Abstract
We’ve known from early on that the inconsistency problem is the central challenge for any theory of truth for fiction constructed in the manner of the one developed here. The inconsistency problem is yet another example of a problem with a huge scholarship in ready wait. Inconsistency has dominated logic since Aristotle’s founding of it. Whatever we end up saying about fiction’s inconsistency problem, will have to have found a way to negotiate the choppy waters of a complex research scholarship rooted in the idea that inconsistency is logic’s most mortal sin.
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Notes
- 1.
For example, if in processing the information contained in some premiss I believe to be true and causally induced to form the belief that q, the inference can have been reliably produced without q’s being true. If, now believing q, I am ledt to believe that r, this inference too r’s could have been well produced without being true.
- 2.
A. Collins and M. Quillam, “Retrieval time from semantic memory”, Journal of Verbal Learning and Verbal Behavior, 8 (1969), 240–249; M. Howe, Introduction to Human Memory, New York: Harper & Row, 1970; A. Klatzley, Human Memory: Structures and Processes, San Francisco, W. H. Freeman, 1975; and P. Lindsay and D. Norman, Human Information Processing, New York: Academic Press, 1977.
- 3.
Christopher Cherniak, “Computational complexity and the universal acceptance of logic”, Journal of Philosophy, 92 (1984), 739–758; pp. 755–756, Minimal Rationality, Cambridge, MA: MIT Press, 1986.
- 4.
But see Carl Hewitt, “Inconsistency robustness in foundations: Mathematics proves its own inconsistency and other matters”, in Hewitt and Woods, Inconsistency Robustness, pages 104–157, 2015.
- 5.
Hewitt has a fondness for caps but tends to slight hyphens. I admit to reverse preferences. But nothing of substance depends on that.
- 6.
Even so, IRDL is amenable to siteness and may have something revealing to say of it.
- 7.
Actually, the reason that Swiss wines don’t travel is that the Swiss would rather drink it themselves than go to the bother of having it drunk in the wine-bars of Bucharest and Manhattan.
- 8.
See for example, Olivier Blanchard, Policy Brief “Do DSGE models have a future?”, Petersen Institute for International Economics, August 2016, 1–4. Blanchard, an MIT economist, is former chief economist at the International Monetary Fund.
- 9.
“Detonation” is a lovely play on the words of the title of Russell’s famous 1905 paper. See Peter Schotch and Ray Jennings, “On detonating”, in Graham Priest, Richard Routley and Jean Norman, editors, Paraconsistent Logic, pages 306–327, Munich: Philosophia Verlag, 1989. Paraconsistent Logic was an authoritative source-book in 1989, and still is. Also important, and more recent, is Jean-Yves Beziau, Walter Carnielli and Dov Gabbay, editors, Handbook of Paraconsistency, volume 9 of Studies in Logic, London: College Publications, 2007. See also Newton da Costa, Décio Krause, and Otávio Bueno, “Paraconsistent logics and paraconsistency’, in Dale Jacquette, editor, Philosophy of Logic, pages 791–911, a volume in Dov Gabbay, Paul Thagard and John Woods, editors, Handbook of the Philosophy of Science, Amsterdam: North-Holland, 2007. An excellent collection on preservationist paraconsistency is Peter Schotch, Bryson Brown and Raymond Jennings, editors, On Preserving: Essays on Preservationism and Paraconsistent Logic, Toronto: University of Toronto Press, 2009. Franz Berto’s How to Sell a Contradiction, 2007, is also impressive reading.
- 10.
A possible exception is Graham Priest in his 1987 book In Contradiction: A Study of the Transconsistent, Dordrecht: Kluwer, second expanded edition, Oxford: Clarendon Press, 2006; pages references to the second edition. At page 19 Priest writes (correctly, in my opinion): “There is, as far as I am aware, no linguistic or grammatical evidence at all that the English predicate ‘is true’ does typically ambiguous duty for an infinite hierarchy of predicates at the deep level.” Even so, he thinks that logic should capture the “deep structure of natural language, or at least part of it.” (p. 74). When we make this investigation, presumably we’ll learn something about the deep structure of natural language truth and, whatever we find, we won’t find it to be transfinitely ambiguous. My chief departure from Priest, and by no means a slighting one, lies among other things in a preference for logic to investigate the deep structure of real-life human cognition, all the while not ignoring how conscious knowing also goes. My thanks to Serban Dragulin for correspondence on this point.
- 11.
Here is Solomon Feferman on a related point: “So far as I know, it has not been determined whether such [inconsistency robust] logics account for ‘ordinary sustained reasoning’, not only in everyday discourse but also in mathematics, determinateness and truth. Feferman, “Axioms for determinateness and truth”, Review of Symbolic Logic, 1 (2008), 204–217.
- 12.
There are some paraconsistent logics in which ex falso holds. Apparently the intention is for it to hold in a limited way, holding for entailment but failing for inference. I thank Gillman Payette for helpful conversations about this.
- 13.
“Begündung einer strengen implikation”, Journal of Symbolic Logic, 21 (1956), 113–128.
- 14.
Indeed, Aristotle’s logic of the syllogism in the fourth century B.C. was paraconsistent at its core and, as we know, implementable by beings like you and me.
- 15.
Symbolic Logic, New York: Dover, 1959, pp. 288–289. First issued in New York by Appleton Century-Croft, 1932.
- 16.
See here Alan Weir’s proof by contradiction that no sentence is unambiguously and concurrently both true and not true, page 385 of “There are no true contradictions”, in Priest et al. 2004.
- 17.
Posted on ResearchGate on September 18th, 2017.
- 18.
Other attempts to crimp the authority of the or-introduction rule are typified by the contextual effects theory of Sperber and Wilson, which banishes all provisions for the introduction of connectives from single premisses. See Dan Sperber and Deirdre Wilson, Relevance, Oxford: Blackwell, 1995. First edition in 1986, 2nd edition, with a new Postface.
- 19.
As opposed to horse-and-buggy conjunction.
- 20.
Advanced by Tarski in “The concept of truth in formalized languages” as a condition of “material adequacy” for any theory of natural language truth. The full condition asserts that “S” is true if and only if S (“Snow is white” is true if and only if snow is white). The condition’s biconditional structure provides that if S then it is true that S, which is the form in which we have line (2).
- 21.
In earlier versions of this proof, there is no mention of truth values, hence no occasion to consider whether “not-” always flips them. Regrettably, this omission helps disguise the fact that flipping is the principal issue of the proof. See, for example, my “Globalization makes inconsistency unrecognizable”, in Payette, editor, “Shut up, he explained”: Essays in Honour of Peter K. Schotch, volume 31 of the Tributes series, pages 137–165, London: College Publications, 2016, p. 152.
- 22.
In the case of information-flows in the cognitive down-below, it matters whether unconceptualized and nonlinguistic items of information can stand to one another in any relation of incompatibility sufficient to trigger ex falso’s proof. Final answers aren’t yet in, but for now I’ll give the nod to an implicit and tacit “Yes”.
- 23.
See, however, Dov M. Gabbay and John Woods, Agenda Relevance: A Study in Formal Pragmatics, volume 1 of their A Practical Logic of Cognitive Systems, Amsterdam: North-Holland, 2003, and my Errors of Reasoning: Naturalizing the Logic of Inference, volume 45 of Studies in Logic, London: College Publications, 2013; second printing with corrections 2014. In both those works, the hypothesis of irrelevance filters was arrived at abductively, using methods most recently reviewed in my “Reorienting the logic of abduction”, in Magnani, and Bertolotti. But in none of those places is there any systematic exposure of how the filter actually works. More recently, I’ve expanded on the idea of filtration systems in Is Legal Reasoning Irrational? This remains an open problem in research programmes in human cognition.
- 24.
See here Alan Weir, “Naïve set theory is innocent!”, Mind, 111 (1998), 763–798.
- 25.
For example, to understand model theory, it is necessary to have some grasp of sets. It is not necessary that it be a post-paradox understanding.
- 26.
Closer to home is the set theory I taught my children when they were in elementary school. I used the old axioms, knowing ex falso to be true. I told them about it, and invited them not to worry about it. They learned a good deal about sets on those Saturday mornings.
- 27.
See Dirk Schlimm, “On Frege’s Begriffsschrift notation for propositional logic” Design principles and trade-offs”, History and Philosophy of Logic, forthcoming. See also Frege’s Lectures on Logic: Carnap’s Student Notes 1910–1914, translated and edited with an introduction, Erich H. Reck and Steve Awodey, based on the German text, edited, with introduction and annotations, by Gottfried Gabriel, Chicago and La Salle: Open Court, 2004.
- 28.
I thank Gottfried Gabriel for helpful discussion of this point.
- 29.
Further details can be found in Errors of Reasoning at pages 290–291. For an indication of how the agenda-relevance approach helps with inference, see pages 292–293, where consideration is given to how agendas operate in the cognitive down-below.
- 30.
How we can abductively advance to knowledge is explained in greater detail in my “Reorienting the logic of abduction”, which prior to editorial excision had been preceded by its main title, “Knowledge without evidence”. See also Is Legal Reasoning Irrational?, pp. 203–209.
- 31.
See, for example, Dale Jacquette’s review of Agenda Relevance, in Studia Logica, 77 (2004) 133–139.
- 32.
Apart from the revelation on page 6, note 21 of Pretense and Pathology that “the idea of a face-value reading … is meant to respect a combination of surface grammatical form and treatment in accepted inferential linguistic practices.” Perhaps the point of semantic redirection is to disrespect what face-value readings respect.
- 33.
Note that “X” here is not a variable but a place-holder for a term. It invites the question of what its substitution options are? If X were us, it would contradict the theory. If it were the theorist, the redirection metaphor loses steam. If the cost of pathological unification were some contrary-to-fact theoretical postulates, why not simply impose them for their instrumental allure?
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Woods, J. (2018). Putting Inconsistency in Its Place. In: Truth in Fiction. Synthese Library, vol 391. Springer, Cham. https://doi.org/10.1007/978-3-319-72658-8_9
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