Results 1  10
of
22
Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
Advice on Abductive Logic
 Logic Journal of the IGPL
"... The action of thought is excited by the initiation of doubt and ceases when belief is attained; so that the production of belief is the sole function of thought. ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
The action of thought is excited by the initiation of doubt and ceases when belief is attained; so that the production of belief is the sole function of thought.
formal and formalized ontologies
 International Journal of HumanComputer Studies
"... 2. Descriptive, formal and formalized ontologies 3. Variants of formalized ontology 4. Some data on formal ontologists 5. A note on Husserl’s conception of formal ontology ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
2. Descriptive, formal and formalized ontologies 3. Variants of formalized ontology 4. Some data on formal ontologists 5. A note on Husserl’s conception of formal ontology
Gödel’s incompleteness theorem and the philosophy of open systems
 7, Centre de Recherches Sémiologiques, Université de Neuchâtel (Neuchâtel
, 1992
"... In recent years a number of criticisms have been raised against the formal systems of ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
In recent years a number of criticisms have been raised against the formal systems of
The MachineAssisted Proof Of Programming Language Properties
, 1996
"... The MachineAssisted Proof of Programming Language Properties Myra VanInwegen Advisor: Carl Gunter The goals of the project described in this thesis are twofold. First, we wanted to demonstrate that if a programming language has a semantics that is complete and rigorous (mathematical), but not to ..."
Abstract
 Add to MetaCart
The MachineAssisted Proof of Programming Language Properties Myra VanInwegen Advisor: Carl Gunter The goals of the project described in this thesis are twofold. First, we wanted to demonstrate that if a programming language has a semantics that is complete and rigorous (mathematical), but not too complex, then substantial theorems can be proved about it. Second, we wanted to assess the utility of using an automated theorem prover to aid in such proofs. We chose SML as the language about which to prove theorems: it has a published semantics that is complete and rigorous, and while not exactly simple, is comprehensible. We encoded the semantics of Core SML into the theorem prover HOL (creating new definitional packages for HOL in the process). We proved important theorems about evaluation and about the type system. We also proved the type preservation theorem, which relates evaluation and typing, for a good portion of the language. We were not able to complete the proof of type prese...
Pocket Mathematics
, 1995
"... Mathematics is in a dramatic and massive process of changing, mainly due to the advent of computers and computer science. Our aim is to present a pocket image of this phenomenon; a "case study" will give us the opportunity to describe some of these new ideas, problems, and techniques. Particularly, ..."
Abstract
 Add to MetaCart
Mathematics is in a dramatic and massive process of changing, mainly due to the advent of computers and computer science. Our aim is to present a pocket image of this phenomenon; a "case study" will give us the opportunity to describe some of these new ideas, problems, and techniques. Particularly, we will be concerned with foreseeable mutations in the interaction between deductive and experimental trends.
Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic—Lambda Calculus and Related Systems; Mechanical Theorem Proving General Terms: Theory
"... We construct a logicenriched type theory LTTw that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTw, including Weyl’s definition of the cardinality of a set and several results from real analysi ..."
Abstract
 Add to MetaCart
We construct a logicenriched type theory LTTw that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTw, including Weyl’s definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a nonconstructive foundation for mathematics.