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)
Dirac’s treatment of his well known Delta function was apparently inconsistent. We show how to reconstruct his reasoning using the inconsistency-tolerant technique of Chunk and Permeate. In passing we take note of limitations and developments of that technique.
A typical theorem of conaexive logics is Aristotle''s Thesis(A), (AA).A cannot be added to classical logic without producing a trivial (Post-inconsistent) logic, so connexive logics typically give up one or more of the classical properties of conjunction, e.g.(A & B)A, and are thereby able to achieve not only nontriviality, but also (negation) consistency. To date, semantical modellings forA have been unintuitive. One task of this paper is to give a more intuitive modelling forA in consistent logics. In addition, while inconsistent (...) but nontrivial theories, and inconsistent nontrivial logics employing prepositional constants (for which the rule of uniform substitution US fails), have both been studied extensively within the paraconsistent programme, inconsistent nontrivial logics (closed under US) do not seem to have been. This paper gives sufficient conditions for a logic containingA to be inconsistent, and then shows that there is a class of inconsistent nontrivial logics all containingA. A second semantical modelling forA in such logics is given. Finally, some informal remarks about the kind of modellingA seems to require are made. (shrink)
The question of the interpretation of impossible pictures is taken up. Penrose's account is reviewed. It is argued that whereas this account makes substantial inroads into the problem, there needs to be a further ingredient. An inconsistent account using heap models is proposed.
Popper's definition looked initially promising provided that the restriction of classical logic was removed. As we have seen, this promise is not fulfilled. The search for a satisfactory verisimilitude ordering must therefore be pursued along more mainstream lines. The present exercise ought, however, to make us aware of the possibility that breakdowns of proposed definitions might only occur because of strictly classical assumptions.
In this article we address the question of how many impossible images Escher produced. To answer requires us first to clarify a range of concepts, including content, ambiguity, illusion, and impossibility. We then consider, and reject, several candidates for impossibility before settling on an answer.
First, we consider an argument due to Popper for maximal strength in choice of logic. We dispute this argument, taking a lead from some remarks by Susan Haack; but we defend a set of contrary considerations for minimal strength in logic. Finally, we consider the objection that Popper presupposes the distinctness of logic from science. We conclude from this that all claims to logical truth may be in equal epistemological trouble.
A principle of continuity due to Leibniz has recently been revived by Graham Priest in arguing for an inconsistent account of motion. This paper argues that the Leibniz Continuity Condition has a reasonable interpretation in a different, though still inconsistent, class of dynamical systems. The account is then applied to the quantum mechanical description of the hydrogen atom.
In this paper, a survey is made of some of the contributionsto the interpretation of Hartle and Hawking's theory of thewave function of the universe and its beginning. It is arguedthat there are considerable difficulties with the interpretationof the theory, but that there is at least one interpretationhitherto not found in the literature which survives existingphilosophical objections.
This paper aims to distinguish and classify sixteen versions of the Necker cube. In particular, it is shown how to describe inconsistent and incomplete theories which correspond in a systematic way to these sixteen diagrams. Concerning two of these sixteen cubes, there is a natural intuition that there is a sense in which they inconsistent. It is seen that this intuition is vindicated by an analysis in which their corresponding theories turn out to be globally inconsistent but not locally inconsistent, (...) while various other cubes of the sixteen are merely locally inconsistent. The Routley functor is seen to be useful in classifying the relations between these diagrams. (shrink)
This paper offers an inconsistent model of motion perception. It was prompted by work on inconsistent motion due to Hegel and, following him, Priest. But the paper skirts Hegel's full scale idealism, by proposing that the inconsistency is with the cognitive contents of motion perception. The paper draws on work in the psychology of perception, and in the theory of inconsistency. I begin by noting the prima facie argument that temporal change threatens inconsistency, and canvassing ways in which this might (...) be avoided. The orthodox reply to the prima facie argument is that one and the same thing can have different incompatible properties at different times. This is plausible when applied to motion in the physical world. However, applied to cognition, it can be seen that the phenomenon of phi or beta, combined with the mechanism of a Reichardt detector, lends support to one key step in the prima facie argument, namely re-identification over time. The inconsistency of the model is then seen to follow from application of the Fade Principle. Advantages of the model are stressed: including a simple explanation of the debilitating condition of akinetopsia, that is, motion blindness; and a suggestion as to how to account for the “moving now.”. (shrink)
This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal logic, intuitionist logic (...) and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T 1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity. (shrink)
This paper is dedicated to Newton da Costa, who, among his many achievements, was the first to aim at dualising intuitionism in order to produce paraconsistent logics, the C-systems. This paper similarly dualises intuitionism to a paraconsistent logic, but the dual is a different logic, namely closed set logic. We study the interaction between the properties of topological spaces, particularly separation properties, and logical theories on those spaces. The paper begins with a brief survey of what is known about the (...) relation between topology and modal logic, intuitionist logic and paraconsistent logic in respect of the incompleteness and inconsistency of theories. Necessary and sufficient conditions which relate the Tⁱ-property to the properties of logical theories, are obtained. The result is then extended to Hausdorff and Normal spaces. In the final section these methods are used to vary the modelling conditions for identity. (shrink)
: Competing accounts of change and motion are given by the seventh-century Buddhist logician Dharmakirti and the contemporary analytical philosopher Graham Priest. They agree on much, but disagree on the issue of the Law of Non-Contradiction. This paper takes Dharmakirti's side, appealing to current space-time theory, while making some qualifications.
We apply linear algebra to the study of the inconsistent figure known as the Crazy Crate. Disambiguation by means of occlusions leads to a class of sixteen such figures: consistent, complete, both and neither. Necessary and sufficient conditions for inconsistency are obtained.
In this paper, a survey is made of some of the contributions to the interpretation of Hartle and Hawking’s theory of the wave function of the universe and its beginning. It is argued that there are considerable difficulties with the interpretation of the theory, but that there is at least one interpretation hitherto not found in the literature which survives existing philosophical objections.
This paper continues the investigation of inconsistent arithmetical structures. In $\S2$ the basic notion of a model with identity is defined, and results needed from elsewhere are cited. In $\S3$ several nonisomorphic inconsistent models with identity which extend the (=, $\S4$ inconsistent nonstandard models of the classical theory of finite rings and fields modulo m, i.e. Z m , are briefly considered. In $\S5$ two models modulo an infinite nonstandard number are considered. In the first, it is shown how to (...) model inconsistently the arithmetic of the rationals with all names included, a strengthening of earlier results. In the second, all inconsistency is confined to the nonstandard integers, and the effects on Fermat's Last Theorem are considered. It is concluded that the prospects for a good inconsistent theory of fields may be limited. (shrink)
The general theory of variable binding term operators is an interesting recent development in logic. It opens up a rich class of semantic and model-theoretic problems. In this paper we survey the recent literature on the topic, and offer some remarks on its significances and on its connections with other branches of mathematical logic.
The paper examines connections between ontology and finance. The ontological debates concerning the role of finance are examined between two opposing schools of thought that can be labelled, very broadly, ‘instrumentalist’ and ‘realist’. These two schools of thought have had momentous repercussions in understanding what is a good society. Each school defines Nature in particular ways which can be explored using ontology and philosophical insight. Our theoretical investigation aims to accommodate Nature in community financial deliberations. A positive role for government (...) is advocated to finance environmental infrastructure initiatives. For example, precautionary strategies to address climate change must be funded. New roles for finance and government are proposed to align human relationships with Nature. Environmental precautionary principles must be developed in conjunction with finance theory to maintain decent standards of living for all. Reliance on impersonal market forces will not be enough to save the planet given the power of some over the many in the neoliberal arena. (shrink)