# consequently.org/writing

## By Greg Restall

To listen to an audio podcast, mouse over the title and click Play. Open iTunes to download and subscribe to podcasts.

#### Description

Publications from Greg Restall. Philosopher, at the University of Melbourne.

Name | Description | Released | Price | ||
---|---|---|---|---|---|

1 | A Cut-Free Sequent System for Two-Dimensional Modal Logic, and why it matters | The two-dimensional modal logic of Davies and Humberstone is an important aid to our understanding the relationship between actuality, necessity and a priori knowability. I show how a cut-free hypersequent calculus for 2d modal logic not only captures the logic precisely, but may be used to address issues in the epistemology and metaphysics of our modal concepts. I will explain how use of our concepts motivates the inference rules of the sequent calculus, and then show that the completeness of the calculus for Davies–Humberstone models explains why those concepts have the structure described by those models. The result is yet another application of the completeness theorem. | 21 11 2010 | Free | View In iTunes |

2 | Always More | A possible world is a point in logical space. It plays a dual role with respect to propositions. (1) A possible world determines the truth value of every proposition. For each world w and proposition p, either at w, p is true, or at w, p is not true. (2) Each set of possible worlds determines a proposition. If S, a subset of W is a set of worlds, there is a proposition p true at exactly the worlds in S. In this paper, I construct a logic, extending classical logic with a single unary operator, which has no complete Boolean algebras as models. If the family of propositions we are talking about in (1) and (2) has the kind of structure described in that logic, then (1) and (2) cannot jointly hold. I then explain what this might mean for theories of propositions and possible worlds. | 15 11 2010 | Free | View In iTunes |

3 | Barriers to Consequence | In this paper we show how the formal counterexamples to Hume’s Law (to the effect that you cannot derive a properly moral statement from properly descriptive statements) are of a piece with formal counterexample to other, plausible “inferential barrier theses”. We use this fact to motivate a uniform treatment of barrier theses which is immune from formal counterexample. We provide a uniform semantic representation of barrier theses which has applications in the case of what we call “Russell’s Law” (you can’t derive a universal from particulars) and “Hume’s Second Law” (you can’t derive a statement about the future from statements about the past). We then finally apply these results to formal treatments of deontic logic to show how to avoid formal counterexamples to Hume’s Law in a plausible and motivated manner. | 15 7 2010 | Free | View In iTunes |

4 | On t and u, and what they can do | This paper shows that once we have propositional constants t (the conjunction of all truths) and u (the disjunction of all untruths), paradox ensues, provided you have a conditional in the language strong enough to give you modus ponens. This is an issue for views like those of Jc Beall, Ross Brady, Hartry Field and Graham Priest. | 16 2 2010 | Free | View In iTunes |

5 | What are we to accept, and what are we to reject, when saving truth from paradox? | In this article, I praise Hartry Field’s fine book Saving Truth From Paradox (Oxford University Press, 2008). I also show that his account of properties is threatened by a paradox, and I explain how we can only avoid this paradox by coming to a clearer understanding the connections between accepting and rejecting (or assertion and denial) and the identity conditions for properties. | 18 1 2010 | Free | View In iTunes |

6 | Models for Substructural Arithmetics | This paper explores models for arithmetics 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. This paper is dedicated to Professor Robert K. Meyer. | 31 12 2009 | Free | View In iTunes |

7 | On Permutation in Simplified Semantics | This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Journal of Philosophical Logic 1993) concerning the modelling conditions for the axioms of assertion A → ((A → B) → B) and permutation (A → (B → C)) → (B → (A → C)). We show that the modelling conditions for assertion and permutation proposed in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. In this note, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose a correction. | 2 11 2009 | Free | View In iTunes |

8 | Anti-Realist Classical Logic and Realist Mathematics | I sketch an application of a semantically anti-realist understanding of the classical sequent calculus to the topic of mathematics. The result is a semantically anti-realist defence of mathematical realism. In the paper, I develop the view and compare it to orthodox positions in the philosophy of mathematics. | 12 10 2009 | Free | View In iTunes |

9 | Assertion, Denial and Non-Classical Theories | In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. Examples discussed include theories of numbers, classes and truth. In the latter two cases, the bitheoretical perspective brings to light some heretofore unconsidered puzzles for friends of naïve theories of classes and truth. | 31 7 2009 | Free | View In iTunes |

10 | A Priori Truths | Philosophers love a priori knowledge: we delight in truths that can be known from the comfort of our armchairs, without the need to venture out in the world for cofirmation. This is due not to laziness, but to two different considerations. First, it seems that many philosophical issues aren’t settled by our experience of the world – the nature of morality; the way concepts pick out objects; the structure of our experience of the world in which we find ourselves – these issues seem to be decided not on the basis of our experience, but in some manner by things prior to (or independently of) that experience. Second, even when we are deeply interested in how our experience lends credence to our claims about the world, the matter remains of the remainder: we learn more about how experience contributes to knowledge when we see what knowledge is available independent of that experience. In this essay we will look at the topic of what can be known a priori. | 16 7 2009 | Free | View In iTunes |

11 | Truth Values and Proof Theory | In this paper I present an account of truth values for classical logic, intuitionistic logic, and the modal logic S5, in which truth values are not a fundamental category from which the logic is defined, but rather, feature as an idealisation of more fundamental logical features arising out of the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical consequence. | 1 5 2009 | Free | View In iTunes |

12 | Using Peer Instruction to Teach Philosophy, Logic and Critical Thinking | We explain how Eric Mazur’s technique of Peer Instruction may be used to teach philosophy, logic and critical thinking — to good effect. | 19 4 2009 | Free | View In iTunes |

13 | Models for Liars in Bradwardine's Theory of Truth | Stephen Read’s work on Bradwardine’s theory of truth is some of the most exciting work on truth and insolubilia in recent years. In this paper, I give models for Read’s formulation of Bradwardine’s theory of truth, and I examine the behaviour of liar sentences in those models. I conclude by examining Bradwardine’s argument to the effect that if something signifies itself to be untrue then it signifies itself to be true as well. We will see that there are models in which this conclusion fails. This should help us elucidate the hidden assumptions required to underpin Bradwardine’s argument, and to make explicit the content of Bradwardine’s theory of truth. | 3 12 2008 | Free | View In iTunes |

14 | Truth Tellers in Bradwardine's Theory of Truth | Stephen Read’s work on Bradwardine’s theory of truth is some of the most exciting work on truth and insolubilia in recent years. Read brings together modern tools of formal logic and Bradwardine’s theory of signification to show that medieval distinctions can give great insight into the behaviour of semantic concepts such as truth. In a number of papers, I have developed a model theory for Bradwardine’s account of truth. This model theory has distinctive features: it serves up models in which every declarative object (any object signifying anything) signifies its own truth. This leads to a puzzle: there are good arguments to the effect that if anything is a truth-teller, it is false. This is a puzzle. What distinguishes paradoxical truth-tellers from benign truth tellers? It is my task in this paper to explain this distinction, and to clarify the behaviour of truth-tellers, given Bradwardine’s account of signification. | 3 10 2008 | Free | View In iTunes |

15 | Truthmakers, Entailment and Necessity 2008 | I update “Truthmakers, Entailment and Necessity” with a response to Stephen Read’s wonderful Mind 2000 argument to the effect that I got the disjunction thesis wrong. I think I didn’t get it wrong, but figuring out why it’s OK to hold that a truthmaker makes a disjunction true iff it makes a disjunct true is not as straightforward as I first thought. Pluralism about truthmaking makes an entrance. | 3 10 2008 | Free | View In iTunes |

16 | Proof Theory and Philosophy | This is my next book-length writing project. I am writing a book which aims to do these things: Be a useable textbook in philosophical logic, accessible to someone who’s done only an intro course in logic, covering at least some model theory and proof theory of propositional logic, and maybe predicate logic. Be a user-friendly, pedagogically useful and philosophically motivated presentation of cut-elimination, normalisation and conservative extension, both (a) why they’re important to meaning theory and (b) how to actually prove them. (I don’t think there are any books like this available, but I’d be happy to be shown wrong.) Present the duality between model theory and proof theory in a philosophically illuminating fashion. Teach both formal philosophical logic in such a way that is not doctrinaire or logically partisan. That is, I will not argue that classical logic, or that intuitionistic logic, or that My Favourite Logic is the One True Logic. (Of course, hearing me say this is not a surprise.) I am (at this stage, at least) planning to make the book available for download as well as published by an academic publisher. The first couple of chapters are now available: pdf with hyperlinks, paper-saving 2up pdf. Comments are most welcome at the wiki. | 3 10 2008 | Free | View In iTunes |

17 | Modal Models for Bradwardine's Theory of Truth | Stephen Read has recently discussed Thomas Bradwardine’s theory of truth, and defended it as an appropriate way to treat paradoxes such as the liar. In this paper, I discuss Read’s formalisation of Bradwardine’s theory of truth, and provide a class of models for this theory. The models facilitate comparison of Bradwardine’s theory with contemporary theories of truth. | 15 9 2008 | Free | View In iTunes |

18 | Molinism and the Thin Red Line | Molinism is an attempt to do equal justice to divine foreknowledge and human freedom. For Molinists, human freedom fits in this universe for the future is open or unsettled. However, God’s middle knowledge – God’s contingent knowledge of what agents would freely do in this or that circumstance – underwrites God’s omniscience in the midst of this openness. In this paper I rehearse Nuel Belnap and Mitchell Green’s argument in ‘Indeterminism and the Thin Red Line’ against the reality of a distinguished single future in the context of branching time, and show that it applies applies equally against Molinism + branching time. In the process, we show how contemporary work in the logic of temporal notions in the context of branching time (specifically, Prior–Thomason semantics) can illuminate discussions in the metaphysics of freedom and divine knowledge. | 7 7 2008 | Free | View In iTunes |

19 | Logic in Australasia | This is an idiosyncratic history of philosophical logic in Australia and New Zealand, highlighting two significant points of research in Australasian philosophical logic: modal logic and relevant/paraconsistent logic. | 3 6 2008 | Free | View In iTunes |

20 | Curry's Revenge: the costs of non-classical solutions to the paradoxes of self-reference | I point out that non-classical “solutions” the paradoxes of self-reference are non-particularly easy to give. Curry’s paradox is very very hard to avoid, if you wish to give a semantically cohrerent picture. | 3 3 2008 | Free | View In iTunes |

21 | Assertion and Denial, Commitment and Entitlement, and Incompatibility (and some consequence) | In this short paper, I compare and contrast the kind of symmetricalist treatment of negation favoured in different ways by Huw Price (in “Why ‘Not’?”) and by me (in “Multiple Conclusions”) with Robert Brandom’s analysis of scorekeeping in terms of commitment, entitlement and incompatibility. Both kinds of account provide a way to distinguish the inferential significance of ”A” and ”A is warranted” in terms of a subtler analysis of our practices: on the one hand, we assert as well as deny; on the other, by distingushing downstream commitments from upstream entitlements and the incompatibility definable in terms of these. In this note I will examine the connections between these different approaches. | 2 3 2008 | Free | View In iTunes |

22 | Proof Theory and Meaning: the context of deducibility | I examine Belnap’s two criteria of existence and uniqueness for evaluating putative definitions of logical concepts in inference rules, by determining how they apply in four different examples: conjunction, the universal quantifier, the indefinite choice operator and the necessity in the modal logic S5. This illustrates the ways that definitions may be evaluated relative to a background theory of consequence, and the ways that different accounts of consequence provide us with different resources for making definitions. | 3 10 2007 | Free | View In iTunes |

23 | Invention is the Mother of Necessity: modal logic, modal semantics and modal metaphysics | Modal logic is a well-established field, and the possible worlds semantics of modal logics has proved invaluable to our understanding of the logical features of the modal concepts such as possibility and necessity. However, the significance of possible worlds models for a genuine theory of meaning–let alone for metaphysics–is less clear. In this paper I shall explain how and why the use of the concepts of necessity and possibility could arise (and why they have the logical behaviour charted out by standard modal logics) without either taking the notions of necessity or possibility as primitive, and without starting with possible worlds. Once we give an account of modal logic we can then go on to give an account of possible worlds, and explain why possible worlds semantics is a natural fit for modal logic without being the source of modal concepts. This paper is an early draft, and comments are most welcome. | 3 10 2007 | Free | View In iTunes |

24 | Decorated Linear Order Types and the Theory of Concatenation | We study the interpretation of Grzegorczyk’s Theory of Concatenation TC in structures of decorated linear order types satisfying Grzegorczyk’s axioms. We show that TC is incomplete for this interpretation. What is more, the first order theory validated by this interpretation interprets arithmetical truth. We also show that every extension of TC has a model that is not isomorphic to a structure of decorated order types. We provide a positive result, to wit, a construction that builds structures of decorated order types from models of a suitable concatenation theory. This construction has the property that if there is a representation of a certain kind, then the construction provides a representation of that kind. | 3 10 2007 | Free | View In iTunes |

25 | Proof Theory and Meaning: on second order logic | Second order quantification is puzzling. The second order quantifiers have natural and compelling inference rules, and they also have natural models. These do not match: the inference rules are sound for the models, but not complete, so either the proof rules are too weak or the models are too strong. Some, such as Quine, take this to be no real problem, since they take “second order logic” to be a misnomer. It is not logic but set theory in sheep’s clothing, so one would not expect to have a sound and complete axiomatisation of the theory. I think that this judgement is incorrect, and in this paper I attempt to explain why. I show how on Nuel Belnap’s criterion for logicality, second order quantification can count as properly logic so-called, since the quantifiers are properly defined by their inference rules, and the addition of second order quantification to a basic language is conservative. With this notion of logicality in hand I then diagnose the incompleteness of the proof theory of second order logic in what seems to be a novel way. | 3 3 2007 | Free | View In iTunes |

26 | Symbolic Logic | 1007 words on symbolic logic – concentrating on the history of 20th Century logic, aimed at an audience of social scientists | 3 3 2007 | Free | View In iTunes |

27 | Assertion, Denial, Accepting, Rejecting, Symmetry and all that | Proponents of a dialethic or “truth-value glut” response to the paradoxes of self-reference argue that “truth-value gap” analyses of the paradoxes fall foul of the extended liar paradox: “this sentence is not true.” If we pay attention to the role of assertion and denial and the behaviour of negation in both “gap” and “glut” analyses, we see that the situation with these approaches has a pleasing symmetry: gap approaches take some denials to not be expressible by negation, and glut approaches take some negations to not express denials. But in the light of this symmetry, considerations against a gap view point to parallel considerations against a glut view. Those who find some reason to prefer one view over another (and this is almost everyone) must find some reason to break this symmetry. This short paper was written for presentation at AAP2004. It’s still very much a polemical draft, and I’m not sure where it’s going. | 5 5 2006 | Free | View In iTunes |

28 | Comparing Modal Sequent Systems | Labelled systems and display systems are very different generalisations of the pure sequent calculus, giving what appear to be quite different accounts of modal deduction. Display sequents are equipped with a rich “structural” vocabulary, allowing us to directly express modal facts in the punctuation of a sequent. Labelled sequents encode into the proof theory the structure of a Kripke model. Formulas are equipped with labels (effectively replacing formulas by predicates of worlds) and the accessibility relation from the model makes its appearance in the syntax of the sequent. In this paper, I show how derivations in display logic may be converted into derivations in a labelled sequent system, lending some support to the claim that a labelled sequent system need be no more expressive than a display system. Using this result, we may we may simplify a labelled proof theory further, so that the labels disappear and we are left with a different, structural sequent system for modal logics. | 3 3 2006 | Free | View In iTunes |

29 | Questions and Answers on Formal Philosophy | My entry in a book of interviews of philosophers who work on the more formal side of the discipline. It gives an account of how I got into this area, what I think logic is good for – when it comes to philosophy – and where I think we should head. | 3 3 2006 | Free | View In iTunes |

30 | Logics, Situations and Channels | The notion of that information is relative to a context is important in many different ways. The idea that the context is small – that is, not necessarily a consistent and complete possible world – plays a role not only in situation theory, but it is also an enlightening perspective from which to view other areas, such as modal logics, relevant logics, categorial grammar and much more. In this article we will consider these areas, and focus then on one further question: How can we account for information about one thing giving us information about something else? This is a question addressed by channel theory. We will look at channel theory and then see how the issues of information flow and conditionality play a role in each of the different domains we have examined. | 3 3 2006 | Free | View In iTunes |

31 | Constant Domain Quantified Modal Logics without Boolean Negation | The paper examines what its title says. Constant domain modal frames seem to be the natural semantics for quantified relevant logics and their cousins. Kit Fine has shown us that things are not that simple, as the natural proof theory is not complete for the natural semantics. In this paper I explore the somewhat simpler case of one-place modal operators. The natural proofs work, but there are a few surprises, such as the need to use intuitionistic implication and its dual, subtraction, in the completeness proof. This paper is dedicated to the memory of Richard Sylvan, who contributed so much to the study of the semantics of relevant logics and their neighbours. | 2 9 2005 | Free | View In iTunes |

32 | Not Every Truth Can Be Known: at least, not all at once | According to the “knowability thesis,” every truth is knowable. Fitch’s paradox refutes the knowability thesis by showing that if we are not omniscient, then not only are some truths not known, but there are some truths that are not knowable. In this paper, I propose a weakening of the knowability thesis (which I call the “conjunctive knowability thesis”) to the effect that for every truth p there is a collection of truths such that (i) each of them is knowable and (ii) their conjunction is equivalent to p. I show that the conjunctive knowability thesis avoids triviality arguments against it, and that it fares very differently depending on one other issue connecting knowledge and possibility. If some things are knowable but false, then the conjunctive knowability thesis is trivially true. On the other hand, if knowability entails truth, the conjunctive knowability thesis is coherent, but only if the logic of possibility is quite weak. Update, April 2005: I’ve added a reference (thanks to Joe Salerno) to a recent paper of Risto Hilpinen, in which he motivates a conjunctive knowability thesis on the grounds of a Peircean pragmatism. | 3 3 2005 | Free | View In iTunes |

33 | Entries "Belnap, Nuel Dinsmore Jr." and "Lambert, J. Karel" from the | Two short entries on two great philosophers of logic (and philosophical logicians) of late 20th and early 21st Century American philosophy, Nuel Belnap and Karel Lambert. | 3 3 2005 | Free | View In iTunes |

34 | Moral Fictionalism versus the rest | In this paper we introduce a distinct meta-ethical position, fictionalism about morality. We clarify and defend the position, showing that it is a way to save the “moral phenomena” while agreeing that there is no genuine objective prescriptivity to be described by moral terms. In particular, we distiguish moral fictionalism from moral quasi-realism, and we show that fictionalism many all of the virtues of quasi-realism but few of the vices. | 3 3 2005 | Free | View In iTunes |

35 | The Geometry of Non-Distributive Logics | In this paper, we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness. We show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic, and we indicate how proofs in this system may be labelled with terms exhibiting a kind of Curry-Howard isomorphism. This natural deduction system is inspired both by Shoesmith and Smiley’s multiple conclusion systems for classical logic and Girard’s proofnets for linear logic. This paper is joint work with Francesco Paoli. | 3 3 2005 | Free | View In iTunes |

36 | One Way to Face Facts | Stephen Neale, in Facing Facts takes theories of facts, truthmakers, and non-extensional connectives to be threatened by triviality in the face of powerful “slingshot” arguments. In this paper I rehearse the most powerful of these arguments, and then show that friends of facts have resources sufficient to not only resist slingshot arguments but also to be untroubled by them. If a fact theory is provided with a model, then the fact theorist can be sure that this theory is secure from triviality arguments. | 3 10 2004 | Free | View In iTunes |

37 | Laws of Non-Contradiction, Laws of the Excluded Middle and Logics | There is widespread agreement that the law of non-contradiction is an important logical principle. There is less agreement on exactly what the law amounts to. This unclarity is brought to light by the emergence of paraconsistent logics in which contradictions are tolerated (in the sense that not everything need follow from a contradiction, and that there are “worlds” in which contradictions are true) but in which the statement ~(A & ~A) (it is not the case that A and not-A) is still provable. This paper attempts to clarify the connection between different readings of the law of non-contradiction, the duality between the law of non-contradiction and the law of the excluded middle, and connections with logical consequence in general. | 20 7 2004 | Free | View In iTunes |

38 | Modelling Truthmaking | Published in a special issue of Logique et Analyse, on Truthmakers, edited by Peter Forrest and Drew Khlentzos. This expands on my paper “Truthmakers, Entailment and Necessity” by showing that the theses on truthmaking given in that paper are mutually consistent. It does this by providing a model in which each thesis is true. This model is an independently motivated model of a weak quantified relevant logic with an existence predicate. | 3 7 2003 | Free | View In iTunes |

39 | Logic | An introduction to the formal and philosophical logic, for a general introductory handbook to philosophy. I introduce the concepts of deductive and inductive validity, formal languages, proofs, models, soundness and completeness, and many other things besides. | 3 3 2003 | Free | View In iTunes |

40 | Just what Full-Blooded Platonism? | Mark Balaguer has, in Platonism and Anti-Platonism in Mathematics, given us an intriguing new brand of platonism, which he calls, plenitudinous platonism, or more colourfully, full-blooded platonism. In this paper, I argue that none of Balaguer’s attempts to characterise full-blooded platonism succeed. They are either too strong, with untoward consequences we all reject, or too weak, not providing a distinctive brand of platonism strong enough to do the work Balaguer requires of it. EXCERPT: “Just What is Full-Blooded Platonism?” | 3 3 2003 | Free | View In iTunes |

41 | Relevance Logic | A revision of the Handbook of Philosophical Logic essay on relevant logics, updating it to include work on display logic, relevant predication, Urquhart’s undecidability and complexity results, and connections with other substructural logics. | 3 3 2002 | Free | View In iTunes |

42 | Carnap’s Tolerance, Meaning and Logical Pluralism | In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap’s Principle of Tolerance from a pluralism which maintains that there are different, equally “good” logical consequence relations on the one language. I will argue that this second form of pluralism does more justice to the contemporary state of logical theory and practice than does Carnap’s more moderate pluralism. | 3 3 2002 | Free | View In iTunes |

43 | Constructive Logic, Truth and Warranted Assertibility | Shapiro and Taschek have argued that simply using intuitionistic logic and its Heyting semantics, one can show that there are no gaps in warranted assertibility. That is, given that a discourse is faithfully modelled using Heyting”s semantics for the logical constants, that if a statement S is not warrantedly assertible, then its negation ~S is. Tennant has argued for this conclusion on similar grounds. I show that these arguments fail, albeit in illuminating ways. I will show that an appeal to constructive logic does not commit one to this strong epistemological thesis, but that appeals to semantics of intuitionistic logic nonetheless do give us certain conclusions about the connections between warranted assertibility and truth. | 3 10 2001 | Free | View In iTunes |

44 | Defending Logical Pluralism | A defence of logical pluralism against a number of objections, primarily from Graham Priest in his article “Logic: One or Many” in the same volume. | 3 7 2001 | Free | View In iTunes |

45 | Logical Pluralism | This is our article on logical pluralism. We argue that the notion of logical consequence doesn’t pin down one deductive consequence relation, but rather, there are many of them. In particular, we argue that broadly classical, intuitionistic and relevant accounts of deductive logic are genuine logical consequence relations. We should not search for One True Logic, since there are Many. A bigger version of the argument (with much more detail) is found in our book of the same name. | 20 12 2000 | Free | View In iTunes |

46 | Defining Double Negation Elimination | In his paper Generalised Ortho Negation'' J. Michael Dunn mentions a claim of mine to the effect that there is no condition on `perp frames' equivalent to the holding of double negation elimination (from ~~A to infer A). That claim of mine was wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses. (Ed Mares has pointed out that there’s some overlap between this paper and his earlierA Star-Free Semantics for R” (JSL 1995), which I’d read long before writing this one. The particular modelling condition for DNE is still original to me, however.) | 3 10 2000 | Free | View In iTunes |

47 | Negation in Relevant Logics: How I stopped worrying and learned to love the Routley Star | Negation raises three thorny problems for anyone seeking to interpret relevant logics. The frame semantics for negation in relevant logics involves a point shift operator *. Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A and ~AvB to B to be invalid, because the corresponding relevant conditional A&(~AvB)→B is not a theorem. Yet we often make the inference from A and ~AvB to B, and we seem to be reasoning validly when we do so. Problem number two is explaining what is really going on here. Finally, we can add an operation which Meyer has called Boolean negation to our logic, which is evaluated in the traditional way: x makes A true if and only if x doesn’t make A true. Problem number three involves deciding which is the real negation. How can we decide between orthodox negation and the new, Boolean negation. In this paper, I present a new interpretation of the frame semantics for relevant logics which will allow us to give principled answers to each of these questions. | 3 3 1999 | Free | View In iTunes |

48 | Displaying and Deciding Substructural Logics 1: Logics with Contraposition | Many logics in the relevant family can be given a proof theory in the style of Belnap’s display logic. However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which more closely models relevant logics. In addition, we use this proof theory to show decidability for a large range of substructural logics. | 3 11 1998 | Free | View In iTunes |

49 | Linear Arithmetic Desecsed | In classical and intuitionistic arithmetics, any formula implies a true equation, and a false equation implies anything. In weaker logics fewer implications hold. In this paper we rehearse known results about the relevant arithmetic R#, and we show that in linear arithmetic LL# by contrast false equations never imply true ones. As a result, linear arithmetic is desecsed. A formula A which entails 0=0 is a secondary equation; one entailed by 0≠0 is a secondary unequation. A system of formal arithmetic is secsed if every extensional formula is either a secondary equation or a secondary unequation. We are indebted to the program MaGIC for the simple countermodel SZ7, on which 0=1 is not a secondary formula. This is a small but significant success for automated reasoning. | 3 3 1998 | Free | View In iTunes |

50 | Logical Laws | This is an introductory essay on the notion of a “Logical Law.” In it, I show that there are three important different questions one can ask about logical laws. Firstly, what it means to be a logical law. Secondly, what makes something a logical law, and thirdly, what are the logical laws. Each of these questions are answered differently by different people. I sketch the important differences in views, and point the way ahead for logical research. | 3 3 1998 | Free | View In iTunes |

51 | Extending Intuitionistic Logic with Subtraction | Ideas on the extension of intuitionistic propositional and predicate logic with a ‘subtraction’ connective, Galois connected with disjunction, dual to the implication connective, Galois connected with conjunction. Presented to an audience at Victoria University of Wellington, July 1997. I like this material, but it does not contain any ideas not accessible elsewhere, so it won’t be published anywhere other than here. | 3 4 1997 | Free | View In iTunes |

52 | Combining Possibilities and Negations | Combining non-classical (or ’sub-classical’) logics is not easy, but it is very interesting. In this paper, we combine nonclassical logics of negation and possibility (in the presence of conjunction and disjunction), and then we combine the resulting systems with intuitionistic logic. We will find that some of Marcus Kracht’s results on the undecidability of classical modal logics generalise to a non-classical setting. We will also see conditions under which intuitionistic logic can be combined with a non-intuitionistic negation without corrupting the intuitionistic fragment of the logic. | 3 3 1997 | Free | View In iTunes |

53 | Display Logic and Gaggle Theory | This paper is a revised version of a talk given at the Logic and Logical Philosophy conference in Poland in September 1995. In it, I sketch the connections between Nuel Belnap’s Display Logic and J. Michael Dunn’s Gaggle Theory. | 4 3 1996 | Free | View In iTunes |

54 | Truthmakers, Entailment and Necessity | A truthmaker for a proposition p is an object such that necessarily, if it exists, then p is true. In this paper, I show that the truthmaker thesis (that every truth has a truthmaker) and the disjunction thesis (that a truthmaker for a disjunction must be a truthmaker for one of the disjuncts) are jointly incompatible with the view that the if in the truthmaking clause is a merely material conditional. I then defend the disjunction thesis, and go on to argue that a relevant conditional will serve well in place of the material conditional in defining what it is to be a truthmaker. Conversely, I argue that a semantics involving truthmakers could shed some light on the interpretation of relevant implication. | 3 3 1996 | Free | View In iTunes |

55 | Realistic Belief Revision | In this paper we consider the implications for belief revision of weakening the logic under which belief sets are taken to be closed. A widely held view is that the usual belief revision functions are highly classical, especially in being driven by consistency. We show that, on the contrary, the standard representation theorems still hold for paraconsistent belief revision. Then we give conditions under which consistency is preserved by revisions, and we show that this modelling allows for the gradual revision of inconsistency. | 8 10 1995 | Free | View In iTunes |

56 | Arithmetic and Truth in Łukasiewicz's Infinitely Valued Logic | Peano arithmetic formulated in Łukasiewicz’s infinitely valued logic collapses into classical Peano arithmetic. However, not all additions to the language need also be classical. The way is open for the addition of a real truth predicate satisfying the T-scheme into the language. However, such an addition is not pleasing. The resulting theory is omega-inconsistent. This paper consists of the proofs and interpretations of these two results. | 3 10 1995 | Free | View In iTunes |

57 | Modalities in Substructural Logics | This paper generalises Girard’s results which embed intuitionistic logic into linear logic by showing how arbitrary substructural logics can be embedded into weaker substructural logics, using a single modality which ‘encodes’ the new structural rules. | 3 4 1995 | Free | View In iTunes |

58 | Logical Pluralism and the Preservation of Warrant | I defend logical pluralism against the charge that One True Logic is motivated by considerations of warrant preservation. On the way I attempt to clarify just a little the connections between deductive validity and epistemology. | 3 3 1995 | Free | View In iTunes |

59 | Proofnets for S5: sequents and circuits for modal logic | In this paper I introduce a sequent system for the propositional modal logic S5. Derivations of valid sequents in the system are shown to correspond to proofs in a novel natural deduction system of circuit proofs (reminiscient of proofnets in linear logic, or multiple-conclusion calculi for classical logic). The sequent derivations and proofnets are both simple extensions of sequents and proofnets for classical propositional logic, in which the new machinery—to take account of the modal vocabulary—is directly motivated in terms of the simple, universal Kripke semantics for S5. The sequent system is cut-free and the circuit proofs are normalising. | 3 3 1995 | Free | View In iTunes |

60 | Relevant Restricted Quantification | The paper reviews a number of approaches for handling restricted quantification in relevant logic, and proposes a novel one. This proceeds by introducing a novel kind of enthymematic conditional. | 3 3 1995 | Free | View In iTunes |

61 | Paraconsistent Logics! | I respond to an interesting argument of Hartley Slater to the effect that there is no such thing as paraconsistent logic. Slater argues that since paraconsistent logics involve interpreting a sentence and its negation as both true at points in a model structure, it is not really negation that is being modelled, since negation is meant to be a contradictory forming operator. I sketch how different paraconsistentists can respond to his argument, and I then defend my own response, that although contradictions are indeed never true (and cannot be true) it does not follow that a semantics ought not evaluate them as true in certain models. | 3 3 1995 | Free | View In iTunes |

62 | A Participatory Theory of the Atonement | We argue that the participatory language, used in the New Testament to describe the the efficacy of Jesus’ death on the cross, is essential for any understanding of the atonement. Purely personal or legal metaphors are incomplete and perhaps misleading on their own. They make much more sense when combined with and undergirded by, participatory metaphors. | 3 3 1995 | Free | View In iTunes |

63 | Four-Valued Semantics for Relevant Logics (and some of their rivals) | This paper gives an outline of three different approaches to the four-valued semantics for relevant logics (and other non-classical logics in their vicinity). The first approach borrows from the ‘Australian Plan’ semantics, which uses a unary operator ‘*’ for the evaluation of negation. This approach can model anything that the two-valued account can, but at the cost of relying on insights from the Australian Plan. The second approach is natural, well motivated, independent of the Australian Plan, and it provides a semantics for the contraction-free relevant logic RW. Unfortunately, its approach seems to model little else. The third approach seems to capture a wide range of formal systems, but at the time of writing, lacks a completeness proof. | 3 3 1995 | Free | View In iTunes |

64 | Minimalists about Truth can (and should) be Epistemicists, and it helps if they are revision theorists too | Minimalists about truth say that the important properties of the truth predicate are revealed in the class of T-biconditionals. Most minimalists demur from taking all of the T-biconditionals of the form “ is true if and only if p”, to be true, because to do so leads to paradox. But exactly which biconditionals turn out to be true? I take a leaf out of the epistemic account of vagueness to show how the minimalist can avoid giving a comprehensive answer to that question. I also show that this response is entailed by taking minimalism seriously, and that objections to this position may be usefully aided and abetted by Gupta and Belnap’s revision theory of truth. | 3 3 1995 | Free | View In iTunes |

65 | Nietzsche, Insight and Immorality | I introduce Nietzsche’s critique of religious belief, for an audience of thinking Christians. I show that his criticism cannot and ought not be simply shrugged off, but rather, it can form the basis of a useful self-critique for the religious believer. I also argue that any response to a Nietzschean critique of religious belief and practice must itself take the form of an embodied believing life, rather than a merely theoretical response. | 3 3 1995 | Free | View In iTunes |

66 | 'Strenge' Arithmetics | We consider the virtues of relevant arithmetic couched in the logic E of entailment as opposed to R of relevant implication. The move to a stronger logic allows us to construct a complete system of true arithmetic, in which whenever an entailment A → B is not true (an example: 0=2 → 0=1 is not provable) then its negation ~(A → B) is true. | 3 3 1995 | Free | View In iTunes |

67 | Paraconsistency Everywhere | Paraconsistent logics are, by definition, inconsistency tolerant: In a paraconsistent logic, inconsistencies need not entail everything. However, there is more than one way a body of information can be inconsistent. In this paper I distinguish contradictions from other inconsistencies, and I show that several different logics are, in an important sense, “paraconsistent” in virtue of being inconsistency tolerant without thereby being contradiction tolerant. For example, even though no inconsistencies are tolerated by intuitionistic propositional logic, some inconsistencies are tolerated by intuitionistic predicate logic. In this way, intuitionistic predicate logic is, in a mild sense, paraconsistent. So too are orthologic and quantum propositional logic and other formal systems. Given this fact, a widespread view that traditional paraconsistent logics are especially repugnant because they countenance inconsistencies is undercut. Many well-understood nonclassical logics countenance inconsistencies as well. (This paper was published in 2004, despite the apparent date of 2002 in the citation. The tale is told in more detail here.) | 3 3 1995 | Free | View In iTunes |

68 | Relevant and Substructural Logics | An historical essay, sketching the development of relevant and substructural logics throughout the 20th Century and into the 21st. | 3 3 1995 | Free | View In iTunes |

69 | Multiple Conclusions | I argue for the following four theses. (1) Denial is not to be analysed as the assertion of a negation. (2) Given the concepts of assertion and denial, we have the resources to analyse logical consequence as relating arguments with multiple premises and multiple conclusions. Gentzen’s multiple conclusion calculus can be understood in a straightforward, motivated, non-question-begging way. (3) If a broadly anti-realist or inferentialist justification of a logical system works, it works just as well for classical logic as it does for intuitionistic logic. The special case for an anti-realist justification of intuitionistic logic over and above a justification of classical logic relies on an unjustified assumption about the shape of proofs. Finally, (4) this picture of logical consequence provides a relatively neutral shared vocabulary which can help us understand and adjudicate debates between proponents of classical and non-classical logics. This paper has now been reprinted in Analysis and Metaphysics, 6, 2007, 14-34. | 3 3 1995 | Free | View In iTunes |

70 | Łukasiewicz, Supervaluations and the Future | In this paper I consider an interpretation of future contingents which motivates a unification of a Łukasiewicz style logic with the more classical supervaluational semantics. This in turn motivates a new non-classical logic modelling what is “made true by history up until now.” I give a simple Hilbert-style proof theory, and a soundness and completeness argument for the proof theory with respect to the intended models. This paper is available at http://www.units.it/~episteme/L&PS_Vol3No1/contents_L&PS_Vol3No1.htm. | 3 3 1995 | Free | View In iTunes |

71 | Ways Things Can't Be | I show that the believer in possible worlds can have impossible worlds for free, as classes of possible worlds. These do exactly the job of ways that things cannot be, and they model the simple paraconsistent logic LP. This motivates a semantics for paraconsistent logic that even a classical logician can love. | 3 3 1995 | Free | View In iTunes |

72 | Envelopes and Indifference | We diagnose the two envelope “paradox”, showing how the indifference principle plays a role in prompting the conflicting assignments of the expected outcomes for switching or keeping. | 3 3 1995 | Free | View In iTunes |

73 | Routes to Triviality | It is well known that contraction-related principles trivialise naïve class theory. It is less well known that many other principles unrelated to contraction also render the theory trivial. This paper provides a characterisation of a large class of formulas which do the job. This class includes all properly implication formulas known in the literature, and adds countably many more. | 3 3 1995 | Free | View In iTunes |

74 | Information Flow and Relevant Logic | John Perry, one of the two founders of the field of situation semantics, indicated in an interview in 1986 that there is some kind of connection between relevant logic and situation semantics. I do know that a lot of ideas that seemed off the wall when I first encountered them years ago now seem pretty sensible. One example that our commentators don’t mention is relevance logic; there are a lot of themes in that literature that bear on the themes we mention. In 1992, in Entailment volume 2, Nuel Belnap and J. Michael Dunn hinted at similar ideas. Referring to situation semantics, they wrote … we do not mean to claim too much here. The Barwise-Perry semantics is clearly independent and its application to natural-language constructions is rich and novel. But we like to think that at least first degree (relevant) entailments have a home there. In this paper I show that these hints and gestures are true. And perhaps truer than those that made them thought at the time. In this paper I introduce the semantics of relevant logics, then I will sketch the parts of situation theory relevant to our enterprise. Finally, I bring the two together in what is hopefully, a harmonious way. | 3 3 1995 | Free | View In iTunes |

75 | A Useful Substructural Logic | I defend the extension of the lambek calculus with a distributive extensional conjunction and disjunction. I show how it independently arises in linguistics, information flow and relevant logics, and relation algebra. I give the logic a cut-free Gentzenisation and show that it is decidable. | 3 3 1994 | Free | View In iTunes |

76 | Subintuitionistic Logics | Once the Kripke semantics for normal modal logics were introduced, a whole family of modal logics other than the Lewis systems S1 to S5 were discovered. These logics were obtained by changing the semantics in natural ways. The same can be said of the Kripke-style semantics for relevant logics: a whole range of logics other than the standard systems R, E and T were unearthed once a semantics was given. In a similar way, weakening the structural rules of the Gentzen formulation of classical logic gives rise to other “substructural” logics such as linear logic. This process of “strategic weakening” is becoming popular today, with the discovery of applications of these logics to areas such as linguistics and the theory of computation. This paper examines what the process of weakening does to the Kripke-style semantics of intuitionistic logic, introducing the family of subintuitionistic logics. | 2 3 1994 | Free | View In iTunes |

77 | On Logics without Contraction | My Ph.D. Thesis, completed in January 1994. I was supervised by Prof. Graham Priest, at the University of Queensland. The thesis is 292 pages of work on logics without the contraction rule. | 5 1 1994 | Free | View In iTunes |

78 | Deviant Logic and the Paradoxes of Self-Reference | I argue that the extant reasons for sticking to classical logic in the face of the paradoxes of self reference are not good reasons. The paradoxes are really difficult and we should use all of the weapons at our disposal. | 3 3 1993 | Free | View In iTunes |

79 | How to be Really Contraction Free | I show that any finitely valued logic of a simple kind fails to support naïve comprehension, if it has a conditional. I then go on to show how some infinitely valued logics also fail to be robustly contraction free. Then I make a bold conjecture that robust contraction freedom is sufficient to support naïve set theory. This conjecture was later proved to be wrong by two graduate students from Monash, Sam Buchart and Su Rogerson, in some delightful work in 1997, which has since also been published in Studia Logica. | 2 3 1993 | Free | View In iTunes |

80 | Simplified Semantics for Relevant Logics (and some of their rivals) | I show how Priest and Sylvan’s simplified semantics extends from basic relevant logics to a large class of stronger logics. The completeness proof is a little tricky, given the different behaviour of the normal world in the models. This is the first paper from my Ph.D. thesis. | 2 3 1993 | Free | View In iTunes |

81 | A Note on Naïve Set Theory in LP | My first publication. It stems from work I did in my Honours year (1989) with Graham Priest, on paraconsistent logic. I explain a particularly simple yet powerful technique for constructing models of naïve set theory in the paraconsistent logic LP. This can be used to show the consistency of the theory, and to construct models invalidating some of the axioms of ZFC. | 3 3 1992 | Free | View In iTunes |

81 Items |