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Formulas as Programs
, 1998
"... We provide here a computational interpretation of firstorder logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts ..."
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We provide here a computational interpretation of firstorder logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts
Worlds, Models, and Descriptions
 Studia Logica, Special Issue Ways of Worlds II
, 2006
"... Abstract. Since the pioneering work by Kripke and Montague, the term possible world has appeared in most theories of formal semantics for modal logics, natural languages, and knowledgebased systems. Yet that term obscures many questions about the relationships between the real world, various models ..."
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Abstract. Since the pioneering work by Kripke and Montague, the term possible world has appeared in most theories of formal semantics for modal logics, natural languages, and knowledgebased systems. Yet that term obscures many questions about the relationships between the real world, various models of the world, and descriptions of those models in either formal languages or natural languages. Each step in that progression is an abstraction from the overwhelming complexity of the world. At the end, nothing is left but a colorful metaphor for an undefined element of a set W called worlds, which are related by an undefined and undefinable primitive relation R called accessibility. For some purposes, the resulting abstraction has proved to be useful, but as a general theory of meaning, the abstraction omits too many significant features. So much information has been lost at each step that many philosophers, linguists, and psychologists have dismissed modeltheoretic semantics as irrelevant to the study of meaning. This article examines the steps in the process of extracting the pair (W,R) from the world and the way people talk about the world. It shows that the Kripke worlds can be reinterpreted as part of a Peircean semiotic theory, which can also include contributions from many other studies in cognitive science. Among them are Dunn's semantics based on laws and facts, the lexical semantics preferred by many linguists, psychological models of how the world is perceived, and philosophies of science that relate theories to the world. A full integration of all those sources is far beyond the scope of this article, but an outline of the approach suggests that Peirce's vision is capable of relating and reconciling the competing sources.
Intuitive Counterexamples for Constructive Fallacies
 Mathematical Foundations of Computer Science 1994  19th International Symposium, MFCS '94, Kosice
, 1994
"... Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formulae are constructively invalid. We argue that the intuitive impact of such countermodels becomes more tr ..."
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Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formulae are constructively invalid. We argue that the intuitive impact of such countermodels becomes more transparent and convincing as we move from Kripke/Beth models based on possible worlds, to Lauchli realizability models. We introduce a new semantics for constructive reasoning, called relational realizability, which strengthens further the intuitive impact of Lauchli realizability. But, none of these model theories provides countermodels with the compelling impact of classical truthtable countermodels for classically unprovable formulae. We prove soundness of the Heyting calculus for relational realizability, and conjecture that there is a constructive choicefree proof of completeness. In this respect, relational realizability improves the metamathematical constructivity of Lauchli realizab...
1 The FMathL mathematical framework
, 2009
"... Please quote the version and date when referring to this version of the document. The newest version will always be available at ..."
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Cited by 1 (1 self)
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Please quote the version and date when referring to this version of the document. The newest version will always be available at
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
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
1 Formulas as Programs
, 1998
"... Abstract. We provide here a computational interpretation of firstorder logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts with the socalled formulas as types approach in w ..."
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Abstract. We provide here a computational interpretation of firstorder logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts with the socalled formulas as types approach in which the proofs of the formulas are typed terms that can be taken as programs. This view of computing is inspired by logic programming and constraint logic programming but differs from them in a number of crucial aspects. Formulas as programs is argued to yield a realistic approach to programming that has been realized in the implemented programming language Alma0 Apt, Brunekreef, Partington & Schaerf (1998) that combines the advantages of imperative and logic programming. The work here reported can also be used to reason about the correctness of nonrecursive Alma0 programs that do not include destructive assignment.
BACKGROUND
"... Received: leave blank. Accepted: leave blank. In this article we are aiming to build cognitive semantics over a first person perspective. Our goal is to specify meanings connected to cognitive agents, rooted in their experience and separable from language, covering a wide spectrum of cognitions rang ..."
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Received: leave blank. Accepted: leave blank. In this article we are aiming to build cognitive semantics over a first person perspective. Our goal is to specify meanings connected to cognitive agents, rooted in their experience and separable from language, covering a wide spectrum of cognitions ranging from living organisms (animals, preverbal children and adult humans) to artificial agents and covering a broad, continuous, spectrum of meanings. As regards the used method, the first person perspective enables a kind of grounding of meanings in cognitions. An ability of cognitive agents to distinguish is a starting point of our approach, distinguishing criteria and schemata are the basic semantic constructs. The resulting construction is based on a projection of the environment into a cluster of current percepts and a similarity function on percepts. Situation schemata, more sophisticated similarity functions, event schemata and distinguishing criteria are built over that basis. Inference rules and action rules are components of our semantics. An interesting property of the proposed semantics is that it enables coexistence of subjective and intersubjective meanings. Subjective (first person perspective) meanings are primary, and we have shown the way from them to collectively accepted (third person perspective) meanings via observable behaviour and feedback about success/failure of actions. An abductive reasoning is an important tool on that way. A construct of an instrument, which represents a measure for using intersubjective meanings, is introduced. The instrument serves as a tool for an inclusion of sophisticated meanings, e.g. of scientific constructs, into our framework. KEY WORDS meaning, cognitive semantics, situated agent, schema, distinguishing criterion CLASSIFICATION