Revisiting the Categories
When we think about giving a “logic” of something, what guides our choices? What makes some concepts logically apt while others seem to resist formalization?
Kant
In the Critique of Pure Reason Kant says the following:
In such a way there arise exactly as many pure concepts of the understanding, which apply to objects of intuition in general a priori, as there were logical functions of all possible judgments in the previous table: for the understanding is completely exhausted and its capacity entirely measured by these functions. Following Aristotle we will call these concepts categories, for our aim is basically identical with his although very distant from it in execution. [A79-80; B105]
Kant proceeds to list off the four types of category—Quantity, Quality, Relation, Modality—that correspond to the four types of judgment. Of the judgments, there are
Quantity: Universal, Particular, Singular
Quality: Affirmative, Negative, Infinite
Relation: Categorical, Hypothetical, Disjunctive
Modality: Problematic, Assertoric, Apodictic
These correspond to the categories:
Quantity: Unity, Plurality, Totality
Quality: Reality, Negation, Limitation
Relation: Substance, Causality, Community
Modality: Possibility, Existence, Necessity
Frege
A few years later, in §4 of the Begriffsschrift, Frege attempts to “explain the significance for our purposes of the distinctions that we introduce among judgments.” Of these distinctions, Frege dismisses the entire class of relational judgments/categories. “The distinction between categoric, hypothetic, and disjunctive judgments seems to me to have only grammatical significance.”
He is likewise dismissive of the class of modal judgments/categories, saying that they have “since this does not affect the conceptual content of the judgment, the form of the apodictic judgment has no significance for us” and that when a speaker frames an assertion as possible, “either the speaker is suspending judgment by suggesting that he knows no laws from which the negation of the proposition would follow or he says that the generalization of this negation is false.”
Frege essentially clears the road of obstacles to a ‘mathematical’ logic by eliminating the super-categories of Relation and Modality as not up for logical analysis. What is left can be given a purely extensional analysis.
Peirce
It was a few years before Frege published the Begriffshrift that Peirce introduced his three fundamental categories: Firstness, Secondness, and Thirdness. Peirce investigated the Kantian categories and concluded “the fundamental categories of thought really have that sort of dependence upon formal logic that Kant asserted.” He says that he “became thoroughly convinced that such a relation really did and must exist.” [CP 1.561]
Peirce went on to consider the possibility that the Kantian categories are “part of a larger system of conceptions.” [CP 1.563] In particular, he thought that they might have a kind of intersecting nature in which, for example:
the categories of relation… are so many different modes of necessity, which is a category of modality; and in like manner, the categories of quality… are so many relations of inherence, which is a category of relation. Thus, as the categories of the third group are to those of the fourth, so are those of the second to those of the third; and I fancied, at least, that the categories of quantity… were, in like manner, different intrinsic attributions of quality. [CP 1.563]
The point of which seems to be that members each of the Kantian fundamental categories relates to or is a form of each of the others. Since there are four fundamental categories each one has three others it can relate to. This reinforces Peirce’s triadic approach to the categories and the idea that the source of the fundamental categories are (the concepts of) the first three ordinals. These are the “modes of being” that Peirce uses to investigate and explain the ‘downstream’ accidental categories: quality, relation, representation.
Questions about the categories are interesting in their own right, but I’m interested in them primarily as the atoms of logical formalism. For Aristotle, the categories are “in no way composite” (Categories §1, part 3). Kant, Frege, and Peirce attempted to classify and supplement the Aristotelian categories. The explosion of intensional logics after C. I. Lewis has given us the tools to represent the logical fine structure of more of the categories than ever. We can represent all kinds of possibility and necessity and all kinds of modal contexts. Graphical models give us formal tools to dig into the structure of relations as well.
I suspect, but can’t yet prove, that the logic of graphical causal models finally puts the categories of action and being acted upon within the scope of purely formal logic. But it seems to me that giving us access to the fine structure of causal relations is what sets graphical modeling apart from purely predicative or probabilistic analyses of causation.
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