Paradox is a situation (saying, statement, proposition or conclusion) that can exist in reality, but does not have a logical explanation. This and aporia should be distinguished. Aporia, unlike the paradox, is fictitious, logically correct situation that can not exist in reality.

 In the broadest sense, a paradox is a statement that is at variance with generally accepted opinion and seems illogical (often only with a superficial understanding). The paradox, unlike the aphorism, surprises with suddenness. Wilde’s "Divorce is done in heaven" can be a good example.  The paradox is always half-truth, and this, as Oscar Wilde said, is "the best thing we can achieve, because there are no absolute truths." The paradox with its stylized form resembles an aphorism. In the paradox, the customary truth crumbles before our eyes and is even mocked. For example: "I've heard so much slander in your address that I have no doubt: you are a fine person!", "Mutual misunderstanding is the most suitable basis for marriage" (O. Wilde).

Paradoxicality is unexpectedness, unfamiliarity, originality, self-contradiction to initial assumptions, conventional wisdom, traditional view or common sense in its content or form. An antonym for paradox is orthodoxy - validity, traditionality. "Orthodox" can be literally understood as "the next dominant tradition".

Paradox is present in many aspects of modern society. Logic is the main adversary of this situation as it is the first measure that is taken by anyone who faces it. 

The paradox in logic is a contradiction that has the status of a logically correct conclusion and is a reasoning leading to mutually exclusive conclusions at the same time. The logical error of the paradox is due to incorrect choice of logical premises, for example, when it comes to subjects that do not have a clear definition. 

There are such varieties of logical paradoxes as aporia and antinomy.

Aporia is characterized by the presence of a point that contradicts the obvious, generally accepted opinion, the common sense as the majority would say. Antinomy essentially stands for the presence of two conflicting and allegedly equally provable judgments.

Science is closely linked to logic and has it at the core of any assumptions or allegations in terms of discoveries and breakthroughs. Therefore, it is inevitably influenced by paradox in various situations. 

Modern sciences that are using logic as an instrument of cognition often encounter theoretical contradictions or contradictions of the consequences that the theory offers with verbalized results of experiment. This is due to logical errors in the construction of judgments, the imperfection of currently existing scientific methods or the insufficient accuracy of the tools used in the experiments, as well as the inadequacy of the accepted idealization, that is, the incorrect axiomatization of theories.

The presence of a paradox stimulates new research, a deeper understanding of the theory, its "obvious" postulates, and often leads to its complete revision.

Paradox Russell, Garay, EPR, Cosmological paradoxes among many others can serve as examples of such issues in science.

Though it may seem that paradox is usually common for logic and science, it is present in other, drastically different spheres. Art is the source of such situations created unintentionally or on purpose to stimulate additional reception from the viewer or listener. 

Paradoxicality is an extremely common quality that is present in works of various genres of art. Because of their unusualness, these statements, names, and content of works invariably attract people's attention. It is widely used in the colloquial genre, theatrical and circus arts, painting and folklore. A good speaker must use this technique in his speeches to maintain a lively interest of listeners. The comedy of most anecdotes is to describe an unusual, original situation. Lewis Carroll was a master of absurd literature and serves a good example in this case. 

Many aphorisms of famous speakers like Voltaire, Nietzsche, and Show include the elements of absurdity as well. Philosophy, or the art of thoughts, is famous for its unsolvable riddles of human existence and eternal mysteries of space and world. 
In classical music, a paradox is usually assigned to exquisite, strange works or fragments that differ from the traditional sound.
Medicine has also got the use of paradox. Such situations appear when the patient shows completely opposite results after treatment than expected. The case when individual is mysteriously defeating the disease after considered dead in advance can be called paradox.