A Logic for Vagueness

This paper presents F, substructural logic designed to treat vagueness. Weaker than Lukasiewicz’s infinitely valued logic, it is presented first in a natural deduction system, then given a Kripke semantics in the manner of Routley and Meyer's ternary relational semantics for R and related systems, but in this case, the points are motivated as degrees to which the truth could be stretched. Soundness and completeness are proved, not only for the propositional system, but also for its extension with first-order quantifiers. The first-order models allow not only objects with vague properties, but also objects whose very existence is a matter of degree

Vague objects

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 2001.Includes bibliographical references (p. 143-145).Peter Unger's puzzle, the problem of the many, is an argument for the conclusion that we are grossly mistaken about what kinds of objects are in our immediate surroundings. But it is not clear what we should make of Unger's argument. There is an epistemic view which says that the argument shows that we don't know which objects are the referents of singular terms in our language. There is a linguistic view which says that Unger's puzzle shows that ordinary singular terms and count nouns are vague. Finally, there is an ontological view which says that the puzzle shows that there are vague objects. The epistemic view offers the simplest solution to the problem of the many, but runs foul of a different problem, the problem of vague reference. The problem of vague reference is that given the presuppositions of the epistemic view there are too many too similar objects that might be the reference of a name such as 'Kilimanjaro' for it to be plausible that the name has a determinate reference. The linguistic view, spelled out in terms of semantic indecision and supervaluation, offers the same solution to the problem of the many and to the problem of vague reference. But it leaves no room for de re beliefs about ordinary material object. The ontological view offers a solution to the problem of the many that avoids the problem of vague reference and the problem of de re beliefs. For these reasons it is preferable to the other two.(cont.) However, ontological vagueness has met strong objections. It has been argued that it is a fallacy of verbalism, that it is inconsistent and that once formulated in a consistent way it is not distinguishable from the linguistic view. These objections can be met, but not without cost. To avoid the charge of being inconsistent, friends of the ontological view have to give up the law of excluded middle. A positive account of vague parthood has two parts. First, parthood is not primitive but dependent on other primitive facts. The most important of the primitive facts are facts about to what kinds objects belong and how objects are causally related. Second, sometimes the primitive facts fail do determine of two objects whether one is part of the other. Given a notion of vague parthood, a notion of vague object can be defined roughly in the following way: An object 0 is vague iff there is an object a such that it is indeterminate whether a is part of 0.by Ólafur Páll Jónsson.Ph.D

Standpoint Logic: A Logic for Handling Semantic Variability, with Applications to Forestry Information

It is widely accepted that most natural language expressions do not have precise universally agreed definitions that fix their meanings. Except in the case of certain technical terminology, humans use terms in a variety of ways that are adapted to different contexts and perspectives. Hence, even when conversation participants share the same vocabulary and agree on fundamental taxonomic relationships (such as subsumption and mutual exclusivity), their view on the specific meaning of terms may differ significantly. Moreover, even individuals themselves may not hold permanent points of view, but rather adopt different semantics depending on the particular features of the situation and what they wish to communicate. In this thesis, we analyse logical and representational aspects of the semantic variability of natural language terms. In particular, we aim to provide a formal language adequate for reasoning in settings where different agents may adopt particular standpoints or perspectives, thereby narrowing the semantic variability of the vague language predicates in different ways. For that purpose, we present standpoint logic, a framework for interpreting languages in the presence of semantic variability. We build on supervaluationist accounts of vagueness, which explain linguistic indeterminacy in terms of a collection of possible interpretations of the terms of the language (precisifications). This is extended by adding the notion of standpoint, which intuitively corresponds to a particular point of view on how to interpret vague terminology, and may be taken by a person or institution in a relevant context. A standpoint is modelled by sets of precisifications compatible with that point of view and does not need to be fully precise. In this way, standpoint logic allows one to articulate finely grained and structured stipulations of the varieties of interpretation that can be given to a vague concept or a set of related concepts and also provides means to express relationships between different systems of interpretation. After the specification of precisifications and standpoints and the consideration of the relevant notions of truth and validity, a multi-modal logic language for describing standpoints is presented. The language includes a modal operator for each standpoint, such that \standb{s}\phi means that a proposition $\phi$ is unequivocally true according to the standpoint $s$ --- i.e.\ $\phi$ is true at all precisifications compatible with $s$. We provide the logic with a Kripke semantics and examine the characteristics of its intended models. Furthermore, we prove the soundness, completeness and decidability of standpoint logic with an underlying propositional language, and show that the satisfiability problem is NP-complete. We subsequently illustrate how this language can be used to represent logical properties and connections between alternative partial models of a domain and different accounts of the semantics of terms. As proof of concept, we explore the application of our formal framework to the domain of forestry, and in particular, we focus on the semantic variability of `forest'. In this scenario, the problematic arising of the assignation of different meanings has been repeatedly reported in the literature, and it is especially relevant in the context of the unprecedented scale of publicly available geographic data, where information and databases, even when ostensibly linked to ontologies, may present substantial semantic variation, which obstructs interoperability and confounds knowledge exchange

Semantics of fuzzy quantifiers

The aim of this thesis is to discuss the semantics of FQs (fuzzy quantifiers), formal semantics in particular. The approach used is fuzzy semantic based on fuzzy set theory (Zadeh 1965, 1975), i.e. we explore primarily the denotational meaning of FQs represented by membership functions. Some empirical data from both Chinese and English is used for illustration. A distinguishing characteristic of the semantics of FQs like about 200 students and many students as opposed to other sorts of quantifiers like every student and no students, is that they have fuzzy meaning boundaries. There is considerable evidence to suggest that the doctrine that a proposition is either true or false has a limited application in natural languages, which raises a serious question towards any linguistic theories that are based on a binary assumption. In other words, the number of elements in a domain that must satisfy a predicate is not precisety given by an FQ and so a proposition con¬ taining one may be more or less true depending on how closely numbers of elements approximate to a given norm. The most significant conclusion drawn here is that FQs are compositional in that FQs of the same type function in the same way to generate a constant semantic pattern. It is argued that although basic membership functions are subject to modification depending on context, they vary only with certain limits (i.e. FQs are motivated—neither completely predicated nor completely arbitrary), which does not deny compositionality in any way. A distinctive combination of compositionality and motivation of FQs makes my formal semantic framework of FQs unique in the way that although some specific values, such as a norm, have to be determined pragmatically, semantic and inferential patterns are systematic and predictable. A number of interdisciplinary implications, such as semantic, general linguistic, logic and psychological, are discussed. The study here seems to be a somewhat troublesome but potentially important area for developing theories (and machines) capable of dealing with, and accounting for, natural languages

Gradable adjectives and the semantics of locatives

This dissertation develops a semantic model of gradable adjectives such as ‘tall’, ‘good’, ‘big’, ‘heavy’, etc., within a formal semantic theory of locatives we call Locative Structure Semantics (LSS). Our central hypothesis is that gradable adjectives are, semantically, a species of locative expression. The view of gradable adjectives as locatives is inspired by the vector-based semantic models of Vector Space Semantics (VSS), as well as the notion of perspective or point of view, as found in Leonard Talmy’s research on spatial expressions (Talmy [153]) and the tradition of Situation Semantics (cf. Barwise and Perry [9, p. 39]). Following Barwise and Seligman [11], we construe the contextual variability that characterises gradable adjectives in terms of shifts in cognitive perspective. We argue that perspectives are a formal part of a semantic representational structure that is shared by expressions from several different domains, which we refer to as a locative structure (L-structure). The notion of an L-structure is influenced by Reichenbach’s notion of tense, and can be thought of as a generalisation of the Reichenbachian notion of tense to the realm of concepts. Reichenbach [134] proposed that each temporal expression is associated with three time points: a speech point, S, an event point, E, and reference point, R, where E refers to the time point corresponding to the event described by the tensed clause, S is (usually) taken to be the speaker’s time of utterance, and R is a temporal reference point relevant to the utterance. In LSS we extend this tripartite scheme to locative expressions in general, to which we assign a ternary structure comprising a Perspective, a Figure, and a Ground, represented symbolically as P, F, and G, and which are generalisations of the Reichenbachian S, E, and R, respectively. We show that a formal semantics based on L-structures enables us to capture important crosscategorial similarities between gradable adjectives, tenses, and spatial prepositions

An ontological analysis of vague motion verbs, with an application to event recognition

This research presents a methodology for the ontological formalisation of vague spatial concepts from natural language, with an application to the automatic recognition of event occurrences on video data. The main issue faced when defining concepts sourced from language is vagueness, related to the presence of ambiguities and borderline cases even in simple concepts such as ‘near’, ‘fast’, ‘big’, etc. Other issues specific to this semantic domain are saliency, granularity and uncertainty. In this work, the issue of vagueness in formal semantics is discussed and a methodology based on supervaluation semantics is proposed. This constitutes the basis for the formalisation of an ontology of vague spatial concepts based on classical logic, Event Calculus and supervaluation semantics. This ontology is structured in layers where high-level concepts, corresponding to complex actions and events, are inferred through mid-level concepts, corresponding to simple processes and properties of objects, and low-level primitive concepts, representing the most essential spatio-temporal characteristics of the real world. The development of ProVision, an event recognition system based on a logic-programming implementation of the ontology, demonstrates a practical application of the methodology. ProVision grounds the ontology on data representing the content of simple video scenes, leading to the inference of event occurrences and other high-level concepts. The contribution of this research is a methodology for the semantic characterisation of vague and qualitative concepts. This methodology addresses the issue of vagueness in ontologies and demonstrates the applicability of a supervaluationist approach to the formalisation of vague concepts. It is also proven to be effective towards solving a practical reasoning task, such as the event recognition on which this work focuses

Multivalued Logic, Neutrosophy and Schrodinger equation

This book was intended to discuss some paradoxes in Quantum Mechanics from the viewpoint of Multi-Valued-logic pioneered by Lukasiewicz, and a recent concept Neutrosophic Logic. Essentially, this new concept offers new insights on the idea of ‘identity’, which too often it has been accepted as given. Neutrosophy itself was developed in attempt to generalize Fuzzy-Logic introduced by L. Zadeh. While some aspects of theoretical foundations of logic are discussed, this book is not intended solely for pure mathematicians, but instead for physicists in the hope that some of ideas presented herein will be found useful. The book is motivated by observation that despite almost eight decades, there is indication that some of those paradoxes known in Quantum Physics are not yet solved. In our knowledge, this is because the solution of those paradoxes requires re-examination of the foundations of logic itself, in particular on the notion of identity and multi-valuedness of entity. The book is also intended for young physicist fellows who think that somewhere there should be a ‘complete’ explanation of these paradoxes in Quantum Mechanics. If this book doesn’t answer all of their questions, it is our hope that at least it offers a new alternative viewpoint for these old questions

The phenomena of vagueness

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1997.Includes bibliographical references (p. 103-105).by Delia Ruby Graff.Ph.D