Kinds Of Quantifiers Research Articles

Quantifiers on languages and codensity monads

AbstractThis paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction hinges on a measure-theoretic characterisation of the profinite monad of the free S-semimodule monad for finite and commutative semirings S, which generalises our earlier insight that the Vietoris monad on Boolean spaces is the codensity monad of the finite powerset functor.

Perception, memory, and imagination as propositional attitudes


 
 
 Jaakko Hintikka started in 1969 the study of the logic of perception as a spe- cial case of his more general approach to propositional attitudes by means of the possible worlds semantics. His students and co-workers extended this study to the logic of memory and imagination. The key elements of this approach are the distinction between physical and perspectival cross-identification and the related two kinds of quantifiers, which allow a formulation of the syntax and semantics of various types of statements about perceiving, re- membering and imagining. This paper surveys the main results of these logical investigations.
 
 

Ways of being and expressivity

In this paper, I present a hermeneutic version of ontological pluralism, addressing the question of the discursive articulation of ways of being. The first section presents the notion of a pluralism of ways of being as a restriction of an ontological monism. The second section puts forward a criticism of Kris McDaniel’s proposal of understanding ways of being as kinds of quantifiers. The third section analyses the notion of way of being as a modal concept, explaining ways of being as internal possibilities endowed with a normative force regarding the identity-conditions of entities. The fourth one is a statement about the need of developing a pluralist account of the propositional reference to entities based on ontological pluralism. The fifth section deals with the issue of the discursive articulation of ways of being. The two last sections present a hypothesis concerning a semantic condition for an adequate articulation of ways of being. I argue for a kind of finitude-sensitivity in the semantics of the discursive articulation of internal possibilities, which implies the requirement of developing a hermeneutic notion of silence that may properly work in the discursive articulation of ways of being.

Mathematics students talking past each other: emergence of ambiguities in linear algebra proof constructions involving the uniqueness quantification

This paper examines proof constructions in group work in the field of linear algebra teaching at the university level. Studies have shown that students at tertiary level have difficulties in understanding different kinds of quantifiers, which are fundamental in linear algebra proof constructions. This study investigates how two student groups, with a tutor involved in one of the groups, construed meaning in the context of proving unique existence of the adjoint endomorphism. The students and the tutor used certain words and phrases in the context of mathematical uniqueness differently. The study analyses from an interactionist standpoint how these ambiguities emerged. The results indicate that due to different background understandings of mathematical uniqueness students attributed different meanings to certain words and expressions, which prevented the students from negotiating a consensus during the proving process.

Nonstandard Functional Interpretations and Categorical Models

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models.

Saber que algo no existe: existencia y lógica epistémica

In this paper we deal with a classical question of western philosophy: Is it possible to know that there are things which do not exist? We begin with a brief analysis of some fragments of Parmenides, and later we study whether these arguments are valid using three kinds of epistemic logic: a classical style epistemic logic, where axiom holds (see PDF), a free epistemic logic, where this axiom does not hold, and finally, we study Lenzen's proposal of using two kinds of quantifiers: possibilistic and realistic quantifiers. We conclude that the last last one is the closest to our intuition and a good way to deal with this kind of problem.

Saber que algo no existe: existencia y lógica epistémica

In this paper we deal with a classical question of western philosophy: Is it possible to know that there are things which do not exist? We begin with a brief analysis of some fragments of Parmenides, and later we study whether these arguments are valid using three kinds of epistemic logic: a classical style epistemic logic, where axiom holds (see PDF), a free epistemic logic, where this axiom does not hold, and finally, we study Lenzen's proposal of using two kinds of quantifiers: possibilistic and realistic quantifiers. We conclude that the last last one is the closest to our intuition and a good way to deal with this kind of problem.

Saber que algo no existe: existencia y lógica epistémica

En este artículo discutimos una pegunta clásica de la filosofía occidental: ¿es posible saber que hay cosas que no existen? Empezamos con un breve análisis de algunos fragmentos de Parménides y a continuación estudiamos si estos argumentos son válidos usando tres tipos de lógica epistémica.

La utilización de fuentes en la realización de diccionarios: ¿Covarrubias (1611), fuente directa en la microestructura de Franciosini (1620)?

The development of the lexicographer's task at the very beginning of the Spanish lexicography activity was based on the conveyance of materials from some works to others, as well as on the imposition of some canons in the way of approaching and developing them. In several occasions, the lexicographers have been accused of a lack of originality, even labels as "copy", "plagiarism" or "source" have been used to hold in contempt their works. The relationship between "Tesoro de la lengua castellana o española" (1960) from Sebastián de Covarrubias and "Vocabulario italiano-español, español-italiano" (1920) from Lorenzo Franciosini can help us to understand the lexicographer's task in that period of the origin and development of the Spanish lexicography, to be exact, with the comparison of their microstructures it can be delimited if the first dictionary is present in the second one and it can also be determined in what way it does. With this kind of quantifier and detailed analysis we can know the true relationships that exist between the dictionaries. In addition, we can avoid remaining in partial appreciations that tend to become difficult to prove truths.

Supposition as Quantification versus Supposition as Global Quantificational Effect

This paper follows up a suggestion by Paul Vincent Spade that there were two Medieval theories of the modes of personal supposition. I suggest that early work by Sherwood and others was a study of quantifiers: their semantics and the effects of context on inferences that can be made from quantified terms. Later, in the hands of Burley and others, it changed into a study of something else, a study of what I call global quantificational effect. For example, although the quantifier in ‘¬∀xPx’ is universal, it can be seen globally as having an existential effect; this is because the formula containing it is equivalent to ‘∃x¬Px’. The notion of global effect can be explained in terms of the modern theory of normal forms. I suggest that early authors were studying quantifiers, and the terminology of the theory of personal supposition is a classification of kinds of quantifiers. In this theory, to say that a term has distributive supposition is to say, roughly, that it is quantified by a universal quantifying sign. Later authors turned this into a theory of global quantificational effect. In the later theory, to say that a term has distributive supposition is to say that the overall effect is as if the term were universally quantified with a quantifier taking (relatively) wide scope. The difference between these two approaches is illustrated by the fact that the term ‘man’ is classified as having distributive ("universal") supposition in ‘Not every man is running’ in the earlier theory, whereas in the later theory that term does not have distributive supposition; it has determinate ("existential") supposition. In the paper I explain these options, and I argue from several texts that the earlier and later medieval theories actually worked like this. In an appendix I make further efforts to clarify the obscure early accounts, as well as the nineteenth century "doctrine of distribution". The last section of the paper discusses the "purpose(s)" of supposition theory.

Frege on Cardinality

Frege invented a theory of quantification in order to provide a means to test validity of proofs in mathematics. A striking innovation in this theory was to construe quantifiers as second-level which applied to first-level concepts. A few years later he introduced his theory of arithmetic in Die Grundlagen der Arithmetik. In Grundlagen, Frege launched an extensive critique of views of others concerning arithmetic. After finding these views unsatisfactory, he concluded that the content of a statement of number is an assertion about a ([i], ?46). Although he presented his view as an improvement on earlier attempts, failure of other views served mainly as independent support for an already motivated theory. This central thesis of Grundlagen follows from earlier work on logic, for attributions of are special cases of Frege's theory of quantification. According to this result, attributions of apply higher-level cardinality concepts to lower-level concepts. However, Frege did not conclude that cardinal numbers are kinds of quantifiers. In Grundlagen he proceeded to argue that cardinal numbers are objects, not concepts. He finally defined numbers as certain objects, extensions of those concepts. The close connection between quantification and has not received attention it deserves. This is not surprising, for it is in apparent conflict with thesis that numbers are objects. Frege's ontology encompassed two ontological categories, function and object.' These ontological categories were mutually exclusive; no function was an object. And were a type of function, namely those which took a single argument, yielding a truth-value. So no concept could be an object. Frege introduced quantifiers as certain second-level concepts, with no

Counting and the Natural Numbers

Early sections of the paper develop a view of the natural numbers and a view of counting which are suggested by the remarks of several modern philosophers. Further investigation of these views leads to one of the main theses of the paper : a special kind of quantifier, the “numerical quantifier” is essential to counting. The remainder of the paper suggests the rudiments of a new view of the natural numbers, a view which maintains that numerical quantifiers are one kind of natural number.