Abstract
We propose a logic of abduction that (i) provides an appropriate formalization of the explanatory conditional, and that (ii) captures the defeasible nature of abductive inference. For (i), we argue that explanatory conditionals are non-classical, and rely on Brian Chellas’s work on conditional logics for providing an alternative formalization of the explanatory conditional. For (ii), we make use of the adaptive logics framework for modeling defeasible reasoning. We show how our proposal allows for a more natural reading of explanatory relations, and how it overcomes problems faced by other systems in the literature.
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Notes
Peirce’s notion of abduction is a difficult one to unravel, for it is entangled with many other aspects of his philosophy and several different conceptions and terminology evolved in his thought. He first used ‘hypothesis’ when his notion had a syllogistic form, changed later to the term ‘retroduction’ and finally to ‘abduction’ for a more general conception of explanatory reasoning. For some key works on Peirce’s abduction, see Fann (1970); Frankfurt (1958); Hintikka (1998).
Note however that abductive inference faces further challenges when applied to first order logic (FOL). Formulations that put forward preconditions \( \varTheta \not \models \varphi \) and \(\varTheta \not \models \lnot \varphi \) to characterize what constitutes an abductive problem [\(\varTheta , \varphi \)] (see Sect. 5) pose a problem for FOL: “it is undecidable even to determine whether a pair [\(\varTheta , \varphi \)] is a genuine abductive problem” (Mayer and Pirri 1993, p. 113). Abduction in FOL may be addressed however without claiming above preconditions, such as it is done in Mayer and Pirri (1993), in which the authors make use of unification and dynamic skolemization of formulae to perform first order abduction. It may also be tackled for some classes of first order formulae using resolution (Marquis 1991) or by adding a restriction to formulae satisfiable in finite domains, as proposed in Aliseda (2006) within the framework of semantic tableaux.
In Woods (2012) it is upheld that a successful abduction does not even allow us to infer the belief that the abduced proposition is true, aiming to capture the subjunctive nature of the conditional.
See (Aliseda 2006, Ch. 3) for a structural characterization of abduction incorporating some of these additional restrictions.
If we drop the assumption that our background knowledge is not subject to revision, or the weaker assumption that our background information is epistemically prior to the explanations derived abductively, the argument no longer holds, since then it is no longer required that the inferred explanations be withdrawn. Moreover, if the underlying logic is weakened to a paraconsistent system, explanations that contradict the background information no longer cause explosion, and abductive inference can be implemented monotonically. See Carnielli (2006) for an example. We thank an anonymous referee for pointing this out to us.
Our use of the term ‘monotonic’ applies to the consequence relation of a logic. The term is sometimes also used in reference to the conditional. In the latter sense, a logic is non-monotonic if its conditional invalidates the inference from \(\psi \rightarrow \varphi \) to \((\psi \wedge \chi )\rightarrow \varphi \), i.e. if it invalidates (SA). The logics to be defined in Sects. 3 and 4 are non-monotonic in both of these senses.
Chellas’s systems are conditional logics, devised with the aim of understanding the nature of “if \(\ldots \) then \(\ldots \)” statements. In Priest (2008), conditional logics are motivated as systems capable of modelling ceteris paribus claims. The connection with explanatory inference is pointed out by van Fraassen, who showed that explanatory conditionals carry a tacit ceteris paribus clause (van Fraassen 1980). This suggests using a conditional logic for modeling explanatory inference. For the semantics for \(\mathbf {CR}\), we refer to Appendix A.
A nice side-effect of the failure of (TRA) is that from circular explanations we cannot infer direct self-explanations. For instance, from the formulas \(p\rightarrow q, q\rightarrow r\), and \(r\rightarrow p\) we cannot infer that \(p\rightarrow p\).
This example is a slightly adapted version of a counterexample to the material conditional given by Priest in 2008.
Thus the rule \(\text{ Prem }\) allows us to introduce premises on the empty condition.
‘\(Dab\)’ is short for ‘disjunction of abnormalities’. If \(\varTheta \) is a singleton \(\{A\}\) for some \(A\in \varOmega \), then \( Dab (\varTheta )=A\).
Note that, when applying \(\text{ RU }\) or \(\text{ RC }\), conditions of the lines used in the derivation are taken over to the conclusion: if at line \(i\) we assume an abnormality \(\varDelta _i\) to be false in deriving a formula \(A_i\) in the proof, and we use line \(i\) in deriving line \(j\), then at line \(j\) too we implicitly assume this abnormality to be false.
The reliability strategy is more cautious still than the minimal abnormality strategy. For instance, from the premise set \(\{r,p\rightarrow r,q\rightarrow r,\lnot (p\wedge q)\}\) this strategy does not allow us to derive \(p\vee q\), as opposed to the minimal abnormality strategy (cfr. infra). Due to this inferential weakness of the reliability strategy, we opt to define only the minimal abnormality strategy.
In the AI-literature, such disjunctions are called floating conclusions, see e.g. Horty (2002); Makinson and Schlechta (1991). An even more ‘skeptical’ strategy which does not allow for the derivation of floating conclusions is the reliability strategy (see footnote 14). For a discussion on minimal abnormality, reliability, and floating conclusions, see (Straßer 2014, Ch. 6).
As noted at the beginning of this section, some authors are not committed to classical logical entailment; others (Aliseda 2006) argue for an “inferential parameter” to be instantiated by any number of consequence relations, but still follow this logical schema.
When interpreting \(\rightarrow \) as a material conditional, these ‘abnormal’ instances of (BMP) have to be filtered out by adding restrictions on the definition of the set of abnormalities. Consequently, the definition of this set in the systems mentioned becomes much more complicated than that of the set \(\varOmega \) of \(\mathbf {CR^{n/m}}\)-abnormalities.
The first-order variant of the (BMP) rule as applied defeasibly in these logics is \((\forall \alpha )(\varPsi (\alpha ) \supset \varPhi (\alpha )), \varPhi (\beta ) / \varPsi (\beta )\), where \(\varPhi \) and \(\varPsi \) are closed first-order formulas with one variable; \(\alpha \) is an individual variable; and \(\beta \) is an individual constant.
The modal variant of the (BMP) rule as applied defeasibly in this logic is \(\Box _n(\psi \rightarrow \varphi ),\varphi / \psi \), where \(\psi \) and \(\varphi \) are \(\mathbf {CL}\)-wffs.
This example was given to us by an anonymous reviewer.
Note however that in some previous proposals of abduction in adaptive logics, a set of ‘abducibles’ was required, as in the logic \(\mathbf {LA^r}\) found in Meheus and Batens (2006).
We thank an anonymous referee for pointing out this connection.
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Acknowledgments
Research for this article was partially supported by the project “Logics of discovery, heuristics and creativity in the sciences” (PAPIIT, IN400514-3) granted by the National Autonomous University of Mexico (UNAM). We are greatly indebted to the Dirección General de Asuntos del Personal Académico (UNAM) and to the Programa de Becas Posdoctorales de la Coordinación de Humanidades (UNAM). We also thank Laura Leonides and two anonymous referees for their many helpful comments and suggestions regarding this paper.
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Appendices
Appendix A: Semantic consequence relations for \(\mathbf {CR}\), \(\mathbf {CR^n}\) and \(\mathbf {CR^m}\)
1.1 Semantics for \(\mathbf {CR}\)
In Chellas (1975) Chellas provides a semantics for \(\mathbf {CR}\) in terms of minimal frames. Let a frame be a structure \(\mathfrak {F} = \langle W,f \rangle \) in which \(W\) is a set and \(f:W\times \wp (W)\mapsto \wp (\wp (W))\). A minimal model on a minimal frame \(\mathfrak {F}\) is a triple \(M = \langle W,f,v_p \rangle \) where \(v_p: \mathcal {P}\mapsto \wp (W)\).
Intuitively, \(W\) can be thought of as a set of possible worlds. \(v_p\) maps proposition letters to sets of worlds. For each possible world and formula, \(f\) selects a class of formulas. Let \(|A|^{M}\) abbreviate the set of worlds at which \(A\) is true on \(M\). The idea is that a conditional \(A\rightarrow B\) is true at a world \(w\) in \(M\) just in case \(f\) picks out the consequent \(B\) relative to the antecedent \(A\):
The classical connectives are defined as usual:
This characterization of \(\rightarrow \) ensures the validity of (RCEA) and (RCEC). For (CM) and (CC), we need two further conditions:
Where \(A\in {\fancyscript{F}}, \vDash ^{M}A\) iff \(\vDash ^{M}_w A\) for every \(w\in W\), and \(\vDash A\) iff \(\vDash ^{M} A\) for every model on \(\mathfrak {F}\). For the full details and for the proof that \(\mathbf {CR}\) is determined by the class of minimal frames for which (CM’) and (CC’) hold, we refer to (Chellas 1975, Sect. 8).
Underlying Chellas’s definition of the function \(f\) is an interpretation of conditionality as relative necessity. A conditional \(A\rightarrow B\) holds if \(B\) is one of the propositions that are necessary relative to \(A\). An alternative interpretation of this idea in our framework proceeds in terms of explanations: \(A\rightarrow B\) if \(B\) is one of the propositions possibly explained by \(A\), where, for each world \(w\) and formula \(A, f\) picks out a class of propositions that are possibly explained by \(A\) at \(w\).
Ben-David and Ben-Eliyahu-Zohary (2000) define a number of systems that bear a family resemblance to \(\mathbf {CR}\), augmenting Chellas’s minimal frames with filtration techniques. Their \(\mathbf {F + CM}\) comes particularly close to Chellas’s \(\mathbf {CK}\), which is in turn slightly stronger than \(\mathbf {CR}\) (see Section 2). \(\mathbf {F + CM}\) is stronger than \(\mathbf {CK}\) since it satisfies the idempotence schema (ID) and validates modus ponens for the conditional ‘\(\rightarrow \)’.Footnote 23
1.2 Semantics for \(\mathbf {CR^n}\) and \(\mathbf {CR^m}\)
\(\mathbf {CR^n}\) and \(\mathbf {CR^m}\) can be characterized as triples consisting of:
-
(i)
A lower limit logic (LLL), the logic \(\mathbf {CR}\),
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(ii)
A set of abnormalities, the set \(\varOmega \) as defined in Sect. 3,
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(iii)
An adaptive strategy: normal selections for \(\mathbf {CR^n}\), minimal abnormality for \(\mathbf {CR^m}\).
The semantics for these logics works by taking the LLL-models and selecting a subset of them. Semantic consequence for \(\mathbf {CR^n}\) is defined as follows:
Definition 9
A (finite) set \(\varDelta \subset \varOmega \) is normal with respect to a set \(\varGamma \subseteq {\fancyscript{F}}\) iff \(\varGamma \not \vDash Dab (\varDelta )\).
Definition 10
\(\varGamma \vDash _{\mathbf {CR^n}} A\) iff, for some \(\varDelta , \varGamma \vDash A \vee Dab (\varDelta )\) whereas \(\varGamma \not \vDash Dab (\varDelta )\).
Where \(M\) is a \(\mathbf {CR}\)-model, let \(Ab(M) = \{A\in \varOmega \mid \vDash ^M A\}\). Semantic consequence for \(\mathbf {CR^m}\) is defined as follows:
Definition 11
A \(\mathbf {CR}\)-model \(M\) of \(\varGamma \) is minimally abnormal iff there is no \(\mathbf {CR}\)-model \(M'\) of \(\varGamma \) such that \(Ab(M')\subset Ab(M)\).
Definition 12
\(\varGamma \vDash _{\mathbf {CR^m}} A\) iff \(A\) is verified by all minimally abnormal models of \(\varGamma \).
Soundness and completeness for \(\mathbf {CR^m}\) follows from the generic soundness and completeness proofs for adaptive logics in standard format from (Batens 2007, Theorem 9). For \(\mathbf {CR^n}\) soundness and completeness follow from the generic adequacy proofs for normal selections in (Straßer 2014, Sect. 2.8).
Appendix B: A problem for adaptive logics for practical abduction
The logic \(\mathbf {LA^r_s}\) from Meheus (2011) extends first-order classical logic (in the lingo of adaptive logics, first-order classical logic is the lower limit logic of \(\mathbf {LA^r_s}\)). Where \(\varPhi \) and \(\varPsi \) are closed first-order formulas with one variable; \(\alpha \) is an individual variable; and \(\beta \) is an individual constant, the set \(\varOmega '\) of \(\mathbf {LA^r_s}\)-abnormalities is defined as follows:
\(\mathbf {LA^r_s}\) makes use of the reliability strategy. Any adaptive logic defined within the standard format from Batens (2007) is fully characterized by these three elements (lower limit logic, set of abnormalities, and adaptive strategy). For more details about this logic, we refer to Meheus (2011).
The following \(\mathbf {LA^r_s}\)-proof illustrates that neither \(Sa\) nor \(Ra\) is derivable from the premise set \(\{(\forall x)(Sx \supset Qx), (\forall x)(Rx\supset Px), Qa, Pa\}\):
We used \(!(\varPsi (\beta ),\varPhi (\beta ))\) as an abbreviation for \((\forall \alpha )(\varPsi (\alpha )\supset \varPhi (\alpha )) \wedge \varPhi (\beta )\wedge \lnot \varPsi (\beta )\). Lines 5 and 6 are marked in view of the minimal \( Dab \)-formulas derived at lines 7 and 8. The proof cannot be extended in a way that would unmark these lines. Hence, neither \(Sa\) nor \(Ra\) is finally derivable.
\(\mathbf {LA^r_s}\) is a minor variant of the logic \(\mathbf {LA^r}\) from Meheus and Batens (2006). \(\mathbf {LATA^r}\) from Aliseda and Leonides (2013) combines \(\mathbf {LA^r_s}\) with the inconsistency-adaptive logic \(\mathbf {CLuN^r}\). The above proof carries over to both \(\mathbf {LA^r}\) and \(\mathbf {LATA^r}\), showing that both of these systems face the same problem.
The logic \(\mathbf {AbL^p}\) from Lycke (2012) uses a bimodal lower limit logic called \(\mathbf {RBK}\), extending (propositional) \(\mathbf {CL}\) with a \(\mathbf {KT}\)-operator \(\Box _e\) for representing empirical background knowledge and an S4-operator \(\Box _n\) for representing nomological background knowledge. Like \(\mathbf {LA^r_s}\), \(\mathbf {AbL^p}\) makes use of the reliability strategy. \(\mathbf {AbL^p}\) is a prioritized adaptive logic, meaning that its set of abnormalities is defined as an ordered sequence. For the full definition of the set of \(\mathbf {AbL^p}\)-abnormalities, and for more details about this logic, we refer to Lycke (2012). Where \(\langle \psi ,\varphi \rangle \) abbreviates \(\Box _n(\psi \supset \varphi ) \wedge \varphi \wedge \lnot \Box _e \varphi \wedge \lnot \psi \), the following \(\mathbf {AbL^p}\)-proof illustrates that neither \(s\) nor \(r\) is derivable from the premise set \(\{\Box _n(s\supset q),\Box _n(r\supset p),q,p\}\):
Lines 7 and 8 are marked in view of the minimal \( Dab \)-formulas derived at lines 9 and 10. The proof cannot be extended in a way that would unmark these lines. Hence, neither \(s\) nor \(r\) is finally derivable.
The reason why the derivations of \(Sa\) and \(Ra\), resp. \(s\) and \(r\), are blocked in the above proofs, has to do with the derivability of a minimal disjunction of abnormalities. This disjunction, in turn, is derivable due to the material conditional used by these logics. In the first proof, for instance, the premise \((\forall x)(Sx \supset Qx)\) allows us to derive \((\forall x)((Sx\wedge \lnot Rx) \supset Qx)\) due to the validity of (SA). Consequently, the logics \(\mathbf {LA^r}\) and \(\mathbf {LA^r_s}\) generate not only \(Sa\) as a possible explanation for \(Qa\), but also \(Sa\wedge \lnot Ra\). The latter ‘explanation’ is incompatible with the explanation \(Ra\) generated for the explanandum \(Pa\). This is what leads to the derivability of the \( Dab \)-formula at line 8 in the proof, which in turn blocks the derivability of \(Ra\). An analogous argument shows why the derivation of \(Sa\) is blocked as an explanation for \(Qa\), and why in the second proof the derivations of \(s\) and \(r\) are blocked as explanations for \(q\) and \(p\) respectively.
It is easily seen why a similar argument does not block the derivation of \(s\) and \(r\) as explanations for \(q\) and \(p\) respectively in a \(\mathbf {CR^{n/m}}\)-proof. The above argument essentially relies on the validity of (SA) for the material conditional of the logics \(\mathbf {LA^r}\), \(\mathbf {LA^r_s}\), and \(\mathbf {AbL^p}\). Since (SA) is invalid for the conditional of \(\mathbf {CR}\) and its adaptive extensions, \(s\rightarrow q\not \vdash (s\wedge \lnot r)\rightarrow q\) and \(r\rightarrow p\not \vdash (r\wedge \lnot s)\rightarrow p\). Hence, no disjunction of abnormalities whatsoever is derivable from the premise set \(\{s\rightarrow q,r\rightarrow p,p,q\}\), and \(r\) and \(s\) are finally derivable in a \(\mathbf {CR^{n/m}}\)-proof.Footnote 24
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Beirlaen, M., Aliseda, A. A conditional logic for abduction. Synthese 191, 3733–3758 (2014). https://doi.org/10.1007/s11229-014-0496-0
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DOI: https://doi.org/10.1007/s11229-014-0496-0