But nathelees, syn I knowe youre delit, / I shal fulfille youre worldly appetit.1 Chaucer’s Wife of Bath centers on a wonderfully fruitful paradox: she claims for women and for herself the right to “maistrie” and “sovereynetee” in marriage, but she does so by articulating the discourse imparted to her by the “auctoritee” of anti-feminism.2 Indeed, this paradoxical challenge to and reiteration of anti-feminist ideas has left Chaucer’s readers debating for decades which way the irony cuts: is the Wife (...) to be understood as a proto-feminist, or is she merely “a delightful buffoon inadvertently lampooning herself for the ironic pleasure of a knowing, male audience”?3 How we choose to answer this question .. (shrink)
In his famous argument for the unreality of time, McTaggart claims that i) being past, being present, and being future are incompatible properties of an event, yet ii) every event admits all these three properties. In this paper, I examine two key concepts involved in the formulation of i) and ii), namely that of “validity” and that of “contradiction”, and for each concept I distinguish a static version and a dynamic version of it. I then arrive at three different (...) ways of formulating McTaggart’s claims that avoid the notorious McTaggart’s Paradox. So long as we demand that McTaggart make clear use/mention and token/type distinctions in his claims, we shall find that it is indeed very difficult for him to get a genuine contradiction from i) and ii). (shrink)
In his review of The Ontology of Time, Thomas Crisp (Notre Dame Philosophical Reviews, 2005a ) argues that Oaklander's version of McTaggart's paradox does not make any trouble for his version of presentism. The aim of this paper is to refute that claim by demonstrating that Crisp's version of presentism does indeed succumb to a version of McTaggart's argument. I shall proceed as follows. In Part I I shall explain Crisp's view and then argue in Part II (...) that his analysis of temporal becoming, temporal properties and temporal relations is inadequate. Finally, in Part III, I shall demonstrate that his presentist ontology of time is susceptible to the paradox he so assiduously sought to avoid. (shrink)
Since McTaggart first proposed his paradox asserting the unreality of time, numerous philosophers have attempted to defend the tensed theory of time against it. Certainly, one of the most highly developed and original is that put forth by Quentin Smith. Through discussing McTaggart's positive conception of time as well as his negative attack on its reality, I hope to clarify the dispute between those who believe in the existence of the transitory temporal properties of pastness, presentness and (...) futurity, and those who deny their existence. We shall see that the debate centers around the ontological status of succession and the B-relations of earlier and later. I shall argue that Smith's tensed theory fails because he cannot account for the sense in which events have their tensed properties successively, and he cannot account for the direction of time. (shrink)
This short article aims to illustrate the mutually question-begging arguments that are often presented in debates between opponents and defenderss of McTaggart’s “proof” that A-properties (pastness, presentness and futurity) are logically incoherent. A sample of such arguments is taken from a recent debate between L. Nathan Oaklander (a defender of McTaggart) and myself (an opponent of McTaggart) and a method of escaping the impasse that is often reached in such debates is suggested.
We shall evaluate two strategies for motivating the view that knowledge is the norm of belief. The first draws on observations concerning belief's aim and the parallels between belief and assertion. The second appeals to observations concerning Moore's Paradox. Neither of these strategies gives us good reason to accept the knowledge account. The considerations offered in support of this account motivate only the weaker account on which truth is the fundamental norm of belief.
In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, (...) but not jointly, lack the problematic feature. (shrink)
A variation of Fitch’s Paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s Paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out (...) of the paradox. (shrink)
In their development of causal decision theory, Allan Gibbard and William Harper advocate a particular method for calculating the expected utility of an action, a method based upon the probabilities of certain counterfactuals. Gibbard and Harper then employ their method to support a two-box solution to Newcomb’s paradox. This paper argues against some of Gibbard and Harper’s key claims concerning the truth-values and probabilities of counterfactuals involved in expected utility calculations, thereby disputing their analysis of Newcomb’s Paradox. If (...) we are right, then Gibbard and Harper’s method of calculating expected utility does not adequately represent rational choice. (shrink)
This article criticises one of Stuart Rachels' and Larry Temkin's arguments against the transitivity of 'better than'. This argument invokes our intuitions about our preferences of different bundles of pleasurable or painful experiences of varying intensity and duration, which, it is argued, will typically be intransitive. This article defends the transitivity of 'better than' by showing that Rachels and Temkin are mistaken to suppose that preferences satisfying their assumptions must be intransitive. It makes cler where the argument goes wrong by (...) showing that it is a version of Zeno's paradox of Achilles and the Tortoise. (shrink)
It is proved that Yablo’s paradox and the Liar paradox are equiparadoxical, in the sense that their paradoxicality is based upon exactly the same circularity condition—for any frame ${\mathcal{K}}$ , the following are equivalent: (1) Yablo’s sequence leads to a paradox in ${\mathcal{K}}$ ; (2) the Liar sentence leads to a paradox in ${\mathcal{K}}$ ; (3) ${\mathcal{K}}$ contains odd cycles. This result does not conflict with Yablo’s claim that his sequence is non-self-referential. Rather, it gives Yablo’s (...)paradox a new significance: his construction contributes a method by which we can eliminate the self-reference of a paradox without changing its circularity condition. (shrink)
“Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox.
In Newcomb’s paradox you can choose to receive either the contents of a particular closed box, or the contents of both that closed box and another one. Before you choose though, an antagonist uses a prediction algorithm to accurately deduce your choice, and uses that deduction to fill the two boxes. The way they do this guarantees that you made the wrong choice. Newcomb’s paradox is that game theory’s expected utility and dominance principles appear to provide conflicting recommendations (...) for what you should choose. Here we show that the conflicting recommendations assume different probabilistic structures relating your choice and the algorithm’s prediction. This resolves the paradox: the reason there appears to be two conflicting recommendations is that the probabilistic structure relating the problem’s random variables is open to two, conflicting interpretations. We then show that the accuracy of the prediction algorithm in Newcomb’s paradox, the focus of much previous work, is irrelevant. We end by showing that Newcomb’s paradox is time-reversal invariant; both the paradox and its resolution are unchanged if the algorithm makes its ‘prediction’ after you make your choice rather than before. (shrink)
Is there a Moore’s paradox in desire? I give a normative explanation of the epistemic irrationality, and hence absurdity, of Moorean belief that builds on Green and Williams’ normative account of absurdity. This explains why Moorean beliefs are normally irrational and thus absurd, while some Moorean beliefs are absurd without being irrational. Then I defend constructing a Moorean desire as the syntactic counterpart of a Moorean belief and distinguish it from a ‘Frankfurt’ conjunction of desires. Next I discuss putative (...) examples of rational and irrational desires, suggesting that there are norms of rational desire. Then I examine David Wall’s groundbreaking argument that Moorean desires are always unreasonable. Next I show against this that there are rational as well as irrational Moorean desires. Those that are irrational are also absurd, although there seem to be absurd desires that are not irrational. I conclude that certain norms of rational desire should be rejected. (shrink)
In this essay I will examine the role that intuition plays in Russell's parado; showing how different appraaches to intuition will license different treatments of the paradox. In addition, I will argue for a specific approach to the paradox, one that follows from the most plausible account of intuition. On this account, intuitions, though fallible, have episternic import. In addition, the intuitions involved in paradoxes point to something wrong with concept that leads to paradox. In the case (...) of Russell's paradox, this is an ambiguity in the notion of a class. (shrink)
There is an argument (first presented by Fitch), which tries to show by formal means that the anti-realistic thesis that every truth might possibly be known, is equivalent to the unacceptable thesis that every truth is actually known (at some time in the past, present or future). First, the argument is presented and some proposals for the solution of Fitch's Paradox are briefly discussed. Then, by using Wehmeier's modal logic with subjunctive marks (S5*), it is shown how the derivation (...) can be blocked if one respects adequately the distinction between the indicative and the subjunctive mood. Essentially, this proposal amounts to the one by Edgington which was formulated with the help of the actuality-operator. Finally it is shown how the criticisms by Williamson against Edgington can be answered by the formulation of a new conception of possible knowledge that \alpha (thereby \alpha being in the indicative mood and thus referring to the actual world). This conception is based on the concept of same de re knowledge in different possible worlds. (shrink)
Propositions such as <It is raining, but I do not believe that it is raining> are paradoxical, in that even though they can be true, they cannot be truly asserted or believed. This is Moore’s paradox. Sydney Shoemaker has recently ar- gued that the paradox arises from a constitutive relation that holds between first- and second-order beliefs. This paper explores this approach to the paradox. Although Shoemaker’s own account of the paradox is rejected, a different account (...) along similar lines is endorsed. At the core of the endorsed account is the claim that conscious beliefs are always partly about themselves; it will be shown to follow from this that conscious beliefs in Moorean propositions are self-contradictory. (shrink)
A number of authors have noted that the key steps in Fitch’s argument are not intuitionistically valid, and some have proposed this as a reason for an anti-realist to accept intuitionistic logic (e.g. Williamson 1982, 1988). This line of reasoning rests upon two assumptions. The first is that the premises of Fitch’s argument make sense from an anti-realist point of view – and in particular, that an anti-realist can and should maintain the principle that all truths are knowable. The second (...) is that we have some independent reason for thinking that classical logic is not appropriate in this area. This paper explores these two assumptions in the context of Michael Dummett’s version of anti-realism, with particular reference to the argument from indefinite extensibility developed at various points in Dummett’s writings (e.g. Dummett 1991 Ch. 24). -/- Dummett argues that certain concepts, the indefinitely extensible concepts, are such that we cannot form a clear and determinate conception of all the objects that fall under them. The most familiar examples of indefinitely extensible concepts are mathematical. Dummett discusses the concepts ordinal number, real number, and natural number, which are indefinitely extensible because any conception that one might form of their complete extension can be extended to a more inclusive conception (as, for example, in Cantor’s proof of the non-denumerability of the set of real numbers). This paper argues that the concept of a truth is indefinitely extensible. This gives a Dummettian anti-realist an independent motivation for rejecting the classical understanding of the quantifiers in this area. At the same time, however, it places in doubt the admissibility of the knowability principle, which seems to involve quantification over the “totality” of truths. As Dummett is at pains to point out (1991: 316), some sentences that purport to quantify over the extension of an indefinitely extensible concept plainly have a truth-value (we can truly say, for example, that every ordinal number has a successor, even though when we say that we are not quantifying over the set of all ordinals). But is the knowability principle one of these sentences? (shrink)
Saul Kripke is struck by a skeptical argument which he says is neither Wittgenstein’s nor his own. I call this new skeptic “Saul Wittgenstein”. SW’s conclusion is that there is no such thing as following a rule. My first aim is to show that Kripke misunderstands the Investigations when he says it offers a “skeptical solution” to SW’s paradox. Wittgenstein’s view of philosophy commits him to a dissolution of the paradox. I show next that LW’s writing contains an (...) implicit dissolution of it. Finally, I point out the main lesson to be derived from Kripke’s discussion--namely, that there is nothing which is common and peculiar to what we call following a rule. (shrink)
The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn't possibly be known. More specifically, if p is a truth that is never known then it is unknowable that p is a truth that is never known. The proof has been used to (...) argue against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge. Still others have taken the proof to be fallacious, since it collapses an apparently moderate brand of anti-realism into an obviously implausible and naive idealism. (shrink)
This paper argues that justification is accessible in the sense that one has justification to believe a proposition if and only if one has higher-order justification to believe that one has justification to believe that proposition. I argue that the accessibility of justification is required for explaining what is wrong with believing Moorean conjunctions of the form, ‘p and I do not have justification to believe that p.’.
We provide an overview of consistent fragments of the theory of Frege’s Grundgesetze der Arithmetik that arise by restricting the second-order comprehension schema. We discuss how such theories avoid inconsistency and show how the reasoning underlying Russell’s paradox can be put to use in an investigation of these fragments.
I offer an interpretation and a partial defense of Kit Fine's ‘Argument from Passage’, which is situated within his reconstruction of McTaggart's paradox. Fine argues that existing A-theoretic approaches to passage are no more dynamic, i.e. capture passage no better, than the B-theory. I argue that this comparative claim is correct. Our intuitive picture of passage, which inclines us towards A-theories, suggests more than coherent A-theories can deliver. In Finean terms, the picture requires not only Realism about tensed (...) facts, but also Neutrality, i.e. the tensed facts not being ‘oriented towards’ one privileged time. However unlike Fine, and unlike others who advance McTaggartian arguments, I take McTaggart's paradox to indicate neither the need for a more dynamic theory of passage nor that time does not pass. A more dynamic theory is not to be had: Fine's ‘non-standard realism’ amounts to no more than a conceptual gesture. But instead of concluding that time does not pass, we should conclude that theories of passage cannot deliver the dynamicity of our intuitive picture. For this reason, a B-theoretic account of passage that simply identifies passage with the succession of times is a serious contender. (shrink)
I show how the 'innersense' (quasiperceptual) view of introspection can be defended against Shoemaker's influential 'argument from selfblindness'. If introspection and perception are analogous, the relationship between beliefs and introspective knowledge of them is merely contingent. Shoemaker argues that this implies the possibility that agents could be selfblind, i.e., could lack any introspective awareness of their own mental states. By invoking Moore's paradox, he rejects this possibility. But because Shoemaker's discussion conflates introspective awareness and selfknowledge, he cannot establish his (...) conclusion. There is thirdperson evidence available to the selfblind which Shoemaker ignores, and it can account for the considerations from Moore's paradox that he raises. (shrink)
Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (...) (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not. The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the property of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat ) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows. The paradox was named after Bertrand Russell, who discovered it in 1901. (shrink)
Moore's paradox pits our intuitions about semantic oddnessagainst the concept of truth-functional consistency. Most solutions tothe problem proceed by explaining away our intuitions. But``consistency'' is a theory-laden concept, having different contours indifferent semantic theories. Truth-functional consistency is appropriateonly if the semantic theory we are using identifies meaning withtruth-conditions. I argue that such a framework is not appropriate whenit comes to analzying epistemic modality. I show that a theory whichaccounts for a wide variety of semantic data about epistemic modals(Update Semantics) (...) buys us a solution to Moore's paradox as a corollary.It turns out that Moorean propositions, when looked at through the lenseof an appropriate semantic theory, are inconsistent after all. (shrink)
The aim of this paper is to distinguish between, and examine, three issues surrounding Humphreys's paradox and interpretation of conditional propensities. The first issue involves the controversy over the interpretation of inverse conditional propensities — conditional propensities in which the conditioned event occurs before the conditioning event. The second issue is the consistency of the dispositional nature of the propensity interpretation and the inversion theorems of the probability calculus, where an inversion theorem is any theorem of probability that makes (...) explicit (or implicit) appeal to a conditional probability and its corresponding inverse conditional probability. The third issue concerns the relationship between the notion of stochastic independence which is supported by the propensity interpretation, and various notions of causal independence. In examining each of these issues, it is argued that the dispositional character of the propensity interpretation provides a consistent and useful interpretation of the probability calculus. (shrink)
Hempel's paradox of the ravens has to do with the question of what constitutes confirmation from a logical point of view; Wason's selection task has been used extensively to investigate how people go about attempting to confirm or disconfirm conditional claims. This paper presents an argument that the paradox is resolved, and that people's typical performance in the selection task can be explained, by consideration of what constitutes an effective strategy for seeking evidence of the tenability of universal (...) or conditional claims in everyday life. (shrink)
One version of Moore’s Paradox is the challenge to account for the absurdity of beliefs purportedly expressed by someone who asserts sentences of the form ‘p & I do not believe that p’ (‘Moorean sentences’). The absurdity of these beliefs is philosophically puzzling, given that Moorean sentences (i) are contingent and often true; and (ii) express contents that are unproblematic when presented in the third-person. In this paper I critically examine the most popular proposed solution to these two puzzles, (...) according to which Moorean beliefs are absurd because Moorean sentences are instances of pragmatic paradox; that is to say, the propositions they express are necessarily false-when-believed. My conclusion is that while a Moorean belief is a pragmatic paradox, it is not just another pragmatic paradox, because this diagnosis does not explain all the puzzling features of Moorean beliefs. In particularly, while this analysis is plausible in relation to the puzzle posed by characteristic (i) of Moorean sentences, I argue that it fails to account for (ii). I do so in the course of an attempt to formulate the definition of a pragmatic paradox in more precise formal terms, in order to see whether the definition is satisfied by Moorean sentences, but not by their third-person transpositions. For only an account which can do so could address (ii) adequately. After rejecting a number of attempted formalizations, I arrive at a definition which delivers the right results. The problem with this definition, however, is that it has to be couched in first-person terms, making an essential use of ‘I’. Thus the problem of accounting for first-/third-person asymmetry recurs at a higher order, which shows that the Pragmatic Paradox Resolution fails to identify the source of such asymmetry highlighted by Moore’s Paradox. (shrink)
Hilary Putnam's argument against metaphysical realism (commonly referred to as the "model theoretic argument") has now enjoyed two decades of discussion.(1) The text is rich and contains variously construable arguments against variously construed philosophical positions. David Lewis isolated one argument and called it "Putnam's Paradox".(2) That argument is clear and concise; so is the paradoxical conclusion it purports to demonstrate; and so is Lewis' paradox-avoiding solution. His solution involves a position I call "anti-nominalism": not only are classes real, (...) but they are divided into arbitrary and 'natural' classes. The natural classes 'carve nature at the joints', being (as other philosophers might say) the extensions of 'real' properties, universals, or Forms.(3) Thus the argument was turned, in effect, into support for a metaphysical realism stronger than Putnam envisaged. (shrink)
In this article I argue that two received accounts of belief and assertion cannot both be correct, because they entail mutually contradictory claims about Moore’s Paradox. The two accounts in question are, first, the Action Theory of Belief (ATB), the functionalist view that belief must be manifested in dispositions to act, and second, the Belief Account of Assertion (BAA), the Gricean view that an asserter must present himself as believing what he asserts. It is generally accepted also that Moorean (...) assertions are absurd, and that BAA explains why they are. I shall argue that ATB implies that some Moorean assertions are, in some fairly ordinary contexts, well justified. Thus BAA and ATB are mutually inconsistent. In the concluding section I explore three possible ways of responding to the dilemma, and what implications they have for the nature of the constitutive relationships linking belief, assent and behavioural dispositions. (shrink)
The principle of indifference is supposed to suffice for the rational assignation of probabilities to possibilities. Bertrand advances a probability problem, now known as his paradox, to which the principle is supposed to apply; yet, just because the problem is ill‐posed in a technical sense, applying it leads to a contradiction. Examining an ambiguity in the notion of an ill‐posed problem shows that there are precisely two strategies for resolving the paradox: the distinction strategy and the well‐posing strategy. (...) The main contenders for resolving the paradox, Marinoff and Jaynes, offer solutions which exemplify these two strategies. I show that Marinoff’s attempt at the distinction strategy fails, and I offer a general refutation of this strategy. The situation for the well‐posing strategy is more complex. Careful formulation of the paradox within measure theory shows that one of Bertrand’s original three options can be ruled out but also shows that piecemeal attempts at the well‐posing strategy will not succeed. What is required is an appeal to general principle. I show that Jaynes’s use of such a principle, the symmetry requirement, fails to resolve the paradox; that a notion of metaindifference also fails; and that, while the well‐posing strategy may not be conclusively refutable, there is no reason to think that it can succeed. So the current situation is this. The failure of Marinoff’s and Jaynes’s solutions means that the paradox remains unresolved, and of the only two strategies for resolution, one is refuted and we have no reason to think the other will succeed. Consequently, Bertrand’s paradox continues to stand in refutation of the principle of indifference. (shrink)
The model-theoretic argument known as Putnam´s paradox threatens our notion of truth with triviality: Almost any world can satisfy almost any theory. Formal argument and intuition are at odds. David Lewis devised a solution according to which the very stucture of the world fixes how it is to be divided into elite classes which determine the reference of any true theory. Three claims are defended: Firstly, Lewis´ proposal must be completed by an account of successful referential intentions. Secondly, contrary (...) to Catherine Elgin´s criticism of Lewis, natural properties corresponding to elite classes may play a role in sound scientific inquiry. Thirdly, despite Bas van Fraassen´s objection that the sceptic cannot consistently maintain doubts about reference, there is a promising sceptical strategy of exploiting Putnam´s results which is answered by Lewis´ account. (shrink)
Curry's paradox, so named for its discoverer, namely Haskell B. Curry, is a paradox within the family of so-called paradoxes of self-reference (or paradoxes of circularity). Like the liar paradox (e.g., ‘this sentence is false’) and Russell's paradox , Curry's paradox challenges familiar naive theories, including naive truth theory (unrestricted T-schema) and naive set theory (unrestricted axiom of abstraction), respectively. If one accepts naive truth theory (or naive set theory), then Curry's paradox becomes a (...) direct challenge to one's theory of logical implication or entailment. Unlike the liar and Russell paradoxes Curry's paradox is negation-free; it may be generated irrespective of one's theory of negation. An intuitive version of the paradox runs as follows. (shrink)
The paper attempts to give a solution to the Fitch’s paradox though the strategy of the reformulation of the paradox in temporal logic, and a notion of knowledge which is a kind of ceteris paribus modality. An analogous solution has been offered in a different context to solve the problem of metaphysical determinism.
Newcomb's Paradox thus serves as an illustrative vindication of the compatibility of divine foreknowledge and human freedom. A proper understanding of the counterfactual conditionals involved enables us to see that the pastness of God's knowledge serves neither to make God's beliefs counterfactually closed nor to rob us of genuine freedom. It is evident that our decisions determine God's past beliefs about those decisions and do so without invoking an objectionable backward causation. It is also clear that in the context (...) of foreknowledge, backtracking counterfactuals are entirely appropriate and that no alteration of the past occurs. With the justification of the one box strategy, the death of theological fatalism seems ensured. *** DIRECT SUPPORT *** A0985044 00003. (shrink)
On the basis of arguments showing that none of the most influential analyses of Moore’s paradox yields a successful resolution of the problem, a new analysis of it is offered. It is argued that, in attempting to render verdicts of either inconsistency or self-contradiction or self-refutation, those analyses have all failed to satisfactorily explain why a Moore-paradoxical proposition is such that it cannot be rationally believed. According to the proposed solution put forward here, a Moore-paradoxical proposition is one for (...) which the believer can have no non-overridden evidence. The arguments for this claim make use of some of Peter Klein’s views on epistemic defeasibility. It is further suggested that this proposal may have important meta-epistemological implications. (shrink)
Assertions of statements such as ‘it’s raining, but I don’t believe it’ are standard examples of what is known as Moore’s paradox. Here I consider moral equivalents of such statements, statements wherein individuals affirm moral judgments while also expressing motivational indifference to those judgments (such as ‘hurting animals for fun is wrong, but I don’t care’). I argue for four main conclusions concerning such statements: 1. Such statements are genuinely paradoxical, even if not contradictory. 2. This paradoxicality can be (...) traced to a form of epistemic self-defeat that also explains the paradoxicality of ordinary Moore-paradoxical statements. 3. Although a simple form of internalism about moral judgment and motivation can explain the paradoxicality of these moral equivalents, a more plausible explanation can be provided that does not rely on this simple form of internalism. 4. The paradoxicality of such statements suggests a more credible understanding of the thesis that those who are not motivated by their moral judgments are irrational. (shrink)
I offer a model of self-knowledge that provides a solution to Moore’s paradox. First, I distinguish two versions of the paradox and I discuss two approaches to it, neither of which solves both versions of the paradox. Next, I propose a model of self-knowledge according to which, when I have a certain belief, I form the higher-order belief that I have it on the basis of the very evidence that grounds my first-order belief. Then, I argue that (...) the model in question can account for both versions of Moore’s paradox. Moore’s paradox, I conclude, tells us something about our conceptions of rationality and self-knowledge. For it teaches us that we take it to be constitutive of being rational that one can have privileged access to one’s own mind and it reveals that having privileged access to one’s own mind is a matter of forming first-order beliefs and corresponding second-order beliefs on the same basis. (shrink)
Recently predominant forms of anti-realism claim that all truths are knowable. We argue that in a logical explanation of the notion of knowability more attention should be paid to its epistemic part. Especially very useful in such explanation are notions of group knowledge. In this paper we examine mainly the notion of distributed knowability and show its effectiveness in the case of Fitch’s paradox. Proposed approach raised some philosophical questions to which we try to find responses. We also show (...) how we can combine our point of view on Fitch’s paradox with the others. Next we give an answer to the question: is distributed knowability factive? At the end, we present some details concerning a construction of anti-realist modal epistemic logic. (shrink)
Sobocinski in his paper on Leśniewski's solution to Russell's paradox (1949b) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a (...) specific reading of the distributive interpretation is unacceptable. It follows from it that every individual is an element of every individual. Finally, another Leśniewskian-style approach which uses so-called higher-order epsilon connectives is used and its weakness is indicated. (shrink)
This paper explores Wittgenstein's attempts to explain the peculiarities of the first-person use of 'believe' that manifest themselves in Moore's paradox, discussed in Philosophical Investigations, Part II, section x. An utterance of the form 'p and I do not believe that p' is a kind of contradiction, for the second conjunct is not, as it might appear, just a description of my mental state, but an expression of my belief that not-p, contradicting the preceding expression of my belief that (...) p. Thus, 'I believe that p' is just a stylistic variant of 'p'; the word 'believe' doesn't seem to have a substantial role to play in such an utterance. Following Wittgenstein, I discuss why there could not be a first-person present-tense use of the word that was more akin to its use in the third person: why it is impossible to describe one's own current beliefs in a detached manner without thereby expressing them. In the final section, I try to develop Wittgenstein's suggestion that the non-epistemic authority we have regarding the contents of our beliefs can be clarified by considering its link with intention and action. (shrink)
Bangu (2010) claims that Bertrand’s paradox rests on a hitherto unrecognized assumption, which assumption is sufficiently dubious to throw the burden of proof back onto ‘objectors to [the principle of indifference]’ (2010: 31). We show that Bangu’s objection to the assumption is ill-founded and that the assumption is provably true.
This is a defense and extension of Stephen Yablo's claim that self-reference is completely inessential to the liar paradox. An infinite sequence of sentences of the form 'None of these subsequent sentences are true' generates the same instability in assigning truth values. I argue Yablo's technique of substituting infinity for self-reference applies to all so-called 'self-referential' paradoxes. A representative sample is provided which includes counterparts of the preface paradox, Pseudo-Scotus's validity paradox, the Knower, and other enigmas of (...) the genre. I rebut objections that Yablo's paradox is not a genuine liar by constructing a sequence of liars that blend into Yablo's paradox. I rebut objections that Yablo's liar has hidden self-reference with a distinction between attributive and referential self-reference and appeals to Gregory Chaitin's algorithmic information theory. The paper concludes with comments on the mystique of self-reference. (shrink)
The first case is usually referred to as omissive and the second as commissive. What is traditionally perceived as paradoxical is that although such statements may well be true, asserting them is clearly absurd. An account of Moore’s Paradox is an explanation of the absurdity. In the last twenty years, there has also been a focus on the incoherence of judging or believing such propositions.
Hume argued that inductive inferences do not have rational justification. My aim is to reject Hume’s argument. The discussion is partly motivated by an analogy with Carroll’s Paradox, which concerns deductive inferences. A first radically externalist reply to Hume (defended by Dauer and Van Cleve) is that justified inductive inferences do not require the subject to know that nature is uniform, though the uniformity of nature is a necessary condition for having the justification. But then the subject does not (...) have reasons for believing what she believes. I defend a moderate externalist account that seeks to partly accommodate that objection to the radical externalist proposal. It is based on an extension of Peacocke’s theory of concepts: possession conditions for predicative concepts standing for natural properties include (fallible) dispositions to project them to new cases in accordance with inductive inferential patterns. (shrink)
Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.
We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet (...) less numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding. (shrink)
Hartry Field's revised logic for the theory of truth in his new book, Saving Truth from Paradox , seeking to preserve Tarski's T-scheme, does not admit a full theory of negation. In response, Crispin Wright proposed that the negation of a proposition is the proposition saying that some proposition inconsistent with the first is true. For this to work, we have to show that this proposition is entailed by any proposition incompatible with the first, that is, that it is (...) the weakest proposition incompatible with the proposition whose negation it should be. To show that his proposal gave a full intuitionist theory of negation, Wright appealed to two principles, about incompatibility and entailment, and using them Field formulated a paradox of validity (or more precisely, of inconsistency). The medieval mathematician, theologian and logician, Thomas Bradwardine, writing in the fourteenth century, proposed a solution to the paradoxes of truth which does not require any revision of logic. The key principle behind Bradwardine's solution is a pluralist doctrine of meaning, or signification, that propositions can mean more than they explicitly say. In particular, he proposed that signification is closed under entailment. In light of this, Bradwardine revised the truth-rules, in particular, refining the T-scheme, so that a proposition is true only if everything that it signifies obtains. Thereby, he was able to show that any proposition which signifies that it itself is false, also signifies that it is true, and consequently is false and not true. I show that Bradwardine's solution is also able to deal with Field's paradox and others of a similar nature. Hence Field's logical revisions are unnecessary to save truth from paradox. (shrink)
Doris Olin's Paradox is a very helpful book for those who want to be introduced to the philosophical treatment of paradoxes, or for those who already have knowledge of the general area and would like to have a helpful resource book.
It is argued that Yablo’s Paradox is not strictly paradoxical, but rather ‘ω-paradoxical’. Under a natural formalization, the list of Yablo sentences may be constructed using a diagonalization argument and can be shown to be ω-inconsistent, but nonetheless consistent. The derivation of an inconsistency requires a uniform fixed-point construction. Moreover, the truth-theoretic disquotational principle required is also uniform, rather than the local disquotational T-scheme. The theory with the local disquotation T-scheme applied to individual sentences from the Yablo list is (...) also consistent. (shrink)
The Lowenheim-Skolem theorems say that if a first-order theory has infinite models, then it has models which are only countably infinite. Cantor's theorem says that some sets are uncountable. Together, these two theorems induce a puzzle known as Skolem's Paradox: the very axioms of (first-order) set theory which prove the existence of uncountable sets can themselves be satisfied by a merely countable model.
Over the years, Skolem’s Paradox has generated a fairly steady stream of philosophical discussion; nonetheless, the overwhelming consensus among philosophers and logicians is that the paradox doesn’t constitute a mathematical problem (i.e., it doesn’t constitute a real contradiction). Further, there’s general agreement as to why the paradox doesn’t constitute a mathematical problem. By looking at the way firstorder structures interpret quantifiers—and, in particular, by looking at how this interpretation changes as we move from structure to structure—we can (...) give a technically adequate “solution” to Skolem’s Paradox. So, whatever the philosophical upshot of Skolem’s Paradox may be, the mathematical side of Skolem’s Paradox seems to be relatively straightforward. (shrink)
Many studies have considered the probability that a pairwise majority rule (PMR) winner exists for three candidate elections. The absence of a PMR winner indicates an occurrence of Condorcet's Paradox for three candidate elections. This paper summarizes work that has been done in this area with the assumptions of: Impartial Culture, Impartial Anonymous Culture, Maximal Culture, Dual Culture and Uniform Culture. Results are included for the likelihood that there is a strong winner by PMR, a weak winner by PMR, (...) and the probability that a specific candidate is among the winners by PMR. Closed form representations are developed for some of these probabilities for Impartial Anonymous Culture and for Maximal Culture. Consistent results are obtained for all cultures. In particular, very different behaviors are observed for odd and even numbers of voters. The limiting probabilities as the number of voters increases are reached very quickly for odd numbers of voters, and quite slowly for even numbers of voters. The greatest likelihood of observing Condorcet's Paradox typically occurs for small numbers of voters. Results suggest that while examples of Condorcet's Paradox are observed, one should not expect to observe them with great frequency in three candidate elections. (shrink)
We argue that Anselm’s ontological argument (or at least one reconstruction of it) is based on an empirical version of Berry’s paradox. It is invalid, but it takes some understanding of trivalence to see why this is so. Under our analysis, Anselm’s use of the notion of existence is not the heart of the matter; rather, trivalence is.
Yablo’s paradox is generated by the following (infinite) list of sentences (called the Yablo list): (s1) For all k > 1, sk is not true. (s2) For all k > 2, sk is not true. (s3) For all k > 3, sk is not true. . . . . . . . .
Plato invokes the Theory of Recollection to explain both ordinary and philosophical learning. In a new reading of Meno’s Paradox and the Slave-Boy Interrogation, I explain why these two levels are linked in a single theory of learning. Since, for Plato, philosophical inquiry starts in ordinary discourse, the possibility of success in inquiry is tied to the character of the ordinary comprehension we bring to it. Through the claim that all learning is recollection, Plato traces the knowledge achievable through (...) inquiry back to our pretheoretical comprehension, showing not just that knowledge is in us, but that it is inchoate in the grasp of a property—akin to a concept—that enables us to speak and think about it ordinarily. Plato acknowledges in the Meno that a second step of argument, and a second application of Recollection, is needed to explain how knowledge comes to be inchoate in our ordinary grasp of a property. Though this second argument is provided most fully in the Phaedo, the evidence of the Meno is sufficient to outline Recollection as a two-stage theory of learning, beginning in ordinary speech and thought and extending, through philosophical reflection, to knowledge. (shrink)
Moore's paradox arises from the logicaloddity of sentences of the form`P and I do not believe that P'or `P and I believe that not-P'. Thiskind of sentence is logically peculiarbecause it is absurd to assert it, although it isnot a logical contradiction. In this paperI offer a new proposal. I argue that Moore's paradox arises because there is a defaultprocedure for evaluating a self-ascribed belief sentence and one is presumptivelyjustified in believing that one believes a sentence when one (...) sincerely assents to it. (shrink)
Skolem's Paradox involves a seeming conflict between two theorems from classical logic. The Löwenheim Skolem theorem says that if a first order theory has infinite models, then it has models whose domains are only countable. Cantor's theorem says that some sets are uncountable. Skolem's Paradox arises when we notice that the basic principles of Cantorian set theory—i.e., the very principles used to prove Cantor's theorem on the existence of uncountable sets—can themselves be formulated as a collection of first (...) order sentences. How can the very principles which prove the existence of uncountable sets be satisfied by a model which is itself only countable? How can a countable model satisfy the first order sentence which says that there are uncountably many mathematical objects—e.g., uncountably many real numbers? (shrink)
Fitch’s paradox shows, from fairly innocent-looking assumptions, that if there are any unknown truths, then there are unknowable truths. This is generally thought to deliver a blow to antirealist positions that imply that all truths are knowable. The present paper argues that a probabilistic version of antirealism escapes Fitch’s result while still offering all that antirealists should care for.
Colin Cheyne's ‘paradox of reasonable believing’ poses a problem for both internalist and externalist theories of rationality. Cheyne suggests that externalists will more easily solve it. I argue the opposite.
It is widely thought that Bayesian confirmation theory has provided a solution to Hempel's Paradox (the Ravens Paradox). I discuss one well-known example of this approach, by John Mackie, and argue that it is unconvincing. I then suggest an alternative solution, which shows that the Bayesian approach is altogether mistaken. Nicod's Condition should be rejected because a generalisation is not confirmed by any of its instances if it is not law-like. And even law-like non-basic empirical generalisations, which are (...) expressions of assumed underlying causal regularities, are not so confirmed if they are absurd in the light of our causal background knowledge or if their instances are not also possible instances of the relevant causal claim. (shrink)
The blame for the semantic and set-theoretic paradoxes is often placed on self-reference and circularity. Some years ago, Yablo [1985; 1993] challenged this diagnosis, by producing a paradox that's liar-like but does not seem to involve circularity. But is Yablo's paradox really non-circular? In a recent paper, Beall [2001] has suggested that there are no means available to refer to Yablo's paradox without invoking descriptions, and since Priest [1997] has shown that any such description is circular, Beall (...) concludes that Yablo's paradox itself is circular. In this paper, we argue that Beall's conclusion is unwarranted, given that (i) descriptions are not the only way to refer to Yablo's paradox, and (ii) we have no reason to believe that because the description involves self-reference, the denotation of that description is also circular. As a result, for all that's been said so far, we have no reason to believe that Yablo's paradox is circular. (shrink)
This is a response to a paper “Paradox without satisfaction”, Analysis 63, 152-6 (2003) by Otavio Bueno and Mark Colyvan on Yablo’s paradox. I argue that this paper makes several substantial mathematical errors which vitiate the paper. (For the technical details, see [12] below.).
Zermelo once wrote that he had anticipated Russell's contradiction of the set of all sets that are not members of themselves. Is this sufficient for having anticipated Russell's Paradox — the paradox that revealed the untenability of the logical notion of a set as an extension? This paper argues that it is not sufficient and offers criteria that are necessary and sufficient for having discovered Russell's Paradox. It is shown that there is ample evidence that Russell satisfied (...) the criteria and that Zermelo did not. (shrink)
Grelling’s Paradox is the paradox which results from considering whether heterologicality, the word-property which a designator has when and only when the designator does not bear the word-property it designates, is had by ‘ ȁ8heterologicality’. Although there has been some philosophical debate over its solution, Grelling’s Paradox is nearly uniformly treated as a variant of either the Liar Paradox or Russell’s Paradox, a paradox which does not present any philosophical challenges not already presented by (...) the two better known paradoxes. The aims of this paper are, first, to offer a precise formulation of Grelling’s Paradox which is clearly distinguished from both the Liar Paradox and Russell’s Paradox; second, to offer a solution to Grelling’s Paradox which both resolves the paradoxical reasoning and accounts for unproblematic predications of heterologicality; and, third, to argue that there are two lessons to be drawn from Grelling’s Paradox which have not yet been drawn from the Liar or Russell’s Paradox. The first lesson is that it is possible for the semantic content of a predicate to be sensitive to the semantic context; i.e., it is possible for a predicate to be an indexical expression. The second lesson is that the semantic content of an indexical predicate, though unproblematic for many cases, can nevertheless be problematic in some cases. (shrink)
In order to refute the widely held belief that the game known as ‘Newcomb's paradox’ is physically nonsensical and impossible to imagine (e.g. because it involves backward causation), I tell a story in which the game is realized in a classical, deterministic universe in a physically plausible way. The predictor is a collection of beings which are by many orders of magnitude smaller than the player and which can, with their exquisite measurement techniques, observe the particles in the player's (...) body so accurately that they can predict his choice (in much the same way as we can predict the motion of celestial bodies). I argue that the player, by choosing whether to take only one box or both boxes, influences whether or not, in the past, the predictor put a million pounds into the second box. Yet, I establish that no causal paradox can arise in this set-up. (shrink)
Working in the fragment of Martin-Löfs extensional type theory [12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's paradox showing that each type possesses a closed term and (hence) that all terms (...) of each type are provably equal. We consider the kind of category theoretic structure which corresponds to this kind of type theory and obtain a categorical version of the paradox. A special case of this result is the degeneracy of a locally cartesian closed category with a morphism which is generic in the sense that every other morphism in the category can be obtained from it via pullback. (shrink)
On the basis of arguments showing that none of the most influential analyses of Moore's paradox yields a successful resolution of the problem, a new analysis of it is offered. It is argued that, in attempting to render verdicts of either inconsistency or self-contradiction or self-refutation, those analyses have all failed to satisfactorily explain why a Moore-paradoxical proposition is such that it cannot be rationally believed. According to the proposed solution put forward here, a Moore-paradoxical proposition is one for (...) which the believer can have no non-overridden evidence. The arguments for this claim make use of some of Peter Klein's views on epistemic defeasibility. It is further suggested that this proposal may have important meta-epistemological implications. (shrink)
English translation of a paper intially publisdhed in French in Dialogue under the title 'Une solution pour le paradoxe de Goodman'. In the classical version of Goodman's paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of n-universes, allowing the distinction, among the criteria of a given n-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) (...) in an n-universe the variables of which are colour and time; (ii) in an n-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution. (shrink)
We propose contractionless constructive logic which is obtained from Nelson's constructive logic by deleting contractions. We discuss the consistency of a naive set theory based on the proposed logic in relation to Curry's paradox. The philosophical significance of contractionless constructive logic is also argued in comparison with Fitch's and Prawitz's systems.
Goodman’s paradox gives rise to a cluster of problems, problems that are in need of different answers. I will discuss some variants of the grue hypothesis applied to the technological context. One conclusion in this paper is that there is room for rational decisions, and that solutions to the paradoxes in technology can be found in the practical choice situation. *Received April 2008. †To contact the author, please write to: Department of Medical and Health Sciences, Linköping University, SE‐581 83 (...) Linköping, Sweden; e‐mail: ingemar.nordin@liu.se. (shrink)
Popper's paradox of ideal evidence has long been viewed as a telling criticism of Keynes's logical theory of probability and its associated concept of the weight of argument. This paper shows that a simple addition to Keynes's definitions of irrelevance enables his theory to elude the paradox with ease. The modified definition draws on ideas already present in Keynes's Treatise on Probability (1973). As a consequence, relevant evidence and the weight of argument may increase, even when new evidence (...) leaves the probability unaltered. (shrink)
Moore’s paradox occurs with sentences, such as (1) It’s raining and I don’t think it’s raining. which are self-defeating in a way that prevents one from making an asser- tion with them.1 But Mark Crimmins has given us a case of a sentence that is syntactically just like (1) but is nonetheless assertible. Suppose I know somebody, and know or have excellent reason to believe that I know that very person under some other guise. I do not know what (...) that other guise is, though I do know that I believe that the person I know under that other guise is an idiot. (shrink)
Russell's "new contradiction" about "the totality of propositions" has been connected with a number of modal paradoxes. M. Oksanen has recently shown how these modal paradoxes are resolved in the set theory NFU. Russell's paradox of the totality of propositions was left unexplained, however. We reconstruct Russell's argument and explain how it is resolved in two intensional logics that are equiconsistent with NFU. We also show how different notions of possible worlds are represented in these intensional logics.
If it is certain that performing an observation to determine whether P is true will in no way influence whether P is tme, then the proposition that the observation is performed ought to be probabilistically independent of P. Applying the notion of "observation" liberally, so that a wide variety of actions are treated as observations, this proposed new principle of belief revision yields the result that simple utihty maximization gives the correct solution to the Fisher smoking paradox and the (...) two-box solution to Newcomb's paradox. Contrary intuitions are explained as arising from mistakenly treating subjective probability as a measure of the intensity of conscious assent, whereas it ought to be regarded as measuring dispositions to action. (shrink)
A coherent theory of relations was a critical part of Russell’s metaphysics. In Appearance and Reality Bradley posed a problem that sits squarely in the way of any doctrine of “external” relations. Russell, determined to advance such a doctrine, tried several times to find a way around the paradox and apparently believed he had succeeded by making use of one of his inventions, the theory of logical types.Gilbert Ryle and Alan Donagan have advanced an argument that I read, over (...) the objections of its authors, as a special case of Bradley’s. In this paper I argue that the ad hoc solution suggested by Donagan to the special problem is one that Russell had already indicated a willingness to accept but that the general problem of the paradox remains.What finally prevents Russell from solving the paradox is a combination of his refusal to abandon the claim that relations are constituents of facts and the necessity of distinguishing a relational fact from its converse. Following some hints that Russell left, I do some reconstruction, showing how the theory of types would (and should) have been applied had Russell followed through on his own insights. The result, I suggest, is a truly Russellian theory that escapes Bradley’s paradox. (shrink)
It is shown that the Fisher smoking problem and Newcomb's problem are decisiontheoretically identical, each having at its core an identical case of Simpson's paradox for certain probabilities. From this perspective, incorrect solutions to these problems arise from treating them as cases of decisionmaking under risk, while adopting certain global empirical conditional probabilities as the relevant subjective probabihties. The most natural correct solutions employ the methodology of decisionmaking under uncertainty with lottery acts, with certain local empirical conditional probabilities adopted (...) as the relevant subjective probabilities. (shrink)
Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ?the most important event? in ?the most important year in my intellectual life?. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell (...) was thus vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory. (shrink)
This article shows that although Fitch’s paradox has been extremely widely studied, up to now no correct formalization of the problem has been proposed. The purpose of this article is to present the paradox front the viewpoint of combining logics. It is argued that the correct minimal logic to state the paradox is composed by a fusion of modal frames, and a fusion of modal languages and logics.
Richard Otte (1985) has recently criticized the resolution of Simpson's paradox given by Nancy Cartwright (1979). He argues that there are difficulties with the version of the theory of probabilistic causality that Cartwright has developed, and that there is a way in which Simpson's paradox can arise that Cartwright's theory cannot handle. And Otte develops his own theory of probabilistic causality. I defend Cartwright's solution, and I argue that there are difficulties with the theory of probabilistic causality that (...) Otte proposes. (shrink)
The paper is devoted to the problem of formal representation of prescriptive obligation, i.e., the obligation concerning the way in which an action is to be performed. Improper representation of prescriptive obligation leads to Forrester's Paradox. In the paper I first present a new version of Forrester's Paradox that generalizes the observation on which the original version is based. Then I challenge the two existing solutions to the paradox. I reject the solution of H.-N. Castañeda and (...) analyze problems to which the solution of W. Sinnott-Armstrong leads. I conclude the paper with a proposal for how to improve Sinnott-Armstrong's solution. (shrink)
This paper discusses Simpson's paradox and the problem of positive relevance in probabilistic causality. It is argued that Cartwright's solution to Simpson's paradox fails because it ignores one crucial form of the paradox. After clarifying different forms of the paradox, it is shown that any adequate solution to the paradox must allow a cause to be both a negative cause and a positive cause of..
Moore’s paradox in belief is the fact that beliefs of the form ‘ p and I do not believe that p ’ are ‘absurd’ yet possibly true. Writers on the paradox have nearly all taken the absurdity to be a form of irrationality. These include those who give what Timothy Chan calls the ‘pragmatic solution’ to the paradox. This solution turns on the fact that having the Moorean belief falsifies its content. Chan, who also takes the absurdity (...) to be a form of irrationality, objects to this solution by arguing that it is circular and thus incomplete. This is because it must explain why Moorean beliefs are irrational yet, according to Chan, their grammatical third-person transpositions are not, even though the same proposition is believed. But the solution can only explain this asymmetry by relying on a formulation of the ground of the irrationality of Moorean beliefs that presupposes precisely such asymmetry. I reply that it is neither necessary nor sufficient for the irrationality that the contents of Moorean beliefs be restricted to the grammatical first-person. What has to be explained is rather that such grammatical non-first-person transpositions sometimes, but not always, result in the disappearance of irrationality. Describing this phenomenon requires the grammatical first-person/non-first person distinction. The pragmatic solution explains the phenomenon once it is formulated in de se terms. But the grammatical first-person/non-first-person distinction is independent of, and a fortiori, different from, the de se /non- de se distinction presupposed by pragmatic solution, although both involve the first person broadly construed. Therefore the pragmatic solution is not circular. Building on the work of Green and Williams I also distinguish between the irrationality of Moorean beliefs and their absurdity. I argue that while all irrational Moorean beliefs are absurd, some Moorean beliefs are absurd but not irrational. I explain this absurdity in a way that is not circular either. (shrink)
I show how an almost exclusive focus on the simplest case - the case of a single particle - along with the commonplace conception of the single-particle wave function as a scalar field on spacetime contributed to the perception, first brought to light by I. Bloch, that there existed a contradiction between quantum theory with instantaneous state collapses and special relativity. The incompatibility is merely apparent since treating wave-function values as hypersurface dependent avoids the contradiction. After clarifying confusions which fueled (...) the perception of a paradox, I elaborate on an analysis of the wave function due to Wayne Myrvold to show that nothing special, or ad hoc, is required in treating wave-function values, even in the single-particle case, as hypersurface-dependent; rather, the hypersurface dependence of these values is the natural development of nonlocal entanglement in the context of the relativity of simultaneity. Properly understood, what Bloch's paradox reveals is that the combination of nonlocal entanglement together with a hypersurface-dependent process of state collapse conflicts with the thesis of spatiotemporal separability and, in particular, with the idea that chances are local matters of fact. (shrink)
The interpretation of tests of a point null hypothesis against an unspecified alternative is a classical and yet unresolved issue in statistical methodology. This paper approaches the problem from the perspective of Lindley's Paradox: the divergence of Bayesian and frequentist inference in hypothesis tests with large sample size. I contend that the standard approaches in both frameworks fail to resolve the paradox. As an alternative, I suggest the Bayesian Reference Criterion: (i) it targets the predictive performance of the (...) null hypothesis in future experiments; (ii) it provides a proper decision-theoretic model for testing a point null hypothesis and (iii) it convincingly accounts for Lindley's Paradox. (shrink)
According to Fisher, a hypothesis specifying a density function for X is falsified (at the level of significance ) if the realization of X is in the size- region of lowest densities. However, non-linear transformations of X can map low-density into high-density regions. Apparently, then, falsifications can always be turned into corroborations (and vice versa) by looking at suitable transformations of X (Neyman's Paradox). The present paper shows that, contrary to the view taken in the literature, this provides no (...) argument against a theory of statistical falsification. (shrink)
