The collapse models of arithmetic are inconsistent, nontrivial models obtained from ℕ and set out in the Logic of Paradox (LP). They are given a general treatment by Priest (Priest, 2000). Finite collapse models are decidable, and thus axiomatizable, because finite. LP, however, is ill-suited to normal axiomatic reasoning, as it invalidates Modus Ponens, and almost all other usual conditional inferences. I set out a logic, A3, first given by Avron (Avron, 1991), and give a first order axiom system for (...) the finite collapse models. I present some standard arithmetical axioms in addition to a cyclic axiom and prove that these axioms are sound and complete for the cyclic models, reporting a similar result for the heap models. The state of the situation for the each of the kinds of infinite collapse model is, however, left an open question. (shrink)

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what (...) mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity. (shrink)

This paper sets out three programmes that attempt to use relevant logic as the basis for a philosophy of mathematics. Although these three programmes do not exhaust the possible approaches to mathematics through relevant logic, they are fairly representative of the current state of the field. The three programmes are compared and their relative strengths and weaknesses set out. At the end of the paper I examine the consequences of adopting each programme for the realist debate about mathematical objects.

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...) this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

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)

The paper concerns interpretations of the paraconsistent logic LP which model theories properly containing all the sentences of first order arithmetic. The paper demonstrates the existence of such models and provides a complete taxonomy of the finite ones.

The present PhD dissertation aims to examine the relation between modality and change in Aristotle’s metaphysics. -/- On the one hand, Aristotle supports his modal realism (i.e., worldly objects have modal properties - potentialities and essences - that ground the ascriptions of possibility and necessity) by arguing that the rejection of modal realism makes change inexplicable, or, worse, banishes it from the realm of reality. On the other hand, the Stagirite analyses processes by means of modal notions (‘change is the (...) actuality of what is virtual insofar as it is virtual’). In other words: to grasp what change is, one has to resort to the modal idiom of potentialities, while the fact that there is change is indicative of the fact that nature is full of modal properties. -/- Aristotle’s modal and kinetic realism finds a negative in the figure of the Megaric Diodorus Kronus. The polemical situation of Greek philosophy has indeed the dialectical advantage of not opposing the Aristotelian position to a sui generis straw man. Both in its reduction of modal judgements to temporal quantifications and in its dissolution of the reality of the state-of-being-in-motion in favour of a cinematographic view of processes, the philosophical figure of reductionist antirealism embodied by Diodorus constitutes an alternative that takes the opposite view of Aristotle’s realism. -/- The present study is structured as a discussion between these two positions. In the face of Diodorus’ antirealist challenges, Aristotle articulates modal and kinetic considerations: the answers he gives to Diodorus’ puzzles thus provide valuable insight into his metaphysics. Moreover, the examination of Aristotle’s and Diodorus’ metaphysics, because of their insight and originality, will not fail to interest the philosopher concerned with the metaphysical foundations of modern physics, insofar as the entanglement between modalities and processes is nowadays at the core of mechanics (phase spaces, path integrals, etc.). (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)

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)

Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of Gödel’s proof (or Rosser’s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest’s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Gödel sentence to the inconsistent and complete theory of arithmetic. We argue, (...) however, that the alternative argument raises a circularity problem. In sum, Gödel’s and its related theorem merely show the relation between a complete and a consistent theory. A contradiction derived by the application of Gödel sentence has the value of true sentences, i.e. the both-value, only under the inconsistent models for arithmetic. Without having the assumption of inconsistency or completeness, a true contradiction is not derivable from the application of Gödel sentence. Hence, Gödel’s and its related theorem never can be a ground for Dialetheism. (shrink)

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)

This paper criticises necessitarianism, the thesis that there is at least one necessary truth; and defends possibilism, the thesis that all propositions are contingent, or that anything is possible. The second section maintains that no good conventionalist account of necessity is available, while the third section criticises model theoretic necessitarianism. The fourth section sketches some recent technical work on nonclassical logic, with the aim of weakening necessitarian intuitions and strengthening possibilist intuitions. The fifth section considers several a prioristic attempts at (...) demonstrating that there is at least one necessary proposition and finds them inadequate. The final section emphasises the epistemic aspect of possibilism. (shrink)

This paper clarifies and discusses Imre Lakatos’ claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that Lakatos’ motivation for classifying mathematics as a quasi-empirical theory is epistemological; what can be called the quasi-empirical epistemology of mathematics is not correct; analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics.

Restall set forth a "consecution" calculus in his "An Introduction to Substructural Logics." This is a natural deduction type sequent calculus where the structural rules play an important role. This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus (...) so as to validate the metainferential rule of reasoning by cases, as well as certain theory-dependent rules. (shrink)

The paradoxes of self reference have to be dealt with by anyone seeking to give a satisfactory account of the logic of truth, of properties, and even of sets of numbers. Unfortunately, there is no widespread agreement as to how to deal with these paradoxes. Some approaches block the paradoxical inferences by rejecting as invalid a move that classical logic counts as valid. In the recent literature, this deviant logic analysis of the paradoxes has been called into question.This disagreement motivates (...) a re-examination of the philosophy of formal logic and the status of logical truths and rules. In this paper I do some of this work, and I show that this gives us the means to defend the deviant approaches against such criticisms. As a result I hope to show that these analyses of the paradoxes are worthy of more serious consideration than they have so far received. (shrink)

Strict finitism posits a largest natural number. The view is usually thought to be objectionably arbitrary. After all, there seems to be no apparent reason as to why the natural numbers should ‘stop’ at a specific point and not a bit later on the natural line. Drawing on how arguments from arbitrariness are employed in mereology, I propose several ways of understanding this objection against strict finitism. No matter how it is understood, I argue that it is always found wanting.

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.

The system R## of "true" relevant arithmetic is got by adding the ω-rule "Infer VxAx from AO, A1, A2, ...." to the system R# of "relevant Peano arithmetic". The rule ⊃E (or "gamma") is admissible for R##. This contrasts with the counterexample to ⊃E for R# (Friedman & Meyer, "Whither Relevant Arithmetic"). There is a Way Up part of the proof, which selects an arbitrary non-theorem C of R## and which builds by generalizing Henkin and Belnap arguments a prime theory (...) T which still lacks C. (The key to the Way Up is a Witness Protection Program, using the ω-rule.) But T may be TOO BIG, whence there is a Way Down argument that produces a better theory TR, such that R## ⊆ TR ⊆ T. (The key to the Way Down is a Metavaluation, on which membership in T is combined with ordinary truth-functional conditions to determine TR.) The result is a theory that is Just Right, whence it never happens that A ⊃ C and A are theorems of R## but C is a non-theorem. (shrink)

In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the (...) usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quanti- fiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. (shrink)