Consequence relations and admissible rules
... If the theory is given to us via a proof system, this might be quite straightforward, but if the theory is characterized in another way, say via a set of models or algebras, it is less clear what is meant. In this paper we describe what this means in detail, in a way that is applicable in many setti ...
... If the theory is given to us via a proof system, this might be quite straightforward, but if the theory is characterized in another way, say via a set of models or algebras, it is less clear what is meant. In this paper we describe what this means in detail, in a way that is applicable in many setti ...
Hilbert`s Program Then and Now
... Mathematical Society entitled “Axiomatic Thought” [1918a]. It is his first published contribution to mathematical foundations since 1905. In it, he again emphasized the requirement of consistency proofs for axiomatic systems: “The chief requirement of the theory of axioms must go farther [than merel ...
... Mathematical Society entitled “Axiomatic Thought” [1918a]. It is his first published contribution to mathematical foundations since 1905. In it, he again emphasized the requirement of consistency proofs for axiomatic systems: “The chief requirement of the theory of axioms must go farther [than merel ...
A joint logic of problems and propositions, a modified BHK
... Atomic formulas: problem symbols, propositional symbols (possibly depending on variables that all range over the same domain of discourse) and the constant ⊥ Formulas are of two types: problems (denoted by Greek letters) and propositions (denoted by Roman letters) Classical connectives: propositions ...
... Atomic formulas: problem symbols, propositional symbols (possibly depending on variables that all range over the same domain of discourse) and the constant ⊥ Formulas are of two types: problems (denoted by Greek letters) and propositions (denoted by Roman letters) Classical connectives: propositions ...
logic, programming and prolog (2ed)
... proved correct. Some of its properties are discussed. The unification algorithm is the basis for SLD-resolution which is the only inference rule needed for definite programs. Soundness and completeness of this rule are discussed. The use of negation in logic programming is discussed in Chapter 4. It ...
... proved correct. Some of its properties are discussed. The unification algorithm is the basis for SLD-resolution which is the only inference rule needed for definite programs. Soundness and completeness of this rule are discussed. The use of negation in logic programming is discussed in Chapter 4. It ...
... [QuiSO). Indeed, an approach sometimes taken t o defining a logic is to specify its set of theorems, or valid formulas. This was, for example, the original approach taken to defining various modal logics [LL59]. However, logic originally started as the study of sound arguments. In fact, Hacking defi ...
KURT GÖDEL - National Academy of Sciences
... into a whole." A set N is a subset of a set M if each member of N is a member of M. For example, the set {0, 1, 2} with the three members shown has the following 8 (= 23) subsets: {0, 1, 2}, {1, 2}, {0, 2}, {0, 1}, {0}, {1}, {2}, { }. Cantor showed that there is no way of pairing all the sets of nat ...
... into a whole." A set N is a subset of a set M if each member of N is a member of M. For example, the set {0, 1, 2} with the three members shown has the following 8 (= 23) subsets: {0, 1, 2}, {1, 2}, {0, 2}, {0, 1}, {0}, {1}, {2}, { }. Cantor showed that there is no way of pairing all the sets of nat ...
CS 208: Automata Theory and Logic
... – Introduced by Alan Turing as a simple model capable of expressing any imaginable computation – Turing machines are widely accepted as a synonyms for algorithmic computability (Church-Turing thesis) – Using these conceptual machines Turing showed that first-order logic validity problem a is non-com ...
... – Introduced by Alan Turing as a simple model capable of expressing any imaginable computation – Turing machines are widely accepted as a synonyms for algorithmic computability (Church-Turing thesis) – Using these conceptual machines Turing showed that first-order logic validity problem a is non-com ...
A Judgmental Reconstruction of Modal Logic
... An alternative way to understand local completeness is to reconsider our meaning explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verification of B. Local completeness entails that it is always possible to bring the verification of A ∧ B int ...
... An alternative way to understand local completeness is to reconsider our meaning explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verification of B. Local completeness entails that it is always possible to bring the verification of A ∧ B int ...
Hilbert`s Program Then and Now - Philsci
... the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new found ...
... the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new found ...
Notions of Computability at Higher Type
... §1. Introduction. This article is essentially a survey of fifty years of research on higher type computability. It was a great privilege to present much of this material in a series of three lectures at the Paris Logic Colloquium. In elementary recursion theory, one begins with the question: what do ...
... §1. Introduction. This article is essentially a survey of fifty years of research on higher type computability. It was a great privilege to present much of this material in a series of three lectures at the Paris Logic Colloquium. In elementary recursion theory, one begins with the question: what do ...
The Premiss-Based Approach to Logical Aggregation Franz Dietrich & Philippe Mongin
... (a ! b), and so on. Assume further that the collective judgments are obtained by aggregating the individual judgments - this is the judgment aggregation problem. A natural approach consists in singling out those propositions - to be called premisses - which are in some sense basic to the other judgm ...
... (a ! b), and so on. Assume further that the collective judgments are obtained by aggregating the individual judgments - this is the judgment aggregation problem. A natural approach consists in singling out those propositions - to be called premisses - which are in some sense basic to the other judgm ...
Proof, Sets, and Logic - Boise State University
... that the Axiom of Rank holds in this interpretation. After that comes a new section on implementation of mathematical concepts in Zermelo set theory with the Axiom of Rank. The Scott trick is introduced; the von Neumann definitions are also given. Material in the old version about an axiomatic cumul ...
... that the Axiom of Rank holds in this interpretation. After that comes a new section on implementation of mathematical concepts in Zermelo set theory with the Axiom of Rank. The Scott trick is introduced; the von Neumann definitions are also given. Material in the old version about an axiomatic cumul ...
HOW TO DEFINE A MEREOLOGICAL (COLLECTIVE) SET
... properties of sets in mathematics and differences between them and so called conglomerates in Section 1, we go on to explicate informally in Section 2 what it means to join many objects into a single entity from point of view of mereology, the theory of part of (parthood) relation. In Section 3 we pr ...
... properties of sets in mathematics and differences between them and so called conglomerates in Section 1, we go on to explicate informally in Section 2 what it means to join many objects into a single entity from point of view of mereology, the theory of part of (parthood) relation. In Section 3 we pr ...
Modular Construction of Complete Coalgebraic Logics
... systems. The image-finite simple probabilistic automata are called probabilistic transition systems in [12]. Note that all the endofunctors in the previous example arise as combinations of a small number of simple functors (constant, identity, powerset and probability distribution functor) using pro ...
... systems. The image-finite simple probabilistic automata are called probabilistic transition systems in [12]. Note that all the endofunctors in the previous example arise as combinations of a small number of simple functors (constant, identity, powerset and probability distribution functor) using pro ...
Dialectica Interpretations A Categorical Analysis
... the papers, preprints or technical reports is preceded by a short declaration that summarizes the current status (published, accepted or submitted), contributions of the authors if more than one author, main results and relation to other work. To help the reader, each declaration also contains a sho ...
... the papers, preprints or technical reports is preceded by a short declaration that summarizes the current status (published, accepted or submitted), contributions of the authors if more than one author, main results and relation to other work. To help the reader, each declaration also contains a sho ...
Proof, Sets, and Logic - Department of Mathematics
... differently). [this was actually done before July 19 but I didn’t update the version notes or post to the web]. Added material on forcing in type theory. Note that I have basic constructions for consistency and independence set up in the outline in to be in the first instance in type theory rather t ...
... differently). [this was actually done before July 19 but I didn’t update the version notes or post to the web]. Added material on forcing in type theory. Note that I have basic constructions for consistency and independence set up in the outline in to be in the first instance in type theory rather t ...
Preferences and Unrestricted Rebut
... preferences over arguments, and proving that the rationality postulates of [2] are still satisfied [12,15] under certain conditions on the preference relation. They then show that various ways of defining preferences based on a prioritisation of defeasible knowledge, satisfy these conditions. Howeve ...
... preferences over arguments, and proving that the rationality postulates of [2] are still satisfied [12,15] under certain conditions on the preference relation. They then show that various ways of defining preferences based on a prioritisation of defeasible knowledge, satisfy these conditions. Howeve ...
CATEGORICAL MODELS OF FIRST
... as an infinitary conjunction or disjunction over the domain of quantification. While it does not directly address the other notions of model above, it is worth noting that techniques developed in this thesis should adapt successfully to them. The issue of finding a notion of equality on proofs is se ...
... as an infinitary conjunction or disjunction over the domain of quantification. While it does not directly address the other notions of model above, it is worth noting that techniques developed in this thesis should adapt successfully to them. The issue of finding a notion of equality on proofs is se ...
MoL-2013-07 - Institute for Logic, Language and Computation
... I would also like to thank the members of my committee, Dr Jakub Szymanik, Dr Benno van den Berg, Prof Dr Stefan Geschke, Dr Yurii Khomskii, Prof Dr Jouko Väänänen, for first agreeing to be on my committe, and then, allowing me to submit after my deadline, and while I am at it, also my supervisor ...
... I would also like to thank the members of my committee, Dr Jakub Szymanik, Dr Benno van den Berg, Prof Dr Stefan Geschke, Dr Yurii Khomskii, Prof Dr Jouko Väänänen, for first agreeing to be on my committe, and then, allowing me to submit after my deadline, and while I am at it, also my supervisor ...
An argumentation framework in default logic
... to earlier approaches to formalize so-called nonmonotonic reasoning, in which conclusions can be invalidated by adding new information to the premises. This kind of reasoning is motivated by the fact that in real life people do not always have sufficient information to make a safe step towards the c ...
... to earlier approaches to formalize so-called nonmonotonic reasoning, in which conclusions can be invalidated by adding new information to the premises. This kind of reasoning is motivated by the fact that in real life people do not always have sufficient information to make a safe step towards the c ...
Programming in Logic Without Logic Programming
... as model-theoretic structures. It is this second, model-theoretic interpretation that underpins the logical semantics of KELPS. States and events can be represented with or without timestamps. The representation without timestamps (e.g. outdoors, reliable(bob)) facilitates destructive updates, becau ...
... as model-theoretic structures. It is this second, model-theoretic interpretation that underpins the logical semantics of KELPS. States and events can be represented with or without timestamps. The representation without timestamps (e.g. outdoors, reliable(bob)) facilitates destructive updates, becau ...
beliefrevision , epistemicconditionals andtheramseytest
... you receive new evidence consistent with your old beliefs, some of the old beliefs have to go — not because they are logically incompatible with the new information but because this information disconfirms them. They are no longer sufficiently probable given the new evidence. If the assumption here ...
... you receive new evidence consistent with your old beliefs, some of the old beliefs have to go — not because they are logically incompatible with the new information but because this information disconfirms them. They are no longer sufficiently probable given the new evidence. If the assumption here ...
On two problems with the Theory of the Creating Subject
... The stance that Brouwer assumes is of a kind with Turing’s, who devised his theoretical analysis of (mechanical) computation in terms of an idealised human, not a machine; Gandy proposed the term ‘computor’ [20, sections 10.1 and 10.3]. Brouwer’s theoretical stance is also of a kind with Chomsky’s, ...
... The stance that Brouwer assumes is of a kind with Turing’s, who devised his theoretical analysis of (mechanical) computation in terms of an idealised human, not a machine; Gandy proposed the term ‘computor’ [20, sections 10.1 and 10.3]. Brouwer’s theoretical stance is also of a kind with Chomsky’s, ...
PDF - University of Kent
... of representing graphically the laws of logic, and in particular the propositional calculus or Boolean algebra. They provide an extremely elegant and parsimonious form of representation for both the algebra and the arithmetic underlying it. In fact, LoF uses only a single explicit symbol, the cross ...
... of representing graphically the laws of logic, and in particular the propositional calculus or Boolean algebra. They provide an extremely elegant and parsimonious form of representation for both the algebra and the arithmetic underlying it. In fact, LoF uses only a single explicit symbol, the cross ...
Non-Classical Logic
... same row, which is impossible. The obvious conclusion is Completeness: If ∆ A then ∆ ` A, for all ∆ and A. that no truth value assignment (interpretation) can make Together these results entail the equivalence of semantic the formula false, so it must be logically valid. and deductive validity. Th ...
... same row, which is impossible. The obvious conclusion is Completeness: If ∆ A then ∆ ` A, for all ∆ and A. that no truth value assignment (interpretation) can make Together these results entail the equivalence of semantic the formula false, so it must be logically valid. and deductive validity. Th ...
Jesús Mosterín
Jesús Mosterín (born 1941) is a leading Spanish philosopher and a thinker of broad spectrum, often at the frontier between science and philosophy.