dicher, B. (2019). Reflective equilibrium on the fringe: The tragic threefold story of a failed methodology for logical theorising. Dialectica (special issue of formalisation, forthcoming).
Dicher, B., & Paoli, F. (2018). The original sin of proof-theoretic semantics. Synthese (Online First: 1–26). https://doi.org/10.1007/s11229-018-02048-x.
Dicher, B., & Paoli, F. (2020). ST, LP and tolerant metainferences. In C. Başkent & T. Ferguson (Eds.), Graham Priest on Dialetheism and Paraconsistency. Berlin: Springer.
Dunn, J. M., & Hardgree, G. (2001). Algebraic methods in philosophical logic. Oxford: Oxford University Press.
Dutilh Novaes, C. (2011). The different ways in which logic is (said to be) formal. History and Philosophy of Logic, 32(4), 303–332.
Dutilh Novaes, C. (2012). Formal languages in logic: A philosophical and cognitive analysis. Cambridge: Cambridge University Press.
Eklund, M. (1996). How logic became first-order. Nordic Journal of Philosophical Logic, 1, 147–167.
Estrada-González, L. (2011). On the meaning of connectives (Apropos of a non-necessitarianist challenge). Logica Universalis, 5, 115–126.
Estrada-González, L. (2012). Models of possibilism and trivialism. Logic and Logical Philosophy, 21, 175–205.
Estrada-González, L. (2015). Fifty (more or less) shades of logical consequence. In P. Arazim & M. Dančák (Eds.), Logica Yearbook 2014. London: College Publications.
Etchemendy, J. (1999). On the concept of logical consequence. Stanford: CSLI.
Etchemendy, J. (2008). Reflections on consequence. In Douglas Patterson (Ed.), New essays on Tarski and philosophy (pp. 263–299). Oxford: Oxford University Press.
Fjellstad, A. (2019). Logical nihilism and the logic of ‘prem’ (Unpublished manuscript).
Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle: Louis Nebert. (English translation in J. van Heijenoort (ed), From Frege to Gödel: A source book in mathematical logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967).
Franks, C. (2016). Logical nihilism. In P. Rush (Ed.), The metaphysics of logic (pp. 109–127). Cambridge: Cambridge University Press.
French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo: An Open Access Journal of Philosophy, 3.
Gentzen, G. (1935). Untersuchungen über das logische Schließen I, II. Mathematische Zeitschrift, 39(1): 176–210; 405–431.
Georgi, G. (2015). Logic for languages containing referentially promiscuous expressions. Journal of Philosophical Logic, 44, 429–451.
Glanzberg, M. (2015). Logical consequence and natural language. In C. Caret & O. T. Hjortland (Eds.), The foundations of logical consequence (pp. 71–120). Oxford: Oxford University Press.
Haack, S. (1978). Philosophy of logics. Cambridge: Cambridge University Press.
Humberstone, L. (1996). Valuational semantics of rule derivability. Journal of Philosophical Logic, 25(5), 451–461.
Jansana, R. (2016). Algebraic propositional logic. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy, (Winter 2016 Edition). https://plato.stanford.edu/archives/win2016/entries/logic-algebraic-propositional/.
Kaplan, D. (1979). On the logic of demonstratives. Journal of Philosophical Logic, 8(1), 81–98.
Kaplan, D. (1989). Demonstratives. In J. Almog, J. Perry, & H. Wettstein (Eds.), Themes from Kaplan (pp. 481–563). Oxford: Oxford University Press.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.
Macbeth, D. (2005). Frege’s logic. Cambridge, MA: Harvard University Press.
MacFarlane, J. (2000). What does it mean to say that logic is formal? Dissertation, University of Pittsburgh. Available at https://johnmacfarlane.net/dissertation.pdf.
MacFarlane, J. (2017). Logical constants. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Winter 2017 Edition). https://plato.stanford.edu/archives/win2017/entries/logical-constants/.
Mares, E. (2012). Relevant logic and the philosophy of mathematics. Philosophy Compass, 7, 481–494.
Mares, E., & Paoli, F. (2014). Logical consequence and the paradoxes. Journal of Philosophical Logic, 43, 439–469.
Mortensen, C. (1989). Anything is possible. Erkenntnis, 30(3), 319–337.
Omori, H., & Wansing, H. (2017). 40 years of FDE: An introductory overview. Studia Logica, 105, 1021–1049.
Paoli, F. (2001). Substructural logics. A primer. Berlin: Springer.
Partee, B.H. (2013). On the history of the question of whether natural language is “illogical”. History and Philosophy of the Language Sciences. https://hiphilangsci.net/2013/05/01/on-the-history-of-the-question-of-whether-natural-language-is-illogical/. Retrieved September 27, 2019.
Peregrin, J., & Svoboda, V. (2017). Reflective equilibrium and the principles of logical analysis: Understanding the laws of logic. London: Routledge.
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8, 219–241.
Priest, G. (2006). Doubt truth to be a liar. Oxford: Clarendon Press.
Putnam, H. (1968). Is logic empirical? In R. Cohen & M. Wartofsky (Eds.), Boston studies in the philosophy of science (Vol. 5, pp. 216–241). Dordrecht: Reidel.
Quine, W. V. O. (1986). Philosophy of logic (2nd ed.). Cambridge, MA: Harvard University Press.
Restall, G. (2000). An introduction to substructural logics. London: Routledge.
Ripley, D. (2015). ‘Transitivity’ of consequence relations. In W. van der Hoek, W. H. Holliday, & W. Wang (Eds.), Logic, rationality, and interaction (pp. 328–340). Berlin, Heidelberg: Springer.
Russell, G. (2017). An introduction to logical nihilism. In H. Leitgeb, I. Niiniluoto, P. Seppälä & E. Sober (Eds) Logic, methodology and philosophy of science. Proceedings of the 15th International Congress. London: College Publications.
Russell, G. (2018). Logical nihilism. Philosophical Issues: A Supplement to Noûs, 2018.
Russell, G. (2019). Varieties of logical consequence by their resistance to logical nihilism. In N. Pedersen, N. Kellen, & J. Wyatt (Eds.), Pluralisms in truth and logic (pp. 331–261). Cham: Palgrave MacMillan.
Scott, D. (1971). On engendering an illusion of understanding. Journal of Philosophy, 68(21), 787–807.
Shapiro, S. (1991). Foundations without foundationalism: A case for second-order logic. Oxford: Oxford University Press.
Shapiro, S. (2014). Varieties of logic. Oxford: Oxford University Press.
Strawson, P. F. (1950). On referring. Mind, 59, 320–344.
Strawson, P. F. (1952). Introduction to logical theory. London: Methuen (Reprinted 2011, Routledge).
Tarski, A. (1956). Logic, semantics, metamathematics: Papers from 1923 to 1938. Oxford: Clarendon Press.
Wyatt, N., & Payette, G. (2019). Against logical generalism. Synthese (Online First). https://doi.org/10.1007/s11229-018-02073-w.
Yagisawa, T. (1993). Logic purified. Noûs, 27, 470–486.
Zardini, E. (2014). Context and consequence. An intercontextual substructural logic. Synthese, 191, 3473–3500.
Zardini, E. (2019). The underdetermination of the meaning of logical constants by rules of inference (Unpublished manuscript).
Acknowledgements
I am grateful to audiences at Babeş-Bolyai University of Cluj, the University of Buenos Aires, and the University of Salzburg where I have presented versions of this paper. I owe special thanks to the participants at the LanCog Group’s M(etaphysics), E(pistemology), L(language), and L(ogic) Work in progress seminar at the Centre of Philosophy of the University of Lisbon, and in particular to Diogo Santos, Ricardo Santos, and Elia Zardini, who read a version of this paper and provided detailed comments and observations. For comments and suggestions I am also grateful to Eduardo Barrio, Natalia Buacar, Bruno Da Ré, Andreas Fjellstad, Roberto Giuntini, Bruno Jacinto, Adrian Luduşan, Francesco Paoli, Greg Restall, Lucas Rosenblat, Mariela Rubin, Federico Pailos, Gigi Ştefanov, Diego Tajer, Kai Tanter, Iulian Toader, Damian Szmuc, and Isis Urgell. I am grateful to two anonymous referees for this journal, whose constructive comments went well beyond what I could acknowledge explicitly in the text. Finally I owe both thanks and apologies to all those whom I forgot to acknowledge explicitly. This work was financially supported by the FCT – Fundação para a Ciência e a Tecnologia, Portugal, grant SFRH/BPD/116125/2016.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dicher, B. Requiem for logical nihilism, or: Logical nihilism annihilated. Synthese 198, 7073–7096 (2021). https://doi.org/10.1007/s11229-019-02510-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-019-02510-4