In recent years there has been a resurgence of interest in logicism as a viable philosophy of mat... more In recent years there has been a resurgence of interest in logicism as a viable philosophy of mathematics, stemming in great part from Crispin Wright’s Frege’s Conception of Numbers as Objects [1984] and the formal and philosophical work of George Boolos. Before this work it was generally accepted that Frege’s project of reducing mathematics to pure logic was devastated by
This chapter is concerned with domain-specific alethic pluralism and domain-specific logical plur... more This chapter is concerned with domain-specific alethic pluralism and domain-specific logical pluralism. If domain-specific alethic pluralism entails domain-specific logical pluralism, and vice versa, then in some sense we really only have one pluralism, not two. If, however, the two sorts of pluralism are independent of each other, then we truly have two distinct kinds of pluralism—that is, we have a plurality of pluralisms. The purpose of this chapter is to argue that domain-specific alethic pluralism does not entail domain-specific logical pluralism (contrary to arguments given by Lynch and Pedersen), nor does domain-specific logical pluralism entail domain-specific alethic pluralism, and hence we do have such a plurality of pluralisms. To accomplish this, in Sect. 2 I show how one can be a domain-specific logical pluralist while being a truth monist, and how one can be a domain-specific truth pluralist while being a logical monist. I will then, in Sect. 3, use the argument of Sect. 2 to identify the mistake in the arguments of Lynch and Pedersen. Section 4 will then further flesh out the model, distinguishing between different senses in which a domain might be epistemically constrained.
ABSTRACT:Issue #500 of Detective Comics contains a short two-page story titled "Once Upon a ... more ABSTRACT:Issue #500 of Detective Comics contains a short two-page story titled "Once Upon a Time," based on Snoopy's fictional story "A Dark and Stormy Night." This essay explores questions about actual and fictional authorship raised by the multi-layered intertextual narrative structure that results, where in some sense Snoopy authored a Batman comic.
The majority of disucssions of Benardete’s Paradox conclude that the traveller approaching the in... more The majority of disucssions of Benardete’s Paradox conclude that the traveller approaching the infinite series of gods will be mysteriously halted despite none of the gods erecting any barriers. Using a revision-theoretic analysis of Benardete’s puzzle, four distinct possible outcomes that might occur given Benardete’s set-up are distinguished. This analysis provides additional insight into the puzzle at hand, via identifying heretofore unnoticed possible outcomes, but it also serves as an example of how the revision theoretic framework can be used to construct exhaustive taxonomies of potential outcomes in apparently contradictory situations.
The Embracing Revenge account of semantic paradox avoids the expressive limitations of previous a... more The Embracing Revenge account of semantic paradox avoids the expressive limitations of previous approaches based on the Kripkean fixed point construction by replacing a single language with an indefinitely extensible sequence of languages, each of which contains the resources to fully characterize the semantics of the previous languages. In this paper we extend the account developed in Cook (2008), Cook (2009), Schlenker (2010), and Tourville and Cook (2016) via the addition of intensional operators such as ``is paradoxical''. In this extended framework we are able to characterize the difference between sentences, such as the Liar and the Truth-teller, that receive the same semantic value in minimal fixed points yet seem to involve distinct semantic phenomena.
In this paper we compare the propositional logic of Frege’s Grundgesetze der Arithmetik to modern... more In this paper we compare the propositional logic of Frege’s Grundgesetze der Arithmetik to modern propositional systems, and show that Frege does not have a separable propositional logic, definable in terms of primitives of Grundgesetze, that corresponds to modern formulations of the logic of “not”, “and”, “or”, and “if…then…”. Along the way we prove a number of novel results about the system of propositional logic found in Grundgesetze, and the broader system obtained by including identity. In particular, we show that the propositional connectives that are definable in terms of Frege’s horizontal, negation, and conditional are exactly the connectives that fuse with the horizontal, and we show that the logical operators that are definable in terms of the horizontal, negation, the conditional, and identity are exactly the operators that are invariant with respect to permutations on the domain that leave the truth-values fixed. We conclude with some general observations regarding how ...
It is sometimes suggested that impure sets are spatially co-located with their members (and hence... more It is sometimes suggested that impure sets are spatially co-located with their members (and hence are located in space). Sets, however, are in important respects like numbers. In particular, sets are connected to concepts in much the same manner as numbers are connected to concepts—in both cases, they are fundamentally abstracts of (or corresponding to) concepts. This parallel between the structure of sets and the structure of numbers suggests that the metaphysics of sets and the metaphysics of numbers should parallel each other in relevant ways. This entails, in turn, that impure sets are not co-located with their members (nor are they located in space).
We shall assume throughout this essay that platonism1 regarding the subject matter of mathematics... more We shall assume throughout this essay that platonism1 regarding the subject matter of mathematics is correct. This is not to say that arguments for platonism–that is, arguments for the existence of abstract objects such as numbers, sets, Hilbert spaces, and so on that comprise the subject matter of mathematics–are either uninteresting or unneeded. Nevertheless, I am not personally interested in such arguments. The reason for this is simple: I have never2 doubted that there are abstract objects, and I have never doubted ...
From Ontos Verlag Publications of the Austrian Ludwig Wittgenstein Society New Series, Nov 3, 2013
The Liar Paradox is constructed within Frege's Grundgesetze using a variant of Gödel's diagonaliz... more The Liar Paradox is constructed within Frege's Grundgesetze using a variant of Gödel's diagonalization lemma. The particular instance of Basic Law V that triggers the Liar paradox is identified, and it is observed that this is exactly the principle that Frege himself identified as the root of Russell's paradox in the appendix to Volume II of the Grundgesetze. Unfortunately, the amended version of Basic Law V which Frege suggests as a patch to his system blocks neither the derivation of the diagonalization theorem nor the construction of the Liar paradox.
The Revenge Problem threatens every approach to the semantic paradoxes that proceeds by introduci... more The Revenge Problem threatens every approach to the semantic paradoxes that proceeds by introducing nonclassical semantic values. Given any such collection Δ of additional semantic values, one can construct a Revenge sentence:This sentence is either false or has a value in Δ.TheEmbracing Revengeview, developed independently by Roy T. Cook and Phlippe Schlenker, addresses this problem by suggesting that the class of nonclassical semantic values is indefinitely extensible, with each successive Revenge sentence introducing a new ‘pathological’ semantic value into the discourse. The view is explicitly motivated in terms of the idea that every notion thatseemsto be expressible (e.g., “has a value in Δ”, for any definite collection of semantic values Δ) should, if at all possible,beexpressible. Extant work on the Embracing Revenge view has failed to live up to this promise, since the formal languages developed within such work are expressively impoverished. We rectify this here by develop...
In recent years there has been a resurgence of interest in logicism as a viable philosophy of mat... more In recent years there has been a resurgence of interest in logicism as a viable philosophy of mathematics, stemming in great part from Crispin Wright’s Frege’s Conception of Numbers as Objects [1984] and the formal and philosophical work of George Boolos. Before this work it was generally accepted that Frege’s project of reducing mathematics to pure logic was devastated by
This chapter is concerned with domain-specific alethic pluralism and domain-specific logical plur... more This chapter is concerned with domain-specific alethic pluralism and domain-specific logical pluralism. If domain-specific alethic pluralism entails domain-specific logical pluralism, and vice versa, then in some sense we really only have one pluralism, not two. If, however, the two sorts of pluralism are independent of each other, then we truly have two distinct kinds of pluralism—that is, we have a plurality of pluralisms. The purpose of this chapter is to argue that domain-specific alethic pluralism does not entail domain-specific logical pluralism (contrary to arguments given by Lynch and Pedersen), nor does domain-specific logical pluralism entail domain-specific alethic pluralism, and hence we do have such a plurality of pluralisms. To accomplish this, in Sect. 2 I show how one can be a domain-specific logical pluralist while being a truth monist, and how one can be a domain-specific truth pluralist while being a logical monist. I will then, in Sect. 3, use the argument of Sect. 2 to identify the mistake in the arguments of Lynch and Pedersen. Section 4 will then further flesh out the model, distinguishing between different senses in which a domain might be epistemically constrained.
ABSTRACT:Issue #500 of Detective Comics contains a short two-page story titled "Once Upon a ... more ABSTRACT:Issue #500 of Detective Comics contains a short two-page story titled "Once Upon a Time," based on Snoopy's fictional story "A Dark and Stormy Night." This essay explores questions about actual and fictional authorship raised by the multi-layered intertextual narrative structure that results, where in some sense Snoopy authored a Batman comic.
The majority of disucssions of Benardete’s Paradox conclude that the traveller approaching the in... more The majority of disucssions of Benardete’s Paradox conclude that the traveller approaching the infinite series of gods will be mysteriously halted despite none of the gods erecting any barriers. Using a revision-theoretic analysis of Benardete’s puzzle, four distinct possible outcomes that might occur given Benardete’s set-up are distinguished. This analysis provides additional insight into the puzzle at hand, via identifying heretofore unnoticed possible outcomes, but it also serves as an example of how the revision theoretic framework can be used to construct exhaustive taxonomies of potential outcomes in apparently contradictory situations.
The Embracing Revenge account of semantic paradox avoids the expressive limitations of previous a... more The Embracing Revenge account of semantic paradox avoids the expressive limitations of previous approaches based on the Kripkean fixed point construction by replacing a single language with an indefinitely extensible sequence of languages, each of which contains the resources to fully characterize the semantics of the previous languages. In this paper we extend the account developed in Cook (2008), Cook (2009), Schlenker (2010), and Tourville and Cook (2016) via the addition of intensional operators such as ``is paradoxical''. In this extended framework we are able to characterize the difference between sentences, such as the Liar and the Truth-teller, that receive the same semantic value in minimal fixed points yet seem to involve distinct semantic phenomena.
In this paper we compare the propositional logic of Frege’s Grundgesetze der Arithmetik to modern... more In this paper we compare the propositional logic of Frege’s Grundgesetze der Arithmetik to modern propositional systems, and show that Frege does not have a separable propositional logic, definable in terms of primitives of Grundgesetze, that corresponds to modern formulations of the logic of “not”, “and”, “or”, and “if…then…”. Along the way we prove a number of novel results about the system of propositional logic found in Grundgesetze, and the broader system obtained by including identity. In particular, we show that the propositional connectives that are definable in terms of Frege’s horizontal, negation, and conditional are exactly the connectives that fuse with the horizontal, and we show that the logical operators that are definable in terms of the horizontal, negation, the conditional, and identity are exactly the operators that are invariant with respect to permutations on the domain that leave the truth-values fixed. We conclude with some general observations regarding how ...
It is sometimes suggested that impure sets are spatially co-located with their members (and hence... more It is sometimes suggested that impure sets are spatially co-located with their members (and hence are located in space). Sets, however, are in important respects like numbers. In particular, sets are connected to concepts in much the same manner as numbers are connected to concepts—in both cases, they are fundamentally abstracts of (or corresponding to) concepts. This parallel between the structure of sets and the structure of numbers suggests that the metaphysics of sets and the metaphysics of numbers should parallel each other in relevant ways. This entails, in turn, that impure sets are not co-located with their members (nor are they located in space).
We shall assume throughout this essay that platonism1 regarding the subject matter of mathematics... more We shall assume throughout this essay that platonism1 regarding the subject matter of mathematics is correct. This is not to say that arguments for platonism–that is, arguments for the existence of abstract objects such as numbers, sets, Hilbert spaces, and so on that comprise the subject matter of mathematics–are either uninteresting or unneeded. Nevertheless, I am not personally interested in such arguments. The reason for this is simple: I have never2 doubted that there are abstract objects, and I have never doubted ...
From Ontos Verlag Publications of the Austrian Ludwig Wittgenstein Society New Series, Nov 3, 2013
The Liar Paradox is constructed within Frege's Grundgesetze using a variant of Gödel's diagonaliz... more The Liar Paradox is constructed within Frege's Grundgesetze using a variant of Gödel's diagonalization lemma. The particular instance of Basic Law V that triggers the Liar paradox is identified, and it is observed that this is exactly the principle that Frege himself identified as the root of Russell's paradox in the appendix to Volume II of the Grundgesetze. Unfortunately, the amended version of Basic Law V which Frege suggests as a patch to his system blocks neither the derivation of the diagonalization theorem nor the construction of the Liar paradox.
The Revenge Problem threatens every approach to the semantic paradoxes that proceeds by introduci... more The Revenge Problem threatens every approach to the semantic paradoxes that proceeds by introducing nonclassical semantic values. Given any such collection Δ of additional semantic values, one can construct a Revenge sentence:This sentence is either false or has a value in Δ.TheEmbracing Revengeview, developed independently by Roy T. Cook and Phlippe Schlenker, addresses this problem by suggesting that the class of nonclassical semantic values is indefinitely extensible, with each successive Revenge sentence introducing a new ‘pathological’ semantic value into the discourse. The view is explicitly motivated in terms of the idea that every notion thatseemsto be expressible (e.g., “has a value in Δ”, for any definite collection of semantic values Δ) should, if at all possible,beexpressible. Extant work on the Embracing Revenge view has failed to live up to this promise, since the formal languages developed within such work are expressively impoverished. We rectify this here by develop...
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