 According to Anderson and Belnap “the inference from \(\neg A\) and \(A\vee B\) to \(B\) is an error: […] a fallacy of relevance” (Anderson and Belnap 1962: 19). Formally, the inference is rejected because according to Anderson and Belnap’s notion of tautological entailment (see their system E) \(\neg A \wedge (A\vee B)\) entails \(B\) if and only if \((\neg A \wedge A) \vee (\neg A \wedge B)\) entails \(B\), only if \((\neg A \wedge A)\) entails \(B\), which fails since the premise \((\neg A \wedge A)\) is not relevant to the conclusion \(B\) (no shared propositional variables). Anderson and Belnap further argued that we can still have a version of disjunctive syllogism involving an intensional notion of disjunction \(\vee_i\), for which addition wouldn’t hold. Intensional disjunction is defined as follows: \((A\vee_i B):= (\neg A \to_i B) \), where \(\to_i\) is an intensional implicational connective, i.e., one satisfying the formal criterion of relevance introduced above. The English or would then be ambiguous between intensional and extensional (truth-functional) disjunction, and only for the latter would addition hold. Intuitively, the intensional variety of or would be one requiring “relevance” between the disjuncts. On the intensional reading, “\(A\) or \(B\)” would entail that \(A\) and \(B\) are so related that we are entitled to say “If \(A\) had not been true, \(B\) would have been true” or “If \(B\) had not been true, \(A\) would have been true” or the like. A disjunction like Either Napoleon was born in Corsica or else the number of the beast is perfect clearly fails to have this property and therefore is of the truth-functional kind. Whereas That is either Drosophilia Melanogaster or D. virilis, I’m not sure which appears to entail that if it is not the one then it is the other, and thus is of the intensional kind. The distinction between intensional and extensional disjunction has been criticized by various scholars, including Burgess (1981, 1983), but has been defended by Read (1981, 1983) who proposed the following example as a clear case of truth-functional disjunction, for which addition is valid: You qualify for a grant if either you are over 65 or you can earn less than 2000 pounds a year (Read 1981: 68). Now if English or had an intensional and an extensional sense, it should be possible to find both in a large variety of linguistic contexts. On a closer inspection however it appears that it is quite hard to find clear cases of intensional disjunction in embedded positions like in the antecedent of a conditional (as in Read’s example), or under negation. For example, That is not Drosophilia Melanogaster or D. virilisit, on its most natural reading, simply means that that is not either, but then or must be extensional here because to derive “not \(A\) and not \(B\)” from “not (\(A\) or \(B\))” addition is needed. This seems to suggest that phenomena of relevance should be captured on a pragmatic level rather than on the level of logic (Grice 1989). An explicit Gricean account of relevance of disjunctive sentences has been indeed proposed by Simons (2001), who observes that felicitous utterances of “\(A\) or \(B\)” require the disjuncts to be relevant alternatives, and argues that such condition, which she calls the relatedness condition, can be derived from general principles of conversation interacting with truth functional interpretation of or. In “Further Notes on Logic and Conversation” Grice had suggested that to felicitously use a disjunctive sentence “\(A\) or \(B\)”, a speaker must be in possession of a reasonable argument with “\(A\) or \(B\)” as conclusion [by Quality] which does not contain one of the disjuncts as a step [by Quantity] (Grice 1989: 44). According to Simons to arrive at such a kind of evidence is almost certainly impossible in case the disjuncts are not (contextually) related to each other (as they were for example in Belnap and Anderson’s Napoleon example, which indeed requires a special context to be acceptable). Simons (2001) also discusses a second requirement that a disjunction needs to satisfy in order to constitute a felicitous contribution to an ordinary conversation, namely what she calls the distinctness condition (aka known as Hurford’s constraint, from Hurford 1974), which states that the disjuncts in a clausal disjunction must be distinct alternatives. The following, which does not satisfy such a constraint since one of the disjunct entails the other, is indeed highly anomalous (Simons 2001: example (2)):