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The speculative logic project
Speculative logic, as envisaged here, is a theory of concepts, roughly in the tradition of Frege's Grundgesetze der Arithmetik, which has a two-fold aim:
The principal clue to answering this question, and at the same time the starting point for the speculative logic project, is provided by the emergence of paradoxes in the foundations of logic, semantics, and set theory which sank Frege's original project of a logical foundation of arithmetic, but at the same time opened up a new alley for a logical foundation of dialectic and speculative philosophy, by showing that the elementary building blocks of (higher order) logic, abstraction and predication, already give rise to contradictions. These contradictions are a source of possible content for logical insight. In this way truth-conserving expansions of a deductive apparatus may become possible without recourse to experience.
This means that one of the cornerstones of speculative logic is unrestricted abstraction, as originally introduced into logic by Frege. The other cornerstone is a restriction of classical logic which becomes necessary if one wants to work consistently with unrestricted abstraction. Apart from that, speculative logic is characterized by an extreme parsimony with regard to theoretical constants. Ideally, besides unrestricted abstraction there should be nothing more than a form of inclusion. At most, identity or alternatively predication may be admitted.
The idea of a speculative logic goes back to the German idealist philosopher
Georg Wilhelm Friedrich Hegel who already tried --- albeit with dubious success
-- to enact this idea in his Science of Logic. It came as a reaction to
Immanuel Kant's Critique of Pure Reason, in which an argument was
presented to the effect that we cannot know what things are in themselves, but
only how they appear to us, essentially through the forms of intuition, space
and time, and the categories of the understanding, grouped in the four classes
of quantity, quality, relation, and modality. Kant's argumentation is
essentially based on the construction of certain antinomies in pure reason in a
section entitled ``Antithetik der reinen Vernunft'' (Antithetic of Pure
Reason). Unfortunately, the standard of the philosophical discussion --- be
that in Kant or Hegel, or almost all of the secondary literature --- is poor and
this is why the philosophical background hardly plays a role in the project of
a development of speculative logic.