Properties of primes and natural mathematics: a minimalist algorithm for prime numbers

Este trabalho discute a quantificação precisa por meios de sistemas numéricos em analogia à análise anterior de Jaspers (2005) a respeito da quantificação comparativamente vaga expressa por operadores do cálculo de predicados {todos, todo, cada, algum, nenhum}. É defendido que números oferecem um interessante teste-base para a validade da abordagem Booleana aos quantificadores (Jaspers, 2005). Mais detidamente, esta excursão na matemática é realizada para mostrar que o mesmo sistema lógico-cognitivo de oposições subjacente na língua natural também governa a matemática natural. O ponto de partida concreto do artigo é o problema dos “twin primes” de Popper, que é seguido por uma discussao de sistemas de números, sobretudo a distinção entre sistema dos numeros naturais {(0,) 1, 2,...} e o sistema de números primos. Em relação ao primeiro será defendido que é organizado pela operação de adição/subtração. A sequência de números primos é diferente, porque é mais multiplicativa/divisional que aditiva. É geralmente reconhecido em círculos matemáticos que o último tipo de sequência é mais complexo que o primeiro. Este fato acompanha bem (e, portanto, oferece suporte indireto para) as descobertas linguísticas em Jaspers (2005), cujo o núcleo foi a defesa que disjunção na língua natural _ conhecida por ser isomórfica à adicao na álgebra álgebra - é cognitiva e lexicalmente mais complexa que a conjunção, que é isomórfica à multiplicação

The English Tenses, Blanché and the Logical Kite

There exists a very systematic limitation on natural compositional concept formation in natural language whose full complexity has not been laid bare. The first section of this chapter will revisit the lexical domains of logical constants, where conceptual “kinship and contrast” define a pattern of opposition first represented in the form of a kite in [21]. Next, I propose a kite analysis for the system of English finite tenses which has a partial precursor in a now-forgotten little article by Robert Blanché on the one hand and in the much more detailed system of binary temporal relations proposed by Vikner [23] on the other. From the former the kite structure for the finite tenses differs in that it focuses on the inviolability of prior binary divisions for later ones – which is the operation in this realm of progressive universe restriction (PUR) as defined in [21] – and on resulting asymmetries in the tense system; from the latter it differs in proposing a system of two pairs of binary oppositions, a temporal pair and an aspectual pair, yielding four binary relations in all (rather than three), with a surprising similarity between the internal architecture of the temporal and the aspectual pair of relations. On the whole, the pattern suggests that basic conceptual oppositions and lexicalization principles are guided by innate linguistic patterns of which binarity, opposition, and asymmetry are the central properties. If correct, the analysis proposed is striking confirmation in yet another semantic domain of the asymmetry approach to the square of opposition and its extensions of our earlier work.status: publishe

Logic and Colour in Cognition, Logic and Philosophy

Colour has been on the minds of philosophers, logicians and linguists for a very long time: its connection with logic; the relation between percepts and concepts; the influence of colour language on colour thought, and the question whether colour is in the world or purely mental. The present contribution starts from the age-old observation that the four logical opposition relations (contradiction, (sub)contrariety, entailment) as embodied in the square of opposition and extensions thereof such as the Blanché hexagon are not particularly choosy about the actual conceptual content of the lexical fields they organize. They generalize over modal operators, propositional operators, predicate calculus operators, tense operators, etc. Why would that be? In the first place, it must mean that the relations are located at a level of generality which has no regard for the concrete conceptual differences between those fields, latching on to some very general substrate. Actually, taking that idea further, I have argued (Jaspers, Logica Universalis 6: 227–248, 2012) that the level of abstraction away from concrete incarnations in the vertices of the hexagon even needs to break out of the conceptual realm into that of colour percepts, where a homologous patterning among primary and secondary colour percepts obtains.status: publishe

Properties of primes and natural mathematics

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Properties of primes and natural mathematics A minimalist algorithm for prime numbers

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Logic and colour

In this paper evidence will be provided that Wittgenstein’s intuition about the logic of colour relations is to be taken near-literally. Starting from the Aristotelian oppositions between propositions as represented in the logical square of oppositions on the one hand and oppositions between primary and secondary colors as represented in an octahedron on the other, it will be shown algebraically how definitions for the former carry over to the realm of colour categories and describe very precisely the relations obtaining between the known primary and secondary colours. Linguistic evidence for the reality of the resulting isomorphism will be provided. For example, the vertices that resist natural single-item lexicalization in logic (such as the O-corner, for which there is no natural lexicalization *nall (=not all)) are not naturally lexicalized in the realm of colour terms either. From the perspective of the architecture of cognition, the isomorphism suggests that the foundations of logical oppositions and negation may well be much more deeply rooted in the physiological structure of human cognition than is standardly assumed.status: publishe

On Full Interpretation

This paper defines domains for bounding in terms of an argument-taking lexical category which, in order to be able to assign all the theta-roles associated with it, requires the presence of a minimal syntactic environment. Depending on the nature of the functional heads INFL and COMP the minimal node bounding a finite clause is either S or CP. The existence of Domains and their delineation derives from the Principle of Full Interpretation. With respect to the format of a uniform locality condition for both binding and bounding, a proposal will be made which redefines Jan Koster's (1978) relation R as a function which takes a designated XP (minimally) counting the 'dependent' constituent as its value, rather than the dependent element itself.status: publishe

Categories and recursion

In this paper a relation is established between the number of categories a system uses and the number of recursive steps required to generate a designated representation. This category/recursion-ratio is proposed on the basis of a comparison between different counting systems. The nature of syntactic concatenation is such that it involves not only merger but also movement of constituents, which sets off ordinary natural language from counting systems and leads to the elaboration of a strictly derivational grammar model.status: publishe

The Language of Science and the Science of Language. On Natural Words and Nonnatural Terminology

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