Everybody knows the Epimenides paradox, also known as the Liar. It goes like this: Epimenides, who is a Cretan, says “all Cretans are liars”–is he telling the truth?
Impurities may be discerned in the above presentation that prevent it from being a perfect paradox. For this reason a purer, “strengthened” version of the liar may be formulated: “This statement is false.” If the statement is false, then it is true. And if it is true, then it is false. Pure paradox.
How do we formalize this? Paradoxes are clearly contradictory. But the form of the simple contradiction, A = ¬A, is not paradoxical; i.e., “this statement is not this statement” fits the form of simple contradiction but it is not clearly paradoxical.
The difference between contradiction and paradox seems to be that the latter involves conditional logic. Now the form of the contradictory conditional is A → ¬A. But there are still non-paradoxical statements that fit that form: if it rains then it does not rain. Why is this not as paradoxical as the liar? Because it is one-sided: if it rains then it doesn’t, and if it doesn’t then it still doesn’t. Circularity does not occur. The statement is always false.
The perfect paradox has the circular quality so described. We want a form such that if it rains then it doesn’t, and if it doesn’t then it does. This is the contradictory material equivalence. The paradox arises when one side of the materially equivalent connective is the negative of the other side, i.e., A ↔︎ ¬A. This is the pure form of the paradox, viz., contradictory material equivalence. If A is, then it is not. And if it is not, then it is.
A paradox is a contradiction, the sides of which are external to each other or held asunder by causality. So even A = ¬A is a paradox, although its paradoxicality vanishes into simple contradiction because the = cancels the mediation.
The material equality (↔︎) is like logical equality (=), except the paradox is more pronounced because the two sides of the equivalence mediate each other. The truth is that, just as the = separates the A from itself in simple equality (A = A), so does the ↔︎ separate the equality itself from itself in material equality (A ↔︎ A). This is what Hegel means by externality.
Now Kant has shown that these logical operations have natural counterparts or Doppelgänger. The conditional corresponds to temporal causality or succession. For example, if there are dark clouds, then it will rain. The dark clouds cause the rain. If there is an egg, then there will be a chicken. The egg precedes and implies the promise of a chicken. We express this as A → ¬A. The egg (A) gives rise to something other than itself (¬A).
But it is also true that this other (i.e. the chicken, ¬A) gives rise to an egg (A). Therefore (A → ¬A) ∧ (¬A → A), which is exactly the contradictory material equivalence: A ↔︎ ¬A.
From this we determine that a paradox, or chicken-and-egg problem as it is called, is a simple contradiction or contradictory equivalence (A = ¬A) that has fallen into time, i.e., where the equality is mediated by causality. It therefore becomes contradictory material equivalence, which is the form of paradox generally. Paradoxes are contradictions, the sides of which are held apart by causal succession.
There are also conjunctive contradictions, e.g., A ∧ ¬A. There is a chicken at one time, then I eat it, and it is no more. Now the logical form of this change is the conjunctive ∧ because in conjunction the two sides are brought together externally. The sides sit side-by-side in their independence and they are brought together by some third element (me when I ate the chicken). But if on the other hand the chicken dies of old age, then the chicken’s death was in a sense suicidal, which is the simple contradiction or A = ¬A, or more specifically A → ¬A.
Now insofar as God, that is the Father, is conceived of as the causa sui, then he is A ↔︎ A. And inasmuch as the Son is the finite Father he is the contradictory Father, i.e., the paradoxical A ↔︎ ¬A.
We can go further here. When we negate the universal, the particular results; e.g., if not all men are tall then only some men are. And likewise when we negate the necessary, the possible results. The universal (all) is to the necessary (must), as the particular (some) is to the possible (can). If mortality is necessary, then all men will be mortal. If mortality is a possibility, then some men may choose mortality and others immortality.
If we begin with ∧, that is
quantityquality (analog of unity). And if we negate the ∧ or conjunction we get ∨ or disjunction, which is qualityquantity (analog of diversity). And from ∨ we can derive → which is causality and then ↔︎ which is what Hegel calls Wechselwirkung or reciprocity. [Update April 5th, 2022: In fact it’s the conjunction is quality and the disjunction is quantity. This is because unity is to quality as diversity is to quantity. Such much is demonstrated in René Guénon’s book, The Reign of Quantity & the Signs of the Times, as it is of course in Hegel’s Logic.]
Finally it is clear that Man and Woman are the two sides of the Concept of the Human Being, and that therefore this Human Concept has the paradoxical form A ↔︎ ¬A. Hence why love is so difficult: it is at bottom—pure paradox.