The aim of this paper is to outline a rst-order model for ampliative reasoning that fruitfully combines the inference patterns of inductive generalization and factual abduction. The pattern of inductive generalization is the archetype pattern of inductive inference by which we arrive at a universally quanti ed statement (All P s are Q) given one or more instances (Some P s are Q). In factual abduction, we reason from a universally quanti ed statement (All P s are Q) and an instance of its consequent (object a is Q) to an instance of its antecedent (object a is P ). It is shown how these patterns can be combined in such a way that inductively inferred generalizations can be used as premises in abductive inferences, and that conclusions of abductive inferences in turn can be used to inductively infer new generalizations. This process is formally explicated within the adaptive logics framework in terms of a preferential model semantics.
This is an exploratory investigation into combinations of ampliative reasoning patterns. Ampliative reasoning occurs whenever we draw inferences the conclusions of which cannot be deduced from the available premises by means of one's preferred standard of deduction. Examples of ampliative reasoning patterns include inductive generalization, abduction or inference to the best explanation, causal discovery, and reasoning by analogy. The study of these patterns is of interest to philosophers investigating the foundations of defeasible reasoning, to logicians investigating the formalization of defeasible reasoning, to computer scientists investigating the automation of defeasible reasoning, and to psychologists investigating defeasible reasoning in the wild.
The focus of this paper is on the formalization of two speci c patterns of ampliative reasoning and their combination. The rst is that of inductive generalization, the archetype pattern of inductive inference by which we reason to a universally quanti ed statement (\All P s are Q") given one or more instances of it. The second pattern is that of factual abduction, by which we reason from a universally quanti ed statement (\All P s are Q") and an instance of its consequent (\a is Q") to an instance of its antecedent (\a is P "). The inference patterns studied here are sub-patterns of the larger classes of inductive inferences and abductive inferences. For a comprehensive taxonomy of patterns of inductive inference, see [17]. For a comprehensive taxonomy of patterns of abductive inference, see [19]. The pattern of factual abduction is also known as simple abduction [23] or plain abduction [1].
The technical implementation and combination of inductive generalization and factual abduction is realized within the adaptive logics framework for modelling patterns of defeasible reasoning. There are two main reasons for choosing this framework. The rst is that both inductive generalization and factual abduction are well-studied within this framework { see [8, 6, 7, 9, 12, 14, 16]. The second is that di erent means are available for combining adaptive logics { see [24, 25, 21].
Section 3 provides a short introduction to the adaptive logics framework, tailored to the aim of this paper. In Section 4 the logic for inductive generalization LIr from [4, 6, 8] is presented and illustrated. In Section 5 the logic for factual abduction FAr is introduced. The latter system is a close cousin of an adaptive logic for factual abduction de ned within the framework of [7] (see footnote 9 below). The logics presented in Sections 4 and 5 are then sequentially combined (Section 6), resulting in the system SIAr.
The modest contribution of this paper is that it provides a full formal explication of how inductive generalization and factual abduction can be combined within a single system, and that this combination is fruitful in the following sense: inductively obtained conclusions can be used as premises in abductive inferences, and vice versa. From this, no conclusions should be drawn yet regarding the normative or descriptive adequacy of this system: more work remains to be done. For instance, an adequate formalization of these inference patterns and their combination requires a detailed study of their alternative logical characterizations, and a richer formal language. Some of these alternatives and enrichments are discussed in Section 7, alongside a number of design choices which are best motivated after de ning SIAr. 2
Let L be a rst-order language built using a set P of unary predicates, a set C of individual constants, a set V of individual variables, and the logical symbols >; ?; :; _; ^; ; ; 9; 8. In what follows, CL refers to rst-order classical logic without identity, and restricted to L (no n-ary predicates for n > 1, no function symbols).
Upper case letters P; Q; R, etc., lower case letters a; b; c, etc., respectively lower case letters x; y; z, denote members of P; C, respectively V . For all 2 C [ V, L = f ; : j 2 Pg, and F is the set of truth-functions of formulas in L . For instance, P a 2 La and :P x _ (Qx Rx) 2 F x. Where 2 C [ V, A( ); B( ), etc. denote members of F , unless further speci ed.
Where M is a CL-model, A 2 L and L, M A means that M veri es A; M is a model of i M A for all A 2 . Relative to a logic L, ML( ) denotes the set of L-models of , and L A means that A is veri ed by all M 2 ML( ). CnL( ) is the set of L-consequences of . 3
Adaptive logics are tools for explicating defeasible reasoning patterns. They were originally developed by Batens, who also de ned a standard format for adaptive logics [3, 5, 7]. Systems de ned within this format are equipped with a dynamic proof theory and a selection semantics in the vein of Shoham's preferred models [20], KLM's preferential models [13], or Makinson's default valuations [15, Ch. 3].1 For conciseness of presentation, the adaptive logics presented here are de ned only from a semantic point of view.
Adaptive logics strengthen a core logic called the lower limit logic. The adaptive semantics is a mechanism for selecting a preferred subset among the models of the lower limit logic relative to a premise set. The selected set contains models that are minimal with respect to a set of abnormalities : a set of formulas characterized by some logical form. The exact way in which an adaptive logic minimizes abnormalities veri ed by its lower limit models varies with the adaptive strategy used. Depending on the strategy used in the minimization process, di erent sets of lower limit models may be selected relative to a premise set, giving rise to possibly di erent sets of logical consequences. An adaptive logic de ned within the standard format is fully characterized in terms of three elements: a lower limit logic, a set of abnormalities, and an adaptive strategy.
Below two adaptive logics will be presented: the logic for inductive generalization LIr, and the logic of factual abduction FAr. These logics have CL as their lower limit logic. They di er with respect to their respective sets of abnormalities. The superscript r is the rst letter of the adaptive strategy used by these logics: the reliability strategy.
Adaptive logics provide a exible framework for studying di erent types of defeasible reasoning patterns and their combinations. This makes them very suitable for the present exploration of combining inductive generalization and factual abduction. The format for combination used here is that of sequential superposition [21, Ch. 3], [22].
Given a premise set L, the logics LIr and FAr are sequentially combined
in the following way:
: : : CnFAr (CnLIr (CnFAr (CnLIr ( )))) : : :
(
In a rst step, LIr is applied to check which generalizations can be inferred from the premise set . Next, FAr is applied to infer new predictions via factual abduction. These new predictions can in turn be used to check for new generalizations by means of LIr, and so on. 4
The adaptive logic LIr strengthens its lower limit logic, CL, by interpreting the world `as uniformly as possible'. It does so by taking as its set of abnormalities a set of falsi ed universally quanti ed statements, so that its least abnormal models are those in which these universally quanti ed statements hold true. The set i of LIr-abnormalities is de ned as follows:2 i = f:8 (A1( ) _ : : : _ An( )) j 2 V; A1( ); : : : ; An( ) 2 L g In the remainder the term generalization refers to formulas of the form 8 (A1( )_ : : : _ An( )), so that i is the set of negated generalizations.
To complete the characterization of LIr, a mechanism is needed for selecting a `preferred' subset of the CL-models of a given premise set relative to the set i. This mechanism is provided by the reliability strategy, which selects a set Mir( ) of i-reliable models of L. The characterization of this set requires some more terminology. `Dab' is an acronym for `disjunction of abnormalities'. Where i, Dabi( ) = W .3 Dabi( ) is a Dabi-consequence of i
CL Dabi( ), and Dabi( ) is a minimal Dabi-consequence of i Dabi( )
is a Dabi-consequence of and there is no 0 such that Dabi( 0) is a
Dabi-consequence of . Where Dabi(
r Mi ( ) = fM 2 MCL( ) j Abi(M )
Ui( )g
LIr A i
M
r
A for all M 2 Mi ( ).
As an illustration of the workings of LIr, consider the premise set 1 = fP a ^ Qa ^ :Ra ^ :Sa; Qb ^ Rb; P b :Sb; :P c ^ :Qc ^ :Rc ^ Sc; :P d ^ Qd ^ :Rd ^ Sd; :P e ^ :Qe ^ :Re ^ :Se; P f ^ Qf ^ :Rf ^ Sf; P g ^ Qg ^ Rg; :P h ^ :Qh ^ Rh; :P i ^ Qi ^ Ri; :P j ^ Qj ^ :Rj ^ :Sjg.
For future reference, it is convenient to list all i-abnormalities that can be formed using only the four predicates occurring in 1 (see Table 1).
The set of Dabi-consequences of 1 contains, amongst others, all disjunctions between formulas listed in Table 1 that are CL-derivable from 1, including `single-disjunct' disjunctions. The minimal Dabi-consequences of 1 are all minimal such disjunctions. They include
2In [26, Sec. 4.2.2] it is shown that the same logic is obtained if i is de ned as the set of formulas of the form :8 A( ), where 2 V and A( ) 2 F .
3If is a singleton fAg, Dabi( ) = A. 1. :8x(P x) 28. :8x(:Qx _ :Sx) 55. :8x(:P x _ :Rx _ Sx) 2. :8x(:P x) 29. :8x(Rx _ Sx) 56. :8x(:P x _ :Rx _ :Sx) 3. :8x(Qx) 30. :8x(Rx _ :Sx) 57. :8x(Qx _ Rx _ Sx) 4. :8x(:Qx) 31. :8x(:Rx _ Sx) 58. :8x(Qx _ Rx _ :Sx) 5. :8x(Rx) 32. :8x(:Rx _ :Sx) 59. :8x(Qx _ :Rx _ Sx) 6. :8x(:Rx) 33. :8x(P x _ Qx _ Rx) 60. :8x(Qx _ :Rx _ :Sx) 7. :8x(Sx) 34. :8x(P x _ Qx _ :Rx) 61. :8x(:Qx _ Rx _ Sx) 8. :8x(:Sx) 35. :8x(P x _ :Qx _ Rx) 62. :8x(:Qx _ Rx _ :Sx) 9. :8x(P x _ Qx) 36. :8x(P x _ :Qx _ :Rx) 63. :8x(:Qx _ :Rx _ Sx) 10. :8x(P x _ :Qx) 37. :8x(:P x _ Qx _ Rx) 64. :8x(:Qx _ :Rx _ :Sx) 11. :8x(:P x _ Qx) 38. :8x(:P x _ Qx _ :Rx) 65. :8x(P x _ Qx _ Rx _ Sx) 12. :8x(:P x _ :Qx) 39. :8x(:P x _ :Qx _ Rx) 66. :8x(P x _ Qx _ Rx _ :Sx) 13. :8x(P x _ Rx) 40. :8x(:P x_:Qx_:Rx) 67. :8x(P x _ Qx _ :Rx _ Sx) 14. :8x(P x _ :Rx) 41. :8x(P x _ Qx _ Sx) 68. :8x(P x _ Qx _ :Rx _ :Sx) 15. :8x(:P x _ Rx) 42. :8x(P x _ Qx _ :Sx) 69. :8x(P x _ :Qx _ Rx _ Sx) 16. :8x(:P x _ :Rx) 43. :8x(P x _ :Qx _ Sx) 70. :8x(P x _ :Qx _ Rx _ :Sx) 17. :8x(P x _ Sx) 44. :8x(P x _ :Qx _ :Sx) 71. :8x(P x _ :Qx _ :Rx _ Sx) 18. :8x(P x _ :Sx) 45. :8x(:P x _ Qx _ Sx) 72. :8x(P x _ :Qx _ :Rx _ :Sx) 19. :8x(:P x _ Sx) 46. :8x(:P x _ Qx _ :Sx) 73. :8x(:P x _ Qx _ Rx _ Sx) 20. :8x(:P x _ :Sx) 47. :8x(:P x _ :Qx _ Sx) 74. :8x(:P x _ Qx _ Rx _ :Sx) 21. :8x(Qx _ Rx) 48. :8x(:P x_:Qx_:Sx) 75. :8x(:P x _ Qx _ :Rx _ Sx) 22. :8x(Qx _ :Rx) 49. :8x(P x _ Rx _ Sx) 76. :8x(:P x _ Qx _ :Rx _ :Sx) 23. :8x(:Qx _ Rx) 50. :8x(P x _ Rx _ :Sx) 77. :8x(:P x _ :Qx _ Rx _ Sx) 24. :8x(:Qx _ :Rx) 51. :8x(P x _ :Rx _ Sx) 78. :8x(:P x _ :Qx _ Rx _ :Sx) 25. :8x(Qx _ Sx) 52. :8x(P x _ :Rx _ :Sx) 79. :8x(:P x _ :Qx _ :Rx _ Sx) 26. :8x(Qx _ :Sx) 53. :8x(:P x _ Rx _ Sx) 80. :8x(:P x_:Qx_:Rx_:Sx) 27. :8x(:Qx _ Sx) 54. :8x(:P x _ Rx _ :Sx) { the abnormalities 1{10, 12{30, 33{36, 39{44, 47{50, 53, 54, 57, 58, 61, 62, 65, 66, 69, 70, 77, and 78 from Table 1, and { the disjunctions listed in Table 2.4
Importantly, the abnormalities 11, 37, 38, 45, 46, 73{76 do not occur as disjuncts
in any minimal Dabi-consequence of 1. Indeed, for any Dabi-consequence of 1
containing one of these abnormalities as one of its disjuncts, there is a strictly
shorter disjunction which is a minimal Dabi-consequence of 1 and which does
not contain the abnormality in question as one of its disjuncts. Thus the set
Ui(
The logic LIr, like all adaptive logics de ned within the standard format, inherits a number of meta-theoretical properties such as { CnCL(CnLIr ( )) = CnLIr ( ) (CL-closure) { CnLIr (CnLIr ( )) = CnLIr ( ) ( xed point) { If M 2 MCL( ) n Mir( ), then there is an M 0 2
Abi(M 0) Abi(M ) (smoothness) MLIr ( ) such that For the generic proofs of these properties for adaptive logics in standard format, see [5, Sec. 6-8]. For a slower-paced introduction to LIr, and for more illustrations of its workings, see [4, 8]. 5
The inference pattern of factual abduction is a defeasible version of the backward modus ponens (BMP) rule. Where 2 V, 2 C, A( ); B( ) 2 F , A( ); B( ) 2 4The tedious exercise of verifying that 1 has no minimal Dabi-consequences of three or more disjuncts is safely left to the interested reader.
F : In order to prevent that the adaptive logic FAr over- or undergenerates abductive consequences, a number of further technical requirements must be imposed on inferences of the form (BMP), as the following examples illustrate.5 Example 1. Let 2 = f8x(Sx Qx); 8x(Rx P x); P a; Qag. Sa is derivable by (BMP) applied to 8x(Sx Qx) and Qa. But 8x(Rx P x) CL 8x((Rx ^ :Sx) P x), so by the same token (BMP) can be applied to 8x((Rx ^ :Sx) P x) and P a so as to infer Ra ^ :Sa, which contradicts the earlier inference of Sa.
This example motivates a restriction according to which (BMP) is not applicable to universally quanti ed conditionals the antecedents of which have been strengthened, such as 8x((Rx ^ :Sx) P x).
Example 2. Let 3 = f8x(P x Qx); Rag. One would not expect P a to be derivable via (BMP). But 8x(P x Qx) CL 8x(P x (Qx _ Rx)) and Ra CL Qa _ Ra. So P a can be inferred by applying (BMP) to 8x(P x (Qx _ Rx)) and Qa _ Ra. The resulting logic overgenerates.
This example motivates a restriction according to which (BMP) is not applicable to universally quanti ed conditionals the consequents of which have been weakened, such as 8x(P x (Qx _ Rx)).
A single technical requirement su ces to ensure that the problems in Examples 1 and 2 are avoided. Note that the universally quanti ed conditional in arguments of the form (BMP) can be expressed equivalently as a (conjunction of) universally quanti ed disjunction(s). For instance, 8x(P x Qx), respectively 8x((P x _ Rx) Qx, 8x(P x (Qx ^ Rx)) are equivalent to 8x(:P x _ Qx), respectively 8x(:P x _ Qx) ^ 8x(:Rx _ Qx), 8x(:P x _ Qx) ^ 8x(:P x _ Rx). If we do the same in Examples 1 and 2, it is immediate that in the undesirable applications of factual abduction the universally quanti ed premise results from weakening a logically stronger generalization. In Example 1, the generalization 8x(:Rx _ P x) was weakened to 8x(:Rx _ Sx _ P x). In Example 2, the generalization 8x(:P x _ Qx) was weakened to 8x(:P x _ Qx _ Rx). These weakened generalizations or their conditional equivalents cause trouble when used as premises in abductive inferences. This motivates a restriction of applications of factual abduction to generalizations from which no disjuncts can be removed. Such generalizations will be called starred generalizations. They make use of a starred quanti er `8', expressing that the generalization in question cannot be shortened. Where 2 V and A1; : : : ; An; B1; : : : ; Bk 2 L : 8 (A1 _ : : : _ An) = 8 (A1 _ : : : _ An) ^ : _f8 (B1 _ : : : _ Bk) j ; =6 fB1; : : : ; Bkg fA1; : : : ; Angg 5Both examples presuppose CL in the background. The rst example is adopted from the technical appendix in [9]. The second example is by Frederik Van De Putte (personal communication).
The logic FAr allows for the defeasible application of the factual abduction pattern to starred generalizations. More precisely, it implements a defeasible version of the `backward disjunctive syllogism' rule obtained by replacing 8 (A( ) B( )) with 8 (:A( ) _ B( )) in (BMP).
The lower limit logic of FAr is CL. Its set of abnormalities is the set Where 2 V; 2 C; A1( ); : : : ; Ai( ); B1( ); : : : ; Bj ( ) 2 L ; i 1; j 1: Given a premise set , a CL-model M of , and a set a, the sets Mra( ), Daba( ), the set of (minimal) Daba-consequences of , and the set Ua( ) of a-unreliable formulas of are de ned exactly like their inductive counterparts: just replace subscripts `i' with `a' in their respective counterpart de nitions in Section 4.
FAr A i
M
r
A for all M 2 Ma( ).
Given a premise set , FAr selects the CL-models of which verify no
aabnormalities except for those in Ua( ), just like LIr would select the CL-models
of which verify no i-abnormalities except for those in Ui( ). By way of
illustration, let 4 = CnLIr (
4 has three minimal Daba-consequences:
(
Let 5 = CnFAr (CnLIr (
So far, new information was obtained at each `round' of application of the
logics LIr and FAr: 8x(:P x _ Qx) 2 CnLIr (
8x(:P x _ :Rx _ :Sx) 62 CnLIr (CnFAr (CnLIr (
The iterative process of applying inductive generalization and factual abduction can be repeated ad in nitum. The consequence operation CnSIAr is de ned as follows:
CnSIAr ( ) = : : : CnFAr (CnLIr (CnFAr (CnLIr ( )))) : : :
(
6This i-abnormality is number 32 in Table 1. It is a member of
Ui(CnFAr (CnLIr (
1: Mj ( ) = M0( ) = MCL( ) 8>fM 2 Mj 1( ) j Abi(M ) Ui(fA j M 0 > >< for all M 0 2 Mj 1( )g)g if j is odd, A >>fM 2 Mj 1( ) j Aba(M ) Ua(fA j M 0 :> for all M 0 2 Mj 1( )g)g if j is even. A De nition 4. Where j 2 N,
M
A for all M 2 Mj ( ).
It was shown generically (for adaptive logics using the reliability strategy) in
[21, Sec. 3.2.1] that, at each step in the construction, the resulting logics are
semantically adequate with respect to the sequence in (
M1( ) = lijm!i1nf Mj ( ) = \ Mj ( ) j2N
SIAr A i
M
A for all M 2 M1( ).
In [24, Sec. 3.3.2] the generic semantic adequacy result from [21] is extended to
the in nite case. Applied to the present setting, (
For languages with a nite signature the logic SIAr is decidable. It remains an open question whether this is also the case for languages of in nite signature. Another open issue is that of determining the computational complexity of SIAr. In [18] it was shown that for adaptive logics de ned within the standard format { such as LIr and FAr { the complexity upper bound in the arithmetical hierarchy is 30. There are currently no published results on the computational complexity of sequentially combined adaptive logics such as SIAr. 7
Here is a di erent way of writing the outcomes obtained for 1 in Sections 4-6:
(
1 SIAr3 8x(:P x _ :Rx _ :Sx) The example shows how information obtained via factual abduction can in turn serve to inductively infer generalizations not previously derivable from the premise set. Stretching things a bit, this example logically explicates and conrms the view { revived by Douglas in [10] { that part of what makes (abduced) explanations useful is their help in generating new predictions (in this case, via inductive generalization). The `stretch' here concerns the use of the term `explanation' for referring to formulas inferred via factual abduction. Arguably, conclusions drawn via factual abduction classify at best as mere potential explanations, and a richer formalism is needed to adequately represent their epistemic status as opposed to e.g. observations in the premise set, cfr. infra. In this respect, the logic SIAr oversimpli es matters.
Besides factual abduction, the logic SIAr goes some way towards explicating another `pattern' of abductive inference, namely the pattern of law-abduction, which has the following logical form [19]: Following an illustration given in [19], let P; Q; R denote respectively `contains sugar', `tastes sweet', and `is a pineapple'. Our background knowledge includes 8x(P x Qx). Some things contain sugar while others don't, and some things taste sweet while others don't, so 8x(:P x _ Qx). The aim is to explain 8x(Rx Qx) { which we obtained by inductive generalization from a number of instances Ra ^ Qa, Rb ^ Qb, etc. Via factual abduction applied to 8x(:P x _ Qx) and Qa; Qb; : : :, the formulas P a; P b; : : : can be inferred. And by inductive generalization applied to Ra ^ P a; Rb ^ P b, etc., we obtain 8x(Rx P x). When asked why pineapples taste sweet, we can now answer by telling that pineapples contain sugar.7
An important design choice in the construction of SIAr is the preference for a sequential combination of the patterns of inductive generalization and factual abduction. As is clear from the characterization of SIAr-consequence in De nitions 3 and 4, SIAr-models are selected sequentially or stepwise relative to either Ui or Ua. At each step in the sequence we select either exclusively with respect to i-unreliable formulas, or we select exclusively with respect to a-unreliable formulas.
A di erent, `parallel' rather than sequential, combination strategy would be
to look at both i-unreliable formulas and a-unreliable formulas in one single step.
To this end, we could de ne a unique set of abnormalities ia = i [ a. The
sets Mia, Dabia( ), Uia, etc. are then rede ned accordingly in terms of ia.
In the resulting logic, (minimal) Dabia-consequences may consist of disjunctions
between one or more members of i and/or one or more members of a. For
7Flach & Kakas thought of law-abduction as a hybrid inference pattern combining
inductive generalization and factual abduction [11, pp. 21-22]. This view was criticized
by Schurz on the grounds that this decomposition of law-abduction is \somewhat
arti cial. Law-abductions are usually performed in one single conjectural step" [19,
p. 212]. For an adaptive logic explicating the latter view, see [12].
instance, the disjunction
:8x(:P x _ Qx) _ (8x(:P x _ Qx) ^ Qd ^ :P d)
(
The disjunction in (
In SIAr the logic LIr is applied rst in the sequence in (
There is a `chicken or egg' reason in favor of the `inductive generalization rst' approach. Inductive generalization has priority over factual abduction in the sense that we need to generalize before we can even start abducing (every application of factual abduction requires a generalization among its premises). A `factual abduction rst' approach would require some explanation as to how generalizations are attained prior to abduction, if not by inductive generalization. No such explanation is required in an `inductive generalization rst' approach of the kind adopted here.
The logic SIAr is instructive in explicating what a combination of the inference patterns of inductive generalization and factual abduction could (and could not) look like. It was used to show how these patterns of ampliative reasoning can be fruitfully combined to infer new predictions and generalizations, and how they can shed light on a di erent pattern, law-abduction. Still, it is too early to make bold claims regarding the adequacy of SIAr in capturing these patterns, for at least two reasons. First, there are many alternative ways to model these ampliative inferences. And second, a fully adequate model requires additional expressive resources.
In [6] Batens considers a number of alternative ways of modeling inductive generalization via an adaptive logic. Various roads for variation are open here. A rst is to change the adaptive strategy.8 A second is to vary the set of abnormalities. Instead of taking negated generalizations such as :8x(P x _ Qx) as members of i, one could take, for example, conjunctions of instances and counterinstances of a generalization, such as 9x(P x _ Qx) ^ 9x:(P x _ Qx). As shown in [6], this gives rise to a slightly di erent logic. A third, unexplored, road for variation is to change the lower limit logic from CL to some non-classical logic. More complex variations still can be obtained by coupling these roads, or even by moving to a combined adaptive logic for inductive generalization { see [6] for some examples.
The inference pattern of factual abduction too can be modeled in a variety of ways. The technical issues discussed in in Examples 1 and 2 can be avoided by means other than the restriction of applications of (BMP) to starred generalizations.9 More generally, a richer framework with more expressive power is required for suitably representing factual abduction. Inferred explanations do not generally have the same epistemic status as observations in our premise set, and generalizations used as premises in an application of factual abduction often have a law-like status which separates them from mere regularities in the explanatory framework. These distinctions are too subtle to make in the rstorder language used in this paper. One of the main open research questions for the present investigation is how we can enrich this formal language with additional expressive resources while preserving the fruitful sequential application of inductive generalization and factual abduction.
Acknowledgments Research for this article was sponsored by a Humboldt fellowship for postdoctoral researchers, and by a Sofja Kovalevskaja award, both granted by the Alexander von Humboldt Foundation, funded by the German Ministry for Education and Research. I am indebted to Frederik Van De Putte, Diderik Batens, and two anonymous referees for valuable comments on an earlier draft of this article.
8Two strategies are currently de ned within the standard format for adaptive logics:
reliability and minimal abnormality. Using the minimal abnormality strategy for
sequential combinations of adaptive logics has the disadvantage that semantic adequacy
results as in (
9As mentioned in Section 1, FAr is closely related to the system AALr de ned in [7]. In the latter logic, the technical issues discussed in Examples 1 and 2 are likewise avoided by admitting only a restricted set of generalizations as candidate premises for abductive inference: FAr admits only `starred' generalizations, while AALr admits only universally quanti ed conditionals the antecedent [consequent] of which has a restricted conjunctive [disjunctive] normal form. In FAr conclusions of abductive inferences are members of L for some 2 C. In AALr conclusions of abductive inferences are formulas of the form ( ) B 0( ) where ( ); 0( ) 2 L for some 2 C.
( ) B 0( ) denotes that ( ) is a `potential explanation' for 0( ).
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Applications in Deontic Logic, Abduction and Belief Revision. PhD thesis, Ghent
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comparative study. Logic Journal of the IGPL, 21:127{159, 2013.
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