There is a hidden internal architecture of logic that is obscured by the implicit paradigms governing syntax, semantics, and consequence. By explicating and reformulating these paradigms in two distinct directions, this hidden architecture becomes visible.

First, modern (post-Fregean) logic has typically approached traditional (pre-Fregean) logic by reformulating its categories within modern logical frameworks. To make the internal architecture of logic visible, however, this direction must be reversed: modern logic must be reconstructed on the basis of the categorical distinctions already present in traditional logic.

Second, the internal architecture of logic has typically been sought through its algebraisation. Yet algebraic abstraction, while structurally powerful, necessarily suppresses certain features specific to logical construction, and therefore cannot render the full internal architecture of logic visible. To make this architecture fully explicit, it is not logic that must be algebraised, but algebra that must be logified.

As a result of this reformulation, the completeness theorem emerges in a new light. It is a universal property of the internal architecture of logic itself, rather than a result tied to particular logical calculi, and it can be formally proved within the new paradigms that make this architecture explicit. In other words, the absence of a completeness theorem in higher-order logics does not reflect an intrinsic limitation of logic itself; it reveals instead that the implicit paradigms governing syntax, semantics, and consequence are insufficient to support completeness in higher orders.

Of course, I cannot present the entire UL during the seminar lecture, as it is a mathematical construction of over 100 pages. The aim of the lecture is to show the basic ideas on which UL is based.

Finally, an important note: I will give the lecture in Hungarian.

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