Logic for People

Leading programming language theorist Robert Harper refers to so-called constructive or intuitionistic logic as “logic as if people mattered”. There is a fascinating convergence of ideas here. In the early 20th century, Dutch mathematician L. E. J. Brouwer developed a philosophy of mathematics called intuitionism. He emphasized that mathematics is a human activity, and held that every proof step should involve actual evidence discernible to a human. By contrast, mathematical Platonists hold that mathematical objects exist independent of any thought; formalists hold that mathematics is a meaningless game based on following rules; and logicists argue that mathematics is reducible to logic.

For Brouwer, a mathematical theorem is true if and only if we have a proof of it that we can exhibit, and each step of that proof can also be exhibited. In the later 19th century, many new results about infinity — and infinities of infinities — had been proved by what came to be called “classical” means, using proof by contradiction and the law of excluded middle. But from the time of Euclid, mathematicians have always regarded reproducible constructions as a better kind of proof. The law of excluded middle is a provable theorem in any finite context. When the law of excluded middle applies, you can conclude that if something is not false it must be true, and vice versa. But it is not possible to construct any infinite object.

The only infinity we actually experience is what Aristotle called “potential” infinity. We can, say, count a star and another and another, and continue as long as you like, but no actually infinite number or magnitude or thing is ever available for inspection. Aristotle famously defended the law of excluded middle, but in practice only applied it to finite cases.

In mathematics there are conjectures that are not known to be true or false. Brouwer would say, they are neither true nor false, until they are proved or disproved in a humanly verifiable way.

The fascinating convergence is that Brouwer’s humanly verifiable proofs turn out also to exactly characterize the part of mathematics that is computable, in the sense in which computer scientists use that term. Notwithstanding lingering 20th century prejudices, intuitionistic math actually turns out to be a perfect fit for computer science. I use this in my day job.

I am especially intrigued by what is called intuitionistic type theory, developed by Swedish mathematician-philosopher Per Martin-Löf. This is offered simultaneously as a foundation for mathematics, a higher-order intuitionistic logic, and a programming language. One might say it is concerned with explaining ultimate bases for abstraction and generalization, without any presuppositions. One of its distinctive features is that it uses no axioms, only inference rules. Truth is something emergent, rather than something presupposed. Type theory has deep connections with category theory, another truly marvelous area of abstract mathematics, concerned with how different kinds of things map to one another.

What especially fascinates me about this work are its implications for what logic actually is. On the one hand, it puts math before mathematical logic– rather than after it, as in the classic early 20th century program of Russell and Whitehead — and on the other, it provides opportunities to reconnect with logic in the different and broader, less formal senses of Aristotle and Kant, as still having something to say to us today.

Homotopy type theory (HoTT) is a leading-edge development that combines intuitionistic type theory with homotopy theory, which explores higher-order paths through topological spaces. Here my ignorance is vast, but it seems tantalizingly close to a grand unification of constructive principles with Cantor’s infinities of infinities. My interest is especially in what it says about the notion of identity, basically vindicating Leibniz’ thesis that what is identical is equivalent to what is practically indistinguishable. This is reflected in mathematician Vladimir Voevodsky’s emblematic axiom of univalence, “equivalence is equivalent to equality”, which legitimizes much actual mathematical practice.

So anyway, Robert Harper is working on a variant of this that actually works computationally, and uses some kind of more specific mapping through n-dimensional cubes to make univalence into a provable theorem. At the cost of some mathematical elegance, this avoids the need for the univalence axiom, saving Martin-Löf’s goal to avoid depending on any axioms. But again — finally getting to the point of this post — in a 2018 lecture, Harper says his current interest is in a type theory that is in the first instance computational rather than formal, and semantic rather than syntactic. Most people treat intuitionistic type theory as a theory that is both formal and syntactic. Harper recommends that we avoid strictly equating constructible types with formal propositions, arguing that types are more primitive than propositions, and semantics is more primitive than syntax.

Harper disavows any deep philosophy, but I find this idea of starting from a type theory and then treating it as first of all informal and semantic rather than formal and syntactic to be highly provocative. In real life, we experience types as accessibly evidenced semantic distinctions before they become posited syntactic ones. Types are first of all implicit specifications of real behavior, in terms of distinctions and entailments between things that are more primitive than identities of things.

Distinction

So, I want to say that distinction is something good, not a defect we ought to remedy. It is a fundamental symptom of life. Stoics, Buddhists and others remind us that it is best not to be too attached to particular forms. This is a wise counsel, but not the whole truth. I am tempted to say there is no compassion without some passion. Caring about anything inevitably involves distinction. It is better to care than not to care.

Everything flows, Heraclitus said. But in order to make distinctions, it has to be possible to compare things. Things must have a character, even if they do not quite ever stay still within their frames. Having a character is being this way and not that. Real being is always being some way or other. Its diversity is something to celebrate.

It is not immoral to prefer one thing to another. We can’t be who we are without definite commitments. Perfect apathy would lead to many sins of omission. It is better to have lived fully. We are not apart from the world, but inhabit the oceans of difference, and sometimes must take a side.

Nondualism?

As far as I know, the explicit term “nondualism” was first used in certain strands of Mahayana Buddhism. I believe it later was adopted by the Vedanta school of Hindu scholastic philosophy. I was fascinated with these as a young man, and was for a time much absorbed in developing a sort of Alan Watts style interpretation of Plotinus’ emphasis on the One as a similar kind of radical nondualism.

Radical nondualism goes beyond the rejection of sharply dualist views like those of Descartes on mind and world, and the different religious dualisms like those of Augustine, the Zoroastrians, the Gnostics, the Manichaeans, or the Samkhya school of Hinduism. Each of these latter has important differences from the others, but what unites them is the strong assertion of some fundamental duality at the heart of things. Radical nondualism aims to consistently reject not only these but any vestige of duality in the basic account of things.

The point of view I would take now is that many useful or arguably necessary distinctions are often formulated in naive, overly blunt ways. We should strive to overcome our naivete and our excessive bluntness, but that does not in any way mean we should try to overcome distinction per se. There can be no meaning — even of the most spiritual sort — without some sort of distinction between things. “All is One” is at best only a half-truth, even if it is a profoundly spiritual one.

Pure Difference?

A common theme here is the conceptual priority of difference over identity. I think that identity is a derived concept, and not a primitive one (see also Aristotelian Identity).

The French philosopher Gilles Deleuze (1925-1995) in Difference and Repetition and other works argued that a pure notion of difference is by itself sufficient for a general account of things. In information theory, information is explained as expressing difference. In Saussurean structural linguistics, we are said to recognize spoken words by recognizing elementary differences between sounds. In both cases, the idea is that we get to meaning by distinguishing and relating.

Deleuze initially cites both of these notions of difference, but goes on to develop arguments grounded largely in Nietzsche and Kierkegaard, whom he uses to argue against Plato and Hegel. His very interesting early work Nietzsche and Philosophy was marred by a rather extreme polemic against Hegel, and in Difference and Repetition he announces a program of “anti-Platonism” that reproduces Nietzsche’s intemperate hostility to Plato. Nietzsche blamed Plato for what I regard as later developments. Neither Plato nor Aristotle made the kind of overly strong assertions about identity that became common later on.

In The Sophist and elsewhere, Plato had his characters speak of Same, Other, and the mixing of the two as equally primordial. Hegel took great pains to elaborate the notion of a “difference that makes a difference”. But Deleuze wants to argue that Plato and Hegel both illegitimately subordinate difference to identity. His alternative is to argue that what is truly fundamental is a primitive notion of difference that does not necessarily “make a difference”, and that come before any “making a difference”. (I prefer the thesis of Leibniz that indiscernibility of any difference is just what identity consists in.)

This is related to Deleuze’s very questionable use of Duns Scotus’ notion of the univocity of being, both in general and more particularly in his interpretation of Spinoza. For Deleuze, pure difference interprets Scotist univocal being.

I frankly have no idea what led to Deleuze’s valorization of Scotus. Deleuze is quite extreme in his opposition to any kind of representationalism, while Scotus made representability the defining criterion of his newly invented univocal being. It is hard to imagine views that are further apart. I can only speculate that Deleuze too hastily picked out Scotus because he wanted to implicitly oppose Thomist orthodoxy, and Scotus is a leading medieval figure outside the Thomist tradition.

For Deleuze, univocal being is pure difference without any identity. Difference that doesn’t make a difference seems to take over the functional role that identity has in theories that treat it as something underlying that exceeds any discernibility based on criteria. I don’t see why we need either of these.

I think Deleuze’s bête noir Hegel actually did a better job of articulating the priority of difference over identity. Hegel did this not by appealing to a putative monism of difference and nothing else, but by developing correlative notions of “difference that makes a difference”, and a kind of logical consequence or entailment that we attribute to real things as we interpret them, independent of and prior to any elaboration of logic in a formal sense.

In Hegel’s analysis as explicated by Brandom, any difference that makes a difference expresses a kind of “material” incompatibility of meaning that rules out some possible assertions. This is just what “making a difference” means. Meanwhile, all positive assertions can be more specifically analyzed as assertions of some consequence or entailment or other at the level of meaning (see Material Consequence). Every predication is analyzable as an assertion of consequence or entailment between subject and predicate, as Leibniz might remind us. It is always valid to interpret, e.g., “a cat is a mammal” as an inference rule for generating conclusions like if Garfield is a cat, then Garfield is a mammal.

What is missing from Deleuze’s account is anything like entailment, the idea of something following from something else. This notion of “following”, I am convinced, is prior to any notion of identity applicable to real things. Without presupposing any pre-existing identities of things, we can build up an account of the world based on the combination of differences that make a difference, on the one hand, and real-world entailments, on the other. Identity is then a result rather than an assumption. Meanings (and anything like identity) emerge from the interplay of practical real-world entailments and distinctions. It is their interplay that gives them definition in terms of one another.

Deleuze was a sort of ontological anarchist, who wanted being to be free of any pre-existing principles. While I agree that we can’t legitimately just assume such principles, I think this is very far from meaning that principles are irrelevant, or actually harmful. On the contrary, as Kant might remind us, principles are all-important. They aren’t just “given”. We have to do actual work to develop them. But if we have no principles — if nothing truly follows from anything else, or is ruled out by anything else — then we cannot meaningfully say anything at all.

Simple Substance?

I tremendously admire Leibniz, but have always been very puzzled by his notion of “substance”. Clearly it is different from that of Aristotle, which I still ought to develop more carefully, based on the hints in my various comments on Aristotle’s very distinctive approaches to “dialectic” and “being”. (See also Form, Substance.)

Leibniz compounds a criterion of simplicity — much emphasized in the neoplatonic and scholastic traditions — with his own very original notion of the complete concept of a thing, which is supposed to notionally encompass every possible detail of its description. He also emphasizes that every substance is “active”. Leibniz’ famous monads are identified by him with substances.

A substance is supposed to be simple. He explicitly says this means it has no parts. In part, he seems to have posited substances as a sort of spiritual atoms, with the idea that it is these that fundamentally make up the universe. The true atoms, Leibniz says, are fundamentally spiritual rather than material, though he also had great interest in science, and wanted to vindicate both mathematical and Aristotelian physics. Leibniz’ notion of spiritual atoms seems to combine traditional attributes of the scholastic “intellectual soul” (which, unlike anything in Aristotle, was explicitly said by its advocates to be a simple substance) with something like Berkeley’s thesis that what can truly be said to exist are just minds.

On the other hand, a substance is supposed to be the real correlate of a “complete” concept. The complete concept of a thing for Leibniz comprises absolutely everything that is, was, or will be true of the thing. This is related to his idea that predicates truly asserted of a grammatical subject must be somehow “contained” within the subject. Leibniz also famously claimed that all apparent interaction between substances is only an appearance. The details of apparent interaction are to be explained by the details contained within the complete concept of each thing. This is also related to his notions of pre-established harmony and possible worlds, according to which God implicitly coordinates all the details of all the complete concepts of things in a world, and makes judgments of what is good at the level of the infinite detail of entire worlds. One of Kant’s early writings was a defense of real interaction against Leibniz.

Finally, every monad is said by Leibniz to contain both a complete microcosm of the world as expressed from its distinctive point of view, and an infinite series of monads-within-monads within it. Every monad has or is a different point of view from every other, but they all reflect each other.

At least in most of his writings, Leibniz accordingly wanted to reduce all notions of relation to explanations in terms of substances. In late correspondence with the Jesuit theologian Bartholomew Des Bosses, he sketched an alternate view that accepted the reality of relations. But generally, Leibniz made the logically valid argument that it is far simpler to explain the universe in terms of each substance’s unique relation to God, rather than in terms of infinities of infinities of relations between relations. For Leibniz all those infinities of infinities are still present, but only in the mind of God, and in reflection in the interior of each monad.

Leibniz’ logically simpler account of relations seems like an extravagant theological fancy, but however we may regard that, and however much we may ultimately sympathize with Kant over Leibniz on the reality of interaction and relations, Leibniz had very advanced intuitions of logical-mathematical structure, and he is fundamentally right that from a formal point of view, extensional properties of things can all be interpreted in an “intensional” way. Intension in logic refers to internal content of a concept, and to necessary and sufficient conditions that constitute its formal definition. This is independent of whatever views we may have about minds. (See also Form as a Unique Thing.)

So, there is much of interest here, but I don’t see how these ultra-rich notional descriptions can be true of what are also supposed to be logical atoms with no parts. In general, I don’t see how having a rich description could be compatible with being logically atomic. I think the notion of logical atomicity is only arrived at through abstraction, and doesn’t apply to real things.

Activity, Embodiment, Essence

I think any finite activity requires some sort of embodiment, and consequently that anything like the practically engaged spirits Berkeley talks about must also have some embodiment. On the other hand, the various strands of activity from which our eventual essence is precipitated over time — commitments, thoughts, feelings — are not strictly tied to single individuals, but are capable of being shared or spread between individuals.

Most notably, this often happens with parents and their children, but it also applies whenever someone significantly influences the commitments, thoughts, and feelings of someone else. I feel very strongly that I partially embody the essence and characters of both my late parents — who they were as human beings — and I see the same in my two sisters. Aristotle suggests that this concrete transference of embodied essence from parents to children is a kind of immortality that goes beyond the eternal virtual persistence of our essence itself.

Our commitments, thoughts, and feelings are not mere accidents, but rather comprise the activity that constitutes our essence. I put commitments first, because they are the least ephemeral. In mentioning commitments I mean above all the real, effective, enduring commitments embodied in what we do and how we act.

Ideas Are Not Inert

In the British empiricist tradition, “ideas” are supposed to be inert contents of an active “mind”, and to be either identical with sensible contents or derived from sensory experience. They are supposed to have content that just “is what it is”, but is nonetheless sufficient to serve as a basis for our conclusions and motivations.

I want to argue instead that the only possible basis for our conclusions and motivations is other conclusions and motivations. As individuals we always start in the middle, with some already existing conclusions and motivations that were not necessarily individually ours to begin with. Language and culture and upbringing provide us with a stock of pre-existing conclusions and particularly shaped motivations.

Further, I don’t see ideas as inert. The notion that ideas are completely inert comes from an extreme polarization between active mind and passive idea that results from entirely subordinating this relation to the grammatical model of subject and predicate. Aristotle’s rather minimalist account of these matters effectively treats objects and ideas as having some activity of their own. For Aristotle, “we” do not hold a monopoly on activity. There is also activity in the world that is independent of us, and much of our activity is our particular reflection of the world’s activity. Indeed for Aristotle I take it to be thought rather than an assumed “thinker” that is primarily active.

Hegel has often been criticized for speaking as if “the Idea” had life of its own, independent of us humans. If one holds an empiricist view of ideas, this can only sound like nonsense, or some kind of animism. But with an Aristotelian view of thoughts as a kind of intrinsically active “contents”, that is not the case. If thoughts are intrinsically active, we need not posit a separate mental “subject” distinct from any actual thought or perception or content as the source of all activity, behind thought.

Plato compared the human soul to a city — a kind of unity to be sure, but a weak one consisting of a federated community and relatively specific “culture” of thoughts and perceptions, subject to varying degrees of coherence. Only under the influence of later theology did it come to be assumed that the soul must necessarily have the far stronger unity of a simple substance. A looser unity of the soul is very compatible with a view of thoughts and perceptions as multiple fibers of activity, from which the overall activity we attribute to the soul or mind is constituted.

Berkeley on Perception

George Berkeley (1685-1753) is most famous for his provocative claim that material objects don’t really exist. Positively, he claimed that “to be is to be perceived”. Berkeley took as a starting point the view of Descartes and Locke that perceptions are “ideas” in the mind, but took issue with the further assumption of Descartes and Locke that ideas nonetheless also “represent” things that exist independent of the mind. It seems to me that the implicit concept of mind in this kind of usage assumes way too much, but for now I won’t dwell on that.

Berkeley has been the subject of superficial ridicule as a poster child for extreme subjectivism, but that is a caricature. Famously, he is supposed to have maintained, e.g., that a tree falling in the woods and heard by no one makes no sound. As 20th century analytic philosophers have noted, however, even if his positions are ultimately untenable, the quality of his arguments is actually quite high. Apart from the abstract “metaphysical” question of the actual existence of external objects, he also generally wanted to vindicate common sense.

Far from denying the existence of any objective reality, what he really wanted to do was articulate an alternate account of objectivity, based on something other than the independent existence of discrete objects. He had two different kinds of responses on the falling tree. One invokes counterfactual conditions; all that is of practical relevance to us are the conditions under which a perception would occur. The other invokes God as a universal witness.

From within the tradition of British empiricism, Berkeley partially anticipates the non-representationalist accounts of objectivity developed by Kant and Hegel, using the resources of a kind of Christian Platonism. Unlike Kant and Hegel, he flatly asserts that what really exists are what he calls spirits, which combine Christian-Platonic attributes with those of minds in a broadly Cartesian-Lockean sense.

A bit like the monads of Leibniz but without the infinite nesting and mutual inclusion Leibniz posited, Berkeley’s spirits are inherently active, and inherently endowed with perception. Spirits have experience that is expressed in purely immanent and immediate — but entirely passive and inert — contentful ideas.

Berkeley wrote an important early work on the theory of vision, arguing that what we really see is immediate phenomena of light and color, rather than inferred “things”. This was an important source for phenomenalism in early 20th century philosophy of science. Like the later phenomenalists, he tried to explain all cognitive error as bad inference from good immediate perception. From this point of view, “ideas” cannot be wrong, because they are purely immediate and purely inert; the possibility of error depends on the actions of finite spirits.

The common tradition of Cartesianism and British empiricism insists that there is a layer of immediate apprehension that is immune to error, and wants to ground knowledge and science by more authentically getting back to that immediate layer. I think Kant and Hegel convincingly showed that everything we experience as immediate actually has a prehistory, so that immediacy itself is only an appearance, and all immediacy that we experience is really what Hegel called mediated immediacy. Mediated immediacy has the same general kind of explanation as what is called “habit” in translations of Aristotle. We “just know” how to ride a bicycle once we have already learned. We don’t have to think about it; we just spontaneously do it. Similarly, I think “immediate” perception involves a complex unconscious application of categories that is affected by large bodies of previous experience.

Thus I want to say that there is no layer of human experience that is immune to error. On the other hand, through reflection and well-rounded judgment, we genuinely but fallibly participate in objectivity. Objectivity is not something that is simply “out there”; it is a real but always finite and relative achievement.

Shaftesbury on Moral Feeling

Anthony Ashley Cooper (1671-1713), third Earl of Shaftesbury, was personally tutored by John Locke as a young man, and the two remained friends in spite of various philosophical differences. Shaftesbury was sympathetic to the Cambridge Platonists, and attracted to aspects of Stoic ethics. He is especially known, however, for his emphasis on the role of feeling in ethics. Rejecting pessimistic Hobbesian and Calvinist views of human nature, he regarded the sense of right and wrong as a kind of second-order feeling — a feeling about other feelings. It is reflective, and while grounded in nature requires the right kind of upbringing and education for its development. The much more rationalistic Leibniz was very impressed by Shaftesbury’s work.

The main role of philosophy for Shaftesbury is to help us “regulate our governing Fancys, Passions, and Humours”, rather than to elaborate a system of the world. Goodness for Shaftesbury is to be understood mainly in terms of motivation rather than results. More objectively, it is grounded in a kind of natural teleology of order and harmony in the world. Something is good if it contributes to the “Existence or Well-Being” of a larger whole such as a species or a world. A virtuous human cultivates “equal, just, and universal Friendship” with humanity as a whole.

Shaftesbury believed in a perfectly good God, and in the argument from design. He opposed voluntarist views that made what is good depend on divine will, and advocated religious tolerance. Motivation by reward and punishment he deemed inadequate as a basis for morality.

Human motivation for Shaftesbury depends entirely on feeling or sentiment, not on reason or belief. He is considered to be a source for Hume’s famous view that in real life, human reason always serves human passions.

Scholars debate the extent to which Shaftesbury’s views should be considered subjectivist, and the extent to which he can be assimilated to the generally egoistic tradition of Hobbes, Locke, and the later Utilitarians. As I have noted previously, “self” has many meanings, from crude to cosmic. Shaftesbury clearly rejects what we would call selfishness, but in other passages promotes a positive view of a broader notion of self. His de-emphasis on reason is tempered by his sense of natural order and purpose in the world and his emphasis on a kind of reflection.

Kant’s emphasis on principles in ethics and his critique of subtler kinds of selfishness in spontaneous moral feeling represent a strong criticism of views like those of Shaftesbury. I think Kant sometimes goes too far in criticizing feeling, but Shaftesbury also goes too far in identifying reason with sterile abstraction. With Aristotle, I see human feeling and human reason as cooperating with one another in producing well-rounded valuations.

Animal Imagination

We talking animals have a unique perspective on what it means to be sentient. For us, any nonverbal awareness is always already implicitly informed by our linguistic abilities. We don’t have to mentally say words to ourselves; language-based understanding unconsciously permeates our elementary perceptions of things.

Nonetheless we share nonlinguistic perception with all animals, and also share emotion and Aristotelian “imagination” with many of them. This kind of “imagination” is an organic production and experiencing of “images” that can play a role somewhat analogous to that of thought based on language in shaping responses to things. I won’t worry for now exactly what an “image” is. Animals clearly anticipate events and consequences that are not immediately present to sensation, based on some kind of experiential learning. This seems to be related to what some of the Latin scholastics tried to explain in terms “natural signification”.

The most obvious interpretation of this kind of imagination is by a kind of analogy with sensation. We and other animals remember sensations that are no longer present, and imaginatively anticipate sensations in advance. This seems to imply somehow imagining certain things to be true, but without any explicit discursive reasoning. What is truth for my puppy?

I think emotion may be a big part of the answer. Emotion is in part a kind of spontaneous valuation of things. Specialists in human social psychology have found that simple emotional valuations of different things are surprisingly good statistical predictors of what ways of combining them people will regard as realistic or unrealistic, or true or false. I’m inclined to speculate that many animals live mainly by this kind of emotionally based valuation and classification (see also Ethos, Hexis; Parts of the Soul; Reasonableness; Feeling; Emotional Intelligence; Aristotle on the Soul; Aristotelian Subjectivity Revisited; Vibrant Matter).