### Some articles on *implies*:

... Using Euler's identity this conjecture

**implies**the transcendence of many numbers involving e and π. 2, y1 = iπ, and y2 = iπ√2, the conjecture — if true —

**implies**that one of the following four numbers is transcendental The first of these is just −1, and the fourth is 1, so the ... By setting x1 = 1, x2 = t, y1 = log2, and y2 = log3, the four exponentials conjecture

**implies**that if t is irrational then one of the following four ...

... The diamond principle, a consequence of V=L,

**implies**that there is a Suslin tree, and Martin's axiom MA(ℵ1)

**implies**that there are no Suslin trees ... Whether the Generalized Continuum Hypothesis

**implies**the existence of an ℵ2-Suslin tree, is a longstanding open problem ...

... In particular continuity with probability one

**implies**continuity in probability continuity in mean-square

**implies**continuity in probability continuity with probability one neither

**implies**...

... ("if A

**implies**B then not-B

**implies**not-A", and vice versa), which expresses the law of contraposition ... ("if not-A

**implies**both B and its negation not-B, then not-A must be false, then A must be true"), which is the principle known as reductio ad absurdum ... ("if A

**implies**B and B

**implies**C, then A

**implies**C"), which is the principle known as syllogism ...

... These relations are transitive ,

**implies**,

**implies**and satisfy

**implies**(this follows trivially from the definition) ,

**implies**,

**implies**...

### Famous quotes containing the word implies:

“Lovers have a way of using this word “nothing” which *implies* exactly the opposite.”

—Honoré De Balzac (1799–1850)

“I’m a priest, not a priestess.... “Priestess” *implies* mumbo jumbo and all sorts of pagan goings-on. Those who oppose us would love to call us priestesses. They can call us all the names in the world—it’s better than being invisible.”

—Carter Heyward (b. 1946)

“Religion means goal and way, politics *implies* end and means. The political end is recognizable by the fact that it may be attained—in success—and its attainment is historically recorded. The religious goal remains, even in man’s highest experiences, that which simply provides direction on the mortal way; it never enters into historical consummation.”

—Martin Buber (1878–1965)