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.
A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline). Consider the following argument: 1. If we charge high fees for university, only the rich will enroll. We charge high fees for university. Therefore, only the rich (...) will enroll. There are many different things one can say about this argument, but many agree that if we do not equivocate (if the terms mean the same thing in the premises and the conclusion) then the argument is valid, that is, the conclusion follows deductively from the premises. This does not mean that the conclusion is true. Perhaps the premises are not true. However, if the premises are true, then the conclusion is also true, as a matter of logic. This entry is about the relation between premises and conclusions in valid arguments. (shrink)
Warren Goldfarb, Deductive Logic, Hackett Publishing Company, 2003. : 0872206602. Deductive Logic is an introductory textbook in formal logic. The book is divided into four parts covering (i) truth-functional logic, (ii) monadic quantifi- cation, (iii) polyadic quantification and (iv) names and identity, and there are exercises for all these topics at the end of the book. In the truth-functional logic part, the reader learns to produce paraphrases of English statements and arguments in logical notation (this subsection (...) is called “analysis”), then about the semantic properties of such paraphrased statements and arguments, such as satisfiability, implication and equivalence (“logical assessment”) and finally (“reflection”) there is an axiomatic proof method and some important extras such as disjunctive normal form and expressive adequacy. Parts two and three mirror this analysis/assessment/reflection structure for monadic and polyadic quantification, though this time the proof system is a natural deduction one, and part three contains a completeness proof for that system. The fourth part of the book introduces names, the identity predicate and descriptions and examines the additional expressive power which these provide. I do not think there should be any doubt that this is an excellent book; it presents the essential topics of a first logic course with accuracy, clarity and attention to detail, and it makes material that can be confusing the first time round—say, the translation of conditionals—transparent and easy to understand. But there are a lot of good introductory logic textbooks out there, so I will say something about how it resembles and differs from some other books, and then discuss one minor irritation with this one. (shrink)
: 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.
In this note I respond to Hartley Slater's argument 12 to the e ect that there is no such thing as paraconsistent logic. Slater's argument trades on the notion of contradictoriness in the attempt to show that the negation of paraconsistent logics is merely a subcontrary forming operator and not one which forms contradictories. I will show that Slater's argument fails, for two distinct reasons. Firstly, the argument does not consider the position of non-dialethic paraconsistency which rejects the possible truth (...) of any contradictions. Against this position Slater's argument has no bite at all. Secondly, while the argument does show that for dialethic paraconsistency according to which contradictions can be true, certain other contradictions must be true, I show that this need not deter the dialethic paraconsistentist from their position. (shrink)
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 episte-.
In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the (...) usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quanti- fiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. (shrink)
Implication barrier theses deny that one can derive sentences of one type from sentences of another. Hume’s Law is an implication barrier thesis; it denies that one can derive an ‘ought’ (a normative sentence) from an ‘is’ (a descriptive sentence). Though Hume’s Law is controversial, some barrier theses are philosophical platitudes; in his Lectures on Logical Atomism, Bertrand Russell claims: You can never arrive at a general proposition by inference particular propositions alone. You will always have to have at least (...) one general proposition in your premises. (Russell, 1918, p. 206) We will refer to this claim—that one cannot derive general sentences from particular sentences—as Russell’s Law.1 A third barrier thesis claims that one cannot derive sentences about the future from sentences about the past or present. Hume’s endorsement of this barrier thesis is well-known: all inferences from experience suppose, as their foundation, that the future will resemble the past . . . if there be any suspicion that the course of nature may change, and that the past may be no rule for the future, all experience becomes useless, and can give rise to no inference or conclusion. It is impossible, therefore, that any argument from experience can prove this resemblance of the past to the future; since all these arguments are founded on the supposition of that resemblance. (Hume, EHU 4.21/37) We will refer to this barrier thesis as Hume’s Second Law. A fourth barrier thesis says that one cannot derive a necessary sentence from one about the actual world and we will refer to this last thesis Kant’s Law. Such implication barrier theses present a problem. (shrink)
I do know that a lot of ideas that seemed o the wall when I rst 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. Barwise and Perry 1985.
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 [2], 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. (shrink)
Substructural logics are non-classical logics weaker than classical logic, notable for the absence of structural rules present in classical logic. These logics are motivated by considerations from philosophy (relevant logics), linguistics (the Lambek calculus) and computing (linear logic). In addition, techniques from substructural logics are useful in the study of traditional logics such as classical and intuitionistic logic. This article provides a brief overview of the field of substructural logic. For a more detailed introduction, complete with theorems, proofs and examples, (...) the reader can consult the books and articles in the Bibliography. (shrink)
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, an idealisation of more fundamental logical features in 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.
I’m delighted to have the opportunity to respond to Hartry Field’s Saving Truth From Paradox [3]. This is a wonderful book: it’s clear and precise, interesting and engaging, and deep and important all at once. Truth and the paradoxes..
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. (shrink)
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 6= 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 signi cant success for automated reasoning. (shrink)
We are pluralists about logical consequence [1]. We hold that there is more than one sense in which arguments may be deductively valid, that these senses are equally good, and equally deserving of the name deductive validity. Our pluralism starts with our analysis of consequence. This analysis of consequence is not idiosyncratic. We agree with Richard Jeffrey, and with many other philosophers of logic about how logical consequence is to be defined. To quote Jeffrey.
In Entailment, Anderson and Belnap motivated their modification E of Ackermann’s strenge Implikation Π Π’ as a logic of relevance and necessity. The kindred system R was seen as relevant but not as modal. Our systems of Peano arithmetic R# and omega arithmetic R## were based on R to avoid fallacies of relevance. But problems arose as to which arithmetic sentences were (relevantly) true. Here we base analogous systems on E to solve those problems. Central to motivating E is the (...) rejection of fallacies of modality. Our slogan here for this is, “No diamonds entail any boxes.” Form the strenge Peano arithmetic E# like R#, adding appropriate forms of the Peano axioms to Ackermann’s E.. (shrink)
What would morality have to be like in order to answer to our everyday moral concepts'? What are we committed to when we make moral claims such as "female infibulation is wrong"; or "we ought give money to famine relief"; or "we have a duty to not to harm others", and when we go on to argue about these sorts of claims'? It has seemed to many ââ¬â and it seems plausible to us ââ¬â that when we assert and argue (...) about things such as these we presuppose at least the following. (shrink)
Consider this situation: Here are two envelopes. You have one of them. Each envelope contains some quantity of money, which can be of any positive real magnitude. One contains twice the amount of money that the other contains, but you do not know which one. You can keep the money in your envelope, whose numerical value you do not know at this stage, or you can exchange envelopes and have the money in the other. You wish to maximise your money. (...) What should you do?1 Here are three forms of reasoning about this situation, which we shall call.. (shrink)
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 a kind of mathematical realism. In the paper, I begin the development of the view and compare it to orthodox positions in the philosophy of mathematics.
Perhaps such a proposition is not expressible in any language that you or I speak, but – so a familiar story goes – it is decided by each world, so it plays just the role that other propositions do, so it counts as a proposition in the same way. In fact, we can see just how it counts as a proposition: given all the worlds in S, our proposition p says that the world is one of the worlds in S. (...) It describes a way the world is, even if we have no means of picking out the set S, so it is a proposition.1.. (shrink)
In this short paper, I compare and contrast the kind of symmetric 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 are what Brandom calls a normative pragmatics. They are both semantic anti-realist accounts of meaning in the significance of vocabulary is explained in terms of our rule-governed (normative) practice (pragmatics). These accounts differ (...) from intuitionist semantic anti-realism by providing a way to distinguish the inferential significance of “A” and “A is warranted.” Although proof plays a central role, in neither accont is verification the primary bearer of meaning. Our accounts make these distinctions in terms of a subtle analysis of our practices. On the one hand according to Price and me, we assert as well as deny; on the other, Brandom distingushes downstream commitments from upstream entitlements and the notion of incompatibility definable in terms of these. In this paper I will examine a number connections between these different approaches, and end with a discussion of the kind of account of proof that might emerge from these considerations. (shrink)
This is an exploratory and expository paper, comparing display logic formulations of normal modal logics with labelled sequent systems. We provide a translation from display sequents into labelled sequents. The comparison between different systems gives us a different way to understand the difference between display systems and other sequent calculi as a difference between local and global views of consequence. The mapping between display and labelled systems also gives us a way to understand labelled systems as properly structural and not (...) just as systems encoding modal logic into first-order logic. (shrink)
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.
In his paper “Generalised Ortho Negation” [2] 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 ∼∼A A. That claim is 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.1..
A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline). Consider the following argument: If we charge high fees for university, only the rich will enroll. We charge high fees for university. Therefore, only the rich will (...) enroll. There are many different things one can say about this argument, but many agree that if we do not equivocate (if the terms mean the same thing in the premises and the conclusion) then the argument is valid, that is, the conclusion follows deductively from the premises. This does not mean that the conclusion is true. Perhaps the premises are not true. However, if the premises are true, then the conclusion is also true, as a matter of logic. This entry is about the relation between premises and conclusions in valid arguments. (shrink)
There is widespread acknowledgement that the law of non-contradiction is an important logical principle. However, there is less-than-universal 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: From the point of view of proofs, not everything need follow from a contradiction — from the point of view of models, there are “worlds” in which contradictions are true. In this sense, the law of non-contradiction is (...) violated in these logics. However, in many paraconsistent logics, statement (it is not the case that ¢ and ¡£¢ ¤ ¢¦¥ not- ¢ ) is still provable. In this sense, the law of non-contradiction is upheld. This paper attempts to clarify the different readings of the law of non-contradiction, in particular taking cues from the tradition of relevant logics. A further guiding principle will be the natural duality between the law of non-contradiction and rejection on the one hand and the law of the excluded middle and acceptance on the other. (shrink)
I am a logical pluralist. I think that logical consequence is not just a manysplendoured thing, but that logical consequence is many different things. There is no one true logic but rather, many. Logic is a matter of “truth preservation in all cases” in the sense that..
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. (shrink)
Our topic is the notion of logical consequence: the link between premises and conclusions, the glue that holds together deductively valid argument. How can we understand this relation between premises and conclusions? It seems that any account begs questions. Painting with very broad brushtrokes, we can sketch the landscape of disagreement like this: “Realists” prefer an analysis of logical consequence in terms of the preservation of truth [29]. “Anti-realists” take this to be unhelpful and o:er alternative analyses. Some, like Dummett, (...) look to preservation of warrant to assert [9, 36]. Others, like Brandom [5], take inference as primitive, and analyse other notions in terms of it. There is plenty of disagreement on the “realist” side of the fence too. It is one thing to argue that logical consequence involves preservation of truth. It is another to explain how far truth must be preserved. Is the preservation essentially modal (in all circumstances [25]) or analytic (vouchsafed by.. (shrink)
As has been made clear in many of the papers in this volume, the crucial feature in Bradwardine’s theory of truth is the notion of signification. Expressed by a ‘connecticate’, which I shall write with the simple infix colon “:”, whenever t is a singular term and p is a sentence..
There are many different approaches to the logic of truth. We could agree with Tarski, that the appropriate way to formalise a truth predicate is in a hierarchy, in which the truth predicate in one language can apply only to sentences from another language. Or, we could attempt to do without type restrictions on the truth predicate. Bradwardine’s theory of truth takes the second of these options: it is type-free, and admits sentences which say of themselves that they are not (...) true to be well-formed. We could take the behaviour of the paradoxes such as the liar to motivate a revision of the basic logic of propositional inference, to allow for truth-value gaps or gluts [9, 11, 15]. On the other hand, we could take it that the paradoxes are no reason to revise our account of the basic laws of logic: a novel account of the behaviour of the truth predicate is what is required. Bradwardine’s account, as elaborated by Read, takes this second option.1 Finally. (shrink)
According to one tradition in realist philosophy, ‘truthmaking’ amounts to necessitation. That is, an object x is a truthmaker for the claim A if x exists, and the existence of x necessitates the truth of A. In symbols: E!x ∧ (E!x ⇒ A). I argued in my paper “Truthmakers, Entailment and Necessity” [14], that if we wish to use this account of truthmaking, we ought understand the entailment connective “⇒” in such a claim as a relevant entailment, in the tradition..
First, a few words of introduction, setting the scene. IÕm not a Nietzsche scholar. IÕm not even an historian of philosophy of any stripe. I am one of the fortunate few who are paid to Ôdo philosophyÕ, but the areas I tend to do most of my work in are logic, philosophy of language and some philosophy of religion. So why am I presenting a paper on Nietzsche? Well, there are at least two reasons. Firstly, I teach philosophy of religion, (...) and in the course I have a section about distinctively modern critics of religious belief. Nietzsche, together with Freud, Feuerbach and Marx present important criticisms which form a part of the fabric of contemporary philosophy of religion, and any student of the area needs to know something about it. So, what better way for me to learn about it than to force myself to write a paper on it? However, my reasons are not just selfish Ñ I do believe that the way that Christians (and other religious believers) respond to these contemporary critics of religion is very important. So, my aim in this paper is not only to give a short introduction to what Nietzsche has to say about Christian faith, but also to examine what an appropriate response for believers might be. This then has consequences for what we take the task of ÔChristian PhilosophyÕ to be. (shrink)
I suppose the natural way to interpret this question is something like “why do formal methods rather than anything else in philosophy” but in my case I’d rather answer the related question “why, given that you’re interested in formal methods, apply them in philosophy rather than elsewhere?” I started off my academic life as an undergraduate student in mathematics, because I was good at mathematics and studying it more seemed like a good idea at the time. I enjoyed mathematics a (...) great deal. At the University of Queensland, where I was studying, there was a special cohort of “Honours” students right from the first year. You were taught more research-oriented and rigourous subjects than were provided for the “Pass” students. This meant that we had a small cohort of students, who knew each other pretty well, studied together and learned a lot. I could see myself making an academic career in mathematics. (I surely couldn’t see myself doing anything other than an academic career. Being around the university was too much fun.) However, there was a fly in the ointment. I was doing well in my studies, but I was losing the feel for a great deal of the mathematics I was doing. Applied mathematics went first, and analysis soon after. I could do the work, but I didn’t understand it. I wrote assignments by matching patterns from what I had written in my lecture notes, or what was in the text with what we were asked. In exams, I just bashed away at the problem, sometimes when asked in an exam to prove that A = B, I’d work at A from the top of a page and keep manipulating it until I’d got stuck. Then I’d work backwards from B, hoping to meet at somewhere rather like where I’d got stuck. If I was honest, I’d write “I don’t know how to get from here to there”. If I was dishonest, I’d just leave the transition unexplained. Knowing what I know now about marking assignments, it doesn’t suprise me that I did very well. The areas where intuition and understanding lasted the longest (and which were most fun) were topology, probability theory, combinatorics, set theory and logic.. (shrink)
This essay is structured around the bifurcation between proofs and models: The first section discusses Proof Theory of relevant and substructural logics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, (...) Dunn’s algebraic models [76, 77] Urquhart’s operational semantics [267, 268] and Routley and Meyer’s relational semantics [239, 240, 241] arrived decades after the initial burst of activity from Alan Anderson and Nuel Belnap. The same goes for work on the Lambek calculus: although inspired by a very particular application in linguistic typing, it was developed first proof-theoretically, and only later did model theory come to the fore. Girard’s linear logic is a different story: it was discovered though considerations of the categorical models of coherence.. (shrink)
Symbolic logic is sited at intersection of philosophy, mathematics, linguistics and computer science. It deals with the structure of reasoning, and the formal features of information. Work in symbolic logic has almost exclusively treated the deductive validity of arguments: those arguments for which it is impossible for the premises to be true and the conclusion false. However, techniques from twentieth-century logic have found a place in the study of inductive or probabilistic reasoning, in which premises need not render their conclusions (...) certain. (shrink)
wardine’s account from Buridan’s [5, 6]. What are we to make of this? If the argument fails, what distinguishes problematic truth-tellers (such as a sentence that explicitly says of itself that it is true) from benign truth tellers? It is my task in this paper to explain this distinction, and to clarify the behaviour of truth-tellers, given my the contemporary formal treatment of Bradwardine’s account of signification.
Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic Belnap 1982 . However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modi ed proof theory which more closely models relevant logics. In addition, we use this proof theory to provide decidability proofs for a large range of substructural logics.
A B S T R AC T: 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.
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.
One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley–Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...) a general conception of conditionality that may unify the three given conceptions. (shrink)
Machine generated contents note: -- Series Editors' PrefaceAcknowledgementsNotes on ContributorsHow Things Are Elsewhere; W. Schwarz Information Change and First-Order Dynamic Logic; B.Kooi Interpreting and Applying Proof Theories for Modal Logic; F.Poggiolesi & G.Restall The Logic(s) of Modal Knowledge; D.Cohnitz On Probabilistically Closed Languages; H.Leitgeb Dogmatism, Probability and Logical Uncertainty; B.Weatherson & D.Jehle Skepticism about Reasoning; S.Roush, K.Allen & I.HerbertLessons in Philosophy of Logic from Medieval Obligations; C.D.Novaes How to Rule Out Things with Words: Strong Paraconsistency and the Algebra of Exclusion; (...) F.Berto Lessons from the Logic of Demonstratives; G.RussellThe Multitude View on Logic; M.Eklund Index. (shrink)
Machine generated contents note: -- Series Editors' PrefaceAcknowledgementsNotes on ContributorsHow Things Are Elsewhere; W. Schwarz Information Change and First-Order Dynamic Logic; B.Kooi Interpreting and Applying Proof Theories for Modal Logic; F.Poggiolesi & G.Restall The Logic(s) of Modal Knowledge; D.Cohnitz On Probabilistically Closed Languages; H.Leitgeb Dogmatism, Probability and Logical Uncertainty; B.Weatherson & D.Jehle Skepticism about Reasoning; S.Roush, K.Allen & I.HerbertLessons in Philosophy of Logic from Medieval Obligations; C.D.Novaes How to Rule Out Things with Words: Strong Paraconsistency and the Algebra of Exclusion; (...) F.Berto Lessons from the Logic of Demonstratives; G.RussellThe Multitude View on Logic; M.Eklund Index. (shrink)
In this paper we develop a participatory model of the Christian doctrine of the atonement, according to which the atonement involves participating in the death and resurrection of Christ. In part one we argue that current models of the atonement—exemplary, penal, substitutionary and merit models—are unsatisfactory. The central problem with these models is that they assume a purely deontic conception of sin and, as a result, they fail to address sin as a relational and ontological problem. In part two we (...) argue that a participatory model of the atonement is both exegetically and philosophically plausible, and should be taken seriously within philosophical theology.i.. (shrink)
Peer Instruction (or PI for short) is a simple and effective technique you can use to make lectures more interactive, more engaging, and more effective learning experiences.
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 confirmation. 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. (shrink)
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 e:ect 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 di:erently depending on another thesis connecting knowledge and possibility. If there are two propositions, inconsistent with one another, but both knowable, 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 weak. (shrink)
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, an idealisation of more fundamental logical features in 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.
Peer Instruction is a simple and effective technique you can use to make lectures more interactive, more engaging, and more effective learning experiences. Although well known in science and mathematics, the technique appears to be little known in the humanities. In this paper, we explain how Peer Instruction can be applied in philosophy lectures. We report the results from our own experience of using Peer Instruction in undergraduate courses in philosophy, formal logic, and critical thinking. We have consistently found it (...) to be a highly effective method of improving the lecture experience for both students and the lecturer. (shrink)
This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993 ) concerning the modelling conditions for the axioms of assertion A → (( A → B ) → B ) (there called c 6) and permutation ( A → ( B → C )) → ( B → ( A → C )) (there called c 7). 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. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001 ), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections. (shrink)
The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. If one (...) or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes. (shrink)
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.
Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline. In this book JC Beall and Greg Restall present and defend what thay call logical pluralism, the view that there is more than one genuine deductive consequence relation, a (...) position which has profound implications for many linguists as well as for philosophers. We should not search for one true logic, since there are many. (shrink)
In this paper we introduce a distinct metaethical 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 distinguish moral fictionalism from moral quasi-realism, and we show that fictionalism possesses the virtues of quasi-realism about morality, but avoids its vices.
Propositional logic -- Propositions and arguments -- Connectives and argument forms -- Truth tables -- Trees -- Vagueness and bivalence -- Conditionality -- Natural deduction -- Predicate logic -- Predicates, names, and quantifiers -- Models for predicate logic -- Trees for predicate logic -- Identity and functions -- Definite descriptions -- Some things do not exist -- What is a predicate? -- What is logic?
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, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. 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. (shrink)
Stephen Neale presents, in Facing Facts (Oxford: Clarendon Press, 2001), one convenient package containing his reasoned complaints against theories of facts and non-extensional connectives. The slingshot is a powerful argument (or better, it is a powerful family of arguments) which constrains theories of facts, propositions and non-extensional connectives by showing that some of these theories are rendered trivial. This book is the best place to find the state of the art on the slingshot and its implications for logic, language and (...) metaphysics. It provides a useful starting point for anyone who has wondered what all of the fuss about the slingshot amounts to. Neale shows that the fuss does amount to something, and that theories of facts must “face facts” and present an adequate response to the slingshot. However, Neale’s evaluation of the state of play for theories of facts is too pessimistic. As the book draws near to a close, Neale writes: As I have stressed, Russell’s Theory of Facts, according to which facts have properties as components, is safe. It is certainly tempting to draw the moral that if one wants non-collapsing facts one needs properties as components of facts. I have not attempted to prove this here, but I suspect it will be proved in due course. (page 210) Neale concludes that while theories which take facts to be structured entities are safe from slingshot arguments, and he suspects that this is the only kind of fact theory safe from slingshot-style collapse. If this were the case, then theories such as situation theories or accounts of truthmakers may well be threatened. However, Neale’s suspicion is ill-founded, as I shall soon show. Not only do Russellian theories of facts survive the slingshot unscathed, but so can theories of facts which take them to be unstructured entities. Furthermore, the way that this may be not only argued for, but proved can provide a new weapon in the armoury of the theorist investigating fact theories. (shrink)
It is known that a number of inference principles can be used to trivialise the axioms of naïve comprehension – the axioms underlying the naïve theory of sets. In this paper we systematise and extend these known results, to provide a number of general classes of axioms responsible for trivialising naïve comprehension.
Mark Balaguer's Platonism and Anti-Platonism in Mathematics presents an intriguing new brand of platonism, which he calls plenitudinous platonism, or more colourfully, full-blooded platonism. In this paper, I argue that Balaguer's attempts to characterise full-blooded platonism fail. 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.
“Paraconsistent” means “beyond the consistent” [3, 15]. Paraconsistent logics tolerate inconsistencies in a way that traditional logics do not. In a paraconsistent logic, the inference of explosion A, ∼AB is rejected. This may be for any of a number of reasons [16]. For proponents of relevance [1, 2] the argument has gone awry when we infer an irrelevant B from the inconsistent premises. Those who argue that inconsistent theories may have some logical content but do not commit us to everything, (...) have reason to think that these theories are closed under a relation of paraconsistent logical consequence [12, 18]. Another reason to adopt a paraconsistent logic is more extreme. You may take the world to be inconsistent [14], and a true theory incorporating this inconsistency must be governed by a paraconsistent logic. (shrink)
Shapiro and Taschek [7] 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 is not warrantedly assertible, then its negation ¡ is. Tennant [8] 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. (shrink)
Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline. In this book JC Beall and Greg Restall present and defend what thay call logical pluralism, the view that there is more than one genuine deductive consequence relation, a (...) position which has profound implications for many linguists as well as for philosophers. We should not search for one true logic, since there are many. (shrink)
This is the first book to systematically survey new areas of substructural logics. This book is geared to introduce the topic to advanced students. An Introduction to Substructural Logics covers the area of logic that is crucial to developments in computing, philosophy and linguistics.
Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic (Belnap, 1982). 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.
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 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. (shrink)
Possible worlds semantics has been very useful in modeling not only the intensionality of necessity and possibility, future and past. It has also found its place in modeling the intentionality of propositional attitudes like belief and knowledge. There is something fruitful in analyzing a belief as a set of possible worlds. The belief is the set of possible worlds in which the belief is true. The belief is true if and only if the actual world is in the corresponding set (...) of propositions. The possible worlds in the set corresponding to the belief represent how the agent per- ceives the world to be. If the belief is false, then the world isn’t how the agent sees the world to be, and so the actual world isn’t in the set of worlds corresponding to the belief (see Lewis [4] and Stalnaker [9]). The same can be said of whole belief states just as much as it can be said of individual beliefs. My belief state is the set of worlds consistent with what I believe. This view has been very fruitful, not least because the set-theoretic structure of sets of possible worlds corresponds nicely with the logical structure of entailment relations among propositions and the behavior of propositional connectives like conjunction, disjunction, and negation. However, the story does not deal well with inconsistent belief. Inconsistent beliefs are true in no possible worlds, so they are each modeled by the same set of worlds—the empty set. My beliefs are often inconsistent, and so are those of many.. (shrink)
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 logicC (orRW). 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. (shrink)
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 (cf. Priest and Sylvan [6], Restall [7], and Routley et al. (...) [8]). 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 (as in Girard [4]). 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 (cf. van Benthem [1]). Until now no-one has (to my knowledge) examined what the process of weakening does to the Kripke-style semantics of intuitionistic logic. This paper remedies the deficiency, introducing the family of subintuitionistic logics. These systems have some appealing features. Unlike other substructural logics such as linear logic (which lack distribution of extensional disjunction over conjunction) they have a very natural Kripke-style worlds semantics. Also, the difficulties with regard to modelling quantification in these systems may be able to shed some light on the difficulties in naturally modelling quantification in relevant logics, as it must be admitted that the semantics currently available for quantified relevant logics are rather baroque (cf. Fine [3]). But most importantly, delving in the undergrowth of logics such as intuitionistic logic gives us a ‘feel’ for how such systems are put together, and what job is being done by each aspect of the modelling conditions in.. (shrink)
The paradoxes of self reference have to be dealt with by anyone seeking to give a satisfactory account of the logic of truth, of properties, and even of sets of numbers. Unfortunately, there is no widespread agreement as to how to deal with these paradoxes. Some approaches block the paradoxical inferences by rejecting as invalid a move that classical logic counts as valid. In the recent literature, this deviant logic analysis of the paradoxes has been called into question.This disagreement motivates (...) a re-examination of the philosophy of formal logic and the status of logical truths and rules. In this paper I do some of this work, and I show that this gives us the means to defend the deviant approaches against such criticisms. As a result I hope to show that these analyses of the paradoxes are worthy of more serious consideration than they have so far received. (shrink)
A logic is said to becontraction free if the rule fromA (A B) toA B is not truth preserving. It is well known that a logic has to be contraction free for it to support a non-trivial naïve theory of sets or of truth. What is not so well known is that if there isanother contracting implication expressible in the language, the logic still cannot support such a naïve theory. A logic is said to berobustly contraction free if there is (...) no such operator expressible in its language. We show that a large class of finitely valued logics are each not robustly contraction free, and demonstrate that some other contraction free logics fail to be robustly contraction free. Finally, the sublogics of (with the standard connectives) are shown to be robustly contraction free. (shrink)
This paper continues the work of Priest and Sylvan inSimplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such asB + andB. We show that the simplified semantics can also be used for a large number of extensions of the positive base logicB +, and then add the dualising* operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.
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 nd that 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.
Proponents of “truth-value glut” responses to the paradoxes of self-reference, such as Priest [6, 7] argue that “truth-value gap” analyses of the paradoxes fall foul of the strengthened 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 fail to 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. (shrink)
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 “ p 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. (shrink)
