I attempt to show, via considering Schlesinger’s device of putting the word ‘now’ in capitals, that the transient view of time can explicate temporal extensivity without presupposing it, and the static view can’t. The argument hinges on the point that duration is generated by continuance of the present—such that ‘the present’ here is used in a nontechnical, nonindexical, and nonreflexive sense, which Schlesinger and others unknowingly give to the word ‘now’ (by “NOW” or “Now” or “’now’”).
We look at two recent accounts of the indefinite extensibility of set, and compare them with a linguistic model of the indefinite extensibility. I suggest the linguistic model has much to recommend over extant accounts of the indefinite extensibility of set, and we defend it against three prima facie objections.
This paper will defend the claim that, under certain circumstances, the material vehicles responsible for an agent’s conscious experience can be partly constituted by processes outside the agent’s body. In other words, the consciousness of the agent can extend. This claim will be supported by the Extended Mind Thesis (EMT) example of the artist and their sketchpad (Clark 2001, 2003). It will be argued that if this example is one of EMT, then this example also supports an argument for consciousness (...) extension. Clark (2009) rejects claims of consciousness extension. This paper will challenge Clark and argue that he fails to show that the material vehicles responsible for consciousness must be internal to the agent. (shrink)
This essay critically reviews Andy Clark’s new book Supersizing the Mind: Embodiment, Action, and Cognitive Extension, in which he argues that there are circumstances in which the mind, properly considered, is found to supervene on not only the brain, but the body and the external environment as well. This review summarizes Clark’s major contributions to this viewpoint for the general reader, then raises a few critical points that help to contextualize Clark’s claims, aims, and methods, while highlighting the book’s strengths (...) and weaknesses. (shrink)
Assuming the indefinite extensibility of any domain of quantification leads to reasoning with extensible domain semantics. It is showed that some theorems (e.g. Thomson's) in conventional semantics logic are not theorems in a logic provided with this new semantics.
The Monist’s call for papers for this issue ended: “if formalism is true, then it must be possible in principle to mechanize meaning in a conscious thinking and language-using machine; if intentionalism is true, no such project is intelligible”. We use the Grelling-Nelson paradox to show that natural language is indefinitely extensible, which has two important consequences: it cannot be formalized and model theoretic semantics, standard for formal languages, is not suitable for it. We also point out that object-object mapping (...) theories of semantics, the usual account for the possibility of non intentional semantics, doesn’t seem able to account for the indefinitely extensible productivity of natural language. (shrink)
In Humanity’s End: Why We Should Reject Radical Enhancement, Nicholas Agar presents a novel argument against the prospect of radical life-extension. Agar’s argument hinges on the claim that extended lifespans will result in people’s lives being dominated by the fear of death. Here we examine this claim and the surrounding issues in Agar’s discussion. We argue, firstly, that Agar’s view rests on empirically dubious assumptions about human rationality and attitudes to risk, and secondly, that even if those assumptions are granted, (...) the fears that Agar adverts to are unlikely to dominate people’s lives if and when radical life-extension is made possible. Further, we claim that the structure of the decision-making process around life-extension is unlikely to be the way that it would have to be in order for Agar’s claims about fear of death to make sense. Finally, we argue that Agar is implicitly committed to a narrow conception of human value. In response, we suggest that the pursuit of life-extension can itself be seen as an expression of certain important aspects of our distinctively human nature. (shrink)
A number of authors have noted that the key steps in Fitch’s argument are not intuitionistically valid, and some have proposed this as a reason for an anti-realist to accept intuitionistic logic (e.g. Williamson 1982, 1988). This line of reasoning rests upon two assumptions. The first is that the premises of Fitch’s argument make sense from an anti-realist point of view – and in particular, that an anti-realist can and should maintain the principle that all truths are knowable. The second (...) is that we have some independent reason for thinking that classical logic is not appropriate in this area. This paper explores these two assumptions in the context of Michael Dummett’s version of anti-realism, with particular reference to the argument from indefinite extensibility developed at various points in Dummett’s writings (e.g. Dummett 1991 Ch. 24). -/- Dummett argues that certain concepts, the indefinitely extensible concepts, are such that we cannot form a clear and determinate conception of all the objects that fall under them. The most familiar examples of indefinitely extensible concepts are mathematical. Dummett discusses the concepts ordinal number, real number, and natural number, which are indefinitely extensible because any conception that one might form of their complete extension can be extended to a more inclusive conception (as, for example, in Cantor’s proof of the non-denumerability of the set of real numbers). This paper argues that the concept of a truth is indefinitely extensible. This gives a Dummettian anti-realist an independent motivation for rejecting the classical understanding of the quantifiers in this area. At the same time, however, it places in doubt the admissibility of the knowability principle, which seems to involve quantification over the “totality” of truths. As Dummett is at pains to point out (1991: 316), some sentences that purport to quantify over the extension of an indefinitely extensible concept plainly have a truth-value (we can truly say, for example, that every ordinal number has a successor, even though when we say that we are not quantifying over the set of all ordinals). But is the knowability principle one of these sentences? (shrink)
Would be fairer to call Peirce’s philosophy of language “extensionalist” or “intensionalist”? The extensionalisms of Carnap and Quine are examined, and Peirce’s view is found to be prima facie similar, except for his commitment to the importance of “hypostatic abstraction”. Rather than dismissing this form of abstraction (famously derided by Molière) as useless scholasticism, Peirce argues that it represents a crucial (though largely unnoticed) step in much working inference. This, it is argued, allows Peirce to transcend the extensionalist-intensionalist dichotomy itself, (...) through his unique triadic analysis of reference and meaning, by transcending the distinction between (as Quine put it) “things” and “attributes”. (shrink)
Extension is probably the most general natural property. Is it a fundamental property? Leibniz claimed the answer was no, and that the structureless intuition of extension concealed more fundamental properties and relations. This paper follows Leibniz's program through Herbart and Riemann to Grassmann and uses Grassmann's algebra of points to build up levels of extensions algebraically. Finally, the connection between extension and measurement is considered.
This paper evaluates the Natural-Kinds Argument for cognitive extension, which purports to show that the kinds presupposed by our best cognitive science have instances external to human organism. Various interpretations of the argument are articulated and evaluated, using the overarching categories of memory and cognition as test cases. Particular emphasis is placed on criteria for the scientific legitimacy of generic kinds, that is, kinds characterized in very broad terms rather than in terms of their fine-grained causal roles. Given the current (...) state of cognitive science, I conclude that we have no reason to think memory or cognition are generic natural kinds that can ground an argument for cognitive extension. (shrink)
Introduction : brainbound versus extended -- From embodiment to cognitive extension -- The active body -- The negotiable body -- Material symbols -- World, Incorporated -- Boundary disputes -- Mind re-bound -- The cure for cognitive hiccups (HEMC, HEC, HEMC ...) -- Rediscovering the brain -- The limits of embodiment -- Painting, planning, and perceiving -- Disentangling embodiment -- Conclusions : mind-sized bites.
An overview of the problem of constructing extension combinatorially from qualities cum dispositional powers. In the model recommended here, Grassmann's algebra provides the combinatorial structure while Machian elements give the content.
This paper investigates the role of a pre-existing body-model that is an enabling constraint for the incorporation of objects into the body. This body-model is also a basis for the distinction between body extensions (e.g., in the case of tool-use) and incorporation (e.g., in the case of successful prosthesis use). It is argued that, in the case of incorporation, changes in the sense of body-ownership involve a reorganization of the body-model, whereas extension of the body with tools does not involve (...) changes in the sense of body-ownership. (shrink)
In a well-known passage in the last chapter of Frege: Philosophy of Mathematics Michael Dummett suggests that Frege’s major “mistake”—the key to the collapse of the project of Grundgesetze—consisted in “his supposing there to be a totality containing the extension of every concept defined over it; more generally [the mistake] lay in his not having the glimmering of a suspicion of the existence of indefinitely extensible concepts” (Dummett [1991, 317]). Now, claims of the form, Frege fell into paradox because……. are (...) notoriously difficult to assess even when what replaces the dots is relatively straightforward. Offerings have included, for instance, that — (A) Unrestricted quantification: Frege fell into paradox because he allowed himself to quantify over a single, all-inclusive domain of objects (Russell, Dummett). (shrink)
In Kant’s logical texts the reference of the form of the judgment to an “unknown = x” is well known, but its understanding remains far from consensual. Due to the universality of all concepts, the subject as much as the predicate, in the form S is P, is regarded as predicate of the x, which, in turn, is regarded as the subject of the judgment. In the CPR, particularly in the text on the “logical use of the understanding”, this Kantian (...) interpretation of the subject-predicate relation leads to the question about the relations that must hold between intuition and concept in the judgment. In contrast to intuition, if no concept, due to its universal character, refers immediately to an object, how should we understand the relations of subject and predicate to one another, as well as their relations to intuition, which corresponds to the very special individuality of that object in general = x? In the Kant-Literatur, the relations between intuition and concept in the judgment have been considered in diverse theoretical backgrounds, mainly in Fregean logic and in the logic of Port-Royal. Although so markedly different, these two solutions to the problem above seem to share a common thesis, in so far as they claim, though in different ways, a predicative character to those relations. If the analytic tradition recognizes in the relation between x and the concept S the marks of a propositional function Sx, in turn, the interpretation elaborated from the background of Port-Royal recognizes in this relation the minor premise x is S implicit in the judgment every S is P. This being the case, if it were possible to prove, on the contrary, that the relations between intuition and concept in the judgment could only be of a non-predicative character, then a third solution would be open to us, a solution that could enable us to track down the sense of the conceptions of judgment and logical form in the CPR. In applying this argumentative strategy, it is of the utmost importance to insist on the specificity of Kant’s notion of extension, in order to prove its irreducibility to the Port-Royal notion of extension as well as to the modern one. (shrink)
Précis of Supersizing the mind: embodiment, action, and cognitive extension (Oxford University Press, NY, 2008) Content Type Journal Article DOI 10.1007/s11098-010-9597-x Authors Andy Clark, Philosophy, University of Edinburgh, Dugald Stewart Building, 3 Charles Street, Edinburgh, EH8 9AD Scotland (UK) Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116.
This paper explores Leibniz's conception of body and extension in the 1680s and 1690s. It is argued that one of Leibniz's central aims is to undermine the Cartesian conception of extended substance, and replace it with a conception on which what is basic to body is force. In this way, Leibniz intends to reduce extension to something metaphysically more basic in just the way that the mechanists reduce sensible qualities to size, shape and motion. It is also argued that this (...) move is quite distinct from the reduction of body to monads and their appetitions and perceptions, so prominent in his later writings. (shrink)
Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from (...) MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses. (shrink)
The Hypothesis of Extended Cognition holds that the mind need not be constrained within biological boundaries. However, conditions must be provided to set a principled outer limit on cognitive extension, or implausibly many cases will be implicated. I argue that, for the case of extended beliefs at least, such conditions must pay attention to a mental state's causal history, in addition to its current functional poise. Extended resources can house an individual's beliefs, I propose, only if she has taken responsibility (...) for their sources in a suitable way. (shrink)
New light is shed on Leibniz’s commitment to the metaphysical priority of the intensional interpretation of logic by considering the arithmetical and graphical representations of syllogistic inference that Leibniz studied. Crucial to understanding this connection is the idea that concepts can be intensionally represented in terms of properties of geometric extension, though significantly not the simple geometric property of part-whole inclusion. I go on to provide an explanation for how Leibniz could maintain the metaphysical priority of the intensional interpretation while (...) holding that logically the intensional and the extensional stand in strictly inverse relation to each other. (shrink)
In his “The Foundations of Mathematics”, Ramsey attempted to marry the Tractarian idea that all logical truths are tautologies and vice versa, and the logicism of the Principia. In order to complete his project, Ramsey was forced to introduce propositional functions in extension (PFEs): given Ramsey's definitions of 1 and 2, without PFEs even the quantifier-free arithmetical truth that 1 ≠ 2 is not a tautology. However, a number of commentators have argued that the notion of PFEs is incoherent. This (...) response was first given by Wittgenstein but has been best developed by Sullivan. While I agree with Wittgenstein and Sullivan's common conclusion, I believe that even the most compelling of Sullivan's arguments is importantly mistaken and that Wittgenstein's remarks are too opaque to be left as the end of the matter. In this article I uncover the fault in Sullivan's argument and present an alternative criticism of PFEs which is Wittgensteinian in spirit without being too mystifying. (shrink)
The worst possible way to resolve this issue is to leave it up to individual choice. There is no known social good coming from the conquest of death (Bailey, 1999). - Daniel Callahan Dramatically extending the human lifespan seems increasingly possible. Many bioethicists object that life-extension will have Malthusian consequences as new Methuselahs accumulate, generation by generation. I argue for a Life-Years Response to the Malthusian Objection. If even a minority of each generation chooses life-extension, denying it to them deprives (...) them of many years of extra life, and their total extra life-years are likely to exceed the total life-years of a majority who do not want life-extension. This is a greater harm to those who want extended life than the Malthusian harms to those who refuse extended life, both because losing an extra year of life is worse than enduring a year of Malthusian conditions, and because the would-be Methuselahs have more life-years at stake. Therefore, even if life-extension seems likely to cause severe overcrowding and resource shortages, that threat is not sufficient to justify society in restricting the development or availability of life-extension. (shrink)
Although the notion of logical object plays a key role in Frege's foundational project, it has hardly been analyzed in depth so far. I argue that Marco Ruffino's attempt to fill this gap by establishing a close link between Frege's treatment of expressions of the form ‘the concept F’ and the privileged status Frege assigns to extensions of concepts as logical objects is bound to fail. I argue, in particular, that Frege's principal motive for introducing extensions into his logical theory (...) is not to be able to make in-direct statements about concepts, but rather to define all numbers as logical objects of a fundamental kind in order to ensure that we have the right cognitive access to them qua logical objects via Axiom V. Contrary to what Ruffino claims, reducibility to extensions cannot be the ‘ultimate criterion’ for Frege of what is to be regarded as a logical object. (shrink)
The purpose of this study is to expand our understanding of the factors that influence ethical behavioral intentions of public accountants. Recent scandals have dominated the news and have caused legislators, regulators and the public to question the role of the accounting profession. Legislative changes have brought about major structural changes in the profession and continued scrutiny will surely lead to further changes. Thus, developing an understanding of the personal and contextual factors that influence ethical decisions is critical. An extension (...) of the theory of planned behavior [Ajzen, I.: 1985, Action Control-From Cognition to Behavior (Springer, Heidelberg)], the model used in this study examined the influence of personal, social and organizational factors on ethical intentions. Specifically, the individual level model tested direct effects of attitudes, subjective norms, perceived behavioral control, moral sensitivity and ethical climate. Professionals from five accounting firms completed a survey that measured responses to ethical dilemmas related to the public accounting domain. To minimize the potential impact of common method bias, the survey instrument was administered in two phases. Hypotheses were evaluated using a structural modeling technique, partial least squares. Results show strong support for a direct relationship between attitudes and ethical intentions. The proposed direct effect of subjective norms was not supported. However, a significant relationship between subjective norms and attitudes was found. Professionals’ attitudes towards ethical issues clearly influence intentions. Moreover, this study illustrates the potential influence of social factors in attitude formation. Future research should explore the factors in the public accounting domain that most strongly influence attitude formation. This study suggests that the theory of reasoned action offers a useful framework for exploring these issues. (shrink)
In trying to characterize the relationship between psychology and neuroscience, the trend has been to argue that reductionism does not work without suggesting a suitable substitute. I offer explanatory extension as a good model for elucidating the complex relationship among disciplines which are obviously connected but which do not share pragmatic explanatory features. Explanatory extension rests on the idea that one field can "illuminate" issues that were incompletely treated in another. In this paper, I explain how this "illumination" would work (...) between psychology and neuroscience. (shrink)
Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influences other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within (...) a broadly Dummettian framework. The conclusions are mostly negative: Dummett's views on analyticity and the logical/non-logical boundary leave little room for logicism. Dummett's considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true - as the logicist contends - then there is tension between this conservation requirement and the ontological commitments of arithmetic. It follows from Dummett's manifestation requirements that if a sentence S is composed entirely of logical terminology, then there is a formal deductive system D such that S is analytic, or logically true, if and only if S is a theorem of D. There is a deep conflict between this result and the essential incompleteness, or as Dummett puts it, the indefinite extensibility, of arithmetic truth. (shrink)
This dissertation extends John Rawls’s mature theory of justice out to address the environmental challenges that citizens of liberal democracies now face. Specifically, using Rawls’s framework of political liberalism, I piece together a theory of procedural justice to be applied to a constitutional democracy. I show how citizens of pluralistic democracies should apply this theory to environmental matters in a four stage contracting procedure. I argue that, if implemented, this extension to Rawls’s theory would secure background environmental justice. I explain (...) why the theory can be viewed as a partially specified political conception of environmental pragmatism, and how it relates to public environmental policy and discourse. While the framework is anthropocentric, it is one that reasonable non-anthropocentrists can endorse. Using this theory, I argue that liberal democracies must take measures to secure basic environmental rights for all presently existing and future citizens. Measures must also be in place to secure a minimum of social goods (including environmental goods) that guarantees that all citizens (present and future) can exercise their basic rights and liberties. Moreover, disparities in environmental goods should only be tolerated if they arise in accord with Rawls’s principle of fair equality of opportunity. I discuss carbon taxes, as well as carbon allocation trading schemes. I also argue that free democracies should employ precautionary reasoning when attempting to meet the demands of background environmental justice. (shrink)
In contrast to rigid conceptions of nonhuman animals, several philosophers have put forth ideas that suggest a more flexible and extended vision of other animals. In articulating the condition of humans in the world, philosophers have referenced ideas that necessarily bring other beings in common with humanity. Significantly, conceptions of movement and biological transformation have played a central role in these ruminations, thereby suggesting the importance of geographical variables in human/nonhuman relations. By drawing out the connections between these perspectives, this (...) paper outlines an ethics of extension. (shrink)
Patrick Grim has put forward a set theoretical argument purporting to prove that omniscience is an inconsistent concept and a model theoretical argument for the claim that we cannot even consistently define omniscience. The former relies on the fact that the class of all truths seems to be an inconsistent multiplicity (or a proper class, a class that is not a set); the latter is based on the difficulty of quantifying over classes that are not sets. We first address the (...) set theoretical argument and make explicit some ways in which it depends on mathematical Platonism. Then we sketch a non Platonistic account of inconsistent multiplicities, based on the notion of indefinite extensibility, and show how Grim’s set theoretical argument could fail to be conclusive in such a context. Finally, we confront Grim’s model theoretical argument suggesting a way to define a being as omniscient without quantifying over any inconsistent multiplicity. (shrink)
The purpose of this paper is to assess the prospects for a neo-logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): PQ[Ext(P) = Ext(Q) [(BAD(P) & BAD(Q)) x(Px Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’. 1 Background: what and why? (...) 2 Framework 3 GOOD candidates, indefinite extensibility 4 The framework of (RV) alone, or almost alone 5 The axioms 6 Brief closing. (shrink)
During the past decade, the so-called “hypothesis of cognitive extension,” according to which the material vehicles of some cognitive processes are spatially distributed over the brain and the extracranial parts of the body and the world, has received lots of attention, both favourable and unfavourable. The debate has largely focussed on three related issues: (1) the role of parity considerations, (2) the role of functionalism, and (3) the importance of a mark of the cognitive. This paper critically assesses these issues (...) and their interconnections. Section 1 provides a brief introduction. Section 2 argues that some of the most prominent objections against the appeal to parity considerations fail. Section 3 shows that such considerations are nevertheless unsuitable as an argument for cognitive extension. First, the actual argumentative burden is carried by an underlying commitment to functionalism, not by the parity considerations themselves. Second, in the absence of an independently motivated mark of the cognitive, the argument based on parity considerations does not get off the ground, but given such a mark, it is superfluous. Section 4 argues that a similar dilemma arises for the attempt to defend cognitive extension by a general appeal to functionalism. Unless it can be independently settled what it is for a process to be cognitive, functionalism itself will be undermined by the possibility of cognitive extension. Like parity considerations, functionalism is thus either unable to support cognitive extension or superfluous. Hence, nothing short of the specification of an appropriate mark of the cognitive that can be fulfilled not only by intracranial but also by extended processes will do as an argument for cognitive extension. (shrink)
The Anselmian Thesis is the thesis that God is that than which nothing greater can be thought. In this paper, I argue that such a notion of God is incoherent due to greatness being indefinitely extensible: roughly, for any great being that can be, there is another one that is greater, so there cannot be a being than which nothing greater can be. Someone will say that it is impossible to produce the best, because there is no perfect creature, and (...) that it is always possible to produce one which would be more perfect.’ G.W. Leibniz. Theodicy. Edited by A. Farrer (Chicago, IL: Open Court, 1985. Pp. 249.). (shrink)
Contemporary philosophers and bioethicists argue that life extension is bad for the individual. According to the agency objection to life extension, being constrained as an agent adds to the meaningfulness of human life. Life extension removes constraints, and thus it deprives life of meaning. In the paper, I concede that constrained agency contributes to the meaningfulness of human life, but reject the agency objection to life extension in its current form. Even in an extended life, decision-making remains constrained, and many (...) obstacles to the fulfilment of an agent’s goals are preserved. Agents with longer lives are also presented with new challenges: for instance, it might be harder for them to avoid chronic boredom, and sustain their motivation to act in the pursuit of their goals. Although objections from agency and boredom are often used in combination to support the view that a much longer life is likely to bring misery or become meaningless, I argue that the acceptance of the boredom objection undermines the persuasiveness of the agency objection. (shrink)
We develop a logical system that captures two different interpretations of what extensive games model, and we apply this to a long-standing debate in game theory between those who defend the claim that common knowledge of rationality leads to backward induction or subgame perfect (Nash) equilibria and those who reject this claim. We show that a defense of the claim à la Aumann (1995) rests on a conception of extensive game playing as a one-shot event in combination with a principle (...) of rationality that is incompatible with it, while a rejection of the claim à la Reny (1988) assumes a temporally extended, many-moment interpretation of extensive games in combination with implausible belief revision policies. In addition, the logical system provides an original inductive and implicit axiomatization of rationality in extensive games based on relations of dominance rather than the usual direct axiomatization of rationality as maximization of expected utility. (shrink)
The main task of this paper is to understand if and how static images like photographs can represent and/or depict temporal extension (duration). In order to do this, a detour will be necessary to understand some features of the nature of photographic representation and depiction in general. This important detour will enable us to see that photographs (can) have a narrative content, and that the skilled photographer can 'tell a story' in a very clear sense, as well as control and (...) guide the attention of the spectator of the photograph. The understanding and defence of this claim is a secondary aim of this paper, and it will then allow us to provide a good treatment of the particular case of photographic representation and depiction of temporal extension. (shrink)
Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ? -rule in his 1931 paper ?Die Grundlegung der elementaren Zahlenlehre?. The main question we discuss here is whether the finitist (meta-)mathematician would (...) be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ?Die Widerspruchsfreiheit der reinen Zahlentheorie? (1936). As to Hilbert and Bernays 1939, we end on a ?critical? note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original finitism have fallen victim to sheer proof-theoretic pragmatism. (shrink)
In this paper, I address both the interpretive and philosophical issues concerning prime matter. My aim is to show that a philosophically interesting account of prime matter can be articulated that strongly coheres with, even if it is not necessitated by, Aristotle’s texts. In articulating the interpretation, I first examine a view defended by both Richard Sorabji and Robert Sokolowski according to which prime matter is extension. Such a view, I argue, is problematic for a number of reasons. Nonetheless, it (...) provides a convenient starting point for the view I defend according to which prime matter is intimately linked to, though not identical with, extension. (shrink)
Sensory substitution devices provide through an unusual sensory modality (the substituting modality, e.g., audition) access to features of the world that are normally accessed through another sensory modality (the substituted modality, e.g., vision). In this article, we address the question of which sensory modality the acquired perception belongs to. We have recourse to the four traditional criteria that have been used to define sensory modalities: sensory organ, stimuli, properties, and qualitative experience (Grice, 1962), to which we have added the criteria (...) of behavioral equivalence (Morgan, 1977), dedication (Keeley, 2002), and sensorimotor equivalence (O’Regan & Noe¨, 2001). We discuss which of them are fulfilled by perception through sensory substitution devices and whether this favors the view that perception belongs to the substituting or to the substituted modality. Though the application of a number of criteria might be taken to point to the conclusion that perception with a sensory substitution device belongs to the substituted modality, we argue that the evidence leads to an alternative view on sensory substitution. According to this view, the experience after sensory substitution is a transformation, extension, or augmentation of our perceptual capacities, rather than being something equivalent or reducible to an already existing sensory modality. We develop this view by comparing sensory substitution devices to other ‘‘mind-enhancing tools’’ such as pen and paper, sketchpads, or calculators. An analysis of sensory substitution in terms of mind-enhancing tools unveils it as a thoroughly transforming perceptual experience and as giving rise to a novel form of perceptual interaction with the environment. (shrink)
A semantical proof of Craig's interpolation theorem for the intuitionistic predicate logic and some intermediate prepositional logics will be given. Our proof is an extension of Henkin's method developed in [4]. It will clarify the relation between the interpolation theorem and Robinson's consistency theorem for these logics and will enable us to give a uniform way of proving the interpolation theorem for them.
George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...) fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)
Ninety-one right brain-damaged patients with left neglect and 43 right brain-damaged patients without neglect were asked to extend horizontal segments, either left- or rightward, starting from their right or left endpoints, respectively. Earlier experiments based on similar tasks had shown, in left neglect patients, a tendency to overextend segments toward the left side. This seemingly paradoxical phenomenon was held to undermine current explanations of unilateral neglect. The results of the present extensive research demonstrate that contralesional overextension is also evident in (...) most right brain-damaged patients without contralesional neglect. Furthermore, they show that in a minority of left neglect patients, the opposite behavior, i.e., right overextension can be found. The paper also reports the results of correlational analyses comprising the parameters of line-extension, line-bisection, and cancellation tasks, as well as the parameters relative to the Milner Landmark Task, by which a distinction is drawn between perceptual and response biases in unilateral neglect. A working hypothesis is then advanced about the brain dysfunction underlying neglect and an attempt is made at finding an explanation of neglect and the links between the mechanisms of space representation and consciousness through the study of the changes induced by unilateral brain lesions in the characteristics of space-coding neurons. Abbreviations: C, control group;GN+91,full group of neglect patients;GN+27,group of neglect patients with relative left overextension;GN+14,group of neglect patients with relative right overextension;GN-43,full group of non-neglect patients;GN-9,group of non-neglect patients with relative left overextension; H canc, H cancellation task; LE, left extension; LE/RE, ratio of left-right extension; N+, neglect patients; N-, non-neglect patients; PB Land-M, perceptual bias on Landmark motor task; PB Land-V, perceptual bias on Landmark verbal task; RB Land-M, response bias on Landmark motor task; RB Land-V, response bias on Landmark verbal task; RE, right extension. (shrink)
Taking our inspiration from modal correspondence theory, we present the idea of correspondence analysis for many-valued logics. As a benchmark case, we study truth-functional extensions of the Logic of Paradox (LP). First, we characterize each of the possible truth table entries for unary and binary operators that could be added to LP by an inference scheme. Second, we define a class of natural deduction systems on the basis of these characterizing inference schemes and a natural deduction system for LP. Third, (...) we show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics. (shrink)
Extensive measurement theory is developed in terms of theratio of two elements of an arbitrary (not necessarily Archimedean) extensive structure; thisextensive ratio space is a special case of a more general structure called aratio space. Ratio spaces possess a natural family of numerical scales (r-scales) which are definable in non-representational terms; ther-scales for an extensive ratio space thus constitute a family of numerical scales (extensive r-scales) for extensive structures which are defined in a non-representational manner. This is interpreted as involving (...) arelational theory of quantity which contrasts in certain respects with thequalitative theory of quantity implicit in standard representational extensive measurement theory. The representational properties of extensiver-scales are investigated, and found to coincide withweak extensive measurement in the sense of Holman. This provides support for the thesis (developed in a separate paper) that weak extensive measurement is a more natural model of actual physical extensive scales than is the standard model using strong extensive measurement. Finally, the present apparatus is applied to slightly simplify the existing necessary and sufficient conditions for strong extensive measurement. (shrink)
Matthias Schirn has argued on a number of occasions against the interpretation of Frege's ``objects of a quite special kind'' (i.e., the objects referred to by names like `the concept F') as extensions of concepts. According to Schirn, not only are these objects not extensions, but also the idea that `the concept F' refers to objects leads to some conclusions that are counter-intuitive and incompatible with Frege's thought. In this paper, I challenge Schirn's conclusion: I want to try and argue (...) that the assumption that `the concept F' refers to the extension of F is entirely consistent with Frege's broader views on logic and language. I shall examine each of Schirn's main arguments and show that they do not support his claim. (shrink)
A rigorous extension of the full Lorentz group is found which is parameterized by interframe velocities v(t) and which reduces to Special Relativity for acceleration-free cases and to Galilean relativity for low velocity cases. Full group properties are exhibited. Four-momentum is defined and particle masses are shown to be invariants. Four-force is introduced and pseudoforces are shown to enter the equations of particle dynamics. Maxwell's equations are shown to take on pseudocurrent terms in accelerating frames. A four-vector Green function solution (...) to the modified Maxwell equations is presented. Finally, a discussion is offered concerning philosophical questions such as the operational definition of time. (shrink)
The notion of perfect recall in extensive games was introduced by Kuhn (1953), who interpreted it as "equivalent to the assertion that each player is allowed by the rules of the game to remember everything he knew at previous moves and all of his choices at those moves''. We provide a characterization and axiomatization of perfect recall based on two notions of memory: (1) memory of past knowledge and (2) memory of past actions.
We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.
Extension of the system that includes the key substrates for sensation, perception, emotion, volition, and cognition, and all representational sources for cognition, supports the view that there is an extended mind and an extended body. These intellectual views can be made practical in a humanist system based on extensions and in religious systems based on extensions. Independently, there is also an institutional extension of secularism. Hence, I maintain, there are five principal forms of extension.
Scientists, bioethicists, and policy makers are currently engaged in a contentious debate about the scientific prospects and morality of efforts to increase human longevity. Some demographers and geneticists suggest that there is little reason to think that it will be possible to significantly extend the human lifespan. Other biodemographers and geneticists argue that there might well be increases in both life expectancy and lifespan. Bioethicists and policy makers are currently addressing many of the ethical, social, and economic issues raised by (...) life extension research. However, the emphasis on philosophical argument supporting or condemning efforts to increase human longevity means that much less attention is currently being given to the factors that might play a role in generating interest in efforts to increase human longevity. This analysis considers three factors that might play a role in heightening public interest in efforts to develop biomedical technologies capable of retarding or reversing aging processes. While discussions of life extension research can seem quite futuristic and impractical, there are some powerful existential factors that might well generate considerable public support for life extension strategies if effective biomedical interventions emerge. Rather than providing philosophical justifications supporting or condemning efforts to increase human longevity, this essay seeks to promote a better understanding of the factors generating contemporary interest in prolonging life and postponing death. (shrink)
I present here a modal extension of T called KTLM which is, by several measures, the simplest modal extension of T yet presented. Its axiom uses only one sentence letter and has a modal depth of 2. Furthermore, KTLM can be realized as the logical union of two logics KM and KTL which each have the finite model property (f.m.p.), and so themselves are complete. Each of these two component logics has independent interest as well.
A semantics may be compositional and yet partial, in the sense that not all well-formed expressions are assigned meanings by it. Examples come from both natural and formal languages. When can such a semantics be extended to a total one, preserving compositionality? This sort of extension problem was formulated by Hodges, and solved there in a particular case, in which the total extension respects a precise version of the fregean dictum that the meaning of an expression is the contribution it (...) makes to the meanings of complex phrases of which it is a part. Hodges' result presupposes the so-called Husserl property, which says roughly that synonymous expressions must have the same category. Here I solve a different version of the compositional extension problem, corresponding to another type of linguistic situation in which we only have a partial semantics, and without assuming the Husserl property. I also briefly compare Hodges' framework for grammars in terms of partial algebras with more familiar ones, going back to Montague, which use many-sorted algebras instead. (shrink)
Important scientific, ethical and sociological debates are emerging over the trans-humanist goal to achieve therapeutic treatments to âcureâ the debilitation of age-related illness and extend the healthy life span of individuals through interventive biogerontological research . The scientific and moral discourses surrounding this contentious scientific field are mapped out, followed by a normative argument favouring âstrongâ deliberative democratic control of human life extension research. This proposal incorporates insights from constructive and participatory technology assessment, upstream public engagement and back-casting analysis; to (...) outline a programme of participatory approaches to encourage two-way dialogue between scientific and citizen perspectives, and foster the long-term deliberative democratic governance of this developing field. (shrink)
Life-extension was the focus for the 10th annual Congress of the International Association of Biomedical Gerontology, held last September at Cambridge University. This scientific convention included a panel of several bioethicists, including Art Caplan, John Harris, and others. The presentations on the ethics of life-extension are reviewed here.
. We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle’s square under its modal form has the following four vertices: A is □α, E is , I is and O is , where α is a logical formula and □ is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether is involutive (...) or not) modal logic. [3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of by the logical operations , under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle’s square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures. (shrink)
Abstract Objections to life extension often focus on its effects for individual well-being. Prominent amongst these concerns is the possibility that life extending technologies will extend lifespan without preventing the ageing of the mind. Writers on the subject express the fear that life extending drugs will keep us physically youthful whilst our minds decay, succumbing to dementia, boredom, and loneliness. Generally these fears remain speculative, in part due to the absence of genuine life extending technologies. In this paper, however, I (...) examine the implications of an existing life extension technology. Caloric restriction (CR) and drugs that mimic its effects, such as rapamycin, metformin and resveratrol have been shown to increase average and maximum lifespan in a wide variety of organisms, and seem likely to do so in humans. Moreover, some CR mimetic drugs (CRMs) are already widely used. This means that they present a pressing test case for fears about mental ageing in an extended life. Misgivings about mental ageing can be divided into biomedical factors such as the likelihood of brain ageing, and psychological factors such as loss of meaning and boredom. I argue that studies of CR suggest that brain ageing will be beneficially slowed. However, it is less clear that deleterious aspects of psychological ageing can be similarly retarded. I argue that this reduces the desirability of life extension unless major social changes can be made. (shrink)
If we are to understand why psychoanalysis extends ordinary psychology in the precise ways that it does, we must take account of the existence of, and the interplay between, two distinct kinds of explanatory concern: functional and idiographic. The form and content of psychoanalytic explanation and its unusual methodology can, at least in part, be viewed as emerging out of Freud's attempt to reconcile these two types of explanatory concern. We must also acknowledge the role of the background theoretical context (...) that shapes Freud's functional thinking about the mind. A neglect of the role of the background theory in shaping the extension of ordinary psychology leaves us with puzzles about the nature and direction of the psychoanalytic extension and gives rise to an unbelievable history of psychoanalysis. (shrink)
We are encouraged that the majority of commentators endorse our evolutionary framework for studying culture, and several suggest extensions. Here we clarify our position, dwelling on misunderstandings and requests for exposition. We reiterate that using evolutionary biology as a model for unifying the social sciences within a single synthetic framework can stimulate a more progressive and rigorous science of culture. (Published Online November 9 2006).
The hypothesis of extended cognition holds that mental states and processes need not be wholly contained within biological confines. Yet the theory is plausible, and informative, only when it can set principled outer limits upon cognitive extension: it should not permit unrestricted expansion of the mental into the material environment. I argue that true cognitive extension occurs only when the subject takes responsibility for the contribution made by a non-neural resource, in a manner that can be illuminated by appeal to (...) theoretical resources from contemporary virtue epistemology. (shrink)
Strictly speaking, intuitionistic logic is not a modal logic. There are, after all, no modal operators in the language. It is a subsystem of classical logic, not [like modal logic] an extension of it. But... (thus Fitting, p. 437, trying to justify inclusion of a large chapter on intuitionist logic in a book that is largely about modal logics).
We show that there are continuum many different extensions of SCI (the basic theory of non-Fregean propositional logic) that lie below WF (the Fregean extension) and are closed under substitution. Moreover, continuum many of them are independent from WB (the Boolean extension), continuum many lie above WB and are independent from WH (the Boolean extension with only two values for the equality relation), and only countably many lie between WH and WF.
Interpreting the diamond of modal logic as the derivative, we present a topological canonical model for extensions of K4 and show completeness for various logics. We also show that if a logic is topologically canonical, then it is relationally canonical.
This work is part of a wider investigation into lattice-structured algebras and associated dual representations obtained via the methodology of canonical extensions. To this end, here we study lattices, not necessarily distributive, with negation operations. We consider equational classes of lattices equipped with a negation operation ¬ which is dually self-adjoint (the pair (¬,¬) is a Galois connection) and other axioms are added so as to give classes of lattices in which the negation is De Morgan, orthonegation, antilogism, pseudocomplementation or (...) weak pseudocomplementation. These classes are shown to be canonical and dual relational structures are given in a generalized Kripke-style. The fact that the negation is dually self-adjoint plays an important role here, as it implies that it sends arbitrary joins to meets and that will allow us to define the dual structures in a uniform way. (shrink)
This paper presents completeness and conservative extension results for the boolean extensions of the relevant logic T of Ticket Entailment, and for the contractionless relevant logics TW and RW. Some surprising results are shown for adding the sentential constant t to these boolean relevant logics; specifically, the boolean extensions with t are conservative of the boolean extensions without t, but not of the original logics with t. The special treatment required for the semantic normality of T is also shown along (...) the way. (shrink)
In both axiomatic theories and the practice of extensive measurement, it is assumed that a series of replicas of any given object can be found. The replicas give rise to a standard series, the "multiples" of the given object. The numerical value assigned to any object is determined, approximately, by comparisons with members of a suitable standard series. This prescription introduces unspecified errors, if the comparison process is somewhat insensitive, so that "replicas" are not really equivalent. In this paper, it (...) is assumed that the comparison process leads only to a semiorder, which allows for such insensitivity. It is shown that, nevertheless, extensive measurement can be carried out, provided that a certain set of (plausible) axioms is valid. Approximate measures, and their limits of error, can be derived from finite sets of semiorder observations. These approximate measures converge to ratio-scale exact measurement. (shrink)
The formal methods of the representational theory of measurement (RTM) are applied to the extensive scales of physical science, with some modifications of interpretation and of formalism. The interpretative modification is in the direction of theoretical realism rather than the narrow empiricism which is characteristic of RTM. The formal issues concern the formal representational conditions which extensive scales should be assumed to satisfy; I argue in the physical case for conditions related to weak rather than strong extensive measurement, in the (...) sense of Holman 1969 and Colonius 1978. The problem of justifying representational conditions is addressed in more detail than is customary in the RTM literature; this continues the study of the foundations of RTM begun in an earlier paper. The most important formal consequence of the present interpretation of physical extensive scales is that the basic existence and uniqueness properties of scales (representation theorem) may be derived without appeal to an Archimedean axiom; this parallels a conclusion drawn by Narens for representations of qualitative probability. It is concluded that there is no physical basis for postulation of an Archimedean axiom. (shrink)
Over recent years, various semantics have been proposed for dealing with updates in the setting of logic programs. The availability of different semantics naturally raises the question of which are most adequate to model updates. A systematic approach to face this question is to identify general principles against which such semantics could be evaluated. In this paper we motivate and introduce a new such principle the refined extension principle. Such principle is complied with by the stable model semantics for (single) (...) logic programs. It turns out that none of the existing semantics for logic program updates, even though generalisations of the stable model semantics, comply with this principle. For this reason, we define a refinement of the dynamic stable model semantics for Dynamic Logic Programs that complies with the principle. (shrink)
In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.
In a recent volume of this journal, L. Angel ([2002]) proposed a collision mechanics leading to such strange results as the possibility that a particle may be in several places at the same time, or the existence of unprepared spatially-separated correlations. I will here show that neither of these results follows from his theory or, if it does, the theory, contrary to what Angel claims, is not a plausible extension of Newtonian collision dynamics. No bilocation No quantum leap No unprepared (...) spatially-separated correlations. (shrink)
We propose a new, rather simple and short proof of Kripke-completeness for the predicate variant of Dummett's logic. Also a family of Kripke-incomplete extensions of this logic that are complete w.r.t. Kripke frames with equality (or equivalently, w.r.t. Kripke sheaves [8]), is described.
Bloom provides a masterful synthesis of recent advances in word-learning, placing them within the framework of abiding theoretical issues. I will augment and challenge his approach by underscoring the significance of word extension for questions concerning (a) the origin and evolution of infants' expectations, and (b) domain-specificity in word-learning.
We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly-̰Σ₃¹ absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for Σ₃¹ absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact (...) equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions. (shrink)
This paper examines the view held by Francesco Piccolomini (1523-1607) on the relation between prime matter and extension. In his discussion of prime matter in the Libri ad scientiam de natura attinentes Piccolomini develops a theory of prime matter that incorporates crucial elements of the viewpoint adhered to by the Neoplatonist Simplicius. The originality of Piccolomini's undertaking is highlighted by contrasting it with the ideas found in Jacopo Zabarella's De rebus naturalibus . The case of Piccolomini shows that, in order (...) to classify early modern metaphysical theories of prime matter, the category `prime matter as sheer dimensionality' is indispensable. (shrink)
The paper studies first order extensions of classical systems of modal logic (see (Chellas, 1980, part III)). We focus on the role of the Barcan formulas. It is shown that these formulas correspond to fundamental properties of neighborhood frames. The results have interesting applications in epistemic logic. In particular we suggest that the proposed models can be used in order to study monadic operators of probability (Kyburg, 1990) and likelihood (Halpern-Rabin, 1987).
If the language is extended by new individual variables, in classical first order logic, then the deduction system obtained is a conservative extension of the original one. This fails to be true for the logics with infinitary predicates. But it is shown that restricting the commutativity of quantifiers and the equality axioms in the extended system and supposing the merry-go-round property in the original system, the foregoing extension is already conservative. It is shown that these restrictions are crucial for an (...) extension to be conservative. The origin of the results is algebraic logic. (shrink)
In [1], D. W. Hart and C. Mcginn considered two logics A1 and A2. These logics embody part of a tradition about a priori knowledge and necessity. They proved that A2 is a conservative extension of a well-known modal logic S5 but left the problem whether A1 is a conservative extension of S4 open. In this note, we shall show that A1 is not a conservative extension of S4 but of S5, and also correct an inadequate proof.
Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free (...) formula. Thus it would seem that any detour going via formulas with conjunction can be avoided, which raises a puzzle: how is this consistent with the point from Section 1? Sections 3 and 4 raise and discuss this puzzle. In fact, the puzzle turns out on closer inspection not to be so puzzling after all, but it does serve as a convenient centrepiece around which to organize a discussion of the phenomenon illustrated by the Bull?Prior example. Section 5 notes that Prior's observation can be extended to the case of the result of adding disjunction to Bull's logic, while Section 6 includes some further remarks aimed at diagnosing one source of possible residual puzzlement. A subtext of our discussion?spanning several of the notes?is that this work by Bull and Prior has been overlooked, their results having to be rediscovered, by many algebraists and logicians in more recent years. (shrink)
This paper introduces a generalization of Reiter’s notion of “extension” for default logic. The main difference from the original version mainly lies in the way conflicts among defaults are handled: in particular, this notion of “general extension” allows defaults not explicitly triggered to pre-empt other defaults. A consequence of the adoption of such a notion of extension is that the collection of all the general extensions of a default theory turns out to have a nontrivial algebraic structure. This fact has (...) two major technical fall-outs: first, it turns out that every default theory has a general extension; second, general extensions allow one to define a well-behaved, skeptical relation of defeasible consequence for default theories, satisfying the principles of Reflexivity, Cut, and Cautious Monotonicity formulated by D. Gabbay. (shrink)
That Kuhn is mistaken in drawing a sharp line between normal and revolutionary phases of science is shown by re-examining the role of models in extending theories to new phenomenal domains. In the light of this revision of the role of models, theory extension, which Kuhn includes in normal science, is shown to be continuous with theory replacement, which Kuhn includes in revolutionary science. Both involve language changes and the 'gestalt switches' associated with revolutionary science. These characteristics cannot be used (...) to demarcate the two phases as Kuhn seems to think. (shrink)
Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (R, V) where $\mathscr{R} \models T$ and V ≠ R is the convex hull of an elementary substructure of R. We deduce (...) that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs (R, N) with R a model of T and N a proper elementary substructure that is Dedekind complete in R. We deduce that the theory of such "tame" pairs is complete. (shrink)
Stakeholder engagement is a crucial conceptof extension education. Engagement expressesdemocratic values of the land-grant mission byproviding opportunities for stakeholders to influenceprogram planning, including setting the agenda andnegotiating resource allocations. In practice, theconcept of engagement guides the formation ofpartnerships among extension, communities, industry,and government. In the area of sustainableagriculture, however, stakeholders may conflict,presenting challenges to the engagement process.Results from a study of a Canadian sustainableagriculture program, produced using culturalanthropology and participatory action research, detailchallenges of the engagement process that led toreconstruction of (...) a farmer-extension partnership.Notable in the early phase of the reconstructionprocess were critical reflection, stakeholder forums,exclusion through caucusing, and coalition building.An argument for a neo-pragmatist view provides atheoretical basis for understanding counterintuitivedimensions of engagement revealed by the study. (shrink)
This article discusses the impact of Descartes’s substance-dualism on his account of discursive reason. Taking the presentation of deduction in the Rules as a paradigmatic case of thought’s extension and movement in time, I analyze the relation between intuitive and discursive understanding and that between intellect and imagination. I focus specifically on the mediation of corporeal impressions and of intellectual ideas by ingenium. As intellectual, ingenium is a faculty of understanding; as joining with phantasia, ingenium has access to corporeal affections, (...) images, and memory. Deduction involves both of these aspects of ingenium, and Descartes’s dualism complicates efforts to clarify the operations and nature of ingenium. Thus the dynamics of dualistic psychology account for some of the limitations of deduction in particular and discursive rationality in general. (shrink)
In this paper we will study the properties of the least extension n() of a given intermediate logic by a strong negation. It is shown that the mapping from to n() is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n (). This summarizes results that can be found already in [13,14] and [4]. Furthermore, we determine the structure of (...) the lattice of extensions of n(LC). (shrink)
Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms (...) of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructed. (shrink)
Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let M be a countable transitive model of ZF. The method of forcing extends M to another model M[ G] of ZF (a "generic extension"). If the axiom of choice holds in M it also holds in M[ G], that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive (...) an application to Boolean toposes. (shrink)
We study extensions of Presburger arithmetic with a unary predicate R and we show that under certain conditions on R, R is sparse (a notion introduced by A. L. Semenov) and the theory of $\langle\mathbb{N}, +, R\rangle$ is decidable. We axiomatize this theory and we show that in a reasonable language, it admits quantifier elimination. We obtain similar results for the structure $\langle\mathbb{Q},+, R\rangle$.
We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B V, thenA×B is a homomorphic image of some well-connected algebra ofV.We prove:• each varietyV of Nelson algebras with PQWC lies in the fibre –1(W) for some varietyW of Heyting algebras having PQWC, • for any varietyW of Heyting algebras with PQWC the least and the (...) greatest varieties in –1(W) have PQWC, • there exist varietiesW of Heyting algebras having PQWC such that –1(W) contains infinitely many varieties (of Nelson algebras) with PQWC. (shrink)
In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic and show that LZF is a conservative extension of ZF– i.e., the Zermelo-Fraenkel set theory without the axiom of regularity. We formulate LZF as a sequent calculus with abstraction terms and prove the partial cut-elimination theorem for it. The cut-elimination result ensures the subterm property for those formulas which contain only terms corresponding to sets in ZF–. This implies that LZF is (...) a conservative extension of ZF– and therefore the former is consistent relative to the latter. (shrink)
In this paper, we prove the correspondence between complete extensions in abstract argumentation and 3-valued stable models in logic programming. This result is in line with earlier work of [6] that identified the correspondence between the grounded extension in abstract argumentation and the well-founded model in logic programming, as well as between the stable extensions in abstract argumentation and the stable models in logic programming.
The paper presents predicate logical extensions of some subintuitionistic logics. Subintuitionistic logics result if conditions of the accessibility relation in Kripke models for intuitionistic logic are dropped. The accessibility relation which interprets implication in models for the propositional base subintuitionistic logic considered here is neither persistent on atoms, nor reflexive, nor transitive. Strongly complete predicate logical extensions are modeled with a second accessibility relation, which is a partial order, for the interpretation of the universal quantifier.
This paper suggests a way of formalizing the amount of information that can be conveyed to each player along every possible play of an extensive game. The information given to each player i when the play of the game reaches node x is expressed as a subset of the set of terminal nodes. Two definitions are put forward, one expressing the minimum amount of information and the other the maximum amount of information that can be conveyed without violating the constraint (...) represented by the information sets. Our definitions provide intuitive characterizations of such notions as perfect recall, perfect information and simultanetty. (shrink)
Restricting attention to the class of extensive games defined by von Neumann and Morgenstern with the added assumption of perfect recall, we specify the information of each player at each node of the game-tree in a way which is coherent with the original information structure of the extensive form. We show that this approach provides a framework for a formal and rigorous treatment of questions of knowledge and common knowledge at every node of the tree. We construct a particular information (...) partition for each player and show that it captures the notion of maximum information in the sense that it is the finest within the class of information partitions that satisfy four natural properties. Using this notion of “maximum information” we are able to provide an alternative characterization of the meet of the information partitions. (shrink)
In certain finite extensive games with perfect information, Cristina Bicchieri (1989) derives a logical contradiction from the assumptions that players are rational and that they have common knowledge of the theory of the game. She argues that this may account for play outside the Nash equilibrium. She also claims that no inconsistency arises if the players have the minimal beliefs necessary to perform backward induction. We here show that another contradiction can be derived even with minimal beliefs, so there is (...) no paradox of common knowledge specifically. These inconsistencies do not make play outside Nash equilibrium plausible, but rather indicate that the epistemic specification must incorporate a system for belief revision. Whether rationality is common knowledge is not the issue. (shrink)
This paper states two sets of axioms sufficient for extensive measurement. The first set, like previously published axioms, requires that each of the objects measured must be classifiable as either greater than, or less than, or indifferent to each other object. The second set, however, requires only that any two objects be classifiable as either indifferent or different, and does not need any information about which object is greater. Each set of axioms produces an extensive scale with the usual properties (...) of additivity and uniqueness except for unit. Moreover, the axioms imply Weber's Law: whether two objects are indifferent depends only upon the ratio of their scale values. (shrink)
