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.

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)

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)

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)

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)

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.

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)

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 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)

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)

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.

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)

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)

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)

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)

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.

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.

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)

Argues that natural kind terms are token-reflexive, with reference ultimately fixed to the underlying explanatory properties of the surface qualities of local matter.

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)

Peter Anstey presents a thorough and innovative study of John Locke's views on the method and content of natural philosophy. Focusing on Locke's Essay concerning Human Understanding, but also drawing extensively from his other writings and manuscript remains, Anstey argues that Locke was an advocate of the Experimental Philosophy: the new approach to natural philosophy championed by Robert Boyle and the early Royal Society who were opposed to speculative philosophy. On the question of method, Anstey shows how Locke's (...) pessimism about the prospects for a demonstrative science of nature led him, in the Essay, to promote Francis Bacon's method of natural history, and to downplay the value of hypotheses and analogical reasoning in science. But, according to Anstey, Locke never abandoned the ideal of a demonstrative natural philosophy, for he believed that if we could discover the primary qualities of the tiny corpuscles that constitute material bodies, we could then establish a kind of corpuscular metric that would allow us a genuine science of nature. It was only after the publication of the Essay, however, that Locke came to realize that Newton's Principia provided a model for the role of demonstrative reasoning in science based on principles established upon observation, and this led him to make significant revisions to his views in the 1690s. On the content of Locke's natural philosophy, it is argued that even though Locke adhered to the Experimental Philosophy, he was not averse to speculation about the corpuscular nature of matter. Anstey takes us into new terrain and new interpretations of Locke's thought in his explorations of his mercurialist transmutational chymistry, his theory of generation by seminal principles, and his conventionalism about species. (shrink)

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity).We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to (...) a wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic. (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)

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)

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)

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)

Knowledge, Nature, and the Good brings together some of John Cooper's most important works on ancient philosophy. In thirteen chapters that represent an ideal companion to the author's influential Reason and Emotion, Cooper addresses a wide range of topics and periods--from Hippocratic medical theory and Plato's epistemology and moral philosophy, to Aristotle's physics and metaphysics, academic scepticism, and the cosmology, moral psychology, and ethical theory of the ancient Stoics.Almost half of the pieces appear here for the first time or are (...) presented in newly expanded, extensively revised versions. Many stand at the cutting edge of research into ancient ethics and moral psychology. Other chapters, dating from as far back as 1970, are classics of philosophical scholarship on antiquity that continue to play a prominent role in current teaching and scholarship in the field. All of the chapters are distinctive for the way that, whatever the particular topic being pursued, they attempt to understand the ancient philosophers' views in philosophical terms drawn from the ancient philosophical tradition itself.Through engaging creatively and philosophically with the ancient texts, these essays aim to make ancient philosophical perspectives freshly available to contemporary philosophers and philosophy students, in all their fascinating inventiveness, originality, and deep philosophical merit. This book will be treasured by philosophers, classicists, students of philosophy and classics, those in other disciplines with an interest in ancient philosophy, and anyone who seeks to understand philosophy in philosophical terms. (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)

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)

RMO -> is the result of adding the ‘mingle principle’ (viz. A-> (A -> A)) to Anderson and Belnap’s implicative logic of relevance R->. The aim of this paper is to provide all possible axiomatizations with independent axioms of RMO -> formulable with Anderson and Belnap’s list extended with three characteristic minglish principles.

RMO -> is the result of adding the ‘mingle principle’ ) to Anderson and Belnap’s implicative logic of relevance R->. The aim of this paper is to provide all possible axiomatizations with independent axioms of RMO -> formulable with Anderson and Belnap’s list extended with three characteristic minglish principles.

"La Nouvelle Alliance (1979), by physicist Ilya Prigogine and philosopher Isabelle Stengers, is both a book on the history and philosophy of thermodynamics and a philosophical essay on man's place in nature. It has given rise to lively interdisciplinary controversy.".

Kleene [5] mentions two ways of extending the constructive ordinals. The first is by relativizing the setOof notations for the constructive ordinals, using fundamental sequences which are partial recursive inO. In this way we obtain the setOOwhich provides notations for the ordinals less than ω1O. Continuing the process, the sequenceO,OO,, … and the corresponding ordinalsare obtained. A second possibility is to define higher number classes in which partial recursive functions are used at limit ordinals to provide an “accessibility” mapping from (...) a previously defined number class. The relationship between the ordinals obtained by the two methods of extension has been an open problem. Methods developed in this article are used to show that the two ways of extending the constructive ordinals are equivalent, provided the sets of notations for the higher number classes satisfy certain natural conditions. Equivalence is obtained, not only with respect to ordinals, but also with respect to the forms of the sets of notations for the higher number classes. Specifically, the fundamental fact that the sets of notations for the constructive ordinals are complete Π11sets generalizes to suitably defined higher number classes. As an application we prove that the ordinals of the Addison and Kleene [1] constructive third number class are exactly the ordinals less than ω1Oand the setof notations for their third number class is recursively isomorphic toOO. (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)

A genome of a living organism consists of a long string of symbols over a finite alphabet carrying critical information for the organism. This includes its ability to control post natal growth, homeostasis, adaptation to changes in the surrounding environment, or to biochemically respond at the cellular level to various specific regulatory signals. In this sense, a genome represents a symbolic encoding of a highly organized system of information whose functioning may be revealed as a natural multilayer structure in (...) terms of complexity and prominence. In this paper we use the mathematical theory of symbolic extensions as a framework to shed light onto how this multilayer organization is reflected in the symbolic coding of the genome. The distribution of data in an element of a standard symbolic extension of a dynamical system has a specific form: the symbolic sequence is divided into several subsequences (which we call layers) encoding the dynamics on various “scales”. We propose that a similar structure resides within the genomes, building our analogy on some of the most recent findings in the field of regulation of genomic DNA functioning. (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)

The fifth chapter provides a detailed discussion of Buridan’s strategy of identifying the conceptual structures discussed in the chapter 4 by means of the various “syntactical clues” provided by spoken and written natural languages. The chapter compares the Buridanian strategy of “regimentation” with the modern strategy of formalization, and argues that for the purposes of a “natural logic” the former is not inferior to the latter. But in order to bridge the conceptual gap between the two approaches, the (...) chapter also discusses a simple extension of standard quantification theory that is capable of faithfully representing a large part of Buridan’s logic. Indeed, the chapter argues that the resulting formal system is in accord not only with Buridan’s logic, but with our natural language intuitions in general, providing more faithful representation of natural language reasoning than standard quantification theory. (shrink)