We introduce an independence notion for choice functions, which we call ‘epistemic independence’ following the work by De Cooman et al. [17] for lower previsions, and study it in a multivariate setting. This work is a continuation of earlier work of one of the authors [29], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [11]. We obtain the many-to-one independent natural extension in (...) this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less expressive imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [28], which De Bock and De Cooman [2] have called S-independence. We show that neither is more general than the other. (shrink)

We investigate epistemic independence for choice functions in a multivariate setting. This work is a continuation of earlier work of one of the authors [23], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [7]. We obtain the independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we (...) are aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less informative imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [22], which De Bock and De Cooman [1] have called S-independence. We show that neither is more general than the other. (shrink)

The proofs of some results of abstract algebraic logic, in particular of the transfer principle of Czelakowski, assume the existence of so-called natural extensions of a logic by a set of new variables. Various constructions of natural extensions, claimed to be equivalent, may be found in the literature. In particular, these include a syntactic construction due to Shoesmith and Smiley and a related construction due to Łoś and Suszko. However, it was recently observed by Cintula and (...) Noguera that both of these constructions fail in the sense that they do not necessarily yield a logic. Here we show that whenever the Łoś–Suszko construction yields a logic, so does the Shoesmith–Smiley construction, but not vice versa. We also describe the smallest and the largest conservative extension of a logic by a set of new variables and show that contrary to some previous claims in the literature, a logic of cardinality \ may have more than one conservative extension of cardinality \ by a set of new variables. In this connection we then correct a mistake in the formulation of a theorem of Dellunde and Jansana. (shrink)

Transfer theorems are central results in abstract algebraic logic that allow to generalize properties of the lattice of theories of a logic to any algebraic model and its lattice of filters. Their proofs sometimes require the existence of a natural extension of the logic to a bigger set of variables. Constructions of such extensions have been proposed in particular settings in the literature. In this paper we show that these constructions need not always work and propose a wider (...) setting in which they can still be used. (shrink)

The framework of natural deduction is extended by permitting rules as assumptions which may be discharged in the course of a derivation. this leads to the concept of rules of higher levels and to a general schema for introduction and elimination rules for arbitrary n-ary sentential operators. with respect to this schema, (functional) completeness "or", "if..then" and absurdity is proved.

This note calls attention to the fact that the natural extensions of standard logics appear to be an application of the method of inductive generation of logics . One can generalize our observations beyond the range of algebraic logics under suitable conditions on cardinals of certain involved sets.

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $$\kappa $$, where $$\kappa $$ is a regular cardinal. The corresponding new notion is called $$\kappa $$ -filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different $$\kappa $$ -filter pairs give rise to a fixed logic of cardinality $$\kappa $$. To make the (...) question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality $$\kappa $$. Along the way we use $$\kappa $$ -filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair. (shrink)

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)

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)

Darwin’s claim that Natural Selection, through optimization of fitness, explains complex biological design has not yet been properly formalized. Alan Grafen’s Formal Darwinism Project aims at providing such a formalization and at demonstrating that fitness maximization is coherent with results from Population Genetics, usually interpreted as denying it. We suggest that Grafen’s proposal suffers from some limitations linked to its concept of design as optimized fitness. In order to overcome these limitations, we propose a classification of evolutionary facts based (...) on a bi-dimensional complexity space, which adds robustness to fitness as measure of the quality of a design. In this space, each point represents an organismic architecture, and movement between two points would be an evolutionary fact. Natural Selection explains movement along the fitness axis, while non-selective mechanisms explain movement on the robustness axis. Moreover, we propose a thermodynamic metaphor to draw parallelisms between notions of fitness and entropy, robustness and energy, and movement in this space and reversible and irreversible cycles. We argue that this metaphor illustrates different evolutionary processes usually overlooked by proposals of formalization of Darwinian theory, such Grafen’s Formal Darwinism Project. (shrink)

ABSTRACT The emerging discourse concerning the desirability of intervention in senescence to achieve radical life extension for persons has featured some striking blurring in traditional liberal and conservative commitments and positions. This affords an opportunity for re‐evaluation of these same. The canonical conservative view of the intrinsic value of life is re‐examined and found primarily to involve a denial of human prerogative, rather than an active underwriting of the value of life extension. A critique is offered of an attempted argument (...) against aging intervention from a proto‐conservative worry about a purported threat to human nature. Immortality is found to be a red herring, but a revealing one. Further, the classic liberal view is examined and found wanting in terms of the gravity of its own commitment to, and fullness of its account of the value of life, and the value of life extension. An analysis of the liberal conception of personhood is proposed that both defines persons necessarily as processes, and demonstrates the inalienable quality of the value of life extension to persons so defined. (shrink)

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety $${\mathcal {A}}$$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory (...) to show that topological and discrete dualities for $${\mathcal {A}}$$ exist in partnership. (shrink)

This article argues for a different outlook on the concept of extension, especially for the reference of general terms in scientific practice. Scientific realist interpretations of the two predominant theories of meaning, namely Descriptivism and Causal Theory, contend that a stable cluster of descriptions or an initial baptism fixes the extension of a general term such as a natural kind term. This view in which the meaning of general terms is presented as monosemantic and the referents as stable, homogeneous, (...) and unchangeable, however, does not reflect the various practices involved in the investigation of research materials and the related application of general terms in scientific practice. By drawing on the taxonomic diversity, particularly of structure-based classifications in chemical databases, this article illustrates the limited utility of such a concept of extension. Research materials often exhibit a plurality of material dimensions that, within different research contexts, allow for various and often equally significant taxonomic demarcations. In light of this, the extension of a general term cannot be uniquely determined by a supposedly independent nature of the referent but is relative to the model context under which the materials are investigated. This significance and plurality of the model context, I claim, needs to be mirrored in an account of meaning that is supposed to reflect scientific reality. On this account, the aim of this article is to present an alternative perspective on the concept of extension to accommodate the diverse material practices that determine the application of general terms in science. (shrink)

In this chapter I give an account of the epistemology underlying the concept of “extension” in the Mengzi, an early Confucian text written in the fourth century BCE. Mengzi suggests in a conversation with King Xuan of Qi that a solution to the King’s problem of how one comes to act in a kingly manner is that one engages in “extension”. I argue that a long-standing scholarly debate on the exact nature of Mengzian “extension” can be resolved by closely investigating (...) the epistemological assumptions that must be in place for “extension” to be a viable solution to King Xuan’s problem. More specifically, my argument is that knowledge of a certain kind, namely knowing-to, is both necessary and sufficient for extension to take place. In other words, for a person S to extend X, where X is a capacity for action, S at least needs to have knowing-to. (shrink)

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.

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)

We offer a new outlook on the vexed question of the reference of natural-kind terms. Since Kripke and Putnam, there is a widespread assumption that natural-kind terms function just like proper names: they designate their referents directly and they are rigid designators: their reference is unchanged even in worlds in which the referent lacks some or all the properties associated with it in the actual world, and which are useful to us in identifying that referent. There have, however, (...) been heated debates about what should be taken as a natural-kind term. Some challenge the very existence of a separate category of natural-kind terms ; some their being directly referential ; some raise the possibility that direct reference extends to terms beyond those usually assumed to fall under the category, e.g. to ‘polemical’ terms. When these debates turn on the question of what natural kinds are, they take on a strong epistemological or metaphysical dimension. We think the issues can be clarified within the limits of the philosophy of language: by looking into what ranges of general terms are perceived by speakers as rigid designators of natural kinds. The first step to take is to ground the various kinds of semantic externalism in distinct brands of semantic deference. This we define as speakers’ being disposed to use words in line with the norms of their linguistic community and as consenting to being corrected when it is manifest that their use and understanding of a word does not match common practice. When those dispositions are present, speakers defer semantically to something beyond themselves. Here, our focus is on spotting the words for which speakers would defer not to the current usage of the word in the linguistic community, nor to the current experts of the field to which the word pertains, but ultimately to the very nature of the referent of the term. When speakers’ deference conforms to that pattern, we argue, that is evidence that indexical externalism provides the right metasemantic account of how the meaning of the word is determined. In other words, one can say that the word is treated like a natural-kind term. But how can patterns of deference be measured? In an ongoing survey, participants are confronted with conditions that may prompt them to revise certain classificatory statements, e.g. An emu is a bird. Each condition makes salient one of the targets we have identified for deference: the community usage, the experts, the ‘world as it is’. In the condition that seeks to tap into the latter kind of deference, participants are presented with a scenario in which future scientific discoveries result in excluding from the extension of a term certain members currently thought to fall under that extension, e.g. discoveries that require excluding certain species now thought to be birds from the Aves class. The scenario is such that it is clear that the ‘discoveries’ bring us closer to the actual essence, if there is one, of birds. Our reasoning is that, if participants significantly modify their statements in the light of that scenario, they can be taken to ‘defer’ to the nature of the referent, thus vindicating indexical externalism. We test if words not normally assumed to be natural-kind terms – for instance summer, contract or rape – exhibit patterns of deference similar to those for bird. If so, then there’s a case for an extension of indexical externalism beyond the usual set of terms. What would be shown in this way is that speakers have something like realist intuitions with respect to words whose meaning is usually taken to be purely conventional or polemical. We are at the pre-test stage for the survey. We cannot yet report on our results. We should, however, have initial results by March, which, we believe, will enrich the observations made by previous empirical studies. References: Braisby, N., Franks, B. & Hampton, J. 1996. Essentialism, word use, and concepts. Cognition 59, 247-74./Genone, J. & Lombrozo, T. 2012. Concept possession, experimental semantics, and hybrid theories of reference. Philosophical Psychology 25, 717-742./Häggqvist, S. & Wikforss, Å. 2015. Experimental semantics: the case of natural kind terms, in J. Haukioja, Experimental Philosophy of Language, London, Bloomsbury./Jylkkä, J., Railo, H. & Haukioja, J. 2009. Psychological essentialism and semantic externalism: Evidence for externalism in lay speakers’ language use. Philosophical Psychology 22, 37-60./Mallon, R., Machery, E., Nichols, S. & Stich, S. 2009. Against arguments from reference, Philosophy and Phenomenological Research 79, 332-356./ Marconi, D. 1997. Language, Speech, and Communication. Lexical Competence. Cambridge : MIT Press./Moravcsik, J. 2016. Meaning, Creativity, and the Partial Inscrutability of the Human Mind, 2nd ed. Stanford: CSLI./Schroeter, L. & Schroeter, F. 2014. Normative concepts: a connectedness model, Philosophers’ Imprint 14, 1-26./ Wikforss, Å. 2010. Are natural kind terms special?, in H. Beebee and N. Sabbarton-Leary, The Semantics and Metaphysics of Natural Kinds, New York, Routledge, 64-83. (shrink)

Charles Darwin's On the Origin of Species is unquestionably one of the chief landmarks in biology. The Origin (as it is widely known) was literally only an abstract of the manuscript Darwin had originally intended to complete and publish as the formal presentation of his views on evolution. Compared with the Origin, his original long manuscript work on Natural Selection, which is presented here and made available for the first time in printed form, has more abundant examples and illustrations (...) of Darwin's argument, plus an extensive citation of sources. (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 article explores Mencius’ extension of moral feelings and its potential to address a key challenge in cosmopolitan education: how to motivate students to expand their existing affection and obligations towards their family and community to the rest of the world. Rather than strong universalism, a Mencian orientation is aligned with rooted cosmopolitanism that takes into account localised and cultural contexts that underpin, determine and give value to social practices. Mencius’ approach, as argued in this essay, highlights the spontaneous human (...) reactions to ethical situations, innate human goodness, analogical reasoning, moral character and agency. A Confucian form of cosmopolitan education gives emphasis not only to the students’ moral imagination but also their instinctual responses, self-reflection and rationality. Such an education seeks to integrate emotions and reasoning through the utilisation of paradigm scenarios that identify and reinforce the natural compassionate feelings in the learner. (shrink)

Public extension services play a key role in the implementation of strategies for rural development based on the sustainable management of natural resources. However, the sector suffers from restricted financial and human resources. Using experiences from participatory action research, a strategy for rural extension in the Amazon was defined to increase the efficiency and the relevance of external support for local resource users. This strategy considered activities initiated and coordinated by local people. Short-term facilitation visits provided continuous external support (...) for the purpose of establishing locally based planning and learning mechanisms. In collaboration with the municipality of Muaná in the Eastern Amazon, the strategy was tested in two traditional communities – Monte Moriat and Boa Esperança. Both communities recognized as most important the need to reduce their dependence on açaí (Euterpe oleracea Mart.), the principal source of food and income. This forest resource has been overexploited, seriously affecting açaí stands and diminishing the forest benefits fundamental for survival. Two local groups decided to raise chickens as a way to reduce nutritional problems and as an alternative source of income. Supported by monthly, short-term planning and evaluation visits, the groups managed to achieve significant progress in their endeavor. The groups identified (1) the control of decision-making and information, and (2) the absence of external obligations to be the greatest advantages of the development initiative. The study confirmed the potential of participatory strategies for public extension in the Amazon. Special attention was given to realistically define the role of forests for local development. However, drastic measures are necessary to smooth the way for real participation in governmental and non-governmental organizations acting in the Brazilian Amazon. (shrink)

This paper investigates a limitation of the model of belief and knowledge prevailing in mainstream economics, namely the state-space model. Because of its set-theoretic nature, this model has difficulties in capturing the difference between expressions that designate the same object but have different meanings, i.e., expressions with the same extension but different intensions. This limitation generates puzzling results concerning what individuals believe or know about the world as well as what individuals believe or know about what other individuals believe or (...) know about the world. The paper connects these puzzling results to two issues that are relevant for economic theory beyond the state-space model, namely, framing effects and the distinction between the model-maker and agents that appear in the model. Finally, the paper discusses three possible solutions to the limitations of the state-space model, and concludes that the two alternatives that appear practicable also have significant drawbacks. (shrink)

Abstract: This paper provides a phenomenological interpretation of technological and natural childbirth. By using Heidegger’s ontology of technology to think about childbirth I argue that these two types of contemporary childbirth present us with a false dilemma as both reflect the same norms Heidegger associates with modernity, namely order, control, and efficiency. The paper briefly explains Heidegger’s concept of the enframing as the essence of the technological age while focusing on how it helps us to avoid falling into a (...) technophilic or technophobic trap. Although the technophobic approach popularized by Lamaze gained some favor with feminists who saw the increased use of reproductive technology as an extension of patriarchal control over women’s bodies, I argue that this natural birthing technique incorporated order and control in ways that are similar to its technophilic counterpart. In order to move beyond what I call the reproductive enframing, it is necessary to recognize the false dilemma presented by the technological and natural alternatives. (shrink)

The Cooperative Extension System serves as the conduit through which scientific knowledge generated by the 130 land-grant colleges and universities in the United States is translated and delivered directly to its constituents. Since its inception over 100 years ago, Extension has been integral in developing, delivering, and applying cutting-edge knowledge in agriculture and natural resources, youth development, family and consumer sciences, and community and rural development. Today, more than ever, Extension will need to lead the way in building and (...) maintaining sustainable partnerships across disciplines and with organizations at the local, state, and national levels to tackle complex issues considering diminishing resources. Bringing together leading and emerging scholars, this volume discusses how Extension is addressing issues and opportunities relevant to children, youth, families, and communities across the country both now and in the future. Topics include Extension's role in supporting childcare, social media use, entrepreneurship, rural communities, and underserved audiences. (shrink)

This major contribution to the history of philosophy provides the most comprehensive guide to modern natural law theory available, sets out the full background to liberal ideas of rights and contractarianism, and offers an extensive study of the Scottish Enlightenment. The time span covered is considerable: from the natural law theories of Grotius and Suarez in the early seventeenth century to the American Revolution and the beginnings of utilitarianism. After a detailed survey of modern natural law theory, (...) the book focuses on the Scottish Enlightenment and its European and American connections. Knud Haakonssen explains the relationship between natural law and civic humanist republicanism, and he shows the relevance of these ideas for the understanding of David Hume and Adam Smith. The result is a completely revised background to modern ideas of liberalism and communitarianism. (shrink)

Contemporary Natural Philosophy is understood here as a project of the pursuit of the integrated description of reality distinguished by the precisely formulated criteria of objectivity, and by the assumption that the statements of this description can be assessed only as true or false according to clearly specified verification procedures established with the exclusive goal of the discrimination between these two logical values, but not with respect to any other norms or values established by the preferences of human collectives (...) or by the individual choices. This distinction assumes only logical consistency, but not completeness. Completeness (i.e., the feasibility to assign true or false value to all possible statements) is desirable, but may be impossible. This paper is not intended as a comprehensive program for the development of the Contemporary Natural Philosophy but rather as a preparation for such program advocating some necessary revisions and extensions of the methodology currently considered as the scientific method. This is the actual focus of the paper and the reason for the reference to Baconian _idola mentis_. Francis Bacon wrote in _Novum Organum_ about the fallacies obstructing progress of science. The present paper is an attempt to remove obstacles for the Contemporary Natural Philosophy project to which we have assigned the names of the Idols of the Number, the Idols of the Common Sense, and the Idols of the Elephant. (shrink)

Paul Pietroski develops an iconoclastic account of linguistic meaning. Here, I invite him to say more about what it implies about the relations between language, truth, and conceptual content. Readers concerned with securing the objectivity of conceptual thought may be worried about his claims that typical concepts “have no extensions” and that they “fit one another better than they fit the world.” Others might applaud his anti‐extensionalism in natural‐language semantics but fear that his account re‐raises familiar problems about (...) extensions at the level of psychology. (shrink)

We enrich intuitionistic logic with a lax modal operator and define a corresponding intensional enrichment of Kripke models M = (W, , V) by a function T giving an effort measure T(w, u) {} for each -related pair (w, u). We show that embodies the abstraction involved in passing from true up to bounded effort to true outright. We then introduce a refined notion of intensional validity M |= p : and present a corresponding intensional calculus iLC-h which gives a (...) natural extension by lax modality of the well-known G: odel/Dummett logic LC of (finite) linear Kripke models. Our main results are that for finite linear intensional models L the intensional theory iTh(L) = {p : | L |= p : } characterises L and that iLC-h generates complete information about iTh(L).Our paper thus shows that the quantitative intensional information contained in the effort measure T can be abstracted away by the use of and completely recovered by a suitable semantic interpretation of proofs. (shrink)

We enrich intuitionistic logic with a lax modal operator ○ and define a corresponding intensional enrichment of Kripke models M = (W, ⊑, V) by a function T giving an effort measure T(w, u) ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{N}} \cup$$ \end{document} {∞} for each ⊑-related pair (w, u). We show that ○ embodies the abstraction involved in passing from “ϕ true up to bounded effort” to “ϕ true outright”. We then introduce a refined (...) notion of intensional validityM |= p : ϕ and present a corresponding intensional calculus iLC-h which gives a natural extension by lax modality of the well-known G: odel/dummett logic LC of (finite) linear Kripke models. Our main results are that for finite linear intensional models L the intensional theory iTh(L) = {p : ϕ | L |= p : ϕ} characterises L and that iLC-h generates complete information about iTh(L).Our paper thus shows that the quantitative intensional information contained in the effort measure T can be abstracted away by the use of ○ and completely recovered by a suitable semantic interpretation of proofs. (shrink)

This volume contains finalized versions of papers presented at an international workshop on extensions of logic programming, held at the Seminar for Natural Language Systems at the University of Tübingen in December 1989. Several recent extensions of definite Horn clause programming, especially those with a proof-theoretic background, have much in common. One common thread is a new emphasis on hypothetical reasoning, which is typically inspired by Gentzen-style sequent or natural deduction systems. This is not only of (...) theoretical significance, but also bears upon computational issues. It was one purpose of the workshop to bring some of these recent developments together. The volume covers topics such as the languages Lambda-Prolog, N-Prolog, and GCLA, the relationship between logic programming and functional programming, and the relationship between extensions of logic programming and automated theorem proving. It contains the results of the first conference concentrating on proof-theoretic approaches to logic programming. (shrink)

Bringing together the latest scientific advances and some of the most enduring subtle philosophical puzzles and problems, this book collects original historical and contemporary sources to explore the wide range of issues surrounding the nature of life. Selections ranging from Aristotle and Descartes to Sagan and Dawkins are organised around four broad themes covering classical discussions of life, the origins and extent of natural life, contemporary artificial life creations and the definition and meaning of 'life' in its most general (...) form. Each section is preceded by an extensive introduction connecting the various ideas discussed in individual chapters and providing helpful background material for understanding them. With its interdisciplinary perspective, this fascinating collection is essential reading for scientists and philosophers interested in astrobiology, synthetic biology and the philosophy of life. (shrink)

In proof-theoretic semantics, meaning is based on inference. It may be seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems K, KT , (...) K4, and S4, with □ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between □ and a natural presentation of ♢. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

Two common claims in environmental moral philosophy are that nature is worthy of respect and that we respect ourselves in respecting nature. In this paper, I articulate two modes of practical reasoning that help make sense of these claims. The first is analogical extension, which understands the respect due human life as the source of a like respect for nature. The second is analogical implication, which involves nature in human life to show us what we are like. These forms of (...) reasoning are relevant to environmental virtue ethics in that both help us conceptualize how respect for nature can be part of our sense of humanity, and not opposed to our sense of humanity. (shrink)

The ‘causal-constitution’ fallacy, the ‘cognitive bloat’ worry, and the persisting theoretical confusion about the fundamental difference between the hypotheses of embedded (HEMC) and extended (HEC) cognition are three interrelated worries, whose common point—and the problem they accentuate—is the lack of a principled criterion of constitution. Attempting to address the ‘causal-constitution’ fallacy, mathematically oriented philosophers of mind have previously suggested that the presence of non-linear relations between the inner and the outer contributions is sufficient for cognitive extension. The abstract idea of (...) non-linearity, however, can be easily misunderstood and has, in the past, led to incorrect and counterintuitive conclusions about what may count as part of one’s overall cognitive system. In order to prevent any further mistakes I revisit dynamical systems theory to study the nature of the continuous mutual interactions that give rise to the aforementioned non-linear relations. Moreover, focusing on these interactions will allow us to provide two distinct arguments in support of the ontological postulation of extended cognitive systems, as well as an objective criterion of constitution. Accordingly, I put forward a version of HEC that treats continuous mutual interactions (and the resultant non-linear relations) not just as sufficient but also as necessary for cognitive extension. Such a qualified version of HEC may exclude certain alleged cases of cognitive extension where the agent does not mutually interact with his artifacts (e.g., shopping lists and directory services), but it is immune both to the ‘causal-constitution’ fallacy and the ‘cognitive bloat’ worry, and it can be sharply distinguished from HEMC. (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)

God and Goodness takes the experience of value as a starting point for natural theology. Mark Wynn argues that theism offers our best understanding of the goodness of the world, especially its beauty and openness to the development of richer and more complex material forms. We also see that the world's goodness calls for a moral response: commitment to the goodness of the world represents a natural extension of the trust to which we aspire in our dealings with (...) human beings. (shrink)

INTRODUCTION This study arose from involvement with the works of Thomas Aquinas (/5-) that was not only intensive, but also extensive in the time devoted to ...

I defend what I take to be a genuinely Kantian view on temporal extension: time is not an object but a human horizon of concrete particulars. As such, time depends on the existence of embodied human subjects. It does not, however, depend on those subjects determined as spatial objects. Starting with a realist notion of “apperception” as applied to indexical space, I proceed with the need for external criteria of temporal duration. In accordance with Kant’s Second Analogy of Experience, these (...) criteria are found in concepts and laws of motion and change. I then see what follows from this for a reasonable notion of transcendental idealism. Finally, in support of my Kantian conclusions, I argue for the transcendentally subjective nature of particular temporal extension. (shrink)

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofKwithin *K. For eachxЄ *K–Kand each natural numberd, YKis defined to be the number of algebraic extensions ofKof degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK= 0 for alldЄ N,d> 1; in other words,Khas no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the (...) existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω are Hilbertian elements in*Q, where pωis theωth prime number. (shrink)

It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.

In this article, I argue for the redundancy of Whitehead’s Platonic notion of the extensive continuum, counterposing it to his related notion of an atomic “ether of events.” I argue that Whitehead’s atomic ether is more compatible with orthodox general relativity than generally supposed and remarkably close to the contemporary idea of a discrete manifold in the causal set theory of quantum gravity. I argue that the method of extensive abstraction complements Whitehead’s atomic hypothesis by demonstrating the ultimately fictive nature (...) of any continuum. (shrink)

In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems |$K$|⁠, |$KT$|⁠, |$K4$| and (...) |$S4$|⁠, with |$\square $| as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between |$\square $| and a natural presentation of |$\lozenge $|⁠. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)