Linnebo and Shapiro have recently given an analysis of potential infinity using modal logic. A key technical component of their account is to show that under a suitable translation ◊ of nonmodal language into modal language, nonmodal sentences ϕ 1, …, ϕ n entail ψ just in case ϕ 1 ◊, …, ϕ n ◊ entail ψ ◊ in the modal logic S4.2. Linnebo and Shapiro establish this result in nonfree logic. In this note I argue that their analysis of (...) potential infinity should be carried out in a free logic. I then extend their key theorems to the setting of negative free logic. (shrink)

The dominance of string theory in the research landscape of quantum gravity physics (despite any direct experimental evidence) can, I think, be justified in a variety of ways. Here I focus on an argument from mathematical fertility, broadly similar to Hilary Putnam’s ‘no miracles argument’ that, I argue, many string theorists in fact espouse in some form or other. String theory has generated many surprising, useful, and well-confirmed mathematical ‘predictions’—here I focus on mirror symmetry and the mirror theorem. These (...) predictions were made on the basis of general physical principles entering into string theory. The success of the mathematical predictions are then seen as evidence for the framework that generated them. I shall attempt to defend this argument, but there are nonetheless some serious objections to be faced. These objections can only be evaded at a considerably high (philosophical) price. (shrink)

Since Leibniz's time, Cartesian mental causation has been criticized for violating the conservation of energy and momentum. Many dualist responses clearly fail. But conservation laws have important neglected features generally undermining the objection. Conservation is _local_, holding first not for the universe, but for everywhere separately. The energy in any volume changes only due to what flows through the boundaries. Constant total energy holds if the global summing-up of local conservation laws converges; it probably doesn't in reality. Energy conservation holds (...) if there is symmetry, the sameness of the laws over time. Thus, if there are time-places where symmetries fail due to nonphysical influence, conservation laws fail there and then, while holding elsewhere, such as refrigerators and stars. Noether's converse first theorem shows that conservation laws imply symmetries. Thus conservation trivially nearly entails the causal closure of the physical. But expecting conservation to hold in the brain simply assumes the falsehood of Cartesianism. Hence Leibniz's objection begs the question. Empirical neuroscience is another matter. So is Einstein's General Relativity: far from providing a loophole, General Relativity makes mental causation _harder_. (shrink)

Demonstrates the validity of the measure presented in "Estimating Semantic Content" on textbook examples using (binary) resolution [a generalization of disjunctive syllogism] theorem proving; the measure is based on logical probability and is the mirror image of logical form; it dates to Popper.

It is shown in this article in how far one has to have a clear picture of Gödel’s philosophy and scientific thinking at hand (and also the philosophical positions of other philosophers in the history of Western Philosophy) in order to interpret one single Philosophical Remark by Gödel. As a single remark by Gödel (very often) mirrors his whole philosophical thinking, Gödel’s Philosophical Remarks can be seen as a philosophical monadology. This is so for two reasons mainly: Firstly, because it (...) pictures a monadology already via its form. And secondly, because Gödel wanted to establish in his Philosophical Remarks a modern monadology in adapting the philosophical principles of Leibniz’s monadology to modern science. This article will only deal with the first aspect that has been mentioned namely that a single remark by Gödel (very often) mirrors his whole philosophy (at the time) for example: set theory, perception and intuition of mathematical objects and axioms, the nature of the human mind, the concept and existence of God, and so on. This renders it extremely difficult to interpret Gödel’s philosophical remarks but interpreting it provides also a deep insight in Gödel’s philosophical thoughts. (shrink)

In “Models and Reality”, H. Putnam formulated his model-theoretic argument against “metaphysical realism”. The article proposes a meta-reconstruction of Putnam’s model-theoretic argument in the light of two mutually compatible interpretations of it–elaborated by Manuel Garcia-Carpintero and Igor van Douven. A critical reflection on these interpretations and their adequacy for Putnam’s argument allows us to expose new theses coherent with Putnam’s reasoning and indicate new paths to improve this argument for our reconstruction task. In particular, we show that Putnam’s position may (...) be coherent with van Douven’s versions of Global Descriptivism under some conditions, but Putnam cannot reject realism as quickly as Carpintero suggests. We show that Suszko’s canonic axiomatic system and Sneed’s concept of theory may provide valuable support for Putnam’s argument. Finally, we critically evaluate Carpintero’s theses about the genesis of unintended interpretations of our languages, adopting the machinery of the Upward Skolem–Loewenheim Theorem and Knight’s Theorem. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

This paper defends the communitarian account of meaning against Boghossian’s (Wittgensteinian) arguments. Boghossian argues that whilst such an account might be able to accommodate the infinitary characteristic of meaning, it cannot account for its normativity: he claims that, since the dispositions of a group must mirror those of its members, the former cannot be used to evaluate the latter. However, as this paper aims to make clear, this reasoning is fallacious. Modelling the issue with four (justifiable) assumptions, it shows that (...) Condorcet’s ‘Jury Theorem’ can be used to prove that the dispositions of the majority of the members of a group can differ from those of any individual member in a way that makes it possible to use communal dispositions as a standard with which individual dispositions can be assessed. Moreover, the argument of the paper is also shown to have general implications for the use of formal methods in the explanation of the nature of certain fallacious inferences. (shrink)

soft servants of durable material: they live without pretension in complicated relays and electrical circuits. Speed, docility are their strength. One asks: “What is 2 × 2?”—“Are you a machine?” They answer or refuse to answer, depending on what you demand. There are, however, other machines as well, more abstract automatons, bolder and more inaccessible, which eat their tape in mathematical formulae. They imitate in language. In infinite loops, farther and farther back in their retreat towards more subtle algorithms, more (...) recursive functions. They are logical and describe themselves. As when a man with a hand-mirror pressed against his nose in front of a mirror sees in infinite rows the same image multiplied in a shrinking, darkening corridor of glass. It is a Gödel theorem as good as any. He sees infinity, but what he does not see is his face. (From Göran Printz-Påhlson´s poem “The Turing Machine” published in Säg minns du skeppet Refanut? Samlade dikter 1950–1983 (1984) Bonniers, Stockholm). (shrink)

This paper introduces a new framework, based on the notion of compatibility space, obtained by adding a primitive incompatibility relation to a state space in the sense of Fine. The key idea inspiring the framework is to modify Fine's truthmaker semantics by taking the notion of incompatibility as primitive, and use it to define other notions. We discuss some interesting features of the framework and explore its advantages over the standard framework of state spaces. We review some applications of the (...) framework, including proofs of soundness and completeness theorems for a number of logics, one way to use compatibility states to mirror Kripke frames, and an exploration of a way to use the new framework to provide truthmaking clauses for modal formulas. (shrink)

An important open problem in Reverse Mathematics is the reduction of the first-order strength of the base theory from IΣ1IΣ1 to IΔ0+expIΔ0+exp. The system ERNA, a version of Nonstandard Analysis based on the system IΔ0+expIΔ0+exp, provides a partial solution to this problem. Indeed, weak Königʼs lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA, while preserving the equivalences, but at the price of replacing equality with ‘≈’, i.e. infinitesimal proximity . The logical principle (...) corresponding to weak Königʼs lemma is the universal transfer principle from Nonstandard Analysis. Here, we consider the intermediate and mean value theorem and their uniform generalizations. We show that ERNAʼs Reverse Mathematics mirrors the situation in classical Reverse Mathematics. This further supports our claim from Sanders [19] that the Reverse Mathematics of ERNA plus universal transfer is a copy up to infinitesimals of that of WKL0. We discuss some of the philosophical implications of our results. (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)

Four important arguments for probabilism—the Dutch Book, representation theorem, calibration, and gradational accuracy arguments—have a strikingly similar structure. Each begins with a mathematical theorem, a conditional with an existentially quantified consequent, of the general form: if your credences are not probabilities, then there is a way in which your rationality is impugned. Each argument concludes that rationality requires your credences to be probabilities. I contend that each argument is invalid as formulated. In each case there is a mirror-image (...) theorem and a corresponding argument of exactly equal strength that concludes that rationality requires your credences not to be probabilities. Some further consideration is needed to break this symmetry in favour of probabilism. (shrink)

Four important arguments for probabilism—the Dutch Book, representation theorem, calibration, and gradational accuracy arguments—have a strikingly similar structure. Each begins with a mathematical theorem, a conditional with an existentially quantified consequent, of the general form: if your credences are not probabilities, then there is a way in which your rationality is impugned.Each argument concludes that rationality requires your credences to be probabilities.I contend that each argument is invalid as formulated. In each case there is a mirror-image theorem (...) and a corresponding argument of exactly equal strength that concludes that rationality requires your credences not to be probabilities. Some further consideration is needed to break this symmetry in favour of probabilism. I discuss the extent to which the original arguments can be buttressed.1. Introduction2. The Dutch Book Argument2.1. Saving the Dutch Book argument2.2. ‘The Dutch Book argument merely dramatizes an inconsistency in the attitudes of an agent whose credences violate probability theory’3. Representation Theorem-based Arguments4. The Calibration Argument5. The Gradational Accuracy Argument6. Conclusion. (shrink)

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

The aim of this paper is to study the n-variable fragment of first order logic from a modal perspective. We define a modal formalism called cylindric mirror modal logic, and show how it is a modal version of first order logic with substitution. In this approach, we can define a semantics for the language which is closely related to algebraic logic, as we find Polyadic Equality Algebras as the modal or complex algebras of our system. The main contribution of the (...) paper is a characterization of the intended ‘mirror cubic’ frames of the formalisms and, a consequence of the special form of this characterization, a completeness theorem for these intended frames. As a consequence, we find complete finite yet unorthodox derivation systems for the equational theory of finite-dimensional representable polyadic equality algebras. (shrink)

It is widely assumed in philosophy that there is a tight connection between semantics and metaphysics. Semantic theories about the meanings of natural language terms and phrases are taken to provide evidence for and against various metaphysical theses about the nature of non-linguistic parts of the world. Call this view the widespread thesis. I argue that the widespread thesis is mistaken: semantic theories do not generally have robust metaphysical consequences. I contend that the best arguments for the widespread thesis turn (...) on an interpretation of semantic theories according to which they are metaphysically illuminating: the metaphysical structure of the world is directly mirrored in the structure of the theory. Terms and existential quantification in the theory signal the existence of entities in the world, predicates signal the existence of properties, and the structures of set theoretic objects invoked by the theory directly correspond to structural features of the world. Such an interpretation is not the only one available, however, and it is a substantive philosophical (and linguistic) issue which interpretation is correct. Accordingly, I develop and defend an alternative interpretation of semantic theories according to which the structural features of these theories and of their theorems reflect biological and computational constraints on the architecture of the language faculty and historical contingencies in its evolutionary development. Supposedly metaphysically committing features of the theory—including the appearance of quantification and (ostensibly) referring terms—may thus be arepresentational artifacts of these constraints, rather than representational features that reflect metaphysical reality. To determine the metaphysical consequences of a semantic theory then, we must determine which of its features represent language-independent characteristics of the world and which are arepresentational consequences of the structure of the language faculty. To do this, however, we must have some prior idea what the language-independent world is like—that is, we must engage in prior metaphysical theorizing that is not beholden to the semantic theory itself. (shrink)

To what extent can constructive mathematics based on intuitionistc logic recover the mathematics needed for spacetime physics? Certain aspects of this important question are examined, both technical and philosophical. On the technical side, order, connectivity, and extremization properties of the continuum are reviewed, and attention is called to certain striking results concerning causal structure in General Relativity Theory, in particular the singularity theorems of Hawking and Penrose. As they stand, these results appear to elude constructivization. On the philosophical side, it (...) is argued that any mentalist-based radical constructivism suffers from a kind of neo-Kantian apriorism. It would be at best a lucky accident if objective spacetime structure mirrored mentalist mathematics. the latter would seem implicitly committed to a Leibnizian relationist view of spacetime, but is it doubtful if implementation of such a view would overcome the objection. As a result, an anti-realist view of physics seems forced on the radical constructivist. (shrink)

The black hole information problem provides important clues for trying to piece together a quantum theory of gravity. Discussions on this topic have generally assumed that in a consistent theory of gravity and quantum mechanics, quantum theory is unmodified. In this review, we discuss the black hole information problem in the context of generalisations of quantum theory. In this preliminary exploration, we examine black holes in the setting of generalised probabilistic theories, in which quantum theory and classical probability theory are (...) special cases. We are able to calculate the time it takes information to escape a black hole, assuming that information is preserved. In quantum mechanics, information should escape pure state black holes after half the Hawking photons have been emitted, but we find that this get’s modified in generalisations of quantum mechanics. Likewise the black-hole mirror result of Hayden and Preskill, that information from entangled black holes can escape quickly, also get’s modified. We find that although information exits the black hole as predicted by quantum theory, it is fairly generic that it fails to appear outside the black hole at this point—something impossible in quantum theory due to the no-hiding theorem. The information is neither inside the black hole, nor outside it, but is delocalised. (shrink)

This chapter tracks the development of the concept of self-awareness (jikaku, 自覚) in the thought of the Japanese modern philosopher Nishida Kitarō (西田幾多郎) (Kitarō Nishida) (1870–1945), founder of the Kyoto School. Nishida’s oeuvre can be divided into distinct periods, from the 1910s to the 1940s until his passing, during which he thematized and focused on different issues. Nevertheless, self-awareness is a unifying theme throughout. In the chapter we trace how Nishida develops this concept through distinct periods in his career. His (...) initial focus upon pure experience gives way to his focus upon its articulation in reflection vis-à-vis intuition as the unfolding of the will. This then leads to his thematization of the place wherein judgment or propositional thought unfolds and the broader contextual place wherein that place is situated. That thematization of place, in turn, develops into a look at the world where we interact as historical and social bodies with others (nature, other people, etc.). Nishida understands self-awareness in accordance with each of these themes in the respective periods of his oeuvre. But common to all of them is how Nishida understands self-awareness as a self-mirroring – self-reflection as reflexivity – founded upon the premise assumed by the articulated image or object of awareness, a premise necessarily irreducible to, and in excess of, the image or object. This is the a priori that Nishida for the most part calls the “absolute nothing” (zettai mu, 絶対無). To explain this Nishida makes use of Josiah Royce’s notion of a self-representative system as well as Richard Dedekind’s notion of an infinite system in his set theory. On the basis of Nishida’s references to the mathematics of set theory, I further compare his understanding of self-awareness to Kurt Gödel’s Incompleteness Theorems as well as to more recent mathematical theorists, influenced by Gödel, whose understanding of the infinite resonates with Nishida’s understanding of the absolute nothing as assumed by self-awareness. (shrink)

The purpose of this paper is to discuss the role of financial practice in the development of mathematics as applied in human judgement. The basis of the paper is in historical research from the 1990s that argues that the monetisation of western commerce, which abstracted value into quantified price, was synthesised with scholastic analysis resulting in a “mathematical mechanistic world picture” that led to the widespread use of mathematics in science from the seventeenth century. An aspect of this process was (...) the quantification of chance that led to the development of mathematical probability, the branch of mathematics most relevant to judgement and the focus of this paper. Ideas from this historical research are related to the fundamental theorem of asset pricing, the foundational theory of contemporary financial mathematics. The paper observes that vestiges of medieval scholastic attitudes to financial ethics can be discerned in the FTAP, offering a novel interpretation of the mathematical theorem. The paper then considers the Dutch book argument, the most popular justification for subjective probability. The paper’s main contribution is in describing the significance of financial practice in validating the DBA and the paper explains how the FTAP addresses some criticisms of how the DBA represents beliefs. The conclusions emphasise the distinction between pure and practical reasoning and that this should be mirrored in a distinction between the mathematics of physica and practica. This point is important as mathematics is becoming more widespread in modelling, and directing, social systems. (shrink)

Summary The presentation will proceed as follows: (§ 3:) For the truth of an affirmative present‐tensed proposition Boethius required that its terms have actual referents, he would not accept any uninstantiated essence as a verifier. He also denied that any proposition about corruptible beings can be strictly necessary. He thus had a problem explaining how a theorem of one of the natural sciences differs from an ordinary contingent proposition. His rejection of uninstantiated essences also (§ 4) raised the question (...) how words can stay significant independently of the existence of referents, and more generally, to which degree language mirrors external reality. After a sketch of his solution to the semantic problems, § 5 returns to the question about scientific propositions. It is claimed that the core of Boethius' solution is a substitution of causal relationships for ordinary things (existents or essences) as an answer to the question what scientific knowledge is knowledge of and what scientific propositions are about. Since (§ 6) to Boethius axiomatized geometry provided the model for all sciences, consistency of a science's set of theorems should be important for him. A preoccupation with consistency can be seen in the rules he lays down for the dialogical game of “obligation” that was used as an academic exercise. An analysis (§ 7) of his description of another such game, the dialectical disputation, shows that he operated with a notion of truth‐within‐a‐game (requiring only conformity with the rules) as as opposed to absolute truth. § 8 claims that Boethius operated with a similar notion of truth‐within‐a‐science as opposed to absolute truth, and that his reason for so doing was the inaccessibility to human minds of the totality of causes. Doing science, then, (§ 9) is much like participating in a scholastic disputational game, and there may be scientifically‐true theorems conflicting with absolute truth known through revelation. § 10 epilogizes about Boethius' views about the highest good. (shrink)

The Brownian motion of small particles interacting with a field at a finite temperature is a well-known and well-understood phenomenon. At zero temperature, even though the thermal fluctuations are absent, quantum fields still possess vacuum fluctuations. It is then interesting to ask whether a small particle that is interacting with a quantum field will exhibit Brownian motion when the quantum field is assumed to be in the vacuum state. In this paper, we study the cases of a small charge and (...) an imperfect mirror interacting with a quantum scalar field in (1 + 1) dimensions. Treating the quantum field as a classical stochastic variable, we write down a Langevin equation for the particles. We show that the results we obtain from such an approach agree with the results obtained from the fluctuation-dissipation theorem. Unlike the finite temperature case, there exists no special frame of reference at zero temperature and hence it is essential that the particles do not break Lorentz invariance. We find that that the scalar charge breaks Lorentz invariance, whereas the imperfect mirror does not. We conclude that small particles such as the imperfect mirror will exhibit Brownian motion even in the quantum vacuum, but this effect can be so small that it may prove to be difficult to observe it experimentally. (shrink)

We consider a “polarized” version of bi-intuitionistic logic [5, 2, 6, 4] as a logic of assertions and hypotheses and show that it supports a “rich proof theory” and an interesting categorical interpretation, unlike the standard approach of C. Rauszer’s Heyting-Brouwer logic [28, 29], whose categorical models are all partial orders by Crolard’s theorem [8]. We show that P.A. Melliès notion of chirality [21, 22] appears as the right mathematical representation of the mirror symmetry between the intuitionistic and co-intuitionistc (...) sides of polarized bi-intuitionism. Philosophically, we extend Dalla Pozza and Garola’s pragmatic interpretation of intuitionism as a logic of assertions [10] to bi-intuitionism as a logic of assertions and hypotheses. We focus on the logical role of illocutionary forces and justification conditions in order to provide “intended interpretations” of logical systems that classify inferential uses in natural language and remain acceptable from an intuitionistic point of view. Although Dalla Pozza and Garola originally provide a constructive interpretation of intuitionism in a classical setting, we claim that some conceptual refinements suffice to make their “pragmatic interpretation” a bona fide representation of intuitionism. We sketch a meaning-asuse interpretation of co-intuitionism that seems to fulfil the requirements of Dummett and Prawitz’s justificationist approach. We extend the Brouwer-Heyting-Kolmogorov interpretation to bi-intuitionism by regarding co-intuitionistic formulas as types of the evidence for them: if conclusive evidence is needed to justify assertions, only a scintilla of evidence suffices to justify hypotheses. (shrink)

Quantum contextuality and its ontological meaning are very controversial issues, and they relate to other problems concerning the foundations of quantum theory. I address this controversy and stress the fact that contextuality is a universal topological property of quantum processes, which conflicts with the basic metaphysical assumption of the definiteness of being. I discuss the consequences of this fact and argue that generic quantum potentiality as a real physical indefiniteness has nothing in common with the classical notions of possibility and (...) counterfactuality, and that also it reverses, in a way, the classical mirror-like relation between actuality and definite possibility. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2601\PINCREPL.261 : 2006-06-05 11:54
iscussion REJOINDER TO SOAMES C P Philosophy / Purdue U. West Lafayette,  ,  @. y goal in reviewing Soames’ book was to help readers of this journal evalMuate his contribution to the history of analytic philosophy, with a special focus on his discussion of Russell. Soames charges both that I misrepresent the contents of his book and that I (...) make mistakes in the interpretation of various aspects of Russell’s philosophy. If I had committed any errors of the former sort, I would certainly apologize and thank Soames for bringing such mistakes to my attention. After explaining why I do not believe I have misrepresented the contents of his book, I will turn to the one substantive issue that he raises in his reply, namely the need for unperceived sense-data in Russell’s external world program. While disagreement here is more understandable, nothing Soames says in his book or in his reply has led me to revise my original remarks. My review began with a one-page summary of Soames’ -page book. I said that “Moore, and nearly every other figure that Soames discusses, is [said to be] guilty of confusing necessity, analyticity and apriority” and in the next paragraph mentioned “Moore’s views on ethics” and “Russell’s conception of analysis ” (“Reply”, p. ). Soames replies: “Not so. Although I do show that such conflation had negative consequences for the views of several philosophers, Moore and Russell are not among them” (p. ). I could have added more page references to my discussion. On Moore, after a long list of twelve “fundamental methodological notions” which include analysis, analyticity and necessity, Soames claims that “unclarity about these fundamental methodological notions eviscerated his most important meta-ethical conclusions” (: ). For  Scott Soames, Philosophical Analysis in the Twentieth Century,  vols. (Princeton: Princeton U. P., ). Christopher Pincock, “History of Philosophical Analysis”, Russell, n.s.  (winter – ): –. Scott Soames, “Reply to Pincock”, Russell, n.s.  (winter –): –. References preceded by “:” or “:” refer to the volumes of Soames’ book. Other unlabelled references in the text will be either to my review or to Soames’ reply. russell: the Journal of Bertrand Russell Studies n.s.  (summer ): – The Bertrand Russell Research Centre, McMaster U.  - _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2601\PINCREPL.261 : 2006-06-05 11:54  Discussion Russell, Soames notes that Russell’s “view that ordinary proper names mean the same as descriptions associated with them by speakers … was ultimately shown to be deeply problematic by Saul Kripke in Naming and Necessity” (: ). A reference is then made to Soames’ discussion of Kripke in Volume  where Soames explains how Kripke’s “modal argument” undermined Russell’s approach (: ). Soames also complains that I object to his “characterization of Russellian propositions as complexes of properties and objects whose structure mirrors the logical forms of sentences that express them” (p. ). What I said was that it was “extremely misleading” to say that (quoting Soames again) “propositions constitute the information encoded by sentences … [and of] the information encoded by a sentence (in a logically perfect language) as a complex entity the structure of which mirrors the structure of sentences” (p.  and : –). So, Soames did not say that propositions are “complexes of properties and objects” and if he had said this I would not have objected. Soames goes on to use a settheoretic construction to explain exactly what propositions are. But there is a big difference between a set-theoretic construction and a complex of the sort Russell had in mind, and a reader of Soames would have no awareness of the gap. That is, she is likely to be misled. Finally, Soames says that “Pincock foolishly speculates” (p. ) in ascribing confusion to him about the relevance of Gödel’s incompleteness theorems to Principia Mathematica. Now, when a philosopher gives an argument described as the “final nail in the coffin” (: ), and this argument depends on a tacit assumption, I submit that it is far from foolish to think that the philosopher believes that this assumption is correct. This is, after all, just the methodology that Soames employs when he reconstructs the commitments of the philosophers that he talks about. And, in this case... (shrink)

This article proposes a periodisation of the narrative work of Jorge Luis Borges (1899-1986). Two meta-literary notions guide the first period: that of an absolute text and that of a text governed by chance, in which the former is linked to the Kabbalah and the latter to Gnosticism. The next period, which is the focal topic of this study, is governed by the notion of figure in the biblical sense of the term. Borges introduces this notion in his 1949 essay (...) "The Mirror Of Enigmas" that summons texts by Léon Bloy, Thomas de Quincey and Novalis. Through these influences, Borges sketches a figurative poetics whose apocalyptic character situates the work thus produced in the indecision between the eschatology and the scatology. (shrink)

This research file’s object is Fénelon’s correspondence, the complete edition of which, achieved in 2007, has considerably enriched our knowledge of both the life and works of the archbishop of Cambrai, by shedding new lights on its multiple facets. Mirror and memory of the published body of work, when it is not in itself an essential and constitutive part of it, the correspondence enhances Fénelon’s published body of work’s understanding through the discovery of its process of elaboration, the obstacles it (...) met, or the circumstances which favored it. The correspondence also reveals that which is not shown by the work : aborted projects and renouncements, lastly and most importantly, the societal environment, surrounding Fénelon, which allows him to weave the fabric and network in which is set this exceptional life made of radical commitments, of battles, of glory but also of disappointments, disgraces and submissions. (shrink)

Imitation, deliberation, and mindreading are characteristically human sociocognitive skills. Research on imitation and its role in social cognition is flourishing across various disciplines; it is here surveyed under headings of behavior, subpersonal mechanisms, and functions of imitation. A model is then advanced within which many of the developments surveyed can be located and explained. The shared circuits model explains how imitation, deliberation, and mindreading can be enabled by subpersonal mechanisms of control, mirroring and simulation. It is cast at a (...) middle, functional level of description, between the level of neural implementation and the level of conscious perceptions and intentional actions. The shared circuits model connects shared informational dynamics for perception and action with shared informational dynamics for self and other, while also showing how the action/perception, self/other and actual/possible distinctions can be overlaid on these shared informational dynamics. It avoids the common conception of perception and action as separate and peripheral to central cognition. Rather, it contributes to the situated cognition movement by showing how mechanisms for perceiving action can be built on those for active perception. (shrink)

This collection of essays by Alvin Goldman explores an array of topics in the philosophy of cognitive science, ranging from embodied cognition to the metaphysics of actions and events.

In this paper, we show that Arrow’s well-known impossibility theorem is instrumental in bringing the ongoing discussion about verisimilitude to a more general level of abstraction. After some preparatory technical steps, we show that Arrow’s requirements for voting procedures in social choice are also natural desiderata for a general verisimilitude definition that places content and likeness considerations on the same footing. Our main result states that no qualitative unifying procedure of a functional form can simultaneously satisfy the requirements of (...) Unanimity, Independence of irrelevant alternatives and Non-dictatorship at the level of sentence variables. By giving a formal account of the incompatibility of the considerations of content and likeness, our impossibility result makes it possible to systematize the discussion about verisimilitude, and to understand it in more general terms. (shrink)

The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues.

A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.

In this work we consider the problem of the approximate hedging of a contingent claim in the minimum mean square deviation criterion. A theorem on martingale representation in case of discrete time and an application of the result for semi-continuous market model are also given.

Numerous studies indicate that observing or knowing about another's action automatically activates the same motor representations that are active when we perform the other's action by ourselves. We investigated how affect influences this mirror mechanism. Based upon findings that positive affect encourages and negative affect impairs spreading activation, we hypothesised that positive affect should increase and negative affect decrease the automatic co-representation of other individuals' actions during jointly performed tasks. Recent research has shown that joint-action effects in a go/no-go variant (...) of the Simon task provide a good index for co-representing another's action. Participants performed such a Simon task together with another person after the induction of neutral, positive, or negative affect. Consistent with our predictions, the joint Simon-like effect was strongest after positive affect induction and absent after negative affect induction. These results represent the first evidence of affect-induced differences in the mirroring of another's action and suggest that affect influences one of our essential abilities for successful social interactions. (shrink)

Following a long-standing philosophical tradition, impartiality is a distinctive and determining feature of moral judgments, especially in matters of distributive justice. This broad ethical tradition was revived in welfare economics by Vickrey, and above all, Harsanyi, under the form of the so-called Impartial Observer Theorem. The paper offers an analytical reconstruction of this argument and a step-wise philosophical critique of its premisses. It eventually provides a new formal version of the theorem based on subjective probability.

In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.

We prove that every countable non-standard model of WKL0 has a proper initial part isomorphic to itself. This theorem enables us to carry out non-standard arguments over WKL0.

Body movement is a primary nonverbal communication channel in humans. Coordinated social behaviors, such as dancing together, encourage multifarious rhythmic and interpersonally coupled movements from which observers can extract socially and contextually relevant information. The investigation of relations between visual social perception and kinematic motor coupling is important for social cognition. Perceived coupling of dyads spontaneously dancing to pop music has been shown to be highly driven by the degree of frontal orientation between dancers. The perceptual salience of other aspects, (...) including postural congruence, movement frequencies, time‐delayed relations, and horizontal mirroring remains, however, uncertain. In a motion capture study, 90 participant dyads moved freely to 16 musical excerpts from eight musical genres, while their movements were recorded using optical motion capture. A total from 128 recordings from 8 dyads maximally facing each other were selected to generate silent 8‐s animations. Three kinematic features describing simultaneous and sequential full body coupling were extracted from the dyads. In an online experiment, the animations were presented to 432 observers, who were asked to rate perceived similarity and interaction between dancers. We found dyadic kinematic coupling estimates to be higher than those obtained from surrogate estimates, providing evidence for a social dimension of entrainment in dance. Further, we observed links between perceived similarity and coupling of both slower simultaneous horizontal gestures and posture bounding volumes. Perceived interaction, on the other hand, was more related to coupling of faster simultaneous gestures and to sequential coupling. Also, dyads who were perceived as more coupled tended to mirror their pair's movements. (shrink)

Between 1964 and 1990, the notion of nonlocality in Bell's papers underwent a profound change as his nonlocality theorem gradually became detached from quantum mechanics, and referred to wider probabilistic theories involving correlations between separated beables. The proposition that standard quantum mechanics is itself nonlocal became divorced from the Bell theorem per se from 1976 on, although this important point is widely overlooked in the literature. In 1990, the year of his death, Bell would express serious misgivings about (...) the mathematical form of the local causality condition, and leave ill-defined the issue of the consistency between special relativity and violation of the Bell-type inequality. In our view, the significance of the Bell theorem, both in its deterministic and stochastic forms, can only be fully understood by taking into account the fact that a fully Lorentz-covariant version of quantum theory, free of action-at-a-distance, can be articulated in the Everett interpretation. (shrink)

The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem’s dependency regarding Gödel numberings. I introducedeviantnumberings, yielding provability predicates satisfying Löb’s conditions, which result in provable consistency sentences. According to the main result of this (...) paper however, these “counterexamples” do not refute the theorem’s prevalent interpretation, since once a natural class ofadmissiblenumberings is singled out, invariance is maintained. (shrink)

A search is under way for a theory that can accommodate our intuitions in population axiology. The object of this search has proved elusive. This is not surprising since, as we shall see, any welfarist axiology that satisfies three reasonable conditions implies at least one of three counter-intuitive conclusions. I shall start by pointing out the failures in three recent attempts to construct an acceptable population axiology. I shall then present an impossibility theorem and conclude with a short discussion (...) of how it might be extended to pluralist axiologies, that is, axiologies that take more values than welfare into account. (shrink)

In several applications of sequent calculi going beyond pure logic, an introduction of suitably defined rules seems to be more profitable than addition of extra axiomatic sequents. A program of formalization of mathematical theories via rules of special sort was developed successfully by Negri and von Plato. In this paper a general theorem on possible ways of transforming axiomatic sequents into rules in sequent calculi is proved. We discuss its possible applications and provide some case studies for illustration.