Abstract
The aim of this paper is twofold. First, I want to argue that the distinction between induction and deduction is less clear-cut than traditionally assumed, and that, moreover, most reasoning processes in the sciences involve an integration of inductive and deductive steps. Next, I want to show how so-called adaptive logics may lead to a better understanding of this integrated use of induction and deduction.
The author is a Postdoctoral Researcher of the Fund for Scientific Research — Flanders (Belgium) and is indebted to Matti Sintonen for interesting questions and comments on a previous version.
The research for this paper was supported by the Fund for Scientific Research — Flanders (Belgium), and indirectly by the Flemish Minister responsible for Science and Technology (contract BIL01/80). Unpublished papers referred to in this paper are available at http://logica.UGent.be/centrum/writings.
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Notes
The premises of an individual inference in a reasoning process need not form part of the background premises. Often, the former are themselves conclusions arrived at by previous inferences.
We shall see such an example from the history of the sciences in Section 2.
See, for instance, Thomas Nickles, “What is a Problem that we may Solve It?”, in: Synthese 47, 1981, pp. 85–118,
Paul Thagard, Computational Philosophy of Science. Cambridge, MA: MIT Press/Bradford Books 1988,
Lindley Darden, Theory Change in Science: Strategies from Mendelian Genetics. Oxford: Oxford University Press 1991,
Ronald Giere, Explaining Science: A Cognitive Approach. Chicago: University of Chicago Press 1988,
Nancy Nersessian, Faraday to Einstein: Constructing Meaning in Scientific Theories. Dordrecht: Martinus Nijhoff Publishers 1984,
David Gooding, Experiment and the Making of Meaning. Dordrecht: Kluwer 1990. Jaakko Hintikka and his associates, especially Matti Sintonen, were among the few philosophers of science that kept trying to apply logic — see also below.
A reasoning process that exhibits an internal dynamics is not necessarily non-monotonic. It has been shown, for instance, that the pure logic of relevant implication can be characterized by a dynamic proof theory (see Diderik Batens, “A Dynamic Characterization of the Pure Logic of Relevant Implication”, in: Journal of Philosophical Logic 30, 2001, pp. 267–280).
Among other things, I mean by this that it should be possible to formulate the metatheory in precise terms.
For an analysis of this particular reasoning process, see Joke Meheus, Liza Verhoeven, Maarten Van Dyck & Dagmar Provijn, “Ampliative Adaptive Logics and the Foundation of Logic-Based Approaches to Abduction”, in: Lorenzo Magnani, Nancy Nersessian & Claudio Pizzi (eds.), Logical and Computational Aspects of Model-Based Reasoning. Dordrecht: Kluwer 2002, pp. 39–71.
See Thomas Kuhn, The Essential Tension. Chicago: University of Chicago Press 1977, pp. 167–171.
Sadi Carnot, Reflexions stir la Puissance Motrice du Feu et sur les Machines propres à dé velopper cette Puissance. Paris: Bachelier 1824.
For a global reconstruction of this particular episode in the history of the sciences, see Peter Clark, “Atomism versus Thermodynamics”, in: Colin Howson (ed.), Method and Appraisal inthe Physical Sciences. The Critical Background to Modern Science, 1880–1905. Cambridge: Cambridge University Press 1976, pp. 41–105,
Joke Meheus, “Clausius’ Discovery of the First Two Laws of Thermodynamics. A Paradigm of Reasoning from Inconsistencies”, in: Philosophica 63, 1999, pp. 89–117 (appeared in 2001). An important difference between these two reconstructions concerns the question whether Clausius actually reasoned from an inconsistent set of premises. In the former, the position seems to be that Clausius first resolved the inconsistencies, on the basis of non-logical grounds, and only then continued reasoning. In the latter, it is argued that localizing and resolving the inconsistencies required reasoning from an inconsistent set of premises.
Rudolf Clausius, “Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen”, reprinted in: Max Planck (ed.), Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen. Leipzig: Verlag von Wilhelm Engelmann 1989, pp. 1–52.
See Rudolf Clausius, “Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen”, revised edition, reprinted and translated in: F. Folie (ed.), Théorie Mécanique de la Chaleur par R. Clausius. Paris: Librairie scientifique, industrielle et agricole 1868, pp. 17–106.
See Joke Meheus, “Adaptive Logic in Scientific Discovery: the case of Clausius” in: Logique et Analyse 165–166, 1993, pp. 359–389 (appeared in 1996)
Joke Meheus, “Inconsistencies in Scientific Discovery. Clausius’s Remarkable Derivation of Carnot’s Theorem”, in: Helge Krach, Geert Vanpaemel & Paul Marage (eds.), History of Modern Physics., Turnhout: Brepols 2002, pp. 143–154.
For the purposes of the present paper, it is not important to see what D and E stand for. The concrete form of the proofs can be found in Meheus, “Inconsistencies in Scientific Discovery. Clausius’s Remarkable Derivation of Carnot’s Theorem”, op. cit.
Some readers may have problems with this. Can one sensibly reason from an inconsistent set of premises, and thus, commit oneself to the simultaneous acceptance of inconsistent statements? It is important to remember, however, that the truth of a statement A not only depends on the state of the world, but also on the language L that is used to express A, and on the correspondence relation R that links L to the world. Evidently, no matter how the world looks like, it is possible to design a language and a correspondence relation such that the true description of the world is inconsistent (see also Diderik Batens, “In Defence of a Programme for Handling Inconsistencies”, in: Joke Meheus (ed.), Inconsistency in Science. Dordrecht: Kluwer 2002, pp. 129–150). Moreover, as the history of the sciences teaches, this is not merely a theoretical possibility.
A logic is called paraconsistent if it does not validate Ex Falso Quodlibet — to derive B from A and — A.
In line with this, almost all (monotonic) paraconsistent logics invalidate Reductio ad Absurdum. The only exception known to me is the logic AN (see Joke Meheus “An Extremely Rich Paraconsistent Logic and the Adaptive Logic Based on It”, in: Diderik Batens, Chris Mortensen, Graham Priest & Jean Paul Van Bendegem (eds.), Frontiers of Paraconsistent Logic. Baldock: Research Studies Press 2000, pp. 189–201).
In Section 6, we shall see that the suggested replacement is not sophisticated enough to lead to an adequate logic. This simplified version is instructive, however, for the argument in the present section.
The consequence relation of CL is undecidable, but it has a positive test. Hence, if I - A , there is bound to ‘exist’ a CL-proof of A from F , even if we may never find it.
Diderik Batens “Dynamic Dialectial Logics” in: Graham Priest, Richard Routley & Jean Norman (eds.), Paraconsistent Logic. Essays on the Inconsistent. München: Philosophia Verlag 1989, pp. 187–217.
Consequence relations that proceed in terms of maximal consistent subsets were first proposed by Rescher and Manor (Nicholas Rescher & Ruth Manor, “On Inference from Inconsistent Premises” in: Theory and Decision 1, 1970, pp. 179–217), and are today very popular in Alapplications. Reconstructing these consequence relations in terms of adaptive logics has the advantage that it thus becomes possible to design a proof theory for them.
See Joke Meheus, “Deductive and Ampliative Adaptive Logics as Tools in the Study of Creativity”, in: Foundations of Science 4, 3, 1999, pp. 325–336.
See http://logica.UGent.be/adlog/albib.html for an overview of the results on adaptive logics.
What is meant with the ambiguous term ‘part’ will become clear below.
See also Meheus, “Adaptive Logic in Scientific Discovery: the Case of Clausius”, op. cit.
Prioritized adaptive logics have important applications in the analysis of discussions and their dynamics — see, for instance, Diderik Batens, “Aspects of the Dynamics of Discussions and Logics Handling Them.” forthcoming. The formal machinery of prioritized adaptive logics is a bit more complicated than that of flat ones, but not fundamentally different.
I follow the characterization of adaptive logics as presented in Diderik Batens, “The Need for Adaptive Logics in Epistemology”, forthcoming.
Occasionally, the two strategies coincide and may be replaced by a simpler strategy. This is the case if, for every minimal Dab-consequence Dab(Δ) of every premise set F , Δ is a singleton.
One of the requirements is that the language of the lower limit logic should be rich enough to express that some formula behaves normally, respectively abnormally. I refer to Batens, “The Need for Adaptive Logics in Epistemology”, op. cit. for further details and for the proofs to the theorems listed below.
The rules PREM, RU and RC are the same if another strategy is applied. The marking definition for the Minimal Abnormality Strategy is a bit more complicated than that for the Reliability Strategy.
For a presentation of the logic P’ at the predicative level, see Diderik Batens, “Inconsistencyadaptive Logics”, in: Ewa Orlowska, Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa. Heidelberg: Physica Verlag 1999, pp. 445–472. In many papers on the subject, the logic is called “ACLuN1”.
To illustrate the dynamical character of the proofs, I shall each time give the complete proof, but, for reasons of space, omit lines that are not central to see the dynamics.
If q ⋀ q would be unconditionally derived at some stage in the proof, then p ⋀ -p would no longer be unreliable (it would not be a disjunct of a minimal Dab-formula at that stage), and hence, line 7 would be unmarked. This, however, is the only way to unmark line 7.
It follows from this that the logic P’ is not sensitive to the formulation of the premises, as is the case, for instance, for consequence relations that proceed in terms of maximal consistent subsets. Compare also with footnote 23, to see what was meant there by the ambiguous ‘parts’.
Diderik Batens, “On a Logic of Induction”, forthcoming.
Diderik Batens & Lieven Haesaert, “On Classical Adaptive Logics of Induction”, to appear.
I use this abbreviation instead of the usual Dab(Δ) to keep the examples within the margins.
See Batens & Haesaert, op. cit.
See Meheus, Verhoeven, Van Dyck & Provijn, op. cit. and Joke Meheus “ Empirical progress and ampliative adaptive logics”, to appear.
The results presented in this section concern work in progress. At this moment, the soundness and completeness proofs have not been formulated. However, in view of the format of the proof theory presented here, there is reason to believe that these proofs will be analogous to that for other adaptive logics.
It is in view of this problem that CL-based approaches to abduction have to require that the explanatory hypothesis is as parsimonious as possible — for a discussion of this problem and its possible solutions, see Atocha Aliseda, Seeking Explanations. Abduction in Logic, Philosophy of Science and Artificial Intelligence. Amsterdam: Institute for Logic, Language and Computation 1997.
See, for example, Jaakko Hintikka, “What is Abduction? The Fundamental Problem of Contemporary Epistemology.” In: Transactions of the Charles S. Peirce Society 34, 1998, pp. 503–533,
Jaakko Hintikka, Ilpo Halonen, & Arto Mutanen, “Interrogitive Logic as a General Theory of Reasoning”, in: Jaakko Hintikka (ed.), Inquiry as Inquiry: A Logic of Scientific Discovery. Dordrecht: Kluwer 1999, pp. 47–90
Matti Sintonen, “In Search of Explanations: From Why-Questions to Shakespearean Questions”, in: Philosophica 51, 1993, pp. 55–81.
The difference between the two approaches is discussed at some length in Diderik Batens & Joke Meheus, “On the Logic and Pragmatics of the Process of Explanation”, in: Mika Kiikeri & Petri Ylikoski (eds.), Explanatory Connections. Electronic Essays Dedicated to Maui Sintonen. http://www.valt.helsinki.fi/kfil/matti/, 2001, 22 p.
See also Batens, “The Need for Adaptive Logics in Epistemology”, op. cit.
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Meheus, J. (2004). Adaptive Logics and the Integration of Induction and Deduction. In: Stadler, F. (eds) Induction and Deduction in the Sciences. Vienna Circle Institute Yearbook, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2196-1_7
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