A theory of necessary and sufficient conditions is presented, as well as a theory of necessary and sufficient causes and effects, viewed as a particular case of the former. Ambiguities of the terms 'condition' and 'necessary condition' are explored, and a neutral meaning for 'condition' is favoured. The relation between necessary and sufficient conditions and implicative conditionals (including counterfactuals) is also discussed. Two problems of counterfactual theories of causal explanation are indicated, concerning (i) how to account for causes that are (...) not necessary and (ii) how to interpret the counterfactual from the negation of the effect to the negation of the cause. Application of the theory of necessary and sufficient conditions to Euler’s explanation of the Königsberg bridge problem is discussed. The theory of necessary and sufficient causes and effects is then applied to the flagpole shadow explanation and to two biological explanations, one involving a sufficient but not necessary cause and the other a necessary but not sufficient cause. Causes that are neither sufficient nor necessary are also recognized. Finally, a probabilistic approach to causes that are either not sufficient or not necessary is explored, including its application to examples of such cases. (shrink)

Necessity and sufficiency are the building blocks of all successful explanations. Yet despite their importance, these notions have been conceptually underdeveloped and inconsistently applied in explainable artificial intelligence, a fast-growing research area that is so far lacking in firm theoretical foundations. In this article, an expanded version of a paper originally presented at the 37th Conference on Uncertainty in Artificial Intelligence, we attempt to fill this gap. Building on work in logic, probability, and causality, we establish the central role of (...) necessity and sufficiency in XAI, unifying seemingly disparate methods in a single formal framework. We propose a novel formulation of these concepts, and demonstrate its advantages over leading alternatives. We present a sound and complete algorithm for computing explanatory factors with respect to a given context and set of agentive preferences, allowing users to identify necessary and sufficient conditions for desired outcomes at minimal cost. Experiments on real and simulated data confirm our method’s competitive performance against state of the art XAI tools on a diverse array of tasks. (shrink)

The concept of an evidential conditional If A then C that can be defined by the conjunction of A>C and ¬C>¬A, where > is a conditional of the kind introduced by Stalnaker and Lewis, has recently been studied in a series of papers by Vincenzo Crupi and Andrea Iacona. In this paper I argue that Crupi and Iacona’s central idea that contraposition captures the idea of evidential support cannot be maintained. I give examples showing that contraposition is neither necessary nor (...) sufficient for a conditional’s antecedent supporting its consequent. Crupi and Iacona’s alternative account of evidential conditionals that is based on a probabilistic measure of evidential support cannot add to the credentials of their modal account, because both the theoretical role of contraposition and the resulting logic are different in this account. (shrink)

In some recent works, Crupi and Iacona have outlined an analysis of ‘if’ based on Chrysippus’ idea that a conditional holds whenever the negation of its consequent is incompatible with its antecedent. This paper presents a sound and complete system of conditional logic that accommodates their analysis. The soundness and completeness proofs that will be provided rely on a general method elaborated by Raidl, which applies to a wide range of systems of conditional logic.

This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent. $${p\Rightarrow q}$$ p ⇒ q is thus defined as $${\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}$$ ¬ ◊ (...) ( p ∧ ¬ q ) ∧ ◊ p ∧ ◊ ¬ q. We explore the logical properties of this conditional in a reflexive normal Kripke semantics, provide an axiomatic system and prove it to be sound and complete for our semantics. The implicative conditional validates transitivity and contraposition, which we take to be integral parts of reasoning and communication. But it only validates restricted versions of strengthening the antecedent, right weakening, simplification, and rational monotonicity. Apparent counterexamples to some of these properties are explained as due to contextual factors. Finally, the implicative conditional avoids the paradoxes of material and strict implication, and validates some connexive principles such as Aristotle’s theses and weak Boethius’ thesis, as well as some highly entrenched principles of conditionals, such as conjunction of consequents, disjunction of antecedents, modus ponens, cautious monotonicity and cut. (shrink)

This paper develops a probabilistic analysis of conditionals which hinges on a quantitative measure of evidential support. In order to spell out the interpreta- tion of ‘if’ suggested, we will compare it with two more familiar interpretations, the suppositional interpretation and the strict interpretation, within a formal framework which rests on fairly uncontroversial assumptions. As it will emerge, each of the three interpretations considered exhibits specific logical features that deserve separate consideration.

This paper outlines an account of conditionals, the evidential account, which rests on the idea that a conditional is true just in case its antecedent supports its consequent. As we will show, the evidential account exhibits some distinctive logical features that deserve careful consideration. On the one hand, it departs from the material reading of ‘if then’ exactly in the way we would like it to depart from that reading. On the other, it significantly differs from the non-material accounts which (...) hinge on the Ramsey Test, advocated by Adams, Stalnaker, Lewis, and others. (shrink)

Once upon a time, some thought that indicative conditionals could be effectively analyzed as material conditionals. Later on, an alternative theoretical construct has prevailed and received wide acceptance, namely, the conditional probability of the consequent given the antecedent. Partly following critical remarks recently ap- peared in the literature, we suggest that evidential support—rather than conditional probability alone—is key to understand indicative conditionals. There have been motivated concerns that a theory of evidential conditionals (unlike their more tra- ditional counterparts) cannot generate (...) a sufficiently interesting logical system. Here, we will describe results dispelling these worries. Happily, and perhaps surprisingly, appropriate technical variations of Ernst Adams’s classical approach allow for the construction of a logic of evidential conditionals with distinctive fea- tures, which is also well-behaved and reasonably strong. (shrink)