Kant's antinomies and the square of opposition
Marcel Quarfood


Aims & changes

The project's aim is to use the square of opposition and its recent extension to a hexagon to investigate Kant's antinomies. In his Critique of Pure Reason Kant presents four pairs of opposed theses about fundamental metaphysical topics. According to Kant, given traditional (pre-critical) philosophical assumptions, both alternatives of an antinomy are provable, giving rise to contradiction. Kant claims that if we adopt transcendental idealism, the opposed propositions are no longer contradictory alternatives, but instead either both false or both true. This solution relies on an implicit use of the Aristotelian square of opposition: in the first two antinomies Kant treats the alternatives as contraries, while the alternatives of the third and fourth antinomies are treated as subcontraries. But whereas the standard square maps the relations between universal and particular propositions, the project uses a square of opposition for singular propositions, which also has Aristotelian roots, but hasn't been employed in Kant studies.

The recent extension of the traditional square of opposition to a hexagon has generated a good deal of interest lately. One aim of the project is to take advantage of the resources provided in this field for mapping Kant's antinomies; so, for instance, the hexagon offers an explanation to the otherwise somewhat puzzling claim that a contradictory pair can be transformed into contraries.

During the research, the focus shifted somewhat: instead of just a narrow focus on the oppositional structures of the antinomies, a more general study was undertaken of areas in Kant's philosophy where squares and hexagons of opposition may be useful tools. In particular, modality is a field that lends itself naturally to such an investigation, given the possible relations between Kant's theory and the traditional modal square of opposition. Kant's pre-critical text "Negative Magnitudes" (1763), on the relation between real oppositions in nature and logical contradiction, also was found to be very relevant to study from the point of view of the square of opposition. A further spin-off (though not connected to Kant) is a three-dimensional diagrammatic representation of the binary connectives of propositional logic, different from those already presented in the literature.

Apart from the intrinsic interest of these additional topics, a reason for the changes in the project's orientation was that the initial focus on the oppositional structure of the antinomies turned out to be a bit narrow when it comes to giving a general interpretation of the antinomies. Though the project's procedure of separating the antinomies' oppositional structure from both the formal features of the proofs of the theses and their metaphysical content is illuminating with regard to the first pair of antinomies (the mathematical antinomies), the advantages of this method of isolation is not as obvious with respect to the second pair (the dynamical antinomies), even if there is some indication that the solution for the first pair was taken by Kant to constitute a necessary first stage for the solution of the dynamical antinomies. For the analysis of the antinomies Kant presents in his other Critiques it is less clear that the hexagon is helpful. The antinomies' intricate interconnections of form and content are perhaps not so easily disentangled by means of one single tool after all.


Results & new research questions

A first important result of the project is that for the mathematical antinomies, the initial aim of modelling their relations of opposition with the hexagon of opposition turns out to be quite illuminating. The transformation of a contradictory opposition (as envisaged from the traditional point of view) to a relation of contrariety (according to transcendental idealism) is shown to be derivable from properties of the triangles of contrariety and subcontrariety that make up the hexagon of opposition. A nice additional feature of this model (in terms of a hexagon for singular propositions) is that Kant's distinction between ordinary negation and infinite judgment (predicate negation) find a natural use.

A second interesting result is that the Kantian theory of modality can be represented as a hexagon of opposition of a special type (a so-called Sherwood-Czezowski hexagon). As Kant distinguishes three modal categories, he is often taken to break with the Aristotelian tradition of representing modality as a square of opposition. However, the three Kantian modal concepts together with their contradictory opposites constitute six concepts whose relations fit this type of hexagon. Since this hexagon is an extension of the square of opposition, a successful mapping of Kant's theory of modality to it would indicate that there is no break with the tradition. But there is much work left to do here. One problem is that Kant seems to use different concepts of contingency. Another difficulty is that the interdefinability of modal concepts appears to speak against their status as separate categories. This latter problem is a case of a general question in Kant's philosophy concerning the relation between transcendental concepts and their logical counterparts. The first problem may be possible to attack by combining the Sherwood-Czezowski hexagon with the standard hexagon. The result of this combination is an octagon, in which different places can be assigned for two senses of "contingent" (non-necessary and neither necessary nor impossible, respectively).

A third result I would point to is an alternative three-dimensional representation of the binary connectives of propositional logic. There is a simple way to find the logical relation between two connectives (contradictoriness, contrariety, subcontrariety, implication or indifference). Excepting tautology and contradiction, this has been used to arrange the remaining 14 connectives three-dimensionally by assigning them to the vertices of a polyhedron, in which hexagons of oppositions between connectives are incorporated. Suitable polyhedra described in the literature are the tetraicosahedron and the rhombic dodecahedron. The new model I propose for this purpose is the elongated hexagonal bipyramid, which has 14 vertices that can be assigned to the connectives. Though this result is of no deep significance for logical theory, the proposed model has attractive heuristic features that may be of some interest, in that three hexagons of oppositions are visible on three of the polyhedron's surface sides.

As for new research questions, some are mentioned above: a wider look at areas of Kant's philosophy where oppositional structures come to the fore, such as "Negative Magnitudes"; an investigation of Kant's modal theory from the project's particular perspective; and a study of 3-D models of propositional logic. A further area to be studied, the importance of which was already pointed out by one of the project's referees, is how the traditional square of opposition was presented in the logical works of Kant's German contemporaries.

International connections & popular outreach

The papers that the project will result in will be published in international journals. I have refereed two papers concerning the square of opposition for international publishers. In 2016 I was invited to the Centre for Advanced Study in Oslo where I presented my account of Kant's antinomies for a group of prominent Kant scholars and logicians. I have also visited an international conference on Kant and the natural sciences in Dortmund (2015).

I have done nothing particular in terms of popular outreach in connection to this project (except for presenting some basic ideas concerning the square of opposition to beginning philosophy students).

Most important publications

The project's most important publications will be "Modelling the First Antinomy with the Hexagon of Opposition" and "Kant's Theory of Modality: Hexagon or Octagon?". I have already given some indication above of why I think these topics are important. The first paper will present my account of the mathematical antinomies, focusing on the first antinomy. It will also contain some discussion of the relation between mathematical and dynamical antinomies. The paper on modality will attempt to show that Kant's theory can be modelled as a hexagon and thus taken as an extension of (rather than as a break with) the tradition. Both of these papers are in the making: the first one is almost finished, the second one is still in need of additional research.

Publication strategy

The project's results will be published as articles in international journals. There are now a number of open access journals of good quality in philosophy, and also the ordinary journals are increasingly inclined to publish papers with open access, so I see no great obstacle to getting them published in conformance with RJ:s policy.