It is shown that there are categorical differences between sentences and statements, which have the consequence in particular that there are no paradoxical cases of self-reference with the latter as there are with the former. The point corrects an extensive train of thought that Graham Priest has pursued over recent years, but also a much wider tradition in logic and the foundations of mathematics that has been dominant for over a century. That tradition might be broadly characterized as Formalist, or (...) Nominalist, and the improved understanding of statements leads us instead into a more Realist approach and thereby contentful logic and mathematics. (shrink)

A logic focusing on the analytic a priori and explicitly rejecting the synthetic a priori developed in the early decades of the 20th century, largely through the efforts of the Logical Empiricists. This group was very influenced by Wittgenstein's early work Tractatus Logico-Philosophicus. But Wittgenstein himself, later on, departed from the Tractatus in significant ways that the Logical Empiricists did not follow. Wittgenstein came later to accept the synthetic a priori, and out of this insight comes a non-analytic logic that (...) differs from standard 20th century logic in many distinct ways. This paper details these differences. (shrink)

Modern Logic has generated a lot of problems for itself through inattention to natural forms of speech. In particular it has had difficulties with a large group of ‘logical paradoxes’ through its preoccupation with the Predicate Calculus and related structures to the exclusion of other formal structures that represent natural language more fully, and thereby escape these paradoxes. In natural speech the unrecognized forms involved are principally individual referring terms with a non-specific or fictional reference. For, under the influence of (...) the Logical Empiricists, the focus in formal logic has been on individual terms with a specific, factual reference. But a similar point arises with certain predicates, and indeed even the proper concept of a predicate has been lost, producing the most notorious of the standard puzzles: Russell's Paradox. The paper inevitably makes reference to contemporary logical symbolism in places, but its purpose is to show that the only way forward is to use a symbolism that is abbreviatory only, i.e. maps directly onto natural speech. (shrink)

It is often taken for granted by writers who propose--and, for that matter, by writers who oppose--'justifications' of inductions, that deduction either does not need, or can readily be provided with, justification. The purpose of this paper is to argue that, contrary to this common opinion, problems analogous to those which, notoriously, arise in the attempt to justify induction, also arise in the attempt to justify deduction.

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of 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. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)

This study provides a unified theory of properties, relations, and propositions (PRPs). Two conceptions of PRPs have emerged in the history of philosophy. The author explores both of these traditional conceptions and shows how they can be captured by a single theory.