Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2015 | OriginalPaper | Buchkapitel

1. Tutorial on Inconsistency-Adaptive Logics

verfasst von : Diderik Batens

Erschienen in: New Directions in Paraconsistent Logic

Verlag: Springer India

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This paper contains a concise introduction to a few central features of inconsistency-adaptive logics. The focus is on the aim of the program, on logics that may be useful with respect to applications, and on insights that are central for judging the importance of the research goals and the adequacy of results. Given the nature of adaptive logics, the paper may be read as a peculiar introduction to defeasible reasoning.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel Round Squares Are No Contradictions (Tutorial on Negation Contradiction and Opposition)
Fußnoten
1
See, for example, [17] for many other real-life examples of reasoning forms for which there is no positive test. The import of a positive test is discussed further in the text.
 
2
Uniform Substitution is rule of propositional logic. Predicative classical logic is traditionally axiomatized in terms of a finite set of rules and axiom schemata, rather than axioms. So no substitution rule is then required. Substitution rules in predicate logic have been studied [56] and the outcome is very instructive.
 
3
The \(\mathbf {L}\)-consequence set of \(\Gamma \) is defined as \(\textit{Cn}_{\mathbf {L}}(\Gamma ) =_{\textit{df}}\{ A \mid \Gamma \vdash _\mathbf {L} A \}\).
 
4
Just think about usual proofs. Every formula in the proof is a consequence of the premise set and every proof may be extended into a longer proof by applications of the rules.
 
5
A logic \(\mathbf {L}\) is paracomplete (with respect to a negation \(\lnot \)) iff some A may false together with its negation \(\lnot A\); syntactically: iff there are \(\Gamma \), A and B such that \(\Gamma , A \vdash _\mathbf {L} B\) and \(\Gamma , \lnot A \vdash _\mathbf {L} B\), but \(\Gamma \nvdash _\mathbf {L} B\).
 
6
In the context of \(\mathbf {CL}^+\), Excluded Middle together with Ex Falso Quodlibet define the classical negation.
 
7
So \(p\wedge \lnot q \vDash _{\mathbf {CL}^+} \lnot q\), \(\forall x\lnot Px \vDash _\mathbf {CL^+} \lnot Pa\), and \(a=b, Px \vDash _\mathbf {CL^+} Pb\), but \(a=b, \lnot Px \nvDash _\mathbf {CL^+} \lnot Pb\).
 
8
Names and notation may obviously be different and the model may be more complex.
 
9
Take conjunction as an example. The clause allowing for gluts: \(v_M(A\wedge B)=1\) iff (\(v_M(A)=1\) and \(v_M(B)=1\)) or \(v(A\wedge B)=1\); the one allowing for gaps: \(v_M(A\wedge B)=1\) iff (\(v_M(A)=1\) and \(v_M(B)=1\)) and \(v(A\wedge B)=1\); the one allowing for both: \(v_M(A\wedge B)=v(A\wedge B)\).
 
10
I heard the claim that restricting the formation rules of natural language so as to classify “this sentence is false” as non-grammatical is illegitimate because the sentence is ‘perfect English’. I also heard the claim that invalidating Disjunctive Syllogism is illegitimate because this reasoning form is ‘perfectly sound’.
 
11
This formula is \(\mathbf {CL}\)-equivalent to A but not \(\mathbf {CLuN}\)-equivalent to it.
 
12
As q is \(\mathbf {CLuN}\)-derivable from the premises, so is \(\lnot p\vee q\). However, relying on p to repeat the move described in the text delivers a formula that was already derivable, viz. q. The same story may be retold for every \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\) and each time the move will be harmless because nothing new will come out of it.
 
13
Do not read the “not derived” as “not derivable”. Indeed, a formula may be derivable in several ways from the same premise set.
 
14
A more accurate wording requires that one adds: in a proof from \(\Gamma _{1}\) that extends the present stage 8. Indeed, the logic we are considering is non-monotonic. So extending the premise set may result in line 6 being marked.
 
15
The reader might think that, as p is also a \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\), \((p\wedge q)\wedge \lnot (p\wedge q)\) is also a \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\). This however is mistaken. \(\lnot q \nvdash _\mathbf {CLuN} \lnot (p\wedge q)\).
 
16
Names like \(\mathbf {LLL}\), \(\mathbf {AL}\), \({\mathbf {AL}^\textit{r}}\), and \(\mathbf {ULL}\) are used as generic names to define the standard format and to study its features. The names refer to arbitrary logics that stand in a certain relation to each other.
 
17
Similarly for those models together with the trivial model—the model that verifies all formulas.
 
18
The notion played a rather central role in discussions on scientific heuristics. A very clear and argued position was for example proposed by Dudley Shapere [60].
 
19
This obviously does not mean that \(\mathbin {\hat{\vee }}\) is a symbol of the language. It is a conventional name to refer to a symbol of the language that has the meaning of classical disjunction. It may even refer ambiguously: if there are several classical disjunctions, \(\mathbin {\hat{\vee }}\) need not always refer to the same one.
 
20
Axioms are suppose to be closed formulas. So \(A\in \mathcal {W}_s\). The idea is that \(\mathbf {CLuN}\)-valid rules are fully retained in the extension. One of these rules is: from \(\vdash A(a)\supset B\) to derive \(\vdash \exists xA(x)\supset B\) provided a does not occur in B.
 
21
The axiom schema may be restricted to \(A \in \mathcal {W}_s^a\), but there is no need to do so.
 
22
The exception may be caused by the logic, which is then called a flip-flop, or by the premise set—for example if the premise set comprises the formulas verified by a \(\mathbf {LLL}\)-model.
 
23
Infinite stages can be extended by inserting lines in the sequence.
 
24
It is ironic that the study of the computational complexity of adaptive logics started with a paper arguing that they are too complex [41]. The philosophical complaints and misunderstandings in that paper were answered in [26]; a mistaken theorem was corrected in [68]. Extremely interesting and more detailed studies followed [54, 55].
 
25
Stepwise: the language \(\mathcal {L}_s\) of \(\mathbf {CLuN}\) is extended with the symbol \(\mathord {\sim }\) and \(\mathbf {CLuN}\) is extended with axioms or rules that give \(\mathord {\sim }\) its classical meaning—for example the schemas \(A\supset (\mathord {\sim }A\supset B)\) and \((A\supset \mathord {\sim }A)\supset \mathord {\sim }A\).
 
26
The classical symbols were actually superimposed on \(\mathcal {L}\): in the extended language, they never occur within the scope of the original logical symbols of \(\mathcal {L}\).
 
27
The distinction warrants that the reference to a finite proof stage in Definition 1.5 is all right.
 
28
The mistake is caused by a confusion between symbols and concepts. If \(\check{\vee }\) occurs in a premise, and so in \(\mathcal {L}\), then \(\check{\vee }\) is not a new symbol of the extended language. So one needs to extend the language with another symbol, say \(\tilde{\vee }\), and call that the checked disjunction.
 
29
All that is new in the restored version is the notion of an inferred \(\textit{Dab}\)-formula.
 
30
Adaptive versions of \(\mathbf {D2}\) and other Jaśkowski logics were extensively studied [48‐50].
 
31
The low computational complexity of the consequence set is rather artificial. We suppose that at least one \(\varphi \cap \Delta = \emptyset \) is given, but precisely locating a \(\varphi \) may be a very complex task.
 
32
The logic-like entity has a rather limited application field. For some \(\Gamma \), \(\Phi (\Gamma )\) is not only infinite but also uncountable.
 
33
That is (i) \(\mathcal {W} \subseteq \mathcal {W}^+\) and (ii) if \(A,B \in \mathcal {W}^+\), then \((A\mathbin {\hat{\vee }}B) \in \mathcal {W}^+\).
 
34
Classical theories, which have \(\mathbf {CL}\) as underlying logic, fail to define such a theory. Their consequence relation is much less complex. If A is not a theorem of a classical theory, humans or Turing machines may never find this out. However, if A is a theorem of the classical theory, humans or Turing machines will find that out at a finite point in time. As this point may be two million years from here, the point is slightly theoretical.
 
Literatur
1.
2.
3.
Avron, A.: Non-deterministic matrices and modular semantics of rules. In: J.-Y. Béziau (ed.) Logica Universalis, pp. 149–167. Birkhäuser Verlag, Basel–Boston–Berlin (2005)
4.
Avron, A., Konikowska, B.: Multi-valued calculi for logics based on non-determinism. J. Interest Group Pure Appl. Logic 10, 365–387 (2005)MathSciNet
5.
6.
Avron, A., Zamansky, A.: Non-deterministic semantics for logical systems. In: Gabbay, D., Guenthner, F. (ed.) Handbook of Philosophical Logic, vol. 16, pp. 227–304. Springer (2011)
7.
8.
Batens, D.: Paraconsistent extensional propositional logics. Logique et Analyse 90–91, 195–234 (1980)MathSciNet
9.
Batens, D.: Dialectical dynamics within formal logics. Logique et Analyse 114, 161–173 (1986)MathSciNet
10.
Batens, D.: Dynamic dialectical logics. In: Priest, G., Routley, R., Norman, J. (eds.) Paraconsistent Logic. Essays on the Inconsistent, pp. 187–217. Philosophia Verlag, München (1989)
11.
Batens, D.: Blocks. The clue to dynamic aspects of logic. Logique et Analyse, 150–152, pp. 285–328 (1995). Appeared 1997
12.
Batens, D.: Inconsistency-adaptive logics. In: Orłowska, E. (ed.) Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pp. 445–472. Physica Verlag (Springer), Heidelberg, New York (1999)
13.
14.
Batens, D.: Towards the unification of inconsistency handling mechanisms. Logic Logical Philos. 8, 5–31 (2000). Appeared 2002
15.
Batens, D.: A general characterization of adaptive logics. Logique et Analyse, 173–175, 45–68 (2001). Appeared 2003
16.
Batens, D.: On a partial decision method for dynamic proofs. In: Decker, H., Villadsen, J., Waragai, T. (eds.) PCL 2002. Paraconsistent Computational Logic, pp. 91–108. (=Datalogiske Skrifter vol. 95) (2002). Also available as cs.LO/0207090 at http://​arxiv.​org/​archive/​cs/​intro.​html
17.
Batens, D.: The need for adaptive logics in epistemology. In: Gabbay, D., Rahman, S., Symons, J., Van Bendegem, J.P. (eds.) Logic. Epistemology and the Unity of Science, pp. 459–485. Kluwer Academic Publishers, Dordrecht (2004)
18.
Batens, D.: A procedural criterion for final derivability in inconsistency-adaptive logics. J. Appl. Logic 3, 221–250 (2005)CrossRefMathSciNetMATH
19.
Batens, D.: Narrowing down suspicion in inconsistent premise sets. In Malinowski and Pietruszczak [42], pp. 185–209
20.
21.
Batens, D.: Towards a dialogic interpretation of dynamic proofs. In Cédric D., Laurent K., Helge R. (eds.) Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman, pp. 27–51. College Publications, London (2009). 558 pp
22.
Batens, D.: New arguments for adaptive logics as unifying frame for the defeasible handling of inconsistency. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds.) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol. 26, pp. 101–122. Springer, Dordrecht (2013)CrossRef
23.
Batens, D.: The consistency of Peano Arithmetic. A defeasible perspective. In: Allo, P., Van Kerkhove, B. (eds.) Modestly Radical or Radically Modest. Festschrift for Jean Paul van Bendegem on the Occasion of His 60th Birthday, volume 24 (sometimes 22) of Tributes, pp. 11–59. College Publications, London (2014)
24.
Batens, D.: Adaptive Logics and Dynamic Proofs. Mastering the Dynamics of Reasoning. 201x. Forthcoming
25.
Batens, D., De Clercq, K.: A rich paraconsistent extension of full positive logic. Logique et Analyse. 185–188, 227–257 (2004). Appeared 2005
26.
Batens, D., De Clercq, K., Verdée, P., Meheus, J.: Yes fellows, most human reasoning is complex. Synthese 166, 113–131 (2009)CrossRefMATH
27.
Batens, D., Meheus, H.: A tableau method for inconsistency-adaptive logics. In Roy D. (ed.) Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Artificial Intelligence, vol. 1847, pp. 127–142. Springer (2000)
28.
Batens, D., Meheus, Joke: Shortcuts and dynamic marking in the tableau method for adaptive logics. Stud. Logica 69, 221–248 (2001)CrossRefMathSciNetMATH
29.
Batens, D., Vermeir, T.: Direct dynamic proofs for the Rescher-Manor consequence relations: the flat case. J. Appl. Non-Class. Logics 12, 63–84 (2002)CrossRefMathSciNetMATH
30.
31.
Carnap, R.: Logical Foundations of Probability. University of Chicago Press, Chicago (1950)MATH
32.
Carnielli, W.A., Coniglio, M.E., Marcos, J.: Logics of formal inconsistency. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 14, pp. 1–93. Springer (2007)
33.
Carnielli, W.A., Marcos, J., de Amo, S.: Formal inconsistency and evolutionary databases. Logic Logical Philos. 8, 115–152 (2001). Appeared 2002
34.
De Clercq, K.: Logica in communicatie. Cahiers du Centre de Logique, vol. 14. Academia-Bruylant, Louvain-la-Neuve (2005)
35.
D’Ottaviano, I.M.L.: Sobre uma Teoria de Modelos Trivalente (in Portuguese). PhD thesis, State University of Campinas (Brazil), 1982
36.
D’Ottaviano, I.M.L.: The completeness and compactness of a three-valued first-order logic. In: Proceedings of the 5th Latin American Symposium on Mathematical Logic, pp. 77–94. Revista Colombiana de Matemáticas, 1–2, 1985
37.
D’Ottaviano, I.M.L.: The model extension theorems for J \(_{3}\)-theories. In: Di Prisco C.A. (ed.) Methods in Mathematical Logic: Proceedings of the 6th Latin American Symposium on Mathematical Logic, Lecture Notes in Mathematics, vol. 1130, pp. 157–173. Springer (1985)
38.
D’Ottaviano, I.M.L.: Definability and quantifier elimination for J\(_3\)-theories. Stud. Logica 46(1), 37–54 (1987)
39.
D’Ottaviano, I.M.L., Epstein, R.L.: A paraconsistent many-valued propositional logic: J\(_3\). Rep. Math. Logic 22, 89–103 (1988)
40.
41.
42.
Jacek, M., Andrzej, P. (eds.): Essays in Logic and Ontology. Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. 91. Rodopi, Amsterdam/New York (2006)
43.
Meheus, J.: Adaptive logic in scientific discovery: the case of Clausius. Logique et Analyse 143–144, 359–389 (1993). Appeared 1996
44.
Meheus, J.: Clausius’ discovery of the first two laws of thermodynamics. A paradigm of reasoning from inconsistencies. Philosophica 63, 89–117 (1999). Appeared 2001
45.
Meheus, J.: Adaptive logics for question evocation. Logique et Analyse 173–175, 135–164 (2001). Appeared 2003
46.
Meheus, J.: An adaptive logic for pragmatic truth. In: Carnielli, W.A., Coniglio, M.E., D’Ottaviano, I.M.L. (eds.) Paraconsistency. The Logical Way to the Inconsistent, pp. 167–185. Marcel Dekker, New York (2002)
47.
Meheus, J.: Inconsistencies in scientific discovery. Clausius’s remarkable derivation of Carnot’s theorem. In: Krach, H., Vanpaemel, G., Marage, P. (eds.) History of Modern Physics. Acta of the XXth International Congress of History of Science, pp. 143–154. Brepols, Turnhout (Belgium) (2002)
48.
Meheus, J.: Discussive adaptive logics: handling internal and external inconsistencies. In: Malinowski and Pietruszczak [42], pp. 211–223
49.
Nasieniewski, M.: An adaptive logic based on Jaśkowski’s logic \(\mathbf{D}_2\). Logique et Analyse 185–188, 287–304 (2004). Appeared 2005
50.
Nasieniewski, M.: Wprowadzenie do logik adaptywnych. Universytetu Mikołaja Kopernika, Toruń, Wydawnictwo Naukowe (2008)
51.
Nersessian, N.: Inconsistency, generic modeling, and conceptual change in science. In: Meheus, J. (ed.) Inconsistency in Science, pp. 197–211. Kluwer, Dordrecht (2002)
52.
Norton, J.: The logical inconsistency of the old quantum theory of black body radiation. Philos. Sci. 54, 327–350 (1987)CrossRefMathSciNet
53.
Norton, J.: A paradox in Newtonian gravitation theory. PSA 1992(2), 420–421 (1993)
54.
Odintsov, S.P., Speranski, S.O.: On algorithmic properties of propositional inconsistency-adaptive logics. Logic Logical Philos. 21, 209–228 (2012)MathSciNetMATH
55.
56.
Pogorzelski, W.A., Prucnal, T.: The substitution rule for predicate letters in the first-order predicate calculus. Rep. Math. Logic 5, 77–90 (1975)MathSciNetMATH
57.
Priest, G.: In Contradiction. A Study of the Transconsistent. Nijhoff, Dordrecht (1987)
58.
Rescher, N., Manor, R.: On inference from inconsistent premises. Theory Decis. 1, 179–217 (1970)CrossRefMATH
59.
60.
Shapere, D.: Logic and the philosophical interpretation of science. In: Weingartner, P. (ed.) Alternative Logics. Do sciences need them?, pp. 41–54. Springer, Berlin, Heidelberg (2004)
61.
Smirnova, E.D.: An approach to the justification of semantics of paraconsistent logics. In: Batens, D., Mortensen, C., Priest, G., Van Bendegem, J.P. (eds.) Frontiers of Paraconsistent Logic, pp. 255–262. Research Studies Press, Baldock, UK (2000)
62.
Smith, J.: Inconsistency and scientific reasoning. Stud. History Philos. Sci. 19, 429–445 (1988)CrossRef
63.
Straßer, C.: Adaptive Logic and Defeasible Reasoning. Applications in Argumentation, Normative Reasoning and Default Reasoning, Trends in Logic, vol. 38. Springer (2014)
64.
Van De Putte, F., Straßer, C.: Extending the standard format of adaptive logics to the prioritized case. Logique et Analyse 220, 601–641 (2012)
65.
Van Kerckhove, B., Vanackere, G.: Vagueness-adaptive logic: a pragmatical approach to Sorites paradoxes. Stud. Logica 75, 383–411 (2003)CrossRef
66.
Vanackere, G.: Ambiguity-adaptive logic. Logique et Analyse 159, 261–280 (1997). Appeared 1999
67.
Vanackere, G.: The role of ambiguities in the construction of collective theories. Logique et Analyse, 173–174–175, 189–214 (2001). Appeared 2003
68.
69.
Verdée, P.: Strong, universal and provably non-trivial set theory by means of adaptive logic. Logic J. IGPL 21, 108–125 (2012)CrossRef
70.
Verdée, P.: Non-monotonic set theory as a pragmatic foundation of mathematics. Found. Sci. 18, 655–680 (2013)CrossRefMathSciNet
71.
Metadaten
Titel
Tutorial on Inconsistency-Adaptive Logics
verfasst von
Diderik Batens
Copyright-Jahr
2015
Verlag
Springer India
Buch
New Directions in Paraconsistent Logic

Print ISBN: 978-81-322-2717-5

Electronic ISBN: 978-81-322-2719-9

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-81-322-2719-9_1