Boghossian argued that externalism is incompatible with a transparency thesis according to which we can know a priori whether any two of our occurrent thoughts have the same or distinct content, and that this transparency thesis is integral to our commonsense conception of rationality, which requires the apriority of our logical abilities. Stalnaker offered a detailed compatibilist response to Boghossian. Boghossian criticized this response, and Stalnaker replied. But the outcome of that important discussion remained unclear, partly because it was unclear (...) how to understand the amended transparency thesis at which Stalnaker was gesturing. My aim in this paper is to settle the matter. I wish to clarify the terms of the debate, also by highlighting the relevant commitments of Stalnaker’s two-dimensionalist account, and to show that his compatibilist response, which appeals to diagonal propositions, is unsuccessful. (shrink)

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a (...) Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers. (shrink)

Speakers assert in order to communicate information. It is natural, therefore, to hold that the content of an assertion is whatever information it communicates to its audience. In cases involving uncertainty about the semantic values of context-sensitive lexical items, moreover, it is natural to hold that the information an assertion communicates to its audience is whatever information audience members are in a position to recover from it by assuming that the proposition it semantically determines is true. This sort of (...) picture corresponds to an influential and widely endorsed theory of assertoric content: diagonalism. I begin by arguing that, despite its intuitive appeal, diagonalism should be rejected because it conflicts with our intuitive judgments about the circumstances in which the contents of speakers’ assertions would be true or false. I then show that the failure of diagonalism requires us to either abandon a familiar way of thinking about information and rational assertion or hold that the content of an assertion is not always the information it communicates. I suggest that we choose the latter horn of this dilemma — assertoric content is better characterized in terms of the commitments speakers undertake than in terms of the information they communicate. (shrink)

We consider natural strengthenings of H. Friedman's Borel diagonalization propositions and characterize their consistency strengths in terms of the n -subtle cardinals. After providing a systematic survey of regressive partition relations and their use in recent independence results, we characterize n -subtlety in terms of such relations requiring only a finite homogeneous set, and then apply this characterization to extend previous arguments to handle the new Borel diagonalization propositions.

Plato’s Theaetetus sets the problem of the definition of science; moreover, what there is in question is the problem of the definition in general. Defining means measuring, referring to definite parameters what is initially indefinite. But it is not a case that the dialogue opens with the discussion about the commensurable and incommensurable numbers: the search for what is common to all sciences is the search for their common measure, for the term to which various elements are or can be (...) commensurated. The apories Plato is showing in refuting the Protagorean thesis appear clearly as an objection against the absolute commensurability of all things: each sense is a parameter of a determinate sensible object and then results as quite incommensurable with another sense; a present sensation is incommensurable with a non present one, either past or future; all these facts question the possibility of the definition, for they reduce the knowledge, and the reality, to a set of atomic and quite unrelated elements. In the same way, the other definitions of science are rejected because of their incompleteness. But the negative conclusion of the Theaetetus regarding the definition of science must be assumed in a positive way: every operation of defining constantly presents an excess which belongs to the incommensurability and leaves every definition in a state of incompleteness. Through a comparison with the problem of the commensurable and incommensurable numbers, what is eventually shown is that the Being itself, as a mean between subject and predicate in the proposition, constitutes the diagonal element of every process of definition, irreducible to the elements that come into play. Being is, literally said, the incommensurable. (shrink)

In the propositional modal treatment of two-variable first-order logic equality is modelled by a ‘diagonal’ constant, interpreted in square products of universal frames as the identity relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is (...) similarly harmless. We show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first-order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them. (shrink)

Robert Stalnaker contrasts two interpretations, semantic and metasemantic, of the two-dimensionalist framework. On the semantic interpretation, the primary intension or diagonal proposition associated with an utterance is a semantic value that the utterance has in virtue of the actual linguistic meaning of the corresponding sentence, and that primary intension is both what a competent speaker grasps and what determines different secondary intensions or horizontal propositions relative to different possible worlds considered as actual. The metasemantic interpretation reverses the order (...) of explanation: an utterance has the primary intension it has because it yields the secondary intensions it yields relative to different possible worlds considered as actual. In these possible worlds, the semantic facts can be different: the metasemantic interpretation is metasemantic in the sense that the secondary intensions are determined relative to possible worlds considered as actual given the meanings the expressions have there. Stalnaker holds a causal picture of the reference of names, according to which names have no meaning over and above their unique referent, and therefore maintains that the semantic interpretation is not an option. He thus endorses the metasemantic interpretation, while insisting that this interpretation does not, contrary to what he originally thought, yield any account of a priori truth and knowledge. My double aim in this paper is to show (i) that the metasemantic interpretation, as sketched by Stalnaker, is not compatible with one natural understanding of the causal picture of reference, on which names are rigid because they have their original bearers essentially , and (ii) that a third kind of interpretation of the framework is available, the metasyntactic interpretation, which grants that names have their bearers essentially and yields some account of a priori knowledge. (shrink)

Written in the form of a dialogue with Brussels sociologist Dieter Vandebroeck, this article retraces the social and intellectual trajectory of Loïc Wacquant as stepping stone to reviewing and discussing the major concepts coined and theoretical propositions elaborated in the course of his research on comparative urban marginality, racial domination, the ghetto, the penal state, neoliberalism, and carnality. This provides an opportunity to specify the relationships between ethnography, history and theory; the dialectic of domination and resistance; the role of public (...) (dis)honor in social life; the uses of Bourdieu's bureaucratic field; and the social and academic conditions of incubation, diffusion, and death of scholarly myths such as the “underclass.” The article closes on a call to clearly distinguish the rhetorical, metaphorical, and analytical usages of concepts and reaffirms the need for epistemic reflexivity as sine qua non for the articulation of robust scientific problematics. (shrink)

This dissertation addresses the question of whether externalism and privileged access are compatible. I defend the view according to which they are, indeed, incompatible. However, this follows unproblematically from a distinction between two notions of mental content that I introduce and develop. My main suggestion is that a confusion between two notions of mental content is responsible for the view that the incompatibility of privileged access and externalism is problematic, since these are views about, strictly speaking, different varieties of mental (...) content. ;The discussion is divided into an introduction and four chapters. In the second chapter, I discuss the main argument for the position according to which externalism and privileged access are compatible. In the third chapter, I distinguish two kinds of mental content by using the two-dimensional semantics framework developed by David Kaplan, Robert Stalnaker and others. I then move to argue that externalism and privileged access are views about different sorts of content. Externalism is correct only when it is understood as a view about "objective content", essentially constituted by truth-conditions, whereas privileged access is correct only when it is understood as a view about "subjective content", a kind of "narrow" content essentially constituted by Stalnaker's "diagonal proposition". In the fourth chapter, I focus on privileged access and I offer an explanation of it. The main tenet of my account of privileged access is that our grounds for our first-order beliefs are the same as our grounds for the appropriate second-order beliefs. In the fifth chapter, I discuss the two main arguments for the position according to which externalism and privileged access are incompatible. By using the distinction regarding mental content drawn on chapter three, I argue that neither argument shows the incompatibility of externalism about objective content and privileged access about subjective content. Thus, I conclude that the incompatibility of privileged access and externalism is not problematic. (shrink)

ABSTRACTTwo-dimensional accounts of speech and thought make use of so-called ‘diagonal’ propositions. If diagonals are indeed propositions, they can be negated: an ‘anti-diagonal’ is the negation o...

Horizontalism is the thesis that what a speaker asserts in literally and sincerely uttering an indicative sentence is some horizontal proposition of her utterance; diagonalism is the thesis that what a speaker asserts in literally and sincerely uttering an indicative sentence is some diagonal proposition of her utterance. Recent work on assertion has reached no clear consensus favoring either horizontalism or diagonalism. I explore a novel strategy for adjudicating between the two views by considering the advantages and (...) disadvantages which would accrue to a linguistic community as a result of adopting different committal practices—that is, practices of associating utterances with the propositions to which speakers undertake assertoric commitments in uttering them—ultimately concluding that a horizontalist practice has important advantages over its competitors. (shrink)

In this paper I put forward a representationalist theory of conscious experience based on Robert Stalnaker's version of two-dimensional modal semantics. According to this theory the phenomenal character of an experience correlates with a content equivalent to what Stalnaker calls the diagonal proposition. I show that the theory is closely related both to functionalist theories of consciousness and to higher-order representational theories. It is also more compatible with an anti-Cartesian view of the mind than standard representationalist theories.

Cantor's diagonal proof is sig­nificant both because the central method of proof used in it has been subsequently applied in a number of other proofs, and because it is considered to confirm the existence of infinite sets whose size fun­ damentally and by an order of magnitude exceeds the size of the "classical" infinite set represented by all natural numbers, while their size can theoretically exceed every conceivable limit. Although Can­tor's proof is generally accepted by the scientific community, some (...) experts are somewhat reserved about it. The aim of this paper is to present Cantor's proof in an accessible way, while pointing out its (hidden) assumptions and possible problematic points, and pointing out that some of its underlying assumptions are not indisputable mathematical truths, but rather postulated propositions that may or may not be accepted. (shrink)

This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and (...) Principia Mathematica. (shrink)

An 8 by 8 checkerboard with two diagonally opposite squares removed cannot be covered by dominoes each of which covers two rectilinearly adjacent squares. present a set theory description of the proposition and an informal proof that the covering is impossible. While no present system that I know of will accept either formal description or the proof, I claim that both should be admitted in any heavy duty set theory.

This paper argues that the classification of propositional attitudes into the de re, de dicto, and de se is incomplete. De se attitudes are widely agreed to be closely connected to de re attitudes. But there is a species of belief that is linked to agent-centered action in the way that de se beliefs are, but is also associated with entities, places, and especially times, under a description. These mark out a fourth kind. One way to think about what makes (...) them distinctive is that, despite being ‘essentially indexical’ they can be retained across different contexts of evaluation without having diagonal content. The paper also discusses the connection between such beliefs and the kinds of utterances (like ‘Don't put off for tomorrow what you can do today’) that are contested as examples of what David Kaplan called ‘monsters.’. (shrink)

This paper will annoy modern logicians who follow Bertrand Russell in taking pleasure in denigrating Aristotle for [allegedly] being ignorant of relational propositions. To be sure this paper does not clear Aristotle of the charge. On the contrary, it shows that such ignorance, which seems unforgivable in the current century, still dominated the thinking of one of the greatest modern logicians as late as 1831. Today it is difficult to accept the proposition that Aristotle was blind to the fact (...) that, for example, incommensurability is a relation and not a property: that the proposition “In every square, the diagonals are incommensurable with the sides” is relational and not categorical. This paper asks the reader to do something more difficult: to accept the proposition that as late as 1831 De Morgan was blind to the same fact. This paper shows conclusively that in 1831, De Morgan was still in the grips of the allegedly Aristotelian paradigm. (shrink)

Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following: Let T be any first-order theory with a predicate Pr satisfying the canonical derivability conditions, including Löb's property. Then any formula in T built up from the propositional variables $q,p_{1},...,p_{n}$ , using logical connectives and the predicate Pr, has the same "fixed-points" relative to q (that is, formulas $\psi (p_{1},...,p_{n})$ for which for all $p_{1},...,p_{n}\vdash _{T}\phi (\psi (p_{1},...,p_{n}),p_{1},...,p_{n})\leftrightarrow \psi (...) (p_{1},...,p_{n})$ ) of a formula $\phi ^{\ast}$ of the same kind, obtained from φ in an effective way. Moreover, such $\phi ^{\ast}$ is provably equivalent to the formula obtained from φ substituting with $\phi ^{\ast}$ itself all the occurrences of q which are under Pr. In the particular case where q is always under Pr in φ, $\phi ^{\ast}$ is the unique (up to provable equivalence) "fixed-point" of φ. Since this result is proved only assuming Pr to be canonical, it can be deduced that Löb's property is, in a sense, equivalent to Gödel's diagonalization lemma. All the results are proved more generally in the intuitionistic case. (shrink)

The paper studies two formal schemes related to -completeness.LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where (x) is a formula with the only free variable x): (a) (for anyn) ( T (n)), (b) T x Pr T (–(x)–) and (c) T x(x) (the notational conventions are those of Smoryski [3]). The aim of this paper is to examine the (...) meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) (c) and (a) (b), where ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) (c) is equivalent to ¬Cons T and 2. the schema corresponding to (a) (b) is not consistent with 1-CON T. The former result follows from a simple adaptation of the -incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma. (shrink)

I can say that a building is tall and you can understand me, even if neither of us has any clear idea exactly how tall a building must be in order to count as tall. This mundane fact poses a problem for the view that successful communication consists in the hearer’s recognition of the proposition a speaker intends to assert. The problem cannot be solved by the epistemicist’s usual appeal to anti-individualism, because the extensions of vague words like ‘tall’ (...) are contextually fluid and can be constrained significantly by speakers’ intentions. The problem can be seen as a special case of a more general problem concerning what King has called “felicitous underspecification.” Traditional theories of vagueness offer nothing that can help with this problem. Appeals to diagonalization do not help either. A more radical solution is needed. (shrink)

The logic of a physical system consists of the elementary observables of the system. We show that for chaotic systems the logic is not any more the classical Boolean lattice but a kind of fuzzy logic which we characterize for a class of chaotic maps. Among other interesting properties the fuzzy logic of chaos does not allow for infinite combinations of propositions. This fact reflects the instability of dynamics and it is shared also by quantum systems with diagonal singularity. (...) We also generalize the fuzzy implication to a probabilistic implication following the hint of von Neumann. In this way we can evaluate the probability of the validity of the logical inference. (shrink)

Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth, and the other not a truth at all. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not at all (...) what they appear. The historical record shows that Russell pursued both these options, but that the struggle with the logical paradoxes pushed him away from the first kind of response and toward the second. An object cannot itself have a kind of inner logical complexity that makes a proposition have a different logical form merely in virtue of being about it, nor can their representatives in logical forms be single things different for different forms, at least not without postulating too many such objects and thereby creating Cantorian diagonal paradoxes. There are only apparent objects which are actually fragments of logical forms, different in different cases. (shrink)

Quotation has been much studied in philosophy. Given that quotation allows one to diagonalize out of any grammar, there have been comparatively few attempts within the linguistic literature to develop an account within a formal linguistic theory. Nonetheless, given the ubiquity of quotation in natural language, linguists need to explicate the formal mechanisms it employs. The central claim of this paper is that once one assumes a dialogical perspective on language such as provided by the KoS (KoS is not an (...) acronym, despite emphasizing a Konversationally Oriented Semantics) framework, formalized in a rich type theory like Type Theory with Records, much of the mystery evaporates. In particular, one can utilize as denotations for quotative constructions entities that are independently motivated for dialogue processing—utterance types and locutionary propositions, Austinian propositions about speech events. (shrink)

It would be an understatement to say that Russell was interested in Cantorian diagonal paradoxes. His discovery of the various versions of Russell’s paradox—the classes version, the predicates version, the propositional functions version—had a lasting effect on his views in philosophical logic. Similar Cantorian paradoxes regarding propositions—such as that discussed in §500 of The Principles of Mathematics—were surely among the reasons Russell eventually abandoned his ontology of propositions.1 However, Russell’s reasons for abandoning what he called “denoting concepts”, and his (...) rejection of similar “semantic dualisms” such as Frege’s theory of sense and reference—at least in “On Denoting”—made no explicit mention of any Cantorian paradox. My aim in this paper is to argue that such paradoxes do pose a problem for certain theories such as Frege’s, and early Russell’s, about how definite descriptions are meaningful. My first aim is simply to lay out the problem I have in mind. Next, I shall turn to arguing that the theories of descriptions endorsed by Frege and by Russell prior to “On Denoting” are susceptible to the problem. Finally, I explore what responses a.. (shrink)

The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as "propositional" logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of (...) subsets so there is a dual concept of logical entropy which is the normalized counting measure on distinctions of partitions. Thus the logical notion of information is a measure of distinctions. Classical logical entropy naturally extends to the notion of quantum logical entropy which provides a more natural and informative alternative to the usual Von Neumann entropy in quantum information theory. The quantum logical entropy of a post-measurement density matrix has the simple interpretation as the probability that two independent measurements of the same state using the same observable will have different results. The main result of the paper is that the increase in quantum logical entropy due to a projective measurement of a pure state is the sum of the absolute squares of the off-diagonal entries ("coherences") of the pure state density matrix that are zeroed ("decohered") by the measurement, i.e., the measure of the distinctions ("decoherences") created by the measurement. (shrink)

This dissertation provides a careful reading of the later Wittgenstein’s philosophy of mathematics centered around three major themes: reality, determination, and infinity. The reading offered gives pride of place to Wittgenstein’s therapeutic conception of philosophy. This conception views questions often taken as fundamental in the philosophy of mathematics with suspicion and attempts to diagnose the confusions which lead to them. In the first essay, I explain Wittgenstein’s approach to perennial issues regarding the alleged reality to which mathematical truths or propositions (...) correspond. Wittgenstein diagnoses exotic pictures of mathematical reality as stemming from misleading analogies formed across empirical and mathematical propositions. The second essay explains Wittgenstein’s treatment of perplexity regarding the ability of a mathematical rule to determine its applications in advance. This too is found to depend on a misleading analogy, in this case across behavioral and mathematical senses of ‘determine’. The third and final essay discusses Wittgenstein’s general critical approach to “the infinite”. Philosophical perplexity about infinity is shown to depend on the verbal imagery regularly associated with proofs and their power to take hold of one’s imagination. Wittgenstein dampens the imaginative excesses often associated with “the infinite” by offering a sober redescription of Cantor’s famous method of diagonalization. (shrink)

There is a long history of philosophical intuition that the human mind must be more than physical or mechanical. I argue that this intuition arises from the perfect “transparency” of physical and mechanical states, in the sense that such states have no obscure or occult elements, but are fully intelligible in mathematical terms. In the paper, I derive a contradiction from the claim that such a physical system has genuine intentionality, comparable with an intelligent human. The contradiction arises from the (...) fact that, according to physicalism, the physical properties of a brain state determine the narrow propositional content of any conscious thought occurring in that state. This fact allows a physical property of brain states to be defined using Cantor’s diagonal construction, and then a contradiction results if a physical system is assumed to form thoughts involving that property. (shrink)

O autor analisa a crítica que fez Wittgenstein aos fundamentos de Matemática na dupla fase do seu pensamento lógico e filosófico. Começa por situá-lo em relação às 3 Escolas que discutiam sobre a fundamentação lógica da matemática: o logicismo, o intuicionismo e o formalismo. Na 1.a fase do Tractatus, vê-se que Wittgenstein é logicista. Mas é original porque não deriva a aritmética do cálculo de classes, como fazia Russell, mas do cálculo proposicional, que generaliza. Considera a matemática como um simples (...) "método lógico". Por isso define o Número como "expoente duma função proposicional"! Mas neste caso trata-se implicitamente, duma nova extensão analógica de princípios e conceitos da aritmética à lógica pura. Por exemplo: noções de número, operação, exponenciação, etc. E se Wittgenstein quer criar, por analogia, números lógicos (símbolos puros), então tem de fazer uma interpretação semântica para passar da lógica formal pura para a aritmética! Só admite o infinito potencial na matemática. Infere-se, pois, que na l.a fase do Tractatus, Wittgenstein, é nominalista e finista. Na 2a fase das Notas (PB ou Remarks) que são manuscritos e fotocópias, (entre 1927 e 1944) Wittgenstein foi variando de pensamento. Agora julga que a matemática se funda a si mesma, como autónoma de lógica pura. Descobre que para além dos "Símbolos" há relações para os "significados" e "usos". E a matemática pode ser teórica e aplicada aos cálculos das ciências. Mas pela sua crítica intuicionista rejeita o valor da teoria dos números irracionais de Dedekind, do teorema e processo da diagonal com que Cantor prova a teoria dos números transfinitos... Wittgenstein aborda ainda os célebres teoremas de Fermat e de Gödel, quando analisa as proposições indecidíveis dos Principia mathematica (de Russell-Whitehead). O autor mostra, no artigo, que Wittgenstein não tem razão na sua crítica pouco profunda ao conceito de número e suas generalizações. .. Na 2.a fase ele continua a ser finitista, mas agora é conceptualista. /// L'auteur analyse la critique que fit Wittgenstein aux fondements des Mathématiques dans la double phase de sa pensée logique et philosophique. Il commence par situer cette pensée en relation avec les trois écoles qui discutaient sur la fondation logique des mathématiques : le logicisme, l'intuitionisme et le formalisme. Dans la première phase du Tractatus, on voit que Wittgenstein est lo-giciste. Mais sa position est originale, parece qu'il ne dérive pas les mathématiques du calcul des classes, comme faisait Russell, mais du calcul propo-sitionnel, qu'il généralise. Il considère les mathématiques comme une simple "méthode logique". C'est pourquoi il définit le Nombre comme "exposant d'une fonction propositionnelle". Mais en ce cas il s'agit implicitement d'une nouvelle extension analogique de principes et de concepts de l'arithmétique à la logique pure. Par exemple : les notions de nombre, d'opération, d'opération exponentionelle etc. Et si Wittgenstein veut créer, par analogie, des nombres logiques (symboles pures), il doit alors faire une interprétation sémantique pour passer de la logique formelle pure aux mathématiques! Il n'admet que l'infini potentiel en mathématiques. Il s'ensuit alors que dans la première phase du Tractatus Wittgenstein est nominaliste et finitiste. Dans la seconde phase des Notes Philosophische Bemerkungen ou Remarks1 qui sont des manuscrits et photocopies (entre 1927 et 1944), Wittgenstein diversifia sa pensée. Il pense alors que les mathématiques se fondent elles-mêmes, de façon autonome par rapport à la logique pure. Il découvre que au-delà des "Symboles" il y a des relations pour les "sens" et les "usages". Et les mathématiques peuvent être théoriques et appliquées aux calculs des sciences. Mais par sa critique intutiuniste, il rejette la valeur de la théorie des nombres irrationnels de Dedekind, du théorème et du procédé de la diagonale par lesquels Cantor prouve la théorie des nombres transfinis. Wittgenstein aborde encore les célèbres théorèmes de Fermât et de Gô-del quand il analyse les propositions indécidables des Principia Mathematica (de Russell – Whitehead). L'auteur montre dans l'article que Wittgenstein n'a pas raison dans sa critique peu profonde au concept de nombre et ses généralisations... Dans la seconde phase, il continue à être finitiste, mais à présent il est conceptualiste. /// The author examines Wittgenstein's criticism of the foundations of mathematics in the two phases of his logical and philosophical thought. He begins by situating it in relation to the three schools which discussed the logical foundations of mathematics: logicism, intuicionism, and formalism. In the first phase of the Tractatus, we can see Wittgenstein as a logicist. His originality resides in his not deriving arithmetic from the calculus of classes, as did Russell, but from propositional calculus that he generalizes. Wittgenstein considers mathemathics as a simple "logical method". This is why he defines "number" as a "exponent of a propositional function". But in this case it is implicitly a matter of a new analogical extension of arithmetical principles and concepts to pure logic. For example, the notions of number, operation, and exponentiation, etc And if Wittgensteinwishes to create by analogy logical numbers (pure symbols), then he has to effectuate a semantical interpretation to pass from pure formal logic to arithmetic! He only allows potencial infinity in mathematics. One can infer that in the Tractatus phase, Wittgenstein is nominalist and finitist. In the second phase of Remarks, manuscripts and copies, dating between 1927 and 1944, Wittgenstein changed his thinking. Now he believes that mathematics founds itself independently of pure logic. He discovers beyond "symbols" there are relations for "meanings" and "uses". And mathematics can be theoretical and applicable to the calculus of sciences. But by his intuitional criticism he can reject the value of the theory of irracional numbers of Dedekind, of the theorem and the process of the diagonal, with wich Cantor proves the theory of transfinit numbers... Wittgenstein refers still the famous theorems of Fermat and Godel when he examines the indecidable propositions of the Prinvipia Mathemativa (of Russell-Whitehead). In this article the author shows that Wittgenstein is not correct in his not too profound criticism of the concept of numbers and its generalizations. In the second phase, he continues to be finitist but now also conceptualist. (shrink)

This book is about one of the most baffling of all paradoxes – the famous Liar paradox. Suppose we say: 'We are lying now'. Then if we are lying, we are telling the truth; and if we are telling the truth we are lying. This paradox is more than an intriguing puzzle, since it involves the concept of truth. Thus any coherent theory of truth must deal with the Liar. Keith Simmons discusses the solutions proposed by medieval philosophers and offers (...) his own solutions and in the process assesses other attempts to solve the paradox. Unlike such attempts, Simmons' 'singularity' solution does not abandon classical semantics and does not appeal to the kind of hierarchical view found in Barwise's and Etchemendy's The Liar. Moreover, Simmons' solution resolves the vexing problem of semantic universality – the problem of whether there are semantic concepts beyond the expressive reach of a natural language such as English. (shrink)

According to contextualist theories in metaethics, when you use a moral term in a context, the context plays an ineliminable part in determining what natural property will be the semantic value of the term. Furthermore, on subjectivist and relativist versions of these views, it is either the speaker's own moral code or her moral community's moral code that constitutes the reference-fixing context. One standard objection to views of this type is that they fail to enable us to disagree in ordinary (...) conversations. In this chapter, I develop a new response to this objection on the basis of Kai von Fintel and Anthony Gillies' notion of proposition clouds. I argue that, because we live in a multicultural society, the conversational contexts we face will fail to disambiguate between all the things we could mean. This is why we can at best put into play proposition clouds when we make moral utterances. All the propositions in such clouds are then available for rejection and acceptance on the behalf of our audiences. The norms of conversation then guide us to make informative contributions to the conversation - accept and reject propositions in a way that leads to co-ordination of action and choice. (shrink)

Richard Gaskin presents a work in the philosophy of language.

What makes a visually appealing landscape? How can the design and use of a landscape be harmonised? In this significantly revised and updated third edition of Simon Bell's seminal text, he further explores the answers to these questions by interrogating a range of design principles, applications and ideas. Written for students, instructors and professionals, the book unveils a visual design vocabulary for anyone involved with landscape aesthetics including landscape architects, architects, planners, urban designers, landscape managers, foresters, geographers and ecologists. Structured (...) around key design terms, which are explained and illustrated using an extensive range of examples from around the world, including North America, Europe and Asia, this book enables you to describe, debate and design the visual landscape. It starts with basic elements, before moving onto variable design components, and then the ways these elements are organized into compositions, in order to demonstrate how landscapes are created and how meanings and patterns are perceived within them. This new full colour edition contains over 240 images; an updated introduction; examples from China, Vietnam, and central Asia; a chapter on how to read and understand visual design elements in the landscape; a teaching model for instructors; and expanded appendix materials including a glossary, references and further reading. (shrink)

The word pro-tasis is etymologically a near equivalent of pre-mise, pro-position, and ante-cedent—all having positional, relational connotations now totally absent in contemporary use of proposition. Taking protasis for premise, Aristotle’s statement (24a16) -/- A protasis is a sentence affirming or denying something of something…. -/- is not a definition of premise—intensionally: the relational feature is absent. Likewise, it is not a general definition of proposition—extensionally: it is too narrow. This paper explores recent literature on these issues.

Stalnaker’s ‘Assertion’ (1978 [1999]) offers a classic account of diagonalization as an approach to the meaning of a declarative sentence in context. Here I explore the relationship between diagonalization and some puzzles in Mahtani’s book The Objects of Credence. Diagonalization can influence how we think about both credence and desirability, so it influences both components of a standard expected utility equation. In that vein, I touch on two of Mahtani’s case-studies, chance and the finite version of the Two Envelope Paradox.