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 the categorical logic and higher topos theory 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 logical 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 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) as the single most important kind of formal operation in reasoning, as distinct from the material inference involved in Aristotelian proposition formation.

Aristotelian Causes

I’ve explained each of the four classic Aristotelian “causes” as playing what Brandom would call an expressive role, helping to explain other meaning, and pointed out how different this is from standard modern notions of what I’ve been calling univocal causality. An Aristotelian cause (aitia) is much more like a nonexclusive reason than it is like anything expressed by mechanical metaphors.

There is another very important modern way of thinking about these matters, inspired by Hume’s critique of realism about causes in the modern sense. Hume pointed out that modern-style talk about cause and effect involves a kind of inferential extrapolation from observed regular patterns of succession. Implicitly influenced by this, much work in the sciences relies directly on statistical correlations observed in data from controlled experiments. What particular causes are said to be at work then becomes a matter of optional statistical inference, subject to possible debate.

Then, too, from the side of subject matter, in fields concerned with complex dynamical systems that can only be modeled in a very tentative way — like ecology, economics, and medicine — it has come to be widely recognized that many causes combine to produce the results we see.

Both the statistical approach and what I’m gesturing at as a “complex systems” approach to causality avoid reliance on mechanical metaphors. Neither of these perspectives rules out underdetermination or overdetermination, or the simultaneous presence of both.

Aristotelian “causality” is simultaneously underdetermining and overdetermining. That is to say, in advance it leaves room for varying outcomes, but in hindsight it provides multiple rationales for a given outcome. Its purpose is to provide not certain prediction, but intelligibility and reasonableness.

In principle, nothing would stop us from combining this with statistical or complex-systems views, but these are still very different approaches. The statistical approach is quantitative and relies on counting minimally interpreted facts, where the Aristotelian approach is qualitative and puts the whole emphasis on rational interpretation. The complex-systems view relativizes causes in the modern event-based sense, without making them like any of the Aristotelian ones, none of which corresponds to an event. It is also not interpretive in the sense developed here.

One might consider mathematical-physical law as a kind of formal cause. Statistics and things like dynamic models could be taken as modern, quantitatively oriented descriptions of what I have called material tendencies. (See also Form; Aristotelian Matter; Efficient Cause; Ends; Natural Ends; Aristotelian Identity; Aristotelian Demonstration.)

Natural Ends

Early modern science sought to banish consideration of ends from the empirical world, in favor of purely mathematical and factual description. Kant recovered a heuristic use of teleology, especially in biology (see Kant’s Recovery of Ends), and numerous more recent biological researchers have followed suit.

It is relatively easy to see that any kind of desire (say on the part of an animal) is a desire for something that is usually more general than a concrete object that satisfies the desire. More broadly, living things can also plausibly be said to have indwelling tendencies of nutrition and reproduction.

The case of inorganic nature is a bit more challenging for us to understand this way, but where modern science sees abstract mathematical-physical laws in operation, the effect of which may be modified by various circumstances, Aristotle saw concrete material tendencies for things to develop in certain ways, subject to similar modifications. At a certain level of abstraction, observable material tendencies can be viewed as “moving” things in a way broadly analogous to desire. The heavy object “wants” to fall. This just refers back to the observed fact that heavy objects have a tendency to fall, when not impeded by something else. At a level of common-sense interpretation of experience, this does not lead to any false conclusions.

There is no reason why the mathematical-law description and the material-tendency description cannot coexist. The predictive power of mathematically formulated laws makes them invaluable for engineering applications. But for ordinary life, what we are usually interested in are qualitative distinctions that have practical significance.

Thus, in addition to rational ends, there seem to be three kinds or degrees of natural ends or endlike things: ends of desire; primitive vital ends; and endlike tendencies associated with Aristotelian matter of various kinds and descriptions. (See also Ends; Aristotelian Matter.)

Aristotelian Equality

Aristotle explains justice as a kind of proportionality, or equality of relations between people and similar objects of concern. The pseudo-Aristotelian treatise Magna Moralia virtually identifies justice with equality between people, but then disappointingly goes on to say that since, e.g., there is no equality between father and son or master and servant, the concept of a justice between them does not apply. Aristotle himself was careful to point out that empirically existing distinctions between people in the positions of masters and servants do not necessarily reflect inherent ones between people, and this ought to be generalized. Surviving texts do not explicitly put the same caveat on, e.g., existing inequalities between the sexes, but it seems to me the same logic should apply.

It also seems to me that equality of relations between people and similar objects of concern actually implies effective equality between people. A generalized equality between people would have been a highly controversial assertion in Aristotle’s time, and it seems to me he should be commended for implying it, rather than criticized for failing to make it explicit. It is in this spirit that I consider the Kantian emphasis on ethical univerality a welcome addition, complementing rather than conflicting with Aristotle’s highly cultivated sensitivity to the nuances of particular situations.

Theory and Practice

Unlike Plato, Aristotle did not make theoretical knowledge an ethical criterion, although he played a great role in the development of many fields of inquiry. Nonetheless, he placed intellectual “virtue” alongside friendship or love as an essential component of the highest ethical development.

I previously suggested there is an indirect way in which any inquiry can help make us better deliberators. Knowledge can of course help, but only if it is relevant to the question at hand. But the ethical value of inquiry lies more in a kind of theoretical practice than in the particular knowledge that may result. Intellectual virtue, I want to say, has to do especially with practices of free and well-rounded interpretation.

Aristotelian Virtue

Aristotelian ethics is not just about cultivating virtue as a kind of good character. Although he does emphasize it a lot, ultimately character is just a means and a potentiality for actualization of the real goal of that part of living well that is under our power. The actualization itself comes from good practice (praxis), grounded in sound, well-rounded deliberation and choice. Aristotle also says the very best life is that of the philosopher. Not only is philosophy valuable in itself, but it also helps us deliberate.

The best deliberation and choice is supported by intellectual virtues, a concern for justice, and a spirit of friendship or love, but it would not even get off the ground without progress in the classic “moral” virtues, which are all said to pertain to our emotions or passions, and to how we are affected by pleasure and pain. Aristotle characterizes each of these emotional virtues as a kind of mean, or balanced emotional state.

Thus, courage is presented as a mean between rashness and cowardice. Temperance is a mean between self-indulgence and insensibility. Interestingly, justice and prudence are also included among emotional virtues, and subjected to a similar analysis. Other emotional virtues are presented as following the same pattern. Unlike the Stoics, neither Plato nor Aristotle advocated suppressing the emotions.

Auspicious Grafts

Ideas of different philosophers — or interpretations of them — cannot just be arbitrarily combined, but they may turn out to be compatible if there is some common basis, or if they have a different scope of application. My semantic Aristotle has a substantial common basis with Brandom’s Kant and Hegel, which is not too surprising, since my reading of Aristotle has among other things benefited from thinking about Brandom’s work, and at a deeper historical level, independent of Brandom, there is a common basis in guiding notions of reason and rational ethics.

I’ve recently commented on connections between Aristotelian and Hegelian dialectic, and previously on Aristotle’s partial anticipation (in his discussion of friendship) of Hegel’s key notion of mutual recognition. It also seems to me that in a very broad way, something like the idea of Kantian synthesis was implicit in Aristotle, so a fuller account can be treated as a welcome addition. Related to this, Kant’s notion of unity of apperception — particularly with Brandom’s way of reading it as an ethical goal — sounds very good from an Aristotelian point of view concerned with ends. (See also Aristotle and Kant; Hegelian Genealogy; Retrospective Interpretation.)