Both I and Belnap, motivated the “Belnap-Dunn 4-valued Logic” by talk of the reasoner being simply “told true” ( T ), and simply “told false” ( F ), which leaves the options of being neither “told true” nor “told false” ( N ), and being both “told true” and “told false” ( B ). Belnap motivated these notions by consideration of unstructured databases that allow for negative information as well as positive information (even when they conflict). We now experience this (...) on a daily basis with the Web. But the 4-valued logic is deductive in nature, and its matrix is discrete: there are just four values. In this paper I investigate embedding the 4-valued logic into a context of probability. Jøsang’s Subjective Logic introduced uncertainty to allow for degrees of belief, disbelief, and uncertainty. We extend this so as to allow for two kinds of uncertainty—that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflicted). Jøsang’s “Opinion Triangle” becomes an “Opinion Tetrahedron” and the 4-values can be seen as its vertices. I make/prove various observations concerning the relation of non-classical “probability” to non-classical logic. (shrink)
(2013). A Theory of Knowledge and Belief Change: Formal and Experimental Perspectives. Australasian Journal of Philosophy. ???aop.label???. doi: 10.1080/00048402.2012.759242.
Consider the Evidence Question: When and under what conditions is proposition P evidence for some agent S? Silins (Philos Perspect 19:375–404, 2005) has recently offered a partial answer to the Evidence Question. In particular, Silins argues for Evidential Internalism (EI), which holds that necessarily, if A and B are internal twins, then A and B have the same evidence. In this paper I consider Silins’s argument, and offer two response on behalf of Evidential Externalism (EE), which is the denial of (...) Evidential Internalism. The first response claims that the allegedly unattractive consequence for EE is not so unattractive. The second response takes the form of a tu quoque, demonstrating that a structurally similar argument can be constructed against EI. The two responses play off one another: objecting to the first puts pressure on one to accept the other. Taken together, the two responses have important ramifications for how we answer the Evidence Question, and how we think about evidence in general. (shrink)
The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$ (...) . This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$ , extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment . We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$ . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$ . These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$ , which we present in another paper. (shrink)
Simple versions of Reliabilism about justification say that S's believing that p is justified if and only if the belief was produced by a belief-forming process that is reliable above some high threshold. Alvin Goldman, in Epistemology and Cognition, argues for a more complex version of the view according to which it is total epistemic systems that are assessed for reliability, rather than individual processes. Why prefer this more complex version of Reliabilism? Two reasons suggest themselves. First, it seems that (...) the interaction of various processes of belief formation is often important. The more complex version appears to account for the interaction of processes. Second, one might doubt whether individual processes will have determinate truth-ratios. If not, the simple version of Reliabilism is a nonstarter. In this paper I argue that, despite these two apparent advantages, the complex version of Reliabilism is untenable. I conclude by arguing that the simple version is actually fine as it is. (shrink)
David Lewis ([1986b]) gives an attractive and familiar account of counterfactual dependence in the standard context. This account has recently been subject to a counterexample from Adam Elga ([2000]). In this article, I formulate a Lewisian response to Elga’s counterexample. The strategy is to add an extra criterion to Lewis’s similarity metric, which determines the comparative similarity of worlds. This extra criterion instructs us to take special science laws into consideration as well as fundamental laws. I argue that the Second (...) Law of Thermodynamics should be seen as a special science law, and give a brief account of what Lewisian special science laws should look like. If successful, this proposal blocks Elga’s counterexample. (shrink)
Bayesian Epistemology is a general framework for thinking about agents who have beliefs that come in degrees. Theories in this framework give accounts of rational belief and rational belief change, which share two key features: (i) rational belief states are represented with probability functions, and (ii) rational belief change results from the acquisition of evidence. This dissertation focuses specifically on the second feature. I pose the Evidence Question: What is it to have evidence? Before addressing this question we must have (...) an understanding of Bayesian Epistemology. The first chapter argues that we should understand Bayesian Epistemology as giving us theories that are evaluative and not action-guiding. I reach this verdict after considering the popular ‘ought’-implies-‘can’ objection to Bayesian Epistemology. The second chapter argues that it is important for theories in Bayesian Epistemology to answer the Evidence Question, and distinguishes between internalist and externalist answers. The third and fourth chapters present and defend a specific answer to the Evidence Question. The account is inspired by reliabilist accounts of justification, and attempts to understand what it is to have evidence by appealing solely to considerations of reliability. Chapter 3 explains how to understand reliability, and how the account fits with Bayesian Epistemology, in particular, the requirement that an agent’s evidence receive probability 1. Chapter 4 responds to objections, which maintain that the account gives the wrong verdict in a variety of situations including skeptical scenarios, lottery cases, scientific cases, and cases involving inference. After slight modifications, I argue that my account has the resources to answer the objections. The fifth chapter considers the possibility of losing evidence. I show how my account can model these cases. To do so, however, we require a modification to Conditionalization, the orthodox principle governing belief change. I present such a modification. The sixth and seventh chapters propose a new understanding of Dutch Book Arguments, historically important arguments for Bayesian principles. The proposal shows that the Dutch Book Arguments for implausible principles are defective, while the ones for plausible principles are not. The final chapter is a conclusion. (shrink)
Symmetric generalized Galois logics (i.e., symmetric gGl s) are distributive gGl s that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGl s by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGl s. We generalize the weak (...) distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGl s. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGl s and classes of structures for them. (shrink)
This book by leading international scholars in the fields of history, philosophy and politics restores the subject to a place at the very centre of political theory and practice.
Finding disjunctivist versions of direct realism unexplanatory, Mark Johnston [(2004). Philosophical Studies, 120, 113–183] offers a non-disjunctive version of direct realism in its place and gives a defense of this view from the problem of hallucination. I will attempt to clarify the view that he presents and then argue that, once clarified, it either does not escape the problem of hallucination or does not look much like direct realism.
In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.
We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn's original one. Ku is the minimal logic that has a (...) characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic Ku. Finally, we unite the two basic logics Ki and Ku together to get a negative modal logic K-, which is dual to the positive modal logic K+ in [7]. Shramko has suggested an extension of Dunn's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations. (shrink)
John Locke (1632-1704) one of the greatest English philosophers of the late seventeenth and early eighteenth century, argued in his masterpiece, An Essay Concerning Human Understanding, that our knowledge is founded in experience and reaches us principally through our senses; but its message has been curiously misunderstood. In this book John Dunn shows how Locke arrived at his theory of knowledge, and how his exposition of the liberal values of toleration and responsible government formed the backbone of enlightened European thought (...) of the eighteenth century. (shrink)
This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these (...) systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity". (shrink)
In this collection of recent essays (several appearing in English for the first time), John Dunn brings his characteristically acute and penetrative insight to a wide range of political issues. In the first essay, 'The history of political theory', Professor Dunn argues for the importance of a historical perspective in the study of political thought. Other pieces engage with central concepts of political philosophy such as obligation, trust, freedom of conscience and property. A group of studies tackle specific contemporary problems (...) and future dangers, for example racism and the dilemma of humanitarian intervention. The volume as a whole articulates the many dangers, but also the huge importance of, contemporary politics, and provides a representative collection of work by one of the most astute political commentators writing today. (shrink)
We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are.
We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutatively and associatively are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, (...) similar to Curry's subtraction operator, which is resituated with Linear Logic's contensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic. (shrink)
The first to focus exclusively on implicit memory research, this book documents the proceedings of a meeting held in Perth, Australia where leading researchers ...
The validity of an entailment has nothing to do with whether or not the components are true, false, necessary, or impossible; it has to do solely with whether or not there is a necessary connection between antecedent and consequent. Hence it is a mistake (we feel) to try to build a sieve which will strain out entailments from the set of material or strict implications present in some system of truth-functions, or of truth-functions with modality. Anderson and Belnap (1962, p. (...) 47). (shrink)
This paper explores the development of mathematics on a quantum logical base when mathematical postulates are taken as necessary truths. First it is shown that first-order Peano arithmetic formulated with quantum logic has the same theorems as classical first-order Peano arithmetic. Distribution for first-order arithmetical formulas is a theorem not of quantum logic but rather of arithmetic. Second, it is shown that distribution fails for second-order Peano arithmetic without extensionality. Third, it is shown that distribution holds for second-order Peano arithmetic (...) (second-order quantum logic) with extensionality. Some remarks about extensions to quantum set theory are made. (shrink)
Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for (...) number theory was obtained earlier by R. K. Meyer and suggested the present abstract development). (shrink)
In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in (...) relevant arithmetic. (shrink)
This study provides a comprehensive reinterpretation of the meaning of Locke's political thought. John Dunn restores Locke's ideas to their exact context, and so stresses the historical question of what Locke in the Two Treatises of Government was intending to claim. By adopting this approach, he reveals the predominantly theological character of all Locke's thinking about politics and provides a convincing analysis of the development of Locke's thought. In a polemical concluding section, John Dunn argues that liberal and Marxist interpretations (...) of Locke's politics have failed to grasp his meaning. Locke emerges as not merely a contributor to the development of English constitutional thought, or as a reflector of socio-economic change in seventeenth-century England, but as essentially a Calvinist natural theologian. (shrink)
