Vagueness, logic, and definite truth
<p>This thesis brings together five essays which develop the theme of boundarylessness as presented in Mark Sainsbury’s ‘Concepts without boundaries’ (1990). According to Sainsbury’s picture, vague concepts classify, not by drawing set-theoretic boundaries, but rather in a manner akin to magne...
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| Format: | Disertacija |
| Jezik: | English |
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2023
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| _version_ | 1826313426695094272 |
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| author | Wall, L |
| author2 | Rumfitt, I |
| author_facet | Rumfitt, I Wall, L |
| author_sort | Wall, L |
| collection | OXFORD |
| description | <p>This thesis brings together five essays which develop the theme of boundarylessness as presented in Mark Sainsbury’s ‘Concepts without boundaries’ (1990). According to Sainsbury’s picture, vague concepts classify, not by drawing set-theoretic boundaries, but rather in a manner akin to magnetic poles—certain paradigm instances of vague concepts behave like magnetic poles, and exert various degrees of influence on a domain of objects. The question then arises as to the logic of vagueness that such an account recommends. The five chapters of this thesis explore different ways of developing this idea. Chapters 1-3 investigate how Sainsbury’s conception can be developed in support of classical logic—in particular, by motivating an Edgington-style verity semantics. Chapter 4 uses the conception to motivate a quantum logic, and Chapter 5 an intuitionistic logic. In each case, Sainsbury’s conception constitutes the motivation for the logical system, and thus is the central uniting theme of the thesis.</p>
<p>The verity-theoretic approach is also a central component of the thesis, which features in four of the five chapters. The approach is forwarded by Dorothy Edgington in her seminal paper ‘Vagueness by degrees’ (1996) and is based on a tradition of probabilistic approaches to semantics. Edgington’s suggestion is, however, that, although the semantics for vague language has the same algebraic structure as probability, the values are not to be interpreted as probabilities, but rather in a purely semantic manner as the degree to which a statement is close to being a clear case of truth.</p>
<p>Chapters 1-3 use Sainsbury’s conception to develop a classical verity semantics, which I take to be the most natural and promising development of Sainsbury’s view. It is argued that verities are the degrees to which statements are close to a clear case of truth (which, for atomic predications, are the degrees to which items fall under the sphere of influence of paradigms). A notion of <em>substantiveness</em> is introduced to distinguish the notion of truth from the more prosaic, bivalent notion of truth governed by the Tarski schemes: a statement is substantively true, it is argued, just in case its truth is fixed by a metasemantic process of content fixing. A metasemantic theory is presented according to which verity assignments are fixed via their role underwriting the success of a probabilistic theory of communication.</p>
<p>In the second chapter, the paradox of higher-order vagueness is discussed with particular attention to the paradox presented by Elia Zardini (2013). The verity semantics is extended to a language equipped with a <em>Def</em> operator where the statement Def(p) has the interpretation “p has verity 1”. It is argued that fine-grained modal verity semantics afford an alternative solution to the paradox: in setting up his paradox, Zardini relies on the assumption that there are clear borderline cases of all orders in a sorites series, but in the classical verity semantics proposed, although there are clear borderline cases, it is increasingly less clear that there are borderline cases of order <em>n</em>— such claims have lower verity—the greater the value of <em>n</em>.</p>
<p>In chapter three, the notion of conditional verity is then addressed. Two initial attempts to elucidate the notion are rejected. The first, a theory of conditional verity based on the supervaluationist approach, is rejected on the grounds that it predicts the wrong values for conditional statements in a conditional sorites argument insofar as it fails to capture the notion of tolerance. The second, an account involving hypothetical rescaling, is initially rejected on the grounds that, despite yielding the correct values for the conditional premises of a sorites, it succumbs to a problem of indeterminacy with respect to other conditional statements. It is argued that this problem is insurmountable. However, it is concluded that some degree of indeterminacy is acceptable for the verity semanticist.</p>
<p>In chapter four, the basis of a quantum logic of vagueness is presented. It is argued that Sainsbury’s conception can be seen as giving rise to a proximity space and thus can be used as the basis of quantum semantics. It is suggested that interpreting the connectives of the language relative to operations on the proximity space is plausible on the grounds that they capture the conditions under which one could permissibly make various assertions on Sainsbury’s account. Hence, there is reasonable ground for taking the logic to be quantum. This notion of permissibility can be made explicit via a modal translation into the modal system KTB (defined in Appendix 1) which provides the epistemic conditions for permissible assertion.</p>
<p>In the final chapter, it is suggested that Sainsbury’s polar conception could serve as the basis of an intuitionistic logic of vagueness. In a series of papers published throughout his career, Crispin Wright establishes himself as the central proponent of the intuitionistic account of vagueness. His argument for intuitionism is based on a Brouwer-Heyting-Kolmogorov (BHK) style, knowledge-theoretic semantic theory for vague language. In this final chapter, it is argued that Wright’s account of definiteness, in combination with his semantics, commits him to a columnar theory of higher-order vagueness—in contrast to the theory presented in chapter three. It is then argued that both the columnar account and Wright’s BHK-style semantics are unsuitable. However, it is suggested that a semantics for intuitionistic logic can be given via a verity semantics, namely because the intuitionistic conditions are the strongest rules which can be justified on the basis of a plausible verity semantics when motivated by the polar conception. This strategy is proffered to the intuitionist without endorsement from the author.</p> |
| first_indexed | 2024-09-25T04:14:33Z |
| format | Thesis |
| id | oxford-uuid:dda6387d-5363-401d-a0a3-62794c94e6ca |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:14:33Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | oxford-uuid:dda6387d-5363-401d-a0a3-62794c94e6ca2024-07-11T09:05:50ZVagueness, logic, and definite truthThesishttp://purl.org/coar/resource_type/c_db06uuid:dda6387d-5363-401d-a0a3-62794c94e6caEnglishHyrax Deposit2023Wall, LRumfitt, IMagidor, O<p>This thesis brings together five essays which develop the theme of boundarylessness as presented in Mark Sainsbury’s ‘Concepts without boundaries’ (1990). According to Sainsbury’s picture, vague concepts classify, not by drawing set-theoretic boundaries, but rather in a manner akin to magnetic poles—certain paradigm instances of vague concepts behave like magnetic poles, and exert various degrees of influence on a domain of objects. The question then arises as to the logic of vagueness that such an account recommends. The five chapters of this thesis explore different ways of developing this idea. Chapters 1-3 investigate how Sainsbury’s conception can be developed in support of classical logic—in particular, by motivating an Edgington-style verity semantics. Chapter 4 uses the conception to motivate a quantum logic, and Chapter 5 an intuitionistic logic. In each case, Sainsbury’s conception constitutes the motivation for the logical system, and thus is the central uniting theme of the thesis.</p> <p>The verity-theoretic approach is also a central component of the thesis, which features in four of the five chapters. The approach is forwarded by Dorothy Edgington in her seminal paper ‘Vagueness by degrees’ (1996) and is based on a tradition of probabilistic approaches to semantics. Edgington’s suggestion is, however, that, although the semantics for vague language has the same algebraic structure as probability, the values are not to be interpreted as probabilities, but rather in a purely semantic manner as the degree to which a statement is close to being a clear case of truth.</p> <p>Chapters 1-3 use Sainsbury’s conception to develop a classical verity semantics, which I take to be the most natural and promising development of Sainsbury’s view. It is argued that verities are the degrees to which statements are close to a clear case of truth (which, for atomic predications, are the degrees to which items fall under the sphere of influence of paradigms). A notion of <em>substantiveness</em> is introduced to distinguish the notion of truth from the more prosaic, bivalent notion of truth governed by the Tarski schemes: a statement is substantively true, it is argued, just in case its truth is fixed by a metasemantic process of content fixing. A metasemantic theory is presented according to which verity assignments are fixed via their role underwriting the success of a probabilistic theory of communication.</p> <p>In the second chapter, the paradox of higher-order vagueness is discussed with particular attention to the paradox presented by Elia Zardini (2013). The verity semantics is extended to a language equipped with a <em>Def</em> operator where the statement Def(p) has the interpretation “p has verity 1”. It is argued that fine-grained modal verity semantics afford an alternative solution to the paradox: in setting up his paradox, Zardini relies on the assumption that there are clear borderline cases of all orders in a sorites series, but in the classical verity semantics proposed, although there are clear borderline cases, it is increasingly less clear that there are borderline cases of order <em>n</em>— such claims have lower verity—the greater the value of <em>n</em>.</p> <p>In chapter three, the notion of conditional verity is then addressed. Two initial attempts to elucidate the notion are rejected. The first, a theory of conditional verity based on the supervaluationist approach, is rejected on the grounds that it predicts the wrong values for conditional statements in a conditional sorites argument insofar as it fails to capture the notion of tolerance. The second, an account involving hypothetical rescaling, is initially rejected on the grounds that, despite yielding the correct values for the conditional premises of a sorites, it succumbs to a problem of indeterminacy with respect to other conditional statements. It is argued that this problem is insurmountable. However, it is concluded that some degree of indeterminacy is acceptable for the verity semanticist.</p> <p>In chapter four, the basis of a quantum logic of vagueness is presented. It is argued that Sainsbury’s conception can be seen as giving rise to a proximity space and thus can be used as the basis of quantum semantics. It is suggested that interpreting the connectives of the language relative to operations on the proximity space is plausible on the grounds that they capture the conditions under which one could permissibly make various assertions on Sainsbury’s account. Hence, there is reasonable ground for taking the logic to be quantum. This notion of permissibility can be made explicit via a modal translation into the modal system KTB (defined in Appendix 1) which provides the epistemic conditions for permissible assertion.</p> <p>In the final chapter, it is suggested that Sainsbury’s polar conception could serve as the basis of an intuitionistic logic of vagueness. In a series of papers published throughout his career, Crispin Wright establishes himself as the central proponent of the intuitionistic account of vagueness. His argument for intuitionism is based on a Brouwer-Heyting-Kolmogorov (BHK) style, knowledge-theoretic semantic theory for vague language. In this final chapter, it is argued that Wright’s account of definiteness, in combination with his semantics, commits him to a columnar theory of higher-order vagueness—in contrast to the theory presented in chapter three. It is then argued that both the columnar account and Wright’s BHK-style semantics are unsuitable. However, it is suggested that a semantics for intuitionistic logic can be given via a verity semantics, namely because the intuitionistic conditions are the strongest rules which can be justified on the basis of a plausible verity semantics when motivated by the polar conception. This strategy is proffered to the intuitionist without endorsement from the author.</p> |
| spellingShingle | Wall, L Vagueness, logic, and definite truth |
| title | Vagueness, logic, and definite truth |
| title_full | Vagueness, logic, and definite truth |
| title_fullStr | Vagueness, logic, and definite truth |
| title_full_unstemmed | Vagueness, logic, and definite truth |
| title_short | Vagueness, logic, and definite truth |
| title_sort | vagueness logic and definite truth |
| work_keys_str_mv | AT walll vaguenesslogicanddefinitetruth |