The phenomenon of the paradoxes of the largest remainders methods has been studied by numerous authors. Nevertheless, the examples presented in their studies do not deal with the case where a party’s possible additional votes can directly lead to a loss in the party’s number of representatives. This paradox, which can be called the no-show apportionment paradox, has not previously been mentioned in the literature. It is based on the assumption that a voter’s favourite party may lose (...) a seat if the voter votes honestly but get more seats if he or she abstains. The article gives simple examples illustrating the no-show paradox in a party-list proportional representation electoral system using the quota method of apportionment. In these situations it would be better for the party if some of their voters abstained, otherwise it would lose one or more seats. (shrink)

We consider two no-show paradoxes, in which a voter obtains a preferable outcome by abstaining from a vote. One arises when the casting of a ballot that ranks a candidate in first place causes that candidate to lose the election, superseded by a lower-ranked candidate. The other arises when a ballot that ranks a candidate in last place causes that candidate to win, superseding a higher-ranked candidate. We show that when there are at least four candidates and when (...) voters may express indifference, every voting rule satisfying Condorcet’s principle must generate both of these paradoxes. (shrink)

In “The Train Paradox”(Philosophia (2006) 34: 437–438) Gwiazda proposes the use of the relativity of simultaneity to formulate a new paradox. My purpose here is to show that there is no Train Paradox in Gwiazda’s sense.

A number of authors have argued that Peano Arithmetic supplemented with a logical validity predicate is inconsistent in much the same manner as is PA supplemented with an unrestricted truth predicate. In this paper I show that, on the contrary, there is no genuine paradox of logical validity—a completely general logical validity predicate can be coherently added to PA, and the resulting system is consistent. In addition, this observation lead to a number of novel, and important, insights into (...) the nature of logical validity itself. (shrink)

We propose a Condorcet-consistent voting method that we call Split Cycle. Split Cycle belongs to the small family of known voting methods satisfying the anti-vote-splitting criterion of independence of clones. In this family, only Split Cycle satisfies a new criterion we call immunity to spoilers, which concerns adding candidates to elections, as well as the known criteria of positive involvement and negative involvement, which concern adding voters to elections. Thus, in contrast to other clone-independent methods, Split Cycle mitigates both “spoiler (...) effects” and “strong no show paradoxes.”. (shrink)

Since Montague’s work it is well known that treating a single modality as a predicate may lead to paradox. In their paper “No Future”, Horsten and Leitgeb show that if the two temporal modalities are treated as predicates paradox might arise as well. In our paper we investigate whether paradoxes of multiple modalities, such as the No Future paradox, are genuinely new paradoxes or whether they “reduce” to the paradoxes of single modalities. In order to address (...) this question we develop a notion of reducibility based on a version of Smoryński Diagonalized Operator Logic. We show that there are reducible multimodal paradoxes as well as irreducible paradoxes of interaction. In particular, we show the No Future paradox to be an irreducible paradox according to our notion of reducibility. (shrink)

In ‘A Consistent Way with Paradox’, Laurence Goldstein (2009) clarifies his solution to the liar, which he touts as revenge immune . In addition, he (Ibid.) responds to one of the objections that Armour-Garb and Woodbridge (2006) raise against certain solutions to the open pair and argues that his proffered solution to the liar family of paradoxes undermines what they (Ibid.) call the dialetheic conjecture . In this paper, after critically evaluating Goldstein’s response to A-G&W, I turn to his (...) proposed solution to the liar paradox, where I show that it is difficult to see how it manages to avoid that conjecture. (shrink)

We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. _Reckless_ theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; _timid_ theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; _non-transitive_ theories deny the principle that, if A is better than B and B is better than C, then A must be better than (...) C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty. (shrink)

Epiphenomenalist dualists hold that certain physical states give rise to non-physical conscious experiences, but that these non-physical experiences are themselves causally inefficacious. Among the most pressing challenges facing epiphenomenalists is the so-called “paradox of phenomenal judgment”, which challenges epiphenomenalism’s ability to account for our knowledge of our own conscious experiences. According to this objection, we lack knowledge of the very thing that epiphenomenalists take physicalists to be unable to explain. By developing an epiphenomenalist theory of subjects and mental states, (...) this paper argues that there is nothing paradoxical or problematic about the epiphenomenalist’s understanding of phenomenal judgments or phenomenal self-knowledge. The appearance of paradox emerges from inconsistently combining (epiphenomenalist) dualism about qualia with a physicalistic conception of subjects of experience. The lesson we should take from this is not that there is anything wrong with epiphenomenalism, but that epiphenomenalist dualists should be “dualists all the way down”—embracing a picture of mind that gives phenomenology a central place, in its understanding of both subjects and their knowledge of their own minds. Epiphenomenalist-friendly accounts of reference and memory are also developed, showing that neither of these issues creates a paradox for the epiphenomenalist. (shrink)

Cook regards Sorenson’s so-called ‘the no-no paradox’ as only a kind of ‘meta-paradox’ or ‘quasi-paradox’ because the symmetry principle that Sorenson imposes on the paradox is meta-theoretic. He rebuilds this paradox at the object-language level by replacing the symmetry principle with some ‘background principles governing the truth predicate’. He thus argues that the no-no paradox is a ‘new type of paradox’ in that its paradoxicality depends on these principles. This paper shows that any (...) theory is inconsistent with the T-schema instances for the no-no sentences, plus the T-schema instance for a Curry sentence associated with the symmetry of the no-no sentences. It turns out that the no-no paradox still depends on the problematic instances of the T-schema in a way that the liar paradox does. What distinguishes the no-no paradox is the T-schema instance for the above Curry sentence, which encodes Sorensen’s symmetry principle at the object-language level. (shrink)

A result from a standard linear model course is that the variance of the ordinary least squares (OLS) coefficient of a variable will never decrease when including additional covariates into the regression. The variance inflation factor (VIF) measures the increase of the variance. Another result from a standard linear model or experimental design course is that including additional covariates in a linear model of the outcome on the treatment indicator will never increase the variance of the OLS coefficient of the (...) treatment at least asymptotically. This technique is called the analysis of covariance (ANCOVA), which is often used to improve the efficiency of treatment effect estimation. So we have two paradoxical results: adding covariates never decreases the variance in the first result but never increases the variance in the second result. In fact, these two results are derived under different assumptions. More precisely, the VIF result conditions on the treatment indicators but the ANCOVA result averages over them. Comparing the estimators with and without adjusting for additional covariates in a completely randomized experiment, I show that the former has smaller variance averaging over the treatment indicators, and the latter has smaller variance at the cost of a larger bias conditioning on the treatment indicators. Therefore, there is no real paradox. (shrink)

Whether we're buying a pair of jeans, ordering a cup of coffee, selecting a long-distance carrier, applying to college, choosing a doctor, or setting up a 401(k), everyday decisions ; both big and small ; have become increasingly complex due to the overwhelming abundance of choice with which we are presented. As Americans, we assume that more choice means better options and greater satisfaction. But beware of excessive choice: choice overload can make you question the decisions you make before you (...) even make them, it can set you up for unrealistically high expectations, and it can make you blame yourself for any and all failures. In the long run, this can lead to decision-making paralysis, anxiety, and perpetual stress. And, in a culture that tells us that there is no excuse for falling short of perfection when your options are limitless, too much choice can lead to clinical depression. In The Paradox of Choice, Barry Schwartz explains at what point choice ; the hallmark of individual freedom and self-determination that we so cherish ; becomes detrimental to our psychological and emotional well-being. In accessible, engaging, and anecdotal prose, Schwartz shows how the dramatic explosion in choice ; from the mundane to the profound challenges of balancing career, family, and individual needs ; has paradoxically become a problem instead of a solution. Schwartz also shows how our obsession with choice encourages us to seek that which makes us feel worse. By synthesizing current research in the social sciences, Schwartz makes the counter intuitive case that eliminating choices can greatly reduce the stress, anxiety, and busyness of our lives. He offers eleven practical steps on how to limit choices to a manageable number, have the discipline to focus on those that are important and ignore the rest, and ultimately derive greater satisfaction from the choices you have to make."--provided by publisher. (shrink)

May's Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May's axioms, we can uniquely determine how to vote on three alternatives. In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential (...) voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, agrees with Minimax voting in all three-alternative elections, except perhaps in some improbable knife-edged elections in which ties may arise and be broken in different ways. (shrink)

The paradox of increase in an ancient argument purporting to show that nothing can grow by acquiring new parts. If it is sound, similar reasoning leads to the more general conclusion that nothing can ever change its parts. After discussing the implicationsof this principle, the paper lays out the paradox in a way that reveals the premises that figure in it. It emerges that the paradox has no easy solution, and can be resisted only by taking (...) on one of five serious metaphysical commitments. (shrink)

The Liar paradox arises when we consider a sentence that says of itself that it is not true. If such self-referential sentences exist? and examples like?This sentence is not true? certainly suggest this?, then our logic and standard notion of truth allow to infer a contradiction: The Liar sentence is true and not true. What has gone wrong? Must we revise our notion of truth and our logic? Or can we dispel the common conviction that there are such self-referential (...) sentences? The present study explores the second path.0After comparing the Liar reasoning in formal and informal logic and showing that there are no Gödelian Liar sentences, the study moves on from the semantics of self-reference to the metaphysics of expressions and proposes a novel solution to the Liar paradox: Meaningful expressions are distinct from their syntactic bases and exist only relative to contexts. Detailed semantico-metaphysical arguments show that in this dynamic setting, an object can be referred to only after it has started to exist. Hence the circular reference needed in the Liar paradox cannot occur, after all. As this solution is contextualist, it evades the expressibility problems of other proposals. (shrink)

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 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. (shrink)

Given any proposition, is it possible to have rationally acceptable attitudes towards it? Absent reasons to the contrary, one would probably think that this should be possible. In this paper I provide a reason to the contrary. There is a proposition such that, if one has any opinions about it at all, one will have a rationally unacceptable set of propositional attitudes—or if one doesn’t, one will end up being cognitively imperfect in some other manner. The proposition I am concerned (...) with is a self-referential propositional attitude ascription involving the propositional attitude of rejection. Given a basic assumption about what constitutes irrationality, and a few assumptions about the nature of cognitively ideal agents, a paradox results. This paradox is superficially like the Liar, but it is importantly different in that no alethic notions are involved at all. As such, it stands independent of the Liar and is not a ‘revenge’ version of it. After setting out the paradox I discuss possible responses. After considering several I argue that one is best off simply accepting that the paradox shows us something surprising and interesting about rationality: that some cognitive shortfall is unavoidable even for ideal agents. I argue that nothing disastrous follows from accepting this conclusion. (shrink)

The Borel–Kolmogorov Paradox is typically taken to highlight a tension between our intuition that certain conditional probabilities with respect to probability zero conditioning events are well defined and the mathematical definition of conditional probability by Bayes’ formula, which loses its meaning when the conditioning event has probability zero. We argue in this paper that the theory of conditional expectations is the proper mathematical device to conditionalize and that this theory allows conditionalization with respect to probability zero events. The conditional (...) probabilities on probability zero events in the Borel–Kolmogorov Paradox also can be calculated using conditional expectations. The alleged clash arising from the fact that one obtains different values for the conditional probabilities on probability zero events depending on what conditional expectation one uses to calculate them is resolved by showing that the different conditional probabilities obtained using different conditional expectations cannot be interpreted as calculating in different parametrizations of the conditional probabilities of the same event with respect to the same conditioning conditions. We conclude that there is no clash between the correct intuition about what the conditional probabilities with respect to probability zero events are and the technically proper concept of conditionalization via conditional expectations—the Borel–Kolmogorov Paradox is just a pseudo-paradox. (shrink)

Sometimes people desire that their lives go badly, take pleasure in their lives going badly, or believe that their lives are going badly. As a result, some popular theories of welfare are paradoxical. I show that no attempt to defend those theories from the paradox fully succeeds.

It can be shown by means of a paradox that, given the Principle of Sufficient Reason, there is no conjunction of all contingent truths. The question is, or ought to be, how to interpret that result: _Quid sibi velit?_ A celebrated argument against PSR due to Peter van Inwagen and Jonathan Bennett in effect interprets the result to mean that PSR entails that there are no contingent truths. But reflection on parallels in philosophy of mathematics shows it can equally (...) be interpreted either as a proof that there are "too many" contingent truths to combine in a single conjunction or as a proof that the concept _contingent truth_ is indefinitely extensible and there is no such thing as "all contingent truths." Either interpretation would reconcile PSR with contingent truth, but the natural rationales of those interpretations are at odds. This essay argues that the second interpretation is a more satisfactory explanation of why, if PSR is true, there should be no conjunction of all contingent truths. This sheds new light on the nature of the explanatory demand embedded in PSR and uncovers a number of surprising implications for the commitments of rationalism. (shrink)

There are few things in our lives we dislike or hate so strongly as pain. On the other hand, the symptoms of the patients suffered from so-called congenital analgesia teach us that if we are deprived of the ability to feel pain, our lives would be unbelievably miserable and disastrous. In this way, if we try to understand the meaning and the value of pain, it seems we immediately find a paradox. The paradoxical character of pain is reflected in (...) philosophical discussions between subjectivist’s and objectivist’s view of pain. In this paper, I try to show that the seemingly paradoxical character emerges because we are obsessed by groundless prejudice that the concept of pain must have one definite single meaning and that if phenomena of pain can be understood as multidimensional, the various characteristics of pain can be interpreted as various aspects of multi-dimensional pain experiences.Hay pocas cosas en nuestras vidas que nos desagraden u odiemos tan fuertemente como el dolor. Por otro lado, los síntomas de los pacientes que padecen la llamada analgesia congénita nos enseñan que, si nos privamos de la capacidad de sentir dolor, nuestras vidas serían increíblemente miserables y desastrosas. De esta manera, si tratamos de comprender el significado y el valor del dolor parece que encontramos inmediatamente una paradoja. El carácter paradójico del dolor se refleja en discusiones filosóficas entre la visión del dolor subjetivista y objetivista. En este trabajo, trato de mostrar que el carácter aparentemente paradójico surge porque estamos obsesionados por el prejuicio infundado de que el concepto de dolor debe tener un único significado definido; y que, si los fenómenos de dolor pueden entenderse como multidimensionales, las diversas características del dolor pueden ser interpretadas como varios aspectos de expe-riencias de dolor multidimensionales.Palabr. (shrink)

The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which (...) are employed in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if φ is true, then there exists a proof of φ Building on the work of Artemov (Bull Symb Log 7(1): 1-36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence. (shrink)

We introduce a variant of pointer structures with denotational semantics and show its equivalence to systems of boolean equations: both have the same solutions. Taking paradoxes to be statements represented by systems of equations (or pointer structures) having no solutions, we thus obtain two alternative means of deciding paradoxical character of statements, one of which is the standard theory of solving boolean equations. To analyze more adequately statements involving semantic predicates, we extend propositional logic with the assertion operator and (...) give its complete axiomatization. This logic is a sub-logic of statements in which the semantic predicates become internalized (for instance, counterparts of Tarski’s definitions and T-schemata become tautologies). Examples of analysis of self-referential paradoxes are given and the approach is compared to the alternative ones. (shrink)

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the (...) same time, the philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

Paradox of the Evil and "Family Ressemblances". The paper tackles the problem of Matter and evil in Plotinus. monistic metaphysics, especially in theperspective of the following aparent inconsistency: if there is no other principle but the Good, then the Good creates the Matter which is the absolute evil. Itfollows that the Good is bad, according to a certain axiom of Proclus, which states that the creator is to a higher degree all what the creature is. The authorshows that, despite (...) what Proclus and then many modern critics believed, Plotinus is consistent within his system. He relies on the axiom that the creature is not all what the creator is, i.e. that the creator also gives what he has not. Therefore, the One gives the Intellect multiplicity and thought which He is deprived of and also gives the Matter the evil which He is also deprived of. The paper also shows that Plotinus developed a logic of ontological procession which is not Aristotelian. This logic does not work by formind classes, but chains of partially intransitive ressemblances. So, the Intellect ressembles the One (the Good), the Soul ressembles the Intellect and the Matter ressembles the Soul; yet the Matter resembles the One no more. Yet, the unity of the world is assured, because of the continuity of the chain. The extreme terms are contrary, though not in the Aristotelian sense of sharing in the same genus. A certain similarty with Wittgenstein's logic of "family resemblances" is striking, which means that not only Wittgenstein, but Plotinus also went beyond the Platonic-Aristotelian Vulgata, even while he was sticking to its linguage. (shrink)

In this paper we show how Dummett-Prawitz-style proof-theoretic semantics has to be modified in order to cope with paradoxical phenomena. It will turn out that one of its basic tenets has to be given up, namely the definition of the correctness of an inference as validity preservation. As a result, the notions of an argument being valid and of an argument being constituted by correct inference rules will no more coincide. The gap between the two notions is accounted for (...) by introducing the distinction between sense and denotation in the proof-theoretic-semantic setting. (shrink)

– The paradoxes of analysis and identity each consist of a pair of statements sharing the same referents, but differing in their informativeness properties. Carnap employs a different solution for each of these paradoxes. Church, Davidson, and others have maintained that the two paradoxes can, and should, be resolved by a single method, viz. one based on the Fregean distinction between sense and reference.The present paper argues that Carnap's solution for the paradox of analysis is unsatisfactory on several counts, (...) but primary among these is that Carnap is driven to hold that no semantic distinction can be drawn between the informativeness properties occurring in this paradox. Nevertheless, the paper shows how the required distinction can be formulated within the general framework of Carnap's extensional‐intensional system of semantics, and that such a distinction provides a single solution for both paradoxes. (shrink)

A paradox is constructed employing four languages L1-L4, such that L1 is a metalanguage for L3, L3 for L2, and L2 for L1; L4 functions as the semantic meta-metalanguage for each of L1-L3. The paradox purports to show that no omniscient being can exist, given that there is a set of true sentences (each true within its respective language) from L1, L2, and L3 that no omniscient being can believe.The remainder of the paper consists in an examination (...) of some attempts at challenging the paradox on syntactic, semantic and pragmatic grounds. Just which of these attempts are the most promising for the religious person is a question which is left open. (shrink)

Paradox of the Evil and "Family Ressemblances". The paper tackles the problem of Matter and evil in Plotinus. monistic metaphysics, especially in theperspective of the following aparent inconsistency: if there is no other principle but the Good, then the Good creates the Matter which is the absolute evil. Itfollows that the Good is bad, according to a certain axiom of Proclus, which states that the creator is to a higher degree all what the creature is. The authorshows that, despite (...) what Proclus and then many modern critics believed, Plotinus is consistent within his system. He relies on the axiom that the creature is not all what the creator is, i.e. that the creator also gives what he has not. Therefore, the One gives the Intellect multiplicity and thought which He is deprived of and also gives the Matter the evil which He is also deprived of. The paper also shows that Plotinus developed a logic of ontological procession which is not Aristotelian. This logic does not work by formind classes, but chains of partially intransitive ressemblances. So, the Intellect ressembles the One (the Good), the Soul ressembles the Intellect and the Matter ressembles the Soul; yet the Matter resembles the One no more. Yet, the unity of the world is assured, because of the continuity of the chain. The extreme terms are contrary, though not in the Aristotelian sense of sharing in the same genus. A certain similarty with Wittgenstein's logic of "family resemblances" is striking, which means that not only Wittgenstein, but Plotinus also went beyond the Platonic-Aristotelian Vulgata, even while he was sticking to its linguage. (shrink)

The “paradox of forgiveness” can be described as follows: Forgiving, unlike forgetting, is tied to reasons. It is a response to considerations that lead us to think that we ought to forgive. On the other hand, acts of forgiveness, unlike excuses, are responses to instances of culpable wrongdoing. If, however, the wrongdoing is culpable, there is (or seems to be) no reason to forgive it. So two mutually exclusive theses about forgiveness both seem to be equally warranted: Forgiveness is (...) related to reasons, but there can be no reasons for forgiveness. In this paper, I attempt to dissolve this paradox. I argue that the paradox arises as a result of a too narrow conception of “reason” and that it can be dissolved if we acknowledge different kinds of reasons for forgiveness. More specifically, I examine three kinds of reasons for forgiving an act of wrongdoing: (1) Moral reasons that make forgiveness morally mandatory. (2) Prudential reasons for forgiveness. (3) Moral reasons that pertain to the character of the forgiver and that favor forgiveness without making it morally mandatory. I show that while the paradox of forgiveness arises when we consider reasons of the first kind, it can be dissolved with recourse to reasons of the second and third kind. The upshot of the argument is that we can be rational in deciding to overcome our feelings of resentment towards an act of unjustified and unexcused wrongdoing—and this is a strong point in favor of forgiveness. (shrink)

In this paper, I present a novel paradox that pertains to a variety of representational states and activities. I begin by proving that there are certain contingently true propositions that no one can occurrently believe. Then, I use this to develop a further proof by which I derive a contradiction, thus giving us the paradox. Next, I differentiate the paradox from the Liar Paradox, and I show how a common response to the different variations of (...) the Liar Paradox fails to avoid the type of paradox provided in this paper. Finally, I demonstrate how the general ideas behind the paradox regarding occurrent belief can be extended to a wide range of other representational states and activities. (shrink)

In his paper, “On paradox without self-reference”, Neil Tennant proposed the conjecture for self-referential paradoxes that any derivation formalizing self-referential paradoxes only generates a looping reduction sequence. According to him, the derivation of the Liar paradox in natural deduction initiates a looping reduction sequence and the derivation of the Yablo's paradox generates a spiral reduction. The present paper proposes the counterexample to Tennant's conjecture for self-referential paradoxes. We shall show that there is a derivation of the (...) Liar paradox which generates a spiraling reduction procedure. Since the Liar paradox is a self-referential paradox, the result is a counterexample to his conjecture. Tennant has believed that classical reductio has no essential role to formalize paradoxes. As our counterexample applies the rule of classical reductio, he may reject the counterexample. In this sense, it will be briefly argued that classical reductio and his rules for the liar sentence share some inferential role. If classical reductio should not be used in paradoxical reasoning, neither should be his rules for the liar sentence. (shrink)

In his paper, “On paradox without self-reference”, Neil Tennant proposed the conjecture for self-referential paradoxes that any derivation formalizing self-referential paradoxes only generates a looping reduction sequence. According to him, the derivation of the Liar paradox in natural deduction initiates a looping reduction sequence and the derivation of the Yablo's paradox generates a spiral reduction. The present paper proposes the counterexample to Tennant's conjecture for self-referential paradoxes. We shall show that there is a derivation of the (...) Liar paradox which generates a spiraling reduction procedure. Since the Liar paradox is a self-referential paradox, the result is a counterexample to his conjecture. Tennant has believed that classical reductio has no essential role to formalize paradoxes. As our counterexample applies the rule of classical reductio, he may reject the counterexample. In this sense, it will be briefly argued that classical reductio and his rules for the liar sentence share some inferential role. If classical reductio should not be used in paradoxical reasoning, neither should be his rules for the liar sentence. (shrink)

The ?no???no? paradox (so-called by Sorensen) consists of a pair of propositions each of which says of the other that it is false. It is not immediately paradoxical, since it has a solution in which one proposition is true, the other false. However, that is itself paradoxical, since there is no clear ground for determining which is which. The two propositions should have the same truth-value. The paper shows how a proposal by the medieval thinker Thomas Bradwardine solves not (...) only the Liar paradox, but also symmetric paradoxes like the ?no???no?, the descending ?no???no?, and the Truth-teller paradoxes. (shrink)

Conceptual development requires learning. It requires learning to make discriminations that were previously unavailable to the subject. Notwithstanding the descriptions of learning available in the psychological and educational literature, there is no account available that shows that it is so much as possible. There can be no such account unless there is an answer to Jerry Fodor’s paradox of learning. On our current understanding of concept acquisition, there is no such thing as learning. In this paper I explore a (...) way of avoiding this conclusion. The enquiry is foundational, an enquiry into the very possibility of learning and development. The account of learning that I sketch has, however, clear consequences for our basic ideas about education. (shrink)

In this essay (for undergraduates) I introduce three of the famous semantic paradoxes: the Liar, Grelling’s, and the No-No. Collectively, they seem to show that the notion of truth is highly paradoxical, perhaps even contradictory. They seem to show that the concept of truth is a bit akin to the concept of a married bachelor—it just makes no sense at all. But in order to really understand those paradoxes one needs to be very comfortable thinking about how lots (...) of interesting sentences talk about not dogs or cats or elections or baseball but sentences. That is, we need to get familiar analyzing sentences that talk about sentences. (shrink)

An important suggestion of objective Bayesians is that the maximum entropy principle can replace a principle which is known to get into paradoxical difficulties: the principle of indifference. No one has previously determined whether the maximum entropy principle is better able to solve Bertrand’s chord paradox than the principle of indifference. In this paper I show that it is not. Additionally, the course of the analysis brings to light a new paradox, a revenge paradox of the (...) chords, that is unique to the maximum entropy principle. (shrink)

The paradoxes of self reference have to be dealt with by anyone seeking to give a satisfactory account of the logic of truth, of properties, and even of sets of numbers. Unfortunately, there is no widespread agreement as to how to deal with these paradoxes. Some approaches block the paradoxical inferences by rejecting as invalid a move that classical logic counts as valid. In the recent literature, this deviant logic analysis of the paradoxes has been called into question.This disagreement motivates (...) a re-examination of the philosophy of formal logic and the status of logical truths and rules. In this paper I do some of this work, and I show that this gives us the means to defend the deviant approaches against such criticisms. As a result I hope to show that these analyses of the paradoxes are worthy of more serious consideration than they have so far received. (shrink)

Deflationist accounts of truth are widely held in contemporary philosophy: they seek to show that truth is a dispensable concept with no metaphysical depth. However, logical paradoxes present problems for deflationists that their work has struggled to overcome. In this volume of fourteen original essays, a distinguished team of contributors explore the extent to which, if at all, deflationism can accommodate paradox. The volume will be of interest to philosophers of logic, philosophers of language, and anyone working on (...) truth. Contributors include Bradley Armour-Garb, Jody Azzouni, JC Beall, Hartry Field, Christopher Gauker, Michael Glanzberg, Dorothy Grover, Anil Gupta, Volker Halbach, Leon Horsten, Paul Horwich, Graham Priest, Greg Restall, and Alan Weir. (shrink)

A growing body of empirical evidence suggests the existence of a Paradox of Choice, whereby a larger choice set leads to a lower expected payoff to the decisionmaker. These empirical findings contradict traditional choice-theoretic results in microeconomics and even social psychology, suggesting profound yet unexplained aspects of choice settings/behavior. The Paradox remains largely as a theoretically-rootless empirical phenomenon. We neither understand the mechanisms and conditions that generate it, nor whether the Paradox stems from choice behavior, choice setting, (...) or both. We present a general search model of consumer choice under incomplete information to examine the theoretical existence and conditions for the Paradox. The model departs from traditional microeconomic choice theory in assuming no “cognitive free lunches.” Search and evaluative efforts require time, focus, and opportunity cost in the model, as in real life. The model shows that conditions for Paradox readily exist, diminishing returns to potential value of the choice in choice set size is a necessary condition for a Paradox outcome, that a Paradox outcome can occur with or without a “choice overload” response by the decisionmaker, and that avoidance of a Paradox outcome is an impossibility in some choice settings (primitive to setting regardless of choice behavior). Lastly, we develop a pareto-improving choice set reduction algorithm that eliminates weakly dominated choice set elements and presents the restricted choice set to the decisionmaker. As such, it is shown that the algorithm acts to treat or mitigate the Paradox of Choice, as well as to pareto-improve choice-supporting evaluation more generally. Like established AI recommender systems, the novel algorithm filters choices. In the present case, the filtering is based upon identification of weakly dominated choice set elements. (shrink)

In this dissertation I examine the sceptical problem of rule following presented by Saul Kripke in his interpretation of Ludwig Wittgenstein's later works: Do any facts determine what rule we were following in our apparently rule-following activities such as the use of language? I distinguish three ways of understanding this question--modest scepticism, radical scepticism, and metascepticism--and address them in Parts 1, 2 and 3 of the dissertation, respectively. ;Part 1 discusses modest scepticism, which asserts that no finite facts about humans (...) can determine rules which have infinitely many possible applications. I resolve this apparent conflict between the finite and the infinite by incorporating the recursiveness of rule applications into the theory of rules. ;Part 2 discusses radical scepticism, which asserts that no facts determine what normative rules we were following. I show first that this challenge has a wider scope than the Humean Thesis of the underivability of ought from is, and that as a result, the radical sceptic can refute attitudinal theories of norms, which have been considered a possible response to the Humean challenge. I then propose a solution to radical scepticism which explains our practice and beliefs about normative rules without assuming the existence of rule-determining facts. Unlike the solution Kripke ascribes to Wittgenstein, it offers no justification of our practice and beliefs about normative rules. Some methodological implications of the solution are also explored. ;Part 3 discusses metascepticism about rule following, whose challenge extends to the rules used for the very discussion of rule following. I examine first whether such a challenge is self-undermining in that it makes the sceptical argument itself impossible. I propose an extended form of reductio ad absurdum argument by which the metasceptic can challenge rules in general, including those used in the sceptical argument. Then I argue that the solution to radical scepticism given in Part 2 survives metascepticism as well. (shrink)

There has been much discussion on the two-envelope paradox. Clark and Shackel (2000) have proposed a solution to the paradox, which has been refuted by Meacham and Weisberg (2003). Surprisingly, however, the literature still contains no axiomatic justification for the claim that one should be indifferent between the two envelopes before opening one of them. According to Meacham and Weisberg, "decision theory does not rank swapping against sticking [before opening any envelope]" (p. 686). To fill this gap in (...) the literature, we present a simple axiomatic justification for indifference, avoiding any expectation reasoning, which is often considered problematic in infinite cases. Although the two-envelope paradox assumes an expectation-maximizing agent, we show that analogous paradoxes arise for agents using different decision principles such as maximin and maximax, and that our justification for indifference before opening applies here too. (shrink)

It is often said that in a purely formal perspective, intuitionistic logic has no obvious advantage to deal with the liar-type paradoxes. In this paper, we will argue that the standard intuitionistic natural deduction systems are vulnerable to the liar-type paradoxes in the sense that the acceptance of the liar-type sentences results in inference to absurdity (⊥). The result shows that the restriction of the Double Negation Elimination (DNE) fails to block the inference to ⊥. It is, however, not the (...) problem of the intuitionistic approaches to the liar-type paradoxes but the lack of expressive power of the standard intuitionistic natural deduction system. We introduce a meta-level negation for a given system and a meta-level absurdity, ⋏, to the intuitionistic system. We shall show that in the system, the inference to ⊥ is not given without the assumption that the system is complete. Moreover, we consider the Double Meta-Level Negation Elimination rules (DMNE) which implicitly assume the completeness of the system. Then, the restriction of DMNE can rule out the inference to ⊥. (shrink)

We give a general proof showing that if the evolution from one state to another is not reversible, then the projective measurements on the superposition of these two states are impossible. Applying this no-go result to the Schrödinger’s cat paradox implies that if something is claimed to be a real Schrödinger’s cat, there will be no measurable difference between it and a trivial classical mixture of ordinary cats in any physically implementable process, unless raising the dead becomes reality. Other (...) similar macroscopic quantum superpositions cannot be observed either, due to the lack of non-commuting measurement bases. Our proof does not involve any quantum interpretation theory and hypothesis. (shrink)

ABSTRACTMany discussions of the ‘preface paradox’ assume that it is more troubling for deductive closure constraints on rational belief if outright belief is reducible to credence. I show that this is an error: we can generate the problem without assuming such reducibility. All that we need are some very weak normative assumptions about rational relationships between belief and credence. The only view that escapes my way of formulating the problem for the deductive closure constraint is in fact itself (...) a reductive view: namely, the view that outright belief is credence 1. However, I argue that this view is unsustainable. Moreover, my version of the problem turns on no particular theory of evidence or evidential probability, and so cannot be avoided by adopting some revisionary such theory. In sum, deductive closure is in more serious, and more general, trouble than some have thought. (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)

IN THE beginning Parmenides sought to deny the void. But he found himself trapped by his language and his thought into admitting what he sought to deny. Wisely, he counseled others to avoid the whole region in which the problem arises, lest they too be unwarily ensnared. Plato, being less easily intimidated and grasping for the first time the urgency of the paradox, unearthed each snare in turn until he felt he had found a safe path through the forbidden (...) terrain in a new conception of being and the derivation of its linguistic consequences in the Sophist. Aristotle evidently took Parmenides’ advice; and save for a few groping scholastics, perhaps Leibniz, Brentano, and Meinong, and Frege only in passing, no one else attempted the crossing before Russell made his spectacular dash through the posted ground from the completely new direction of linguistic reference. Again the problem lay dormant for half a century until Strawson constructed a new low road through ordinary language and Quine improved Russell’s high algebraic pass. Refinements of these routes have been forthcoming, especially from Searle and Kripke, until today it might appear that there are two super highways through Parmenides’ forbidden country of nonbeing. In this essay I will first argue that these new linguistic highways are no more than flimsy camouflage hiding but not resolving the old paradoxes. I will then show how Plato’s ontological way out, though more difficult, is the straight and narrow path. (shrink)

A longstanding paradox was first reported by David Miller in 1975 and highlighted by Karl Popper in 1979. Miller showed that the ranking of predictions from two theories, in terms of closeness to observation, appears to be reversed when the problem is transformed into a different mathematical space. He concluded that “… no false theory can … be closer to the truth than is another theory”. This flies in the face of normal scientific practice and is thus paradoxical; it (...) is named here the “Miller-Popper paradox”. This paper proposes a resolution of the paradox, through consideration of the inevitable errors and uncertainties in both observations and predictions. It is proved that, for linear transformations and Gaussian error distributions, the transformation between spaces creates no change in quantitative measures of “closeness-to-observation” when these measures are based in probability theory. The extension of this result to nonlinear transformations and to non-Gaussian error distributions is also discussed. These results demonstrate that concepts used in comparison of predictions with observations—concepts of “closeness”, “consistency”, “agreement”, “falsification”, etc.—all imply some knowledge of the uncertainty characteristics of both predictions and observations. (shrink)

For each of the many goals of an agent it is true that the agent wants its realization. Given further very plausible assumptions, one can show that there is no good reason for an agent not to want the realization of all of his goals. However, it seems also true that reaching all of one’s goals would be extremely boring; most human beings would consider such a life not worth living. In this respect, leading a life is like playing (...) some game: A game loses its point if one always easily wins. Human agents not only pursue goals but they also have the second-order desire to lead the life of a goal-pursuer. More precisely, they want to be goal-pursuers such that many but not all of their goals are being realized. I argue here that they have a good reason not to want the realization of all of their goals. Now, all this constitutes an antinomy: Apparently, there are very plausible arguments both for affirming as well as denying that an agent has good reasons not to want the realization of his goals. This antinomy is similar to the preface-paradox about beliefs which has a convincing solution (due to Frank Ramsey). I argue, however, that there is no such solution in the case of goals. This leaves us with the question what a solution could look like. (shrink)

In response to the so-called “paradox of deontology,” many have argued that the agent-relativity of deontological constraints accounts for why an agent may not kill one in order to prevent five others from being killed. Constraints provide reasons for particular agents not to kill, not reasons to minimize overall killings. In this paper, I tease out the significance of an underappreciated aspect of this agent-relative position, i.e. it provides no guidance as to what an agent ought to do when (...) faced with the prospect of killing one in order to prevent herself from killing five. After rejecting mere agent-relativity, the view that agents are morally permitted to violate constraints in order to minimize their overall violations, I offer a view that this is both agent- and time-relative, and show how this view exemplifies the underlying motivations for deontological constraints while successfully responding to both the inter- and intra-personal paradoxes of deontology. (shrink)