I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

The extensions of vague predicates like ‘is bald’, ‘is tall’, and ‘is a heap’ apparently lack sharp boundaries, and this makes such predicates susceptible to soritical reasoning, i.e. reasoning that leads to some version of the notorious sorites paradox. This essay is concerned with a certain kind of theory of vagueness, according to which the symptoms and puzzles of vagueness should be accounted for in terms of a particular species of context sensitivity exhibited by vague expressions. The basic idea is (...) that the extensions of vague predicates vary with certain contextual factors, and that this fact can explain why they appear to lack sharp boundaries. This kind of view is referred to as contextualism about vagueness. A detailed characterisation of contextualism about vagueness is given in chapter two and three. In chapter two, a generic version of contextualism about vagueness is developed, and some alternative forms of context sensitivity are introduced. In chapter three, the specific contextual factors appealed to by different contextualists are discussed. In chapter four, different contextualist diagnoses of the sorites paradox are considered, and found to be problematic in various ways. It is argued that contrary to what some of its proponents have claimed, contextualism about vagueness is not superior to other comparable theories of vagueness when it comes to explaining the appeal of soritical reasoning. In chapter five, a certain version of the sorites paradox, known as the forced march sorites, is discussed. It is argued that “data” about how speakers would behave in the forced march cannot lend any firm support to contextualism about vagueness. In chapter six, some problems concerning the instability of the contextual factors are considered. One problem is that contextualist diagnoses of the sorites which locate a fallacy of equivocation in the reasoning seem to render non-soritical reasoning fallacious as well. A model for treating this problem is suggested, but on closer consideration, it turns out to be problematic. Moreover, this model is of no help in solving the more general problem that even if classical logic remains valid for vague language on some contextualist views, the instability of the extensions of vague predicates makes it difficult to know when a certain piece of reasoning instantiates a valid argument form. Other difficulties arise with respect to speech reports and belief contents. Chapter seven concludes with a summary and some methodological remarks. (shrink)