A conditional logic for abduction

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.

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 $A$:

$$\begin{aligned} \vDash ^{M}_{w} A\rightarrow B \text{ iff } |B|^{M}\in f(w,|A|^{M}) \end{aligned}$$

The classical connectives are defined as usual:

$$\begin{aligned} \mathrm If ~p\in \fancyscript{P}~\mathrm then ~&\vDash ^{M}_{w} p ~\mathrm iff ~p\in v_p(w)\\&\vDash ^{M}_{w}\top \\&\vDash ^{M}_{w}\lnot A ~\mathrm iff ~~\mathrm not ~\vDash ^{M}_{w} A\\&\vDash ^{M}_{w} A\vee B ~\mathrm iff ~\vDash ^{M}_{w} A ~\mathrm or ~\vDash ^{M}_{w} B \end{aligned}$$

This characterization of $\rightarrow$ ensures the validity of (RCEA) and (RCEC). For (CM) and (CC), we need two further conditions:

$$\begin{aligned} \mathrm{If}~ Y, Y^\prime \, \in \,\, f(w, X), \hbox {then}\, Y \cap Y^\prime \in f (w, X) \end{aligned}$$

(CM')

$$\begin{aligned} \mathrm{If}~Y \cap Y^\prime \, \in \,\, f(w, X), \hbox {then}\, Y, Y^\prime \in f (w, X) \end{aligned}$$

(CC')

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 ‘$\rightarrow$’.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 $\mathbf {CR}$,

• (ii) A set of abnormalities, the set $\varOmega$ as defined in Sect. 3,

• (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:

$$\begin{aligned} \varOmega '&= \{(\forall \alpha )(\varPsi (\alpha )\supset \varPhi (\alpha )) \wedge \varPhi (\beta )\wedge \lnot \varPsi (\beta ) \mid \text{ no } \text{ predicate } \text{ that } \text{ occurs }\\&\qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \quad \text{ in } \varPhi (\beta ) \text{ occurs } \text{ in } \varPsi (\beta )\} \end{aligned}$$

$\mathbf {LA^r_s}$ 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.²⁴