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, a difference in true wind creates a different apparent wind at different heights.

" 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.

Thus, there is substantial debate regarding the identity of the book's author.

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, Boyarin has argued that " Jewishness disrupts the very categories of identity, because it is not national, not genealogical, not religious, but all of these, in dialectical tension.

This identifies the tangent space T < sub > e </ sub > at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by = < sub > e </ sub >.

Thus, for Rand, the fact that consciousness must itself possess identity implies the rejection of both universal skepticism based on the " limits " of consciousness and any claim to revelation, emotion or faith based belief.

Thus token identity holds between colours and physical / brain states at best.

Thus, for a language to be used at all it must have some public criterion of identity.

Thus, the aggregate term Third World was challenged as misleading even during the Cold War period because it had no consistent or collective identity among the countries it supposedly encompassed.

Thus, it is not correct to establish a strict common origin, but a common cultural identity.

Thus, by combinatorial specifying the identity of particular body regions, Hox genes determine where limbs and other body segments will grow in a developing embryo or larva.

Thus, although other economists focus on the identity or type of legal system of the colonizers to explain institutions, these authors look at the environmental conditions in the colonies to explain institutions.

Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered an operator that takes zero arguments.

Thus the sine law is equivalent to the angle sum identity when the angles sum is between 0 and 180 degrees and when each individual angle is between 0 and 90 degrees.

Thus, individuals with little or no direct ancestry from the Arabian Peninsula could identify themselves or be considered as Arabs, partially by virtue of their home language ( see Arab identity ).

Thus Alexander is attempting to " naturalize " language and national identity via the metaphor of " blood.

Thus the only two field automorphisms of that leave the real numbers fixed are the identity map and complex conjugation.

Thus, groups with a strong sense of unity and identity can benefit from kin selection behavior such as common property and shared resources.

Thus the Orthodox emphasis would be that Christ died, not to appease an angry and vindictive Father, or to avert the wrath of God, but to change people so that they may become divine, that is to say, become God in his energies or operations but not in his essence or identity as God ( see theosis ).

Thus, critics argue that Eurocommunism does not have a well-defined identity and cannot be regarded as a separate movement in its own right.

The notion is closely related to the idea of a multiplicative character in group theory, which is a homomorphism χ from a group G to the multiplicative group of a field F. Thus χ: G → F < sup >×</ sup > satisfies χ ( e ) = 1 ( where e is the identity element of G ) and

Thus, the Jacobi identity for Lie algebras simply becomes the assertion that the action of any element on the algebra is a derivation.

Thus, if the structure of one member of a family is known, a reliable prediction may be made for a second member of the family, and the higher the identity level, the more reliable the prediction.

Thus, the principle of nationalism serves to preserve the cultural traditions of La Familia de La Raza and promotes our identity as a Chicana / Chicano Gente .’ MEChA members themselves differ in their interpretations of " La Raza ".

If the domain has a multiplicative identity ( which we may call 1 ), this is equivalent to saying that 1 ≠ 0 Thus a domain is a nontrivial ring without left or right zero divisors.