Dialectics is the name of an advanced kind of thinking.
It is a kind of thinking that goes beyond the EITHER/OR absolute rigidity [never BOTH/AND] of formal-logic thinking.
Dialectics is a kind of thinking that the plurality of humanity will need to master before becoming capable of advancing from the state-capitalist, totalitarian, humanocidal dictatorship into which capitalisms converge, to equitarian political-economic democracy -- i.e., if humanity is to, first, survive, and then to thrive.
Call dialectic the kind of thinking that emerges in "the dialectical operations stage of adult human cognitive development", after the "formal operations stage of adult cognitive development" is surpassed.
The best way to briefly express the core principle of dialectic is via the Seldonian -- purely qualitative -- category notation.
The reality represented by a kind-of-thing category, call it "a", interacts with itself, or 'intra-acts' -- in the internal mind-world, or in the external world, or in both, e.g., concurrently -- because of the "internal strife" within that reality.
[using "a^2" to stand for " acts on a", "a interacts with a", "a squared", "a in the 2nd degree", "a times itself", or "a OF a"] --
a^2 = a x a = a + b
-- where "b" names the contra-category to category "a".
The self-negation of "a", modeled by its self-multiplication, produces "a" again, but also produces its opposite ["a", "plus" its opposite, "b", like apples "plus" oranges].
For example, "a" might stand for the "primitive communal" kind of society, with which humanity began.
Then, "b" might stand for its historical successor and opposite kind of society, class self-divided society.
The "formal operations stage of adult cognitive development" stops its kind of thinking with the irreconcilable, "eternal" opposition of a and b as idea[l]s, and, as realities, demands the complete dominance of one, and the complete obliteration of the other by the one.
But the "dialectical operations stage" thought-process expects to see the "karmic" negation again of that first negation --
a^3 = a x ( a x a) = a x (a + b) = a + b + C(a,b)
-- where C(a,b) stands for the combination, "complex unity", or "dialectical synthesis" of the erstwhile opposites a and b.
Note: This "category combinatorics" can continue beyond the third term, C(a,b).
But the simplest "image" of dialectics I know of is:
a ---> a(a(a)) = ~(~(a)) = a + b + C(a,b).