Sven Rosenkranz’s Justification as Ignorance shows how a strongly internalist conception of justification can be derived from a strongly externalist conception of knowledge, given an identification of justification with second-order ignorance and a set of structural principles concerning knowing and being in a position to know. Among these principles is an epistemic analogue of the Geach modal schema which states that one is always in a position to know that one doesn’t know p or in a position to know (...) that one doesn’t know not-p. Even suitably refined, the principle faces a range of counterexamples in which it misleadingly and persistently seems to one that one knows whether p without it seeming to one that one knows p nor that one knows not-p. These cases also threaten the book’s case for the luminosity of second-order ignorance, which is in turn central to its derivation of strongly internalist principles of justification. (shrink)

We use quotation marks when we wish to refer to an expression. We can and do so refer even when this expression is composed of characters that do not occur in our alphabet. That's why Tarski, Quine, and Geach's theories of quotation don't work. The proposals of Davidson, Frege, and C. Washington, however, do not provide a plausible account of quotation either. (Section I). The problem is to construct a Tarskian theory of truth for an object language that contains (...) quotation marks, without appealing to quotation marks in the metalanguage. I propose to supply Tarski's truth definition with one axiom that determines the denotation of all expressions containing quotation marks. According to this axiom, quotation marks create a non-extensional context. Since admitting such contexts does not lead to any difficulties in the recursive truth characterization, we may indeed dispense with extensionalism. (Section II). Finally, I argue that we classify and denote expressions in the very same way that we classify and denote extralinguistic entities. Both tokens and types of written signs can be easily incorporated into the naturalist's worldview. (Section III). (shrink)

Propositional identity is not expressed by a predicate. So its logic is not given by the ordinary first order axioms for identity. What are the logical axioms governing this concept, then? Some axioms in addition to those proposed by Arthur Prior are proposed.

Geach is best known for his contributions to theoretical philosophy: Most of his more than one hundred papers and a dozen books are on logic, philosophy of language and metaphysics. But he also made significant contributions to ethics. Particularly influential were a series of short metaethics papers, which are small masterpieces, both in terms of philosophical content and style. In usually less than ten pages, Geach delivers sharp analyses and powerful objections against influential schools. His arguments are always (...) so clear and his examples so simple that they leave the reader wondering why no one before Geach detected the problems he points out. (shrink)

In his celebrated 'Good and Evil' (l956) Professor Geach argues as against the non-naturalists that ‘good’ is attributive and that the predicative 'good', as used by Moore, is senseless.. 'Good' when properly used is attributive. 'There is no such thing as being just good or bad, [that is, no predicative 'good'] there is only being a good or bad so and so'. On the other hand, Geach insists, as against non-cognitivists, that good-judgments are entirely 'descriptive'. By a consideration (...) of what it is to be an A, we can determine what it is to be a good A, even where the ‘A’ in question is ‘human being’. These battles are fought on behalf of naturalism, indeed, of an up-to-date Aristotelianism. Geach plans to 'pass' from the 'purely descriptive' man to good/bad man, and from human act to good/bad human act. I argue: (l) That the predicative 'good' does have a genuine sense and that it is a mistake to suppose that ‘good’ is a purely attributive adjective. This does not entail that the predicative good (as used by Moore) denotes a non-natural property, but his mistake, if any is metaphysical or ontological not conceptual. (2) That the attributive 'good' cannot be used to generate a naturalistic ethic. It is difficult to extract a set of biologically based requirements out of human nature that are a) reasonably specific; b) rationally binding or at least highly persuasive; and c) morally credible. -/- On the way I protest against Geach’s tendency to try to win arguments by affecting not to understand things. -/- My views to some extent anticipate those of Kraut in *Against Absolute Goodness*. (shrink)

This is an opinionated overview of the Frege-Geach problem, in both its historical and contemporary guises. Covers Higher-order Attitude approaches, Tree-tying, Gibbard-style solutions, and Schroeder's recent A-type expressivist solution.

Aquinas’ theory of being has received a growing amount of attention from contemporary scholars, both from a historic and a philosophical point of view. An important source of this attention is Geach’s seminal Form and Existence. In it, Geach argues that Aquinas subscribes to a tensed notion of existence and a theory of time according to which past and future entities do not exist in act. Subsequent commentators, such as Kenny in his Aquinas on Being, have agreed with (...) Geach on both points. In this paper, I argue that in several passages of his corpus, most notably those in which he is concerned with God’s knowledge of future contingents, Aquinas implicitly subscribes to a theory of being and time according to which: past and future entities are attributed existence in act, there is theoretical need for introducing a tenseless notion of existence. (shrink)

Dies bietet einen knappen Überblick über Leben und Werk des britischen Philosophen P. T. Geach. Unter anderem wird vorgeführt, mithilfe welches Arguments Geach die Sein/Sollen-Schranke zu überwinden beanspruchte.

This paper discusses the first incarnation of what came to be known as the “Frege-Geach” point. The point was made by Peter Geach in his 1960 essay “Ascriptivism”, and developed in “Assertion”, a 1965 piece. Geach’s articles launch a wholesale attack on theories of non-descriptive performances advanced by “some Oxford philosophers” whom he accuses of ignoring “the distinction between calling a thing ‘P’ and predicating ‘P’ of a thing”. One view that Geach specifically targets is H. (...) L. A. Hart’s claim that sentences of the form “X φ-ed” are not primarily descriptive but ascriptive of responsibility for actions. Hart explicitly accepted Geach’s criticism, and disowned his essay. I argue that he was wrong to do so. Perhaps the essay was worth retracting, but not because of Geach’s objections. I begin by restating and refining Geach’s arguments, in order to bring out the flaw he claimed to have detected in the “pattern of philosophising” that he took Hart’s essay to exemplify. I go on to argue that Geach’s original point poses no obstacle either to non-descriptivism in general, or to Hart’s sui generis non-descriptivist claim in particular. (shrink)

According to the established understanding of the Frege-Geach problem, it is a challenge exclusively for metaethical expressivism. In this paper, I argue that it is much wider in scope: The problem applies generally to views according to which moral sentences express moral judgments entailing that one is for or against something, irrespective of what mental states the judgments consist in. In particular, it applies to motivational internalism about moral judgments. Most noteworthy, it applies to cognitivist internalism according to which (...) moral judgments consist in motivating beliefs. Hence, in order for a metaethical view to evade the Frege-Geach problem, it should avoid stating that moral judgments are motivating. (shrink)

In the 1960s, Peter Geach and John Searle independently posed an important objection to the wide class of 'noncognitivist' metaethical views that had at that time been dominant and widely defended for a quarter of a century. The problems raised by that objection have come to be known in the literature as the Frege-Geach Problem, because of Geach's attribution of the objection to Frege's distinction between content and assertoric force, and the problem has since occupied a great (...) deal of the attention both of defenders of broadly noncognitivist views, and of their critics. In this article I explain Geach and Searle's historical objections, and put the subsequent discussion into dialectical context, paying some attention to the developments along the way and how they have enhanced our overall understanding of the problem. The article covers a lot of territory, so we will only be able to see the highlights, along the way. For further reading, see the Works Cited. (shrink)

Hybrid expressivists claim to solve the Frege-Geach problem by offloading the explanation of the logico-semantic properties of moral sentences onto beliefs that are components of hybrid states they express. We argue that this strategy is undermined by one of hybrid expressivism’s own commitments: that the truth of the belief-component is neither necessary nor sufficient for the truth of the hybrid state it composes. We articulate a new approach. Instead of explaining head-on what it is for, say, a pair of (...) moral sentences to be inconsistent, expressivists should “sidestep” and explain what it is _to think that_ a pair of moral sentences is inconsistent. To think so is to think they cannot both be true – a modal notion. Since expressivists have given accounts of such modals, we illustrate how sentences like ‘“lying is wrong” and “lying is not wrong” are inconsistent’ express sensible – and rationally compelling – states of mind. (shrink)

Several philosophers have recently appealed to predication in developing their theories of cognitive representation and propositions. One central point of difference between them is whether they take predication to be forceful or neutral and whether they take the most basic cognitive representational act to be judging or entertaining. Both views are supported by powerful reasons and both face problems. Many think that predication must be forceful if it is to explain representation. However, the standard ways of implementing the idea give (...) rise to the Frege-Geach problem. Others think that predication must be neutral, if we’re to avoid the Frege-Geach problem. However, it looks like nothing neutral can explain representation. In this paper I present a third view, one which respects the powerful reasons while avoiding the problems. On this view predication is forceful and can thus explain representation, but the idea is implemented in a novel way, avoiding the Frege-Geach problem. The key is to make sense of the notion of grasping a proposition as an objectual act where the object is a proposition. (shrink)

I follow the distinction of predicative versus attributive adjectives introduced by P.T. Geach in 1956 and discuss Atina Knowles' arguments contra the revisions of Geach's original conception.

According to judgment internalism, there is a conceptual connection between moral judgment and motivation. This paper offers an argument against that kind of internalism that does not involve counterexamples of the amoralist sort. Instead, it is argued that these forms of judgment internalism fall prey to a Frege-Geach type argument.

I resolve the major challenge to an Expressivist theory of the meaning of normative discourse: the Frege–Geach Problem. Drawing on considerations from the semantics of directive language (e.g., imperatives), I argue that, although certain forms of Expressivism (like Gibbard’s) do run into at least one version of the Problem, it is reasonably clear that there is a version of Expressivism that does not.

Mark Eli Kalderon has argued for a fictionalist variant of non-cognitivism. On his view, what the Frege–Geach problem shows is that standard non-cognitivism proceeds uncritically from claims about use to claims about meaning; if non-cognitivism's claims were solely about use it would be on safe ground as far as the Frege–Geach problem is concerned. I argue that Kalderon's diagnosis is mistaken: the problem concerns the non-cognitivist's account of the use of moral sentences too.

Despite its many advantages as a metaethical theory, moral expressivism faces difficulties as a semantic theory of the meaning of moral claims, an issue underscored by the notorious Frege-Geach problem. I consider a distinct metaethical view, inferentialism, which like expressivism rejects a representational account of meaning, but unlike expressivism explains meaning in terms of inferential role instead of expressive function. Drawing on Michael Williams’ recent work on inferential theories of meaning, I argue that an appropriate understanding of the pragmatic (...) role of moral discourse—the facilitation of coordinated social behavior—suggests the kind of inferences we should expect terms with this function to license. I offer a sketch of the inferential roles the moral ‘ought’ plays, and argue that if we accept that the relevant inferential roles are meaning-constitutive, we will be in a position to solve the Frege-Geach problem. Such an inferentialist solution has advantages over those forwarded by expressivists such as Blackburn and Gibbard. First, it offers a more straightforward explanation of the meaning of moral terms. It also gives simple answers to at least two semantic worries that have vexed contemporary expressivists—the “problem of permissions” and the commitment to “mentalism”, both of which I argue are problems that don’t get traction with an inferentialist approach. I conclude by considering ways in which this approach can be expanded into a more robust semantic account. (shrink)

Axiom weakening is a technique that allows for a fine-grained repair of inconsistent ontologies. Its main advantage is that it repairs on- tologies by making axioms less restrictive rather than by deleting them, employing the use of refinement operators. In this paper, we build on pre- viously introduced axiom weakening for ALC, and make it much more irresistible by extending its definitions to deal with SROIQ, the expressive and decidable description logic underlying OWL 2 DL. We extend the (...) definitions of refinement operator to deal with SROIQ constructs, in particular with role hierarchies, cardinality constraints and nominals, and illustrate its application. Finally, we discuss the problem of termi- nation of an iterated weakening procedure. (shrink)

Although the concept of uncertainty is as old as Epicurus’s writings, and an excellent quantitative theory, with entropy as the measure of uncertainty having been developed in recent times, there has been little exploration of the qualitative theory. The purpose of the present paper is to give a qualitative axiomatization of uncertainty, in the spirit of the many studies of qualitative comparative probability. The qualitative axioms are fundamentally about the uncertainty of a partition of the probability space of events. Of (...) course, it is common to speak of the uncertainty, or randomness, of a random variable, but only the partition defined by the values of the random variable enter into the definition of uncertainty, not the actual values. It is straightforward to add axioms for decision making following the general line of Savage from the 1950s. Indeed, in the spirit of Epicurus, it is really our intuitive feeling about the uncertainty of the future that motivates much of our thinking about decisions. Here, the distinction between the concepts of probability and uncertainty can be made by citing many familiar examples. Without spelling out the technical details, the axiomatization of qualitative probability with uncertainty as the most important primitive concept, it is possible to raise a different kind of question about bounded rationality. This new question is whether or not one should bound the uncertainty in thinking and investigating any detailed framework of decision making. Discussion of this point is certainly different from the question of bounding rationality by not maximizing expected utility. In practice, we naturally bound uncertainty in our analysis of decision-making problems. As in the case of formulating an alternative for maximizing expected utility, so is the case of rational alternatives to maximizing uncertainty. There are several issues to consider. In the spirit of my other work in qualitative probability, I explore alternatives rather than attempt to give a definitive argument for one single solution. (shrink)

Spinoza's causal axiom is at the foundation of the Ethics. I motivate, develop and defend a new interpretation that I call the ‘causally restricted interpretation’. This interpretation solves several longstanding puzzles and helps us better understand Spinoza's arguments for some of his most famous doctrines, including his parallelism doctrine and his theory of sense perception. It also undermines a widespread view about the relationship between the three fundamental, undefined notions in Spinoza's metaphysics: causation, conception and inherence.

Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: (...) the parallel postulate. -/- Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. -/- There are other differences that are or might become important. -/- Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. -/- Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. -/- The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content. (shrink)

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us, all directions are the same to us and all units of length we use to create geometric figures are the same to us. On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s (...) system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: it supports the thesis that Euclidean geometry is a priori, it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics. (shrink)

Según varios filósofos del lenguaje contemporáneos, el argumento Frege-Geach plantea una objeción genuina para el expresivismo semántico. En el presente texto se sostiene que una manera eficaz de enfrentar y superar este argumento es modificando la concepción expresivista clásica. Se examinan el expresivismo clásico y el expresivismo mínimo y se propone una versión de este último que, a la vez que supera la objeción derivada del argumento Frege-Geach, tiene la ventaja de incluir dentro del espectro del expresivismo teorías (...) que claramente tienen este “espíritu” pero que la caracterización clásica deja fuera. (shrink)

The problem of identity is central to epistemic transference. However, relative identity appears to be the only way to work out an epistemic useful notion of identity. Relative identity, on its part, is either parasitic on strict identity or not identity at all. If, on the contrary, we ought for a strict concept of identity capable of satisfying its requirements, we end up with a tautologic and epistemic worthless category. The paper provides an answer to this problem, which, while working (...) with a strict notion, shows how it might still serve epistemic purposes. In doing so, it shows how a formal reconstruction of our objectual world and the identities we refer to poses a workable model upon which our messy epistemic one acquires stability. The paper focuses primarily on the Geach-Quine discussion on identity. (shrink)

Chains of causes appear when the existence of God is discussed. It is claimed by some that these chains must be finite and terminated by God. But these chains seem endless through our knowledge search. This endlessness for the physical reasons for any world event expresses the greatness and complexity of God’s creation and so the transcendence of God. So, only we can put our hands on physical reasons in an endless forage for knowledge. Yet, the endlessness of the physical (...) causes chains confirms the existence of God, and this is what our paper tries using Zorn’s lemma, an equivalent for one of the most celebrated axioms of mathematics, the Axiom of Choice. (shrink)

The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...) (H-axioms) in countable transitive models, the collection of which constitutes the `hyperuniverse' (H), has remarkable 1st-order consequences, some of which we review in section 5. (shrink)

Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face (...) complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology. (shrink)

In quantum theory every state can be diagonalized, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This elementary structure plays an ubiquitous role in quantum mechanics, quantum information theory, and quantum statistical mechanics, where it provides the foundation for the notions of majorization and entropy. A natural question then arises: can we reconstruct these notions from purely operational axioms? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that (...) every state can be diagonalized. The first axiom is Causality, which ensures that the marginal of a bipartite state is well defined. Then, Purity Preservation states that the set of pure transformations is closed under composition. The third axiom is Purification, which allows to assign a pure state to the composition of a system with its environment. Finally, we introduce the axiom of Pure Sharpness, stating that for every system there exists at least one pure effect occurring with unit probability on some state. For theories satisfying our four axioms, we show a constructive algorithm for diagonalizing every given state. The diagonalization result allows us to formulate a majorization criterion that captures the convertibility of states in the operational resource theory of purity, where random reversible transformations are regarded as free operations. (shrink)

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in Maudlin and Malament. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of Tarski : a predicate of betwenness and a four place (...) predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane—which obeys the Euclidean axioms in Tarski and Givant, 175–214 1999)—and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagoras’ theorem. We conclude with a Representation Theorem relating models \ of our system \ that satisfy second order continuity to the mathematical structure \, called ‘Minkowski spacetime’ in physics textbooks. (shrink)

Ontology engineering is a hard and error-prone task, in which small changes may lead to errors, or even produce an inconsistent ontology. As ontologies grow in size, the need for automated methods for repairing inconsistencies while preserving as much of the original knowledge as possible increases. Most previous approaches to this task are based on removing a few axioms from the ontology to regain consistency. We propose a new method based on weakening these axioms to make them less restrictive, employing (...) the use of refinement operators. We introduce the theoretical framework for weakening DL ontologies, propose algorithms to repair ontologies based on the framework, and provide an analysis of the computational complexity. Through an empirical analysis made over real-life ontologies, we show that our approach preserves significantly more of the original knowledge of the ontology than removing axioms. (shrink)

Our question, in this paper, is about the plausibility of the expressivist account of one’s self- attribution of mental states. More to the point, we will strictly follow the principle of charity as a mean to show that an expressivist philosopher can have good and reasonable answers to the set of objections put together in so called “Geach’s point”. Using this method, we hope to give enough evidences that an expressivist philosopher has enough resources to build a plausible explanation (...) for one’s attribution of mental states to herself. (shrink)

Peirce and Dewey were generally more concerned with the process of scientific activity than purely mathematical work. However, their accounts of knowledge production afford some insights into the epistemology of mathematical postulates, especially definition and axioms. Their rejection of rationalist metaphysics and their emphasis on continuity in inquiry provides the pretext for the pragmatic a priori – hypothetical and operational assumptions whose justification relies on their fruitfulness in the long run. This paper focuses on the application of this idea to (...) the epistemology of definitions and an account of progress in mathematics, although it has broader implications for the study of conceptual change and the function and basis of presuppositions in the sciences. (shrink)

In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations (...) for, Russell’s method of analysis, which are intended to shed light on his view about the status of mathematical axioms. I describe the position Russell develops in consequence as “immanent logicism,” in contrast to what Irving (1989) describes as “epistemic logicism.” Immanent logicism allows Russell to avoid the logocentric predicament, and to propose a method for discovering structural relationships of dependence within mathematical theories. (shrink)

A description of consciousness leads to a contradiction with the postulation from special relativity that there can be no connections between simultaneous event. This contradiction points to consciousness involving quantum level mechanisms. The Quantum level description of the universe is re- evaluated in the light of what is observed in consciousness namely 4 Dimensional objects. A new improved interpretation of Quantum level observations is introduced. From this vantage point the following axioms of consciousness is presented. Consciousness consists of two distinct (...) components, the observed U and the observer I. The observed U consist of all the events I is aware of. A vast majority of these occur simultaneously. Now if I were to be an entity within the space-time continuum, all of these events of U together with I would have to occur at one point in space-time. However, U is distributed over a definite region of space-time (region in brain). Thus, I is aware of a multitude of space-like separated events. It is seen that this awareness necessitates I to be an entity outside the space-time continuum. With I taken as such, a new concept called concept A is introduced. With the help of concept A a very important axiom of consciousness, namely Free Will is explained. Libet s Experiment which was originally seen to contradict Free will, in the light of Concept A is shown to support it. A variation to Libet s Experiment is suggested that will give conclusive proof for Concept A and Free Will. (shrink)

Ontologies represent principled, formalised descriptions of agents’ conceptualisations of a domain. For a community of agents, these descriptions may differ among agents. We propose an aggregative view of the integration of ontologies based on Judgement Aggregation (JA). Agents may vote on statements of the ontologies, and we aim at constructing a collective, integrated ontology, that reflects the individual conceptualisations as much as possible. As several results in JA show, many attractive and widely used aggregation procedures are prone to return inconsistent (...) collective ontologies. We propose to solve the possible inconsistencies in the collective ontology by applying suitable weakenings of axioms that cause inconsistencies. (shrink)

Axiom weakening is a novel technique that allows for fine-grained repair of inconsistent ontologies. In a multi-agent setting, integrating ontologies corresponding to multiple agents may lead to inconsistencies. Such inconsistencies can be resolved after the integrated ontology has been built, or their generation can be prevented during ontology generation. We implement and compare these two approaches. First, we study how to repair an inconsistent ontology resulting from a voting-based aggregation of views of heterogeneous agents. Second, we prevent the generation (...) of inconsistencies by letting the agents engage in a turn-based rational protocol about the axioms to be added to the integrated ontology. We instantiate the two approaches using real-world ontologies and compare them by measuring the levels of satisfaction of the agents w.r.t. the ontology obtained by the two procedures. (shrink)

This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of the set-theoretic community---set-theoretic (...) practitioners with a pluralist view and set-theoretic practitioners who aim for ZFC-proofs. By describing this, the paper gives a more complete picture of new axioms in set-theoretic practice. These observations, for instance, show that set-theoretic practitioners interested in ZFC-proofs use tools that go beyond ZFC. The analysis is based on empirical data that was collected in an extensive interview study with set-theoretic practitioners. (shrink)

The basic axioms or formal conditions of decision theory, especially the ordering condition put on preferences and the axioms underlying the expected utility formula, are subject to a number of counter-examples, some of which can be endowed with normative value and thus fall within the ambit of a philosophical reflection on practical rationality. Against such counter-examples, a defensive strategy has been developed which consists in redescribing the outcomes of the available options in such a way that the threatened axioms or (...) conditions continue to hold. We examine how this strategy performs in three major cases: Sen's counterexamples to the binariness property of preferences, the Allais paradox of EU theory under risk, and the Ellsberg paradox of EU theory under uncertainty. We find that the strategy typically proves to be lacking in several major respects, suffering from logical triviality, incompleteness, and theoretical insularity. To give the strategy more structure, philosophers have developed “principles of individuation”; but we observe that these do not address the aforementioned defects. Instead, we propose the method of checking whether the strategy can overcome its typical defects once it is given a proper theoretical expansion. We find that the strategy passes the test imperfectly in Sen's case and not at all in Allais's. In Ellsberg's case, however, it comes close to meeting our requirement. But even the analysis of this more promising application suggests that the strategy ought to address the decision problem as a whole, rather than just the outcomes, and that it should extend its revision process to the very statements it is meant to protect. Thus, by and large, the same cautionary tale against redescription practices runs through the analysis of all three cases. A more general lesson, simply put, is that there is no easy way out from the paradoxes of decision theory. (shrink)

Suppose one has a system, the infinite set of positive integers, P, and one wants to study the characteristics of a subset (or subsystem) of that system, the infinite subset of odd positives, O, relative to the overall system. In mathematics, this is done by pairing off each odd with a positive, using a function such as O=2P+1. This puts the odds in a one-to-one correspondence with the positives, thereby, showing that the subset of odds and the original set of (...) positives are the same size, or have the same cardinality. This counter-intuitive result ignores the “natural” relationship of one odd for every two positives in the sequence of positive integers, which would suggest that O is one-half the size of P. However, in the set of axioms that constitute mathematics, it is considered valid. Fair enough. In the physical universe (i.e., the starting system), though, relationships between entities matter. In biochemistry, if you start with an organism, say a rat, and you want to study its heart, you can do this by removing some heart cells and studying them in isolation in a cell culture system. But, the results often differ compared to what occurs in the intact animal because studying the isolated cultured cells ignores the relationships in the intact body between the cells, the rest of the heart tissue and the rest of the rat. In chemistry, if a copper atom were studied in isolation, it would never be known that copper atoms in bulk can conduct electricity because the atoms share their electrons. In physics, the relationships between inertial reference frames in relativity, and observer and observed in quantum physics can't be ignored. Relationships matter in the physical world, but the mathematics of infinite sets is still used to describe it. Does this matter? It seems to, at least in physics. Infinities cause numerous problems in theoretical physics such as non-renormalizability and problems in unifying quantum mehanics and general relativity . This suggests that the pairing off method and the mathematics of infinite sets based on it are analogous to a cell culture system or studying a copper atom in isolation if they are used in studying the physical universe because they ignore the inherent relationships between entities. In the real, physical world, the natural, inherent, relationships between entities can't be ignored. Said another way, the set of axioms that constitute abstract mathematics may be similar but not identical to the set of physical axioms by which the real, physical universe runs. This suggests that the results from abstract mathematics about infinities may not apply to or should be modified for use in physics. (shrink)

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of (...) infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part. (shrink)

Spinoza's Ethics self-consciously follows the example of Euclid and other geometers in its use of axioms and definitions as the basis for derivations of hundreds of propositions of philosophical si...

The purpose of this paper is to challenge some widespread assumptions about the role of the modal axiom 4 in a theory of vagueness. In the context of vagueness, axiom 4 usually appears as the principle ‘If it is clear (determinate, definite) that A, then it is clear (determinate, definite) that it is clear (determinate, definite) that A’, or, more formally, CA → CCA. We show how in the debate over axiom 4 two different notions of clarity (...) are in play (Williamson-style "luminosity" or self-revealing clarity and concealeable clarity) and what their respective functions are in accounts of higher-order vagueness. On this basis, we argue first that, contrary to common opinion, higher-order vagueness and S4 are perfectly compatible. This is in response to claims like that by Williamson that, if vagueness is defined with the help of a clarity operator that obeys axiom 4, higher-order vagueness disappears. Second, we argue that, contrary to common opinion, (i) bivalence-preservers (e.g. epistemicists) can without contradiction condone axiom 4 (by adopting what elsewhere we call columnar higher-order vagueness), and (ii) bivalence-discarders (e.g. open-texture theorists, supervaluationists) can without contradiction reject axiom 4. Third, we rebut a number of arguments that have been produced by opponents of axiom 4, in particular those by Williamson. (The paper is pitched towards graduate students with basic knowledge of modal logic.). (shrink)