Thursday, June 24, 2021

Evening Note for Thursday, June 24

 Thought for the Evening: Reduction to Box

If we set aside some marginal forms, the most obvious feature of modal logic is the division of modalities into a strong modality (Box) and a weak modality (Diamond). The exact interpretation of each can vary considerably, since we can take the pair to be necessity-possibility, always-sometimes, everywhere-somewhere, everyone-someone, required-allowed, obligatory-permissible, and many more.

One of the things that is noticeable about our explanatory practice is our tendency to try to trace everything back to (some kind of) Box. We try to explain the variable in terms of the invariable, the possible in terms of the necessary, the permissible in terms of the obligatory. If someone asks why John is doing something, "Everyone is doing it" would be a perfectly reasonable answer for a wide variety of cases; if someone asks why Mary is doing something right now, "She always does it" will often work as a response. Or we may trace it back to some Box that is less obviously connected to our starting case; for instance, if someone asks why John is taking a given route, I might say, "Because it is necessary for him to get back home." This seems to be quite general; the weakly modal case is traced back to the strongly modal rule.

The implications will differ according to the modality and how it works. I've previously suggested that anything that can be put into a square of opposition can be treated as a modal operator. But there are different kinds of square of opposition, and how you reduce to Box differs according to which one describes your modal operator.

One kind, the semidegenerate square, collapses Diamond and Box into each other; it's not actually a 'square' at all, just a line that divides things into two categories Box/Diamond and Not-Box/Not-Diamond. If you have a weakly modal case, like this possible thing, you can immediately recognize it as necessary. Systems like this in which you can go directly from Diamond to Box are very boring. If "Sometimes A" implies "Always A" or "X is Permissible" implies "X is Obligatory", reduction to Box is trivial; you have a flat, unstructured system. Total determinism gives you such a flat, unstructured system; everything possible is necessary, and all your explanation ends where it started. Nonetheless, I suspect that this is why determinism  is a kind of recurring dream; if it's all necessary already, there's not much left to explain. It cuts the Gordian knot, gets you where you want to be, even if it is by the lazy shortcut of moving the finish line to the starting line. Nonetheless, you do get real modalities like this, particularly in mathematics and logic: sometimes in those fields you can prove that something is necessary just by proving that it is possible. If it's genuinely possible that there is an odd number higher than four, then it is necessary for there to be an odd number higher than four.

In a Boolean or modern square of opposition, Diamond and Box are only minimally related: Box is contradictory opposite to Diamond-Not and Diamond is contradictory opposite to Box-Not, and that's it. This makes reduction to Box extremely difficult; it's like trying to explain an actual event with a bare hypothetical -- even if the bare hypothetical is relevant, it doesn't have a lot of explanatory heft. Perhaps because of this, we don't actually use Boolean-square modalities all that much.

The other major square of opposition, of course, is the Aristotelian, classical, or traditional square, which has not only contradiction but also contrariety, subcontrariety, and subalternation. Subalternation seems particularly important: Box implies Diamond. This gives us a full set of resources (beginning from one corner of the square we can get to any other corner in several ways), so Aristotelian-square modalities seem to be the ideal kind of modality for reduction to Box. And indeed most of the modalities we use in an explanatory way are quite naturally seen as subalternating: if something happens always, it happens sometimes; if something happens everywhere, it happens somewhere; if something is necessary, it is possible; if something is obligatory, it is permissible; and so forth.

If we start with Diamond, how do we reduce to Box. Well, one way is that you can divide up all the Diamond-states precisely and then sum over them. For instance, if you want to move from 'sometimes' to 'always', one way you can do it, is by taking all the particular 'sometimeses', that is, the individual times, and identify what is true of all of them. If something is true at every time, it is always true. You could also, however, get to Box without summing if you could rule out the right kinds of things; in practice, I think, we tend to do this by appealing to causes.

This is all immensely simplified. In fact, we run into any number of complications. One of the obvious ones is that strong and weak modalities are not equally strong and weak, so you can reduce Box to stronger Box; and by the same token, there are always many different Boxes to which you could reduce Diamond. For instance, if our Diamond is interpreted in terms of time, This happens sometimes, we can explain this by a temporal Box, This happens always; but it also seems that we could explain it by an alethic Box, This happens necessarily. Those are obviously two very different explanations, and they would not be equally useful in every situation. The relations are also not always straightforward. There are situations in which it is perfectly legitimate to reduce "This happens sometimes" to "It is obligatory that this happens"; but how "It is obligatory that this happens" relates to "This happens always" is less than lucid. And, of course, we are rarely in a position to reduce directly; we often have to do it in qualified ways (e.g., we might not be able to reduce directly to "This happens always" but only to "This happens always unless that does")  or with scaffolding that gives us additional information we need to maneuver (like causal reasoning or analogy or physical theory), all of which introduce their own complications.

Various Links of Interest

* Asya Passinsky, Norm and Object: A Hylomorphic Theory of Social Objects (PDF)

* Zion Lights, The Sad Truth about Traditional Environmentalism

* Kenneth L. Pearce, Astell and Masham on Epistemic Authority and Women's Individual Judgment in Religion (PDF)

* Shane Gassaway, Just Silence in Plato's Clitophon (PDF)

* Edmund Waldstein, Common Good Eudemonism

* Tanner Greer, How I Taught the Iliad to Chinese Teenagers

* Philip Pilkington, Monetary Faith

* Ljiljana Radenovic, From Deficient Liberalism Toward a Deeper Sense of Freedom

* Rafael de Arizaga, Matrimony Doesn't Exist

* Hrishiskesh Joshi, Dare to Speak Your Mind and Together We Flourish

* Nur Banu Simsek, Arise to Wisdom, on Ibn Tufayl

* The Hannah Arendt Papers at the Library of Congress

* Chiara Marletto, Our Little Life is Rounded with Possibility, on the importance of counterfactuals to understanding the world

* An interesting look at a particular case of agro-mining, in which plants are grown in metal-heavy soil and then the metal harvested from the plants.

* Christopher Frey, Plato and Aristotle between Autonomy and Oppression

* Zachary Micah Gartenberg, On the causal role of privation in Thomas Aquinas's metaphysics (PDF)

* Avery Hurt, What's Really Happening when You Experience Deja Vu?, discusses the current speculations (for speculations are all we actually have at present) about the phenomenon. It's a topic I'm interested because I have a long history of very intense deja-vu experiences.

* Umberto Eco's private library is being moved to Bologna University, where it will be preserved as an archive, i.e., it will continue to be organized along the same plan Eco himself used, as a distinct part of the library. They will also eventually digitize all his notes and comments on the books.

* Thornton Lockwood, In Praise of Solon: Aristotle on Classical Greek Democracy (PDF)

Currently Reading

The Vinland Sagas
Peter Martyr Vermigli, Commentary on Aristotle's Nicomachean Ethics
Richard Courant & Herbert Robbins, What is Mathematics?