We here make preliminary investigations into the model theory of DeMorgan logics. We demonstrate that Łoś's Theorem holds with respect to these logics and make some remarks about standard model-theoretic properties in such contexts. More concretely, as a case study we examine the fate of Cantor's Theorem that the classical theory of dense linear orderings without endpoints is $\aleph_{0}$-categorical, and we show that the taking of ultraproducts commutes with respect to previously established methods of constructing nonclassical structures, namely, Priest's Collapsing (...) Lemma and Dunn's Theorem in 3-Valued Logic. (shrink)

This paper is about the concept of semantically closed languages. Roughly speaking, those are languages which can name their own sentences and apply to them semantic predicates, such as the truth or satisfaction predicates. Hence, they are “self-referential languages,” in the sense that they are capable of producing sentences about themselves or other sentences in the same language. In section one, we introduce the concept informally; in section two, we provide the formal definition of first-order semantically closed languages, which is (...) the Tarskian definition with some technical modifications. Then, we construct a semantic for this kind of language, and prove that the language is indeed semantically closed (according to Definition 1). Finally, we discuss whether the logic underlying the construction is classical, and future goals of this research. (shrink)

In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wittgenstein's view given by Alan Turing, who attended the lectures, the so-called ‘falling bridges’-objection. Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of that practice. If we then assume that we have adopted (...) a paraconsistent logic, Wittgenstein's answer to Turing is that if we run into trouble building our bridge, it is either because we have made a calculation mistake or our calculus does not actually describe the phenomenon it is intended to model. The possibility of either kind of error is not particular to contradictions nor inconsistency, and thus contradictions do not have any special status as a thing to be avoided. (shrink)

The phenomenological movement begins in the Prolegomena to Husserl’s Logical Investigations as a philosophy of logic. Despite this, remarkably little attention has been paid to Husserl’s arguments in the Prolegomena in the contemporary philosophy of logic. In particular, the literature spawned by Gilbert Harman’s work on the normative status of logic is almost silent on Husserl’s contribution to this topic. I begin by raising a worry for Husserl’s conception of ‘pure logic’ similar to Harman’s challenge to explain the connection between (...) logic and reasoning. If logic is the study of the forms of all possible theories, it will include the study of many logical consequence relations; by what criteria, then, should we select one consequence relation as correct? I consider how Husserl might respond to this worry by looking to his late account of the ‘genealogy of logic’ in connection with Gurwitsch’s claim that ‘[i]t is to prepredicative perceptual experience … that one must return for a radical clarification and for the definitive justification of logic’. Drawing also on Sartre and Heidegger, I consider how prepredicative experience might constrain or guide our selection of a logical consequence relation and our understanding of connectives like implication and negation. (shrink)

Ultralogic as Universal? is a seminal text in non-classcial logic. Richard Routley presents a hugely ambitious program: to use an 'ultramodal' logic as a universal key, which opens, if rightly operated, all locks. It provides a canon for reasoning in every situation, including illogical, inconsistent and paradoxical ones, realized or not, possible or not. A universal logic, Routley argues, enables us to go where no other logic—especially not classical logic—can. Routley provides an expansive and singular vision of how a universal (...) logic might one day solve major problems in set theory, arithmetic, linguistics, physics, and more. It circulated in typescript in the late 1970s before appearing as the Appendix to Exploring Meinong's Jungle and Beyond. With engaging, forceful prose, unsparing criticism of entrenched institutions, and many tantalizing proof sketches, Ultralogic? has had a major influence on the development of paraconsistent and relevant logic. This new edition makes this work available for a modern audience, newly typeset and corrected, along with extensive notes, and new commentary essays. (shrink)

In this paper I’ll explore the relation between ω-inconsistency and plain inconsistency, in the context of theories that intend to capture semantic concepts. In particular, I’ll focus on two very well known inconsistent but non-trivial theories of truth: LP and STTT. Both have the interesting feature of being able to handle semantic and arithmetic concepts, maintaining the standard model. However, it can be easily shown that both theories are ω- inconsistent. Although usually a theory of truth is generally expected to (...) be ω-consistent, all conceptual concerns don’t apply to inconsistent theories. Finally, I’ll explore if it’s possible to have an inconsistent, butω-consistent theory of truth, restricting my analysis to substructural theories. (shrink)

This paper explores early Australasian philosophy in some detail. Two approaches have dominated Western philosophy in Australia: idealism and materialism. Idealism was prevalent between the 1880s and the 1930s, but dissipated thereafter. Idealism in Australia often reflected Kantian themes, but it also reflected the revival of interest in Hegel through the work of ‘absolute idealists’ such as T. H. Green, F. H. Bradley, and Henry Jones. A number of the early New Zealand philosophers were also educated in the idealist tradition (...) and were influential in their communities, but produced relatively little. In Australia, materialism gained prominence through the work of John Anderson, who arrived in Australia in 1927, and continues to be influential. John Anderson had been a student of Henry Jones, who might therefore be said to have influenced both main strands of Australian philosophical thought. (shrink)

As far as disputes in the philosophy of pure mathematics goes, these are usually between classical mathematics, intuitionist mathematics, paraconsistent mathematics, and so on. My own view is that of a mathematical pluralist: all these different kinds of mathematics are equally legitimate. Applied mathematics is a different matter. In this, a piece of pure mathematics is applied in an empirical area, such as physics, biology, or economics. There can then certainly be a disputes about what the correct pure mathematics to (...) apply is. Such disputes may be settled by the standard criteria of scientific theory selection (adequacy of empirical predications, simplicity, etc.) But what, exactly is it to apply a piece of pure mathematics? How is mathematics applied? By and large, philosophers of mathematics have cared more about pure mathematics than applied mathematics, and not a lot of thought has gone into this question. In this paper I will address the issue and some of its implications. (shrink)

A theorem from Archimedes on the area of a circle is proved in a setting where some inconsistency is permissible, by using paraconsistent reasoning. The new proof emphasizes that the famous method of exhaustion gives approximations of areas closer than any consistent quantity. This is equivalent to the classical theorem in a classical context, but not in a context where it is possible that there are inconsistent innitesimals. The area of the circle is taken 'up to inconsistency'. The fact that (...) the core of Archimedes's proof still works in a weaker logic is evidence that the integral calculus and analysis more generally are still practicable even in the event of inconsistency. (shrink)

Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.

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)