The Australasian Journal of Logic
http://ojs.victoria.ac.nz/ajl
<span lang="EN-US">The Australasian Journal of Logic is an online, open access journal run under the auspices of the Australasian Association of Logic and the Centre for Logic, Language and Computation at Victoria University. The journal is peer reviewed. The journal was started in 2003, with Greg <span class="SpellE">Restall</span> as its founding editor. The AJL welcomes submissions in any area of logic, either proving technical results or doing conceptual work concerning logic. </span><p><span lang="EN-US">ISSN: </span>1448-5052</p>Victoria University of Wellingtonen-USThe Australasian Journal of Logic1448-5052Burn all your textbooks
http://ojs.victoria.ac.nz/ajl/article/view/4022
The standard propositional exposition of necessary and sufficient conditions, as available in introductory logic texts, leads to a contradiction. It should be abolished.Varol Akman2017-08-082017-08-08143Relevance via decomposition
http://ojs.victoria.ac.nz/ajl/article/view/4009
<p>We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment. To extend it, a natural recursion is built into the procedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition.<br /><br /></p>David Makinson2017-08-082017-08-08143FDE Circumscription
http://ojs.victoria.ac.nz/ajl/article/view/3979
<p>In his article "Reassurance via Translation" Marcel Crabbe proposed a formalism to obtain reassurance and classical recapture in the setting of minimal FDE. His formalism proved to be general enough to be extended in order to formalize other forms of non-monotonic systems based on preference relations. It is the aim of this article to show how his result can be extended in a natural way by combining two diﬀerent reasoning systems, namely minimal FDE and circumscription, in order to get a paraconsistent and paracomplete version of circumscription, which we will call paracomplistent circumscription, which has the advantages of FDE and circumscription but is neither explosive nor lacks modus ponens in consistent contexts. Furthermore, we will complete a proof Crabbe left unﬁnished.</p>Daniel Skurt2017-04-182017-04-18143Hybridized Paracomplete and Paraconsistent Logics
http://ojs.victoria.ac.nz/ajl/article/view/4035
This paper contributes to the study of paracompleteness and paraconsistency. We present two logics that address the following questions in novel ways. How can the paracomplete theorist characterize the formulas that defy excluded middle while maintaining that not all formulas are of this kind? How can the paraconsistent theorist characterize the formulas that obey explosion while still maintaining that there are some formulas not of this kind?Colin Caret2017-04-112017-04-11143Paraconsistent Measurement of the Circle
http://ojs.victoria.ac.nz/ajl/article/view/4034
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.Zach WeberMaarten McKubre-Jordens2017-04-112017-04-11143Neighborhood Semantics for Logics of Unknown Truths and False Beliefs
http://ojs.victoria.ac.nz/ajl/article/view/4033
This article outlines a semantic approach to the logics of unknown truths, and the logic of false beliefs, using neighborhood structures, giving results on soundness, completeness, and expressivity. Relational semantics for the logics of unknown truths are also addressed, specically the conditions under which sound axiomatizations of these logics might be obtained from their normal counterparts, and the relationship between refexive insensitive logics (RI-logics) and logics containing the provability operator as the primary modal operator.David GilbertGiorgio Venturi2017-04-112017-04-11143Fixed-Point Models for Theories of Properties and Classes
http://ojs.victoria.ac.nz/ajl/article/view/4032
There is a vibrant (but minority) community among philosophical logicians seeking to resolve the paradoxes of classes, properties and truth by way of adopting some non-classical logic in which trivialising paradoxical arguments are not valid. There is also a long tradition in theoretical computer science|going back to Dana Scott's fixed point model construction for the untyped lambda-calculus of models allowing for fixed points. In this paper, I will bring these traditions closer together, to show how these model constructions can shed light on what we could hope for in a non-trivial model of a theory for classes, properties or truth featuring fixed points.Greg Restall2017-04-112017-04-11143Skolem Functions in Non-Classical Logics
http://ojs.victoria.ac.nz/ajl/article/view/4031
<p>This paper shows how to conservatively extend theories formulated in non-classical logics such as the Logic of Paradox, the Strong Kleene Logic and relevant logics with Skolem functions. Translations to and from the language extended by Skolem functions into the original one are presented and shown to preserve derivability. It is also shown that one may not always substitute s=f(t) and A(t, s) even though A(x,y) determines the extension of a function and f is a Skolem function for A.<em><br /></em></p><p> </p><p> </p>Tore Fjetland Øgaard2017-04-112017-04-11143Non-Classical Circular Definitions
http://ojs.victoria.ac.nz/ajl/article/view/4030
Circular denitions have primarily been studied in revision theory in the classical scheme. I present systems of circular denitions in the Strong Kleene and supervaluation schemes and provide complete proof systems for them. One class of denitions, the intrinsic denitions, naturally arises in both schemes. I survey some of the features of this class of denitions.Shawn Standefer2017-04-112017-04-11143Metametaphysics and Dialetheism
http://ojs.victoria.ac.nz/ajl/article/view/4029
This paper reflects on metametaphysics and as such develops a metametameta-physical view: that quietist metametaphysics requires dialetheism, and in turn a paraconsistent logic. I demonstrate this using Carnap’s metametaphysical position in his 'Empiricism, Semantics and Ontology' (1950) as an example, with regard to how it exhibits self-reference and results in inconsistency. I show how applying Carnap’s position to itself produces a dilemma, both horns of which lead to a contradiction. Such inconsistency commonly arises from meta-theories with global scope, as the 'meta' approach aims to transcend the scope of that which it is theorizing about, whilst the global nature will place itself back within the scope of that which it is theorizing about, which together result in the theory referring to itself whilst refuting itself. I argue that any global metametaphysical theory that draws a limit to thought will face self-reference problems leading to contradictory realms. My conclusion is conditional: If we want to meta-philosophize in such a way and treat quietist meta-theories as being true, then we need to be dialetheist and utilize a paraconsistent logic in order to accommodate the contradictions that result from such theorizing.Suki Finn2017-04-112017-04-11143What If? The Exploration of an Idea
http://ojs.victoria.ac.nz/ajl/article/view/4028
A crucial question here is what, exactly, the conditional in the naive truth/set comprehension principles is. In 'Logic of Paradox', I outlined two options. One is to take it to be the material conditional of the extensional paraconsistent logic LP. Call this "Strategy 1". LP is a relatively weak logic, however. In particular, the material conditional does not detach. The other strategy is to take it to be some detachable conditional. Call this "Strategy 2". The aim of the present essay is to investigate Stragey 1. It is not to advocate it. The work is simply an extended exploration of the strategy, its strengths, its weaknesses, and the various dierent ways in which it may be implemented. In the first part of the paper I will set up the appropriate background details. In the second, I will look at the strategy as it applies to the semantic paradoxes. In the third I will look at how it applies to the set-theoretic paradoxes.Graham Priest2017-04-112017-04-11143A Remark on Maksimova's Variable Separation Property in Super-Bi-Intuitionistic Logics
http://ojs.victoria.ac.nz/ajl/article/view/4027
We provide a sucient frame-theoretic condition for a super bi-intuitionistic logic to have Maksimova's variable separation property. We conclude that bi-intuitionistic logic enjoys the property. Furthermore, we offer an algebraic characterization of the super-bi-intuitionistic logics with Maksimova's property.Guillermo Badia2017-04-112017-04-11143Resolving Quine's Confict: A Neo-Quinean View of the Rational Revisability of Logic
http://ojs.victoria.ac.nz/ajl/article/view/4026
<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 10.000000pt; font-family: 'CMR10';">There is an apparent conflict in Quine’s work between, on the one hand, his clear commitment to the rational revisability of logic and, on the other, his principle of charitable translation and ‘change of logic, change of subject’ argument. I argue that the apparent conflict is mostly resolved under close exegesis, but that the translation argument normatively rules out collaborative revision and allows only revision by individuals. However, I articulate a Neo-Quinean view that preserves the rational acceptability of collaborative revision. On that view, everything is rationally revisable in some manner or other — it’s just that the logical principles that laypersons find (actually or potentially) obvious and tacitly use to govern their everyday inferences can’t be rationally revised quickly and all at once. Since, in Quine’s view, what most people find obvious resists change, rational changes to laymen’s logic must be gradual.</span></p></div></div></div>Amanda Bryant2017-04-112017-04-11143There is no Logical Negation: True, False, Both, and Neither
http://ojs.victoria.ac.nz/ajl/article/view/4025
In this paper I advance and defend a very simple position according to which logic is subclassical but is weaker than the leading subclassical-logic views have it.Jc Beall2017-04-112017-04-11143Front Matter
http://ojs.victoria.ac.nz/ajl/article/view/4024
<p>Editors' Introduction and List of Contributors</p>Zach WeberMaarten McKubre-JordensPatrick Girard2017-04-112017-04-11143Two-Dimensional Tableaux
http://ojs.victoria.ac.nz/ajl/article/view/3894
We present two-dimensional tableau systems for the actuality, fixedly, and up-arrow operators. All systems are proved sound and complete with respect to a two-dimensional semantics. In addition, a decision procedure for the actuality logics is discussed.David Gilbert2016-12-192016-12-19143Off-Topic: A New Interpretation of Weak-Kleene Logic
http://ojs.victoria.ac.nz/ajl/article/view/3976
This paper offers a new and very simple alternative to Bochvar's well known nonsense -- or meaninglessness -- interpretation of Weak Kleene logic. To help orient discussion I begin by reviewing the familiar Strong Kleene logic and its standard interpretation; I then review Weak Kleene logic and the standard (viz., Bochvar) interpretation. While I note a common worry about the Bochvar interpretation my aim is only to give an alternative -- and I think very elegant -- interpretation, not necessarily a replacement.Jc Beall2016-11-282016-11-28143Multi-sorted version of second order arithmetic
http://ojs.victoria.ac.nz/ajl/article/view/3936
<p>This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.</p>Farida Kachapova2016-09-052016-09-05143Old Wine in (Somewhat Leaky) New Bottles: Some Comments on Beall
http://ojs.victoria.ac.nz/ajl/article/view/3934
Dialetheists concerning the paradoxes of self-refrence have often argued that the phenomeonon provides a choice between inconsistency and expressive incompleteness, and that inconsistency is the correct choice. In a recent paper (<a id="magicparlabel-107" style="font-size: 10px;">`Trivialising Sentences and the Promise of Semantic Completeness', <em>Analysis</em> (2015) 75: 573-84), JC Beall attacks this argument. This paper analyses his arguments, and argues that his paper simply provides a new spin on matters well known.</a>Graham Priest2016-09-052016-09-05143Omega-inconsistency without cuts and nonstandard models
http://ojs.victoria.ac.nz/ajl/article/view/3900
This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee (1985) for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omega-inconsistency and shows thus, pace Cobreros et al.(2013), that the result in McGee (1985) does not rely on transitivity. Finally, it also explains why the omega-inconsistent logics of truth in question do not require nonstandard models of arithmetic.Andreas Fjellstad2016-09-052016-09-05143Solutions to Some Open Problems from Slaney
http://ojs.victoria.ac.nz/ajl/article/view/3923
<p>In response to a paper by Harris & Fitelson, Slaney states several open questions concerning possible strategies for proving distributivity in a wide class of positive sentential logics. In this note, I provide answers to all of Slaney's open questions. The result is a better understanding of the class of positive logics in which distributivity holds.</p>Branden Fitelson2016-06-022016-06-02143Contraction and revision
http://ojs.victoria.ac.nz/ajl/article/view/3935
<p>An important question for proponents of non-contractive approaches to paradox is why contraction fails. Zardini offers an answer, namely that paradoxical sentences exhibit a kind of instability. I elaborate this idea using revision theory, and I argue that while instability does motivate failures of contraction, it equally motivates failure of many principles that non-contractive theorists want to maintain.</p>Shawn Standefer2016-04-162016-04-16143Remarks on Ontological Dependence in Set Theory
http://ojs.victoria.ac.nz/ajl/article/view/3899
In a recent paper, John Wigglesworth explicates the notion of a set's <em>being grounded in</em> or <em>ontologically depending on</em> its members by the modal statement that in any world (possible or impossible), that a set exists in that world entails that its members exist as well. After suggesting that variable-domain S5 captures an appropriate account of metaphysical necessity, Wigglesworth purports to prove that in any set theory satisfying the axiom <strong>Extensionality</strong> this condition holds, that is, that sets ontologically depend on their members with respect to extraordinarily weak notions of set. This paper diagnoses a number of problems concerning Wigglesworth's formal argument. For one, we will show that Wigglesworth's argument is invalid as it requires an appeal to hidden, extralogical theses concerning rigid designation and the persistence of sets across possible worlds. Having demonstrated the indispensability of these principles to Wigglesworth's argument, we will then show that even granted the enthymematic premises, the argument only proves the ontological dependence of <em>singletons</em> on their members and does not extend to sets in general. Finally, we will consider strengthenings of Wigglesworth's reasoning and suggest that even the weakest generalization will bear undesirable consequences.Thomas Macaulay Ferguson2016-04-162016-04-16143Towards Paraconsistent Inquiry
http://ojs.victoria.ac.nz/ajl/article/view/2102
<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>In this paper, we discuss Hintikka’s theory of interrogative approach to inquiry with a focus on bracketing. First, we dispute the use of bracketing in the interrogative model of inquiry arguing that bracketing provides an indispensable component of an inquiry. Then, we suggest a formal system based on strategy logic and logic of paradox to describe the epistemic aspects of an inquiry, and obtain a naturally paraconsistent system. We then apply our framework to some cases to illustrate its use. </span></p></div></div></div>Can Baskent2016-03-022016-03-02143Worlds and Models in Bayart and Carnap
http://ojs.victoria.ac.nz/ajl/article/view/3927
In the early days of the semantics for modal logic the `possible worlds' were thought of as models or interpretations. This was particularly so when the interpretation was of \emph{logical} necessity or possibility, where this was understood in terms of validity. Arnould Bayart in 1958 may have been the first modal logician to argue explicitly against the identification of necessity and validity. This note contrasts his semantics with that provided by Rudolf Carnap in 1946, and examines Bayart's proof that if you identify necessity with validity then certain theorems of S5 are not valid. The proof is then examined using Carnap's semantics.Max Cresswell2016-01-222016-01-22143Relevance Logic: Problems Open and Closed
http://ojs.victoria.ac.nz/ajl/article/view/3926
I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of <strong>R</strong> in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of <strong>R</strong>. Some related problems are discussed along the way.Alasdair Urquhart2016-01-222016-01-22143First degree formulas in quantified S5
http://ojs.victoria.ac.nz/ajl/article/view/3891
<p>This note provides a proof that the formula L(Ex)( Fx & ~LFx) is not equivalent to any first degree formula in the context of the quantified version of the modal logic S5. This solves a problem posed by Max Cresswell.</p><p> </p><p> </p>Alasdair Urquhart2015-07-202015-07-20143Substructural Negations
http://ojs.victoria.ac.nz/ajl/article/view/2225
<p>We present substructural negations, a family of negations (or negative modalities) classified in terms of structural rules of an extended kind of sequent calculus, display calculus. In considering the whole picture, we emphasize the duality of negation. Two types of negative modality, impossibility and unnecessity, are discussed and "self-dual" negations like Classical, De Morgan, or Ockham negation are redefined as the fusions of two negative modalities. We also consider how to identify, using intuitionistic and dual intuitionistic negations, two accessibility relations associated with impossibility and unnecessity.</p>Takuro Onishi2015-06-032015-06-03143Two Temporal Logics of Contingency
http://ojs.victoria.ac.nz/ajl/article/view/2133
This work concerns the use of operators for past and future contingency in Priorean temporal logic. We will develop a system named {\bf{C}}$_t$, whose language includes a propositional constant and prove that (i) {\bf{C}}$_t$ is complete with respect to a certain class of general frames and (ii) the usual operators for past and future necessity are definable in such system. Furthermore, we will introduce the extension {\bf{C}}${_t}lin$ that can be interpreted on linear and transitive general frames. The theoretical result of the current work is that contingency can be treated as a primitive notion in reasoning about temporal modalities.Matteo Pascucci2015-05-222015-05-22143Set-Theoretic Dependence
http://ojs.victoria.ac.nz/ajl/article/view/2131
In this paper, we explore the idea that sets depend on, or are grounded in, their members. It is said that a set depends on each of its members, and not vice versa. Members do not depend on the sets that they belong to. We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members. We extend the modal truth conditions to include impossible worlds and give a more satisfactory account of the dependence of a set on its members. Focusing on the case of singletons, we articulate a logical framework in which to evaluate set-theoretic dependence claims, using a normal first-order modal logic. We show that on this framework the dependence of a singleton on its single members follows from logic alone. However, the converse does not hold.John Wigglesworth2015-05-052015-05-05143On Artifacts and Truth-Preservation
http://ojs.victoria.ac.nz/ajl/article/view/2045
<p>In <em>Saving Truth from Paradox</em>, Hartry Field presents and defends a theory of truth with a new conditional. In this paper, I present two criticisms of this theory, one concerning its assessments of validity and one concerning its treatment of truth-preservation claims. One way of adjusting the theory adequately responds to the truth-preservation criticism, at the cost of making the validity criticism worse. I show that in a restricted setting, Field has a way to respond to the validity criticism. I close with some general considerations on the use of revision-theoretic methods in theories of truth.</p>Shawn Standefer2015-05-052015-05-05143Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic
http://ojs.victoria.ac.nz/ajl/article/view/2066
In <em>An Introduction to Non-Classical Logic: From If to Is </em>Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.Marilynn Johnson2015-04-132015-04-13143Editorial Preface
http://ojs.victoria.ac.nz/ajl/article/view/2197
This first issue of volume 12 introduces a new format for the Australasian Journal of Logic.Edwin Mares2015-01-072015-01-07143A Modal-tense Sortal Logic with Variable-Domain Second-order Quantification
http://ojs.victoria.ac.nz/ajl/article/view/2084
We propose a new intensional semantics for modal-tense second-order languages with sortal predicates. The semantics provides a variable-domain interpretation of the second-order quantifiers. A formal logical system is characterized and proved to be sound and complete with respect to the semantics. A contemporary variant of conceptualism as a theory of universals is the philosophical background of the semantics. Justification for the variable-domain interpretation of the second-order quantifiers presupposes such a conceptualist framework.Max Alberto Freund2015-01-042015-01-04143Two-valued logics for naive truth theory
http://ojs.victoria.ac.nz/ajl/article/view/2082
It is part of the current wisdom that the Liar and similar semantic<br />paradoxes can be taken care of by the use of certain non-classical<br />multivalued logics. In this paper I want to suggest that bivalent logic can do just as well. This is accomplished by using a non-deterministic matrix to define the negation connective. I show that the systems obtained in this way support a transparent truth predicate. The paper also contains some remarks on the conceptual interest of such systems.Lucas Daniel Rosenblatt2015-01-042015-01-04143Paraconsistent games and the limits of rational self-interest
http://ojs.victoria.ac.nz/ajl/article/view/2021
It is shown that logical contradictions are derivable from natural translations into first order logic of the description and background assumptions of the Soros Game, and of other games and social contexts that exhibit conflict and reflexivity. The logical structure of these contexts is analysed using proof-theoretic and model-theoretic techniques of first order paraconsistent logic. It is shown that all the contradictions that arise contain the knowledge operator K. Thus, the contradictions do not refer purely to material objects, and do not imply the existence of inconsistent, concrete, physical objects, or the inconsistency of direct sensory experience. However, the decision-making of rational self-interested agents is stymied by the appearance of such intensional contradictions. Replacing the rational self-interest axioms with axioms for an appropriate moral framework removes the inconsistencies. Rational moral choice in conflict-reflexive social contexts then becomes possible.Arief DaynesPanagiotis AndrikopoulosParaskevas PagasDavid Latimer2015-01-042015-01-04143Remarks on "Random Sequences"
http://ojs.victoria.ac.nz/ajl/article/view/2134
We show that standard statistical tests for randomness of finite sequences are language-dependent in an inductively pernicious way.Branden FitelsonDaniel Osherson2015-01-042015-01-04143Rough Consequence and other Modal Logics
http://ojs.victoria.ac.nz/ajl/article/view/2194
Chakraborty and Banerjee have introduced a rough consequence logic based on the modal logic S5. This paper shows that rough consequence logics, with many of the same properties, can be based on modal logics as weak as K, with a simpler formulation than that of Chakraborty and Banerjee. Also provided are decision procedures for the rough consequence logics and equivalences and independence relations between various systems S and the rough consequence logics, based on them. It also shows that each logic, based on such an S, is theorem equivalent, but not necessarily equivalent, to the modal logic M-S. The paper also shows that rough consequence logic, which was designed to handle rough equality, is somewhat limited for that purpose.Martin Bunder2015-01-042015-01-04143Goedel's Property Abstraction and Possibilism
http://ojs.victoria.ac.nz/ajl/article/view/2145
<p>Gödel’s Ontological argument is distinctive because it is the most sophisticated and formal of ontological arguments and relies heavily on the notion of <em>positive property</em>. Gödel uses a third-order modal logic with a property abstraction operator and property quantification into modal contexts. Gödel describes <em>positive property</em> as "independent of the accidental structure of the world"; "pure attribution," as opposed to privation; "positive in the 'moral aesthetic sense.'" <em>Pure attribution</em> seems likely to be related to the Leibnizian concept of perfection.</p><p>By a careful examination of the formal semantics of third-order modal logic with property abstraction together with a Completeness result for third-order modal logic with property abstraction for faithful models that I previously developed in 2000 in my work, <em>Gödel’s Ontological Argument</em>, I argue that it is not possible to develop a sufficient applied third-order modal semantics for Gödel’s ontological argument. As I explore possible approaches for an applied semantics including anti-Realist accounts of the semantics of modal logic compatible with Actualism, I argue that Gödel makes implicit philosophical assumptions which commit him to both possibilism (the belief in merely possible objects) and modal realism (the belief in possible worlds).</p>Randoph Rubens Goldman2014-11-112014-11-11143Much Ado About Nothing
http://ojs.victoria.ac.nz/ajl/article/view/2144
<p align="LEFT">The point of this paper is to bring together three topics: non-existent objects, mereology, and nothing(ness). There are important inter-connections, which it is my aim to spell out, in the service of an account of the last of these.</p>Graham Priest2014-11-112014-11-11143Representing Counterparts
http://ojs.victoria.ac.nz/ajl/article/view/2143
<p align="LEFT">This paper presents a counterpart theoretic semantics for quantified modal logic based on a fleshed out account of Lewis's notion of a 'possibility'. According to the account a possibility consists of a world and some haecceitistic information about how each possible individual gets represented de re. Following Hazen, a semantics for quantified model logic based on evaluating formulae at possibilities is developed. It is shown that this framework naturally accommodates an actuality operator, addressing recent objections to counterpart theory, and is equivalent to the more familiar Kripke semantics for quantied modal logic with an actuality operator.</p>Andrew Bacon2014-11-112014-11-11143Modal Noneism: Transworld Identity, Identification, and Individuation
http://ojs.victoria.ac.nz/ajl/article/view/2142
<p align="LEFT">Noneism a is form of Meinongianism, proposed by Richard Routley and developed and improved by Graham Priest in his widely discussed book <em>Towards Non-Being</em>. Priest's noneism is based upon the double move of (a) building a worlds semantics including impossible worlds, besides possible ones, and (b) admitting a new comprehension principle for objects, differerent from the ones proposed in other kinds of neo-Meinongian theories, such as Parsons' and Zalta's. The new principle has no restrictions on the sets of properties that can deliver objects, but parameterizes the having of properties by objects to worlds. Modality is therefore explicitly built in - so the approach can be conveniently labeled as "modal noneism". In this paper, I put modal noneism to work by testing it against classical issues in modal logic and semantics. It turns out that - perhaps surprisingly - the theory (1) performs well in problems of transworld identity, which are frequently considered to be the difficult ones in the literature; (2) faces a limitation, albeit not a severe one, when one comes to transworld individuation, which is often taken (especially after Kripke's notorious 'stipulation' solution) as an easy issue, if not a pseudo-problem; and (3) may stumble upon a real trouble when dealing with what I shall call 'extensionally indiscernible entities' - particular nonexistent objects modal noneism is committed to.</p>Francesco Berto2014-11-112014-11-11143Discriminator logics (Research announcement)
http://ojs.victoria.ac.nz/ajl/article/view/2020
<p>A discriminator logic is the <strong>1</strong>-assertional logic of a discriminator variety <em>V </em>having two constant terms <strong>0 </strong>and <strong>1</strong> such that <em>V </em>⊨ <strong>0</strong> <img class="tex" src="http://img3.wikia.nocookie.net/__cb1414788729/latex/en/images/math/3/0/a/30a83899d43b156384be84db854210fd.png" alt="\approx" /><strong>1</strong> iff every member of <em>V</em> is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system <strong>SBPC</strong>, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic <strong>S</strong> can be presented (up to definitional equivalence) as an axiomatic extension of <strong>SBPC</strong> by a set of extensional logical connectives taken from the language of <strong>S</strong>. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work. </p>Matthew SpinksRobert BignallRobert Veroff2014-11-112014-11-11143Merge: In Honour of Robert K. Meyer
http://ojs.victoria.ac.nz/ajl/article/view/1816
Methods for unifying inconsistent pairs of theories, which we call collectively MERGE, are defined and their properties outlined.Chris Mortensen2014-04-292014-04-29143The Completeness of Carnap's Predicate Logic
http://ojs.victoria.ac.nz/ajl/article/view/2017
<p> </p><p> </p><p><span style="font-family: Times; font-size: small;"><span lang="EN-GB">The paper first proves the completeness of the (non-modal) first-order predicate logic presented in Carnap’s 1946 article ‘Modalities and quantification’. By contrast the modal logic defined by the semantics Carnap produces is unaxiomatisable. One can though adapt Carnap’s semantics so that a standard completeness proof for a Carnapian version of predicate S5 turns out to be available.</span></span></p><script id="ncoEventScript" type="text/javascript">// <![CDATA[ function DOMContentLoaded(browserID, tabId, isTop, url) { var object = document.getElementById("cosymantecnisbfw"); if(null != object) { object.DOMContentLoaded(browserID, tabId, isTop, url);} }; function Nav(BrowserID, TabID, isTop, isBool, url) { var object = document.getElementById("cosymantecnisbfw"); if(null != object) object.Nav(BrowserID, TabID, isTop, isBool, url); }; function NavigateComplete(BrowserID, TabID, isTop, url) { var object = document.getElementById("cosymantecnisbfw"); if(null != object) object.NavigateComplete(BrowserID, TabID, isTop, url); } function Submit(browserID, tabID, target, url) { var object = document.getElementById("cosymantecnisbfw"); if(null != object) object.Submit(browserID, tabID, target, url); }; // ]]></script>Max Cresswell2014-04-082014-04-08143Power Matrices and Dunn--Belnap Semantics: Reflections on a Remark of Graham Priest
http://ojs.victoria.ac.nz/ajl/article/view/2044
The plurivalent logics considered in Graham Priest's recent paper of that name can be thought of as logics determined by matrices (in the `logical matrix' sense) whose underlying algebras are power algebras (a.k.a. complex algebras, or `globals'), where the power algebra of a given algebra has as elements \textit{subsets} of the universe of the given algebra, and the power matrix of a given matrix has has the power algebra of the latter's algebra as its underlying algebra, with its designated elements being selected in a natural way on the basis of those of the given matrix. The present discussion stresses the continuity of Priest's work on the question of which matrices determine consequence relations (for propositional logics) which remain unaffected on passage to the consequence relation determined by the power matrix of the given matrix with the corresponding (long-settled) question in equational logic as to which identities holding in an algebra continue to hold in its power algebra. Both questions are sensitive to a decision as to whether or not to include the empty set as an element of the power algebra, and our main focus will be on the contrast, when it is included, between the power matrix semantics (derived from the two-element Boolean matrix) and the four-valued Dunn--Belnap semantics for first-degree entailment a la Anderson and Belnap) in terms of sets of classical values (subsets of {T, F}, that is), in which the empty set figures in a somewhat different way, <br />as Priest had remarked his 1984 study, `Hyper-contradictions', in which what we are calling the power matrix construction first appeared.Lloyd Humberstone2014-04-082014-04-08143Plurivalent Logics
http://ojs.victoria.ac.nz/ajl/article/view/1830
<p>In this paper, I will describe a technique for generating a novel kind of semantics for a logic, and explore some of its consequences. It would be natural to call the semantics produced by the technique in question ‘many-valued'; but that name is, of course, already taken. I call them, instead, ‘plurivalent'. In standard logical semantics, formulas take exactly one of a bunch of semantic values. I call such semantics ‘univalent'. In a plurivalent semantics, by contrast, formulas may take one or more such values (maybe even less than one). The construction I shall describe can be applied to any univalent semantics to produce a corresponding plurivalent one. In the paper I will be concerned with the application of the technique to propositional many-valued (including two-valued) logics. Sometimes going plurivalent does not change the consequence relation; sometimes it does. I investigate the possibilities in detail with respect to small family of many-valued logics.</p>Graham Priest2014-04-082014-04-08143Editor's Introduction
http://ojs.victoria.ac.nz/ajl/article/view/2065
<script id="ncoEventScript" type="text/javascript">// <![CDATA[ function DOMContentLoaded(browserID, tabId, isTop, url) { var object = document.getElementById("cosymantecnisbfw"); if(null != object) { object.DOMContentLoaded(browserID, tabId, isTop, url);} }; function Nav(BrowserID, TabID, isTop, isBool, url) { var object = document.getElementById("cosymantecnisbfw"); if(null != object) object.Nav(BrowserID, TabID, isTop, isBool, url); }; function NavigateComplete(BrowserID, TabID, isTop, url) { var object = document.getElementById("cosymantecnisbfw"); if(null != object) object.NavigateComplete(BrowserID, TabID, isTop, url); } function Submit(browserID, tabID, target, url) { var object = document.getElementById("cosymantecnisbfw"); if(null != object) object.Submit(browserID, tabID, target, url); }; // ]]></script>Edwin Mares2014-04-082014-04-08143Intersection Type Systems and Logics Related to the Meyer–Routley System B+
http://ojs.victoria.ac.nz/ajl/article/view/1762
Some, but not all, closed terms of the lambda calculus have types; these types are exactly the theorems of intuitionistic implicational logic. An extension of these simple (→) types to intersection (or →∧) types allows all closed lambda terms to have types. The corresponding →∧ logic, related to the Meyer–Routley minimal logic B+ (without ∨), is weaker than the →∧ fragment of intuitionistic logic. In this paper we provide an introduction to the above work and also determine the →∧ logics that correspond to certain interesting subsystems of the full →∧ type theory.Martin Bunder2012-09-162012-09-16143Flexibility in Ceteris Paribus Reasoning
http://ojs.victoria.ac.nz/ajl/article/view/1826
Ceteris Paribus clauses in reasoning are used to allow for defeaters of norms, rules or laws, such as in von Wright’s example “I prefer my raincoat over my umbrella, everything else being equal”. In earlier work, a logical analysis is offered in which sets of formulas Γ, embedded in modal operators, provide necessary and sufficient conditions for things to be equal in ceteris paribus clauses. For most laws, the set of things allowed to vary is small, often finite, and so Γ is typically infinite. Yet the axiomatisation they provide is restricted to the special and atypical case in which Γ is finite. We address this problem by being more flexible about ceteris paribus conditions, in two ways. The first is to offer an alternative, slightly more general semantics, in which the set of formulas only give necessary but not (necessarily) sufficient conditions. This permits a simple axiomatisation.Jeremy SeligmanPatrick Girard2011-12-212011-12-21143Translatable Self-Reference
http://ojs.victoria.ac.nz/ajl/article/view/1824
Stephen Read has advanced a solution of certain semantic paradoxes recently, based on the work of Thomas Bradwardine. One consequence of this approach, however, is that if Socrates utters only ‘Socrates utters a falsehood’ (a), while Plato says ‘Socrates utters a falsehood’ (b), then, for Bradwardine two different propositions are involved on account of (a) being self-referential, while (b) is not. Problems with this consequence are first discussed before a closely related analysis is provided that escapes it. Moreover, this alternative analysis merely relies on quantification theory at the propositional level, so there is very little to question about it. The paper is the third in a series explaining the superior virtues of a referential form of propositional quantification.Hartley Slater2011-05-192011-05-19143Fraenkel–Carnap Questions for Equivalence Relations
http://ojs.victoria.ac.nz/ajl/article/view/1825
An equivalence is a binary relational system A = (A,ϱA) where ϱA is an equivalence relation on A. A simple expansion of an equivalence is a system of the form (Aa1…an) were A is an equivalence and a1,…,an are members of A. It is shown that the Fraenkel-Carnap question when restricted to the class of equivalences or to the class of simple expansions of equivalences has a positive answer: that the complete second-order theory of such a system is categorical, if it is finitely axiomatizable.George WeaverIrena Penev2011-05-192011-05-19143Adding to Relevant Restricted Quantification
http://ojs.victoria.ac.nz/ajl/article/view/1823
This paper presents, in a more general setting, a simple approach to ‘relevant restricted generalizations’ advanced in previous work. After reviewing some desiderata for restricted generalizations, I present the target route towards achieving the desiderata. An objection to the approach, due to David Ripley, is presented, followed by three brief replies, one from a dialetheic perspective and the others more general.Jc Beall2011-04-142011-04-14143Proving Induction
http://ojs.victoria.ac.nz/ajl/article/view/1821
The hard problem of induction is to argue without begging the question that inductive inference, applied properly in the proper circumstances, is conducive to truth. A recent theorem seems to show that the hard problem has a deductive solution. The theorem, provable in ZFC, states that a predictive function M exists with the following property: whatever world we live in, M ncorrectly predicts the world’s present state given its previous states at all times apart from a well-ordered subset. On the usual model of time a well-ordered subset is small relative to the set of all times. M’s existence therefore seems to provide a solution to the hard problem. My paper argues for two conclusions. First, the theorem does not solve the hard problem of induction. More positively though, it solves a version of the problem in which the structure of time is given modulo our choice of set theory.Alexander Paseau2011-02-152011-02-15143The Boxdot Conjecture and the Language of Essence and Accident
http://ojs.victoria.ac.nz/ajl/article/view/1822
We show the Boxdot Conjecture holds for a limited but familiar range of Lemmon-Scott axioms. We re-introduce the language of essence and accident, first introduced by J. Marcos, and show how it aids our strategy.Christopher Steinsvold2011-02-152011-02-15143AJL Comment: One Philosopher is Correct (Maybe)
http://ojs.victoria.ac.nz/ajl/article/view/1820
It is argued that there may be a philosopher who is correct.Paul Skokowski2010-12-012010-12-01143Complement-Topoi and Dual Intuitionistic Logic
http://ojs.victoria.ac.nz/ajl/article/view/1819
Mortensen studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complement-topos logic, which throws some light on the problem of giving a dualization for LJ.Luis Estrada-González2010-11-262010-11-26143Cantor’s Proof in the Full Definable Universe
http://ojs.victoria.ac.nz/ajl/article/view/1818
Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the scope of quantifiers reveals a natural way out.Laureano LunaWilliam Taylor2010-11-032010-11-03143Review: Vagueness and Degrees of Truth
http://ojs.victoria.ac.nz/ajl/article/view/1817
A Review of Nicholas J.J. Smith, Vagueness and Degrees of Truth, Oxford University Press, 2008.Christian G. Fermüller2010-11-022010-11-02143A Logic for Vagueness
http://ojs.victoria.ac.nz/ajl/article/view/1815
This paper presents F, substructural logic designed to treat vagueness. Weaker than Lukasiewicz’s infinitely valued logic, it is presented first in a natural deduction system, then given a Kripke semantics in the manner of Routley and Meyer's ternary relational semantics for R and related systems, but in this case, the points are motivated as degrees to which the truth could be stretched. Soundness and completeness are proved, not only for the propositional system, but also for its extension with first-order quantifiers. The first-order models allow not only objects with vague properties, but also objects whose very existence is a matter of degree.John Slaney2010-11-012010-11-01143The D-Completeness of T→
http://ojs.victoria.ac.nz/ajl/article/view/1810
A Hilbert-style version of an implicational logic can be represented by a set of axiom schemes and modus ponens or by the corresponding axioms, modus ponens and substitution. Certain logics, for example the intuitionistic implicational logic, can also be represented by axioms and the rule of condensed detachment, which combines modus ponens with a minimal form of substitution. Such logics, for example intuitionistic implicational logic, are said to be D-complete. For certain weaker logics, the version based on condensed detachment and axioms (the condensed version of the logic) is weaker than the original. In this paper we prove that the relevant logic T[→], and any logic of which this is a sublogic, is D-complete.R. K. MeyerM. W. Bunder2010-09-222010-09-22143Extending Metacompleteness to Systems with Classical Formulae
http://ojs.victoria.ac.nz/ajl/article/view/1811
In honour of Bob Meyer, the paper extends the use of his concept of metacompleteness to include various classical systems, as much as we are able. To do this for the classical sentential calculus, we add extra axioms so as to treat the variables like constants. Further, we use a one-sorted and a two-sorted approach to add classical sentential constants to the logic DJ of my book, Universal Logic. It is appropriate to use rejection to represent classicality in the one-sorted case. We then extend these methods to the quantified logics, but we use a finite domain of individual constants to do this.Ross T. Brady2010-09-222010-09-22143Boolean Conservative Extension Results for some Modal Relevant Logics
http://ojs.victoria.ac.nz/ajl/article/view/1812
This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean negation.Edwin D. MaresKoji Tanaka2010-09-222010-09-22143Logics without the contraction rule and residuated lattices
http://ojs.victoria.ac.nz/ajl/article/view/1813
In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework.Hiroakira Ono2010-09-222010-09-22143Models for Substructural Arithmetics
http://ojs.victoria.ac.nz/ajl/article/view/1814
This paper explores models for arithmetic in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R. Greg Restall2010-09-222010-09-22143Partial Confirmation of a Conjecture on the Boxdot Translation in Modal Logic
http://ojs.victoria.ac.nz/ajl/article/view/1808
The purpose of the present note is to advertise an interesting conjecture concerning a well-known translation in modal logic, by confirming a (highly restricted) special case of the conjecture. Rohan FrenchLloyd Humberstone2009-07-292009-07-29143The Law of Non-Contradiction as a Metaphysical Principle
http://ojs.victoria.ac.nz/ajl/article/view/1806
The goals of this paper are two-fold: I wish to clarify the Aristotelian conception of the law of non-contradiction as a metaphysical rather than a semantic or logical principle, and to defend the truth of the principle in this sense. First I will explain what it in fact means that the law of non-contradiction is a metaphysical principle. The core idea is that the law of non-contradiction is a general principle derived from how things are in the world. For example, there are certain constraints as to what kind of properties an object can have, and especially: some of these properties are mutually exclusive. Given this characterisation, I will advance to examine what kind of challenges the law of non-contradiction faces—the main opponent here is Graham Priest. I will consider these challenges and conclude that they do not threaten the truth of the law of non-contradiction understood as a metaphysical principle.Tuomas E. Tahko2009-06-112009-06-11143A Note on Identity and Higher Order Quantification
http://ojs.victoria.ac.nz/ajl/article/view/1807
It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics (where the variables range over all subsets of the domain) in which the identity relation is not definable. The point is that the definability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed. This paper is a follow-up to (Urbaniak 2006), where it has been proven that no actual axiomatization of Leśniewski’s Ontology determines the standard semantics for the epsilon connective.Rafal Urbaniak2009-06-112009-06-11143Review: H. van Ditmarsch, W. van der Hoek and B. Kooi’s Dynamic Epistemic Logic
http://ojs.victoria.ac.nz/ajl/article/view/1805
Patrick Girard2009-05-292009-05-29143Linear Algebra Representation of Necker Cubes I: The Crazy Crate
http://ojs.victoria.ac.nz/ajl/article/view/1803
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.Chris MortensenSteve Leishman2009-03-272009-03-27143Linear Algebra Representation of Necker Cubes II: The Routley Functor and Necker Chains
http://ojs.victoria.ac.nz/ajl/article/view/1804
In this sequel, linear algebra methods are used to study the Routley Functor, both in single Neckers and in Necker chains. The latter display a certain irreducible higher-order inconsistency. A definition of degree of inconsistency is given, which classifies such inconsistency correctly with other examples of local and global inconsistency.Chris Mortensen2009-03-272009-03-27143Logical dynamics meets logical pluralism?
http://ojs.victoria.ac.nz/ajl/article/view/1801
Where is logic heading today? There is a general feeling that the discipline is broadening its scope and agenda beyond classical foundational issues, and maybe even a concern that, like Stephen Leacock’s famous horseman, it is ‘riding off madly in all directions’. So, what is the resultant vector? There seem to be two broad answers in circulation today. One is logical pluralism, locating the new scope of logic in charting a wide variety of reasoning styles, often marked by non-classical structural rules of inference. This is the new program that I subscribed to in my work on sub-structural logics around 1990, and it is a powerful movement today. But gradually, I have changed my mind about the crux of what logic should become. I would now say that the main issue is not variety of reasoning styles and notions of consequence, but the variety of informational tasks performed by intelligent interacting agents, of which inference is only one among many, involving observation, memory, questions and answers, dialogue, or general communication. And logical systems should deal with a wide variety of these, making information-carrying events first-class citizens in their set-up. The purpose of this brief paper is to contrast and compare the two approaches, drawing freely on some insights from earlier published papers. In particular, I will argue that logical dynamics sets itself the more ambitious diagnostic goal of explaining why substructural phenomena occur, by ‘deconstructing’ them into classical logic plus an explicit account of the relevant informational events.Johan van Benthem2008-12-172008-12-17143Logical Pluralism Hollandaise
http://ojs.victoria.ac.nz/ajl/article/view/1802
Johan van Benthem compares and contrasts two research programmes, which he calls logical pluralism and logical dynamics, stating his ‘preference’ for the second of these ‘alternatives’. In this note I want to put the matter into a slightly different perspective.Graham Priest2008-12-172008-12-17143An abstract approach to reasoning about games with mistaken and changing beliefs
http://ojs.victoria.ac.nz/ajl/article/view/1800
We do not believe that logic is the sole answer to deep and intriguing questions about human behaviour, but we think that it might be a useful tool in simulating and understanding it to a certain degree and in specifically restricted areas of application. We do not aim to resolve the question of what rational behaviour in games with mistaken and changing beliefs is. Rather, we develop a formal and abstract framework that allows us to reason about behaviour in games with mistaken and changing beliefs leaving aside normative questions concerning whether the agents are behaving “rationally”; we focus on what agents do in a game. In this paper, we are not concerned with the reasoning process of the (ideal) economic agent; rather, our intended application is artificial agents, e.g., autonomous agents interacting with a human user or with each other as part of a computer game or in a virtual world. We give a story of mistaken beliefs that is a typical example of the situation in which we should want our formal setting to be applied. Then we give the definitions for our formal system and how to use this setting to get a backward induction solution. We then apply our semantics to the story related earlier and give an analysis of it. Our final section contains a discussion of related work and future projects. We discuss the advantages of our approach over existing approaches and indicate how it can be connected to the existing literature.Benedikt LöweEric Pacuit2008-11-282008-11-28143Collapsing Arguments for Facts and Propositions
http://ojs.victoria.ac.nz/ajl/article/view/1799
Kurt Gödel argues in “Russell’s Mathematical Logic” that on the assumption that, contrary to Russell, definite descriptions are terms, it follows given only several “apparently obvious axioms” that “all true sentences have the same signification (as well as all false ones).” Stephen Neale has written that this argument, and others by Church, Davidson, and Quine to similar conclusions, are of considerable philosophical interest. Graham Oppy, responding to this opinion, says they are of minimal interest. Falling between these is my opinion that implications of these arguments for propositions and facts are of moderate philosophical interest, and that these arguments provide occasions for reflection of possible interest on fine lines of several theories of definite descriptions and class–abstractions.John Howard Sobel2008-11-272008-11-27143Paraconsistent Vagueness: Why Not?
http://ojs.victoria.ac.nz/ajl/article/view/1798
The idea that the phenomenon of vagueness might be modelled by a paraconsistent logic has been little discussed in contemporary work on vagueness, just as the idea that paraconsistent logics might be fruitfully applied to the phenomenon of vagueness has been little discussed in contemporary work on paraconsistency. This is prima facie surprising given that the earliest formalisations of paraconsistent logics presented in Jáskowski and Halldén were presented as logics of vagueness. One possible explanation for this is that, despite initial advocacy by pioneers of paraconsistency, the prospects for a paraconsistent account of vagueness are so poor as to warrant little further consideration. In this paper we look at the reasons that might be offered in defence of this negative claim. As we shall show, they are far from compelling. Paraconsistent accounts of vagueness deserve further attention.Dominic HydeMark Colyvan2008-11-262008-11-26143Reply to Beall and Priest
http://ojs.victoria.ac.nz/ajl/article/view/1797
In my “Deep Inconsistency” (Australasian Journal of Philosophy, 2002), I compared my meaning-inconsistency view on the liar with Graham Priest’s dialetheist view, using my view to help cast doubt on Priest’s arguments for his view. Jc Beall and Priest have recently published a reply to my article (Australasian Journal of Logic, 2007). I here respond to their criticisms. In addition, I compare the meaning–inconsistency view with Anil Gupta and Nuel Belnap’s revision theory of truth, and discuss how best to deal with the strengthened liar.Matti Eklund2008-11-072008-11-07143Mathematical and Physical Continuity
http://ojs.victoria.ac.nz/ajl/article/view/1796
There is general agreement in mathematics about what continuity is. In this paper we examine how well the mathematical definition lines up with common sense notions. We use a recent paper by Hud Hudson as a point of departure. Hudson argues that two objects moving continuously can coincide for all but the last moment of their histories and yet be separated in space at the end of this last moment. It turns out that Hudson’s construction does not deliver mathematically continuous motion, but the natural question then is whether there is any merit in the alternative definition of continuity that he implicitly invokes. Mark ColyvanKenny Easwaran2008-09-162008-09-16143Review: Eckart Menzler-Trott’s — Logic’s Lost Genius: The Life of Gerhard Gentzen
http://ojs.victoria.ac.nz/ajl/article/view/1795
Review of Eckart Menzler-Trott’s book, Logic’s Lost Genius: The Life of Gerhard GentzenJohn N. Crossley2008-09-152008-09-15143Categorical Abstract Algebraic Logic: Equivalential π-Institutions
http://ojs.victoria.ac.nz/ajl/article/view/1790
The theory of equivalential deductive systems, as introduced by Prucnal and Wrónski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-institutions. More precisely, the notion of an N-equivalence system for a given π-institution is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-institution I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-institutions, as they relate to the existence of N-equivalence systems.George Voutsadakis2008-08-042008-08-04143Bayesians sometimes cannot ignore even very implausible theories (even ones that have not yet been thought of)
http://ojs.victoria.ac.nz/ajl/article/view/1791
In applying Bayes’s theorem to the history of science, Bayesians sometimes assume – often without argument – that they can safely ignore very implausible theories. This assumption is false, both in that it can seriously distort the history of science as well as the mathematics and the applicability of Bayes’s theorem. There are intuitively very plausible counter-examples. In fact, one can ignore very implausible or unknown theories only if at least one of two conditions is satisfied: (i) one is certain that there are no unknown theories which explain the phenomenon in question, or (ii) the likelihood of at least one of the known theories used in the calculation of the posterior is reasonably large. Often in the history of science, a very surprising phenomenon is observed, and neither of these criteria is satisfied.Branden FitelsonNeil Thomason2008-08-042008-08-04143Church-Rosser property and intersection types
http://ojs.victoria.ac.nz/ajl/article/view/1792
We give a proof via reducibility of the Church-Rosser property for the system D of λ-calculus with intersection types. As a consequence we can get the confluence property for developments directly, without making use of the strong normalization property for developments, by using only the typability in D and a suitable embedding of developments in this system. As an application we get a proof of the Church-Rosser theorem for the untyped λ-calculus.George KoletsosGeorge Stavrinos2008-08-042008-08-04143A Rejection System for the First-Degree Formulae of some Relevant Logics
http://ojs.victoria.ac.nz/ajl/article/view/1793
The standard Hilbert-style of axiomatic system yields the assertion of axioms and, via the use of rules, the assertion of theorems. However, there has been little work done on the corresponding axiomatic rejection of non-theorems. Such Hilbert-style rejection would be achieved by the inclusion of certain rejection-axioms (r-axioms) and, by use of rejection-rules (r-rules), the establishment of rejection-theorems (r-theorems). We will call such a proof a rejection-proof (r-proof). The ideal to aim for would be for the theorems and r-theorems to bemutually exclusive and exhaustive. That is, if a formula A is a theorem then it is not an r-theorem, and if A is a non-theorem then it is an r-theorem. In this paper, I present a rejecion system for the first-degree formulae of a large number of relevant logics.Ross T. Brady2008-08-042008-08-04143Modal Formulas True at Some Point in Every Model
http://ojs.victoria.ac.nz/ajl/article/view/1794
In a paper on the logical work of the Jains, Graham Priest considers a consequence relation, semantically characterized, which has a natural analogue in modal logic. Here we give a syntactic/axiomatic description of the modal formulas which are consequences of the empty set by this relation, which is to say: those formulas which are, for every model, true at some point in that model.Lloyd Humberstone2008-08-042008-08-04143Forcing with Non-wellfounded Models
http://ojs.victoria.ac.nz/ajl/article/view/1784
We develop the machinery for performing forcing over an arbitrary (possibly non-wellfounded) model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is (countable) transitive. However, for establishing properties of a given (possibly non-wellfounded) model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling the standard steps of presentation found in [19] and [14].Paul Corazza2007-11-302007-11-3014360% Proof Lakatos, Proof, and Paraconsistency
http://ojs.victoria.ac.nz/ajl/article/view/1789
Imre Lakatos’ Proofs and Refutations is a book well known to those who work in the philosophy of mathematics, though it is perhaps not widely referred to. Its general thrust is out of tenor with the foundationalist perspective that has dominated work in the philosophy of mathematics since the early years of the 20th century. It seems to us, though, that the book contains striking insights into the nature of proof, and the purpose of this paper is to explore and apply some of these.Graham PriestNeil Thomason2007-11-282007-11-28143An atomic theory with no prime models
http://ojs.victoria.ac.nz/ajl/article/view/1788
We construct an atomic uncountable theory with no prime models. This contrasts with the countable case.Tarek Sayed Ahmed2007-11-192007-11-19143Not so deep inconsistency: a reply to Eklund
http://ojs.victoria.ac.nz/ajl/article/view/1787
In his “Deep Inconsistency?” Eklund attacks arguments to the effect that some contradictions are true, and especially those based on the liar paradox, to be found in Priest’ In Contradiction. The point of this paper is to evaluate his case. JC BeallGraham Priest2007-11-112007-11-11143Four Variables Suffice
http://ojs.victoria.ac.nz/ajl/article/view/1786
What I wish to propose in the present paper is a new form of “career induction” for ambitious young logicians. The basic problem is this: if we look at the n-variable fragments of relevant propositional logics, at what point does undecidability begin? Focus, to be definite, on the logic R. John Slaney showed that the 0-variable fragment of R (where we allow the sentential con- stants t and f) contains exactly 3088 non-equivalent propositions, and so is clearly decidable. In the opposite direction, I claimed in my paper of 1984 that the five variable fragment of R is undecidable. The proof given there was sketchy (to put the matter charitably), and a close examination reveals that although the result claimed is true, the proof given is incorrect. In the present paper, I give a detailed and (I hope) correct proof that the four variable fragments of the principal relevant logics are undecidable. This leaves open the question of the decidability of the n-variable fragments for n = 1, 2, 3. At what point does undecidability set in?Alasdair Urquhart2007-11-102007-11-10143Reduction in first-order logic compared with reduction in implicational logic
http://ojs.victoria.ac.nz/ajl/article/view/1785
In this paper we discuss strong normalization for natural deduction in the →∀-fragment of first-order logic. The method of collapsing types is used to transfer the result (concerning strong normalization) from implicational logic to first-order logic. The result is improved by a complement, which states that the length of any reduction sequence of derivation term r in first-order logic is equal to the length of the corresponding reduction sequence of its collapse term rc in implicational logic.Tigran M. Galoyan2007-11-082007-11-08143The McKinsey–Lemmon logic is barely canonical
http://ojs.victoria.ac.nz/ajl/article/view/1783
We study a canonical modal logic introduced by Lemmon, and axiomatised by an infinite sequence of axioms generalising McKinsey’s formula. We prove that the class of all frames for this logic is not closed under elementary equivalence, and so is non-elementary. We also show that any axiomatisation of the logic involves infinitely many non-canonical formulas.Robert GoldblattIan Hodkinson2007-11-062007-11-06143Expressive Three-valued Truth Functions
http://ojs.victoria.ac.nz/ajl/article/view/1782
The expressive truth functions of two-valued logic have all been characterized, as have the expressive unary truth functions of finitely-many-valued logic. This paper introduces some techniques for identifying expressive functions in three-valued logics.Stephen Pollard2006-05-292006-05-29143An Analysis of Inconsistent and Incomplete Necker Cubes
http://ojs.victoria.ac.nz/ajl/article/view/1781
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.Chris Mortensen2006-03-302006-03-30143Natural Derivations for Priest, An Introduction to Non-Classical Logic
http://ojs.victoria.ac.nz/ajl/article/view/1779
This document collects natural derivation systems for logics described in Priest, An Introduction to Non-Classical Logic [4]. It provides an alternative or supplement to the semantic tableaux of his text. Except that some chapters are collapsed, there are sections for each chapter in Priest, with an additional, final section on quantified modal logic. In each case, (i) the language is briefly described and key semantic definitions stated, (ii) the derivation system is presented with a few examples given, and (iii) soundness and completeness are proved. There should be enough detail to make the parts accessible to students would work through parallel sections of Priest.Tony Roy2006-03-202006-03-20143Inexpressiveness of First-Order Fragments
http://ojs.victoria.ac.nz/ajl/article/view/1777
It is well-known that first-order logic is semi-decidable. Therefore, first-order logic is less than ideal for computational purposes (computer science, knowledge engineering). Certain fragments of first-order logic are of interest because they are decidable. But decidability is gained at the cost of expressiveness. The objective of this paper is to investigate inexpressiveness of fragments that have received much attention.William C. Purdy2006-01-252006-01-25143Some non-standard interpretations of the axiomatic basis of Leśniewski’s Ontology
http://ojs.victoria.ac.nz/ajl/article/view/1778
We propose an intuitive understanding of the statement: ‘an axiom (or: an axiomatic basis) determines the meaning of the only specific constant occurring in it.’ We introduce some basic semantics for functors of the category s/n,n of Lesniewski’s Ontology. Using these results we prove that the popular claim that the axioms of Ontology determine the meaning of the primitive constants is false.Rafał Urbaniak2006-01-252006-01-25143Logic of Violations: A Gentzen System for Reasoningwith Contrary-To-Duty Obligations
http://ojs.victoria.ac.nz/ajl/article/view/1780
In this paper we present a Gentzen system for reasoning with contrary-to-duty obligations. The intuition behind the system is that a contrary-to-duty is a special kind of normative exception. The logical machinery to formalise this idea is taken from substructural logics and it is based on the definition of a new non-classical connective capturing the notion of reparational obligation. Then the system is tested against well-known contrary-to-duty paradoxes.Guido GovernatoriAntonino Rotolo2006-01-252006-01-25143From Paradox to Judgment: towards a metaphysics of expression
http://ojs.victoria.ac.nz/ajl/article/view/1775
The Liar sentence is a singularly important piece of philosophical evidence. It is an instrument for investigating the metaphysics of expressing truths and falsehoods. And an instrument too for investigating the varieties of conflict that can give rise to paradox. It shall serve as perhaps the most important clue to the shape of human judgment, as well as to the nature of the dependence of judgment upon language use.Mariam Thalos2005-10-062005-10-06143Playing cards with Hintikka An introduction to dynamic epistemic logic
http://ojs.victoria.ac.nz/ajl/article/view/1776
This contribution is a gentle introduction to so-called dynamic epistemic logics, that can describe how agents change their knowledge and beliefs. We start with a concise introduction to epistemic logic, through the example of one, two and finally three players holding cards; and, mainly for the purpose of motivating the dynamics, we also very summarily introduce the concepts of general and common knowledge. We then pay ample attention to the logic of public announcements, wherein agents change their knowledge as the result of public announcements. One crucial topic in that setting is that of unsuccessful updates: formulas that become false when announced. The Moore-sentences that were already extensively discussed at the conception of epistemic logic in Hintikka’s ‘Knowledge and Belief ’ (1962) give rise to such unsuccessful updates. After that, we present a few examples of more complex epistemic updates.H. P. DitmarschW. Van Der HoekB. P. Kooi2005-10-062005-10-06143Review: Frank Markham Brown’s Boolean Reasoning: The Logic of Boolean Equations
http://ojs.victoria.ac.nz/ajl/article/view/1774
Brown, Frank Markham: Boolean Reasoning: The Logic of Boolean Equations. Second edition, New York: Dover, 2003; i–xii, 291 pp. USD$16.95. ISBN: 0486427854.Kari Saukkonen2005-08-122005-08-12143Review: Warren Goldfarb’s Deductive Logic
http://ojs.victoria.ac.nz/ajl/article/view/1773
Warren Goldfarb, Deductive Logic, Hackett Publishing Company, 2003. ISBN: 0872206602.Gillian Russell2005-07-112005-07-11143Justification of Argumentation Schemes
http://ojs.victoria.ac.nz/ajl/article/view/1769
Argumentation schemes are forms of argument that capture stereotypical patterns of human reasoning, especially defeasible ones like argument from expert opinion, that have proved troublesome to view deductively or inductively. Much practical work has already been done on argumentation schemes, proving their worth in A1 [19], but more precise investigations are needed to formalize their structures. The problem posed in this paper is what form justification of a given scheme, as having a certain precise structure of inference, should take. It is argued that defeasible argumentation schemes require both a systematic and a pragmatic justification, of a kind that can only be provided by the case study method of collecting key examples of arguments of the types traditionally classified as fallacies, and subjecting them to comparative examination and analysis. By this method, postulated structures for schemes can be formulated as hypotheses to solve three kinds of problems: (1) how to classify such arguments into different types, (2) how to identify their premises and conclusions, and (3) how to formulate the critical questions used to evaluate each type of argument.Douglas Walton2005-07-082005-07-08143Basic Relevant Theories for Combinators at Levels I and II
http://ojs.victoria.ac.nz/ajl/article/view/1770
The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of ALL B∧T theories provides a good model for the combinators CL at Level-I, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at Level-I, because the set of all B+T theories is NOT a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to PRIME B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at Level-II, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at Level-II also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to Level-II only in part. Specifically, only some of the basic combinator laws are proved here. We accordingly leave some work for the reader.Koushik PalRobert K. Meyer2005-07-082005-07-08143Tonk Strikes Back∗
http://ojs.victoria.ac.nz/ajl/article/view/1771
What is a logical constant? In which terms should we characterize the meaning of logical words like “and”, “or”, “implies”? An attractive answer is: in terms of their inferential roles, i.e. in terms of the role they play in building inferences. More precisely, we favor an approach, going back to Dosen and Sambin, in which the inferential role of a logical constant is captured by a double line rule which introduces it as reflecting structural links (for example, multiplicative conjunction reflects comma on the right of the turnstyle). Rule-based characterizations of logical constants are subject to the well known objection of Prior’s fake connective, tonk. We show that some double line rules also give rise to such pseudo logical constants. But then, we are able to find a property of a double line rules which guarantee that it defines a genuine logical constant. Thus we provide an alternative answer to Belnap’s requirement of conservatity in terms of a local requirement on double line rules. Denis BonnayBenjamin Simmenauer2005-07-082005-07-08143Constant Domain Quantified Modal Logics Without Boolean Negation
http://ojs.victoria.ac.nz/ajl/article/view/1772
his paper provides a sound and complete axiomatisation for constant domain modal logics without Boolean negation. This is a simpler case of the difficult problem of providing a sound and complete axiomatisation for constant-domain quantified relevant logics, which can be seen as a kind of modal logic with a two-place modal operator, the relevant conditional. The completeness proof is adapted from a proof for classical modal predicate logic (I follow James Garson’s 1984 presentation of the completeness proof quite closely), but with an important twist, to do with the absence of Boolean negation. Greg Restall2005-07-082005-07-08143Limiting Cases for Spectrum Closure Results
http://ojs.victoria.ac.nz/ajl/article/view/1768
The spectrum of a first-order sentence is the set of cardinalities of its finite models. Given a spectrum S and a function f, it is not always clear whether or not the image of S under f is also a spectrum. In this paper, we consider questions of this form for functions that increase very quickly and for functions that increase very slowly. Roughly speaking, we prove that the class of all spectra is closed under functions that increase arbitrarily quickly, but it is not closed under some natural slowly increasing functions.Aaron Hunter2004-10-262004-10-26143The Classical and Maximin Versions of the Two-Envelope Paradox
http://ojs.victoria.ac.nz/ajl/article/view/1765
The Two-Envelope Paradox is classically presented as a problem in decision theory that turns on the use of probabilities in calculating expected utilities. I formulate a Maximin Version of the paradox, one that is decision-theoretic but omits considerations of probability. I investigate the source of the error in this new argument, and apply the insights thereby gained to the analysis of the classical version.Bruce Langtry2004-08-022004-08-02143A Poor Concept Script
http://ojs.victoria.ac.nz/ajl/article/view/1766
The formal structure of Frege’s ‘concept script’ has been widely adopted in logic text books since his time, even though its rather elaborate symbols have been abandoned for more convenient ones. But there are major difficulties with its formalisation of pronouns, predicates, and propositions, which infect the whole of the tradition which has followed Frege. It is shown first in this paper that these difficulties are what has led to many of the most notable paradoxes associated with this tradition; the paper then goes on to indicate the lines on which formal logic—and also the lambda calculus and set theory—needs to be restructured, to remove the difficulties. Throughout the study of what have come to be known as first-, second-, and higher-order languages, what has been primarily overlooked is that these languages are abstractions. Many well known paradoxes, we shall see, arose because of the elementary level of simplification which has been involved in the abstract languages studied. Straightforward resolutions of the paradoxes immediately appear merely through attention to languages of greater sophistication, notably natural language, of course. The basic problem has been exclusive attention to a theory in place of what it is a theory of, leading to a focus on mathematical manipulation, which ‘brackets off ’ any natural language reading.Hartley Slater2004-08-022004-08-02143Modal Predicates
http://ojs.victoria.ac.nz/ajl/article/view/1767
Despite the wide acceptance of standard modal logic, there has always been a temptation to think that ordinary modal discourse may be correctly analyzed and adequately represented in terms of predicates rather than in terms of operators. The aim of the formal model outlined in this paper is to capture what I take to be the only plausible sense in which ‘possible’ and ‘necessary’ can be treated as predicates. The model is built by enriching the language of standard modal logic with a quantificational apparatus that is “substitutional” rather than “objectual”, and by obtaining from the language so enriched another language in which constants for such predicates apply to singular terms that stand for propositions.Andrea Iacona2004-08-022004-08-02143Possibility Semantics for Intuitionistic Logic
http://ojs.victoria.ac.nz/ajl/article/view/1764
The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.M. J. Cresswell2004-04-302004-04-30143Propositional Identity and Logical Necessity
http://ojs.victoria.ac.nz/ajl/article/view/1763
In two early papers, Max Cresswell constructed two formal logics of propositional identity, PCR and FCR, which he observed to be respectively deductively equivalent to modal logics S4 and S5. Cresswell argued informally that these equivalences respectively “give … evidence” for the correctness of S4 and S5 as logics of broadly logical necessity. In this paper, I describe weaker propositional identity logics than PCR that accommodate core intuitions about identity and I argue that Cresswell’s informal arguments do not firmly and without epistemic circularity justify accepting S4 or S5. I also describe how to formulate standard modal logics (K, S2, and their extensions) with strict equivalence as the only modal primitive.David B. Martens2004-03-122004-03-12143Editorial
http://ojs.victoria.ac.nz/ajl/article/view/1759
The editorial explains why we have decided to launch the Journal. Greg Restall2003-07-012003-07-01143Semantic Decision Procedures for Some Relevant Logics
http://ojs.victoria.ac.nz/ajl/article/view/1760
This paper proves decidability of a range of weak relevant logics using decision procedures based on the Routley-Meyer semantics. Logics are categorized as F-logics, for those proved decidable using a filtration method, and U-logics, for those proved decidable using a direct (unfiltered) method. Both of these methods are set out as reductio methods, in the style of Hughes and Cresswell. We also examine some extensions of the U-logics where the method fails and infinite sequences of worlds can be generated.Ross Brady2003-07-012003-07-01143Three Schools of Paraconsistency
http://ojs.victoria.ac.nz/ajl/article/view/1761
A logic is said to be paraconsistent if it does not allow everything to follow from contradictory premises. There are several approaches to paraconsistency. This paper is concerned with several philosophical positions on paraconsistency. In particular, it concerns three ‘schools’ of paraconsistency: Australian, Belgian and Brazilian. The Belgian and Brazilian schools have raised some objections to the dialetheism of the Australian school. I argue that the Australian school of paraconsistency need not be closed down on the basis of the Belgian and Brazilian schools’ objections. In the appendix of the paper, I also argue that the Brazilian school’s view of logic is not coherent.Koji Tanaka2003-07-012003-07-01143