Our ability to arrive at knowledge by chains of judgment is constitutive of our rationality, likewise our ability to discern the self-evidence of logical and arithmetical laws. To count an activity as ＂thinking about...Our ability to arrive at knowledge by chains of judgment is constitutive of our rationality, likewise our ability to discern the self-evidence of logical and arithmetical laws. To count an activity as ＂thinking about the physical world＂ is to hold it assessable in the light of the laws of physics; whereas to count an activity as ＂thinking at all＂ is to hold it assessable in the light of the laws of logic. Thus, the kind of generality that distinguishes logic from the special sciences is a generality in the applicability of the norms it provides. Logical laws are more general than laws of the special sciences because they prescribe universally the way in which one ought to think, if one is to think all. Logicism is usually understood to be the thesis that all, or at least large parts of, mathematics can be reduced to logic. This thesis has two sides： （1） all mathematical concepts can be defined in terms of basic logical concepts; （2） all mathematical theorems can be deduced from basic logical truths. According to logicism all terms, including all mathematical terms, are to be given a definite meaning within the basic system. This paper aims at a comparative analysis of the contributions of Frege and Russell to the development of modem logic by reviewing in some detail their essential features and derivations. Without making any pretensions to proffering a definitive resolution of any puzzles, the discussion will, however, raise some fundamental questions, and offer a critical evaluation of the putative success or failure of the logicist programmes of Frege and Russell.展开更多
One of the remarkable results of Frege＇s Logicism is Frege＇s Theorem, which holds that one can derive the main truths of Peano arithmetic from Hume＇s Principle （HP） without using Frege＇s Basic Law V. This result...One of the remarkable results of Frege＇s Logicism is Frege＇s Theorem, which holds that one can derive the main truths of Peano arithmetic from Hume＇s Principle （HP） without using Frege＇s Basic Law V. This result was rediscovered by the Neo-Fregeans and their allies. However, when applied in developing a more advanced theory of mathematics, their fundamental principles--the abstraction principles--incur some problems, e.g., that of inflation. This paper finds alternative paths for such inquiry in extensionalism and object theory.展开更多
This paper aims to examine the general issue of how reference is possible in philosophy of language through a case analysis of the ＂double reference＂ semantic-syntactic structure of ideographic hexagram （guaxiang ...This paper aims to examine the general issue of how reference is possible in philosophy of language through a case analysis of the ＂double reference＂ semantic-syntactic structure of ideographic hexagram （guaxiang 卦象） names in the Yijing text. I regard the case of the ＂hexagram＂ names as being quite representative of the ＂double-reference＂ semantic-syntactic structure of referring names. I thus explore how the general morals drawn from this account of ＂hexagram＂ names can engage two representative approaches, the Fregean and Kripkean ones, and contribute to our understanding and treatment of the issue of reference.展开更多
A very interesting account of the reference of number words in classical Indian philosophy was given by Mahesa Chandra (1836 -1906) in his Brief Notes on the Modern Nyaya System of Philosophy and its Technical Terms (...A very interesting account of the reference of number words in classical Indian philosophy was given by Mahesa Chandra (1836 -1906) in his Brief Notes on the Modern Nyaya System of Philosophy and its Technical Terms (BN), a primer on NavyaNyaya terminology and doctrines. Despite its English title, BN is a Sanskrit work. The section on"number"(samkhya) provides an exposition of a theory of number which can account for both the adjectival and the substantival use of number words in Sanskrit. According to D. H. H. Ingalls (19161999), some ideas about the reference of number words in BN are close to the FregeRussell theory of natural number. Ingalls's comparison refers to a concept of number in Navya-Nyaya which is related to the things numbered via the SO called"circumtaining relation"(paryapti). Although there is no theory of sets in Navya-Nyaya, Navya-Naiyayikas do have a realist theory of properties (dharma) and their theory of number is a theory of properties as constituents of empirical reality, anchored to their system of ontological categories. As shown by George Bealer, properties can serve the same purpose as sets in the Frege -Russell theory of natural number. In the present paper, we attempt a formal reconstruction of Mahesa Chandra's exposition of the Navya-Nyaya theory of number, which accounts for its affinity to George Bealer's neo-Fregean analysis. As part of our appraisal of the momentousness and robustness of the"circumtaining"concept of number, we show that it can be cast into a precise recursive definition of natural number and we prove property versions of Peano's axioms from this definition.展开更多
Semantic relationism is a methodology proposed by Kit Fine for solving the antinomy of variables. Fine proposed the relational semantics for first-order logic, which can be used to solve Frege＇s puzzle of names. In t...Semantic relationism is a methodology proposed by Kit Fine for solving the antinomy of variables. Fine proposed the relational semantics for first-order logic, which can be used to solve Frege＇s puzzle of names. In this paper, I generalize Frege＇s puzzle to other linguistic expressions, including definite descriptions, predicates, quantifiers and modalities. I then apply Fine＇s semantic relationistic approach to these puzzles.展开更多
Frege argued that a predicate was a functional expression and the reference of it a concept, which as a predicative function had one or more empty places and was thus incomplete. Frege＇s view gives rise to what has b...Frege argued that a predicate was a functional expression and the reference of it a concept, which as a predicative function had one or more empty places and was thus incomplete. Frege＇s view gives rise to what has been known as the paradox of the concept ＂horse.＂ In order to resolve this paradox, I argue for an opposite view which retains the point that a predicate is a function, i.e. that a predicative function is complete in a sense. Specifically speaking, a predicate performing the function of a predicate has at least one empty place and has no reference, while a predicate performing the function of a subject does not have any empty place but does have a reference. Frege not only regarded a concept with one or more empty places as the reference of a predicate but also took a set of objects without any empty place to be the extension of a concept with one or more empty places. Thus, it presents a complex relationship between the reference of a predicate and its corresponding extension, leading to disharmony in his theory. I argue that this is because there is a major defect in Frege＇s theory of meaning, namely the neglect of common names. What he called extensions of concepts are actually extensions of common names, and the references of predicates and the extensions of common names have a substantial difference despite being closely related.展开更多
This paper compares Frege＇s philosophy of mathematics with a naturalistic and nominalistic philosophy of mathematics developed in Ye （2010a, 2010b, 2010c, 2011）, and it defends the latter against the former. The pa...This paper compares Frege＇s philosophy of mathematics with a naturalistic and nominalistic philosophy of mathematics developed in Ye （2010a, 2010b, 2010c, 2011）, and it defends the latter against the former. The paper focuses on Frege＇s account of the applicability of mathematics in the sciences and his conceptual realism. It argues that the naturalistic and nominalistic approach fares better than the Fregean approach in terms of its logical accuracy and clarity in explaining the applicability of mathematics in the sciences, its ability to reveal the real issues in explaining human epistemic and semantic access to objects, its prospect for resolving internal difficulties and developing into a full-fledged theory with rich details, as well its consistency with other areas of our scientific knowledge. Trivial criticisms such as ＂Frege is against naturalism here and therefore he is wrong＂ will be avoided as the paper tries to evaluate the two approaches on a neutral ground by focusing on meta-theoretical features such as accuracy, richness of detail, prospects for resolving internal issues, and consistency with other knowledge. The arguments in this paper apply not merely to Frege＇s philosophy. They apply as well to all philosophies that accept a Fregean account of the applicability of mathematics or accept conceptual realism. Some of these philosophies profess to endorse naturalism.展开更多
文摘Our ability to arrive at knowledge by chains of judgment is constitutive of our rationality, likewise our ability to discern the self-evidence of logical and arithmetical laws. To count an activity as ＂thinking about the physical world＂ is to hold it assessable in the light of the laws of physics; whereas to count an activity as ＂thinking at all＂ is to hold it assessable in the light of the laws of logic. Thus, the kind of generality that distinguishes logic from the special sciences is a generality in the applicability of the norms it provides. Logical laws are more general than laws of the special sciences because they prescribe universally the way in which one ought to think, if one is to think all. Logicism is usually understood to be the thesis that all, or at least large parts of, mathematics can be reduced to logic. This thesis has two sides： （1） all mathematical concepts can be defined in terms of basic logical concepts; （2） all mathematical theorems can be deduced from basic logical truths. According to logicism all terms, including all mathematical terms, are to be given a definite meaning within the basic system. This paper aims at a comparative analysis of the contributions of Frege and Russell to the development of modem logic by reviewing in some detail their essential features and derivations. Without making any pretensions to proffering a definitive resolution of any puzzles, the discussion will, however, raise some fundamental questions, and offer a critical evaluation of the putative success or failure of the logicist programmes of Frege and Russell.
文摘One of the remarkable results of Frege＇s Logicism is Frege＇s Theorem, which holds that one can derive the main truths of Peano arithmetic from Hume＇s Principle （HP） without using Frege＇s Basic Law V. This result was rediscovered by the Neo-Fregeans and their allies. However, when applied in developing a more advanced theory of mathematics, their fundamental principles--the abstraction principles--incur some problems, e.g., that of inflation. This paper finds alternative paths for such inquiry in extensionalism and object theory.
文摘This paper aims to examine the general issue of how reference is possible in philosophy of language through a case analysis of the ＂double reference＂ semantic-syntactic structure of ideographic hexagram （guaxiang 卦象） names in the Yijing text. I regard the case of the ＂hexagram＂ names as being quite representative of the ＂double-reference＂ semantic-syntactic structure of referring names. I thus explore how the general morals drawn from this account of ＂hexagram＂ names can engage two representative approaches, the Fregean and Kripkean ones, and contribute to our understanding and treatment of the issue of reference.
文摘A very interesting account of the reference of number words in classical Indian philosophy was given by Mahesa Chandra (1836 -1906) in his Brief Notes on the Modern Nyaya System of Philosophy and its Technical Terms (BN), a primer on NavyaNyaya terminology and doctrines. Despite its English title, BN is a Sanskrit work. The section on"number"(samkhya) provides an exposition of a theory of number which can account for both the adjectival and the substantival use of number words in Sanskrit. According to D. H. H. Ingalls (19161999), some ideas about the reference of number words in BN are close to the FregeRussell theory of natural number. Ingalls's comparison refers to a concept of number in Navya-Nyaya which is related to the things numbered via the SO called"circumtaining relation"(paryapti). Although there is no theory of sets in Navya-Nyaya, Navya-Naiyayikas do have a realist theory of properties (dharma) and their theory of number is a theory of properties as constituents of empirical reality, anchored to their system of ontological categories. As shown by George Bealer, properties can serve the same purpose as sets in the Frege -Russell theory of natural number. In the present paper, we attempt a formal reconstruction of Mahesa Chandra's exposition of the Navya-Nyaya theory of number, which accounts for its affinity to George Bealer's neo-Fregean analysis. As part of our appraisal of the momentousness and robustness of the"circumtaining"concept of number, we show that it can be cast into a precise recursive definition of natural number and we prove property versions of Peano's axioms from this definition.
文摘Semantic relationism is a methodology proposed by Kit Fine for solving the antinomy of variables. Fine proposed the relational semantics for first-order logic, which can be used to solve Frege＇s puzzle of names. In this paper, I generalize Frege＇s puzzle to other linguistic expressions, including definite descriptions, predicates, quantifiers and modalities. I then apply Fine＇s semantic relationistic approach to these puzzles.
文摘Frege argued that a predicate was a functional expression and the reference of it a concept, which as a predicative function had one or more empty places and was thus incomplete. Frege＇s view gives rise to what has been known as the paradox of the concept ＂horse.＂ In order to resolve this paradox, I argue for an opposite view which retains the point that a predicate is a function, i.e. that a predicative function is complete in a sense. Specifically speaking, a predicate performing the function of a predicate has at least one empty place and has no reference, while a predicate performing the function of a subject does not have any empty place but does have a reference. Frege not only regarded a concept with one or more empty places as the reference of a predicate but also took a set of objects without any empty place to be the extension of a concept with one or more empty places. Thus, it presents a complex relationship between the reference of a predicate and its corresponding extension, leading to disharmony in his theory. I argue that this is because there is a major defect in Frege＇s theory of meaning, namely the neglect of common names. What he called extensions of concepts are actually extensions of common names, and the references of predicates and the extensions of common names have a substantial difference despite being closely related.
文摘This paper compares Frege＇s philosophy of mathematics with a naturalistic and nominalistic philosophy of mathematics developed in Ye （2010a, 2010b, 2010c, 2011）, and it defends the latter against the former. The paper focuses on Frege＇s account of the applicability of mathematics in the sciences and his conceptual realism. It argues that the naturalistic and nominalistic approach fares better than the Fregean approach in terms of its logical accuracy and clarity in explaining the applicability of mathematics in the sciences, its ability to reveal the real issues in explaining human epistemic and semantic access to objects, its prospect for resolving internal difficulties and developing into a full-fledged theory with rich details, as well its consistency with other areas of our scientific knowledge. Trivial criticisms such as ＂Frege is against naturalism here and therefore he is wrong＂ will be avoided as the paper tries to evaluate the two approaches on a neutral ground by focusing on meta-theoretical features such as accuracy, richness of detail, prospects for resolving internal issues, and consistency with other knowledge. The arguments in this paper apply not merely to Frege＇s philosophy. They apply as well to all philosophies that accept a Fregean account of the applicability of mathematics or accept conceptual realism. Some of these philosophies profess to endorse naturalism.
