Lionel Curtis, writing in the imperialist journal The Round Table, wrote: " If the British Commonwealth can only be preserved by such means, it would become a negation of the principle for which it has stood ".

Axiom 1: If a property is positive, then its negation is not positive.

If one cannot discern existence from its negation, then the concept of existence has no meaning ; or in other words, does not ' exist ' in any meaningful way.

If P is justified, then another belief Q may be justified by P. If P is not justified, then P cannot be a justifier for any other belief: neither for Q, nor for Qs negation.

If we apply the stratification test known from Datalog, the predicate shaves is exposed as unstratifiable since it is defined recursively over its negation.

If either A or B were true, then the disjunction of A and B would be true, making its negation false.

If one wishes to include a connective for negation rather than consider it an abbreviation for, it is enough to add:

This represents the semantics of double negation ( e. g., " If NOT NOT any matching WMEs, then ...").

If we characterize predicate monism as the view subscribed to by eliminative materialists, who maintain that such intentional predicates as believe, desire, think, feel, etc., will eventually be eliminated from both the language of science and from ordinary language because the entities to which they refer do not exist, then predicate dualism is most easily defined as the negation of this position.

( If Zermelo – Fraenkel set theory ( ZFC ) is consistent, then neither the continuum hypothesis nor its negation can be proven from ZFC.

* If Martin's axiom and the negation of the continuum hypothesis both hold, then there is a non-free Whitehead group.

In propositional logic, double negation is the theorem that states that " If a statement is true, then it is not the case that the statement is not true.

“ If the constitution of a state is democratic, then every exceptional negation of democratic principles, every exercise of state power independent of the approval of the majority, can be called dictatorship .”

A nondeterministic algorithm for determining whether a 2-satisfiability instance is not satisfiable, using only a logarithmic amount of writable memory, is easy to describe: simply choose ( nondeterministically ) a variable v and search ( nondeterministically ) for a chain of implications leading from v to its negation and then back to v. If such a chain is found, the instance cannot be satisfiable.

If there is no one who will speak in negation, a student will be called to affirm.

If the high-order bit is set on a character ( i. e., if the byte value of the character is between 128 and 255 ) the character is generally rendered in the reverse video ( also called " inverse video ") of its counterpart between 0 and 127, using a bitwise negation of the character's glyph.

If there is an element with, then we may define negation by and.

If we fix our attention precisely on the Infinity of God, then, focusing the negation not upon the positive content of any created perfection but upon the fact that, because it is finite it is determined in kind and limited in degree, we may affirm that it is not found in God.

If it were not above and beyond the determinateness, it would not feel it as its negation and would feel no pain.

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

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.

If the computational Diffie – Hellman assumption holds in the underlying cyclic group, then the encryption function is one-way.

* If four points A, B, C, D are given that form a cyclic quadrilateral, then the nine-point circles of ABC, BCD, CDA and DAB concur at the anticenter of the cyclic quadrilateral.

If n is a positive integer, the integers between 1 and n − 1 which are coprime to n ( or equivalently, the congruence classes coprime to n ) form a group with multiplication modulo n as the operation ; it is denoted by Z < sub > n </ sub >< sup >×</ sup > and is called the group of units modulo n or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this group is cyclic if and only if n is equal to 2, 4, p < sup > k </ sup >, or 2 p < sup > k </ sup > where p < sup > k </ sup > is a power of an odd prime number.

If also, the cyclic quadrilateral becomes a triangle and the formula is reduced to Heron's formula.

If the diagonals of a cyclic quadrilateral intersect at P, and the midpoints of the diagonals are M and N, then the anticenter of the quadrilateral is the orthocenter of triangle MNP.

* If a cyclic quadrilateral has side lengths that form an arithmetic progression the quadrilateral is also ex-bicentric.

* If the opposite sides of a cyclic quadrilateral are extended to meet at E and F, then the internal angle bisectors of the angles at E and F are perpendicular.

* If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side.

If you examine the multiples of 1 / 7, you can see that each is a cyclic permutation of these six digits:

* If π < sub > 1 </ sub >( M ) is virtually cyclic but not finite then the geometric structure on M is S < sup > 2 </ sup >× R, and M is compact.

* If π < sub > 1 </ sub >( M ) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.

* If π < sub > 1 </ sub >( M ) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H < sup > 2 </ sup >× R or the universal cover of SL ( 2, R ).

If it is compact, then the 2 geometries can be distinguished by whether or not π < sub > 1 </ sub >( M ) has a finite index subgroup that splits as a semidirect product of the normal cyclic subgroup and something else.

* If π < sub > 1 </ sub >( M ) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or non-compact.

If the sequence b < sub > n </ sub > were periodic in n with period N, then it would be a cyclic convolution of length N, and the zero-padding would be for computational convenience only.

If the random starting point is 3. 6, then the houses selected are 4, 19, 35, 51, 66, 82, 98, and 113, where there are 3 cyclic intervals of 15 and 5 intervals of 16.

If A has a multiplicative identity 1, then it is immediate that the equivalence class ξ in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation.

If ord < sub > n </ sub > a is actually equal to φ ( n ) and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U ( n ) is cyclic and the residue class of a generates it.

If X is the plane with the origin missing, and G is the infinite cyclic group generated by ( x, y )→( 2x, y / 2 ) then this action is wandering but not properly discontinuous, and the quotient space is non-Hausdorff.

If u is a non-zero element of U, u · R = U ( where u · R is the cyclic submodule of U generated by u ).

If rectangles are included in the class of trapezoids then one may concisely define an isosceles trapezoid as " a cyclic quadrilateral with equal diagonals " or as " a cyclic quadrilateral with a pair of parallel sides.