CLAUDIO PIZZI TIMOTHY WILLIAMSON STRONG BOETHIUS’ THESIS AND CONSEQUENTIAL IMPLICATION ABSTRACT. The paper studies the relation between systems of modal logic and systems of consequential implication, a non-material form of implication satisfying “Aristotle’s Thesis” (p does not imply not p) and “Weak Boethius’ Thesis” (if p implies q, then p does not imply not q). Definitions are given of consequential implication in terms of modal operators and of modal operators in terms of consequential implication. The modal equivalent of “Strong Boethius’ Thesis” (that p implies q implies that p does not imply not q) is identified. §0 Consider the following claims about an intuitively conceived relation of implication: (1) No proposition implies its own negation. (2) No proposition implies each of two contradictory propositions. (3) No proposition implies every proposition. (4) No proposition is implied by every proposition. (1) is often called “Aristotle’s Thesis”. (2) will be called here “Weak Boethius’ Thesis”; it has received several names in literature, being called the “Law of Conditional Contrariety” in Angell (1978), “Straw- son’s Principle” in Routley et al. (1982) and simply “Boethius’ Thesis” in Pizzi (1977). As these names and the connection between (3)–(4) and so-called paradoxes of implication suggest, (1)–(4) have often been found plausible in the history of logic, although the historical details cannot be discussed here. Strict implication, the necessitation of mate- rial implication, is well known to violate each of (1)–(4). However, it does so only when it links propositions of different modal status (as necessary, contingent or impossible). We might therefore introduce a notion of analytic consequential implication which holds between a pair of propositions just in case (i) the first strictly implies the sec- ond and (ii) they have the same modal status. This definition corre- sponds to the formula (p ⊃ q) ∧ (p ≡ q) ∧ (♦p ≡ ♦q), which is equivalent to (p ⊃ q) ∧ (q ⊃ p) ∧ (♦q ⊃ ♦p) in any nor- mal modal logic. Moreover, analytic consequential implication is eas- Journal of Philosophical Logic 26: 569–588, 1997. c  1997 Kluwer Academic Publishers. Printed in the Netherlands. VTEX(RO) PIPS No.: 120082 HUMNKAP M49620.tex; 25/07/1997; 11:28; v.7; p.1