Results 11  20
of
34
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
ENGLISH VERBPREPOSITION CONSTRUCTIONS: CONSTITUENCY AND ORDER
"... This article offers a comprehensive analysis of the constituentstructure and linearorder properties of English transitive and intransitive VP constructions involving socalled ‘particles ’ (turn on the lights/the lights on, mess up the song/the song up, shut up, sit down, etc.). Drawing on both s ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
This article offers a comprehensive analysis of the constituentstructure and linearorder properties of English transitive and intransitive VP constructions involving socalled ‘particles ’ (turn on the lights/the lights on, mess up the song/the song up, shut up, sit down, etc.). Drawing on both standard and certain new evidence and arguments, it is proposed that VP constructions generally come in one or both of two varieties: lexical compounds (mess up in mess up the song) and/or discontinuous verbs, that is, lexemes with more than one piece projected as a word or phrase (mess... up in mess the song up), and that the alternation, for those that have both manifestations, reflects different argument structure possibilities for a lexeme with the same overall conceptual semantics. The internal structure of VPs built on VP lexemes is examined in some detail. The popular ‘smallclause ’ approach, according to which the DP of transitive VP structures is the subject of a phrase that has the P as its predicate, is shown to be problematic, primarily because there in fact exists a true smallclause construction that can have aPasitspredicate and the putative small clause of cases like mess the song up systematically lacks the defining properties of this construction. The wordorder restrictions that the smallclause approach is designed, in part, to account for are shown to follow from a set of independently needed linearization constraints, which are motivated by functional principles. *
On Lukasiewicz's fourvalued modal logic
, 2000
"... . # Lukasiewicz's fourvalued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (nonmodal) twovalued logic is presented, and some consequences are drawn concerning its classification and its algebraic behav ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
. # Lukasiewicz's fourvalued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (nonmodal) twovalued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counterintuitive aspects of this logic are discussed under the light of the presented results, # Lukasiewicz's own texts, and related literature. 1 Introduction The Polish philosopher and logician Jan # Lukasiewicz (Lwow, 1878  Dublin, 1956) is one of the fathers of modern manyvalued logic, and some of the systems he introduced are presently a topic of deep investigation. In particular his infinitelyvalued logic belongs to the core systems of mathematical fuzzy logic as a logic of comparative truth, see [5, 15, 14, 16]. However, it must be stressed here that # Lukasiewicz's logical work bears also a special relationship to modal logic. Actually, modal notions were part of #...
On modal logics of partial recursive functions
 Studia Logica
"... The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above inter ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and nondeterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established. Keywords: modal logic, recursive function, CurryHoward isomorphism 1
Inheritance Reasoning and HeadDriven Phrase Structure Grammar
, 1990
"... Inheritance networks are a type of semantic network which represent both strict (classical implication) and defeasible (nonclassical) relationships among entities. We present an established approach to defeasible reasoning which defines inference in terms of the construction of paths through a netw ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Inheritance networks are a type of semantic network which represent both strict (classical implication) and defeasible (nonclassical) relationships among entities. We present an established approach to defeasible reasoning which defines inference in terms of the construction of paths through a network. Much of literature on inheritance is concerned with specifying the most "intuitive" system of path construction. However, when considering a fundamental feature of these approachesthe status accorded to redundant linkswe find that topological considerations espoused in the literature are insufficient for determining the valid inferences of a network. This implies that the "intuitiveness" of a particular method depends upon the domain being represented. Though Touretzky has demonstrated that it is unsound in some cases, the pathpreference algorithm known as shortest path reasoning, is actually the most intuitive algorithm to use when reasoning about the inheritance network which r...
Variants of the Basic Calculus of Constructions
, 2004
"... In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version i ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.
On the Role of Implication in Formal Logic
, 1998
"... Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with no ..."
Abstract
 Add to MetaCart
Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higherorder BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cutelimination...
Linear Arithmetic Desecsed
"... In classical and intuitionistic arithmetics, any formula implies a true equation, and a false equation implies anything. In weaker logics fewer implications hold. In this paper we rehearse known results about the relevant arithmetic R # , and we show that in linear arithmetic LL # by contrast ..."
Abstract
 Add to MetaCart
In classical and intuitionistic arithmetics, any formula implies a true equation, and a false equation implies anything. In weaker logics fewer implications hold. In this paper we rehearse known results about the relevant arithmetic R # , and we show that in linear arithmetic LL # by contrast false equations never imply true ones. As a result, linear arithmetic is desecsed. A formula A which entails 0 = 0 is a secondary equation; one entailed by 0 6= 0 is a secondary unequation. A system of formal arithmetic is secsed if every extensional formula is either a secondary equation or a secondary unequation. We are indebted to the program MaGIC for the simple countermodel SZ7, on which 0 = 1 is not a secondary formula. This is a small but significant success for automated reasoning. LINEAR ARITHMETIC DESECSED John K. Slaney Robert K. Meyer Greg Restall September 12, 1996 Abstract In classical and intuitionistic arithmetics, any formula implies a true equation, and a fal...
Manipulating Proofs
"... The purpose of this paper is to explain an approach to formal logic which is concerned with provability rather than with truth alone, as in the traditional approach. The traditional approach to propositional logic will probably be familiar to programmers, since it has been incorporated into most pro ..."
Abstract
 Add to MetaCart
The purpose of this paper is to explain an approach to formal logic which is concerned with provability rather than with truth alone, as in the traditional approach. The traditional approach to propositional logic will probably be familiar to programmers, since it has been incorporated into most programming languages and spreadsheets. The approach explained in this paper has, however, become important in computer science in recent years. It is characterized by taking proofs as objects which can be manipulated. The idea of treating proofs as mathematical objects to be manipulated goes back to Hilbert, who, in response to the attacks on classical mathematics from the intuitionists, proposed to ground classical mathematics by treating proofs as arrays of meaningless symbols (using a formalized language) and proving by unquestionably valid combinatorial techniques that no proof could end in a formula with the form of a contradiction. Godel's Second Theorem, which shows that it is impossib...
LambdaCalculus and Functional Programming
"... This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, bo ..."
Abstract
 Add to MetaCart
This paper deals with the problem of a program that is essentially the same over any of several types but which, in the older imperative languages must be rewritten for each separate type. For example, a sort routine may be written with essentially the same code except for the types for integers, booleans, and strings. It is clearly desirable to have a method of writing a piece of code that can accept the specific type as an argument. Milner developed his ideas in terms of type assignment to lambdaterms. It is based on a result due originally to Curry (Curry 1969) and Hindley (Hindley 1969) known as the principal typescheme theorem, which says that (assuming that the typing assumptions are sufficiently wellbehaved) every term has a principal typescheme, which is a typescheme such that every other typescheme which can be proved for the given term is obtained by a substitution of types for type variables. This use of type schemes allows a kind of generality over all types, which is known as polymorphism.