self-reference in arithmetic i - Utrecht University Repository
... In philosophical discussions, the notion of self-reference in metamathematics still assumes a prominent role. Here self-reference in formal languages is a topic in its own right. Some authors, among them Heck (2007) and Milne (2007), focus on self-reference in metamathematics and ask, for instance, ...
... In philosophical discussions, the notion of self-reference in metamathematics still assumes a prominent role. Here self-reference in formal languages is a topic in its own right. Some authors, among them Heck (2007) and Milne (2007), focus on self-reference in metamathematics and ask, for instance, ...
Foundations of Logic Programmin:
... These can be characterised (by Gomel's completeness iheorem [tfyj, L99]) as the formulas which are logical consequences of the axioms of the theory, thai is, they are true in every interpretation which is a model of each nf the axioms of the ...
... These can be characterised (by Gomel's completeness iheorem [tfyj, L99]) as the formulas which are logical consequences of the axioms of the theory, thai is, they are true in every interpretation which is a model of each nf the axioms of the ...
Chiron: A Set Theory with Types
... The usefulness of a logic is often measured by its expressivity: the more that can be expressed in the logic, the more useful the logic is. By a logic, we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may al ...
... The usefulness of a logic is often measured by its expressivity: the more that can be expressed in the logic, the more useful the logic is. By a logic, we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may al ...
another essay - u.arizona.edu
... been to a large extent unified in the so-called Standard Model. Contrary to Einstein’s conviction, and despite his scruples, there is a widespread belief today that any plausible candidate for a unified fundamental theory (a “Theory of Everything”) would be a quantum theory. The experimentally succe ...
... been to a large extent unified in the so-called Standard Model. Contrary to Einstein’s conviction, and despite his scruples, there is a widespread belief today that any plausible candidate for a unified fundamental theory (a “Theory of Everything”) would be a quantum theory. The experimentally succe ...
Factoring Out the Impossibility of Logical Aggregation
... of this straightforward point. Its method is to introduce a mapping from pro…les of individual judgments to social judgments, where judgments are formalized as sets of formulas in some logical language, and then investigate the e¤ect of imposing axiomatic conditions on this mapping. Among the resul ...
... of this straightforward point. Its method is to introduce a mapping from pro…les of individual judgments to social judgments, where judgments are formalized as sets of formulas in some logical language, and then investigate the e¤ect of imposing axiomatic conditions on this mapping. Among the resul ...
Axiomatic Set Teory P.D.Welch.
... analytic set satisfied CH. (At the same time they were producing results indicating that such sets were very “regular”: they were all Lebesgue measurable, had a categorical property defined by Baire and many other such properties. Borel in particular defined a hierarchy of sets now named after him, ...
... analytic set satisfied CH. (At the same time they were producing results indicating that such sets were very “regular”: they were all Lebesgue measurable, had a categorical property defined by Baire and many other such properties. Borel in particular defined a hierarchy of sets now named after him, ...
The Complete Proof Theory of Hybrid Systems
... proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s incompleteness theorem shows, however, that hybrid systems do not have a sound and complete calculus that is effective, because both their discrete fragment and their continuous fragment alone are nonaxiomatizable ...
... proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s incompleteness theorem shows, however, that hybrid systems do not have a sound and complete calculus that is effective, because both their discrete fragment and their continuous fragment alone are nonaxiomatizable ...
The Mole - Prairie Science
... Note that the NUMBER is always the same, but the MASS is very different! Mole is abbreviated mol (gee, that’s a lot quicker to write, huh?) ...
... Note that the NUMBER is always the same, but the MASS is very different! Mole is abbreviated mol (gee, that’s a lot quicker to write, huh?) ...
A Nonstandard Approach to the. Logical Omniscience Problem
... What about logical omniscience? Notice that notions like "validity" and "logical consequence" (which played a prominent part in our informal description of logical omniscience) are not absolute notions; their formal definitions depend on how truth is defined and on the class of worlds being consider ...
... What about logical omniscience? Notice that notions like "validity" and "logical consequence" (which played a prominent part in our informal description of logical omniscience) are not absolute notions; their formal definitions depend on how truth is defined and on the class of worlds being consider ...
Godel`s Proof
... conclusions, and here I would like to point out one key difference. In their “Concluding Reflections,” Nagel and Newman argue that from Gödel’s discoveries it follows that computers—“calculating machines,” as they call them—are in principle incapable of reasoning as flexibly as we humans reason, a re ...
... conclusions, and here I would like to point out one key difference. In their “Concluding Reflections,” Nagel and Newman argue that from Gödel’s discoveries it follows that computers—“calculating machines,” as they call them—are in principle incapable of reasoning as flexibly as we humans reason, a re ...
Turner`s Logic of Universal Causation, Propositional Logic, and
... propositional language with a modal operator C. UCL formulas are propositional formulas with unary modal operator C. A UCL theory is a set of UCL formulas. The semantics of UCL is defined through causally explained interpretations. A UCL structure is a pair (I, S) such that I is an interpretation, a ...
... propositional language with a modal operator C. UCL formulas are propositional formulas with unary modal operator C. A UCL theory is a set of UCL formulas. The semantics of UCL is defined through causally explained interpretations. A UCL structure is a pair (I, S) such that I is an interpretation, a ...
Points, lines and diamonds: a two-sorted modal logic for projective
... language MLG 2 is two-sorted as well: we will distinguish point formulas and line formulas. The language then will have two diamonds with respectively the incidence relation and its converse as accessibility relation. These diamonds thus turn respectively line formulas into point formulas, and vice ...
... language MLG 2 is two-sorted as well: we will distinguish point formulas and line formulas. The language then will have two diamonds with respectively the incidence relation and its converse as accessibility relation. These diamonds thus turn respectively line formulas into point formulas, and vice ...
on the Complexity of Quantifier-Free Fixed-Size Bit-Vector
... description into a bit-level circuit, as in hardware synthesis. The result can then be checked by a (propositional) SAT solver. In [1], we gave the following example (in SMT2 syntax) to point out that bit-blasting is not polynomial in general. It checks commutativity of adding two bit-vectors of bit ...
... description into a bit-level circuit, as in hardware synthesis. The result can then be checked by a (propositional) SAT solver. In [1], we gave the following example (in SMT2 syntax) to point out that bit-blasting is not polynomial in general. It checks commutativity of adding two bit-vectors of bit ...
Well-foundedness of Countable Ordinals and the Hydra Game
... However, there are a few weaker results about comparison of countable ordinals and countable ordinal arithmetic that hold in ACA0 , and some that even hold in RCA0 . These will be discussed in subsequent sections. With this in mind, it is quite reasonable to claim that ACA0 is insufficient to develo ...
... However, there are a few weaker results about comparison of countable ordinals and countable ordinal arithmetic that hold in ACA0 , and some that even hold in RCA0 . These will be discussed in subsequent sections. With this in mind, it is quite reasonable to claim that ACA0 is insufficient to develo ...
New insights into soft gluons and gravitons. In
... It is well-known that scattering amplitudes in quantum field theory are beset by infrared divergences. Consider, for example, the interaction shown in figure 1, in which a vector boson splits into a quark pair. Either the final state quark or anti-quark may emit gluon radiation, and the Feynman rule ...
... It is well-known that scattering amplitudes in quantum field theory are beset by infrared divergences. Consider, for example, the interaction shown in figure 1, in which a vector boson splits into a quark pair. Either the final state quark or anti-quark may emit gluon radiation, and the Feynman rule ...
Heyting-valued interpretations for Constructive Set Theory
... during his doctoral studies, and to Harold Simmons and Thierry Coquand for discussions on formal topology. ...
... during his doctoral studies, and to Harold Simmons and Thierry Coquand for discussions on formal topology. ...
On Provability Logic
... The non-existence of “the modal logic” can be explained by the fact that nested modalities are rare in natural language. We seldom say that it is necessary that something is possible and thus there is no agreement whether for instance the modal propositional formula 3p → 23p should be accepted as a ...
... The non-existence of “the modal logic” can be explained by the fact that nested modalities are rare in natural language. We seldom say that it is necessary that something is possible and thus there is no agreement whether for instance the modal propositional formula 3p → 23p should be accepted as a ...
EMBEDDING AN ANALYTIC EQUIVALENCE RELATION IN THE
... space and E0 ⊂ Ω × Ω is a Σ11 equivalence relation, then there are ∆11 equivalence relations E1 and E2 and a ∆11 subset Z of Ω, such that E1 ∪ E2 (Z × Z) is Borel equivalent to E0 . If Ω is the Baire space, then E1 and E2 can be taken to be closed, Z can be taken to be open, and the Borel equivale ...
... space and E0 ⊂ Ω × Ω is a Σ11 equivalence relation, then there are ∆11 equivalence relations E1 and E2 and a ∆11 subset Z of Ω, such that E1 ∪ E2 (Z × Z) is Borel equivalent to E0 . If Ω is the Baire space, then E1 and E2 can be taken to be closed, Z can be taken to be open, and the Borel equivale ...
A Concise Introduction to Mathematical Logic
... One feature of modern logic is a clear distinction between object language and metalanguage. The first is formalized or at least formalizable. The latter is, like the language of this book, a kind of a colloquial language that differs from author to author and depends also on the audience the author ...
... One feature of modern logic is a clear distinction between object language and metalanguage. The first is formalized or at least formalizable. The latter is, like the language of this book, a kind of a colloquial language that differs from author to author and depends also on the audience the author ...
Normal form results for default logic
... In this paper we develop a representation theory for default logic of Reiter ([Rei80]). The question is whether one can find “normal forms” for default theories, that is, if there are syntactical constraints which can be imposed on default theories without changing extensions. In this section we int ...
... In this paper we develop a representation theory for default logic of Reiter ([Rei80]). The question is whether one can find “normal forms” for default theories, that is, if there are syntactical constraints which can be imposed on default theories without changing extensions. In this section we int ...
Algebraizing Hybrid Logic - Institute for Logic, Language and
... on at least one branch of the tableau will be satisfiable by label too. Remark 2.4.2. The systematic tableau construction is defined in [5]. Roughly speaking, this construction is needed in order to prove strong completeness. Theorem 2.4.1. ([5]) Any consistent set of formulas in countable language ...
... on at least one branch of the tableau will be satisfiable by label too. Remark 2.4.2. The systematic tableau construction is defined in [5]. Roughly speaking, this construction is needed in order to prove strong completeness. Theorem 2.4.1. ([5]) Any consistent set of formulas in countable language ...
INTRODUCTION TO THE THEORY OF PROOFS 3A. The Gentzen
... such that a formula χ occurs in both B1 and A2 , we can construct a propositional, Mix-free proof of (68) which uses the same logical rules. Outline of the proof. We define the left Mix rank to be the number of consecutive sequents in the proof which ends with A1 ⇒ B1 starting from the last one and ...
... such that a formula χ occurs in both B1 and A2 , we can construct a propositional, Mix-free proof of (68) which uses the same logical rules. Outline of the proof. We define the left Mix rank to be the number of consecutive sequents in the proof which ends with A1 ⇒ B1 starting from the last one and ...
A Resolution-Based Proof Method for Temporal Logics of
... First, note that is not a quantified language. We shall thus build formulae from a set Φ p q r of primitive propositions. In fact, the language generalizes classical propositional logic, and thus it contains the standard propositional connectives (not) and (or); the remaini ...
... First, note that is not a quantified language. We shall thus build formulae from a set Φ p q r of primitive propositions. In fact, the language generalizes classical propositional logic, and thus it contains the standard propositional connectives (not) and (or); the remaini ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... provide cut-free systems for all logics in the cube, they do not provide cut-free systems for all possible combinations of axioms. For example, the logic S5 can be obtained by adding b and 4, or by adding t and 5, to the modal logic K, but a complete cut-free system could only be obtained by adding ...
... provide cut-free systems for all logics in the cube, they do not provide cut-free systems for all possible combinations of axioms. For example, the logic S5 can be obtained by adding b and 4, or by adding t and 5, to the modal logic K, but a complete cut-free system could only be obtained by adding ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... provide cut-free systems for all logics in the cube, they do not provide cut-free systems for all possible combinations of axioms. For example, the logic S5 can be obtained by adding b and 4, or by adding t and 5, to the modal logic K, but a complete cut-free system could only be obtained by adding ...
... provide cut-free systems for all logics in the cube, they do not provide cut-free systems for all possible combinations of axioms. For example, the logic S5 can be obtained by adding b and 4, or by adding t and 5, to the modal logic K, but a complete cut-free system could only be obtained by adding ...