In the past decades a large number of search algorithms for causal discovery has been developed. Perhaps the most influential theoretical framework for developing such search algorithms is the causal Bayes’ nets framework which has given rise to, among many others, the IC- and IC∗ -algorithms by Pearl (2000, 2009) and the SGS-, the PC- and the IG-algorithms by Spirtes et al. (2000). As is well-known, these algorithms are based on a number of axioms, to wit the Causal Markov Condition and the Faithfulness Condition (which is also known as Stability).^{1}

These conditions or axioms are not absolute. In the words of Spirtes et al. (2000, 9): “The Markov Condition is not given by God; it can fail for various reasons […]. The reliability of inferences based upon the Condition is only guaranteed if substantive assumptions obtain. But the Condition is weak enough that there is often reason to think it applies.”^{2} One domain in which the Causal Markov Condition fails, comprises deterministic causal structures (Baumgartner, 2009, 72). Such deterministic structures may also violate Faithfulness (Spirtes et al., 2000, 81ff).

In our paper we present a logic, called ELIM^{r}, for the discovery of deterministic causal regularities starting from empirical data. It is an adaptive logic (we shall shortly explain what this means) that is inspired by Mackie’s theory of causes as inus-conditions.

Mackie’s theory of causes as inus-conditions is not probabilistic. It focusses on deterministic causal relations at the generic or type level. From Mill, Mackie borrows the idea that causation is seldom, if ever, an invariable sequence or regularity between a single antecedent (e.g. a short circuit) and a single consequent (e.g. a fire). Instead, it is often the case that the effect P occurs when some conjunction of factors (e.g. ABC; a short circuit, the presence of oxygen, the presence of inflammable materials) occurs, but not when any of these conjuncts fails to occur. Moreover, alternative conjunctions of factors (e.g. the conjunctions DGH and JKL) may also be followed invariably by P. A, in this example, is an insufficient but non-redundant part of an unnecessary but sufficient condition for P. In short, using the first letters of the italicized words, it is an inus-condition for P.

Mackie stresses the fact that our knowledge of complex causal regularities is seldom, if ever, complete. “What we know are certain elliptical or gappy universal propositions.” (Mackie, 1974, 66) Moreover, he writes that “the elliptical character of causal regularities as known is closely connected with our characteristic methods of discovering and establishing them: it is precisely for such gappy statements that we can obtain fairly direct evidence from quite modest ranges of observation.” (Mackie, 1974, 68)

The logic to be presented in this paper will serve as an explication for Mackie’s views on these ‘characteristic methods’. As we will show, the gappy or elliptical character of our causal universal propositions gives the discovery of such propositions an interesting dynamics for which adaptive logics are wellsuited.

Adaptive logics are tools for formalizing defeasible reasoning. They have been used to model a wide variety of reasoning patterns including explanatory reasoning, inductive generalization, and reasoning in the presence of inconsistencies. Moreover, Pearl’s IC-algorithm has served as the basis for ALIC, an adaptive logic for causal discovery (Leuridan, 2009). These patterns are nonmonotonic: conclusions drawn from a set of premises may be withdrawn in the light of additional information (new premises). Adaptive logics are particularly suitable for capturing the non-monotonicity of defeasible reasoning. For a general introduction to adaptive logics, see Batens (2001, 2007).

The logic ELIM^{r} consists of two preliminary logics: ELI^{r} and Mr. ELI^{r} allows to derive logical equivalences — of a particular, Mackie-style type — from empirical data. Mr then serves to minimize these equivalences; intuitively, Mr serves to throw out redundant factors.

During our presentation, we will not spend too much time on the technicalities of our approach. Instead, we will focus on:

- giving a brief overview of Mackie’s theory of causes as inus-conditions with a special emphasis on the gappy or elliptical nature of our causal knowledge (and corresponding discovery methods);
- giving a general overview of the preliminary logics ELI
^{r} and M^{r} and the resulting logic for the discovery of causal regularities, ELIM^{r} ;
- discussing the relation between our logic ELIM
^{r} and one recent and very interesting such discovery procedure for deterministic causation that also starts from Mackie’s theory, viz. the Boolean algorithm for coincidence analysis (CNA) proposed by Baumgartner (2009); and
- discussing the relations between our logic ELIM
^{r} and some recent work on qualitative explications of inductive generalization and abductive or explanatory inference, viz. Batens (2011); Beirlaen & Aliseda (2014); Meheus & Batens (2006).

**References**

- Batens, D. (2001). A general characterization of adaptive logics. Logique & Analyse, 173–175:45–68.
- Batens, D. (2007). A universal logic approach to adaptive logics. Logica Universalis, 1:221–242.
- Batens, D. (2011). Logics for qualitative inductive generalization. Studia Logica, 97:61–80.
- Baumgartner, M. (2009). Uncovering deterministic causal structures: a Boolean approach. Synthese, 170(1):71–96.
- Beirlaen, M. and Aliseda, A. (2014). A conditional logic for abduction. Synthese, 191(15):3733–3758.
- Leuridan, B. (2009). Causal discovery and the problem of ignorance. An adaptive logic approach. Journal of Applied Logic, 7(2):188–205.
- Mackie, J. L. (1974). The Cement of the Universe: A Study of Causation. Clarendon Press, Oxford.
- Meheus, J. and Batens, D. (2006). A formal logic for abductive reasoning. Logic Journal of the IGPL, 14:221–236.
- Pearl, J. (2000). Causality. Models, Reasoning, and Inference. Cambridge University Press, Cambridge.
- Pearl, J. (2009). Causality. Models, Reasoning, and Inference. Cambridge University Press, Cambridge, 2nd edition.
- Spirtes, P., Glymour, C., and Scheines, R. (2000). Causation, Prediction, and Search. MIT Press, Cambridge, Massachusetts.

**Footnotes**

1 For the Causal Markov Condition, see Spirtes et al. (2000, 54) and Pearl (2000, 30); for the Faithfulness Condition or Stability, see Spirtes et al. (2000, 56) and Pearl (2000, 48) respectively.

2 In this quote they discuss the Markov Condition, but their claim applies to the Causal Markov Condition as well.