The aim of this paper is to shed new light on the logical roots of arithmetic by presenting a logical framework that takes seriously ordinary locutions like ‘at least n Fs’, ‘n more Fs than Gs’ and ‘n times as many Fs as Gs’, instead of paraphrasing them away in terms of expressions of the form ‘the number of Fs’. It will be shown that the basic concepts of arithmetic can be intuitively defined in the language of ALA, and the (...) Dedekind–Peano axioms can be derived from those definitions by logical means alone. It will also be shown that some fundamental facts about cardinal numbers expressed using singular terms of the form ‘the number of Fs’, including Hume’s Principle, can be derived solely from definitions. (shrink)
ABSTRACTThis paper presents a view of quantities as ‘adverbial’ entities of a certain kind—more specifically, determinate ways, or modes, of having length, mass, speed, and the like. In doing so, it will be argued that quantities as such should be distinguished from quantitative properties or relations, and are not universals but are particulars, although they are not objects, either. A main advantage of the adverbial view over its rivals will be found in its superior explanatory power with respect to both (...) certain fundamental principles of quantity and ordinary quantitative reasoning involving quantitative relations like three times as long as and 2 metres longer than. (shrink)
The neo-Fregeans have argued that definition by abstraction allows us to introduce abstract concepts such as direction and number in terms of equivalence relations such as parallelism between lines and one-one correspondence between concepts. This paper argues that definition by abstraction suffers from the fact that an equivalence relation may not be sufficient to determine a unique concept. Frege’s original verdict against definition by abstraction is thus reinstated.
Frege’s argument against the ancient Greek conception of numbers as 'multitudes of units’ has been hailed as one of the most successful in his "Grundlagen". The aim of this paper is to show that despite Frege’s best efforts, the Euclidean conception remains a viable alternative to the Fregean conception of numbers by arguing that neither a dilemma argument Frege brings against the Euclidean conception nor a possible argument against it based on the truth of what is known as "Hume’s Principle" (...) succeeds. (shrink)
This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs exist n-wise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the so-called Caesar objection will be answered by explaining exactly what kind of manner or mode numbers are. And then what we shall call the Functionality of Cardinality objection will be answered by establishing the fact that for any numbers m and (...) n, if there are exactly m Fs and also there are exactly n Fs, then m = n. (shrink)
In a forthcoming paper in this journal, entitled “Bad company objection to Joongol Kim’s adverbial theory of numbers”, Namjoong Kim presents an ingenious Russell-style paradox based on an analogue of Kim’s definition of the number 1, and argues that Kim’s theory needs to provide a criterion of demarcation between acceptable and unacceptable definitions of adverbial entities. This paper addresses this ‘bad company’ objection and some other related issues concerning Kim’s adverbial theory by clarifying the purposes and uses of the formal (...) framework of the theory. (shrink)
This paper is an exposition and defense of Kant’s philosophy of geometry. The main thesis is that Euclidean geometry investigates the properties of geometrical objects in an inner space that is given to us a priori (pure space) and hence is a priori and synthetic. This thesis is supported by arguing that Euclidean geometry requires certain intuitive objects (Sect. 1), that these objects are a priori constructions in pure space (Sect. 2), and finally that the role of geometrical construction is (...) to provide geometrical objects, not concepts, as some have claimed (Sect. 3). (shrink)
Is it possible to characterize the sortal essence of Fs for a sortal concept F solely in terms of a criterion of identity C for F? That is, can the question ‘What sort of thing are Fs?’ be answered by saying that Fs are essentially those things whose identity can be assessed in terms of C? This paper presents a case study supporting a negative answer to these questions by critically examining the neo-Fregean suggestion that cardinal numbers can be fully (...) characterized as those things whose identity can be assessed in terms of one-one correspondence between concepts. (shrink)
[Author's note: although this paper is written in Korean, it is archived here in the hope of bringing it to the attention of a wider audience including scholars of pragmatics and of Korean linguistics.] Recently, Korean linguists and philosophers of language have engaged in discussions on the meaning and usage of the Korean determiner ‘uri’ as in such phrases as ‘uri manura [our wife]’ which might seem strange given the monogamous marital institution of Korea. The aim of this paper is (...) to provide a new interpretation of such expressions as ‘uri manura’. To that end, various proposals concerning the meaning and usage of the determiner ‘uri’ in those expressions will be critically examined first. Then it will be argued that ‘uri manura’ is a polite form of ‘nae manura [my wife]’ and is used when one needs, or wants, to speak in a polite tone in deference to the hearer or the wider audience. The upshot will be that the ubiquitous use of ‘uri’ in Korean is not due to the collectivist nature of the Korean society, as has often been claimed, but rather results from the linguistic embodiment of the Confucian tradition in Korea that values courteous words and behavior. (shrink)
This paper presents three explanations of why Frege took the universal, rather than the existential, quantifier as primitive in his formalization of logic. The first two explanations provide technical reasons related to how Frege formalizes the logic of truth-functions and the logic of quantification. The third, philosophical explanation locates the reason in Frege's logicist goal of analyzing arithmetical concepts---especially the concepts of 0 and 1---in purely logical terms.
This paper provides a critical examination of three related attempts to defend Composition as Identity (CI), namely the thesis that if some things compose something, then they are it. First, it will be argued against Donald Baxter’s view of composition as ‘loose identity’ that by construing composition as strictly a many-many relation, the view trivializes CI, and cannot be an option for the advocate of CI who takes composition as a genuine many-one relation. Second, it is argued against Baxter’s modified (...) view of composition as ‘cross-count identity’ that the ‘are’ in ‘they are it’ cannot be viewed as expressing cross-count identity. Lastly, it is argued against Aaron Cotnoir’s view of composition as ‘general identity’ that it amounts to resorting back to Baxter’s old view of composition as a many-many relation. (shrink)
Plural identity—the relation of identity between some things xx and some things yy—has been standardly defined in terms of the plural relation one of (or among). This paper challenges that standard view. To that end, it will be argued, first, that the identity relation, singular or plural, can only be defined in a higher-order language, second, that the standard definition of plural identity in terms of the one of (or among) relation should be regarded instead as providing a criterion of (...) identity for pluralities of some particular kind, and third, that there are pluralities of another kind with which a different criterion of identity is associated. The upshot will be that plural identity as standardly defined cannot be the plural counterpart of singular identity. (shrink)
The dissertation is an inquiry into the ontology and epistemology of numbers. As regards the former, the Fregean conception of numbers as objects and the Russellian conception of numbers as higher-level entities are both critically examined. A conception of numbers as modes of existence , that is, ways or manners in which things exist, is introduced and defended instead. As regards the latter, the basic concepts of arithmetic are explicated in terms of pure logic alone, and all the truths of (...) arithmetic are shown to follow from those explications solely by logical means. A new version of logicism in the philosophy of arithmetic is thereby established. (shrink)
It would seem that some statements like ‘There are exactly four moons of Jupiter’ and ‘The number of moons of Jupiter is four’ have the same truth-conditions and yet differ in ontological commitment. One strategy to resolve this paradoxical phenomenon is to insist that the statements have not only the same truth-conditions but also the same ontological commitments; the other strategy is to reject the presumption that they have the same truth-conditions. This paper critically examines some popular versions of these (...) two strategies, and defends a new solution according to which the statements have the same ontological commitments and yet differ in truth-conditions. (shrink)
Byeong-Uk Yi has argued that number words like ‘two’ primarily function as numerical predicates as in ‘Socrates and Hippias are two ’, and other grammatical uses of number words can be paraphrased in terms of the predicative use. This paper critically examines Yi’s paraphrase scheme and also some other alternative schemes, and argues that the adjectival use of number words as in ‘The Scots and the Irish are two peoples’ cannot be paraphrased in terms of the predicative use.