Thus any expression in sentential logic has a pa translation.

Thus, for example, his teaching on the Trinity reveals a subordination of Son to Father that in the later crass form of Arianism the Church rejected as heretical.

Thus the adventure reveals itself in the context of a timber raid, bringing cedar wood to timberless Mesopotamia.

Thus, an issue limit figure never fully reveals the true number actually created.

Thus, being a " wooly " god, Veles was considered to be a protector of shepherds, which reveals one additional trait of his enemity with Perun, who, as a giver of rain, would be god of farmers.

Thus, the range captures the intensity of their feelings for a given item, while the results of analysis of multiple items ( if the items are developed appropriately ) reveals a pattern that has scaled properties of the kind Likert identified.

Thus, he has decided to abdicate his throne, and reveals that Willingham will succeed him.

Thus, she reveals the salvation or realization of true human potential of the highest order.

Thus, the widest view of history reveals three most important stages of development: Oriental imperial ( the stage of oneness, of suppression of freedom ), Greek social democracy ( the stage of expansion, in which freedom was lost in unstable demagogy ), and Christian constitutional monarchy ( which represents the reintegration of freedom in constitutional government ).

Thus the study of liminality in sociology, ritual and theatre reveals the fictional elements underpinning society.

Thus, when more than one distinct form leads to a particular cell in the X-region, a chain of information cells must be created to accommodate the forms, one cell in the chain for each form.

Thus a simple count of the number of distinct symbols is an important clue to the nature of an unknown script.

Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

Thus perceptions must already come parcelled into distinct " bundles " before they can be associated according to the relations of similarity and causality: in other words, the mind must already possess a unity that cannot be generated, or constituted, by these relations alone.

Thus a distinct and legitimate non-Cantonese culture with specifics to Hong Kong grew.

Thus, Christianity acquired an identity distinct from Rabbinic Judaism, but this distinction was not recognised all at once by the Roman Empire, see Split of early Christianity and Judaism for details.

Thus, seawater flow is eastward in the Strait's surface waters, and westward below ; once in the Atlantic, this chemically distinct Mediterranean Intermediate Water can persist thousands of kilometres away from its source.

Thus, if n is a product of powers of distinct primes, say n

Thus, a metaphor creates new links between otherwise distinct conceptual domains, whereas a metonymy relies on the existing links within them.

Thus followed the line of Lomonosov and the contributions of the Russian school became more frequent through his disciples, and in the nineteenth century we have great geographers as Vasily Dokuchaev who performed works of great importance as a " principle of comprehensive analysis of the territory " and " Russian Chernozem " latter being the most important where introduces the geographical concept of soil, as distinct from a simple geological strata, and thus founding a new geographic area of study: the Pedology.

Thus it is assumed that the property is a universal which is distinct from the particular individual who has the property ( MacLeod & Rubenstein, 2006, § 1b ).

Thus, for Christians, Christ himself is the only high priest, and Christians have no priesthood independent or distinct from participation in the priesthood of Christ, the head of the Church.

Thus " pop music " may be used to describe a distinct genre, aimed at a youth market, often characterized as a softer alternative to rock and roll.

Thus, while there are short-term memory components to working memory models, the concept of short-term memory is distinct from these more hypothetical concepts.

Thus the term humanist can mean a humanities scholar, as well as refer to The Enlightenment / Renaissance intellectuals, and those who have agreement with the pre-Socratics, as distinct from secular humanists.

Thus, one way to show two objects of C are distinct ( up to isomorphism ) is to show that their images under f are distinct ( i. e. not isomorphic ).

Thus, most fortified wines are regarded as distinct from dessert wines, but some of the less strong fortified white wines, such as Pedro Ximénez sherry and Muscat de Beaumes-de-Venise, are regarded as honorary dessert wines.

Thus, concupiscence as a distinct term is more likely to be used by Catholics.

Thus the history of English military law up to 1879 may be divided into three periods, each having a distinct constitutional aspect: ( I ) prior to 1689, the army, being regarded as so many personal retainers of the sovereign rather than servants of the state, was mainly governed by the will of the sovereign ; ( 2 ) between 1689 and 1803, the army, being recognized as a permanent force, was governed within the realm by statute and without it by the prerogative of the crown and ( 3 ) from 1803 to 1879, it was governed either directly by statute or by the sovereign under an authority derived from and defined and limited by statute.

Thus, by taking Scott to this territory and keeping him there for two and a half years, Emerson was breaking the law in three distinct ways.

Thus, completing any given task on the Internet, such as retrieving a webpage, relies on the contributions of hundreds of people at a hundred or more distinct companies, each of which may have only very tenuous connections with each other.

Thus, it appears that people's feelings and creative cognitions are interwoven in several distinct ways within the complex fabric of their daily work lives.

" Thus, ecumenism is the promotion of unity or cooperation between distinct religious groups or denominations of Christianity.

Thus, it refers to two or more distinct concepts sharing the " same name " or signifier.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

In a digital implementation, the number of computations performed per sample is proportional to N. Thus the mathematical problem is to obtain the best approximation ( in some sense ) to the desired response using a smaller N, as we shall now illustrate.

Thus, numerical mathematical methods are required, even to solve the system of one nucleus and two electrons.

However, a second definition and usage has historically been in practice in many fields of computer science and information technology, which defines the prefix kilo when used with byte or bit units of data as 1024 ( 2 < sup > 10 </ sup >); this is due to the mathematical coincidence that Thus, in these fields 1 kilobyte is equal to 1 kibibyte, a new unit standardized as part of the binary prefixes to resolve the ambiguity.

Thus, there is a direct mathematical relationship between the pressure and the speed, so if one knows the speed at all points within the airflow one can calculate the pressure, and vice versa.

Thus the actual mathematics of mathematical games may not be apparent to the average player.

Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.

Thus, one would expect to find a factor that roughly corresponds to " left " and " right ", as this is the dominant framing for politics in our society, but the basis of Eysenck's " tough / tender-minded " thesis ( the second, " T "- factor ) may well represent nothing beyond an abstract mathematical construct.

Thus the mathematical formulation of the Alcubierre metric does not contradict the conventional claim that the laws of relativity do not allow a slower-than-light object to accelerate to faster-than-light speeds.

Thus the mathematical formalism used to describe the time evolution of a non-relativistic system proposes two opposed kinds of transformation:

Thus the axioms of ZFC refer only to sets, not to urelements ( elements of sets which are not themselves sets ) or classes ( collections of mathematical objects defined by a property shared by their members ).

Thus during the session 1778 – 1779, in addition to his mathematical work, he delivered an original course of lectures on morals.

Thus, in mathematical physics, the structure coefficients are often written c < sub > i, j </ sub >< sup > k </ sup >, and their defining rule is written using the Einstein notation as

Thus, in mathematical terms, they never appear as indices to the scattering matrix, which is to say, they never appear as the observable inputs and outputs of the physical process being modelled.

Thus, inverse problems are one of the most important, and well-studied mathematical problems in science and mathematics.

Thus, Green's later mathematical contributions, which exhibited knowledge of very modern developments in maths, could not have resulted from his tenure at the Robert Goodacre Academy.

Thus, from the point of view of strict mathematical description, the thesis that everything is a computing system in this second sense cannot be supported ".

Thus the vaccination programme will produce an increase in the average age of infection, another mathematical justification for a result that might have been intuitively obvious.

Thus, it follows by mathematical induction.

Thus, in addition to the calculus topics that one would expect to find in any mathematical software, MacTutor is particularly strong in geometry, algebra ( and in particular, group theory ), graph theory, number theory as well as stacks on statistics, matrices and complex analysis.

Thus, if a fundamental physical law is expressed with a mathematical equation in one inertial frame, it must be expressed by an identical equation in any other inertial frame, provided both frames are parameterised with charts of the same type.

Thus both numbers, 6 and 10, were credited with perfection, both on purely mathematical grounds and on grounds of their relevance in nature.

Thus, Marx's value theory offers an interpretation, generalisation or explanation concerning the " grand averages " of the relative price movements of products, and of economic behaviour in capitalist production as a social system, but it is not possible to deduce specific real product-prices from product-values according to some mathematical function, among other things because, to find labour-values, a relationship between product-prices and labour hours worked must already be assumed.