Delta: [...] He seems engrossed in the production of monstrosities. But monstrosities never foster growth, either in world of nature or in the world of thought.
Gamma: Geneticists can easily refute that. Have you not heard that mutations producing monstrosities play a considerable role in macro-evolution? They call such monstrous mutants ‘hopeful monsters’.
Lakatos, Proofs and refutations, pp. 21–22
Abstract
Arguments, the story goes, have one or more premises and only one conclusion. A contentious generalisation allows arguments with several disjunctively connected conclusions. Contentious as this generalisation may be, I will argue nevertheless that it is justified. My main claim is that multiple conclusions are epiphenomena of the logical connectives: some connectives determine, in a certain sense, multiple-conclusion derivations. Therefore, such derivations are completely natural and can safely be used in proof-theoretic semantics.
This is a preview of subscription content, log in via an institution to check access.
Notes
The exact terminology is irrelevant; ‘formalised argument’ would have been a good alternative to ‘derivation’, which I nonetheless prefer as it frees ‘argument’ to be used as a shortcut for ‘vernacular argument’.
In the literature, the first member is usually called antecedent, but precedent fits better with succedent.
There will be no need to go past the propositional level. Therefore, I shall sometimes use ‘logical constant’ instead of ‘connective’.
Dummett’s preferred formalism is natural deduction, although he seems not to be having any qualms with sequent calculi beyond their (unspecific and potential) appeal to multiple conclusions. Furthermore, following Gentzen, he takes the R-introductions (i.e., the natural deduction introductions) to be meaning conferring. Harmony then boils down to the L-rules (the eliminations) being faithful to this meaning determination. The reverse direction, going from L- to R-rules is usually called stability. But Dummett’s terminology is unclear; ditto for that used in the subsequent literature. Here I shall use harmony in such a way as to include stability. Hence the term builds-in no assumption of priority of one type of rules over the other.
There are other ways of making room for classical logic in the inferentialist world, such as going bilateral and setting denial on a par with assertion (Restall 2005; Rumfitt 2000), or rejecting the proof-theoretic desideratum of purity (or separability) (Milne 2013). (Recall that a logical constant is said to be governed by pure (L/R) rules if and only if, when schematically formulated, these rules mention no other constant than the one they purportedly define.)
This is the terminology consecrated in the linear logic tradition. Sometimes the additives are called extensional or context-sharing, while the multiplicatives are also called intensional or context-independent.
I add the subscript m to record the fact that this rule is multiplicative. Sometimes, I shall append the subscript to the connectives themselves, writing, e.g., \(\vee _{m}\) to denote disjunction as given by multiplicative rules.
Conversely, given the R-introductions, \( \vee\! \uparrow \) is derivable which, in fact, would work as a formal proof that \(\vee \)L, a.k.a \(\vee\! \uparrow \), is invertible). One of the required derivations is:
The derivation of X, B : Y follows the same pattern.
See Humberstone (2011, 789–798) and the references therein for a summary of the debate.
And here one may wish to further argue that there can be no doubt that classical logic allows for a behaviour like that described by the multiplicative rules for ‘or’. If we further assume that the vernacular roughly obeys classical logical laws, then we have a good motivation for accepting multiple conclusions.
I am indebted to an anonymous referee for prompting me to consider this issue in more detail.
This observation is made by way of doing justice to Tennant’s view that falsum is not a logical constant, but rather a punctuation mark, signalling a deduction gone pear-shaped (Tennant 1999). Again, I am indebted to an anonymous referee for bringing this to my attention.
In the sequent calculus, vacuous discharges correspond to weakenings while multiple discharges are expressed as contractions.
Incidentally, this suggests that the efforts to show that multiple conclusions are not part of the vernacular practice are pointless. If appealing to them in proof-theoretic arguments is somehow circular, then the circularity would not be fixed by there being multiple-conclusion derivations in ‘nature’.
For a brief discussion of this kind of inter-conectedness of the connectives see (Dicher 2017).
Besides, the usual tu quoque, that multiple premises, being conjunctive structures, are just as sinful as multiple conclusions, is pretty difficult to dismiss in this setup. (I thank an anonymous referee for noticing this.)
In passing, let us note that the theoretical question of which kind of features should be preferred is not—indeed, cannot be—settled by the Principle of Anwerability.
References
Avron, A. (1991). Simple consequence relations. Information and Computation, 92, 105–139.
Beall, J. C. (2011). Multiple-conclusion LP and default classicality. Review of Symbolic Logic, 4, 326–336.
Beall, J. C. (2014). Strict-choice validities: A note on a familiar pluralism. Erkenntnis, 79, 301–307.
Belnap, N. D. (1962). Tonk, plonk and plink. Analysis, 22, 130–134.
Cintula, P., & Paoli, F. (2016). Is multiset consequence trivial? Synthese. https://doi.org/10.1007/s11229-016-1209-7.
Cook, R. T. (2005). What’s wrong with tonk(?). Journal of Philosophical Logic, 34(2), 217–226.
Dicher, B. (2016a). Weak disharmony: Some lessons for proof-theoretic semantics. Review of Symbolic Logic, 9(3), 583–602.
Dicher, B. (2016b). A proof-theoretic defence of meaning-invariant logical pluralism. Mind, 125(499), 727–757.
Dicher, B. (2017). On a generality condition in proof-theoretic semantics. Theoria, 83(4), 394–418.
Došen, K. (1989). Logical constants as punctuation marks. Notre Dame Journal of Formal Logic, 30(3), 362–381.
Dummett, M. (1991). The logical basis of metaphysics. London: Duckworth.
Francez, N. (2015). Proof-theoretic semantics. London: College Publications.
Francez, N., & Dyckhoff, R. (2011). A note on harmony. Journal of Philosophical Logic, 40, 1–16.
Gentzen, G. (1935). Untersuchungen über das logische Schließen. I, II. Mathematische Zeitschrift, 39(1), 176–210; 405–431.
Gratzl, N., & Orlandelli, E. (2018). Logicality, double-line rules, and modalities. Studia Logica. https://doi.org/10.1007/s11225-017-9778-0.
Gratzl, N., & Orlandelli, E. (2017). Double-line harmony in a sequent setting. In P. Arazim & T. Lvika (Eds.), The logica yearbook 2016 (pp. 157–171). London: College Publications.
Hjortland, O. T. (2013). Harmony and the context of deducibility. In C. D. Novaes & O. T. Hjortland (Eds.), Insolubles and consequences: Essays in honour of Stephen Read (pp. 105–117). London: College Publications.
Humberstone, L. (2007). Investigations into a left-structural right-substructural sequent calculus. Journal of Logic, Language and Information, 16(2), 141–171.
Humberstone, L. (2011). The Connectives. Cambridge: MIT Press.
Kurbis, N. (2015). Proof-theoretic semantics, a problem with negation and prospects for modality. Journal of Philosophical Logic, 44(6), 713–727.
Milne, P. (2013). Harmony, purity, simplicity and a “Seemingly Magical Fact”. The Monist, 85, 498–534.
Paoli, F. (2002). Substructural logics: A primer. Berlin: Springer.
Paoli, F. (2007). Implicational paradoxes and the meaning of logical constants. Australasian Journal of Philosophy, 85(4), 553–579.
Prawitz, D. (1965). Natural deduction: A proof theoretical study. Stockholm: Almqvist and Wiksell.
Prior, A. N. (1960). The runabout inference-ticket. Analysis, 21(2), 38–39.
Read, S. (2000). Harmony and autonomy in classical logic. Journal of Philosophical Logic, 29(2), 123–154.
Read, S. (2010). General-elimination harmony and the meaning of the logical constants. Journal of Philosophical Logic, 39, 557–576.
Restall, G. (2015). Generality and existence (I): quantification and free logic. http://consequently.org/writing/generality-and-existence-1/.
Restall, G. (2005). Multiple conclusions. In P. Hájek, L. Valdés-Villanueva, & D. Westerstahl (Eds.), Logic, methodology and philosophy of science: Proceedings of the twelfth international congress (pp. 189–205). London: KCL Publications.
Restall, G., & Paoli, F. (2005). The geometry of non-distributive logics. Journal of Symbolic Logic, 70(4), 1108–1126.
Rumfitt, I. (2000). “Yes” and “No”. Mind, 109, 61–84.
Rumfitt, I. (2008). Knowledge by deduction. Grazer Philosophische Studien, 77, 61–84.
Schroeder-Heister, P. (2014). Proof-theoretic semantics. In Zalta, E. N. (Eds.), The stanford encyclopedia of philosophy. Summer 2014 edition. http://plato.stanford.edu/archives/sum2014/entries/proof-theoretic-semantics/. First published 2012.
Shoesmith, D. J., & Smiley, T. J. (1978). Multiple-conclusion logic. Cambridge: Cambridge University Press.
Steinberger, F. (2010). Why conclusions should remain single. Journal of Philosophical Logic, 39, 333–355.
Tennant, N. (1999). Negation, absurdity and contrariety. In D. M. Gabbay & H. Wansing (Eds.), What is Negation? (pp. 199–222). Dordrecht: Springer.
Tennant, N. (forthcoming). Inferentialism, logicism, harmony, and a counterpoint. In Miller, A. (Ed.), Essays for Crispin Wright: Logic, language and mathematics. Oxford: Oxford University Press.
Acknowledgements
I am grateful to the audience at the Lancog Seminar Series in Analytic Philosophy (Centre of Philosophy, University of Lisbon) and in particular to Ricardo Santos and Elia Zardini for a very interesting discussion of these issues. Special thanks are due to Francesco Paoli, who read an early version of this paper and suggested many invaluable improvements. I owe a debt of gratitude to Greg Restall for many discussions about multiple conclusions. Last but not least, I thank the two anonymous referees for this journal for their very detailed and tremendously helpful comments. At various stages during the writing of this paper I have benefited from a postdoctoral scholarship financed by the Regione Autonoma Sardegna within the project CRP-78705 (L.R. 7/2007), “Metaphor and argumentation” and from a postdoctoral fellowship funded by the Fundação para a Ciência e a Tecnologia, Portugal (grant SFRH/BPD/116125/2016).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dicher, B. Hopeful Monsters: A Note on Multiple Conclusions. Erkenn 85, 77–98 (2020). https://doi.org/10.1007/s10670-018-0019-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10670-018-0019-3