Why Modality?

Why should we care about something as seemingly esoteric as modality? Without modal concepts like necessary, possible, and should, there could be no knowledge beyond mere acquaintance with particulars, and no ethics at all. Nothing we said could have any force. We could not really even form any general concepts. Nothing would really follow from anything else. The fact that necessary or should is sometimes applied in too strong a way — or that possible is sometimes applied in too abstract a way — does not negate their essential role.

I want to say that modal concepts are properly meaningful only when bound to a context. It is only the lack of proper binding to a context that makes their application too strong or too abstract.

This relation actually goes both ways. Modal assertions not limited by context sound like dogmatism or would-be despotism; but equally, an emphasis on context with no consideration of modality at all would lead to bad relativism or particularism. Modality and context have a kind of complementarity, and need to go together. Either one without the other causes trouble, but the two together ground ethics, knowledge, and wisdom. Their combination is another good example of an Aristotelian mean. (See also Modality and Variation.)

Modality and Variation

David Corfield suggests that modality has to do with ranges of variation. This seems extremely helpful. He connects Brandom’s notion of ranges of counterfactual robustness with mathematical analyses of variation. Corfield approvingly cites Brandom’s argument that in order to successfully apply empirical concepts at all, we must already be able to apply modal concepts like possibility and necessity. This always seemed right to me, but the talk about possible worlds made me worry about what sounded like impossibly strong quantification over infinities of infinities. Corfield also points out that Saul Kripke originally cautioned against uncareful extension of his possible-worlds talk.

It now seems to me that Brandom’s counterfactual robustness and Corfield’s mathematically analyzable variation together can be taken as an explanation for modal notions of necessity that previously seemed to be simply posited, or pulled out of the blue. Modality suddenly looks like a direct consequence of the structure of ranges of variation. Previously, I associated both structure and Brandom’s modally robust counterfactuals with Aristotelian potentiality, so this fits well.

Corfield also relates this to work done by the important 20th century neo-Kantian, Ernst Cassirer, on invariants behind the various systems of Euclidean and non-Euclidean geometry. He points out that Cassirer thought similar concerns of variation and invariance implicitly arise in ordinary visual perception, and connects this with Brandom’s thesis that modality is already there in our everyday application of empirical concepts.

The British Empiricist David Hume famously criticized common-sense assumptions about causality and necessity, preferring to substitute talk about our psychological tendencies to associate things that we have experienced together. Hume pointed out that from particular facts, no knowledge of causality or necessity can ever be derived. This is true; no knowledge of necessity could arise from acquaintance with particular facts. But if necessity and other modalities are structural, as Corfield suggests, they do not need to be inferred from particular facts, or to be arbitrarily posited.

The kind of necessity associated with structural determination is quite different from unconditional predestination. I want to affirm the first, and deny the second. Structural determination only applies within well-defined contexts, so it is bounded. If we step outside of the context where it applies, it no longer has force. (See also New Approaches to Modality; Free Will and Determinism.)

Leibniz is more familiar to me than Kripke, so when I hear “possible worlds”, I have tended to imagine complete alternate universes à la Leibniz. “Worlds”, however, could be read much more modestly as just referring to Corfield’s ranges of variation.

New Approaches to Modality

I periodically peek at the groundbreaking work on formal systems that is going on in homotopy type theory (HoTT), and in doing so just stumbled on an intriguing treatment of modal HoTT that seems much more philosophically promising to me than standard 20th century modal logic.

Types can be taken as formalizing major aspects of the Aristotelian notions of substance and form. Type theory — developed by Swedish philosopher Per Martin-Löf from early 20th century work by the British philosopher Bertrand Russell and the American mathematician Alonzo Church — is the most important thing in the theory of programming languages these days. It is both a higher-order constructive logic and an abstract functional programming language, and was originally developed as a foundation for constructive mathematics. Several variants of type theory have also been used in linguistics to analyze meaning in natural language.

Homotopy type theory combines this with category theory and the categorical logic pioneered by American mathematician William Lawvere, who was also first suggested a category-theory interpretation of Hegelian logic. HoTT interprets types as paths between topological spaces, higher-order paths between paths, and so on, in a hierarchy of levels that also subsumes classical logic and set theory. It is a leading alternative “foundation” or framework for mathematics, in the less epistemologically “foundationalist” spirit of previous proposals for categorical foundations. It is also a useful tool for higher mathematics and physics that includes an ultra-expressive logic, and has a fully computational interpretation.

There is a pretty readable new book on modal HoTT by British philosopher David Corfield, which also gives a nice introductory prose account of HoTT in general and type theory in general. (I confess I prefer pages of mostly prose — of which Corfield has a lot — to forests of symbolic notation.) Corfield offers modal HoTT as a better logic for philosophy and natural language analysis than standard 20th century first-order classical logic, because its greater expressiveness allows for much richer distinctions. He mentions Brandom several times, and says he thinks type theory can formally capture many of Brandom’s concerns, as I previously suggested. Based on admittedly elementary acquaintance with standard modal logic, I’ve had a degree of worry about Brandom’s use of modal constructs, and this may also help with that.

The worry has to do with a concept of necessity that occasionally sounds overly strong to my ear, and is related to my issues with necessity in Kant. I don’t like any universal quantification on untyped variables, let alone applied to all possible worlds, which is the signature move of standard modal logic. But it seems that adding types into the picture changes everything.

Before Corfield brought it to my attention, I was only dimly aware of the existence of modal type theory (nicely summarized in nLab). This apparently associates modality with the monads (little related to Leibnizian ones) that I use to encapsulate so-called effects in functional programming for my day job. Apparently William Lawvere already wrote about geometric modalities, in which the modal operator means something like “it is locally the case that”. This turns modality into a way of formalizing talk about context, which seems far more interesting than super-strong generalization. (See also Modality and Variation; Deontic Modality; Redding on Morals and Modality).

It also turns out Corfield is a principal contributor to the nLab page I previously reported finding, on Hegel’s logic as a modal type theory.

Independent of his discussion of modality, Corfield nicely builds on American programming language theorist Robert Harper’s notion of “computational trinitarianism”, which stresses a three-way isomorphism between constructive logic, programming languages, and mathematical category theory. The thesis is that any sound statement in any one of these fields should have a reasonable interpretation in both of the other two.

In working life, my own practical approach to software engineering puts a high value on a kind of reasoning inspired by a view of fancy type theory and category theory as extensions or enrichments of simple Aristotelian logic, which on its formal side was grounded in the composition of pairs of informally generated judgments of material consequence or material incompatibility. I find the history of these matters fascinating, and view category theory and type theory as a kind of vindication of Aristotle’s emphasis on composition (or what could be viewed as chained function application, or transitivity of implicit implication, since canonical Aristotelian propositions actually codify material inferences) as the single most important kind of formal operation in reasoning.

Redding on Morals and Modality

A recent web draft by Australian philosopher Paul Redding — author of a nice introductory book on analytic readings of Hegel — makes quite a few interesting points about Leibniz, Kant, Hegel, J.N. Findley, and modal logic. Findley was an important 20th century philosopher with analytic training who developed a very this-worldly but still metaphysical reading of Hegel, with strong influence from Wittgenstein. Findley’s student Arthur Prior apparently developed an “actualist” alternative to the more common possible worlds approach to modal logic, which latter is usually said to have an antecdent in Leibniz. Redding argues that there is a similarity between Prior’s criticism of modal possible worlds and Hegel’s criticism of Kantian formalism in ethics.

I take the assertions of Leibniz in a more tentative way than Redding seems to, and sharply distinguish between Leibniz and his Wolffian semi-followers. Leibniz’s thought on possible worlds, though, is one of the parts of his work I agree is less attractive, even though I am sympathetic to its motivation as an alternative to theological voluntarism. It seems to me like a beautiful but very extravagant speculation, related to his thoughts on infinity. Leibniz’s youthful co-discovery of the calculus was but one aspect of a lifelong fascination with the new idea of a mathematical infinity. Explicit reliance on the assumption of this kind of “actual infinity” is removed from later presentations of mathematical analysis, which instead carefully talk about differentials and integrals in terms of limits. For what it’s worth, Aristotle argued against any actual infinity, and Hegel called it “bad infinity”.

Redding attributes to Findley criticism of an ethics of rules in favor of an ethics of values. I like this very much in general, but I make a big distinction between rules that would supposedly just tell us what to do (which I find hideous) and higher-order rules like Kant’s categorical imperative, which merely requires that we aim at universality, without presuming to tell us exactly what we should do. While taking Hegel’s criticism of Kantian formalism a bit more literally than I would, Redding nonetheless concludes that Hegel’s position is an extension of Kant’s.

Redding notes Hegel’s complaint against Kant’s advocacy at one point of “duty for duty’s sake”. I find this formula as unappealing as the categorical imperative is salutary. But it turns out that Kantian “duty” is really a stand-in for the kind of absence of material inconsistency that characterizes a unity of apperception. Redding cites Hegel in the Philosophy of Right as criticizing Kantian duty as mere “absence of contradiction”. He correctly points out that what is at issue is hardly the law of non-contradiction in the usual sense, so Kant’s argument is not really like the Wolffians’ attempt to derive a whole metaphysics from that logical law. But Redding then attributes to Hegel an emphasis on “actualized Sittlichkeit” as opposed to empty formalism. Hegel may have said the words, but I think this is way too simple. It sounds like some actually existing set of norms just taken at face value. I’d take empty formalism over that any day. (See discussion on Pippin’s concern about positivity in Mutual Recognition.) Unfortunately, Redding also moves from unity of apperception to a Fichtean self-identity of a Subject (“I = I”), from which I want to sharply separate Kant and Hegel.

The idea of building logical modality into the actual world rather talking about quantification over possible worlds seems appealing to me, but I would not want to go so far as to deny potentiality, as Kant seemed to in his more Newtonian moments, to which Redding alludes. I think Hegel went a long way toward recovering something like potentiality.

Modality

Modality is a way of formally, logically talking about what I would call the higher-order aspects of the ways of being of things. It is most commonly associated with necessity and possibility, but I think these are actually atypical examples that may give a misleading impression of what modality in general is, because necessity and possibility both have a kind of extreme all-or-nothing character that does not hold for modality in general.

I don’t believe quantification across all possible worlds could be interpretable by any process of interpretation, so I don’t consider it even intelligible in an acceptably strong sense, and I also don’t believe in unconditional necessity. Anything real or any truth about it, as well as anything I would accept as a legitimate formal construction, has conditions, even if they are only implicit. So, standard modal logic concerned with operators for these unconditional things — technical interest aside — does not seem very useful to me, because the resulting propositions would be too strong.

I am a bit surprised that Brandom is so charitable toward formal possible-worlds semantics, given his reservations about formalism in general. Technically innovative as it was, this approach seems like an extravagant extreme of infinitary classical representationalism.

Modality itself should be safe from these concerns. At a handwaving level, I imagine an indefinite number of modalities related to particular specifiable conditions, and expressing structural “degrees” or “flavors” of more specific necessity or possibility based on those conditions.

Wilfrid Sellars suggested that modalities should be understood as specific forms of normative bindingness. This seems very helpful as an alternative to extensionalist possible-worlds formalisms. (See also Why Modality?; Modality and Variation; New Approaches to Modality; Deontic Modality; Redding on Morals and Modality.)