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A Reflection on Russell's Ramified Types and Kripke's Hierarchy of Truths
 Journal of the Interest Group in Pure and Applied Logic 4(2
, 1996
"... Both in Kripke's Theory of Truth ktt [8] and Russell's Ramified Type Theory rtt [16, 9] we are confronted with some hierarchy. In rtt, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can only depend on ..."
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Both in Kripke's Theory of Truth ktt [8] and Russell's Ramified Type Theory rtt [16, 9] we are confronted with some hierarchy. In rtt, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can only depend on objects of lower orders and types. Kripke on the other hand, has a ladder of languages where the truth of a proposition in language Ln can only be made in Lm where m ? n. Kripke finds a fixed point for his hierarchy (something Russell does not attempt to do). We investigate in this paper the similarities of both hierarchies: At level n of ktt the truth or falsehood of all ordernpropositions of rtt can be established. Moreover, there are ordernpropositions that get a truth value at an earlier stage in ktt. Furthermore, we show that rtt is more restrictive than ktt, as some type restrictions are not needed in ktt and more formulas can be expressed in the latter. Looking back at the dou...
GÖDEL AND SET THEORY
"... Kurt Gödel (1906–1978) with his work on the constructible universe L established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of firstorder logic as the framework and a matter of method for set theory and secured the ..."
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Kurt Gödel (1906–1978) with his work on the constructible universe L established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of firstorder logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of settheoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel’s work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges
What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end
Model transformations with constructive type theory
"... Abstract. This report is the culmination of nine months of research into the feasibility of implementing model transformations with constructive type theory. Early signs are quite encouraging. In fact, by virtue of its embodiment of the twin notions of logic and computation, constructive type theory ..."
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Abstract. This report is the culmination of nine months of research into the feasibility of implementing model transformations with constructive type theory. Early signs are quite encouraging. In fact, by virtue of its embodiment of the twin notions of logic and computation, constructive type theory appears to be a natural language for specifying, validating and implementing model transformations. The bulk of this report is taken up by three case studies, of which the third, in particular, gave useful pointers as to the direction of future research. Some big challenges lay ahead, not least in finding ways of managing the scale and complexity of large transformations. However, what has been achieved so far is believed to be a good foundation. This research is being done as part of the Predictive Assembly Laboratory (PALab) group at King’s College, London, under the supervision of Dr. Iman Poernomo. 1.
Vagueness and Truth
"... In philosophy of logic and elsewhere, it is generally thought that similar problems should be solved by similar means. This advice is sometimes elevated to the status of a principle: the principle of uniform solution. In this paper I will explore the question of what counts as a similar problem and ..."
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In philosophy of logic and elsewhere, it is generally thought that similar problems should be solved by similar means. This advice is sometimes elevated to the status of a principle: the principle of uniform solution. In this paper I will explore the question of what counts as a similar problem and consider reasons for subscribing to the principle of uniform solution. 1 Introducing the Principle of Uniform Solution It would be very odd to give different responses to two paradoxes depending on minor, seeminglyirrelevant details of their presentation. For example, it would be unacceptable to deal with the paradox of the heap by invoking a multivalued logic, ̷L∞, say, and yet, when faced with the paradox of the bald man, invoke a supervaluational logic. Clearly these two paradoxes are of a kind—they are both instances of the sorites paradox. And whether the sorites paradox is couched in terms of heaps and grains of sand, or in terms of baldness and the number of hairs on the head, it is essentially the same problem and therefore must be solved by the same means. More generally, we might suggest that similar paradoxes should be resolved by similar means. This advice is sometimes elevated to the status of a principle, which usually goes by the name of the principle of uniform solution. This principle and its motivation will occupy us for much of the discussion in this paper. In particular, I will defend a rather general form of this principle. I will argue that two paradoxes can be thought to be of the same kind because (at a suitable level of abstraction) they share a similar internal structure, or because of external considerations such as the relationships of the paradoxes in question to other paradoxes in the vicinity, or the way they respond to proposed solutions. I will then use this reading of the principle of uniform solution to make a case for the sorites and the liar paradox being of a kind.
PROBABILITY WEIGHTING BY CHILDREN AND ADULTS
"... Experimental and real world evidence show that many aspects of risktaking behavior can be explained by assuming that people weight outcomes by a subjective rather than objective probability. In this paper we explore how these probability weights change with age by looking at risktaking behavior in ..."
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Experimental and real world evidence show that many aspects of risktaking behavior can be explained by assuming that people weight outcomes by a subjective rather than objective probability. In this paper we explore how these probability weights change with age by looking at risktaking behavior in participants from age 5 to 64. Our participants make choices between certain payoffs and simple gambles for which the probability of winning and losing varies. We find strong evidence for probability weighting in children. Children underweight lowprobability events and overweight highprobability ones. Probability weighting tends to diminish with age, and on average adults use the objective probability when evaluating gambles over gains.
Journal of the IGPL
"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."
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In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logicaloperationsasadjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9gfragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and singlevalued relations. However, when enriched with proofsasarrows, this familiar concept must be supplied with an additional conversion rule, conne...
Kingman, category and combinatorics
, 2009
"... Kingman’s Theorem on skeleton limits –passing from limits as n! 1 along nh (n 2 N) for enough h> 0 to limits as t! 1 for t 2 R –is generalized to a Baire/measurable setting via a topological approach. Its affinity with a combinatorial theorem due to Kestelman and to Borwein and Ditor and another due ..."
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Kingman’s Theorem on skeleton limits –passing from limits as n! 1 along nh (n 2 N) for enough h> 0 to limits as t! 1 for t 2 R –is generalized to a Baire/measurable setting via a topological approach. Its affinity with a combinatorial theorem due to Kestelman and to Borwein and Ditor and another due to Bergelson, Hindman and Weiss is established. As applications, a theory of ‘rational’ skeletons akin to Kingman’s integer skeletons, and more appropriate to a measurable setting, is developed, and two combinatorial results in the spirit of van der Waerden’s celebrated theorem on arithmetic progressions are offered.
IFSAEUSFLAT 2009 Noncommutative EQlogics and their extensions
"... Abstract — We discuss a formal manyvalued logic called EQlogic which is based on a recently introduced special class of algebras called EQalgebras. The latter have three basic binary operations (meet, multiplication, fuzzy equality) and a top element and, in a certain sense, generalize residuated ..."
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Abstract — We discuss a formal manyvalued logic called EQlogic which is based on a recently introduced special class of algebras called EQalgebras. The latter have three basic binary operations (meet, multiplication, fuzzy equality) and a top element and, in a certain sense, generalize residuated lattices. The goal of EQlogics is to present a possible direction in the development of mathematical logics in which axioms are formed as identities. In this paper we propose a basic EQlogic and three extensions which end up with a logic equivalent to the MTLlogic. Keywords — EQalgebra, fuzzy equality, residuated lattice, MTLlogic, fuzzy logic. 1