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Applications of Formal Philosophy pp 169–193Cite as

Abduction Logics: Illustrating Pitfalls of Defeasible Methods

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Part of the book series: Logic, Argumentation & Reasoning ((LARI,volume 14))

Abstract

On the one hand this paper offers an introduction to adaptive logics, focussing on properties that are imposed upon adaptive logics by the fact that they explicate defeasible reasoning. On the other hand new adaptive logics of abduction are presented and employed to illustrate those properties. These logics were developed in view of the criticism to existing adaptive logics of abduction.

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Notes

  1. 1.

    Formal logics need not validate Uniform Substitution. Let \(\mathcal {S}\), \(\mathcal {P}^r\), and \(\mathcal {C}\) be respectively the sets of sentential letters, predicates of rank r, and individual constants of \(\mathcal {L}\). Let f be a one-one mapping such that \(f:\mathcal {S}\longrightarrow \mathcal {S}\), \(f:\mathcal {P}^r\longrightarrow \mathcal {P}^r\), and \(f:\mathcal {C}\longrightarrow \mathcal {C}\). Extend f first to formulas, f(A) being the result of replacing in A every \(\xi \in \mathcal {S}\cup \mathcal {P}^r\cup \mathcal {C}\) by \(f(\xi )\); next to sets of formulas: \(f(\varGamma ) = _{ df } \{ f(A) \mid A\in \varGamma \}\). That \(\varGamma \vdash _\mathbf {L} A\) iff \(f(\varGamma ) \vdash _\mathbf {L} f(A)\) is sufficient for \(\mathbf {L}\) to be formal. Incidentally, the Uniform Substitution rule is tiresome at the predicative level [39].

  2. 2.

    For readers not familiar with paraconsistency, if \(A\vee B\) and \(\lnot A\) are true, then so is \((A\wedge \lnot A)\vee (B\wedge \lnot A)\). Only the second disjunct entails B. So, in the paraconsistent case, \(\{ A\vee B, \lnot A\}\) does not entail B but only \(B\vee (A\wedge \lnot A)\). This has been pointed out a long time ago by Newton da Costa, by Alan Anderson and Nuel D. Belnap, and by many others, and it gave rise to inconsistency-adaptive logics.

  3. 3.

    If both B and \(\lnot B\) are true, then so is \(A\supset B\), even if A is true and \(\lnot A\) is false.

  4. 4.

    For present purposes, call an adaptive logic \(\mathbf {AL}\) corrective iff all \(\mathbf {AL}\)-consequences of \(\varGamma \) are \(\mathbf {CL}\)-consequences (classical logic consequences) of \(\varGamma \); ampliative iff the converse holds.

  5. 5.

    Both readers familiar with the literature on explanation and readers familiar with detective stories will remember cases of a person who was poisoned at 4 PM and shot at 6 PM, but actually died of a heart attack at 5 PM.

  6. 6.

    I phrase this in terms of probabilities, but any similar approach may be equally acceptable.

  7. 7.

    People often define their views as the coherent systematization of their explicit views. The connected reasoning displays the internal dynamics. So everyone who has thought about his or her views is familiar with the internal dynamics.

  8. 8.

    The idea behind \(\mathbf {AAL}\) is that some formulas of the form \(A(a)\triangleright E(a)\) are derivable from \(\varGamma \) on top of the \(\mathbf {CL}\)-consequences of \(\varGamma \). So the transition from \(\mathbf {CL}\) to \(\mathbf {AAL}\) cannot have the effect that more generalizations become derivable from \(\varGamma \).

  9. 9.

    Note that defeasibility is not connected to derivability but to one’s insights in derivability.

  10. 10.

    A is a Weak consequence of \(\varGamma \) iff it is a \(\mathbf {CL}\)-consequence of a consistent subset of \(\varGamma \). Incidentally, Weak is paraconsistent, \(p,\lnot p\nvdash _{\mathrm {Weak}} q\), and not reflexive, \(p\wedge \lnot p\nvdash _{\mathrm {Weak}}p\wedge \lnot p\).

  11. 11.

    The reader may safely identify a logic that has static proofs with a compact Tarski logic, viz. a logic that is compact, reflexive, transitive, and monotonic.

  12. 12.

    The symbol \(\mathbin {\mathring{\vee }}\) is a conventional name to refer to a symbol of the language that has the meaning of classical disjunction.

  13. 13.

    For a restricted logical form, see the form defining \(\mathcal {G}\) below in the text.

  14. 14.

    Which applications are validated is determined by \(\mathbf {LLL}\)-consequences of the premise set. So adaptive logics display a form of content-guidance—Dudley Shapere [47] among others stated and defended the function of this feature in scientific methodology.

  15. 15.

    The reasoning in this paragraph applies to language schemas. It applies to languages of which the predicates are conceptually independent.

  16. 16.

    The simplified normal forms of contingent formulas do not contain irrelevant literals. Example: the simplified disjunctive normal form of \((Px\wedge Qx)\vee (Px\wedge \lnot Qx)\vee (Rx\wedge Px)\) is Px.

  17. 17.

    In the present context, \(\lceil \forall x(A(x)\supset B(x))\rceil \) may also be defined as \(\{ \forall x(C(x))\mid C\in s(A(x)\supset B(x))\}\). This however would not suit the generic approach, for example in case causal generalizations are considered.

  18. 18.

    A proof stage is a sequence of lines. A proof is a chain of stages, every stage containing the lines of the previous stage in the same order.

  19. 19.

    The logic under consideration has the unusual property that lines that are marked at a stage remain marked at all subsequent stages.

  20. 20.

    The \(\mathbf {ULL}\)-models are the \(\mathbf {LLL}\)-models that verify no abnormality.

  21. 21.

    The subset is proper, except for some exceptional adaptive logics called flip-flops.

  22. 22.

    A choice set of \(\langle \varDelta _1, \varDelta _2,\ldots \rangle \) comprises one member of every \(\varDelta _i\) (\(i\in \{1,2,\ldots \}\)). A choice set of \(\langle \varDelta _1, \varDelta _2,\ldots \rangle \) is minimal iff none of its proper subsets is a choice set of \(\langle \varDelta _1, \varDelta _2,\ldots \rangle \).

  23. 23.

    The so-called Simple strategy will do for such \(\varGamma \). Marking for the Simple strategy is defined by: line l is marked at stage s iff, at stage s, a member of the condition of line l is derived on the condition \(\emptyset \).

  24. 24.

    The statement should be taken literally in several senses. It pertains to the proofs, to the computational complexity of the consequence sets, to the semantic selection criterion, ....

  25. 25.

    Note that \(\lnot \Box _i\lnot A\) is adaptively derivable whenever A is compatible with \(T_i\).

  26. 26.

    While the premise \(\Box _2 Sa\) allows me to easily make the point, its presence is obviously artificial. Note, however, that \(\Box _2(\lnot Pa\vee Sa\vee Ta)\) is \({{\mathbf {CL}}^{\mathrm {M}}}\)-derivable from \(\varGamma _{2}\) and hence does not have any potential explanations.

  27. 27.

    Actually this too involves adaptive features, just like the study of yes/no questions [30, 31].

  28. 28.

    So inconsistency-adaptive logics assign all \(\mathbf {CL}\)-consequences to consistent premise sets.

  29. 29.

    That insights gained in terms of dynamic proofs may be rephrased in semantic terms does not undermine the avail of dynamic proofs.

  30. 30.

    Several points from this paragraph were clarified by a weird discussion in the literature [12, 21, 52] that started off as an attack on adaptive logics, presumably on the dynamic proofs, although that was never very clear.

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Acknowledgements

I am indebted to Joke Meheus and especially to Frederik Van De Putte for comments on a draft of this paper.

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  1. Centre for Logic and Philosophy of Science, Ghent University, Ghent, Belgium

    Diderik Batens

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  1. Institute of Philosophy, Sociology and Journalism, University of Gdansk, Poland and , Centre for Logic and Philosophy of Science, Ghent University,, Ghent, Belgium

    Rafał Urbaniak

  2. Dalhousie University , Halifax, Nova Scotia, Canada

    Gillman Payette

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Batens, D. (2017). Abduction Logics: Illustrating Pitfalls of Defeasible Methods. In: Urbaniak, R., Payette, G. (eds) Applications of Formal Philosophy. Logic, Argumentation & Reasoning, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-58507-9_8

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