Where the Sum of Our Expectation Fails Us: The Exchange Paradox

In the exchange paradox, two players receive envelopes containing different amounts of money. The assignment of the amounts ensures each player has the same probability of receiving each possible amount. Nonetheless, for each specific amount a player may find in his envelope, there is a positive expectation of gain if the player swaps envelopes with the other player, in apparent contradiction with the symmetry of the game. I consider a variant form of the paradox that avoids problems with improper probabilities and I argue that in it these expectations give no grounds for a decision to swap since that decision must be based on a summation of all the expectations. But this sum yields a non-convergent series that has no meaningful value. The conflicting recommendations – that it is to one or the other player's advantage to swap – arise from different ways of grouping terms in the sum that yield an illusion of convergence. I describe a generalized exchange paradox, explore some of its properties and display another example.     

On Russell's vulnerability to Russell's paradox

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.     

Avoiding and Alternate Possibilities

Greg Janzen has recently criticised my defence of Frankfurt’s counterexample to the Principle of Alternate Possibilities by arguing that Jones avoids killing Smith in the counterfactual scenario. Janzen’s argument consists in introducing a new thought-experiment which is supposed to be analogous to Frankfurt’s and where the agent is supposed to avoid A-ing. Here I argue that Janzen’s argument fails on two counts, because his new scenario is not analogous to Frankfurt’s and because the agent in his new scenario does not avoid A-ing.     

The Importance of Being Actual: Some Reasons for and Against Procreation

In 'Axiological Actualism' Josh Parsons attempts to defend both the intuition that the anticipated welfare of a person cannot constitute a reason to bring him or her into being and the intuition that such considerations can constitute a reason not to. The former, 'basic' intuition he defends by an appeal to the belief that 'ethical theory should refrain from assigning levels of welfare … or anything of the sort to merely possible people'. The latter, 'converse' intuition he defends by an appeal to prudential considerations. I argue that Parsons's attempts to defend these intuitions are unpersuasive. On the one hand, and notwithstanding his attempts to demonstrate the contrary, the basic intuition is undermined by the claim that an actual person could have been worse off if she had never existed. On the other, his grounding of the converse intuition in prudential considerations runs counter to the ought implies can dictum and is also highly counterintuitive.     

ROBUSTNESS AND THE NEW RIDDLE REVIVED

The problem of induction is perennially important in epistemology and the philosophy of science. In response to Goodman's ‘New Riddle of Induction’, Frank Jackson made a compelling case for there being no new riddle, by arguing that there are no nonprojectible properties. Although Jackson's denial of nonprojectible properties is correct, I argue here that he is mistaken in thinking that he thereby shows that there is no new riddle of induction, and demonstrate that his solution to the grue paradox fails to rule out the possibility of equally justified contradictory inductions. More importantly, in illuminating where Jackson's argument fails, the paper casts a new light on the problem of induction, locating the problem not in the nature of the next (unexamined) x, but in the counterfactual robustness of properties of already examined x's.     

Frankfurt-Style Cases User Manual: Why Frankfurt-Style Enabling Cases Do Not Necessitate Tech Support

‘Frankfurt-style cases’ (FSCs) are widely considered as having refuted the Principle of Alternate Possibilities (PAP) by presenting cases in which an agent is morally responsible even if he could not have done otherwise. However, Neil Levy (J Philos 105:223–239, 2008) has recently argued that FSCs fail because we are not entitled to suppose that the agent is morally responsible, given that the mere presence of a counterfactual intervener is enough to make an agent lose responsibility-grounding abilities. Here, I distinguish two kinds of Frankfurt counter-arguments against the PAP: the direct and the indirect counter-arguments. I then argue that Levy’s argument, if valid, can shed doubt on the indirect argument but leaves the direct argument untouched. I conclude that FSCs can still do their job, even if we grant that the mere presence of a counterfactual intervener can modify an agent’s abilities.     

Keep or trade? Effects of pay-off range on decisions with the two-envelops problem

The “two-envelops” problem has stimulated much discussion on probabilistic reasoning, but relatively little experimentation. The problem specifies two identical envelopes, one of which contains twice as much money as the other. You are given one of the envelopes and the option of keeping it or trading for the other envelope. Variables of interest include the possible amounts of money involved, what is known about the process by which the amounts of money were assigned to the envelopes, and whether you are allowed to know how much money is in the envelope in hand before deciding whether to keep or trade. In an earlier study, Butler and Nickerson found that when participants were allowed to know how much was in the envelope in hand, they generally elected to trade if that amount was small relative to the range of possibilities and to keep otherwise. The present experiments showed that this propensity was independent of the amount of money in the envelopes. Participants made decisions with a strong bias for avoiding the risk of losing by trading, particularly when the amount in hand was known and large relative to the range of possible amounts, regardless of the absolute value of the gamble. The results illustrate the dependence of thinking on the context in which it occurs, and demonstrate a tendency to treat quantities that are large or small relative to a particular context in which they are encountered as though they were large or small in a more general sense.     

Platonism and metaphor in the texts of mathematics: Gödel and Frege on mathematical knowledge

In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism on a literal comparison between mathematical intuition and physical perception. He concludes that since we accept sense perception as a source of empirical knowledge, then we similarly should posit a faculty of mathematical intuition to serve as the source of mathematical knowledge. Unlike Gödel, Frege does not posit a faculty of mathematical intuition. Frege talks instead about grasping thoughts about abstract objects. However, despite his hostility to metaphor, he uses the notion of ‘grasping’ as a strategic metaphor to model his notion of thinking, i.e., to underscore that it is only by logically manipulating the cognitive content of mathematical propositions that we can obtain mathematical knowledge. Thus, he construes ‘grasping’ more as theoretical activity than as a kind of inner mental ‘seeing’.

     

The Mere Addition Paradox,Parity and Vagueness*

Derek Parfit’s mere addition paradox has generated a large literature. This paper articulates one response to this paradox—which Parfit himself suggested—in terms of a formal account of the relation of parity. I term this response the ‘parity view’. It is consistent with transitivity of ‘at least as good as’, but implies incompleteness of this relation. The parity view is compatible with critical‐band utilitarianism if this is adjusted to allow for vagueness. John Broome argues against accounts which involve incompleteness. He thinks they are based on an intuition of ‘neutrality’, which is most naturally understood in terms of equality. There is no rationale, on Broome’s view, for seeing it as ‘incommensurateness’ which leads to incompleteness. Parity provides one. Broome’s worries that ‘incommensurateness’ makes neutrality implausibly ‘greedy’, and that ‘incommensurateness’ and vagueness are incompatible do not constitute a knock‐down case against the parity view. Similar worries arise for his preferred vagueness view.     

Are Mathematical Theories Reducible to Non-analytic Foundations?

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process.     

Careful What You Wish

Dilip Ninan has raised a puzzle for centered world accounts of de re attitude reports extended to accommodate what he calls “counterfactual attitudes.” As a solution, Ninan introduces multiple centers to the standard centered world framework, resulting in a more robust semantics for de re attitude reports. However, while the so-called multi-centered world proposal solves Ninan’s counterfactual puzzle, this additional machinery is not without problems. In Section 1, I present the centered world account of attitude reports, followed by the extension to counterfactual attitudes which Ninan targets with his puzzle. In Section 2, I pose the counterfactual puzzle and present Ninan’s multi-centered world solution, emphasizing similarities and differences between multi-centered and centered world accounts of attitude reports. In Section 3, I argue the counterfactual attitude wishing falls under the purview of the multi-centered proposal, but that the proposal generates false predictions for wish reports concerning unsatisfiable content. I canvass responses, ultimately concluding Ninan’s proposal requires substantial revisions.     

Counterfactuals and Epistemic Probability

Philosophers have often attempted to use counterfactual conditionals to analyze probability. This article focuses on counterfactual analyzes of epistemic probability by Alvin Plantinga and Peter van Inwagen. I argue that a certain type of counterfactual situation creates problems for these analyses. I then argue that Plantinga’s intuition about the role of warrant in epistemic probability is mistaken. Both van Inwagen’s and Plantinga’s intuitions about epistemic probability are flawed.     

Frankfurt versus Frankfurt: a new anti-causalist dawn

There is an important anomaly to the causalist/compatibilist paradigm in the philosophy of action and free will. This anomaly, which to my knowledge has gone unnoticed so far, can be found in the philosophy of Harry Frankfurt. Two of his most important contributions to the field – his influential counterexample to the Principle of Alternate Possibilities and his ‘guidance’ view of action – are incompatible. Frankfurt's counterexample to the Principle works only if we do not understand action as Frankfurt does in his guidance account. If, on the other hand, we understand agency in terms of the agent's guidance, then his counterexample to the Principle fails because, then, counterfactual scenarios of Frankfurt-type counterexamples are such that what happens does not count as the relevant agent's action. So Frankfurt-type counterexamples do not show that the agent could not have avoided acting as she did: so they fail to offer a scenario in which the agent is intuitively responsible even though she could not have avoided acting as she did. Therefore, Frankfurt-type counterexamples do not challenge the Principle, according to which ‘a person is morally responsible for what he has done only if he could have done otherwise’. The importance of this inconsistency goes far beyond the issue of coherence within Frankfurt's philosophy. I shall argue that this inconsistency represents an important anomaly within the causalist/compatibilist framework; so that we should start to seriously consider having to move on from the established paradigm.     

The exchange paradox: Probabilistic and cognitive analysis of a psychological conundrum

The term “exchange paradox” refers to a situation in which it appears to be advantageous for each of two holders of an envelope containing some amount of money to always exchange his or her envelope for that of the other individual, which they know contains either half or twice their own amount. We review several versions of the problem and show that resolving the paradox depends on the specifics of the situation, which must be disambiguated, and on the player's beliefs. The latter psychological variables are part and parcel of the resolution. Assuming reasonable subjective distributions, exchanging cannot always be advantageous for both players. We suggest several deep-rooted psychological reasons for the considerable difficulty people demonstrably have in dealing with this problem. Implicit widespread and compelling assumptions—that affect judgement in diverse contexts—obstruct the solution. Analysing this paradox underscores the close connection between psychology and probability theory.     

Radicals versus Moderates: A Critique of Gyekye’s Moderate Communitarianism

Abstract

The communitarian conception of person is a widely accepted view in African thought. Kwame Gyekye thinks there is a distinction between what he calls radical communitarianism and his own version of moderate communitarianism. He is of the view that radical communitarianism is faced with insurmountable problems and ought to be jettisoned in favour of his moderate communitarianism. Gyekye’s strategy is twofold; he firstly seeks to show the shortcomings of radical communitarianism - particularly by attacking Ifeanyi Menkiti’s position. Secondly, he seeks to show the authenticity of his version as well as its serious regard for individual rights as representing a triumph over radical communitarianism. In this paper, I seek to contest both of Gyekye’s strategies.     

What Dennett can't imagine and why

Woven into Dennett's account of consciousness is his belief that certain possibilities are not conceivable. This is manifested in his view that we are not conscious in any sense in which we can imagine that philosophers’ ‘zombies’ might not be conscious, and also in his claims about ‘Hindsight’, and what possibilities this can coherently suggest to us. If the possibilities Dennett denies none the less seem conceivable to us, then if he does not give us reason to think they are actually incoherent, we ought to reject his theory, since it denies the intelligibility of the very notion we should want a theory of consciousness to discuss. I argue that Dennett does not provide us with convincing reasons of the relevant sort, and I suggest that his difficulty with the concept of consciousness is rooted in questionable epistemological assumptions which he fails to justify.     

Gödel And The Intuition Of Concepts

Gödel has argued that we can cultivate the intuition or ‘perception’ of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward ‘objects’ and the meaning of the term ‘abstract’ in the context of a theory of the intentionality of cognition.     

Emptying a Paradox of Ground

Sometimes a fact can play a role in a grounding explanation, but the particular content of that fact make no difference to the explanation—any fact would do in its place. I call these facts vacuous grounds. I show that applying the distinction between-vacuous grounds allows us to give a principled solution to Kit Fine and Stephen Kramer’s paradox of (reflexive) ground. This paradox shows that on minimal assumptions about grounding and minimal assumptions about logic, we can show that grounding is reflexive, contra the intuitive character of grounds. I argue that we should never have accepted that grounding is irreflexive in the first place; the intuitions that support the irreflexive intuition plausibly only require that grounding be non-vacuously irreflexive. Fine and Kramer’s paradox relies, essentially, on a case of vacuous grounding and is thus no problem for this account.     

Counterfactuals and the fixity of the past

I argue that David Lewis’s attempt, in his ‘Counterfactual Dependence and Time’s Arrow’, to explain the fixity of the past in terms of counterfactual independence is unsuccessful. I point out that there is an ambiguity in the claim that the past is counterfactually independent of the present (or, more generally, that the earlier is counterfactually independent of the later), corresponding to two distinct theses about the relation between time and counterfactuals, both officially endorsed by Lewis. I argue that Lewis’s attempt is flawed for a variety of reasons, including the fact that his own theory about the evaluation of counterfactuals requires too many exceptions to the general rule that the past is counterfactually independent of the present. At the end of the paper, I consider a variant of Lewis’s strategy that attempts to explain the fixity of the past in terms of causal, rather than counterfactual, independence. I conclude that, although this variant avoids some of the objections that afflict Lewis’s account, it nevertheless seems to be incapable of giving a satisfactory explanation of the notion of the fixity of the past.     

Updating Thought Theory: Emotion and the Non‐Paradox of Fiction

Over the past four decades, the paradox of fiction has sparked considerable debate among philosophers. Unfortunately, the most promising solution to this puzzle, thought theory, currently earns its plausibility by way of intuition rather than evidence. I aim to address this by updating thought theory in light of recent empirical findings on affect. I will draw upon a wide range of scientific research—on the cognitive mechanisms driving emotion, the role of affect in counterfactual mind wandering and prospection, and the evolutionary function of affect—to substantiate the claim that fictions can give rise to both real and rational emotional states.