Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus it is reasonable to believe that there is a significant difference between the two groups in their performance on this task after a brief `` structuring '' experience.

Thus, according to the Bank of England's Boxall and Gallagher ( 1997 ), "... there was virtually no difference between banks and building society ' listed ' interest rates for home finance mortgage lending between 1984 and 1997.

Thus, one can consider data compression as data differencing with empty source data, the compressed file corresponding to a " difference from nothing ".

Thus the maximum difference will differ depending on which of < math >

Thus, the distinction between micro-and macroevolution is not a fundamental onethe only difference between them is of time and scale.

Thus the difference between the two definitions of real numbers can be thought of as the difference in the interpretation of the statement " for all ... there exists ..."

Thus less energy is needed in the smaller nuclei and the difference to the state before is set free.

Thus, a few volts ' difference at the grid would make a large change in the plate current and could lead to a much larger voltage change at the plate ; the result was voltage and power amplification.

Thus, the main difference between Gregory's conception of ἀποκατάστασις and that of Origen would be that Gregory believes that mankind will be collectively returned to sinlessness, whereas Origen believes that personal salvation will be universal.

Thus, sensitive detection of specific optical frequencies necessitates optical heterodyne detection, in which two different ( close-by ) wavelengths of light illuminate the detector so that the oscillating electrical output corresponds to the difference between their frequencies.

Thus it has been supposed that the difference between the numbers is accounted for a garrison of 12 ships left at Aegina.

Thus, decentralisation opens up opportunities for local people to make a difference by coming together and giving them the resources to effect the local governments ( Larson, 2002 ).

Thus it is only our inability to observe the third group telescopically that makes the difference.

Thus as there is no fixed meaning in things, but they draw their meaning in a referential difference to other things, the wind objection can be incorporated into Theaetetus's claim that " Knowledge is sense perception ".

Thus, the difference of any two solutions to the equation Ax = b lies in the null space of A.

Thus this reflects the difference between the Egyptians who are interconnected with the elemental earth and the Romans in their dominating the hard-surfaced, impervious world.

Thus both the flyer and bobbin rotate to twist the yarn, and the difference in speed continually winds the yarn onto the bobbin.

Thus, as with all heat pumps, the Coefficient of Performance ( amount of thermal energy moved per unit of input work required ) decreases with increasing temperature difference.

Thus, although many religions may not recognize the difference, for Aristotle and many philosophers hubris is altogether an entirely different thing from pride.

Thus, the relative difference in velocity between a source and an observer is what gives rise to the doppler effect.

Thus, when a differential amplifier is used to extract this difference, its output constitutes the low pass filter section.

Thus the Comte de Montlosier, in exile during the French Revolution, equated class difference in 18th-century France with racial difference.

Thus, intuitionists are ready to accept a statement of the form " P or Q " as true only if we can prove P or if we can prove Q:

Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows.

Thus, though knowledge requires justification, truth, and belief, the word " knowledge " can't be, according to Williamson's theory, accurately regarded as simply shorthand for " justified true belief.

Thus a statement of the form is said to be true, under a particular interpretation, if there is some object in the domain of discourse of that interpretation that satisfies the predicate that the interpretation uses to assign meaning to the symbol Phil.

Thus the songs of Uttarakhand are a true reflection of the Cultural Heritage and the way people live their lives in the Himalayas.

Thus for iconoclasts the only true ( and permitted ) " icon " of Jesus was the Eucharist, which was believed to be his body and blood.

Thus Irenaeus provides the earliest witness to the assertion of the four canonical Gospels, possibly in reaction to Marcion's edited version of the Gospel of Luke, which Marcion asserted was the one and only true gospel.

Thus ( D1 ) is both true and false.

Thus, for example, the statement, " It is true that two plus two equals four ", contains no more information than the statement " two plus two equals four ", because the phrase " it is true that ..." is always implicitly there.

Thus for a theory " T ", " G " is true, but not provable in " T ".

Thus, in the cases listed above, the disjunction of and is true in all cases except 4.

Thus it is true in every case above except case 2, because this is the only case when is true but is not.

Thus to Brouwer, we are not justified in asserting " either Goldbach's conjecture is true, or it is not.

Thus, the statement " there are no contradictions in the Principia system " cannot be proven in the Principia system unless there are contradictions in the system ( in which case it can be proven both true and false ).

Thus, according to the law of non-contradiction, " not p " must be false, and so, according to the law of the excluded middle, p is true.

Thus, many scholars will call a body of stories " mythology ", leaving open the question of whether the stories are true or false.

" Thus without holding a proposition up against the real world, we cannot tell whether the proposition is true or false.

Thus in Puget Sound itself, the names given by Wilkes are common and Spanish names rare, while the reverse is true for the San Juan and Gulf Islands ( although the Spanish did not explore Puget Sound as thoroughly as the British and Americans, resulting in fewer Spanish names to start with ).

Thus the Council Fathers meant to say that the being of the Church as such is a broader entity than the Roman Catholic Church, but within the latter it acquires, in an incomparable way, the character of a true and proper subject.

Thus, there is no such thing as a sentence, term, expression or word symbolically representing a single true meaning ; it is underspecified ( which cat sat on which mat?

Thus this report of a feud between Eadwig and Dunstan could either have been based on a true incident of a political quarrel for power between a young king and powerful church officials who wished to control the king and who later spread this legend to blacken his reputation, or it could be an urban legend ; the Chronicle also tells of Odo putting aside the King's marriage on the grounds Eadwig and his wife were " too related ".

Thus, the subaltern relationship between universal and existential also does not hold, since for a nonexistent S, " All S is P " is true but does not entail " Some S is P ", which is false.