Although this model is often named after Hegel, he himself never used that specific formulation. Hegel ascribed that terminology to Kant. Carrying on Kant's work, Fichte greatly elaborated on the synthesis model, and popularized it.
On the other hand, Hegel did use a three-valued logical model that is very similar to the antithesis model, but Hegel's most usual terms were: Abstract-Negative-Concrete. Sometimes Hegel would use the terms, Immediate-Mediated-Concrete. Hegel used these terms hundreds of times throughout his works.
The formula, Thesis-Antithesis-Synthesis, does not explain why the Thesis requires an Antithesis. However, the formula, Abstract-Negative-Concrete, suggests a flaw in any initial thesis—it is too abstract and lacks the negative of trial, error and experience. The same applies to the formula, Immediate-Mediated-Concrete. For Hegel, the Concrete, the Synthesis, the Absolute, must always pass through the phase of the Negative, that is, Mediation. This is the actual essence of what is popularly called Hegelian Dialectics.
To describe the activity of overcoming the negative, Hegel also often used the term Aufhebung, variously translated into English as "sublation" or "overcoming," to conceive of the working of the dialectic. Roughly, the term indicates preserving the useful portion of an idea, thing, society, etc., while moving beyond its limitations. (Jacques Derrida's preferred French translation of the term was relever).
In the Logic, for instance, Hegel describes a dialectic of existence: first, existence must be posited as pure Being (Sein); but pure Being, upon examination, is found to be indistinguishable from Nothing (Nichts). When it is realized that what is coming into being is, at the same time, also returning to nothing (in life, for example, one's living is also a dying), both Being and Nothing are united as Becoming.
As in the Socratic dialectic, Hegel claimed to proceed by making implicit contradictions explicit: each stage of the process is the product of contradictions inherent or implicit in the preceding stage. For Hegel, the whole of history is one tremendous dialectic, major stages of which chart a progression from self-alienation as slavery to self-unification and realization as the rational, constitutional state of free and equal citizens. The Hegelian dialectic cannot be mechanically applied for any chosen thesis. Critics argue that the selection of any antithesis, other than the logical negation of the thesis, is subjective. Then, if the logical negation is used as the antithesis, there is no rigorous way to derive a synthesis. In practice, when an antithesis is selected to suit the user's subjective purpose, the resulting "contradictions" are rhetorical, not logical, and the resulting synthesis is not rigorously defensible against a multitude of other possible syntheses. The problem with the Fichtean "Thesis-Antithesis-Synthesis" model is that it implies that contradictions or negations come from outside of things. Hegel's point is that they are inherent in and internal to things. This conception of dialectics derives ultimately from Heraclitus.
Hegel has outlined that the purpose of dialectics is "to study things in their own being and movement and thus to demonstrate the finitude of the partial categories of understanding"
One important dialectical principle for Hegel is the transition from quantity to quality, which he terms the Measure. The measure is the qualitative quantum, the quantum is the existence of quantity.
As an example, Hegel mentions the states of aggregation of water: "Thus the temperature of water is, in the first place, a point of no consequence in respect of its liquidity: still with the increase or diminution of the temperature of the liquid water, there comes a point where this state of cohesion suffers a qualitative change, and the water is converted into steam or ice". As other examples Hegel mentions the reaching of a point where a single additional grain makes a heap of wheat; or where the bald-tail is produced, if we continue plucking out single hairs."The identity between quantity and quality, which is found in Measure, is at first only implicit, and not yet explicitly realised. In other words, these two categories, which unite in Measure, each claim an independent authority. On the one hand, the quantitative features of existence may be altered, without affecting its quality. On the other hand, this increase and diminution, immaterial though it be, has its limit, by exceeding which the quality suffers change. [...] But if the quantity present in measure exceeds a certain limit, the quality corresponding to it is also put in abeyance. This however is not a negation of quality altogether, but only of this definite quality, the place of which is at once occupied by another. This process of measure, which appears alternately as a mere change in quantity, and then as a sudden revulsion of quantity into quality, may be envisaged under the figure of a nodal (knotted) line".
Another important principle for Hegel is the negation of the negation, which he also terms Aufhebung (sublation): Something is only what it is in its relation to another, but by the negation of the negation this something incorporates the other into itself. The dialectical movement involves two moments that negate each other, a somewhat and an another. As a result of the negation of the negation, "something becomes an other; this other is itself somewhat; therefore it likewise becomes an other, and so on ad infinitum" Something in its passage into other only joins with itself, it is self-related. In becoming there are two moments coming-to-be and ceasing-to-be: by sublation, i.e. negation of the negation, being passes over into nothing, it ceases to be, but something new shows up, is coming to be. What is sublated (aufgehoben) on the one hand ceases to be and is put to an end, but on the other hand it is preserved and maintained. In dialectics, a totality transforms itself; it is self-related.