Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set

It is the inference that if the proposition p is true, and proposition q is true, then the logical conjunction of the two propositions p and q is true.

It is true for real nonzero p, as well as zero, positive and negative infinity p, as defined above.

The Kullback – Leibler divergence ( or information divergence, information gain, or relative entropy ) is a way of comparing two distributions: a " true " probability distribution p ( X ), and an arbitrary probability distribution q ( X ).

If Alice knows the true distribution p ( x ), while Bob believes ( has a prior ) that the distribution is q ( x ), then Bob will be more surprised than Alice, on average, upon seeing the value of X.

However, Aristotle also writes, " since it is impossible that contradictories should be at the same time true of the same thing, obviously contraries also cannot belong at the same time to the same thing " ( Book IV, CH 6, p. 531 ).

~ p ∨ p. Since p → p is true ( this is Theorem 2. 08, which is proved separately ), then ~ p ∨ p must be true.

When the sample is heated, at a certain temperature it will glow from the emission of electrons released from the defects, and this glow can be used to estimate the age of the sample to a threshold of approximately 15 percent of its true age.

This is as true of the Queen the United Kingdom with her as it is of the 54 percent of householders in Britain who are owner-occupiers.

Effectively, the standard is satisfied if there is greater than 50 percent chance that the proposition is true.

Up to that time ( and still true in underdeveloped countries ), five or less percent of the population was over 65.

They take prey from leaves and branches ; true vireos also flycatch, and the Gray Vireo takes 5 percent of its prey from the ground ( Salaman & Barlow 2003 ).

If we use the " relative " definition, then we express this absolute margin of error as a percent of the true value.

Charrière's 1970 best-selling book Papillon, which Papillon said was " 75 percent true ," details his alleged numerous escapes, attempted escapes, adventures and recaptures, from his imprisonment in 1932 to his final escape to Venezuela.

The average French economist may find such arguments 45 percent true, whereas the average American economist may find them 80 percent true.

Arguing an enthymeme is successful when the economist defends the 45 or 80 percent true as ' true enough.

In another sense, it is represented in the arguments of the `` true believers '' who seek to disprove the validity of all other beliefs and ideas in order to retain confidence in theirs.

A statistician computes the sample mean and a confidence interval for the true mean.

* a confidence interval ( or set estimate ); i. e., an interval constructed using a dataset drawn from a population so that, under repeated sampling of such datasets, such intervals would contain the true parameter value with the probability at the stated confidence level,

The estimates may deviate from the true value and the range of values which for some probability will contain the true value is referred to as the confidence interval.

The width is also dependent on the statistical certainty for which the confidence interval values forms a bounded range, thus the statistical certainty that the true value is within that range of values.

This vast increase in numbers diluted troop quality and undermined the confidence inexperienced commanders had in their men ; this was especially true of Rawlinson.

This is especially true if the unexpected shock is one ( like a fall in consumer confidence ) which tends to lower both output and inflation ; in that case, expanding the money supply ( lowering interest rates ) helps by increasing output while stabilizing inflation and inflationary expectations.

The commander-in-chief, it is true, never reposed in him the confidence that he gave to Beresford, Hill and Robert Craufurd.

Dancers are aware of this customer desire for increased confidence, and a dancer therefore will give off the impression that a customer is seeing her true self, when in reality it is just part of her act.

You will give us leave therefore — in whom you have reposed special confidence — as your representative body, to suggest a few things which call for the serious attention of everyone who has the true interest of America at heart.

A 3 % margin of error means that if the same procedure is used a large number of times, 95 % of the time the true population average will be within the 95 % confidence interval of the sample estimate plus or minus 3 %.

For example, if the true value is 50 percentage points, and the statistic has a confidence interval radius of 5 percentage points, then we say the margin of error is 5 percentage points.

As another example, if the true value is 50 people, and the statistic has a confidence interval radius of 5 people, then we might say the margin of error is 5 people.

For example, suppose the true value is 50 people, and the statistic has a confidence interval radius of 5 people.

Note that there is not necessarily a strict connection between the true confidence interval, and the true standard error.

The standard error can be used to create a confidence interval within which the " true " percentage should be to a certain level of confidence.

More specifically, the meaning of the term " confidence level " is that, if confidence intervals are constructed across many separate data analyses of repeated ( and possibly different ) experiments, the proportion of such intervals that contain the true value of the parameter will match the confidence level ; this is guaranteed by the reasoning underlying the construction of confidence intervals.

It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure.

This is true for any fixed time interval.

An indication of the unresolved status of the " nature abhors a vacuum " principle at that time may be taken from Pascal's opinion, expressed in the conclusion of the Nouvelles Experiences, when he writes: " I hold for true the maxims set out below: ( a ) that all bodies possess a repugnance to being separated one from another and from admitting a vacuum in the interval between them-that is to say that nature abhors a void.

This statement for real analytic functions ( with open ball meaning an open interval of the real line rather than an open disk of the complex plane ) is not true in general ; the function of the example above gives an example for x < sub > 0 </ sub > = 0 and a ball of radius exceeding 1, since the power series diverges for | x | > 1.

If the width of the interval is doubled, then probably only 4. 6 % of the true values lie outside the doubled interval, and if the width is tripled, probably only 0. 3 % lie outside.

* Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval 1.

One can prove that this periodogram converges to the true PSD when the averaging time interval T goes to infinity ( Brown & Hwang ) to approach the Power Spectral Density ( PSD ).

The true artist is like one of those scientists who, from a single bone can reconstruct an animal's entire body.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

That is particularly true of sovereignty when it is applied to democratic societies, in which `` popular '' sovereignty is said to exist, and in federal nations, in which the jobs of government are split.

On Fridays, the day when many Persians relax with poetry, talk, and a samovar, people do not, it is true, stream into Chehel Sotun -- a pavilion and garden built by Shah Abbas 2, in the seventeenth century -- but they do retire into hundreds of pavilions throughout the city and up the river valley, which are smaller, more humble copies of the former.

The resulting picture might appear a maze of restless confusions and contradictions, but it is more true to life than a portrait of an artificially contrived order.

`` What is more true than anything else??

To swim is true, and to sink is true.

One is not more true than the other.

that is, he is suspect, guilty, punishable, as is anyone in Mann's stories who produces illusion, and this is true even though the constant elements of the artist-nature, technique, magic, guilt and suffering, are divided in this story between Jacoby and Lautner.

A broader concept of imitation is needed, one which acknowledges that true invention is important, that the artist's creativity in part transcends the non-artistic causal factors out of which it arises.

It is true that New England, more than any other section, was dedicated to education from the start.

`` The man's true reputation is his work ''.

Though it centers around the brilliant and enigmatic figure of Charles 12,, the true hero is not finally the king himself.

Of few authors is this more true than of Heidenstam.

it is true that they are also extremely dull.

Years ago this was true, but with the replacement of wires or runners by radio and radar ( and perhaps television ), these restrictions have disappeared and now again too much is heard.