What does it take for two logics to be mere notational variants? The present paper proposes a variety of different ways of cashing out notational variance, in particular isolating a constraint on any reasonable account of notational variance which makes plausible that the only kinds of translations which can witness notational variance are what are sometimes called definitional translations.

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy, and in which there has been a recent resurgence of interest. In contemporary presentations, it is stated in terms of fundamental languages: as the view that such languages contain more than one quantifier. For example, one ranging over abstract objects, and another over concrete ones. A natural worry, however, is that the languages proposed by the pluralist (...) are mere notational variants of those proposed by the monist, in which case the debate between the two positions would not seem to be substantive. Jason Turner has given an ingenious response to this worry, in terms of a principle that he calls ‘logical realism’. This paper offers a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that the languages in question are notational variants. Indeed, there seems to be every reason to think that they are. (shrink)

There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power, translation, and notational variance. In doing so, I show how (...) these investigations help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language. (shrink)

This paper discusses two notational variance views with respect to indexical singular reference and content: the view that certain forms of Millianism are at bottom notational variants of a Fregean theory of reference, the Fregean Notational Variance Claim; and the view that certain forms of Fregeanism are at bottom notational variants of a direct reference theory, the Millian Notational Variance Claim. While the former claim rests on the supposition that a direct reference (...) theory could be easily turned into a particular version of a neo-Fregean one by showing that it is bound to acknowledge certain senselike entities, the latter claim is based upon the supposition that a neo-Fregean theory could be easily turned into a particular version of a Millian one by showing that De Re senses are theoretically superfluous and hence eliminable. The question how many accounts of singular reference and content are we confronted with here — Two different (and mutually antagonistic) theories? Or just two versions of what is in essence the same theory? — is surely of importance to anyone interested in the topic. And this question should be answered by means of a careful assessment of the soundness of each of the above claims. Before trying to adjudicate between the two accounts, one would naturally want to know whether or not there are indeed two substantially disparate accounts. Grosso modo, if the Fregean Claim were sound then we would have a single general conception of singular reference to deal with, viz. Fregeanism; likewise, if the Millian Claim were sound we would be facing a single general conception of singular reference, viz. Millianism. My view is that both the Fregean Notational Variance Claim and its Millian counterpart are wrong, though naturally on different grounds. I have argued elsewhere that the Fregean Notational Variance Claim - considered in its application to the semantics of propositional-attitude reports involving proper names — is unsound. I intend tosupplement in this paper such a result by trying to show that the Millian Claim - taken in its application to the semantics of indexical expressions — should also be rated as incorrect. I focus on a certain set of arguments for the Millian Claim, arguments which I take as adequately representing the general outlook of the Millian theorist with respect to neo-Fregeanism about indexicals and which involve issues about the cognitive significance of sentences containing indexical terms. (shrink)

According to ontological nihilism there are, fundamentally, no individuals. Both natural languages and standard predicate logic, however, appear to be committed to a picture of the world as containing individual objects. This leads to what I call the \emph{expressibility challenge} for ontological nihilism: what language can the ontological nihilist use to express her account of how matters fundamentally stand? One promising suggestion is for the nihilist to use a form of \emph{predicate functorese}, a language developed by Quine. This proposal faces (...) a difficult objection, according to which any theory in predicate functorese will be a notational variant of the corresponding theory stated in standard predicate logic. Jason Turner (2011) has provided the most detailed and convincing version of this objection. In the present paper, I argue that Turner's case for the notational variance thesis relies on a faulty metasemantic principle and, consequently, that an objection long thought devastating is in fact misguided. (shrink)

This paper is devoted to the topic of indexicality in relation to the problem of cognitive significance. I undertake a critical examination of what I call the Millian Notational Variance Claim; this is the claim that those versions of a neo-Fregean semantics for demonstratives and other indexicals which rest upon the notion of a de re sense are eventually notational variants of a directly referential or Millian semantics for indexicals. I try to show that several lines of (...) reasoning that might be pursued by Millian theorists with a view to establishing the Millian Notational Variance Claim are inconclusive, and hence that the claim is in general unsound. The problem of cognitive significance is tackled in connection with those categories of indexicals concerning which neo-Fregeanism and Millianism are alleged to yield similar results, viz. temporal indexicals, spatial indexicals, and perceptual demonstratives. I argue towards the conclusion that the notions the Millian theorist might invoke to accommodate the phenomena of cognitive significance in this area of indexicality are hardly adequate to the effect, and hence that senses are indispensable also here. (shrink)

The paper presents a demarcation of a “minimalistic” concept of logical form, which nevertheless largely agrees with the way the term “logical form” is commonly used in contemporary logic and philosophy of logic. We see logical forms as formulas of formal languages assigned to (compounds of) sentences of a natural language (perhaps modulo notational variance). We thus reject the views of logical forms as underlying structures of thoughts or of the material reality that surrounds us. The assignment of (...) the forms, we claim, aims at envisaging the logical (especially inferential) properties of the analyzed sentences, arguments, or other texts, as the analyzing formulas wear them, as it were, on their sleeves. Hence we suggest an “expressivist” and a pragmatic understanding of logical forms - they are used to expose (and fix) logical properties of sentences in textual contexts and the way of their employment is determined by the goals of the particular studies. (shrink)

A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI, LV, or LF, and every antitone logic can be extended to LN, LV, or LF, where LI, LN, LV, and LF are logics axiomatized, respectively, by the schemas □α↔α, □α↔¬α, □α↔⊤, and □α↔⊥. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV, LF, and the minimum amphitone logic AM axiomatized by the schema □α→□β. These logics, (...) along with LI, LN, and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the nonmodal connectives. We close by discussing these divergences and conditions under which our results do carry over. (shrink)

This is a book about statistical theory without sample theory. A very substantial portion of the modern theory of statistics can be treated without involving oneself in problems of analysis consequent upon the treatment of sampling. Accordingly, Rozeboom has written a book which, while sophisticated, does not demand any high-powered mathematical knowledge or competence. A good deal of the theories of distribution, statistical regression, factor analysis, variance structure, reliability, and miscellaneous applications of probability theory is covered. The author concentrates (...) upon questions of algebraic structure and foundations of the theory, although this is a "practical" text in the best sense of the word. To that end, Rozeboom has designed a special notation which facilitates comprehension of the text, although it does not mix well with the standard modes of logical and mathematical expression. In general, this book is quite well suited for use by students of the social and behavioral sciences and the philosophy of science who have a special interest in the theory and technique of statistical inference. There are a large number of typographical errors in the notation, which are corrected in a six-page addenda supplied with the book.—H. P. K. (shrink)

This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in requiring nonstandard (...) assumptions about mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature. (shrink)

This dissertation argues for Semantic Variance, the thesis that nearly every utterance is such that there is no proposition that more than one languge user takes to be that utterance's truth-conditional content. I argue that Semantic Variance is problematic for standard theories concerning the nature of communication, the epistemic significance of ordinary disputes, the semantics of speech reports, and the nature of linguistic competence. In response to the problems arising from the truth of Semantic Variance, I develop (...) new accounts of the transmission of relevant information, ordinary disputes, and the semantics of speech reports. Towards the end of the dissertation I outline a pluralistic account about the nature of communication and linguistic competence. (shrink)

Recently a number of works in meta-ontology have used a variant of J.H. Harris's collapse argument in the philosophy of logic as an argument against Eli Hirsch's quantifier variance. There have been several responses to the argument in the literature, but none of them have identified the central failing of the argument, viz., the argument has two readings: one on which it is sound but doesn't refute quantifier variance and another on which it is unsound. The central lesson (...) I draw is that arguments against quantifier variance must pay strict attention to issues of translation and interpretation. The paper also has a substantial appendix in which I prove the equivalence of plural mereological nihilism and standard first-order atomistic mereology; results of this kind are often appealed to in the literature on quantifier variance but without many details on the nature or proof of the result. (shrink)

Quantifier variance faces a number of difficulties. In this paper we first formulate the view as holding that the meanings of the quantifiers may vary, and that languages using different quantifiers may be charitably translated into each other. We then object to the view on the basis of four claims: (i) quantifiers cannot vary their meaning extensionally by changing the domain of quantification; (ii) quantifiers cannot vary their meaning intensionally without collapsing into logical pluralism; (iii) quantifier variance is (...) not an ontological doctrine; (iv) quantifier variance is not compatible with charitable translation and as such is internally inconsistent. In light of these troubles, we recommend the dissolution of quantifier variance and suggest that the view be laid to rest. (shrink)

Quantifier variance is a well-known view in contemporary metaontology, but it remains very widely misunderstood by critics. Here we briefly and clearly explain the metasemantics of quantifier variance and distinguish between modest and strong forms of variance (Section I), explain some key applications (Section II), clear up some misunderstandings and address objections (Section III), and point the way toward future directions of quantifier-variance-related research (Section IV).

The thesis of quantifier variance is consistent and cannot be refuted via a collapse argument.

Quantifier variance holds that different languages can have unrestricted quantifier expressions that differ in meaning, where an expression is a “quantifier” just in case it plays the right inferential role. Several critics argued that J.H. Harris’s “collapse” argument refutes variance by showing that identity of inferential role is incompatible with meaning variance. This standard, syntactic collapse argument has generated several responses. More recently, Cian Dorr proved semantic collapse theorems to generate a semantic collapse argument against variance. (...) The argument is significantly different from standard collapse, so it requires a new response. Here I clarify and analyze the semantic collapse argument, and explain how variantists can and should respond to it. The paper also includes an appendix showing the difficulties of positing identity variance without quantifier variance. The argument in the appendix has yet to appear in print, but is familiar to specialists. (shrink)

This article was first published as “Notation atomique et hypothèses atomistiques”, Revue des questions scientifiques, 31 (1892), 391– 457. It is the second of a series of articles Duhem was to publish in the Catholic journal Revue des questions scientifiques, in which he presents his understanding of what can justifiably be said about the structure of chemical substances as captured by chemical formulas. The argument unfolds following a broadly historical development of events throughout the course of the century which was (...) coming to a close as he wrote. He later reflected in his classic The Aim and Structure of Physical Theories – based in large part on articles which had appeared in the Revue – that “To give the history of a physical principle is at the same time to make a logical analysis of it” (p. 269). Logical analysis clearly dominates in the present article. The historical context was elaborated considerably in a later work, Le mixte et la combinaison chimique: Essai sur l’évolution d’une idée, which did not lead him to retract any aspect of his earlier position but provided a broader setting in which it could be elaborated. In particular, the Aristotelian influence, which is only hinted at here in some of the formulations (see especially the beginning of section VII) without mentioning Aristotle by name, is explicit in the later work, making Duhem’s own ontological conception a little clearer. A discussion of stereoisomerism, conspicuously absent in the present article, is also integrated into the later book. The same holds for Avogadro’s hypothesis. (shrink)

Objects often manifest themselves in incompatible ways across perspectives that are epistemically on a par. The standard response to such cases is to deny that the properties that things appear to have from different perspectives are properties that things really have out there. This type of response seems worrying: too many properties admit of perspectival variance and there are good theoretical reasons to think that such properties are genuinely instantiated. So, we have reason to explore views on which things (...) can have the incompatible properties they appear to have across perspectives. This paper explores the view that things can have incompatible properties if the world is not a metaphysically unified place but is instead fragmented. There is a sensible notion of co-obtainment, on which two facts can each obtain without co-obtaining. Using this notion, we can step back from our embedded perspectives on the world without deeming the contents of those perspectives to be mere appearances. This renders th.. (shrink)

A sense of unity -- Basic objects : a reply to Xu -- Objectivity without objects -- The vagueness of identity -- Quantifier variance and realism -- Against revisionary ontology -- Comments on Theodore Sider's four dimensionalism -- Sosa's existential relativism -- Physical-object ontology, verbal disputes, and common sense -- Ontological arguments : interpretive charity and quantifier variance -- Language, ontology, and structure -- Ontology and alternative languages.

Quantifier variance entails that ‘there exists’ has a variety of meanings. Determining what makes all these meanings quantifier meanings is a problem associated with this view. A reasonable candidate suggested by Hirsch is the set of formal rules governing quantification. However, the collapse argument presents a notorious objection to the viability of the candidate: there cannot be more than one quantifier obeying the same rules up to logical equivalence. It is proposed that a quantifier variantist who intends to retain (...) Hirsch’s proposed solution can admit a many-sorted language in order to block the collapse argument. Within the many-sorted language, a Kaplanian thesis about the rules governing sorted quantifiers saves quantifier variance from meaning variance. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects—those which visually resemble in some way the concepts they represent—offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative (...) relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd's hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance. (shrink)

Expressively equivalent logical languages can enunciate logical notions in notationally diversified ways. Frege’s Begriffsschrift, Peirce’s Existential Graphs, and the notations presented by Wittgenstein in the Tractatus all express the sentential fragment of classical logic, each in its own way. In what sense do expressively equivalent notations differ? According to recent interpretations, Begriffsschrift and Existential Graphs differ from other logical notations because they are capable of “multiple readings.” We refute this interpretation by showing that there are at least three different kinds (...) of such multiple readings. While readings of the first kind do not capture any essential difference among notations but only among vocabularies, corresponding to readings of the second and the third kind two general parameters according to which notations may differ are defined: linearity vs. non-linearity, and tabularity vs. non-tabularity. This answers the question of how there can be substantially different but expressively equivalent logical notations. (shrink)

Attempting to deflate ontological debates, the proponent of Quantifier Variance (QV) claims that there are multiple quantifier meanings of equal metaphysical merit. According to Hirsch—the main proponent of QV—metaphysical merit should be understood intensionally: two languages have equal merit if they allow us to express the same possibilities. I examine the notion of metaphysical merit and its purported link to intensionality. That link, I argue, should not be supported by adopting an intensional theory of semantic content. Rather, I give (...) a general strategy for supporting claims about metaphysical merit and examine whether that strategy can be used to link merit and intensionality. Though I don’t deliver a definitive verdict, the discussion provides a clearer framework for articulating and evaluating claims about metaphysical merit. (shrink)

The bias–variance tradeoff is a theoretical concept that suggests machine learning algorithms are susceptible to two kinds of error, with some algorithms tending to suffer from one more than the other. In this letter, we claim that the bias–variance tradeoff is a general concept that can be applied to human cognition as well, and we discuss implications for research in cognitive science. In particular, we show how various strands of research in cognitive science can be interpreted in light (...) of the bias–variance tradeoff, giving insight into individual differences in learning, the nature of cognitive processes, and debates in cognitive science research. (shrink)

Notations are cognitive systems involving distinctive psychological functions, behaviors, and material forms. Seen through this lens, two main types – semasiography and visible language – are fundamentally differentiated by their material prehistories, emphasis on iconography, and the centrality of language's combinatorial faculty. These fundamental differences suggest that key qualities (iconicity, expressiveness, concision) are difficult to conjoin in a single system.

Multicultural environments require learning multiple number notations wherein some are encountered more frequently than others. This leads to differences in exposure and consequently differences in usage between notations. We find that differential notational usage imposes a significant neurocognitive load on number processing. Despite simultaneous acquisition, forty-two adult binumerate populations, familiar with two positional writing systems namely Hindu Nagari digits and Hindu Arabic digits, reported significantly lower preference and usage for Nagari as compared to Arabic. Twenty-four participants showed significantly increased (...) reaction times and reduced accuracy while performing magnitude comparison tasks in Nagari with respect to Arabic. Functional magnetic resonance imaging revealed that processing Nagari elicited significantly greater activity in number processing and attention networks. A direct subtraction of networks for Nagari and Arabic notations revealed a neural circuit comprising of bilateral intra-parietal sulcus, inferior and mid frontal gyri, fusiform gyrus and the anterior cingulate cortex (FDR p<0.005). Additionally, whole brain correlation analysis showed that activity in the left inferior parietal region was modulated by task performance in Nagari. We attribute the increased activation in Ng to increased task difficulty due to infrequent exposure and usage. Our results reiterate the role of the left intra-parietal sulcus in modulating performance in numeric tasks and highlight that of the attention network for monitoring symbolic notation mode in binumerates. (shrink)

In the work of both Matti Eklund and John Hawthorne there is an influential semantic argument for a maximally expansive ontology that is thought to undermine even modest forms of quantifier variance. The crucial premise of the argument holds that it is impossible for an ontologically "smaller" language to give a Tarskian semantics for an ontologically "bigger" language. After explaining the Eklund-Hawthorne argument (in section I), we show this crucial premise to be mistaken (in section II) by developing a (...) Tarskian semantics for a mereological universalist language within a mereological nihilist language (a case which we, and Eklund and Hawthorne, take as representative). After developing this semantics we step back (in section III) to discuss the philosophical motivations behind the Eklund- Hawthorne argument’s demand for a semantics. We ultimately conclude that quantifier variantists can meet any demand for a semantics that might reasonably be imposed upon them. (shrink)

BackgroundConcerns about research misbehavior in academic science have sparked interest in the factors that may explain research misbehavior. Often three clusters of factors are distinguished: individual factors, climate factors and publication factors. Our research question was: to what extent can individual, climate and publication factors explain the variance in frequently perceived research misbehaviors?MethodsFrom May 2017 until July 2017, we conducted a survey study among academic researchers in Amsterdam. The survey included three measurement instruments that we previously reported individual results (...) of and here we integrate these findings.ResultsOne thousand two hundred ninety-eight researchers completed the survey. Results showed that individual, climate and publication factors combined explained 34% of variance in perceived frequency of research misbehavior. Individual factors explained 7%, climate factors explained 22% and publication factors 16%.ConclusionsOur results suggest that the perceptions of the research climate play a substantial role in explaining variance in research misbehavior. This suggests that efforts to improve departmental norms might have a salutary effect on behavior. (shrink)

This article argues that the much-maligned ‘notational variants’ of a given formal linguistic theory play a role similar to alternative numerical measurement scales. Thus, they can be used to identify the invariant components of the grammar; i.e., those features that do not depend on the choice of empirically equivalent representation. Treating these elements as the ‘meaningful’ structure of language has numerous consequences for the philosophy of science and linguistics. I offer several such examples of how linguistic theorizing can profit (...) from adopting a measurement-theoretic viewpoint. The first concerns a measurement-theoretic response to a famous criticism of Quine's. Others follow from issues of simplicity in the current biolinguistics program. An unexpected similarity with behaviorist practices is also uncovered. I then argue that managable and useful steps can be taken in this area. (shrink)

The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without caution, as the (...) use of anachronistic notation tends to impede, rather than enhance, our ability to recognize the emergent nature of mathematical objects. (shrink)

Kris McDaniel has recently defended a criterion for being an ontological pluralist that classifies the quantifier variantist as one. In this paper, I argue that this is a mistake. There is an important difference between the two views, which is sometimes obscured by a common view in the metaphysics of fundamentality. According to the simple analysis, a language is ideal—it allows for a maximally metaphysically perspicuous description of reality—just in case all its primitives are perfectly natural. I argue that this (...) analysis struggles to distinguish quantifier variance from ontological pluralism, and then I discuss various accounts that can do better. I then propose a criterion for being an ontological pluralist that does not misclassify the quantifier variantist. Finally, I discuss some additional advantages of my proposal. (shrink)

Die Theorie der Notation wurde von Nelson Goodman ([Goodman 1968]: Kap. 4) im Zusam­menhang mit der Frage nach den Iden­titäts­krite­rien für Kunstwer­ke ent­wickelt. Eine Nota­tion ist ein Zeichen­system, das ein syntak­tisches oder seman­tisches Krite­rium dafür ermög­licht, welche Gegen­stände oder Ereig­nisse Einzel­fälle eines bestimm­ten Werks sind. Ein solches Krite­rium ist dann notwen­dig, wenn Werke mehre­re Einzel­fälle zulas­sen, deren Iden­tität nicht durch ihre Entste­hungsge­schichte bestimmt ist. Da dies in para­digma­tischer Weise in der Musik der Fall ist, führe ich den Begriff der Nota­tion (...) für die Musik ein, ehe ich mich kurz der Lite­ratur und ausführ­licher den bilden­den Künsten zuwen­de. Die resul­tieren­den Iden­titäts­krite­rien sollen auch für Musik, Texte und Bilder ohne Kunstwerk­status gelten, auch wenn wir hier in der Regel nicht von Werken sprechen. (shrink)

SummaryHuman height is a heritable trait that is known to be influenced by environmental factors and general standard of living. Individual and population stature is correlated with health, education and economic achievement. Strong sexual selection pressures for stature have been observed in multiple diverse populations, however; there is significant global variance in gender equality and prohibitions on female mate selection. This paper explores the contribution of general standard of living and gender inequality to the variance in global female (...) population heights. Female population heights of 96 nations were culled from previously published sources and public access databases. Factor analysis with United Nations international data on education rates, life expectancy, incomes, maternal and childhood mortality rates, ratios of gender participation in education and politics, the Human Development Index and the Gender Inequality Index was run. Results indicate that population heights vary more closely with gender inequality than with population health, income or education. (shrink)

This Element lays the foundation, introduces a framework, and sketches the program for a systematic study of mathematical notations. It is written for everyone who is curious about the world of symbols that surrounds us, in particular researchers and students in philosophy, history, cognitive science, and mathematics education. The main characteristics of mathematical notations are introduced and discussed in relation to the intended subject matter, the language in which the notations are verbalized, the cognitive resources needed for learning and understanding (...) them, the tasks that they are used for, their material basis, and the historical context in which they are situated. Specific criteria for the design and assessment of notations are discussed, as well as ontological, epistemological, and methodological questions that arise from the study of mathematical notations and of their use in mathematical practice. (shrink)

This paper outlines a critique of the use of the genetic variance–covariance matrix (G), one of the central concepts in the modern study of natural selection and evolution. Specifically, I argue that for both conceptual and empirical reasons, studies of G cannot be used to elucidate so-called constraints on natural selection, nor can they be employed to detect or to measure past selection in natural populations – contrary to what assumed by most practicing biologists. I suggest that the search (...) for a general solution to the difficult problem of identifying causal structures given observed correlation’s has led evolutionary quantitative geneticists to substitute statistical modeling for the more difficult, but much more valuable, job of teasing apart the many possible causes underlying the action of natural selection. Hence, the entire evolutionary quantitative genetics research program may be in need of a fundamental reconsideration of its goals and how they correspond to the array of mathematical and experimental techniques normally employed by its practitioners. (shrink)

Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading time, (...) taking reader preferences into account. We present a representational infrastructure for notations in living mathematical documents. Mathematical notations can be defined declaratively. Author and reader can extensionally define the set of available notation definitions at arbitrary document levels, and they can guide the notation selection function via intensional annotations. We give an abstract specification of notation definitions and the flexible rendering algorithms and show their coverage on paradigmatic examples. We show how to use this framework to render OpenMath and Content-MathML to Presentation-MathML, but the approach extends to arbitrary content and presentation formats. We discuss prototypical implementations of all aspects of the rendering pipeline. (shrink)

ABSTRACT There has been recent interest in the idea that, when metaphysicians disagree over the truth of (say) ‘There are numbers’ or ‘Chairs exist’, their dispute is merely verbal. This idea has been taken to motivate quantifier variance, the view that the meanings of quantifier expressions vary across different ontological languages, and that each of these meanings is of equal metaphysical merit. I argue that quantifier variance cannot be upheld in light of natural language theorists’ analyses of quantifier (...) expressions. The idea that metaphysicians are engaged in verbal disputes can be maintained only through alternative strategies that have nothing to do with quantifier expressions. (shrink)

In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less (...) than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}. We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions. (shrink)

Spekkens has introduced a toy theory (Spekkens in Phys. Rev. A 75(3):032110, 2007) in order to argue for an epistemic view of quantum states. I describe a notation for the theory (excluding certain joint measurements) which makes its similarities and differences with the quantum mechanics of stabilizer states clear. Given an application of the qubit stabilizer formalism, it is often entirely straightforward to construct an analogous application of the notation to the toy theory. This assists calculations within the toy theory, (...) for example of the number of possible states and transformations, and enables superpositions to be defined for composite systems. (shrink)

We provide an overview of the meta-ontological position known as "Quantifier Variance".

This volume contains English translations of Frege's early writings in logic and philosophy and of relevant reviews by other leading logicians. Professor Bynum has contributed a biographical essay, introduction, and extensive bibliography.

Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of logically equivalent (...) formulas, and presenting the notation's close connection to syntax trees. In the second part, Frege's considerations regarding the design principles underlying the Begriffsschrift are presented. Frege was quite explicit about these in his replies to early criticisms and unfavorable comparisons with Boole's notation for propositional logic. This discussion reveals that the Begriffsschrift is in fact a well thought-out and carefully crafted notation that intentionally exploits the possibilities afforded by the two-dimensional medium of writing like none other. (shrink)

Buchholz presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive recursive function its n th predecessor and, e.g. the last rule applied. Here we extend the method to the more general setting of infinite terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, we use the (...) method to 1. give a new proof of a well-known trade-off theorem , which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and 2. construct a continuous normalization operator with an explicit modulus of continuity. (shrink)

Gregory Landini offers a detailed historical account of Frege's notations and the philosophical views that led Frege from Begriffssscrhrift to his mature work Grundgesetze, addressing controversial issues that surround the notations.

This volume contains English translations of Frege's early writings in logic and philosophy and of relevant reviews by other leading logicians. Professor Bynum has contributed a biographical essay, introduction, and extensive bibliography.