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Carnapian Modal and Epistemic Logic and Arithmetic with Descriptions
No full textIn the first chapter I have introduced Carnapian intensional logic again st the background of Frege's and Quine's puzzles. The main body of the dissertation consists of two parts. In the first part I discussed Carnapian modal logic and arithmetic with descriptions. In the second chapter, I have described three Carnapian theories, CCL, CFL, and CNL. All three theories have three things in common. First, they are formulated in languages containing description terms. Second, they contain a system of modal logic. Third, they do not contain the unrestricted classical substitution principle, but they do contain the classical substitution principle restricted to non-modal formulas and the Carnapian substitution principle , which says that two terms can be substituted salva veritate if they are necessarily coreferential. There are two major differences between the three theories. First, CCL and CFL allow universal instantiation with description terms, whereas CNL does not. Moreover, the quantificational theory of CCL is classical, whereas the quantificational theory of CFL is a (negative) free logic. Another difference is that CCL and CFL contain different description principles. Most importantly, the description principle of CCL ensures that even improper descriptions have a denotation, whereas the description principle of CFL does not guarantee this. CNL does not have a description principle. In the third chapter, I have studied collapse arguments for CCL, CFL, and CNL. A collapse argument is an argument for the following statement: if p is true, then it is necessarily true. A crucial role in the proofs of these collapse results was played by so-called 'self-predication principles', which say that under certain conditions the (possibly complex) predicate that expresses the descriptive condition can be combined by the description term formed out of that predicate with the result being a true sentence. In this chapter I have discussed a collapse argument for the extension of CCL with a self-predication principle, I have given a collapse argument for a similarly extended CFL, and most importantly, I have given a collapse argument for the extension of CNL with a self-predication principle. Finally, I have argued that the relevant self-predication principles are unsound under a Carnapian interpretation. In the fourth chapter, I have studied the extension of Peano Arithmetic (PA) with a Carnapian modal logic C, which is a dummy letter standing for either CCL or CFL. One can prove that the principle of the necessity of identity is a theorem of CPA. This implies that one gets a collapse result for CPA. The standard principle of weak induction was crucial for the proof. One can also prove that, if one assumes a particular self-predication principle, and if one assumes the principle of strong induction or, equivalently, the least-number principle, then one gets a partial collapse of de re modal truths in de dicto modal truths. I have argued that, if the box operator is interpreted as a metaphysical necessity operator, then Platonists would not be inimical to the collapse result. But if CPA is extended with a physical theory, then there is a threat that physical truths become physical necessities. It was shown that, under a Carnapian interpretation, the standard principle of weak induction is unsound, and that it can be replaced by a Carnapian principle of weak induction that is sound. The problem of logical and mathematical omniscience prevents ordinary Carnapian intensional logic from being taken seriously as a logic adequate for describing the principles of demonstrability. Yet many of the proof-theoretic results of the first part carry over to the part on Carnapian epistemic arithmetic with descriptions, since proof-theoretic results are independent of the informal reading of the operators. In the fifth chapter, I looked at extensions of arithmetic with a modal logic in which the box operator is interpreted as a demonstrability operator. A first extension in that sense is Shapiro's Epistemic Arithmetic (EA). Shapiro himself offered the problem of mathematical omniscience as a reason why it is difficult to find a model theory for EA. Horsten attempted to provide a model theory via the detour of Modal-Epistemic Arithmetic (MEA). The attention of the reader was drawn to an incoherence in the model theory of (MEA). Two alternative solutions were presented and, after a short discussion of the problem of de re demonstrability one of those alternatives was chosen. The discussion of the problem of de re demonstrability made it clear that it would be interesting to study the epistemic properties of notation systems. Horsten himself provided a framework for this, viz. Carnapian Epistemic Arithmetic (CEA), and he started a systematic study of the epistemic properties of notation systems within that framework. However, he did not provide non-trivial but adequate models. To make a start with solving the problem of finding good models for CEA, I introduced Carnapian Modal-Epistemic Arithmetic (CMEA) In constructing CMEA I incorporated the lesson about the principle of weak induction learnt in the fourth chapter. In the sixth chapter, I gave a critical assessment of an argument (the 'Description Argument') concerning the limits of de re demonstrability about the natural numbers. The conclusion of the Description Argument is that it is undemonstrable that there is a natural number that has a certain property but of which it is undemonstrable that it has that property. A crucial step in the Description Argument involved a self-predication principle. Making good use of one of the results obtained in the third chapter, I proved a collapse result for the background theory against which the Description Argument was formulated. I concluded that either the either the Description Argument is sound but its conclusion is trivial (since all arithmetical truths are demonstrable), or the Description Argument is unsound (since there are some arithmetical truths that are undemonstrable), or it is a cheap shot (since logic should be neutral on the issue). As an appendix I included an article co-authored by prof. dr. Leon Horst en and me. The topic of the article is indirectly related to some other topics (the logical form of descriptions, the notion of strict necessity ) investigated in my dissertation. Also, it backs up one of the addition al theses I might be asked to publicly defend during my doctoral exam. T he topic of the appendix is the set of the so-called 'paradoxes of strict implication'. Jonathan Lowe has argued that a particular variation on C.I. Lewis' notion of strict implication avoids the paradoxes of strict implication. Pace Lowe, it is argued that Lowe's notion of implication does not achieve this aim. Moreover, a general argument is offered to the effect that no other variation on Lewis' notion of constantly strict implication describes the logical behaviour of natural language conditionals in a satisfactory way.status: Publishe
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
The innateness hypothesis and mathematical concepts
Get PDFIn historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical con¬cepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist hypothesis? Second, grant¬ing that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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