A new version of the ontological argument for the existence of God is outlined and examined. After giving a brief account of some traditional ontological arguments for the existence of God, where their defects are identified, it is explained how this new argument is built upon their foundations and surmounts their defects. In particular, this version uses the resources of impossible worlds to plug the common escape route from standard modal versions of the ontological argument. After outlining the nature of (...) impossible worlds, and motivating the need for positing them, the new argument is delineated and its premises justified. It is taken for granted that the argument cannot be sound, since it would prove too much. However, its premises are all plausible, and their denial promises to have significant ramifications. Several intuitive lines of objections are then explored in order to illuminate their shortcomings. The puzzle that the argument poses is therefore not whether the argument is sound, for it clearly cannot be. Rather, it is to place pressure on its plausible premises, so some plausible account of how the argument fails can be identified, and that the devising of such an account promises to be insightful. In the process, we should gain an improved understanding of how such ontological arguments work. (shrink)

A new version of the ontological argument for the existence of God is outlined and examined. After giving a brief account of some traditional ontological arguments for the existence of God, where their defects are identified, it is explained how this new argument is built upon their foundations and surmounts their defects. In particular, this version uses the resources of impossible worlds to plug the common escape route from standard modal versions of the ontological argument. After outlining the nature of (...) impossible worlds, and motivating the need for positing them, the new argument is delineated and its premises justified. It is taken for granted that the argument cannot be sound, since it would prove too much. However, its premises are all plausible, and their denial promises to have significant ramifications. Several intuitive lines of objections are then explored in order to illuminate their shortcomings. The puzzle that the argument poses is therefore not whether the argument is sound, for it clearly cannot be. Rather, it is to place pressure on its plausible premises, so some plausible account of how the argument fails can be identified, and that the devising of such an account promises to be insightful. In the process, we should gain an improved understanding of how such ontological arguments work. (shrink)

According to what has become a standard history of quantum mechanics, von Neumann in 1932 succeeded in convincing the physics community that he had proved that hidden variables were impossible as a matter of principle. Subsequently, leading proponents of the Copenhagen interpretation emphatically confirmed that von Neumann's proof showed the completeness of quantum mechanics. Then, the story continues, Bell in 1966 finally exposed the proof as seriously and obviously wrong; this rehabilitated hidden variables and made serious foundational research (...) possible. It is often added in recent accounts that von Neumann's error had been spotted almost immediately by Grete Hermann, but that her discovery was of no effect due to the dominant Copenhagen Zeitgeist. We shall attempt to tell a more balanced story. Most importantly, von Neumann did not claim to have shown the impossibility of hidden variables tout court, but argued that hidden-variable theories must possess a structure that deviates fundamentally from that of quantum mechanics. Both Hermann and Bell appear to have missed this point; moreover, both raised unjustified technical objections to the proof. Von Neumann's conclusion was basically that hidden-variables schemes must violate the "quantum principle" that all physical quantities are to be represented by operators in a Hilbert space. According to this conclusion, hidden-variables schemes are possible in principle but necessarily exhibit a certain kind of contextuality. As we shall illustrate, early reactions to Bohm's theory are in agreement with this account. Leading physicists pointed out that Bohm's theory has the strange feature that particle properties do not generally reveal themselves in measurements, in accordance with von Neumann's result. They did not conclude that the "impossible was done" and that von Neumann had been shown wrong. (shrink)

Beginning with Zagzebski (The Philosophical Quarterly 44:65–73, 1994), some philosophers have argued that there can be no solution to the Gettier counterexamples within the framework of a fallibilist theory of knowledge. If true, this would be devastating, since it is believed on good grounds that infallibilism leads to scepticism. But I argue here that these purported proofs are mistaken and that the truthmaker solution to the Gettier problems is both cogent and fallibilist in nature. To show this I develop the (...) notion of evidence of a state of affairs, a crucial concept in the truthmaker theory. I also argue that a common principle of the transmission of evidence through entailment is false, and the cause of much of the trouble. (shrink)

John von Neumann's proof that quantum mechanics is logically incompatible with hidden varibales has been the object of extensive study both by physicists and by historians. The latter have concentrated mainly on the way the proof was interpreted, accepted and rejected between 1932, when it was published, and 1966, when J.S. Bell published the first explicit identification of the mistake it involved. What is proposed in this paper is an investigation into the origins of the proof rather (...) than the aftermath. In the first section, a brief overview of the his personal life and his proof is given to set the scene. There follows a discussion on the merits of using here the historical method employed elsewhere by Andrew Warwick. It will be argued that a study of the origins of von Neumann's proof shows how there is an interaction between the following factors: the broad issues within a specific culture, the learning process of the theoretical physicist concerned, and the conceptual techniques available. In our case, the ‘conceptual technology’ employed by von Neumann is identified as the method of axiomatisation. (shrink)

Written for any motivated reader with a high-school knowledge of mathematics, and the discipline to follow logical arguments, this book presents the proofs for revolutionary impossibility theorems in an accessible way, with less jargon and notation, and more background, intuition, examples, explanations, and exercises.

Since Kenneth Arrow showed the general possibility theorem, a number of social choice theorists have provided alternative proofs of it. In a recent article, Geanakoplos (Econ Theory 26:211–215, 2005) has constructed a new proof of the theorem. The present article provides alternative proofs of various Arrovian impossibility results from the 1960s to the 1970s by utilizing Geanakoplos’s method. We prove semi-order impossibility theorems, the quasi-transitive veto theorem, the quasi-transitive dictatorship theorem, the triple acyclic veto theorem, and the (...) impossibility theorem without the Pareto principle. (shrink)

The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived—without any further assumptions about the external world—from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. (...) free from contradiction. A complete proof will rather have to consist of two parts. First, a proof of premiseGod is possible. Second, a demonstration of the “remarkable proposition”If God is possible, then God exists. The present contribution investigates an interesting paper in which Leibniz tries to prove proposition. It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ as follows:\\). Proposition which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows:\) \rightarrow \, E)\). At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to, the possibility of the necessary being has to be understood in the sense of something which possibly exists. However, in other places of his proof, Leibniz interprets the assumption that the necessary being is impossible in the diverging sense of something which involves a contradiction. Furthermore, Leibniz believes that an »impossible thing«, y, is such that contradictory propositions like \\) and \\) might both be true of y. It will be argued that the latter assumption is incompatible with Leibniz’s general views about logic and that the crucial proof is better reinterpreted as dealing with the necessity, possibility, and impossibility of concepts rather than of objects. In this case, the counterpart of turns out to be a theorem of Leibniz’s second order logic of concepts; but in order to obtain a full demonstration of the existence of God, the counterpart of, i.e. the self-consistency of the concept of a necessary being, remains to be proven. (shrink)

In Prior Analytics 1.15, Aristotle attempts to give a proof through the impossible of Barbara, Celarent, Darii, and Ferio with an assertoric first premiss, a contingent second premiss, and a possible conclusion. These proofs have been controversial since antiquity. I shall show that they are valid, and that Aristotle is able to explain them by relying on two meta-syllogistic lemmas on the nature of possibility interpreted as syntactic consistency. It will turn out that Aristotle's proofs are not of the (...) intended schemata. I shall investigate some of the results that the impact of this reconstruction on the modal syllogistic has: the relationship between Aristotle's syllogistic and the logics of relevance; the value of Aristotle's requirement that universal affirmative propositions be taken ‘absolutely’; the destruction of many Aristotelian proofs; the recovery of certain principles of modal opposition from a charge of inconsistency. (shrink)

In this article, I argue that it is impossible to complete infinitely many tasks in a finite time. A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task remaining, when tasks are done 1-by-1. I suggest that the only way to deny this premise is by begging the question, that is, by assuming that supertasks are possible. I go on to present one reason why this conclusion (that supertasks are (...) impossible) is important, namely that it implies a new verdict on a decision puzzle propounded by Jeffrey Barrett and Frank Arntzenius. (shrink)

In his book Philosophy of Logic, Putnam (1971) presents a short argument which reads like—and indeed, can be reconstructed as—a formal proof that a nominalistic physics is impossible. The aim of this paper is to examine Putnam’s proof and show that it is not compelling. The precise way in which the proof fails yields insight into the relation that a nominalistic physics should bear to standard physics and into Putnam’s indispensability argument.

There is an emerging consensus in the literature on probabilistic coherence that such coherence cannot be truth conducive unless the information sources providing the cohering information are individually credible and collectively independent. Furthermore, coherence can at best be truth conducive in a ceteris paribus sense. Bovens and Hartmann have argued that there cannot be any measure of coherence that is truth conducive even in this very weak sense. In this paper, I give an alternative impossibility proof. I provide (...) a relatively detailed comparison of the two results, which turn out to be logically unrelated, and argue that my result answers a question raised by Bovens and Hartmann’s study. Finally, I discuss the epistemological ramifications of these findings and try to make plausible that a shift to an explanatory framework such as Thagard’s is unlikely to turn the impossibility into a possibility. (shrink)

In this paper I prove a theorem which is similar to Arrow's famous impossibility theorem. I show that no social welfare function can be both minimally majoritarian and also independent of irrelevant alternatives. My condition of minimal majoritarianism is substantially weaker than simple majority rule.

The strange story of the von Neumann impossibility proof is recalled, and the even stranger story of later impossibility proofs, and how the impossible was done by de Broglie and Bohm. Morals are drawn.

Existing discussions of legal proof address a host of apparently disparate questions: What does it take to prove a fact beyond a reasonable doubt? Why is the reasonable doubt standard notoriously elusive, sometimes considered by courts to be impossible to define? Can the standard of proof by a preponderance of the evidence be defined in terms of probability thresholds? Why is statistical evidence often insufficient to meet the burden of proof? -/- This paper defends an account of (...) proof that addresses each of these questions. Where existing theories take a piecemeal approach to these puzzles, my theory develops an insight that unifies them—namely, the thesis that legal proof requires knowledge. Although this thesis may seem radical at first, I answer several common objections to it, and I argue that it does better than several competing accounts when it comes to making sense of our intuitions about what legal proof requires. (shrink)

Which rules for aggregating judgments on logically connected propositions are manipulable and which not? In this paper, we introduce a preference-free concept of non-manipulability and contrast it with a preference-theoretic concept of strategy-proofness. We characterize all non-manipulable and all strategy-proof judgment aggregation rules and prove an impossibility theorem similar to the Gibbard--Satterthwaite theorem. We also discuss weaker forms of non-manipulability and strategy-proofness. Comparing two frequently discussed aggregation rules, we show that “conclusion-based voting” is less vulnerable to manipulation than (...) “premise-based voting”, which is strategy-proof only for “reason-oriented” individuals. Surprisingly, for “outcome-oriented” individuals, the two rules are strategically equivalent, generating identical judgments in equilibrium. Our results introduce game-theoretic considerations into judgment aggregation and have implications for debates on deliberative democracy. (shrink)

A probability aggregation rule assigns to each profile of probability functions across a group of individuals (representing their individual probability assignments to some propositions) a collective probability function (representing the group's probability assignment). The rule is “non-manipulable” if no group member can manipulate the collective probability for any proposition in the direction of his or her own probability by misrepresenting his or her probability function (“strategic voting”). We show that, except in trivial cases, no probability aggregation rule satisfying two mild (...) conditions (non-dictatorship and consensus preservation) is non-manipulable. (shrink)

Based on the strict definitions of concepts, such as deduction, the deduction rule and the deduction system, the form axiom, the substantive axiom, this article clearly shows the essence of the deductive reasoning, namely “Related attribute and the related restriction relations, which are conveyed in what the main concept of the deduction refers to, must be contained in those conveyed in what the premise proposition refers to”。Then puts forward the theorem “contradiction can not be derived from the strict deduction system”, (...) and gives the proofs. (shrink)

‘Minimal liberalism’, in Sen’s strict definition, is impossible, because any ‘social state’, once chosen, freezes all of its components, thereby removing any prospect of further assignment of choice-making authority.

The impossibility results in judgement aggregation show a clash between fair aggregation procedures and rational collective outcomes. In this paper, we are interested in analysing the notion of rational outcome by proposing a proof-theoretical understanding of collective rationality. In particular, we use the analysis of proofs and inferences provided by linear logic in order to define a fine-grained notion of group reasoning that allows for studying collective rationality with respect to a number of logics. We analyse the well-known (...) paradoxes in judgement aggregation and we pinpoint the reasoning steps that trigger the inconsistencies. Moreover, we extend the map of possibility and impossibility results in judgement aggregation by discussing the case of substructural logics. In particular, we show that there exist fragments of linear logic for which general possibility results can be obtained. (shrink)

One of the key tenets in Wittgenstein’s philosophy of mathematics is that a mathematical proposition gets its meaning from its proof. This seems to have the paradoxical consequence that a mathematical conjecture has no meaning, or at least not the same meaning that it will have once a proof has been found. Hence, it would appear that a conjecture can never be proven true: for what is proven true must ipso facto be a different proposition from what was (...) only conjectured. Moreover, it would appear impossible that the same mathematical proposition be proven in different ways. — I will consider some of Wittgenstein’s remarks on these issues, and attempt to reconstruct his position in a way that makes it appear less paradoxical. (shrink)

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established (...) to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. -/- Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. -/- However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. -/- The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion. (shrink)

hiS volume in the Synthese Library Series is the result of a conference T held at the University of Roskilde, Denmark, October 31st-November 1st, 1997. The aim was to provide a forum within which philosophers, math ematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called Proof Theory: History and Philosophical Significance. To quote from the conference abstract: Proof theory was developed as part (...) of Hilberts Programme. According to Hilberts Programme one could provide mathematics with a firm and se cure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert's Programme in its original form was unfeasible mainly due to Gtldel's incompleteness theorems. Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite of these problems, Gentzen showed that by extending Hilbert's proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitis tic methods but still by methods of minimal strength. This generalization of Hilbert's original programme has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the philosophy of mathematics. (shrink)

Proof and Disproof in Formal Logic is a lively and entertaining introduction to formal logic providing an excellent insight into how a simple logic works. Formal logic allows you to check a logical claim without considering what the claim means. This highly abstracted idea is an essential and practical part of computer science. The idea of a formal system-a collection of rules and axioms, which define a universe of logical proofs-is what gives us programming languages and modern-day programming. This (...) book concentrates on using logic as a tool: making and using formal proofs and disproofs of particular logical claims. The logic it uses-natural deduction-is very small and very simple; working with it helps you see how large mathematical universes can be built on small foundations. The book is divided into four parts: Part I "Basics" gives an introduction to formal logic with a short history of logic and explanations of some technical words. Part II "Formal Syntactic Proof" show you how to do calculations in a formal system where you are guided by shapes and never need to think about meaning. Your experiments are aided by Jape, which can operate as both inquisitor and oracle. Part III "Formal Semantic Disproof" shows you how to construct mathematical counterexamples to shoe that proof is impossible. Jape can check the counterexamples you build. Part IV " Program Specification and Proof" describes how to apply your logical understanding to a real computer science problem, the accurate description and verification of programs. Jape helps, as far as arithmetic allows. Aimed at undergraduates and graduates in computer science, logic, mathematics and philosophy, the text includes reference to and exercises based on the computer software package Jape, an interactive teaching and research tool designed and hosted by the author that is freely available on the web. (shrink)

I wish to present a proof that vagueness is impossible. Of course, vagueness is possible; and so there must be something wrong with the proof. But it is far from clear where the error lies and, indeed, all of the assumptions upon which the proof depends are ones that have commonly been accepted. This suggests that we may have to radically alter our current conception of vagueness if we are to make proper sense of what it is.

In lieu of an abstract, here is a brief excerpt of the content:Proof and Persuasion in Black Athena:: The Case of K. O. MüllerJosine H. BlokNon tali auxilio.Virgil, Aeneid II, 521When in 1824 the German classical scholar Karl Otfried Müller (1797–1840) set down to write a review of Champollion’s first Letter to M. Dacier (1822), he was profoundly interested. 1 For several years he had been working on Egypt, and as he told his parents in 1820, “I have come (...) to love Egyptian antiquity so much, that, if I were not constricted by the schedule of my classes, I would have set myself to the decipherment of the hieroglyphs, which I would not deem impossible by the clues I have found.” 2 During the same years (1820–25) that he wrote his early books on ancient Greece, he [End Page 705] reviewed sixteen studies on Egypt alone. 3 In his review of the Letter, then, he wanted to draw “the attention of [the] readers to this, certainly not unfounded, discovery,” 4 though he feared the author was inclined to jump to conclusions. But in August 1824 he reviewed Champollion’s more complete publication of his findings of the same year and, recalling his previous anxiety, commented:Now, however, the reading of the present work has totally convinced me, that the usage of hieroglyphs to indicate sounds is as ancient as this writing system itself. [... This discovery should also mean that...] the history of Egyptian religion and state will be reconstructed and expanded. 5From that moment he sided unequivocally with Champollion, politely but clearly reminding German colleagues who still stumbled on with decipherments of their own, that this would not do. 6With these facts in mind, Martin Bernal’s rendering in his Black Athena, volume I, is bound to surprise: “[U]nlike Humboldt, Niebuhr and Bunsen [Müller] disregarded the sensational scholarly developments between 1815 and 1830. There is no indication that he paid any attention to Champollion’s decipherment” (316). The contradiction between Bernal’s statements and the sources on Müller’s life and work turns out not to be an incidental error but part of a larger pattern. To understand this pattern and its objectives, let me first summarize Bernal’s argument and my own.In his challenging book Bernal argues that the ancient Greek world was founded on the colonization of Greece by Phoenicians and Egyptians in the second millennium bc. 7 The material which is to prove his thesis is offered in volume II, though volume I includes a summary. In this first volume Bernal observes that from late antiquity until the eighteenth century, awareness of the Afroasiatic roots of ancient civilization had been retained as the heritage of ancient Greece itself. Hence Bernal labels this perception underlying Western classical scholarship of the early modern period the “Ancient Model.” By or soon after the 1820s, however, it was replaced by the “Aryan Model,” which took Greek civilization to be partly autochthonous and partly [End Page 706] shaped by invasions coming from the North. To reveal that the fall of the Ancient Model was not the result of internal developments in classical scholarship but of externalist influences, notably the belief in progress, the defense of Christianity, Romanticism, and most importantly racism, is the core theme of volume I. Bernal sustains his argument by contending that historical source criticism was not really or entirely an internal development but created to serve the external ends, and by presenting Müller as the embodiment of Romanticist racism, who was responsible for overthrowing the Ancient Model and inaugurating its Aryan successor. Hence, by thus designating Müller as the axis around which the overall turn of classical scholarship revolved, he wants to prove that the fall of the Ancient Model was unjustified in terms of scholarship but only occurred due to dishonorable ideologies.Through a limited number of representative issues I want to argue that Bernal’s rendering of Müller and the context of his work is untenable in the light of the source material and that Bernal’s explanation of the fall of the Ancient Model is untenable as well. My aim is not just to clear Müller’s name... (shrink)

In his "Ontological proof", Kurt Gödel introduces the notion of a second-order value property, the positive property P. The second axiom of the proof states that for any property φ: If φ is positive, its negation is not positive, and vice versa. I put forward that this concept of positiveness leads into a paradox when we apply it to the following self-reflexive sentences: (A) The truth value of A is not positive; (B) The truth value of B is (...) positive. Given axiom 2, sentences A and B paradoxically cannot be both true or both false, and it is also impossible that one of the sentences is true whereas the other is false. (shrink)

One of the main controversies of the Logic Schools of the 12th century centered on the question: What follows from the impossible? In this paper arguments for two diametrically opposed positions are examined. The author of the ‘Avranches Text’ who probably belonged to the school of the Parvipontani defended the view that from an impossible proposition everything follows (‘Ex impossibili quodlibet’). In particular he developed a proof to show that by means of so-called ‘disjunctive syllogism’ any arbitrary proposition B (...) can be logically derived from a pair of contradictory propositions A and Not-A. The author of the Ars Meliduna instead argued that nothing follows from an impossible proposition (‘ex falso nihil sequitur’). This view is supported by various counterexamples which aimed to show that the admission of impossible premises would give rise to inconsistent conclusions. Upon closer analysis these inconsistencies do not, however, have the formal structure of a real contradiction like A and Not-A, but rather the structure of two rivalling conditionals like ‘If B then A’ and ‘If B then Not-A’. Hence these counterexamples rather have to be considered as refutations of the basic principles of ‘connexive logic’. (shrink)

Complex frameworks for defining programming languages aim to generate various tools (e.g. interpreters, symbolic execution engines, deductive verifiers, etc.) using only the formal definition of a language. When used at an industrial scale, these tools are constantly updated, and at the same time, it is required to be trustworthy. Ensuring the correctness of such a framework is practically impossible. A solution is to generate proof objects as correctness artefacts that can be checked by an external trusted checker. A logic (...) suitable for developing such frameworks is matching logic. K framework is a canonical example having matching logic-based foundation. Since the (symbolic) configurations of the programs are represented by matching logic patterns, the algorithms computing the dynamics of these configurations can be seen as pattern transformers and a proof object should be generated for the relationship between these patterns. In this paper, we show that conjunctions and disjunctions of patterns, produced by semantics or analysis rules, can be safely normalized using unification and antiunification algorithms. We also provide a prototype implementation of our proof object generation technique and a checker for certifying the generated objects. (shrink)

Population axiology concerns how to evaluate populations in regard to their goodness, that is, how to order populations by the relations “is better than” and “is as good as”. This field has been riddled with impossibility results which seem to show that our considered beliefs are inconsistent in cases where the number of people and their welfare varies.1 All of these results have one thing in common, however. They all involve an adequacy condition that rules out Derek Parfit’s Repugnant (...) Conclusion: The Repugnant Conclusion: For any perfectly equal population with very high positive welfare, there is a population with very low positive welfare which is better, other things being equal.2 1 The informal Mere Addition Paradox in Parfit (1984), pp. 419ff is the locus classicus. For an informal proof of a similar result with stronger assumptions, see Ng (1989), p. 240. A formal proof with slightly stronger assumptions than Ng’s can be found in Blackorby and Donaldson (1991). For theorems with much weaker assumptions, see my (1999), (2000b), and especially (2000a), (2001), and (2009). 2 See Parfit (1984), p. 388. My formulation is more general than Parfit’s apart from that he doesn’t demand that the people with very high welfare are equally well off. Expressions such as “a population with very high positive welfare”, “a population with very low positive welfare”, etc., are elliptical for the more cumbersome phrases “a population consisting only of lives with.. (shrink)

According to one of Leibniz's theories of contingency a proposition is contingent if and only if it cannot be proved in a finite number of steps. It has been argued that this faces the Problem of Lucky Proof , namely that we could begin by analysing the concept ‘Peter’ by saying that ‘Peter is a denier of Christ and …’, thereby having proved the proposition ‘Peter denies Christ’ in a finite number of steps. It also faces a more general (...) but related problem that we dub the Problem of Guaranteed Proof . We argue that Leibniz has an answer to these problems since for him one has not proved that ‘Peter denies Christ’ unless one has also proved that ‘Peter’ is a consistent concept, an impossible task since it requires the full decomposition of the infinite concept ‘Peter’. We defend this view from objections found in the literature and maintain that for Leibniz all truths about created individual beings are contingent. (shrink)

In order to claim that one has experimentally tested whether a noncontextual ontological model could underlie certain measurement statistics in quantum theory, it is necessary to have a notion of noncontextuality that applies to unsharp measurements, i.e., those that can only be represented by positive operator-valued measures rather than projection-valued measures. This is because any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to (...) arbitrary experimental procedures, it is shown that the outcome of a measurement depends deterministically on the ontic state of the system being measured if and only if the measurement is sharp. Hence for every unsharp measurement, its outcome necessarily has an indeterministic dependence on the ontic state. We defend this proposal against alternatives. In particular, we demonstrate why considerations parallel to Fine’s theorem do not challenge this conclusion. (shrink)

herent and rational way. Several proposals have been made for information merging in which it is possible to encode the preferences of sources (Benferhat, Dubois, Prade, & Williams, 1999; Benferhat, Dubois, Kaci, & Prade, 2000; Lafage & Lang, 2000; Meyer, 2000, 2001; Andreka, Ryan, & Schobbens, 2001). Information merging has much in common with social choice theory, which aims to define operations reflecting the preferences of a society from the individual preferences of the members of the society. Given this connection, (...) frameworks for information merging should provide satisfactory resolutions of problems raised in social choice theory. We investigate the link between the merging of epistemic states and two important results in social choice theory. We show that Arrow’s well-known impossibility theorem (Arrow, 1963) does not hold in merging frameworks when the preferences of sources are represented in terms of epistemic states. This is achieved by providing a consistent set of properties for merging from which Arrow-like properties can be derived. Similarly, by extending these to a consistent framework which includes properties corresponding to the notion of being strategy-proof, we show that results due to Gibbard and Satterthwaite (Gibbard, 1973; Satterthwaite, 1973, 1975) and other (Benoit, 2000; Barber´a, Dutta, & Sen, 2000) do not hold in merging frameworks. (shrink)

The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular latticesP L or partial-Boolean algebrasP A; according to whether the semantic mappings are required to be truth-functional or truth-functional ; and according to whether two-or-higher dimensional Hilbert spaceP structures or three-or-higher dimensional Hilbert spaceP structures are being considered. If the quantumP structures are taken to be (...) orthomodular latticesP L, then bivalent mappings which preserve the operations and relations of aP L must be truth-functional . Then as suggested by von Neumann and Jauch-Piron and as proven in this paper, the mere presence of incompatible elements in aP L is sufficient to rule out any semantical or hidden-variable proposal which imposes this strong condition, for anytwo-or-higher dimensional Hilbert spaceP L structure. Thus from the orthomodular lattice perspective, the peculiarly non-classical feature of quantum mechanics and the peculiarly non-Boolean feature of the quantum propositional structures is the existence of incompatible magnitudes and propositions. However, the weaker truth-functionality condition can instead be imposed upon the semantic or hidden-variable mappings on theP L structures, although such mappings ignore the lattice meets and joins of incompatibles and preserve only the partial-Boolean algebra structural features of theP L structures. Or alternatively, the quantum propositional structures can be taken to be partial-Boolean algebrasP A, where bivalent mappings which preserve the operations and relations of aP A need only be truth-functional (c). In either case, the Gleason, Kochen-Specker proofs show that any semantical or hidden variable proposal which imposes this truth-functionality (c) condition is impossible for anythree-or-higher dimensional Hilbert spaceP A orP L structures. But such semantical or hidden-variable proposals are possible for any two dimensional Hilbert spaceP A orP L structures, in spite of the presence of incompatibles in these structures, in spite of the fact that Heisenberg's Uncertainty Principle applies to the incompatible elements in these structures, and in spite of the fact that these structures are non-Boolean in the Piron sense. (shrink)

St Anselm (1033-1109) devised an ontological “proof” of the existence of God based on the impossibility of conceiving of God's non-existence. This famous argument inspired a much less-widely known atheistic ontological “proof” of God's non-existence by Melbourne philosopher Douglas Gasking (1911-1994). Juxtaposing Gasking’s argument for the non-existence of God with Anselm’s “proof” brings the basic defect of Anselm’s argument into sharp relief.

In “The Impossibility of Pure Libertarianism” Braham and van Hees prove that four conditions on rights—completeness, conclusiveness, non-imposition, and symmetry—cannot be satisfied simultaneously. If Braham and van Hees’s proof is to have any relevance, at least some prominent libertarians must endorse their four conditions, and libertarianism as a philosophical position must in some way be committed to all the axioms. In this paper we demonstrate the irrelevance of Braham and van Hees’s proof by showing that some of (...) the most prominent libertarians do not endorse the completeness and conclusive conditions, and that there is nothing about libertarianism as a philosophical position that commits the libertarian to these two axioms. Indeed, we show that, more generally, there are strong reasons for libertarians to reject both conditions. As such, libertarians should not lose any sleep over Braham and van Hees’s proof. (shrink)

Metalogic is an open-ended cognitive, formal methodology pertaining to semantics and information processing. The language that mathematizes metalogic is known as metalanguage and deals with metafunctions purely by extension on patterns. A metalogical process involves an effective enrichment in knowledge as logical statements, and, since human cognition is an inherently logic–based representation of knowledge, a metalogical process will always be aimed at developing the scope of cognition by exploring possible cognitive implications reflected on successive levels of abstraction. Indeed, it is (...) basically impracticable to maintain logic–and–metalogic without paying heed to cognitive theorizing. For it is cognitively irrelevant whether possible conclusions are deduced from some premises before the premises are determined to be true or whether the premises themselves are determined to be true first and, then, the conclusions are deduced from them. In this paper we consider the term metalogic as inherently embodied under the framework referred to as cognitive science and mathematics. We propose a metalogical interpretation of Arrow’s impossibility theorem and, to that end, choice theory is understood as a mental course of action dealing with logic and metalogic issues, in which a possible mathematical approach to model a mental course of action is adopted as a systematic operating method. Nevertheless, if we look closely at the core of Arrow’s impossibility theorem in terms of metalogic, a second fundamental contribution to this framework is represented by the Nash equilibrium. As a result of the foregoing, therefore, we prove the metalogical equivalence between Arrow’s impossibility theorem and the existence of the Nash equilibrium. More specifically, Arrow’s requirements correspond to the Nash equilibrium for finite mixed strategies with no symmetry conditions. To demonstrate this proof, we first verify that Arrow’s set and Nash’s set are isomorphic to each other, both sets being under stated conditions of non–symmetry. Then, the proof is completed by virtue of category theory. Indeed, the two sets are categories that correspond biuniquely to one another and, thus, it is possible to define a covariant functor that preserves their mutual structures. According to this, we show the proof–dedicated metalanguage as a precursor to a special equivalence theorem. (shrink)

This paper argues that there is only one proof for God’s existence in Avicenna, and only one way for establishing the proof within his metaphysical system. This metaphysical proof is essentially derived from a priori notions, among which the notion of existence has the central role. Avicenna’s proof is structured in such a way that all its concepts are either derived from the meaning of ‘existence’ or are connected with this meaning. In this sense Avicenna’s (...) class='Hi'>proof sets out a scenario of discursive a priori knowledge established purely on considerations of fundamental ontological meanings such as ‘existence’, ‘existent’, ‘thing’, ‘necessary’, ‘possible’, ‘impossible’, ‘one’ and ‘cause’. (shrink)

It is widely assumed among contemporary philosophers that Descartes’ version of ontological proof, among other weaknesses, makes an impossible and unjustified move from the mental world of concepts to the real world of things. Contrary to this opinion I will try to show that Descartes’ famous principle of clear and distinct perception suffices to find an adequate inferential connection between the contents of the human mind and extra-mental reality. In a clear and distinct way we cognitively grasp the concept (...) of supremely perfect being as the concept that we do not construct by an arbitrary definition of the word ”God’. (shrink)

Suppose that the members of a group each hold a rational set of judgments on some interconnected questions, and imagine that the group itself has to form a collective, rational set of judgments on those questions. How should it go about dealing with this task? We argue that the question raised is subject to a difficulty that has recently been noticed in discussion of the doctrinal paradox in jurisprudence. And we show that there is a general impossibility theorem that (...) that difficulty illustrates. Our paper describes this impossibility result and provides an exploration of its significance. The result naturally invites comparison with Kenneth Arrow's famous theorem (Arrow, 1963 and 1984; Sen, 1970) and we elaborate that comparison in a companion paper (List and Pettit, 2002). The paper is in four sections. The first section documents the need for various groups to aggregate its members' judgments; the second presents the discursive paradox; the third gives an informal statement of the more general impossibility result; the formal proof is presented in an appendix. The fourth section, finally, discusses some escape routes from that impossibility. (shrink)

Putnam construed the aim of Carnap’s program of inductive logic as the specification of a “universal learning machine,” and presented a diagonal proof against the very possibility of such a thing. Yet the ideas of Solomonoff and Levin lead to a mathematical foundation of precisely those aspects of Carnap’s program that Putnam took issue with, and in particular, resurrect the notion of a universal mechanical rule for induction. In this paper, I take up the question whether the Solomonoff–Levin proposal (...) is successful in this respect. I expose the general strategy to evade Putnam’s argument, leading to a broader discussion of the outer limits of mechanized induction. I argue that this strategy ultimately still succumbs to diagonalization, reinforcing Putnam’s impossibility claim. (shrink)