on the contrary

adjective

1. opposite in nature or character; diametrically or mutually opposed: contrary to fact; contrary propositions.
2. opposite in direction or position: departures in contrary directions.
3. being the opposite one of two: I will make the contrary choice.
4. unfavorable or adverse.
5. perverse; stubbornly opposed or willful.

noun, plural con·trar·ies.

6. something that is contrary or opposite: to prove the contrary of a statement.
7. either of two contrary things.
8. Logic. a proposition so related to another proposition that both may not be true though both may be false, as with the propositions “All judges are male” and “No judges are male.”

adverb

9. in opposition; oppositely; counter: to act contrary to one’s own principles.

Idioms

10. by contraries, contrary to expectation.
11. on the contrary,
    1. in opposition to what has been stated.
    2. from another point of view: On the contrary, there may be some who would agree with you.
12. to the contrary,
    1. to the opposite effect: I believe he is innocent, whatever they may say to the contrary.
    2. to a different effect.

adjective

1. opposed in nature, position, etccontrary ideas
2. (kənˈtrɛərɪ) perverse; obstinate
3. (esp of wind) adverse; unfavourable
4. (of plant parts) situated at right angles to each other
5. logic (of a pair of propositions) related so that they cannot both be true at once, although they may both be false togetherCompare subcontrary (def. 2), contradictory (def. 3)

noun plural -ries

6. the exact opposite (esp in the phrase to the contrary)
7. on the contrary quite the reverse; not at all
8. either of two exactly opposite objects, facts, or qualities
9. logic a statement that cannot be true when a given statement is true

adverb (usually foll by to)

10. in an opposite or unexpected waycontrary to usual belief
11. in conflict (with) or contravention (of)contrary to nature

adj.mid-14c., from Anglo-French contrarie, from Latin contrarius “opposite, opposed,” from contra “against” (see contra). If we take the statement All men are mortal, its contrary is Not all men are mortal, its converse is All mortal beings are men, & its opposite is No men are mortal. The contrary, however, does not exclude the opposite, but includes it as its most extreme form. Thus This is white has only one opposite, This is black, but many contraries, as This is not white, This is coloured, This is dirty, This is black; & whether the last form is called the contrary, or more emphatically the opposite, is usually indifferent. But to apply the opposite to a mere contrary (e.g. to I did not hit him in relation to I hit him, which has no opposite), or to the converse (e.g. to He hit me in relation to I hit him, to which it is neither contrary nor opposite), is a looseness that may easily result in misunderstanding; the temptation to go wrong is intelligible when it is remembered that with certain types of sentence (A exceeds B) the converse & the opposite are identical (B exceeds A). [Fowler] As a noun from late 13c. Related: Contrarily; contrariwise. It’s the opposite, as in Is his shoulder hurting?—On the contrary, it’s all better, or We thought you didn’t like opera.—On the contrary, I love it. This phrase, at first put as by or for or in the contrary, dates from the late 1300s; on has been used since the mid-1800s. see on the contrary; to the contrary.