Abstract
There are several features of law which rightly draw the interest of philosophers, especially those whose expertise lies in ethics and social and political philosophy. But the law also has features which haven’t stirred much in the way of philosophical investigation. I must say that I find this surprising. For the fact is that a well-run criminal trial is a master-class in logic and epistemology. Below I examine the logical and epistemological properties of greatest operational involvement in a criminal proceedings, concepts such as evidence, proof, argument, inference, relevance, probability, and more. My principal objective is to expose the deep cleavage between establishment norms in epistemology and logic and standard practice in criminal proceedings. This gives us two options to reflect upon. In one, the establishment norms for the correct management of the concepts in question are basically sound. In that case, as I will show, common law criminal practice would be basically unsound; its convictions would be basically false and unjust. Seen from the other perspective, the criminal justice system would be basically sound, and its criminal convictions basically true and just. It turns out to matter that option one meets with widely spread common disbelief and is generally taken as contrary to common knowledge. What is needed here is an epistemology which gives these sentiments some air to breathe. I will argue that on balance it is the logico-epistemic establishment which requires some serious rethinking.
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Notes
Mind you, it can hardly be said that commoners advanced their station at Runnymede. But it was a start.
Think here of R. H. Thouless’, Straight and Crooked Thinking (1930). It is a nice little primer, but its title is letter-perfect. A close runner-up in that same respect is Johnson and Blair’s Logical Self-Defense (1977.)
I am framing the present discussion for criminal trials which, in all its material respects, have been conducted in the manner they’re designed for. So conceived of, wrongful convictions in Canada proceed from human error (or worse), which might affect \(\approx \) 800 in a population of \(\approx \) 172,000 convictions per annum, although mainly reversed on appeal. So the number of fully tested false convictions in a population of 38 million souls is indeed very low. On the other hand, for reasons of public policy, it is artificially difficult for prosecutors to convict, indeed so difficult that the ratio of false acquittals is artificially high. Larry Laudan has written of this in Truth, Error, and Criminal Law (2006), and The Law’s Flaws: Rethinking Trials and Errors (2016). Laudan’s complaint is that (American) criminal law is too highly risk-averse, not that its juries are legally incompetent en large.
In Canada, persons under criminal indictment normally have the option of trial by jury.
In smaller communities, this condition is not enforced. Juries must be random selections of the accused’s peers, that is, by persons at large in the village or township who are free of any material interest in any element of the crime as charged.
In the interests of daintiness, I’ll not concern us with the common woman or the woman in the street, and I shan’t deign to feign acquiescence to the singularity of ‘they’.
Logicians of various theoretical stripes have also turned their attention to the analysis of legal reasoning. Consider, for example, the large body of work by the late Douglas Walton and his other talented colleagues, in which informal logic and the theory of argument combine with techniques in AI and computer science to make computer models of the logical structure of legal reasoning. What is especially interesting about these investigations and also, I would say, somewhat puzzling is that their principals should find legal reasoning interesting enough and important enough to make computer models of it and design computer-assisted programs for its simulation, but no one yet has blown the whistle on the legal incompetents problem. Why ever not, we might wonder? Why leave such low-lying fruit unplucked? The question invitesa conjecture. My own is that these modellers of legal reasoning don’t happen to think that the legal incompetency phenomenon merits logical plucking; in other words, they are parties to the common knowledge that, when properly run, criminal trials in Canada produce safe convictions. See, for example, Douglas Walton and Thomas Gordon, “Critical questions in computational models of legal argument”, P. E. Dunne, et al., editors, Argumentation in Artificial Intelligence and Law, (2005), 103–111, and Douglas Walton, Chris Reed and Fabrizio Macagno, Argumentation Schemes, (2008).
Of course, there is more to legal reasoning than what goes on in criminal trials, and more to what goes on in criminal proceedings than the reasoning done by jurors. I will mention in passing that the only dialogue that is permitted at trial is that between counsel and witnesses in their question-and-answer exchanges, and upon occasion, between counsel and the judge under leave to approach the bench. (Such exchanges are off-the-record and typically unheard by the jury.). Although each counsel has a closing argument, he isn’t in advancing it actually dialoguing with anyone. Rather he is making a stump-speech in which he lays out his side’s theory of the case. It just goes to show that argument-making needn’t be dialogical even when transacted in public and in adversarial settings.
In my use of the term ‘establishment’, intend no imputation of Deep State malfeasance. Quite to the contrary, I mean by ‘establishment epistemology’ what would be found distributed over philosophy’s PhD curricula in today’s universities or in the leading anthologies, companions and handbooks of the associated subjects. “Establishment” here is a purely descriptive term.
“Absurdum” carries two meanings here. In one, it is the pragmatic inconsistency lodged in theoretical norms its supporters do not and cannot comply with (and yet to whom footlessness is a matter of sheer indifference). In an older sense, absurdity is the sheer harshness of a doctrine’s uncompensated implications.
An early version of an epistemology favourable to common knowledge and common sense can be found in Woods (2013/14). A version of it more tailored for the law is Woods (2015, 2018).
Here I part company with Lycan. I disavow justification as a general condition, and will come back to it briefly a bit later. See Woods (2015/2018), “Appendix A”, “Justificationism fights back”.
Consider, in this regard, the law’s own difficulties in qualifying psychiatric experts.
[32], chapter 5, “The poverty of philosophical method: A case study”.
A jury hangs when it cannot fulfill the unanimity requirement for decision, notwithstanding repeated instructions from the bench to keep trying.
Although high probability of guilt is insufficient for criminal conviction, it is possible in criminal law to meet the proof standard if the jury deems some element of the charge to have been proved probabilistically by preponderance of evidence, provided that overall the proof meets the criminal standard. But in none of these instances is there any reason to think that the probability axioms have regulatory effect. See here Woods (2015/18) p. 165. Perhaps I might mention in passing that, within each of the leading schools of establishment epistemology, there are also notable levels of reflective disequilibria. There is scarcely a basic principle of Bayesian epistemology that some Bayesians won’t disavow.
There are policy reasons against over-long trials.
Pennell v. Oklahoma 1982, and Cosco v. Wyoming 1974. See also Woods (2018), 165–167.
US v. Glass 1988. See also J. W. Strong, MacCormick on Evidence, 5th edition, St. Paul, MN: West Group, 1999. As it happens, in putting it this way, the 1988 court missed the boat. Granted that we all understand reasonable doubt without having it defined for us, it does not follow that, without assistance, we will understand “proof of guilt as charged beyond a reasonable doubt”, for guilt, as charged, is a matter for which we lack legal competence.
It is true that jurors are sworn to tell the truth, whether by oath or “affirmation”. Perhaps these undertakings are reassuring. Perhaps they are probative, in each case backed by God’s commandment not to take His name in vain and the state’s crime of perjury. But in each case the assurance rests on self-attestation.
See Woods (2018), chapter 20, “An Epistemology for Law”. See also my “Four grades of ignorance-involvement and how they nourish the cognitive economy”, Synthese, 198 (2021), 3339–3368.
The doctrine of stare decisis is under current duress in Canadian courts by activist judges in courts below. See here [57], “Appendix H” “Weakening Stare Decisis”.
Beyond the intention of the legislators (roughly speaking).
Think here of the classical definition of deductive implication: Propositions S\(_{\mathrm {1}}\), …, S\(_{\mathrm {n}}\) deductively imply proposition S\(\prime \) if and only if it is in no sense possible for the S\(_{\mathrm {i }}\)to be jointly true and S\(\prime \) concurrently not. When read from left to right, the definition is met with widespread communal approval. But when read from right to left, the approval rate goes down; and if attention is called to ex falso, approval rates tumble. Ex falso is the doctrine that a contradiction deductively implies every proposition whatever. This does not, of course, falsify the definition. But it does suggest that when people make the correct consequence-attributions they are implementing the left to right reading.
A franchise coffee shop.
Indeed it follows from the standard definition of following of necessity from: S\(^{\prime }\) follows of necessity from S\(_{\mathrm {1}}\), …, S\(_{\mathrm {n}}\) just in case there is no respect in which it is in any sense possible for the S\(_{i} \)to be true and S\(^{\prime }\) not.
See, for example, da Costa, “On the theory of inconsistent formal systems” (1974); Batens, “Extensional paraconsistent logics” (1980); Priest et al, editors, Paraconsistent Logic (1989); Avron, “Whither relevance logic?” 1992; and Schotch et al., editors, Essays on Preservationism and Paraconsistent Logic (2009).
See for example, [2].
For readers who prefer not to give it a pass, here is the proof. It is conditional on the assumption that for some S, ‘S and not-S’ is true. Then, on the principle that if both are true so is each, it is now implied that S is true. Next, on the principle that if some proposition is true, then for any proposition-pair of which that proposition is a member, it is implied that at least one of {S, X}is true for arbitrary X. We now reapply the both-each principle to “(‘S and not-S’) is true” from which it is implied that ‘not-S’ is true. But this contradicts the S in {S, X}, from which it follows that X is true. Of course we also have the implication that S is true. It doesn’t matter. If ‘not-S’ does not contradict the S of {S, X}, it cannot on pain of vicious equivocation contradict the S of the original assumption ‘S and not-S’. QED. Earlier I mentioned the reluctance of the public at large to acquiesce to the ex falso implication of the classical biconditional definition of deductive implication, which is heavily assented to by that same public. There is more to draw from this than space permits. But at least we can say this: Sometimes our intuitions misadvise us.
Besides, as we said, the bench does its admonitory best to avoid hung juries.
[22].
Here is a quotation concerning a real trial: “Thus, on three of the counts …different jurors could [\(=\) were legally permitted to] find the defendants guilty of the same charge for different reasons.” (Conrad Black, A Matter of Principle, Toronto: McClelland & Stewart, 2011, p. 414. Lord Black was one of the aforementioned defendants.)
The efficaciousness of epistemic transmission—knowing something by being told it—turns to some extent on the state the knowledge-bearing information is in. We all know the story about the joke which after a dozen iterations of re-telling ends up being about a different matter and not funny at all. All I will say about this now is that when information does make its way to common knowledge, it is usually easy to remember and is usually more generally formulated than detailed.
For a discussion of how filtration systems serve as a hypothesis-search device in abductive contexts, readers could consult Dov M. Gabbay and John Woods, “Filtration structures and the cut-down problem for abduction”, in Kent A. Peacock and Andrew D. Irvine, editors, Mistakes of Reason: Essays in Honour of John Woods, pages 398-417, Toronto: University of Toronto Press, 2005.
The classic source of the implicit and tacit in logic is Aristotle’s Rhetoric. An enthymeme is a one-premiss argument with a “missing” but implicated second premiss. Had the second premiss occurred explicitly the argument would have been a syllogism. Aristotle rightly assumes that an enthymeme’s missing premiss is easily made explicit. What interests him is the logical status of unrestored enthymemes. Enthymemes are interesting in other ways and, I would say, important. So would Fabio Paglieri, whom I join in noting that, in some suitably extended and generalized form, enthymematic speech is the human norm. Non-enthymematic speech is pedantic speech. It takes longer to utter and is less good at holding our attention. It offends against the Quantity Law. See Paglieri and Woods, “Enthymematic parsimony”, Synthese (2011a), and “Enthymemes: From resolution to reconstruction”, Argumentation (2011b).
The ampliation desideratum emphasizes the importance of defeasible reasoning in human affairs. It also raises the question of whether truth-preserving inference cam be ampliative. Sextus Empiricus and Mill doubted it. Frege made provision for it by adding purely stipulative definitions to a deductive system’s stock of premiss-eligible propositions. It is not a matter that need deter us here.
Richard Shiffrin, “Automatism and consciousness” (1997), 50–62, and J. St. B. T. Evans, Hypothetical Thinking (2007), 14–15. See more recently, Fabio Paglieri, editor, The Role of the Natural and Social Contexts of Shaping Consciousness (2012), and Dale Jacquette, editor, The Bloomsbury Companion to the Philosophy of Consciousness (2018). Early stirrings of my thinking on these matters can be found in my “Speaking your mind: Inarticulacies constitutional and circumstantial”, in Christian Campolo and Dale Turner, editors, Argumentation and Articulation, a special issue of Argumentation, 16 (2002), 59–78.
See, for example, [10], chapter 9. There is a link between Aristotle’s concept of potentiality (dunamis) and the phase-transitions of modern physics. For Aristotle, dunamis is a thing’s capacity to take on a new form without losing its identity. (Metaphysics, Book 8 1086\(^{\mathrm {a}}\), 15, 27). An item’s haecceity is that in virtue of which it is its self-same thing and not another thing. A thing’s quiddity is that in virtue it is the very kind of thing it is. Then the phase-transition thesis has it that there are ranges of cases in which a thing’s haecceity is unmolested by the loss and subsequent restoration of quiddity.
Again, for an overview see [57], “Appendix A”, “Justificationism Fights Back”.
A welcome byproduct of the causal-response characterization is that it is not a condition on one’s knowing that S at tthat one knows at tthat one knows S at t. In other words, the KK-hypothesis fails in causal-response epistemology. And a good thing too.
Inductive backing gives inductively sound inference. Deductive backing gives deductively sound inference. Abductive backing gives abductively sound inference, whose conclusion is that because, if true, qwould make the surprising \(p_{\mathrm {i}}\) a matter of course, there is reason to suspect that qis true. This last condition is Peirce’s. Peirce’s ideas on ideas on abduction arose in the context of experimental science, and are framed against a JTB epistemological background. Peirce insists that even at its abductively perfect best, abduction provides no reason to believe the abduced hypothesis. However, when placed against a causal-response epistemological background, a good deal of what a being like us knows will be arrived at abductively. For Peirce’s take, see C. S. Peirce, Collected Works, 8 volumes, Cambridge, MA: Harvard University Press, 1931–1958; 5. 189. For the CR-take, see my “Reorienting the logic of abduction”, in Lorenzo Magnani and Tomasso Bertolotti, editors, Handbook of Model-Based Reasoning, pages 137–150, Berlin: Springer, 2017.
Note well: It is not a condition on well-filteredness that a juror’s total evidence-set be consistent. Provided it is given no premissory role in the decision, the inconsistency is harmless.
Bernhard Riemann, “Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse”, Universität Gottingen, 1851. See also, for example, David Hilbert, “Über Flachen von Konstanter Krümmung”, Transactions of the American Mathematical Society, 2 (1901), 87–99.
Farkas and Kra, Riemannian Surfaces (1980).
C. S. Peirce, CP 5.171, and Reasoning and the Logic of Things (1992), 128.
In the beginning, Gaus did not understand the mapping theorem, except for the remote possibility of an implicit and tacit understanding. What he did understand was that Riemann was on to something important and ground-breaking.
I draw here on passages from [60]. In some respects, Lorenzo Magnani’s distinction between computationally tailored and humanly tailored AI programs captures the point at hand. See his (2019a) and (2019b). There is also a question posed by argument schemes such as the one found on page 233 of Douglas Walton’s Methods of Argumentation, New York: Cambridge University Press, 2013. It schematizes “the three stages of a dialogue”. If Bill and Sue were inclined to have a chat, they’d be advised to implement this scheme. The question this poses is whether the implementation conditions can be written down. Can they be computer-modelled? We might note in passing that implicity and tacity are reserved by some authors for knowledge how. I don’t share that reservation. They are often at least equally properties of knowledge that. Both are important sources of the implicit and tacit. Concerning knowledge how, a useful survey is Neil Gasgoine and Tim Thornton’s Tacit Knowledge, Durham: Acumen, 1996.
Further details and citations can be found in Errors of Reasoning, section 9.7, pages 318–320.
Akihiro Kanamori, “Mathematical knowledge: Motley and complexity of proof”, (2013), 32.
See here Woods, Truth in Fiction: Rethinking its Logic, chapter 9, “Putting inconsistency in its place” (2018b).
Carl Hewitt, “Formalizing common sense reasoning for inconsistency-robust coordination.” (2015).
Hewitt to Woods, personal correspondence.
It might be argued that the Newton-Leibniz calculus is strictly speaking, not contradictory. It is true that, by the requirement of division by infinitesmals, \(\alpha \ne \) 0 when \(\alpha \) is an infinitesimal, and that, by the requirement that infinitesmals and their products be discounted in the final value of the derivative, \(\alpha =\) 0. At Metaphysics 1005\(^{\mathrm {b}}\) 19–20, Aristotle gives his fullest formulation of the Law of Noncontradiction: “It is impossible that the same thing belong and not belong to the same thing at the same time and in the same respect.” (Emphasis mine) Perhaps this gives the N-L calculus an out. Perhaps it’s enough to say that \(\alpha \)’s identity and non-identity to zero is in respect of different operational constraints. I really don’t know what to say. For safety’s sake let our present discussion revert to any theory whose contradictoriness is not in doubt.
Mark as false, but not reject. Newton needs the infinitesmals, inconsistency and all, as what Nancy Cartwright would later call lies of convenience. Some lies! Some convenience!
Thus confirming [20]. See also his (1986).
And as a result, greatly complicating the construction of the semantics for having. See, for example, the Routley star semantics for FDE as laid out in Richard Routley and Valerie Routley, “The semantics of first degree entailment” (1972).
It is a good rule, but it is not something we can elect to comply with in the general case. For the most part, our filtration-devices do it for us automatically.
Woods, “Response to Madeleine Ransom”, in Gabbay et al., editors, Natural Arguments, pages 590–592.
References
Alshanetsky, E.: Articulating a Thought. Oxford University Press, New York (2019)
Anderson, A.R., Belnap, N.D. Jr.: Entailment: The Logic of Relevance and Necessity, vol. 1. Princeton University Press, Princeton (1975)
Aristotle.: In: Barnes, J. (ed.), The Complete Works of Aristotle: The Revised English Translation, vol. 2, Princeton University Press, Princeton (1984)
Avron, A.: Whither relevant logic? J. Philos. Log. 21, 243–281 (1992)
Ayer, A.J.: The Problem of Knowledge. Macmillan, London (1958)
Batens, D.: Extensional paraconsistent logics. Logique et Analyse 90–91, 195–234 (1980)
Black, C.: A Matter of Principle. McClelland & Stewart, Toronto (2011)
Brandom, R.B.: Making it Explicit: Reasoning, Representing, and Discussion Commitment. Harvard University Press, Cambridge (1994)
Brandom, R.B.: Articulating Reason: An Introduction to Inferentialism. Harvard University Press, Cambridge (2000)
Callen, H.B.: Thermodynamics as an Introduction to Thermostatistics, 2nd edn. Wiley, New York (1985)
Chisholm, R.M.: Perceiving: A Philosophical Study. Cornell University Press, Ithaca (1957)
da Costa, N.: On the theory of inconsistent systems. Notre Dame J. Formal Log. 15, 497–510 (1974)
Bar-On, D.: Speaking My Mind: Expression and Self-Knowledge. Oxford University Press, Oxford (2004)
Evans, J.S.B.: Hypothetical Thinking: Dual Processes in Reasoning and Judgement. Psychology Press, Hove (2007)
Farkas, H.M.: Riemannian Surfaces. Springer, New York (1980)
Gabbay, D.M., Woods, J.: Filtration structures and the cut-down problem for abduction. In: Peacock, K.A., Irvine, A.D. (eds.) Mistakes of Reason: Essays in Honour of John Woods, pp. 398–417. University of Toronto Press, Toronto (2005)
Gasgoine, N., Thornton, T.: Tacit Knowledge. Acumen, Durham (1996)
Gettier, E.L.: Is justified true belief knowledge? Analysis 23, 121–123 (1963)
Goodman, N.: Fact, Fiction and Forecast. Harvard University Press, Cambridge (1955)
Harman, G.: Induction: a discussion of the relevance of knowledge to the theory of induction. In: Induction, A., Belief, R. (eds.) Marshall Swain, pp. 83–99. Reidel, Dordrecht (1970)
Harman, G.: Change in View: Principles of Reasoning. MIT Press, Cambridge (1986)
Hewitt, C.: Formalizing common sense reasoning for scalable inconsistency-robust information coordination using direct logic\(^{\rm TM}\) reasoning and the actor model. In: Hewitt, C., Woods, J. (eds.), Inconsistency Robustness, Studies in Logic, vol. 52, College Publications,London, pp. 3–103 (2015)
Hilbert, D.: Über Flachen von Konstanter Krümmung. Trans. Am. Math. Soc. 2, 87–99 (1901)
Hintikka, J.: Information, deduction, and the a priori. Nôus 4, 135–152 (1970)
Hintikka, J.: Logic, Language-Games and Information: Kantian Themes in the Philosophy of Logic. Clarendon Press, Oxford (1973)
Johnson, R.H., Anthony, B.J.: Logical Self-Defense. McGraw-Hill Ryerson, Toronto (1977)
Kanamori, A.: Mathematical knowledge: motley and complexity of proof. Ann. Jpn. Assoc. Philos. Sci. 21, 21–35 (2013)
Lackey, J.: Testimonial knowledge. In: Bernecker, S., Pritchard, D. (eds.) The Routledge Companion to Epistemology, pp. 316–324. Routledge, New York (2011)
Laudan, L.: Truth, Error, and Criminal Law: An Essay in Legal Epistemology. Cambridge University Press, New York (2006)
Laudan, L.: The Law’s Flaws: Rethinking Trials and Errors, The Law and Society Series, vol. 3, College Publications, London (2016)
Levelt, W.J.M.: Speaking: From Intention to Articulation. MIT Press, Cambridge (1998)
Lycan, W.G.: Evidence in Philosophy. Oxford University Press, New York (2019)
Magnani, L.: AlphaGo, locked strategies, and eco-cognitive openness. Philosophies (2019a). https://doi.org/10.1007/978-3-030-81447-2_3
Magnani, L.: Computational domestication of ignorant entities: unconventional cognitive embodiments. Synthese. https://doi.org/10.1007/s11229-020-02530-5(2019)
Paglieri, F., Woods, J.: Enthymematic parsimony. Synthese 178, 461–501 (2011)
Paglieri, F., Woods, J.: Enthymemes: from resolution to reconstruction. Argumentation 25, 127–139 (2011)
Peirce, C.S.: In: Hartshorne, C., Weiss, P. (eds.) Collected Papers, vol. 8, I–VI, and Burks, A.W. (ed.) vol. VII and VIIII, Harvard University Press, Cambridge (1931–1958)
Peirce, C.S.: Reasoning and the logic of things: the cambridge conference lectures of 1898. In: Kentner, K.L. (ed.), With an introduction by Kentner and Hilary, Harvard University, Putnam, Cambridge (1992)
Plato.: Theaetetus. In: Hamilton, E., Cairns, H. (eds.) Plato: The Collected Dialogues, Princeton University Press, Princeton, pp. 847–919 (1961)
Priest, G., Routely, R., Norman, J. (eds.): Paraconsistent Logic: Essays on the Inconsistent. Philosophia, Munich (1979)
Ransom, M.: Naturalizing logic: a case study of the ad hominem and implicit bias. In: Gabbay, D., Magnani, L., Park, W., Pietarinen, A.-V. (eds.) Natural Arguments: A Tribute to John Woods, Tributes, vol. 40, pp. 573–589. College Publications, London (2019)
Rawls, J.: A Theory of Justice. Harvard University Press, Cambridge (1971)
Riemann, B.: Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse”, Universitä, Gottingen, 1851 (Reprinted in H. Weber, editor, Bernhard Riemann, Gesammelte Mathematische Werke, pp. 1–43, Leipzig: Druck und Verlag von B. G. Teubner) (1876)
Routley, R., Routley, V.: The semantics of first degree entailment. Noûs 6, 335–395 (1972)
Schotch, P., Brown, B., Jennings, R. (eds.): On Preserving: Essays on Preservationism and Paraconsistent Logic. University of Toronto Press, Toronto (2009)
Shiffrin, R.M.: Automatism and consciousness. In: Cohen, J.D., Schooler, J.W. (eds.) Scientific Approaches to Consciousness, pp. 49–64. Erlbaum, Mahwah (1997)
Strong, J.W.: MacCormick on Evidence, 5th edn. West Group, St. Paul (1999)
Thouless, P.H.: Straight and Crooked Thinking. Hodden, London (1930)
von Kleist, H.: On the Gradual Production of Thoughts During Speech (translated into English and edited by David Constantine), Hackett, Indianapolis (2004)
Walton, D.: Appeal to Expert Opinion. Pennsylvania State University Press, University Park (1997)
Walton, D.: Critical questions in computational models of legal argument. In: Dunne, P.E., et al. (eds.) Argumentation in Artificial Intelligence and Law, IAAIL Workshop Series, pp. 103–111. Wolf Legal Publishers, Nijmegan (2005)
Walton, D., Reed, C.: Fabrizio: Argumentation Schemes. Cambridge University Press, Cambridge (2008)
Walton, D.: Methods of Argumentation. Cambridge University Press, New York (2013)
Wiles, A.: Modern elliptic curves and Fermat’s Last Theorem. Ann. Math. 141, 443–551
Woods, J.: Speaking your mind: inarticulacies constitutive and circumstantial. In: Campolo, C., Turner, D.(eds.), Argumentation and Articulation, A Special Issue of Argumentation, vol. 16, pp. 59–78 (2002)
Woods, J.: Errors of Reasoning: Naturalizing the Logic of Inference, Studies in Logic, vol. 45, College Publications, London (reprinted with corrections in 2014) (2013)
Woods, J.: Is legal reasoning irrational? In: An Introduction to the Epistemology of Law, vol. 2, The Law and Society Series, 2nd edn. (revised and expanded), College Publications, London, 1st edn. vol. 2015 (2018a)
Woods, J.: Truth in Fiction: Rethinking Its Logic, vol. 391, Synthese Library, Springer, Cham (2018b)
Woods, J.: Response to Madeleine Ransom. In: Gabbay et al. (ed.), Natural Arguments, p. 590–592 (2019)
Woods, J.: Four grades of ignorance-involvement and how they nourish the cognitive economy. Synthese 198, 3339–3368 (2021)
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“Our most necessary beliefs may be both unjustified and unjustifiable from our own perspective, and ... the attempt to justify them will lead merely to their loss.” Roger Scruton.
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Appendices
Appendix A: What Riemann Knew in 1851
Consider now the 19th century evolution of the concept of complex functions from Gauss to Riemann. Riemann’s great, and greatly disliked, breakthrough in his doctoral dissertation of 1851, the mapping theorem, was achieved by the use of geometrico-topological constructions that enabled him to investigate essential aspects of the behaviour of functions of complex variables. He did this without performing the prescribed algorithmic computations and beyond the reach of all then-known methods of proof. Riemann, in effect, was doing creative mathematics avant la lettre. The lettre would be written later by Klein, Hilbert, Poincaré and others.Footnote 47 The concept of Riemannian surfaces was an enormously fecund one, giving rise to several distinct but related fields of study: complex manifolds, Lie groups, algebraic number theory, harmonic analysis, abelian varieties, and algebraic topology.Footnote 48 Riemann himself was advancing a theorem of theories that wouldn’t come together for another fifty years. Were Peirce to have been consulted, he might have suggested that Riemann had had a flare, an innate instinct, for guessing right.Footnote 49 Perhaps this is so, but it doesn’t exclude the possibility that Riemann’s guesses were causally supported by what he already implicitly and tacitly had grasped of those theories-to-be.
A good guess is not just a lucky shot in the dark. Not all readers will be familiar with Riemann’s mapping theorem, but everyone will have retained some trace of the binomial theorem in elementary algebra and of some of the basic metatheorems of the metatheory of the first-order predicate calculus. It is clear that when theorems obtain, they do not obtain in a vacuum. Theorems have connections to that in virtue of which they obtain. Those connections lie nested in the theory which proves them. It would be surpassing strange that when a genius grasps a theorem before the theory whose theorem it is has even seen the light of day, he would have no ken whatever—no matter how implicit, tacit, and unformed—of the theory in which the theorem will turn out to have been grounded.
It can be said, of course, that the implicity and tacity of Riemann’s understanding of complex functions was only a temporary limitation. This is true, but beside the present point. What counts is not whether his successors could convert the unspoken to the spoken in a well-developed differential geometry and topology. What counts is that Riemann himself had a cognitive grasp of a part of mathematics before he could speak it. Like his teacher, Gauss, he was thinking in the manner of things not to be found in the settled mathematics of his day, and well before it could be set out chapter and verse.Footnote 50 It is true, of course, that had he lived long enough, he could have had a more fully articulated command of complex function theory. Alas, he died in 1866 at the shocking age of forty. Riemann’s case generalizes nicely:
The intrinsic implicity of progress: As long as a research programme remains in its progressive stage, it will always be the case that some of what is known of it at any given time twill be known implicitly and tacitly at t.
I come back now to the computer modelling of legal reasoning briefly mentioned in footnote 7 check?. It raises a rather fundamental question. Is the data-set of legal reasoning ready for it? Can we reliably model the human cognitive animal if much of the most productive of what he knows at tis beyond his powers to articulate attand often ever after? Does a computer have the power to model the humanly inexpressible, especially when software engineers demand strictly explicit instructions for coding? How, in any event, would we know that the computer got it right? Consider the phenomena of Big Data, Deep Learning, and self-teaching theorem-provers. Computer-assisted proofs now produce theorems whose proofs, it is said, cannot be fathomed even by the best of our ilk. For two millenia and more, proof ruled the roost in mathematics and man ruled the proofs. As we have it now, proofs still call the shots but, in ranges of cases, they appear not to be ruled by us. They are ruled by the computers we’ve built to teach themselves how to prove things beyond human ken of how it’s done. This bespeaks an interesting epistemic future for frontier mathematics. And, I would add, for logic and epistemology too.Footnote 51
The recent proof of Fermat’s most famous theorem also contains much food for related thought. The theorem asserts that no three positive integers x, y, z can satisfy the equation \(x^{n} +y^{n} = z^{n}\) for any value of \(n > 2\). Although attributed to Andrew Wiles, the proof was a team effort, and taxingly long and complex, employing devices of which Fermat could have had no ken, not even implicitly and tacitly. It made abundant use of a SWAC computer, and made its way through the Fey-Katz modularity theorem, Ribet’s epsilon conjecture, the Taniyama-Shimura conjecture, and the Taylor reworking of Horizontal Iwasawa Theory and an updated Euler system. The Wiles and Co. proof runs to 108 printed pages. Its real, as opposed to honorifically nominal, author was a multi-agency chain of individuals dating from Babylonian times, to the makers of Diophantine equations in the third century, to the tenth century sum-of-squares investigators, and on to Fermat’s own partial solutions flowing to the work of Mackell and Kunner and advanced by Gerd Faltings in 1903. With SWAC computer-assistance, Faltings’ results were extended by Wagstaff, Bubler, Crandell, Ewald and Metsäkyla. Then comes the Fey modularity and Kazy’s improvements and the further assistance rendered by Serre, Ribet, Kolyvagin, Flack and Katz and, later, Taylor.Footnote 52 Akihiro Kanamori is moved by this remarkable ensemble to make two observations. The first echoes the one quoted just lines above.
“The weight of [this and other] various examples shows how far away mathematics now is from being comprehended by any formal notion of proof and any [current] theory of mathematical knowledge, and how the limits of human intelligibility are being put to the test.”Footnote 53
The other touches on the substance of the Wiles et al. proof:
“In a substantial sense, [Fermat’s theorem] is dated and unto itself has no intrinsic interest whatsoever and it only grew in historical significance as it withstood more and more techniques [to prove it], techniques that enriched mathematics considerably …. [Wiles] actually established the Simura-Taniyama conjecture about elliptic curves in algebraic geometry through a beautiful synthetic proof, and this among mathematicians has been seen as a great advance.” (idem.)
The real interest lies not in the confirmation of a piece of 17\(^{\mathrm {th}}\) century number theory, as a corollary of a theorem on modular elliptic curves. What matters, among other things, is the complete vindication of a piece of new mathematics which had made its debut as a Peircean guess and in short order was a conjectured working hypothesis, available for provisional premissory engagements, to be judged for its fruitfulness in the demonstration of further results. Until the Wiles proof, the Simura-Taniyama was an instrumentaly helpful working hypothesis. Now it is a theorem whose large importance lies, in substantial part, in the spur it gives to further developments about elliptical curves in algebraic geometry. This takes us straight back to the intrinsic implicity and tacity of progressive science. As long as a theory progresses, it relies upon an implicit and tacit grasp of its own adumbrated future.
Appendix B: Solving the Inconsistency Problem
It is easily proved in English (and every other of humanity’s mother tongues) that from a contradiction every statement of English of necessity follows.Footnote 54 Let us not belabour the point here. For present purposes, it is enough to simply assume that the ex falso quodlibet theorem is true of English. Consider now some plain facts. Newton knew of the inconsistency that dogged the infinitesimal calculus. (Leibniz, too, independently.) But neither he nor anyone else anywhere else thought that there was nothing that Principia Mathematica made known about gravitation. Prior to Russell’s shocking disclosure of 1902, Frege taught his students what would come to be known as set theory. Unbeknownst, his axiom 5 implied a contradiction. Does anyone in the know really think that there wasn’t a single blessed truth about sets that Frege’s lectures made known in Jena? These are not rare cases. If, as I myself think, Carl Hewitt is right in saying that any big information system is, though pervasively and inerradicably inconsistent, nevertheless robust,Footnote 55 the same will have to be said of the big information systems which animate the human cognitive economy—including those in which all that a person knows at any tis explicitly or implicitly lodged; his background information, deep memory, and so on. Accordingly, a robustly inconsistent-information system is one with inbuilt measures for the efficient and productive management of inconsistency. A big information system is one that requires multiples of millions of lines of code to computerize. Initially, Hewitt was thinking of systems such as Five-Eyes (which soon will be Four-Eyes, and also possibly Three-Eyes). He now agrees that this totality of what a neurotypical human adult knows at a time, both explicitly and implicitly, requires big-system anchorage.Footnote 56 Both separately and collectively, these cases leave a rich data-set Dat for some theory or other to hold to account.
Once the tripartite distinction dividing consequence-having, consequence-spotting and consequence-drawing is expressly noted, it is clear for all to see that while a theory of consequence-having might well be a strictly abstract theory, the theories of spotting and drawing couldn’t, with any plausibility, not be naturalistic theories, hence theories that make a good fist of respecting data-set Dat. We can now see a full-service theory of deduction D—a full service deductive logic—as the ordered triple \(\langle \)D\(_{h}\), D\(_{s}\), D\(_{d}\rangle \) of theories of consequence-having, consequence-spotting, and consequence-drawing, two of whose elements are wide-open to circumspect empirical investigation.
Let’s now apply this tripartite distinction to data-set Dat and take Newton as a first example. On present assumptions, every theorem of Principia Mathematica has a validly implied negation.Footnote 57 Yet it is plain that nowhere in Principia do we find any sign of an exhaustive spotting of the consequences had by the inconsistency. There are reasons for this. One is their massive irrelevance. The other is their hefty cardinality. There are too many of them, namely \(\omega \). For these same reasons plus another, we see nothing in Principia to suggest exhaustive drawings of the consequences of the theory’s inconsistency. The further reason has to do with the doxastic structure of inference. When Sue infers S* from premisses S\(_{\mathrm {1}}\), …, S\(_{\mathrm {n}}\), she forms the belief that S* is a consequence of the S\(_{i}\). This is spotting. When she goes on to draw S*, she forms the ancillary belief that because the S\(_{i} \)are true and S* is a consequence of them, S* is also true. By construction of the case, the negation of S* follows of necessity from Principia. But Sue’s belief in S*’s truth motivates her to reject S*’s negation and to mark as false the premisses that imply it.Footnote 58
What we have here is impressive evidence of cognitive filtration in which only proper subsets of the outputs of consequence-having are made available as inputs to consequence-spotting, and only proper subsets of them as inputs for consequence-drawing. To work effectively, the filter requires no adjustment of the having-relation—in particular, no paraconsistentizing of it. What is required is that there be no wholesale reissuing of the conditions on having as rules for drawing. In other words, the logical relation of entailment needs to be sharply distinguished from the cognitive act of inferring.Footnote 59 As we now see, the dead-wrong way of reading ex falso is to have it say that from a contradiction everything whatever is validly inferable. What it actually does say, and is actually true, is that from a contradiction everything whatever is validly entailed.
As we have it so far, the history of logic in its accommodation of data-set Dat reflects a preference for placing conditions on the implication relation thereby constraining the act of inference in something like the ways attributed by the filtration hypothesis, by cutting down drastically the outputs of havingFootnote 60. Seen from either perspective, it should be clear that the natural homes of paraconsistent restraint are the cognitive domains of spotting and drawing. This obviates the need to tamper with implication, and argues instead that we should concentrate on the human reasoner, as a knowledge-seeking processor of information, not on the particularities of relations that obtain in logical space independently of negotiational engagement with actors. It should be clear by now that this processing, this extraction of knowledge from information, and this avoidance of massive inconsistency, is largely subconscious and automatic, and that most of what we know of its operations is implicit and tacit, hence a tricky candidate for express theoretical articulation. Moreover, the idea that it is only idle inconsistencies that the cognitive devices of humanity filter out ascribes to them an unwarranted inefficiency. There is plenty of behavioural evidence that our filtration devices automatically deliver the goods for Harman’s Clutter Avoidance Rule: Do not clutter up your mind with trivialities. (1986, 12).Footnote 61 Clearly irrelevance is on this list, and I think that, forewarned, hidden and unbidden bias may also be.Footnote 62 Nor should we overlook our overall net success in filtering good information from misinformation. Most of what a human being will know in this life he will know by having been told it. In the general case, tellings are epistemically efficacious in the absence of reliability tests. Sometimes, of course, we are taken in by a liar or a fool, but there is no doubting our stout record of not accepting everything we’ve been told. There are noticeable universalities in what we reject. By and large, an adult human being won’t believe a moral pronouncement upon a random telling of it, he’ll be much more inclined to believe on a stranger’s sayso that Station Centraal is at the north end of Spuistraat and diagonally to the right across the canal.
Acknowledgements: I have more people to thank than I can remember, but I am happy to call to mind the following, to whom all my grateful thanks. On matters of law, Dov Gabbay, Matthias Armgardt, Shahid Rahman, Paul Bartha, M. A. Armstrong, and Andrew Irvine; on matters implicit and tacit, Gottfried Gabriel, Selene Arfini, Margaret Schabas, Christopher Mole, Lorenzo Magnani, Woosuk Park: on matters abductive, Gabbay, Ahti-Veikko Pietarinen, the late Jaakko Hintikka, Daniele Chifi, Magnani, Park; on inconsistency-management, Gabbay, Gillman Payette, Carl Hewitt, Jean-Yves Beziau; and on causal-response epistemology, Maurice Finochiario, Magnani, Park, Alirio Rosales, and Fabio Paglieri, Matthieu Fontaine, Cristina Barés-Gómez. Special thanks to philosopher of foundational biology, Alirio Rosales, for instruction in the physics of phase-transitions, and for his wise counsel overall, Hans Vilhelm Hansen.
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Woods, J. The Role of the Common in Cognitive Prosperity: Our Command of the Unspeakable and Unwriteable. Log. Univers. 15, 399–433 (2021). https://doi.org/10.1007/s11787-021-00289-y
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DOI: https://doi.org/10.1007/s11787-021-00289-y
Keywords
- Causal-response epistemology
- Cognitive economics
- Commonness
- Criminal jurisprudence
- Implicity
- Information
- Phase transitions
- Storage
- Tacity