Positivity of Form (a): The most straightforward way a law is positive is through its form. A rule becomes law by virtue of having formal validity within a state. It is declared by a legitimate authority (a legislature, a monarch) and becomes part of the official legal code. The science that studies these valid laws, whatever their content, is the positive science of law (positive Rechtswissenschaft). Its task is to know what is legally binding, not necessarily what is rationally justified.

Positivity of Content (b): This is the more complex and philosophically interesting part. The content of right also necessarily contains positive, or non-universal, elements. I identify three sources for this:

(α) Historical and National Context: The laws of a nation are not created in a vacuum. They are inevitably shaped by the specific character of its people (Nationalcharakter), its unique historical stage of development, and even its climate and geographical situation (what I call the “connection of all the relations conditioned by natural necessity”). For example, laws governing agriculture will differ between a mountainous region and a coastal plain. A nation’s legal code reflects its spirit and its history.
(β) Judicial Application: A universal legal principle (e.g., “property should be respected”) must be applied to an infinite variety of particular cases. This application requires a further determination that the universal concept itself cannot provide. For instance, what punishment is appropriate for a specific theft? How long should a statute of limitations be? This decision, which translates the universal into the particular, is necessarily left to the discretion (Verstand) of a judge or legislator. It introduces a non-deducible, positive element.
(γ) The Finality of Decision: In the real world, a legal process must come to an end. A final decision must be made. At some point, one must simply say: “The punishment is ten years, not nine or eleven.” “The deadline is June 1st, not June 2nd.” This final, concrete determination is a necessary act of positing that cannot be derived purely from the rational concept. It is a moment of pure decision required for the law to be actual.
The Remark (Anmerkung) to § 3:

The Remark here is crucial. It serves as a direct polemic against the confusion between the philosophical and the positive sciences of right.

Its main purpose is to defend philosophy from irrelevant criticisms. One cannot refute a philosophical theory of Right by pointing out that an existing law contradicts it, or by showing the historical origin of that law. That would be to confuse the question of rational justification with the question of positive validity.

I criticize those jurists who, when asked for the reason behind a law, retreat to mere authority—”it is so written in the code,” or “the Roman jurists decided it so.” This, I argue, is a refusal to engage in rational thought.

However, the Remark also contains a profounder point. The “positivity” of law is not a flaw; it is a necessary moment of the Idea of Right itself. For the rational Concept of right to be actual, it must be posited, it must take on a definite, historical, and particular form. A system of right that remained purely a set of abstract, universal principles would not be a real system of right at all. It would be an impotent abstraction. The irrationality lies not in the existence of positive law, but in the belief that the positive aspect (the historical fact, the arbitrary decision) is its ultimate foundation, rather than the rational concept which it serves to actualize.

Insights from the Lectures:

In my lectures, I would place this argument in the context of the great debate with the Historical School of law, championed by figures like Friedrich Carl von Savigny. The Historical School correctly saw that law grows organically from the spirit and history of a people (Volksgeist). I acknowledge this point in § 3 (α). However, they erred in thinking that history alone provides the justification. For me, history is the theater in which the rational Idea of right unfolds, but history itself is not the ultimate judge of what is right. We still need philosophy to provide the standard of reason by which we can judge whether a given historical law is adequate to the Concept of Right.

So, to my students, I would say: Savigny is right that law is historical, but he is wrong to think that makes philosophy irrelevant. Philosophy comprehends the reason within the history.

Connection to the Encyclopaedia of the Philosophical Sciences:

As before, there is not a single paragraph in the Encyclopaedia that maps perfectly onto § 3 of the GPR. The Encyclopaedia presents the compressed, logical skeleton. However, the entire movement from “Abstract Right” (§ 488 ff.) to the later stages is an illustration of this principle.

Abstract Right (§§ 489-490): This begins with the purely abstract command: “Be a person and respect others as persons.” This is the universal, philosophical concept.
The move to Gesetz (Statute Law): For this abstract command to become effective in a society, it must be made into positive law (Gesetz). The Zusatz to § 503 (in the section on Morality) touches upon this, noting that right must be “known” and “have the form of universality” which is given to it by being posited as law.
The entire section on the State, particularly on Legislative Power (§ 529 ff.), presupposes this transition. The legislature’s job is precisely to take the principles of right and give them the positive determination needed for them to function in a specific state.
In essence, the Encyclopaedia shows the logical necessity of moving from the abstract universal to the concrete particular. § 3 of the Philosophy of Right describes the character of this concrete particularity—its “positivity”—as it appears in the sphere of law. It shows that the contingent, historical, and decisional elements of law are not alien to reason, but are the very means by which reason makes itself real in the world.