Francesco Praolini has recently put pressure on the view that justified believability is closed under conjunction introduction. Based on what he calls ‘the hybrid paradox,’ he argues that accepting the principle of conjunction closure for justified believability, quite surprisingly, entails that one must also accept the principle of factivity for justified believability, i.e. that there are no propositions that are justifiably believable and false at the same time. But proponents of conjunction closure can do without (...) factivity, as I argue in this short note. A less demanding principle is available. (shrink)

Some theories of rational belief assume that beliefs should be closed under conjunction. I motivate the rejection of the conjunction closure, and point out that the consequences of this rejection are not as severe as it is usually thought. An often raised objection is that without the conjunction closure people are unable to reason. I outline an approach in which we can – in usual cases – reason using conjunctions without accepting the closure in (...) its whole generality. This solution is based on the notion of confidence levels, which can be defined using probabilities. Moreover, on this approach, reasoning has a scalable computational complexity adaptable to cognitive abilities of both rationally bounded and perfectly rational agents. I perform a simulation to assess its error rate, and compare it to reasoning with conjunction closure. (shrink)

This paper is closely related to investigations of abstract properties of basic logical notions expressible in terms of closure spaces as they were begun by A. Tarski (see [6]). We shall prove many properties of -conjunctive closure spaces (X is -conjunctive provided that for every two elements of X their conjunction in X exists). For example we prove the following theorems:1. For every closed and proper subset of an -conjunctive closure space its interior is empty (i.e. (...) it is a boundary set). 2. If X is an -conjunctive closure space which satisfies the -compactness theorem and [X] is a meet-distributive semilattice (see [3]), then the lattice of all closed subsets in X is a Heyting lattice. 3. A closure space is linear iff it is an -conjunctive and topological space. 4. Every continuous function preserves all conjunctions. (shrink)

According to the principle of Conjunction Closure, if one has justification for believing each of a set of propositions, one has justification for believing their conjunction. The lottery and preface paradoxes can both be seen as posing challenges for Closure, but leave open familiar strategies for preserving the principle. While this is all relatively well-trodden ground, a new Closure-challenging paradox has recently emerged, in two somewhat different forms, due to Backes :3773–3787, 2019a) and Praolini :715–726, (...) 2019). This paradox synthesises elements of the lottery and the preface and is designed to close off the familiar Closure-preserving strategies. By appealing to a normic theory of justification, I will defend Closure in the face of this new paradox. Along the way I will draw more general conclusions about justification, normalcy and defeat, which bear upon what Backes :2877–2895, 2019b) has dubbed the ‘easy defeat’ problem for the normic theory. (shrink)

The principle of epistemic closure is the claim that what is known to follow from knowledge is known to be true. This intuitively plausible idea is endorsed by a vast majority of knowledge theorists. There are significant problems, however, that have to be addressed if epistemic closure – closed knowledge – is endorsed. The present essay locates the problem for closed knowledge in the separation it imposes between knowledge and evidence. Although it might appear that all that stands (...) between knowing the truth of the premises of a valid inference and knowledge of its conclusion is inferring it from the premises, the evidence for each of the premises may jointly count against the conclusion. The intuitive view regarding inferred knowledge says one thing, the evidence says another. One epistemological framework that seems to have the resources to resolve this tension endorses the view that knowledge always requires conclusive evidence. A second framework resolves the tension by limiting the scope of the closure principle. Only inferences drawn directly from propositions contained in the scope of a single knowledge operator are considered closed. The aim of the present essay is to revive the unpopular third option, the idea that knowledge is open. The essay proceeds by arguing that in different ways the two former frameworks only succeed in relocating the problem, not in resolving it. The first framework, the infallibilist view, relocates the problem to a sharp separation between knowledge of the occurrences of events from knowledge of their chance of occurring, a separation leading to several significant additional problems. The fallibilist view, the second framework, in endorsing closure neglects to take into full account the ways in which evidence fails to be transitive. For instance, evidence can count in favor of a conjunction while counting against each of its conjuncts. This fact, which is argued for in the essay on probabilistic as well as non-probabilistic grounds, is used as the foundation of an argument against closed knowledge that can be used as a way to understand several of the most fundamental challenges of epistemology. Not only can an open knowledge view that is based on open evidence resolve all these problems in a simple and natural way, it can also respond to formidable challenges that significantly hinder other open knowledge views. There are good reasons, then, to view both knowledge and evidence as open. (shrink)

It is well-known that versions of the lottery paradox and of the preface paradox show that the following three principles are jointly inconsistent: (Sufficiency) very probable propositions are justifiably believable; (Conjunction Closure) justified believability is closed under conjunction introduction; (No Contradictions) propositions known to be contradictory are not justifiably believable. This paper shows that there is a hybrid of the lottery and preface paradoxes that does not require Sufficiency to arise, but only Conjunction Closure and (...) No Contradictions; and it argues that, given any plausible solution to this paradox, if one is not ready to deny Conjunction Closure (and analogous consistency principles), then one must endorse the thesis that justified believability is factive. (shrink)

We argue that giving up on the closure of rational belief under conjunction comes with a substantial price. Either rational belief is closed under conjunction, or else the epistemology of belief has a serious diachronic deficit over and above the synchronic failures of conjunctive closure. The argument for this, which can be viewed as a sequel to the preface paradox, is called the ‘review paradox'; it is presented in four distinct, but closely related versions.

In our previous paper Algebraic Logic for Classical Conjunction and Disjunction we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA and (...) of all distributive closure operators overA. In the present paper we study the lattice structure of this last set, give a description of its finite and infinite operations, and obtain a topological representation. We also apply the mentioned isomorphism and other results to obtain proofs with a logical flavour for several new or well-known lattice-theoretical properties, like Hashimoto's characterization of distributive lattices, and Priestley's topological representation of the congruence lattice of a bounded distributive lattice. (shrink)

Skeptical puzzles and arguments often employ knowledge-closure principles . Epistemologists widely believe that an adequate reply to the skeptic should explain why her reasoning is appealing albeit misleading; but it’s unclear what would explain the appeal of the skeptic’s closure principle, if not for its truth. In this paper, I aim to challenge the widespread commitment to knowledge-closure. But I proceed by first examining a new puzzle about failing to know—what I call the New Ignorance Puzzle . (...) This puzzle resembles to the Old Ignorance Puzzle , although it does not involve a closure principle. It instead centers on the standard view of ignorance, a highly intuitive principle stating that ignorance is merely a failure to know. In Sects. 2 and 3, I argue that the best way to solve the New Ignorance Puzzle is to reject the standard view of ignorance and to explain away its appeal via conversational implicature. I then use this solution to the New Ignorance Puzzle as a way of explaining why knowledge-closure principles would seem true, and why abominable conjunctions would seem abominable, even if such principles were false . The upshot is a new way of explaining why the skeptic’s reasoning is appealing albeit misleading. (shrink)

Take the following principle (or schema) as the focus of the ensuing discussion (“P” and “Q” are placeholders for propositions): 1 (Closure) If one knows P and competently deduces Q from P, thereby coming to believe Q, while retaining one's knowledge that P, one comes to know that Q. My strategy in outline: first, I want to set out Fred Dretske's classic challenge to (Closure) – a challenge which began in 1970–1971. Then I want to consider a specific, (...) recent counter‐challenge to Dretske's challenge to (Closure) mounted by John Hawthorne, and to defend Dretske's challenge from Hawthorne's counter‐challenge. Doing this is not to invalidate (Closure). In this regard my conclusions are modest: Dretske's challenge to (Closure) is – or, better: can be made – sophisticated and, so far, unmet. (shrink)

In this paper I defend the claim that justification is closed under conjunction, and confront its most alarming consequence — that one can have justification for believing propositions that are unlikely to be true, given one's evidence.

There are two ways of representing rational belief: qualitatively as yes-or-no belief, and quantitatively as degrees of belief. Standard rationality conditions are: consistency and logical closure, for qualitative belief, satisfaction of the probability axioms, for quantitative belief, and a relationship between qualitative and quantitative beliefs in accordance with the Lockean thesis. In this paper, it is shown that these conditions are inconsistent with each of three further rationality conditions: fallibilism, open-mindedness, and invariance under independent conceptual expansions. Restrictions of the (...) Lockean thesis that have been suggested in the literature cannot remove the inconsistency. In the outlook we discuss two alternative ways of dealing with this problem: restricting conjunctive closure or going for a dual system account. (shrink)

Epistemic justification has been widely accepted as both a gradational and relational notion. Given those properties, a natural thought is to take degrees of epistemic justification to be probabilities. In this paper, we present a simple Bayesian framework for justification. In the first part, after putting the model in an evidentialist form, we distinguish different senses of “being evidence for” and “confirming”. Next, we argue that this conception should accommodate the two relevant kinds of qualitative confirmation or evidential support. In (...) the second part of the paper, we discuss the claim that this view is unable to satisfy the modified version of the conjunction closure for beliefs in probabilistically independent propositions. We defend that the underlying assumption on which this objection depends leads to an improper reading of the concept of epistemic probability. After providing a better interpretation of it, we put forward a rationale, which is based on the notion of conditional uncertainty, in support of a more plausible and restricted version of the closure of justification under conjunction. (shrink)

In the literature, one finds two accounts of the normative status of rational belief: the ought account and the permissibility account. Both accounts have their advantages and shortcomings, making it difficult to favour one over the other. Imagine that there were two principles of rational belief or rational degrees of belief commonly considered plausible, but which, however, yielded a paradox together with one account, but not with the other. One of the accounts therefore requires us to give up one of (...) the plausible principles; whereas the other allows us to save them both. The fact that it allows us to save both of the plausible principles might well be considered a strong reason in favour of the relevant account. The permissibility-account-based resolution of the lottery paradox suggests that the permissibility account is a candidate for being supported in this way, since the account seems to save two plausible principles of rational belief and rational degrees of belief. I argue that even if the permissibility account were supported in this way the support would be defeated, since one cannot provide an analogous resolution of the preface paradox. The principles remain unsaved by the permissibility account. (shrink)

What is the epistemological structure of situations where many small risks amount to a large one? Lottery and preface paradoxes and puzzles about quantum-mechanical blips threaten the idea that competent deduction is a way of extending our knowledge. Seemingly, everyday knowledge involves small risks, and competently deducing the conjunction of many such truths from them yields a conclusion too risky to constitute knowledge. But the dilemma between scepticism and abandoning MPC is false. In extreme cases, objectively improbable truths are (...) known. Safety is modal, not probabilistic, in structure, with closure and factiveness conditions. It is modelled using closeness of worlds. Safety is analogous to knowledge. It suggests an interpretation of possible worlds semantics for epistemic logic. To avoid logical omniscience, a relation of epistemic counterparthood between formulas is introduced. This supports a safety conception of knowledge and formalizes how extending knowledge by deduction depends on logical competence. (shrink)

The conclusive reasons view of knowledge entails the “abominable conjunction” that I know that I have hands but I do not know that I am not a brain in a vat. The argument from abomination takes this as a reason to reject the view. This paper aims to buttress the argument from abomination by adding a new sort to this list: the logical abominations. These include: “I know that argument is sound and that sound arguments have true conclusions but (...) I don’t know whether the conclusion of that argument is true”. Two standard replies to the argument from abomination are raised. It is argued that the logical abominations open new holes in both. (shrink)

This paper explores some consequences of Lewis’s (Australas J Philos 74(4):549–567, 1996) understanding of how knowledge is compartmentalized. It argues, first, that he underestimates how badly it impacts his view. When knowledge is compartmentalized, it lacks at least one of two essential features of Lewis’s account: (a) Elusiveness—familiar skeptical possibilities, when relevant, are incompatible with everyday knowledge. (b) Knowledge is a modality—when a thinker knows that p, there is no relevant possibility where p is false. Lewis proposes compartmentalized knowledge to (...) keep treating knowledge as a modality while mitigating one of its unrealistic epistemological implications: In normal modal epistemic logic (and standard possible world semantics), a thinker always counts as knowing the strongest proposition that follows from the set of all the individual propositions that this thinker knows. Lewis’s compartmentalization proposal is that thinkers merely know the conjunctions of propositions that are known in each of, but not across, their compartments. The irony is that in avoiding overblown knowledge the view now allows for thinkers to attend to skeptical error possibilities and yet knowledge is present. The account avoids inflation of knowledge in one sense only to acquire another type of knowledge the view denies subjects can have. This problem motivates an inspection of knowledge accounts whose intra-compartment closure principles are weaker than those that are valid in normal modal-logic. The conclusion is that some formulations of closure can avoid the challenge Lewis’s view faces. Nevertheless, even these closure principles pose a barrier—perhaps an implausible barrier—for knowledge of ignorance. Even when the reasoning supporting the lack of knowledge is sound, a subject cannot always know she doesn’t know. Interestingly, this obstacle is one that knowledge of knowledge doesn’t seem to face. (shrink)

There is a deep tension between logical and probabilistic norms of belief. This article illustrates the normative force that is associated with these frameworks by showing how rather unrestricted belief bases can be improved by undergoing logical and probabilistic reflection. It is argued that probabilistic reasoning accounts for the reliability of the conclusions one can draw from the beliefs. Most importantly, reliability commands us to care for the increasing uncertainty of conjunctions of beliefs. Deductive logic captures the agent's commitment towards (...) all conclusions of her beliefs. In other words, it seems that commitment forces the agent to believe conclusions, which reliability tells her to view with scepticism. This conflict is resolved by arguing that commitment merely forces the agent to find the logical conclusions of her beliefs acceptable, not necessarily to believe them. This demands consistency but no deductive closure. Furthermore, it is shown that by restoring consistency, a straightforward connection between high subjective probability and full belief is undermined. Finally, the article delineates how this approach deals with the lottery paradox and the preface paradox. (shrink)

I explore some issues in the logics and dialectics of practical modalities connected with the Consequence Argument (CA) considered as the best argument for the incompatibility of free will and determinism. According to Lewis (1981) in one of the possible senses of (in)ability, the argument is not valid; however, understood in the other of its possible senses, the argument is not sound. This verdict is based on the assessment of the modal version of the argument, where the crucial notion is (...) power necessity (“no choice” operator), while Lewis analyses the version where the central notion is the locution “cannot render false.”Lewis accepts closure of the relevant (in)ability operator under entailment but not closure under implication. His strategy has a seemingly strange corollary: a free predetermined agent is able (in a strong, causal sense) to falsity the conjunction of history and law. I compare a Moorean position with respect to radical skepticism and knowledge closure with ability closure and propose to explain Lewis’s strategy in the framework of his Moorean stance. (shrink)

Recent work has considered the problem of extending to the case of iterated belief change the so-called `Harper Identity' (HI), which defines single-shot contraction in terms of single-shot revision. The present paper considers the prospects of providing a similar extension of the Levi Identity (LI), in which the direction of definition runs the other way. We restrict our attention here to the three classic iterated revision operators--natural, restrained and lexicographic, for which we provide here the first collective characterisation in the (...) literature, under the appellation of `elementary' operators. We consider two prima facie plausible ways of extending (LI). The first proposal involves the use of the rational closure operator to offer a `reductive' account of iterated revision in terms of iterated contraction. The second, which doesn't commit to reductionism, was put forward some years ago by Nayak et al. We establish that, for elementary revision operators and under mild assumptions regarding contraction, Nayak's proposal is equivalent to a new set of postulates formalising the claim that contraction by ¬A should be considered to be a kind of `mild' revision by A. We then show that these, in turn, under slightly weaker assumptions, jointly amount to the conjunction of a pair of constraints on the extension of (HI) that were recently proposed in the literature. Finally, we consider the consequences of endorsing both suggestions and show that this would yield an identification of rational revision with natural revision. We close the paper by discussing the general prospects for defining iterated revision in terms of iterated contraction. (shrink)

En este artículo, considero algunos supuestos costos intuitivos relativos a la negación de la generalidad del Principio de Cierre para el conocimiento. Usualmente los filósofos descartan tal negación como altamente contra-intuitiva pero argumento que, por lo menos en relación a los supuestos costos aquí considerados, esto es incorrecto: dadas nuestras intuiciones folk, no hay tales costos. Por lo tanto un falibilista que busca detener el argumento escéptico basado en el principio de cierre puede restringir el principio sin sufrir esos supuestos (...) costos intuitivos. (shrink)

The concept of being in a position to know is an increasingly popular member of the epistemologist’s toolkit. Some have used it as a basis for an account of propositional justification. Others, following Timothy Williamson, have used it as a vehicle for articulating interesting luminosity and anti-luminosity theses. It is tempting to think that while knowledge itself does not obey any closure principles, being in a position to know does. For example, if one knows both p and ‘If p (...) then q’, but one dies or gets distracted before being able to perform a modus ponens on these items of knowledge and for that reason one does not know q, one is still plausibly in a position to know q. It is also tempting to suppose that, while one does not know all logical truths, one is nevertheless in a position to know every logical truth. Putting these temptations together, we get the view that being in a position to know has a normal modal logic. A recent literature has begun to investigate whether it is a good idea to give in to these twin temptations—in particular the first one. That literature assumes very naturally that one is in a position to know everything one knows and that one is not in a position to know things that one cannot know. It has succeeded in showing that, given the modest closure condition that knowledge is closed under conjunction elimination, being a position to know cannot satisfy the so-called K axiom of normal modal logics. In this paper, we explore the question of the normality of the logic of being in a position to know in a more far-reaching and systematic way. Assuming that being in a position to know entails the possibility of knowing and that knowing entails being in a position to know, we can demonstrate radical failures of normality without assuming any closure principles at all for knowledge. Moreover, the failure of normality cannot be laid at the door of the K axiom for knowledge, since the standard principle NEC of necessitation also fails for being in a position to know. After laying out and explaining our results, we briefly survey the coherent options that remain. (shrink)

At least since Aristotle’s famous 'sea-battle' passages in On Interpretation 9, some substantial minority of philosophers has been attracted to the doctrine of the open future--the doctrine that future contingent statements are not true. But, prima facie, such views seem inconsistent with the following intuition: if something has happened, then (looking back) it was the case that it would happen. How can it be that, looking forwards, it isn’t true that there will be a sea battle, while also being true (...) that, looking backwards, it was the case that there would be a sea battle? This tension forms, in large part, what might be called the problem of future contingents. A dominant trend in temporal logic and semantic theorizing about future contingents seeks to validate both intuitions. Theorists in this tradition--including some interpretations of Aristotle, but paradigmatically, Thomason (1970), as well as more recent developments in Belnap, et. al (2001) and MacFarlane (2003, 2014)--have argued that the apparent tension between the intuitions is in fact merely apparent. In short, such theorists seek to maintain both of the following two theses: (i) the open future: Future contingents are not true, and (ii) retro-closure: From the fact that something is true, it follows that it was the case that it would be true. It is well-known that reflection on the problem of future contingents has in many ways been inspired by importantly parallel issues regarding divine foreknowledge and indeterminism. In this paper, we take up this perspective, and ask what accepting both the open future and retro-closure predicts about omniscience. When we theorize about a perfect knower, we are theorizing about what an ideal agent ought to believe. Our contention is that there isn’t an acceptable view of ideally rational belief given the assumptions of the open future and retro-closure, and thus this casts doubt on the conjunction of those assumptions. (shrink)

The paper critically discusses two prominent arguments against closure principles for knowledge. The first one is the “argument from aggregation”, claiming that closure under conjunction has the consequence that, if one individually knows i premises, one also knows their i-fold conjunction—yet, every one of the premises might exhibit interesting positive epistemic properties while the i-fold conjunction might fail to do so. The second one is the “argument from concatenation”, claiming that closure under entailment has (...) the consequence that, if one knows a premise, one also knows each of its remote consequences one arrives at—yet, again, the premise might exhibit interesting positive epistemic properties while some of its remote consequences might fail to do so. The paper firstly observes that the ways in which these two arguments try to establish that the relevant closure principle has the relevant problematic consequence are strikingly similar. They both crucially involve showing that, given the features of the case, the relevant closure principle acts in effect as a soritical principle, which is in turn assumed to lead validly to the relevant problematic consequence. There are however nontransitive logics of vagueness where soritical principles do not have any problematic consequence. Assuming that one of these logics is the correct logic of vagueness, the paper secondly observes that both arguments describe situations where knowledge is arguably vague in the relevant respects, so that a tolerant logic should be used in reasoning about it, with the effect that the relevant soritical principle no longer validly leads to the relevant problematic consequence. This shows an interesting respect in which the gap between validity and good inference that arguably arises in a transitive framework can be bridged in a tolerant one, thereby approximating better certain features of our epistemic lives as finite subjects. Moreover, even for those who do not subscribe to tolerant logics, the paper’s two observations jointly indicate that, for all the arguments from aggregation and concatenation show, the status of the relevant closure principles should be no worse than that assigned by one’s favoured theory of vagueness to soritical principles, which only rarely is plain falsity and can indeed get arbitrarily close to full truth. (shrink)

Twin-earth thought experiments, standardly construed, support the externalist doctrine that the content of propositional attitudes involving natural-kind terms supervenes upon properties external to those who entertain them. But this doctrine in conjunction with a common view of self-knowledge might have the intolerable consequence that substantial propositions concerning the environment could be knowable a priori. Since both doctrines, externalism and privileged self-knowledge, appear independently plausible, there is then a paradox facing the attempt to hold them concurrently. I shall argue, however, (...) that externalist claims about the dependence of content on environmental factors presuppose certain theses about the semantics of natural-kind terms that, if sound, would make those claims eligible for empirical justification instead. In fact, that is the only interpretation of their epistemic status that could square with the standard conclusion from twin-earth cases. Furthermore, it can be shown to solve the paradox of externalism and self-knowledge in a more doxastically conservative way--accommodating precisely each of the well-accepted intuitions about knowledge, closure and the individuation of content given up by available competitors. (shrink)

According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A deductive system is a second-order matrix over the formula algebra of some fixed but arbitrary language. A second-order matrix A is (...) Fregean if, for any subset X of A, the set of all pairs a,b such that X{a} and X{b} have the same closure is a congruence relation on A. Hence a deductive system is Fregean if interderivability is compositional. The logics intermediate between the classical and intuitionistic propositional calculi are the paradigms for Fregean logics. Normal modal logics are non-Fregean while quasi-normal modal logics are generally Fregean.The main results of the paper: Fregean deductive systems that either have the deduction theorem, or are protoalgebraic and have conjunction, are completely characterized. They are essentially the intermediate logics, possibly with additional connectives. All the full matrix models of a protoalgebraic Fregean deductive system are Fregean, and, conversely, the deductive system determined by any class of Fregean second-order matrices is Fregean. The latter result is used to construct an example of a protoalgebraic Fregean deductive system that is not strongly algebraizable. (shrink)

The paper explains how Gettier’s conclusion can be reached on general theoretical grounds within the framework of epistemic logic, without reliance on thought experiments. It extends the argument to permissive conceptions of justification that invalidate principles of multi-premise closure and require neighbourhood semantics rather than semantics of a more standard type. The paper concludes by recommending a robust methodology that aims at convergence in results between thought experimentation and more formal methods. It also warns against conjunctive definitions as sharing (...) several of the drawbacks of disjunctive definitions. (shrink)

Dave Ripley has recently argued against the plausibility of multiset consequence relations and of contraction-free approaches to paradox. For Ripley, who endorses a nontransitive theory, the best arguments that buttress transitivity also push for contraction—whence it is wiser for the substructural logician to go nontransitive from the start. One of Ripley’s allegations is especially insidious, since it assumes the form of a trivialisation result: it is shown that if a multiset consequence relation can be associated to a closure operator (...) in the expected way, then it necessarily contracts. We counter Ripley’s objection by presenting an approach to multiset consequence that escapes this trap. This approach is multiple-conclusioned in a heterodox way, for multiple succedents are given a conjunctive, rather than a disjunctive reading. Finally, we address a further objection by French and Ripley to the effect that the informational interpretation of sequents in linear logic does not motivate cut. (shrink)

The aim of this paper is technically to study Belnap's four-valued sentential logic . First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for (...) negation. It is also proved that Belnap's logic is the only sentential logic satisfying the above-mentioned conditions together with Anderson-Belnap's Variable-Sharing Property. Finally, we obtain a finite Hilbert-style axiomatization of this logic. As a consequence, we obtain a finite Hilbert-style axiomatization of Priest's logic of paradox. (shrink)

Two proofs are given which show that if some set of truths fall under finitely many concepts (so-called Collectivity), then they all fall under at least one of them even if we do not know which one. Examples are given in which the result seems paradoxical. The first proof crucially involves Moorean propositions while the second is a reconstruction and generalization of a proof due to Humberstone free from any reference to such propositions. We survey a few solution routes including (...) Tennant-style restriction strategies. It is concluded that accepting Collectivity for some set of truths while also denying that any of the involved concepts in isolation capture all of them requires that one of these concepts cannot be closed under conjunction elimination. This is surprising since the paper surveys several applications in which Collectivity and the latter closure condition seemed jointly satisfiable for concepts of actual philosophical interest. (shrink)

Fitch's Paradox and the Paradox of the Knower both make use of the Factivity Principle. The latter also makes use of a second principle, namely the Knowledge-of-Factivity Principle. Both the principle of factivity and the knowledge thereof have been the subject of various discussions, often in conjunction with a third principle known as Closure. In this paper, we examine the well-known Surprise Examination paradox considering both the principles on which this paradox rests and some formal characterisations of the (...) surprise notion, crucial in this paradox. Standard formalizations of the Surprise Examination paradox in modal logic do not seem, at first glance, to depend on either factivity or knowledge-of-factivity, but we will argue that both factivity and knowledge-of-factivity play a key implicit role in the paradox. Namely, they are implicitly, perhaps unintentionally, used in order to simplify the definition of surprise. We analyze modal logical formalizations of three versions of the paradox concluding that the Surprise Examination paradox is the result of two flaws: the assumption of knowledge-of-factivity, and the over-simplification of the definition of "surprise" accordingly. By fixing these two flaws, the Surprise Examination paradox vanishes. (shrink)

It seems reasonable to seek a definition of naturalism, yet an accurate general definition proves to be elusive. After considering proposals from Quine, Nagel, and Chalmers, I propose that naturalism as understood by the majority of contemporary naturalists is best defined by the conjunction of mind-body supervenience, an understanding of the physical as mechanistic, and the causal closure of the physical domain. I then argue that naturalism so defined is in principle unable to account for the existence of (...) rationality; it follows that naturalism must be rejected. (shrink)

In this chapter I look at some questions around the notion of experimentation in philosophy, science, and the arts, through the thought of Gaston Bachelard and Gilles Deleuze. My argument is articulated around three areas of enquiry – Bachelard’s work on the experimental sciences, Deleuze’s notion of philosophy as an experimental practice, and recent musicological debate around the practical and political stakes of the term ‘experimental music’. By drawing together these three senses of experimentation, I test the possibilities of understanding (...) experimentation as a transdisciplinary concept and/or method. I develop a notion of experimentation as open, fluid, and non-hierarchical, but also consider points where such an idea is short-circuited by the reassertion of disciplinary closure and more top-down forms of method. My frame for discussing this question is a commonly posited distinction between the experiment and the experimental. Here the experiment is something like a controlled and closed environment in which a privileged observer tests predefined hypotheses, while the experimental concerns attempts to relinquish such control and to produce contexts in which the unknown and the unexpected can arise. By turning to Bachelard’s studies of the practice of science, I will question the common conception of a disciplinary split between the experiments of science and experimental art, showing both how such a distinction cannot be so neat and how these terms are often not easily separable. Putting this notion into conjunction with recent critical discourse on experimentation in music, namely regarding the kinds of exclusions and closures that the term ‘experimental music’ has produced, and with Deleuze’s criticisms of scientific method as well as the apparent disciplinary closure of his transdisciplinary project that is present and his and Félix Guattari’s final work, What is Philosophy?, I argue that refining our understanding of experimentation as a pluralistic and fragile concept will help us engage with the difficulties raised in these fields. More generally I point towards a project of mapping out the diverse and divergent relations that a transdisciplinary understanding of experimentation may draw between philosophy, science, and art. (shrink)

We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of _T_, _S4_, and _S5_, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment (...) of such a dynamic epistemic logic, we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of “nested” or “tree-sequent” calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario. (shrink)

I argue that there are living and everyday case in which rationality requires you, as a non-idealized human thinker, to have inconsistent beliefs while recognizing the inconsistency. I defend my argument against classical and insightful objections by Doris Olin, as well as others. I consider three versions of the preface paradox as candidate cases, including Makinson’s original version. None is free from objection. However, there is a fourth version, Modesty, that supposes that you believe that at least one of your (...) beliefs is false. I argue that this version escapes all the objections that could trouble the other versions, including the objection that given certain closure principles for justified belief, justified inconsistent beliefs saddle you with a pair of justified beliefs that are in explicit contradiction. I also argue more tentatively for the same verdict for Modesty*, a version that supposes that you believe that at least one of your beliefs is false. In that case, your belief guarantees that its content is true, in which case your beliefs are inconsistent, because at least one of them must be false. Once you think you’re wrong, you must be right! Crucial to my argument is a distinction between explicitly contradictory beliefs and three forms of inconsistency in belief. In the first of these, you believe contingent propositions each of which is logically independent of the others, yet you also believe the syntactic negation of their conjunction. In the second, your beliefs are inconsistent because of the rigid designation of demonstratives such as ‘this’ embedded in their contents. In the third, as in Modesty and Modesty* your beliefs are inconsistent because of self-reference. (shrink)

The purpose of the paper is to show that by cleaning Classical Logic (CL) from redundancies (irrelevances) and uninformative complexities in the consequence class and from too strong assumptions (of CL) one can avoid most of the paradoxes coming up when CL is applied to empirical sciences including physics. This kind of cleaning of CL has been done successfully by distinguishing two types of theorems of CL by two criteria. One criterion (RC) forbids such theorems in which parts of the (...) consequent (conclusion) can be replaced by arbitrary parts salva validitate of the theorem. The other (RD) reduces the consequences to simplest conjunctive consequence elements. Since the application of RC and RD to CL leads to a logic without the usual closure conditions, an approximation to RC and RD has been constructed by a basic logic with the help of finite (6-valued) matrices. This basic logic called RMQ (relevance, matrix, Quantum Physics) is consistent and decidable. It distinguishes two types of validity (strict validity) and classical or material validity. All theorems of CL (here: classical propositional calculus CPC) are classically or materially valid in RMQ. But those theorems of CPC which obey RC and RD and avoid the difficulties in the application to empirical sciences and to Quantum Physics are separated as strictly valid in RMQ. In the application to empirical sciences in general the proposed logic avoids the well known paradoxes in the area of explanation, confirmation, versimilitude and Deontic Logic. Concerning the application to physics it avoids also the difficulties with distributivity, commensurability and with Bell's inequalities. (shrink)

In my view, meanings are instructions to construct monadic concepts that can be conjoined with others, given a few thematic relations and an operation of existential closure. For example, ‘red ball’ is understood as—and has the semantic property of being—an instruction to fetch and conjoin two concepts that are linked, respectively, to ‘red’ and ‘ball’. Other expressions are more complex. But to a first approximation, ‘I stabbed it violently with this’ is an instruction to construct and existentially close a (...) six-conjunct concept of the form indicated below; AGENT(E, S) & STAB(E) & BEFORE(E, T) & PATIENT(E, 1) & VIOLENT(E) & INSTRUMENT(E, 2) where ‘S’, ‘T’, ‘1’ and ‘2’ stand for concepts of the relevant speaker, time, and things demonstrated. The verb and adverb correspond directly to conjoinable concepts of events. The pronouns correspond to such concepts via certain relational notions, reflected with the tense, preposition, and grammatical relations that ‘stab’ bears to its subject and object. I have argued elsewhere that this neo-Davidsonian conception of semantics is descriptively adequate, and yet. (shrink)

The conceptual history of the term “organization” begins in Medieval times with the reception and transformation of Aristotle’s philosophy of life. It designates the corporeal structure and conditions of identity of natural “organic bodies,” a term that had been used to refer to living beings since antiquity. The term played an important role in specifying the ontological status of living beings. At the same time, it offered a basis for their mechanistic understanding. Starting with mechanistic models of life in the (...) second half of the seventeenth century, “organization” and “life” were increasingly used interchangeably. This conjunction of meaning transformed “living beings” into “organisms.” Within physiological accounts of the eighteenth century, the living organization was compared to a causal cycle of interdependency. Philosophically, this conjunction was adapted at the end of the century in Kant’s philosophy of “organized beings of nature” in which he located the idea of causal cyclicity within a teleological framework and specified an “organized being” in causal terms as a system of interacting and interdependent parts characterized by functional closure. Thus, “organization” refers to the constitution of living beings as a particular kind of causal system. In the nineteenth century, the term achieves the status of a signal word for the life sciences and starts being applied in a wide variety of contexts, from comparative anatomy to physiology and ecology. It was supplemented by two other fundamental notions, namely, “regulation” and “evolution,” the first referring to the stabilization and the second to the long-term transformation of natural organizations. The twentieth century saw a further intensification of the complementarity of the perspectives associated with these three terms. Finally, in recent years, a substantial improvement in understanding the causal structure of “organization” was achieved by analyzing it in terms of the “closure of constraints.”. (shrink)

We will present a new lottery-style paradox on counterfactuals and chance. The upshot will be: combining natural assumptions on the truth values of ordinary counterfactuals, the conditional chances of possible but non-actual events, the manner in which and relate to each other, and a fragment of the logic of counterfactuals leads to disaster. In contrast with the usual lottery-style paradoxes, logical closure under conjunction—that is, in this case, the rule of Agglomeration of counterfactuals—will not play a role in (...) the derivation and will not be entailed by our premises either. We will sketch four obvious but problematic ways out of the dilemma, and we will end up with a new resolution strategy that is non-obvious but less problematic: contextualism about what counts as a proposition. This proposal will not just save us from the paradox, it will also save each premise in at least some context, and it will be motivated by independent considerations from measure theory and probability theory. (shrink)

Let L ω ∞ω be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let p(n) be the edge probability of the random graph on n vertices. It is shown that if p(n) ≪ n -1 satisfies certain simple conditions on its growth rate, then for every σ∈ L ω ∞ω , the probability that σ holds for (...) the random graph on n vertices converges. In fact, if $p(n) = n^{-\alpha}, \alpha > 1$ , then the probability is either smaller than 2 -n d for some $d > 0$ , or it is asymptotic to cn -d for some $c > 0, d \geq 0$ . Results on the difficulty of computing the asymptotic probability are given. (shrink)

The article approaches cut elimination from a new angle. On the basis of an arbitrary inference relation among logically atomic formulae, an inference relation on a language possessing logical operators is defined by means of inductive clauses similar to the operator-introducing rules of a cut-free intuitionistic sequent calculus. The logical terminology of the richer language is not uniquely specified, but assumed to satisfy certain conditions of a general nature, allowing for, but not requiring, the existence of infinite conjunctions and disjunctions. (...) We investigate to what extent structural properties of the given atomic relation are preserved through the extension to the full language. While closure under the Cut rule narrowly construed is not in general thus preserved, two properties jointly amounting to closure under the ordinary structural rules, including Cut, are. -/- Attention is then narrowed down to the special case of a standard first-order language, where a similar result is obtained also for closure under substitution of terms for individual parameters. Taken together, the three preservation results imply the familiar cut-elimination theorem for pure first-order intuitionistic sequent calculus. -/- In the interest of conceptual economy, all deducibility relations are specified purely inductively, rather than in terms of the existence of formal proofs of any kind. (shrink)

Modal arguments for incompatibility of freedom and determinism are typically based on the “transfer principle” for inability to act otherwise (Beta). The principle of agglomerativity (closure under conjunction introduction) is derivable from Beta. The most convincing counterexample to Beta is based on the denial of Agglomeration. The defender of the modal argument has two ways to block counterexamples to Beta: (i) use a notion of inability to act otherwise which is immune to the counterexample to agglomerativity; (ii) replace (...) Beta with a logically stronger principle Beta 2. I argue that the second strategy fails because the strengthened principle and Agglomeration together entail Beta. So this strategy makes sense only if Beta 2 is applied without Agglomeration. But if Beta 2 is used without Agglomeration, then the incompatibilist will undercut the rationale for the premise of his argument. I illustrate this point with the analysis of Warfield (1996) and his use of Beta 2 in the so called direct argument for incompatibilism. (shrink)

This paper aims to argue in support of the neo-Moorean attempt to solve a skeptical paradox. It defends the thesis that neo-Mooreans retain advantages and avoid disadvantages of rival anti-skeptical strategies – namely epistemic contextualism. The puzzle that a radical skeptic poses is exemplified by Nozick’s famous Brain in a Vat thought experiment, which enables construing valid arguments consisting of jointly inconsistent but independently plausible premises. The first and the second part of the paper are devoted to Nozick’s conditional analysis (...) of knowledge and De Rose’s epistemic contextualism, both based on the sensitivity principle. Referring to De Roses’ contextualist theory, we demonstrate that the failure of Nozick’s conditional analysis of knowledge to provide a satisfactory answer to a skeptical paradox does not concern the sensitivity principle but rather closure denial and embracing the so-called “abominable conjunction”. In the third part, we point out the weaknesses of the presumably most successful, contextualist response to the paradox. We explain that even though DeRose’s anti-skeptical strategy is built upon Nozick’s theory, he successfully surmounts its difficulties. Yet it seems that as a contextualist, he necessarily makes some concessions to a radical skeptic. Eventually, the article introduces Black’s neo-Moorean anti-skeptical theory based on the sensitivity principle as a strategy that makes neither concessions, nor counterintuitive proposals. (shrink)

It has been claimed that there is a lottery paradox for justification and an analogous paradox for knowledge, and that these two paradoxes should have a common solution. I argue that there is in fact no lottery paradox for knowledge, since that version of the paradox has a demonstrably false premise. The solution to the justification paradox is to deny closure of justification under conjunction. I present a principle which allows us to deny closure of justification under (...) conjunction in certain kinds of cases, but which still allows that belief in a conjunction on the basis of justified belief in its conjuncts can often be justified. (shrink)

This manuscript presents a topological argumentation framework for modelling notions of evidence-based (i.e., justified) belief. Our framework relies on so-called topological evidence models to represent the pieces of evidence that an agent has at her disposal, and it uses abstract argumentation theory to select the pieces of evidence that the agent will use to define her beliefs. The tools from abstract argumentation theory allow us to model agents who make decisions in the presence of contradictory information. Thanks to this, it (...) is possible to define two new notions of beliefs, grounded beliefs and fully grounded beliefs. These notions are discussed in this paper, analysed and compared with the existing notion of topological justified belief. This comparison revolves around three main issues: closure under conjunction introduction, the level of consistency and their logical strength. (shrink)

n + 1 nested k-ary fixed point operators are more expressive than n. This holds on finite structures for all sublogics of partial fixed point logic PFP that can express conjunction, existential quantification and deterministic transitive closure of binary relations using at most k-ary fixed point operators and that are closed against subformulas. Among those are a lot of popular fixed point logics.

Thomas Bradwardine's solution to the semantic paradoxes, presented in his Insolubilia written in Oxford in the early 1320s, turns on two main principles: that a proposition is true only if things are wholly as it signifies; and that signification is closed under consequence. After exploring the background in Walter Burley's account of the signification of propositions, the question is considered of the extent to which Bradwardine's theory is compatible with the distribution of truth over conjunction, disjunction, negation and the (...) conditional. (shrink)