Abstract
Five major stances on the problems of the possibility and fruitfulness of a debate on the principle of non-contradiction (PNC) are described: Detractors, Fierce supporters, Demonstrators, Methodologists and Calm supporters. We show what Calm supporters have to say on the other parties wondering about the possibility and fruitfulness of a debate on PNC. The main claim is that one can find all the elements of Calm supporters already in Aristotle’s works. In addition, we argue that the Aristotelian refutative strategy, originally used for dealing with detractors of PNC in Metaphysics, has wider implications for the possibility and fruitfulness of an up-to-date debate on PNC, at least in exhibiting some serious difficulties for the other parties.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Change history
14 October 2018
The original version of the book was published with incorrect given and surname for author “M. del Rosario Martínez-Ordaz” in matadata for chapters “The Possibility and Fruitfulness of a Debate on the Principle of Non-contradiction” and “Keeping Globally Inconsistent Scientific Theories Locally Consistent” have been corrected. The correction chapters and book have been updated with the changes.
Notes
- 1.
Advocators of “anti-exceptionalism about logic” —for example, Williamson ([37, 38]) and Hjortland ([23])— can be regarded as methodologists, and their views surely would have implications for the debate here discussed. Note also that Priest appears as a detractor and a demonstrator, a contradiction that, for him, would show that the debate would be better approached through the methodological principles of rational choice. This would be more extensively discussed in Sects. 6 and 7 below.
- 2.
The only caveat was that Aristotle aimed at establishing a version of PNC stronger than his refutations of detractors allowed to conclude but, as we have mentioned, this is not a problem for Calm supporters in general.
- 3.
An ancestor of this is found in Plato, Republic [24] 436b: “It is obvious that the same thing will never do or suffer opposites in the same respect in relation to the same thing and at the same time.”
- 4.
Also found in Plato. See Republic [24] 602e8-9: “And haven’t we said that it is impossible for the same [person] to think/imagine at once/simultaneously opposite [things] about the same [things]?”
- 5.
We do not know a simple expression to name this property, not even in our mother-tongue, and certainly we do not dare piss the owners of English off by inventing one, hence the blank space.
- 6.
And it must not be confused with Non-referential gap: There is no truth value describable as neither-true-nor-false. These are independent principles. Bivalence might not hold because, say, there is a third value, but it is not “neither-true-nor-false” but rather, say, “true-and-false”.
- 7.
A key ingredient in classical logic is Functionality: (a) Every proposition has one value (b) but only one. In a language with the usual ingredients (conjunction, disjunction, etc. as certain functions) this, together with Bivalence, Falsity of at least one contradictory, No values-gluts and either version of (Interpretational) Non-contradiction, suffices to give all the usual valuations of classical logic.
- 8.
For more details on this principle and its importance on Aristotle’s overall argument against Detractors of PNC in Metaphysics \(\varGamma \), see [35], where the principle is taken as saying that the meaning of a term is unique, definite and determinate. Nonetheless, we take the mutual exclusiveness of certain semantic properties as a more appropriate reconstruction of Aristotle’s claims on the uniqueness, definiteness and determinateness of meanings and their role in his refutations.
- 9.
For a guided tour on even more versions of PNC, see [16].
- 10.
Whereas it can be debated whether Aristotle’s notion of deduction allows reflexivity (A is deducible from A) because of the “something different from [the things] supposed results” clause, demonstration certainly does not allow it: A would have to be both more certain (for it is a premise in the demonstration) and less certain (for it is demonstrated) than itself.
- 11.
This was a permanent stance:
No truth does have, and no truth could have, a true negation. Nothing is, and nothing could be, literally both true and false. This we know for certain, and a priori and without any exception for especially perplexing subject matters. (...) That may seem dogmatic. And it is: I am affirming the very thesis that [the foes of the PNC] have called into question and —contrary to the rules of debate— I decline to defend it. Further, I concede that it is indefensible against their challenge. They have called so much into question that I have no foothold on undisputed ground. So much the worse for the demand that philosophers always must be ready to defend their theses under the rules of debate. [21, 101]
And in a letter interceding for the publication of Priest’s In Contradiction:
Many people will think that it is an easy thing to refute Priest’s position, decisively and in accordance with customary rules of debate. It is not an easy thing. I myself think that it is an impossible thing: so much is called into question that debate will bog down into question-begging and deadlock. (Quoted in [27, xix].)
In what follows we will use interchangeably ‘x is debated’, ‘x is disputed’, ‘the possibility that x does not hold is entertained’ and even ‘x is defended’ in this reconstruction.
- 12.
Field [14] and Boghossian [6] call a principle “default reasonable” if it is regarded as being good at leading to true beliefs and avoiding error (hence reasonable) and it is done so without first adducing evidence or argument in its favor (hence default). We do not think PNC would be default reasonable in this sense for Lewis: It is not merely good at leading to true beliefs and avoiding errors, but is the sine qua non to draw a distinction between true beliefs and errors at all; also, it is not merely default, but already the distinction between what counts as evidence and an argument in favor of something presupposes it.
- 13.
See [9, Ch. 5] for further details.
- 14.
- 15.
Priest [26, 14f] does not regard as refutations some arguments where apparently there is no other agent saying anything on the basis that Aristotle required Detractors to say something. This internalization would help to explain the fact that there could be a refutation even if there were nobody saying anything: Detractors’ saying has been internalized in the supporter’s argument.
- 16.
We would like to emphasize that the following reconstruction of Aristotle’s refutation strategy is not an attempt to make an exegetically perfect reconstruction —considering, for example, the fact that it is described in terms of a contemporary semantics. The main purpose of this reconstruction is to show in a comprehensive manner how the strategy works. We leave to the Aristotelian scholars to decide whether it matches perfectly the original proposal.
- 17.
And at least in one case, Aristotle distinguishes between Heracliteanism and dialetheism because they need to be refuted in different ways, cf. Metaphysics 1008a7-12.
- 18.
Whether these views should actually be attributed to the historical Heraclitus is a moot point; see [15, Ch. 5]. However, the interest here lies more in the position itself than in the correctness of Aristotle’s scholarship.
- 19.
The problem is that Priest sometimes conflates trivialism —“Everything is true”— with a version of Heracliteanism —’All contradictions are true’— (cf. [28, 131], although sometimes he says that he is aware that the identification depends on certain assumptions, notoriously ‘and’-elimination (see [26, 56]. Aristotle too thought that semantic Heracliteanism could be equated with trivialism, but he was cautious: “The doctrine of Heraclitus, that all things are and are not, seems to make everything true (...)” (Metaphysics1012a25).
- 20.
In proof-theoretic terms, Heracliteans would not accept conjunction elimination when the premise is a contradiction, and they would accept conjunction introduction when the conjuncts are not true but only if they are contradictories. In model-theoretic terms, it may be that \(v(A) \ne \top \) or \(v(B) \ne \top \) even if \(v(A\wedge B) = \top \). So, when the conjuncts are contradictories, semantic Heracliteanism’s conjunction resembles the relevance logics’ fusion connective, and thus is not so odd by contemporary lights. Besides, in certain sense, semantic Heracliteanism is dual to non-adjunctivism, which is a thesis found in some paraconsistent logics like the earliest one, Jaśkowski’s: In non-adjunctivism, in general \(v(A\wedge \lnot A) \ne \top \) even if \(v(A) = v(\lnot A) = \top \). Non-adjunctivism was used in some of the earliest attempts to make sense of impossible worlds; see [29].
- 21.
An argument similar to this one is presented in Metaphysics K (see 1062b2ff).
- 22.
Again according to Aristotle, Cratylus held that even contradictions over-fix reality so they must be altogether false, like their conjuncts. However, he noticed the problem faced by the Heraclitean and did not make the mistake of trying to philosophize about that, so he attempted to threw the ladder out after stating his position and ceased to do virtually any philosophical statement (only “virtually” because he kept denying with his forefinger and that still causes problems). See Metaphysics 1010a10ff.
- 23.
Note that according to the Aristotelian argument, the option of describing a language in which all contradictions are true but none of their components is true is open to Heracliteans, provided that in its metalanguage either not all contradictions are true or some of the components of a contradiction are true separately from it.
- 24.
That PNC might hold only for restricted versions has been pointed out several times by scholars. Substances and “essential predicates” are the usual candidates for which oPNC certainly holds (cf. [11, 17, 19]), although sometimes a slippery slope from the restricted oPNC about essential predicates to a general oPNC about any predicates is attempted; see for example [33]. Tahko [32] defends that oPNC holds certainly for concreta and its properties in “genuinely possible worlds with macroscopic objects”, but not necessarily for every object of every possible world, not even for every object of every physically possible world. More recently, Coren [10] has recently provided an interpretation like ours, but attempting to show that Aristotle can succeed in defending a principle much stronger than MSPNC.
- 25.
- 26.
In [27, 241ff], Priest explicitly says that some arithmetical statements (in an inconsistent arithmetic) are both provable and not provable.
- 27.
Lewis mentions this:
It is not an easy thing [to refute Priest’s position, decisively and in accordance with customary rules of debate]. I myself think that it is an impossible thing: so much is called into question that debate will bog down into question-begging and deadlock. (On this point, Priest disagrees with me: he thinks that shared principles of methodology might provide enough common ground.) (Quoted in [27, xix].)
- 28.
See also [26, 110].
- 29.
This reveals that Priest underestimated Aristotle’s remark in Posterior Analytics mentioned in Sect. 4. Aristotle says that whenever one wants to demonstrate something but not also something excluded by it, one has to presuppose PNC. If it is presupposed in no demonstration, one always could demonstrate something but also something else excluded by what was supposedly demonstrated, hence all exclusions could be demonstrated. Given MSPNC, the fact that contradictory propositions can be accepted is irrelevant as to the Indemonstrability of (MS)PNC, although perhaps it counts against some form of Universality for especially stronger versions of it.
- 30.
To be fair, dialetheists and other detractors are right in questioning stronger forms of PNC, like “any sentence of the form \(A\wedge \lnot A\) is untrue in any valuation”, and their assimilation to weaker forms like MSPNC.
- 31.
Cf. the reconstruction of the debates in [8, 170–173].
References
Aristotle: Metaphysics. In The Complete Works of Aristotle, ed. J Barnes (vol. 2). Princeton N.J.: Princeton University Press, 1984.
Aristotle: Posterior Analytics, edited and translated with a commentary by Jonathan Barnes, Oxford: Clarendon Press, second edition, 1994.
Aristotle: Prior Analytics, Robert Smith (trans.), Indianapolis: Hackett, 1989.
Berto, Francesco. 2008. Adynaton and material exclusion. Australasian Journal of Philosophy 86 (2): 165–190.
Berto, Francesco. 2012. How to rule out things with words. In New Waves in Philosophical Logic, ed. Greg Restall, and Gillian Russell, 169–189. Great Britain: Palgrave Macmillan.
Boghossian, Paul. 2000. Knowledge of logic. In New Essays on the A Priori, ed. Paul Boghossian, and Christopher Peacocke, 229–254. New York: Oxford University Press.
Boole, George. 1854. An Investigation of the Laws of Thought on which are founded the Mathematical Theories of Logic and Probabilities. Cambridge: Cambridge University Press.
Bueno, Otávio, and Mark Colyvan. 2004. Logical non-apriorism and the "law" of non-contradiction. In The Law of Non-Contradiction: New Philosophical Essays, ed. Jc Beall, Graham Priest, and Bradley Armour-Garb, 156–175. New York: Oxford University Press.
Castagnoli, Luca. 2010. Ancient Self-Refutation: The Logic and History of the Self-Refutation Argument from Democritus to Augustine. Cambridge: Cambridge University Press.
Coren, Daniel. 2018. Why does Aristotle defend the principle of noncontradiction against its contrary? Philosophical Forum 49 (1): 39–59.
Cresswell, Maxwell. 2004. Non-contradiction and substantial predication. Theoria 70: 166–183.
Dancy, Russell M. 1975. Sense and Contradiction: A Study in Aristotle. Dordrecht: Reidel.
Dutilh Novaes, Catarina. 2012. Formal Languages in Logic -A Philosophical and Cognitive Analysis. Cambridge: Cambridge University Press.
Field, Hartry. 2000. Apriority as an evaluative notion. In New Essays on the A Priori, ed. Paul Boghossian, and Christopher Peacocke, 117–149. New York: Oxford University Press.
Graham, Daniel. 2006. Explaining the Cosmos: The Ionian Tradition of Scientific Philosophy. Cambridge: Cambridge University Press.
Grim, Patrick. 2004. What is a contradiction? In The Law of Non-Contradiction: New Philosophical Essays, ed. Jc Beall Graham Priest, and Bradley Armour-Garb, 49–72. New York: Oxford University Press.
Irwin, Terence. 2006. Aristotle’s First Principles. Oxford: Oxford University Press.
Kant, Immanuel. 1755. Concerning the principle of contradiction. In Kant. Theoretical Philosophy 1755-1770, ed. David Walford, and Ralf Meerbote, 6–10. Cambridge: Cambridge University Press.
Kirwan, Christopher. 1993. Aristotle, Metaphysics (Books $\Gamma $, $\Delta $, E), 2nd ed. Oxford: Oxford University Press.
Lear, Jonathan. 1988. Aristotle: The Desire to Understand. Cambridge: Cambridge University Press.
Lewis, David. 1982. Logic for equivocators. Noûs 16 (3): 431–441; Reprinted as Chapter 7 of Lewis. 1998. Papers in Philosophical Logic, 97–110, Cambridge: Cambridge University Press.
Lewis, David. 2004. Letters to Beall and Priest. In The Law of Non-Contradiction: New Philosophical Essays, ed. Jc Beall Graham Priest, and Bradley Armour-Garb, 176–177. New York: Oxford University Press.
Ole Thomassen Hjortland. 2017. Anti-exceptionalism about logic. Philosophical Studies 174 (3): 631–658.
Plato: The Republic, Desmond Lee (trans.), Penguin Classics, 1955.
Priest, Graham. 1989. Reductio ad absurdum et modus tollendo ponens. In Paraconsistent Logic. Essays on the Inconsistent, ed. Richard Routley, Graham Priest, and Jean Norman, 613–626. München: Philosophia Verlag.
Priest, Graham. 2006. Doubt Truth to be a Liar. Oxford: Oxford University Press.
Priest, Graham. 2006. In Contradiction, 2nd ed. Oxford: Oxford Clarendon Press.
Priest, Graham. 2007. Paraconsistency and dialetheism. In The Many Valued and Nonmonotonic Turn in Logic, vol. 8, ed. Dov Gabbay, and John Woods, 129–204., Handbook of the History of Logic The Netherlands: Elsevier.
Rescher, Nicholas, and Robert Brandom. 1980. The Logic of Inconsistency: A Study in Non-Standard Possible Worlds Semantics and Ontology. Oxford: Basil Blackwell.
Russell, Bertrand. 1903. The Principles of Mathematics. London and New York: Routledge.
Russell, Bertrand. 1912. The Problems of Philosophy. New York and Oxford: Oxford University Press.
Tahko, Tuomas E. 2014. The metaphysical interpretation of logical truth. In The Metaphysics of Logic: Logical Realism, Logical Anti-Realism and All Things In Between, ed. Penelope Rush, 233–248. Cambridge: Cambridge University Press.
Wedin, Michael V. 2000. Some logical problems in metaphysics gamma. Oxford Studies in Ancient Philosophy 19: 113–162.
Wedin, Michael V. 2014. The science and axioms of being. In A Companion to Aristotle, ed. Georgios Anagnostopoulos, 123–143. Massachusetts: Wiley-Blackwell.
Whitaker, Charles W. A. 1996. Aristotle’s De Interpretatione: Contradiction and Dialectic. Oxford: Oxford Clarendon Press.
Whitehead, Alfred North, and Bertrand Russell. 1910. Principia Mathematica, vol. I, 2nd ed. Cambridge: Cambridge University Press.
Williamson, Timothy. 2013. Logic, metalogic and neutrality. Erkenntnis 79 (2): 211–231.
Williamson, Timothy. 2017. Semantic paradoxes and abductive methodology. In Reflections on the Liar, ed. Bradley Armour-Garb, 325–346. Oxford: Oxford University Press.
Acknowledgements
This work was supported by the PAPIIT project IA401117 “Philosophical Aspects of Contra-Classical Logics”. We would like to thank Daniel Cohnitz, Alex Davies, Víctor Cantero, Maite Ezcurdia, Eduardo García-Ramírez, Tuomas Tahko and Pedro Stepanenko, as well as the audience at the conference Trends in Logic XVI, for their comments on previous incarnations of this paper, some of them known as “Kantham Priest vs. Aristot Lewis”. The anonymous referees deserve special mention for their extremely useful reports.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Estrada-González, L., Martínez-Ordaz, M.d.R. (2018). The Possibility and Fruitfulness of a Debate on the Principle of Non-contradiction. In: Carnielli, W., Malinowski, J. (eds) Contradictions, from Consistency to Inconsistency. Trends in Logic, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-98797-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-98797-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-98796-5
Online ISBN: 978-3-319-98797-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)