consistent

Crispin Wright (born 1942) is a British philosopher, who has written on neo-Fregean philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge and objectivity. He is Professor of Logic and Metaphysics at the University of St. Andrews. His best-known work is arguably Frege's Conception of Numbers as Objects (1983), where he argues that Frege's logicist project could be revived by removing Basic Law (V) from the formal system. Arithmetic is then d ...

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The cut-elimination theorem is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus that makes use of the cut rule also possesses a cut-free proof ...

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Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. Theory - Etymology. The word ‘theory’ derives from the Greek ‘theorein’, which means ‘to look at’. According to some sources, it was used frequently in terms of ‘looking at’ a theatre stage, which may explain why sometimes the word ‘theory’ is used as something provisional or not completely resembling real. The term ‘theoria’ (a noun) was already used by ...

Including:

• Theory - Etymology
• Theory - Science
• Theory - Models
• Theory - Types of theories
• Theory - Further explanation of a scientific theory
• Theory - Characteristics
• Theory - Mathematics
• Theory - Other fields
• Theory - List of famous theories
• Theory - Reference

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Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. See consistency proof. In statistics, consistency refers to a property of estimators. In decision theory, a voting system is called "consistent" if and only if it guarantees that whenever option A is the winner both in district 1 and district 2 then option A must a

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Consider a sample of size N drawn from a normal distribution of mean μ and unit variance (i.e., ). The sample mean, , of the sample , defined as has variance . This is equal to the reciprocal of the Fisher information from the sample (this is clear from the definition) and thus, by the Cramér-Rao inequality, the sample mean is effici ...

See also:

Efficiency statistics, Efficiency statistics - Efficient estimator, Efficiency statistics - Asymptotic efficiency, Efficiency statistics - Examples, Efficiency statistics - Relative efficiency

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The exponential family arises naturally as the answer to the following question: what is the maximum entropy distribution consistent with given constraints on expected values? The information entropy of a probability distribution dF(x) can only be computed with respect to some other probability distribution (or, more generally, a positive measure), and both measures must be mutually absolutely continuous. Accordingly, we need to pick a reference measure dH(x) with the same support as dF(xSee also:

Exponential family, Exponential family - Examples, Exponential family - Maximum entropy derivation, Exponential family - Role in statistics, Exponential family - Classical estimation: sufficiency, Exponential family - Bayesian estimation: conjugate distributions, Exponential family - Statistical inference, Exponential family - Sampling distributions, Exponential family - Hypothesis testing, Exponential family - Confidence intervals

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A model is formally defined in the context of some language L, which consists of a set of constant symbols, a set of relation symbols each of valence some positive integer, and a set of function symbols each of valence some positive integer. A model of the language L consists of several things: A universe set A which contains all the objects of interest (the "domain of discourse"), and An element of A for each constant symbol of L. A function from ASee also:

Model theory, Model theory - Definition, Model theory - Theorems of model theory

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Certain physical paradoxes defy common sense predictions about physical situations. In some cases, this is the result of modern physics correctly describing the natural world in circumstances which are far outside of everyday experience. For example, special relativity has traditionally yielded two common paradoxes: the twins paradox and the ladder paradox. Both of these paradoxes involve thought experiments which defy traditional common sense assumptions about time and space. In particular, the effects of time dilation and length contractio ...

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Physical paradox, Physical paradox - Paradoxes relating to false assumptions, Physical paradox - Paradoxes relating to unphysical mathematical idealizations, Physical paradox - Quantum mechanical paradoxes, Physical paradox - Causality paradoxes, Physical paradox - Observational paradoxes

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Gödel's second incompleteness theorem can be stated as follows: For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. (Proof of the "if" part:) If T is inconsistent then anything can be proved, including that T is consistent. (Proof of the "only if" part:) If T is consistent then T does not i ...

See also:

Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Gödel's Theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes

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In The Case for Animal Rights, Regan argues that non-human animals are the bearers of moral rights. His philosophy lies broadly within the tradition of Immanuel Kant, though he rejects Kant's idea that respect is due only to rational beings. Regan points out that we routinely ascribe inherent value, and thus the right to be treated with respect, to humans who are not rational, including infants and the severely mentally impaired. The crucial attribute that all humans have in common, he argues, is not rationality, but the fact t ...

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Tom Regan, Tom Regan - Background, Tom Regan - Animal rights, Tom Regan - Works

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Large systems architecture was developed as a way to handle systems too large for one person to conceive of, let alone design. Systems of this size are rapidly becoming the norm, so architectural approaches and architects are increasingly needed to solve the problems of large systems. Systems architect - Users and Sponsors. Engineers as a group do not have a reputation for understanding and responding to human needs comfortably or for developing humanly functional and aesthetically pleasing products. Archi ...

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Systems architect, Systems architect - Background, Systems architect - Users and Sponsors, Systems architect - High level requirements, Systems architect - Cost/benefit analyses, Systems architect - Partitioning and layering, Systems architect - Acceptance test, Systems architect - Providing good communications with users and engineers, Systems architect - People

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Large systems architecture was developed as a way to handle systems too large for one person to conceive of, let alone design. Systems of this size are rapidly becoming the norm, so architectural approaches and architects are increasingly needed to solve the problems of large systems. Hardware architect - Users and Sponsors. Engineers as a group do not have a reputation for understanding and responding to human needs comfortably or for developing humanly functional and aesthetically pleasing products. Arch ...

See also:

Hardware architect, Hardware architect - Background, Hardware architect - Users and Sponsors, Hardware architect - High level requirements, Hardware architect - Cost/benefit analyses, Hardware architect - Partitioning and layering, Hardware architect - Acceptance test, Hardware architect - Providing good communications with users and engineers, Hardware architect - People

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Gödel's second incompleteness theorem can be stated as follows: For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. (Proof of the "if" part:) If T is inconsistent then anything can be proved, including that T is consistent. (Proof of the "only if" part:) If T is consistent then T does not i ...

See also:

Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes

Read more here: » Gödel's incompleteness theorem: Encyclopedia II - Gödel's incompleteness theorem - Second incompleteness theorem

An antichain A of P is a subset such that if p and q are in A, then p and q are incompatible (written p ⊥ q), meaning there is no r in P such that r ≤ p and r ≤ q. In the Borel sets example, incompatibility means p∩q has measure zero. In the finite partial functions example, incompatibility means that p∪q is not a function, in other words ...

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Forcing mathematics, Forcing mathematics - Forcing posets, Forcing mathematics - Countable transitive models and generic filters, Forcing mathematics - Forcing, Forcing mathematics - Consistency, Forcing mathematics - Cohen forcing, Forcing mathematics - The countable chain condition, Forcing mathematics - Easton forcing, Forcing mathematics - Random reals, Forcing mathematics - Boolean-valued models, Forcing mathematics - Meta-mathematical explanation

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Turning to perspectivalism (relativism), a proposition is only ever true relative to something like a perspective. There are four main versions of perspectivalism, and some interesting subdivisions: Epistemic theories of truth - Individual Perspectivalism. Perspectives are the points of view of particular individual persons. So, a proposition is true for a person if and only if it is accepted or believed by that person (i.e., "true for me"). Epistemic th ...

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Epistemic theories of truth, Epistemic theories of truth - Verificationist views, Epistemic theories of truth - Positivism, Epistemic theories of truth - A Priorism, Epistemic theories of truth - Pragmatism, Epistemic theories of truth - Perspectivist views, Epistemic theories of truth - Individual Perspectivalism, Epistemic theories of truth - Discourse Perspectivalism Conventionalism, Epistemic theories of truth - Collectivist Perspectivalism, Epistemic theories of truth - Transcendental Perspectivalism, Epistemic theories of truth - Summary

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In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it does in other contexts. Neither is a scientific theory a fact. Scientific theories are never proven to be true, but can be disproven. All scientific understanding takes the form of hypotheses, theories, or laws. Theories are typically ways of explaining why things happen, often, but not always after their occurrence is no longer in scientific dispute. In referring to the "theory of global warming" for example, the worldwide ...

See also:

Theory, Theory - Etymology, Theory - Science, Theory - Models, Theory - Types of theories, Theory - Further explanation of a scientific theory, Theory - Characteristics, Theory - Mathematics, Theory - Other fields, Theory - List of famous theories, Theory - Reference

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The library implements most of SQL-92 standard, including transactions that are atomic, consistent, isolated, and durable (ACID), triggers and most of the complex queries. No type-checking is done. You can insert a string into an integer column, for example. Some users see this as an innovation that makes the database much more useful, especially when bound to a dynamically typed scripting language. Othe ...

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SQLite, SQLite - Features, SQLite - Language bindings, SQLite - Products

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In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it often does in other contexts. Scientific theories are never proven to be true, but can be disproven. All scientific understanding takes the form of hypotheses, or conjectures. A theory is in this context a set of hypotheses that are logically bound together (See also hypothetico-deductive method). Theories are typically ways of explaining why things happen, often, but not always after their occurrence is no longer in scientific di ...

See also:

Theory, Theory - Etymology, Theory - Science, Theory - Models, Theory - Types of theories, Theory - Further explanation of a scientific theory, Theory - Characteristics, Theory - Mathematics, Theory - Other fields, Theory - List of famous theories, Theory - Reference

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When x and ε are uncorrelated, the second term goes to zero in the limit and the estimator is consistent. When x and ε are correlated, however, the estimator is biased. An instrumental variable is one that is correlated with the independent variable but not with the error term. The estimator is When z and ε are uncorrelated, the final term vanishes in the limit providing a consistent estimator. Not ...

See also:

Instrumental variable, Instrumental variable - Mathematics, Instrumental variable - Applications and problems

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