Transcendent function:

A psychic function that arises from the tension between consciousness and the unconscious and supports their union. (See also opposites and tertium non datur.)
When there is full parity of the opposites, attested by the ego’s absolute participation in both, this necessarily leads to a suspension of the will, for the will can no longer operate when every motive has an equally strong counter-motive. Since life cannot tolerate a standstill, a damming up of vital energy results, and this would lead to an insupportable condition did not the tension of opposites produce a new, uniting function that transcends them. This function arises quite naturally from the regression of libido caused by the blockage.[Ibid., par. 824.]

The tendencies of the conscious and the unconscious are the two factors that together make up the transcendent function. It is called “transcendent” because it makes the transition from one attitude to another organically possible.[“The Transcendent Function,” CW 8, par. 145.]

In a conflict situation, or a state of depression for which there is no apparent reason, the development of the transcendent function depends on becoming aware of unconscious material. This is most readily available in dreams, but because they are so difficult to understand Jung considered the method of active imagination-giving “form” to dreams, fantasies, etc.–to be more useful.

Once the unconscious content has been given form and the meaning of the formulation is understood, the question arises as to how the ego will relate to this position, and how the ego and the unconscious are to come to terms. This is the second and more important stage of the procedure, the bringing together of opposites for the production of a third: the transcendent function. At this stage it is no longer the unconscious that takes the lead, but the ego.[Ibid., par. 181.]

This process requires an ego that can maintain its standpoint in face of the counter-position of the unconscious. Both are of equal value. The confrontation between the two generates a tension charged with energy and creates a living, third essence.

Tertium non datur:

The reconciling “third,” not logically foreseeable, characteristic of a resolution in a conflict situation when the tension between opposites has been held in consciousness.

As a rule it occurs when the analysis has constellated the opposites so powerfully that a union or synthesis of the personality becomes an imperative necessity. . . . [This situation] requires a real solution and necessitates a third thing in which the opposites can unite. Here the logic of the intellect usually fails, for in a logical antithesis there is no third. The “solvent” can only be of an irrational nature. In nature the resolution of opposites is always an energic process: she acts symbolically in the truest sense of the word, doing something that expresses both sides, just as a waterfall visibly mediates between above and below.[“The Conjunction,” CW 14, par. 705.]