No abstract

ÔAristotelian logicÕ, as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classi ed as Ôhypothetical syllogismsÕ. However, Aristotle did not discuss such arguments, nor did he call any arguments Ôhypothetical syllogismsÕ. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them Ôhypothetical syllogismsÕ; nor did they describe them as ponendo ponens, etc.

2. Higher-order borderline vagueness: some preliminary remarks (i) This paper is concerned only with Sorites-vague predicates-that is, predicates that give rise to Sorites paradoxes. Every Sorites paradox runs on what we call its dimension D. Thus, for a paradox built on 'tall', the dimension D is height. 5 Multidimensional Sorites-vague predicates are considered only insofar as they give rise to one-dimensional Sorites paradoxes. The most basic case of (CC) is then CFa → CCFa, with F for a simple, ordinary language, Sorites-vague predicate and a for a designator. An example would be 'if it is clear that Curly is bald, then it is clear that it is clear that Curly is bald.' Given the context sensitivity of Sorites-vague predicates, the semantic value of Fa is always assumed to be relative to a context C.(ii) We consider in the first instance those theories of higher-order vagueness that intend to capture higher-order borderline cases; that is, borderline borderline cases, borderline borderline borderline cases, and so on. This type of higher-order vagueness-which we also call higher-order borderline vagueness-is the one most frequently 4

In this paper I argue that the ÒdiscoveryÓ of the problem of causal determinism and freedom of decision in Greek philosophy is the result of a mix-up of Aristotelian and Stoic thought in later antiquity; more precisely, a (mis-)interpretation of AristotleÕs philosophy of deliberate choice and action in the light of Stoic theory of determinism and moral responsibility. The (con-)fusion originates with the beginnings of Aristotle scholarship, at the latest in the early 2nd century A.D. It undergoes several developments, absorbing Epictetan, Middle-Platonist, and Peripatetic ideas; and it leads eventually to a concept of freedom of decision and an exposition of the Òfree-will problemÓ in Alexander of AphrodisiasÕ On Fate and in the Mantissa ascribed to him.

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist's characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator 'it is clearly the case that'. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus (IPC) and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson's objections to intuitionistic treatments of vagueness.