Existence, knowledge & truth in mathematics

This thesis offers an overview of some current work in the philosophy of mathematics, in particular of work on the metaphysical, epistemological, and semantic problems associated with mathematics, and it also offers a theory about what type of entities numbers are. Starting with a brief look at the historical and philosophical background to the problems of knowledge of mathematical facts and entities, the thesis then tackles in depth, and ultimately rejects as flawed, the work in this area of Hartry Field, Penelope Maddy, Jonathan Lowe, John Bigelow, and also some aspects of the work of Philip Kitcher and David Armstrong. Rejecting both nominalism and physicalism, but accepting accounts from Bigelow and Armstrong that numbers can be construed as relations, the view taken in this work is that mathematical objects, numbers in particular, are universals, and as such are mind dependent entities. It is important to the arguments leading to this conception of mathematical objects, that there is a notion of aspectual seeing involved in mathematical conception. Another important feature incorporated is the notion, derived from Anscombe, of an intentional object. This study finishes by sketching what appears to be a fruitful line of enquiry with some significant advantages over the other accounts discussed. The line taken is that the natural numbers are mind dependent intentional relations holding between intentional individuals, and that other classes of number - the rationals, the reals, and so on - are mind dependent intentional relations holding between other intentional relations. The distinction in type between the natural numbers and the rest, is the intuitive one that is drawn naturally in language between the objects referred to by the so-called count nouns, and the objects referred to by the so-called mass nouns

FUNCTIONAL DEPENDENCIES AND INCOMPLETE INFORMATION

Functional dependencies play an important role in relational database design. They are defined in the context of a single relation which at all times must contain tuples with non-null entries. In this paper we examine an extension of the functional dependency interpretation to handle null values, that is, entries in tuples that represent incomplete information in a relational database. A complete axiomatization of inference rules for extended functional dependencies is also presented. Only after having such results is it possible to talk about decompositions and normalization theory in a context of incomplete information. Finally, we show that there are several practical advantages in using nulls and a weaker notion of constraint satisfiability.Information Systems Working Papers Serie

Fundamentals and applications of order dependencies

Business-intelligence queries often involve SQL functions and algebraic expressions. There can be clear semantic relationships between a column's values and the values of a function over that column. A common property is monotonicity: as the column's values ascend, so do the function's values (or the other column's values). This we call an order dependency (OD). Queries can be evaluated more efficiently when the query optimizer uses order dependencies. They can be run even faster when the optimizer can also reason over known ODs to infer new ones. Order dependencies can be declared as integrity constraints, and they can be detected automatically for many types of SQL functions and algebraic expressions. We present optimization techniques using ODs for queries that involve join, order by, group by, partition by, and distinct. Essentially, ODs can further exploit interesting orders to eliminate or simplify potentially expensive sorts in the query plan. We evaluate these techniques over our prototype implementation in IBM® DB2® using the TPC-DS® benchmark schema and some customer inspired queries. Our experimental results demonstrate a significant performance gain. Dependencies have played an important role in database theory. We study the theoretical aspects of order dependencies-and unidirectional order dependencies (UODs), a proper sub-class of ODs-which describe the relationships among lexicographical orderings of sets of tuples. We investigate the inference problem for order dependencies. We establish the following: (i) a sound and complete axiomatization for UODs which is sound for ODs; (ii) a hierarchy of order dependency classes; (iii) a proof of co-NP-completeness of the inference problem for ODs and for the subclass of UODs; (iv) a proof of co-NP-completeness of the inference problem of functional dependencies (FDs) from ODs in general, but demonstrate linear time complexity for the inference of FDs from UODs; (v) a sound and complete elimination procedure for testing logical implication over ODs; and (vi) a sound and complete polynomial inference algorithm for sets of UODs over natural domains

Lighthouse principle for diffusion in social networks

The article investigates an influence relation between two sets of agents in a social network. It proposes a logical system that captures propositional properties of this relation valid in all threshold models of social networks with the same structure. The logical system consists of Armstrong axioms for functional dependence and an additional Lighthouse axiom. The main results are soundness, completeness, and decidability theorems for this logical system

Propositions

This encyclopedia article explores the basics of what philosophers mean by the word 'proposition.' The term 'proposition' has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other "propositional attitudes" (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of sentences