Inferential bandwidth: civic and public reasoning today, social media included

I

There is a moment, familiar to almost everyone who has tried to argue seriously with another human being, in which the exchange stops being an exchange. Two people sit across a table, or face each other across a screen, and one of them says a sentence that contains three separate claims folded into each other, and the other answers only the first claim, as if the other two had never been spoken. The first person repeats the sentence, louder this time, or with a different opening word, believing the problem was volume or phrasing. It was neither. The sentence never entered the other person whole. Something in the room, in the two bodies present in it, refused to hold more than one thing at a time, and what looks like stubbornness or bad faith is often something more ordinary and more grave, a narrowing that has little to do with intelligence in the sense the word usually carries and everything to do with a capacity that has almost no name in ordinary speech, the capacity to keep several structures alive in the mind at once without one collapsing into the other. That capacity has an old name, worn smooth by centuries of use until it stopped telling us anything about the body it once described, and it deserves to be heard again before it is given a newer one.

The word reason descends from the Latin ratio, from reor, to reckon, to think, and ratio meant before anything else a reckoning, a counting up, the operation a merchant performs when he checks whether what left the warehouse matches what arrived at the port. But ratio also meant relation, proportion, the way one thing stands to another, and a reckoning is only possible once two things have been set into relation with each other. To reason, in this buried sense, is to hold two or more things in a live relation, to let them press against each other, and to let something emerge from that pressure that was contained in neither alone. That is already a claim about the body rather than about an abstract faculty floating above it, because holding things in relation is something a body does, the way a hand holds two ends of a rope taut enough to feel where the rope will break.

The frame through which I have tried for years to understand what a human being is I call antropismo trágico, a tragic anthropism, and its first conviction can be stated simply. There is no thinking that happens outside the flesh, lo encarnado, a body that shows up into a situation and suffers its weight. My word for that showing up is comparecencia, appearance in the strong sense in which a witness appears before a court, present in person, answerable, exposed. Reasoning is no exception carved out of this rule. It is one of the highest and most fragile achievements of comparecencia, and when it fails, it fails for the same reason a body fails at anything demanding, because it was never trained to bear that particular weight, or because something exhausted it before the weight arrived. The structures of reasoning, once seen clearly, explain why so many arguments today go nowhere, and every one of them keeps returning to the body, because there is nowhere else for it to go.

II

The simplest structure human beings ever formalized is the syllogism, and the Greek word behind it, syllogismos συλλογισμός, is built from syn, together, and logízesthai, to reckon, to count through. A syllogism is a reckoning done together, two claims brought into the same room and forced to produce a third that neither claim announces on its own. All music held only on paper waits for a body to sound it. This score is music held only on paper. Therefore this score waits for a body to sound it. Nothing in the third sentence was stated in the first two, and yet the third sentence was already there, waiting, the moment the first two were held together properly. Here lies the most basic form of what this essay calls inferential bandwidth, the capacity to keep two claims present in the mind long enough for their friction to produce something neither one could produce alone. A syllogism asks almost nothing of the body compared to what comes after, and even so, most people, most of the time, drop one of the two premises the instant a third party changes the subject, or raises their voice, or says something that stings. The claim that was held a second ago disappears, and reasoning collapses back into assertion, which is to say into the repetition of a single sentence at increasing volume.

The syllogism has a birth certificate, which is rare for an operation of the mind. Aristotle set it out in the Prior Analytics, and he was entitled to say, as he does at the close of the Sophistical Refutations, that on the subject of reasoning nothing at all had existed before him and that he had had to work it out by long and laborious inquiry. People had reasoned well before him, in courts, in geometry, in the dialogues of Plato, but no one before him had separated the form of a valid inference from its content and shown that the form alone carries the validity, so that any matter poured into the same mould comes out the same way. That separation is the first great act of bandwidth in the history of thought, because it asks the mind to hold two things at once in the strongest sense, the particular claim about rain or music or justice and the empty shape that makes the claim follow.

Before the simple syllogism gives way to anything stranger, it first has to be stretched across more than one step, and this stretching alone already defeats a great deal of ordinary argument. If enforcing a law requires the threat of coercion, and the threat of coercion forecloses a portion of freedom, then enforcing any law forecloses a portion of freedom, and the conclusion sits several inferential steps away from the first premise rather than one. Nothing in that chain is difficult on its own. What is difficult is holding the first link in place while the mind walks out to the last, resisting the pull to answer the conclusion as though it had arrived by itself, stripped of the premises that actually earned it. Watch how an argument dies in real conversation and this is usually the exact place it dies, far from any exotic frontier of logic, at the second link of an ordinary chain, where a listener, already tired, already primed to defend a position rather than trace one, drops the middle term and answers the far end of the chain as if it were a bare assertion shouted at them out of nowhere. A chain of conditionals asks for nothing more than continuous tracking, but continuous tracking is precisely the labor that a mind running low on presence refuses to perform, and it refuses in a manner that feels, from the inside, like clarity rather than exhaustion, since the person who drops the middle term experiences their own shortcut as the obvious response and never as a lapse.

III

Aristotle built an entire architecture on top of this basic operation, and everything that followed him in the history of logic can be read as an attempt to notice what the simple syllogism leaves out. The first thing it leaves out is time and necessity, the difference between what merely happens to be so and what could not have been otherwise. A modal claim carries this difference inside itself. It is raining is one kind of sentence. It must be raining, given what we know of the pressure system, is another kind entirely, and it necessarily rains in a climate like this one during this season is a third kind again, because it reaches for a claim about how the world is built rather than about how the world merely turned out today. Aristotle himself worked out a modal syllogistic in the same Prior Analytics, and the difficulty of it was so real that commentators have argued over its consistency for more than two thousand years, while the Megarian Diodorus Cronus built his famous Master Argument on the claim that the possible is only what is or will be true, a claim that tied possibility to time so tightly that the two could no longer be told apart.

To hold a modal claim properly, a person has to feel the difference between a fact and a necessity in something more than the words used to state it, has to sense that one claim would survive the world being slightly different and the other would not. This is an embodied discrimination before it is a logical one, close cousin to the discrimination a string player makes between a note that happens to be in tune today because the room is warm and a note that is in tune because the interval itself is correctly measured against its neighbors. Musicians who have spent years inside intonation know this difference in their hands before they could ever articulate it in a sentence. A cellist who has played the same Bach suite in a cold church in February and in a crowded hall in July knows that one kind of rightness travels with the instrument and the other stays behind in the room, and something similar has to happen in a mind that reasons well about necessity and possibility, an internal sense for which claims are hostage to circumstance and which are free of it.

IV

Modal logic asks what must be and what might be. Deontic logic, its close relative, asks what ought to be, from the Greek déon δέον, that which is binding, that which is needed, and here the confusion multiplies because human speech collapses is and ought into the same grammatical form constantly, so that a sentence describing how something currently works gets heard as a sentence prescribing how it should continue to work, and a sentence prescribing a different arrangement gets heard as an insult to the people currently living inside the old one. David Hume noticed the slide in 1739, in a short paragraph of the Treatise of Human Nature that has never stopped being quoted. In every system of morality he had met, he wrote, the author proceeds for some time in the ordinary way of reasoning, describing God or human affairs, and then suddenly, instead of the usual copulations of propositions, is and is not, Hume finds no proposition that is not connected with an ought or an ought not, a change imperceptible and yet of the last consequence, since this new relation can never be a deduction from the others. Two centuries later Georg Henrik von Wright gave the ought its own formal grammar, with operators for the obligatory, the permitted and the forbidden standing beside the logical operators of necessity and possibility, and the very fact that a separate logic was needed shows how far the two kinds of sentence lie from each other underneath their shared grammar.

A great deal of what passes for political disagreement in public life is this confusion running at scale, one person making a claim about what is the case and being heard as though they had made a claim about what is right, or the reverse, someone arguing about what ought to happen and being answered with statistics about what currently does happen, as if the statistics settled the question rather than simply describing the world that the argument was trying to change. Untangling is from ought requires holding both sentences apart in the mind at the same moment, seeing clearly that a description leaves an obligation open and an obligation leaves a description standing, and this untangling is exhausting in a way that most public discourse has no patience for, because patience itself is a form of comparecencia, a willingness to remain present to a difficulty rather than resolving it by force. The word patience carries the same lesson, from patior, to suffer, to undergo, and the person who waits for two sentences to separate is undergoing them, letting them act on the body before acting on them.

V

Then there is the trouble that quantifiers cause, and this trouble is old enough that grammarians and medieval logicians, with their doctrines of supposition and of the scope of signs like every and some, noticed it long before Gottlob Frege gave it a formal treatment in 1879, in the small and revolutionary book he called Begriffsschrift, a concept script, where for the first time the order in which the quantifiers stand was written into the very shape of the sentence. Everyone loves someone is a sentence that hides an entire fork in the road inside three words. It might mean that there exists one beloved person whom everyone loves, a single figure at the center of universal affection, or it might mean that each person, taken separately, loves somebody or other, with no single figure shared among them at all. The sentence itself leaves the choice open, and a listener who cannot hold both readings in view at once, testing which one the speaker intended, will simply default to whichever reading arrived first and defend that reading as though it were the only one available.

This happens constantly in disputes that sound substantive and turn out to be disputes about scope, about which quantifier governs which. Arguments about music collapse into shouting for this reason more often than for any other. Every critic misses something in this symphony, says one person, meaning that each critic, taken alone, has a blind spot of their own, and the other hears that there is one thing in the symphony every critic misses, a hidden secret that only the speaker claims to have found, and answers with indignation a boast that was never made. The same sentence carried a modest observation about the limits of listening into one mind and an arrogant claim of private revelation into the other. The failure here belongs to architecture rather than to morals. Nobody paused long enough to ask which structure the sentence was actually built from, because pausing costs something, and the something it costs is exactly the reserve of attention this essay calls bandwidth.

VI

Reasoning becomes stranger still when a claim starts referring back to itself, and this is where recursive and self-referential structures enter, structures that the ordinary mind avoids instinctively because they produce a kind of vertigo, the sensation of standing on a floor that is somehow also the ceiling. A sentence that says of itself that it is false forces the mind into a loop that never resolves into a stable answer. Eubulides of Miletus, a contemporary and rival of Aristotle, is credited with posing it in the fourth century before our era, and it has never been dissolved, only relocated, from the Stoics who wrote whole treatises on it to the medieval logicians who called such sentences insolubilia, the insolubles. Far from a parlor trick invented by clever logicians to amuse themselves, the liar is a warning built into the deep structure of language and thought, a signal that some systems, including systems as large as a whole culture’s image of itself, can fold back on themselves in ways that produce genuine paradox rather than mere confusion.

In 1931 Kurt Gödel turned the warning into a theorem. He showed that any consistent formal system strong enough to express ordinary arithmetic contains true sentences it cannot prove, and, in the second of his two results, that such a system cannot prove its own consistency from inside itself. Wittgenstein found the same wall in a humbler place. There is one thing, he wrote, of which one can say neither that it is one metre long nor that it is not, and that is the standard metre in Paris, because the bar that serves as the measure cannot at the same time be measured by itself. A culture that insists it is the most rational culture in history, purely because it possesses more devices for calculation than any culture before it, is making a self-referential claim of exactly this kind, since the claim of rationality is itself a product of the very faculty being praised, and a faculty can no more certify its own soundness from inside itself than a ruler can measure whether it is the correct length using only itself as the standard.

Recursive reasoning asks the mind to step outside its own operation and look back at that operation from a vantage the operation cannot reach on its own, and very few people are trained, or have ever been asked, to attempt this step, because it requires a kind of doubling of presence, being here and also watching oneself be here, that comparecencia makes possible but never makes easy. Every musician knows a version of it. The singer who sings and at the same time hears herself singing, as if from the back of the hall, is performing a small recursion with her whole body, and anyone who has tried it knows how quickly the second presence devours the first, how easily listening to oneself becomes watching oneself, and how the voice tightens the instant it becomes its own audience.

VII

Dialectical reasoning, older than most of what has been described so far, asks for something different again, the deliberate confrontation of two positions that each contain a portion of the truth while neither contains the whole of it, where the earlier structures had asked for the isolation of a single argument’s internal form. The Greek dialégesthai διαλέγεσθαι means to converse through, to talk something out rather than to talk at someone, and a real dialectic requires that both positions be held in their strongest form at the same time, never weakened into caricature so that one side can be dismissed quickly. This is perhaps the rarest discipline of all the ones under discussion, because it demands that a person feel the pull of a position they do not hold, feel it as a live force rather than as a target, and let that pull do real work on their own position before answering it.

Almost nobody does this anymore in public argument. The far more common move is to construct the weakest possible version of an opponent’s claim and defeat that weak version triumphantly, the maneuver English calls the straw man, after a figure of stuffed straw that is easy to knock down because it never fights back. The opposite discipline also has a name. Philosophers call it the principle of charity, and the hardest form of it asks a person to restate an opponent’s position so well that the opponent would wish to have said it that way. The straw man has become the default mode of disagreement for a reason that has little to do with any growth in dishonesty. Constructing an opponent’s strongest case and holding it alongside one’s own costs a great deal of bandwidth, and bandwidth, in the economy that governs most people’s days now, is the scarcest resource there is.

VIII

Underneath all of this sits a kind of reasoning that almost nobody performs consciously and almost everybody performs badly, the probabilistic updating of belief in the face of new evidence. Thomas Bayes, a Presbyterian minister in Tunbridge Wells with a taste for mathematics, left among his papers at his death in 1761 an essay on how to infer the probability of a cause from the events it produces. His friend Richard Price found it, edited it and read it to the Royal Society, which printed it in 1763, and it was Pierre-Simon Laplace who, working independently from 1774 onward, turned the idea into the general method that now carries Bayes’s name. The core of it is simple to state and brutally hard to practice. A belief should shift in proportion to the strength of new evidence, neither collapsing entirely at the first contrary fact nor remaining frozen despite a mounting weight of contrary facts.

What happens instead, almost universally, is that the first story a person hears about an event becomes an anchor that all subsequent information must struggle against. Amos Tversky and Daniel Kahneman gave the word anchoring its technical sense in 1974, after showing that people asked to estimate a quantity stayed close to an arbitrary number spun on a wheel of fortune in front of them, even when they knew the number meant nothing. The same weight attaches to the first report of an event. A correction issued after a false report barely moves the needle of belief, while the false report itself, arriving first, does nearly all the work of shaping what a person will believe for years afterward, and psychologists who study misinformation have given this persistence a name of its own, the continued influence effect, observed even in people who remember the correction and accept it as true.

Intelligence offers no protection here. Extremely intelligent people anchor on first impressions just as readily as anyone else, sometimes more readily, because intelligence supplies elaborate justifications for staying anchored that a less clever mind would never have thought to construct. Dan Kahan and his colleagues found exactly this when they gave the same table of numbers to subjects once as data about a skin cream and once as data about gun control. The most numerate subjects read the skin cream correctly and then, when the identical table touched their politics, used their skill to read it wrongly in the direction their side preferred, so that the gap between partisans grew wider as mathematical ability rose.

What Bayesian reasoning actually requires is a humility that has nothing to do with cleverness, a willingness to hold a belief loosely enough that evidence can move it, and this willingness is again a bodily posture before it is an intellectual one. It resembles the openness a string quartet needs in rehearsal far more than any mental operation performed in isolation. A player has to be willing to hear that a phrase they have shaped one way for years might need to bend differently once a new colleague joins the group, and the bending happens first in the bow arm, in the breath taken before the upbeat, long before the player could say in words why the old shape no longer holds.

IX

There is a further complication that formal logic only recognized as a formal problem in the twentieth century, though poets and mystics had lived inside it for far longer, the recognition that not every claim resolves cleanly into true or false. The problem itself is ancient. In the ninth chapter of De Interpretatione, Aristotle asked whether the sentence there will be a sea battle tomorrow, spoken today, is already true or already false, and saw that if it were, everything would happen of necessity and deliberation would be pointless. Tomorrow has not happened yet, and the battle depends on choices and contingencies that remain open. Twenty-three centuries later the Polish logician Jan Łukasiewicz built a logic with three values precisely to give that sentence a place to stand, announcing it in 1920 in a two-page paper whose third value he read as possible, and with Alfred Tarski he later extended the idea to logics with infinitely many values along a continuous scale. Ordinary binary logic had no room for claims that are partially true, or true only under conditions that have not yet arrived, or true in a way that depends on a future choice not yet made.

A mind trained only in binary logic, the logic of yes and no, on and off, with us and against us, cannot hold this kind of claim without distorting it, without forcing it prematurely into one bin or the other. A great portion of the coarseness in contemporary argument comes from exactly this forcing, the collapsing of every claim, no matter how genuinely graded or conditional or partial its truth, into one of two bins, because two bins are all the architecture in use can accommodate. A culture saturated with devices that reward instant binary reactions, a thumb pointed up or down, an emoji chosen from a small fixed set, trains its participants continuously toward two-valued thinking and away from the many-valued reasoning that most real situations actually demand. Music has always lived in the third value. A dominant chord, while it sounds, is neither an arrival nor a departure. It is on its way, and a listener who forces it into one bin or the other has stopped hearing the very thing that makes it a dominant, its leaning toward a future that has not yet arrived.

X

Every structure described so far moves downward from what is given to what follows from it, and the mind also has to move the other way, upward from scattered cases toward something it has never been told. Induction is the oldest name for that ascent, the step from this swan is white and that one and that one to all swans are white, and Hume, again, saw that no amount of repetition can make the step necessary, since nothing in a thousand past sunrises guarantees the next one. Induction asks the body for a peculiar kind of holding, the cases kept present one beside the other until their common shape begins to show, and at the same time a readiness to drop that shape the moment a black swan swims into view. Most people manage one half of this. They either refuse to generalize at all, treating every case as an island, or generalize from the first two cases and then stop looking.

Charles Sanders Peirce named a third movement, distinct from both deduction and induction, the one by which a mind confronted with something surprising invents the hypothesis that would make the surprise ordinary. He called it abduction, and gave it a form as plain as a syllogism. The surprising fact C is observed. But if A were true, C would be a matter of course. Hence there is reason to suspect that A is true. Abduction is where new ideas come from, and its demand on bandwidth is the most delicate of all, because the mind must hold the surprising fact open, refusing to explain it away, while also holding several candidate explanations in suspension and letting each one try to make the surprise vanish. A detective, a physician at a bedside and a scholar staring at a manuscript that contradicts every catalogue are all doing the same thing, and all of them fail in the same way when they seize the first hypothesis that fits and stop holding the others.

Analogy is the movement that carries a structure from one domain into another, and it may be the oldest and most fertile of all. Dedre Gentner showed in 1983 that a good analogy maps relations rather than surface features, so that the atom resembles the solar system because a smaller body revolves around a larger one that attracts it, in complete indifference to the fact that the sun is hot and yellow and the nucleus is neither. To reason by analogy is to hold two whole systems in the mind at once and feel which relations travel from one to the other and which stay behind, and this is perhaps the operation in which bandwidth shows most nakedly, since an analogy collapses into a mere likeness the moment one of the two systems slips out of view. The Greeks had a word for it, analogía ἀναλογία, proportion, the same word as the ratio of the beginning in another tongue, and the circle closes on itself, because to reason was always to hold things in proportion to one another.

XI

And finally, sitting above all of these structures, there is metalogical reasoning, the reasoning that examines reasoning itself, that asks whether a particular argument is valid and also whether the method used to reach it could, by its very construction, ever have produced a different answer, whether the game was fair before it was ever played. This is the rarest reasoning of all, because it requires a person to suspect their own tools, to ask whether the syllogism they just built was built to discover a truth or built backward from a conclusion they had already decided to defend. Almost every serious failure of reasoning visible in music criticism, in philosophical debate, in the smaller disputes of daily life, traces back to this final and highest failure, the failure to ask whether the entire apparatus of argument being deployed was ever capable of reaching a conclusion other than the one it reached. A person can be flawless at every level below this one, impeccable with syllogisms, sensitive to modal distinctions, scrupulous about quantifier scope, patient with dialectic, humble in updating belief, alert to graded truth, inventive in hypothesis, and still fail here, because this level asks the mind to doubt itself in a way that feels, to most people most of the time, like a kind of self-betrayal, when it is in fact the highest form of intellectual honesty available to a finite creature.

This ladder, from the simple syllogism up through metalogical doubt, makes visible something that abstract complaints about declining rationality never quite capture. Reasoning is a set of distinct operations, each with its own demands, each capable of failing independently of the others, and a person can be sharp at one level and hopeless at another without any contradiction at all. Most readers will never write a truth table in their lives and lose nothing essential by it. What all these operations share, what binds them together underneath their formal differences, is the requirement that has been circling since the opening scene, the requirement to hold more than one structure alive in the mind at the same time without letting one collapse into the other prematurely. The name for this holding, this capacity, is inferential bandwidth, a word borrowed from engineering because engineering at least has the honesty to admit that a channel can only carry so much at once before something gets dropped.

The word has already been used in a neighboring sense, and the neighbor deserves to be named. Sendhil Mullainathan and Eldar Shafir called bandwidth the general capacity of a mind to compute, attend and resist impulse, and showed that scarcity taxes it. In one of their studies sugarcane farmers in Tamil Nadu performed measurably worse on tests of fluid intelligence before the harvest, when they were poor, than after it, when the same people had been paid, and the poorer shoppers in a New Jersey mall lost ground on the same tests when they were merely asked to think through a large and costly car repair, while the richer ones held steady. Their bandwidth is a reservoir that worry drains. Inferential bandwidth, in the sense used here, is narrower and also deeper, the capacity to hold several relations at once, and it can be drained in exactly the way they describe, by hunger, by fear, by the day, but it can also be built. Cognitive psychologists have measured its ceiling with some precision. Graeme Halford and his colleagues argued that an adult mind can process at most four related variables in a single representation, and that anything more complex has to be broken into pieces or compressed into chunks before it can be held. The sentence at the table, with its three claims folded into one another, sat very close to that ceiling. The person shouting past their interlocutor did not lack information. They lacked the capacity, in that moment, worn down by the day, by the noise, by whatever private exhaustion they carried into the room, to hold their own claim and the other person’s claim in the same field of attention long enough to let the two claims touch, and every failure of reasoning of this kind is a bandwidth failure before it is ever a knowledge failure.

XII

Here antropismo trágico has something distinctive to contribute that a purely cognitive account of reasoning failure lacks. A purely cognitive account treats bandwidth as a fixed hardware limit, a processing speed a person either has or lacks, the way a chip has a certain clock rate stamped on it at the factory. Bandwidth in the sense meant here belongs to a body that shows up into situations across a lifetime, and that body’s capacity to hold structures in relation grows or shrinks according to what it has been asked to bear and how often it has borne it. The ceiling Halford measured holds for raw relations, and yet experts walk past it every day, because years of practice teach them to fold many relations into a single one that occupies a single place in the mind. William Chase and Herbert Simon showed in 1973 that a chess master who glances for five seconds at a position from a real game can rebuild it almost piece for piece, while a beginner recalls a handful of pieces, and that the master’s advantage vanishes when the pieces are scattered at random across the board. The master sees more because the master has learned what belongs together. Anders Ericsson and Walter Kintsch went further and argued that experts build a long-term working memory, retrieval structures laid down through practice that let them keep vast amounts of material within reach of the present moment, as if the narrow window of attention had been fitted with a hinge that opens onto a room behind it.

This is why serious training in any demanding discipline, in philosophy and formal logic and equally in music, medicine, law, craftsmanship of any real depth, produces something that looks from the outside like raw intelligence and is in truth a trained capacity for presence, an enlarged bandwidth built through years of being made to hold difficult structures without collapsing them. A physician learning differential diagnosis is trained, explicitly and painfully, to keep several competing hypotheses about a patient’s condition alive at once, resisting the pull toward what clinicians call premature closure, the acceptance of a diagnosis before it has been verified, which would let her stop thinking and start treating. Studies of diagnostic error find premature closure among the most frequent of the cognitive faults behind it, and it kills patients whose actual condition does not match the first plausible story that presented itself.

Music has kept the oldest and most exact school of this capacity, and it has done so for centuries without ever calling it by that name. When Johann Joseph Fux published his Gradus ad Parnassum in Vienna in 1725, a dialogue in Latin between a master and a pupil, he set out a path of five species of counterpoint that begins with a single note against a single note and ends with free lines moving against a fixed melody, and every step on that path adds exactly one more thing to be held at once. The pupil writing in two voices has to keep the line he is writing, the line already there, and the interval between them alive together, and to feel at each step whether the next interval is a consonance that rests or a dissonance that has to be prepared and resolved, which is a small syllogism sung in time. In the conservatories of Naples, in the same decades, students learned from partimenti, unfigured or lightly figured basses from which they had to realize at the keyboard, in real time, the harmony, the voice leading and often the imitative upper parts that the bass implied, so that a whole composition had to be inferred from a single line and held in the hands while it was being invented.

The ear has its own ceiling, and it has been measured. David Huron asked listeners to say how many voices they heard in polyphonic textures of a single timbre and found that their accuracy fell off steeply once the texture passed three voices, with a growing tendency to hear fewer voices than were really sounding. A fugue in four or five voices therefore asks for more than the untrained ear can carry, and the history of listening to Bach is in large part the history of ears that learned, slowly, to carry it. A conductor studying a symphony has to hold the horn line, the string line, the harmonic structure beneath both and the larger architectural arc of the whole movement simultaneously present in a single act of attention that unfolds in real time and cannot be paused to think one strand through before returning to the others. Reading a full score at the piano, reducing twenty staves to ten fingers as the pages turn, is a daily exercise in exactly the folding that Chase and Simon found in the chess master.

What the musician learns to hold has a precise anatomy. In the vocabulary I use for it, every living sound is woven on a loom, the telar, of three threads that cannot be pulled apart without killing the sound. The first is Carne, flesh, the face of a sound, its breath, its weight, everything it carries from the body that made it. The second is Inclinación, inclination, the intention and the gesture with which a sound leans, the way an appoggiatura presses on the note it belongs to. The third is Hacia, toward, direction, pulse, the pure futurity of a sound already on its way to the next one. A musician who plays a single phrase well is holding all three at once, the weight of each note, its leaning, its direction, and the moment one of them drops out of attention the phrase goes slack, heavy without direction, or directed without weight, or correct and empty. Musical bandwidth is the capacity to keep the whole loom strung.

Neither the physician nor the conductor was born with this capacity. Both built it, slowly, through repeated exposure to situations that demanded it and would not forgive its absence, and the building itself was a bodily process, exhausting in the specific way that comparecencia is always exhausting, because showing up fully to a difficulty costs something real and cannot be faked. The experimental evidence points the same way from the other side. Programs that try to enlarge working memory directly, through hours of abstract training on computer tasks, make people better at the tasks themselves and leave the rest of their thinking almost untouched, as large meta-analyses have repeatedly found. Bandwidth built in a vacuum stays in the vacuum. The bandwidth that survives is the one that grew while the body was trying to do something it cared about.

XIII

What has happened to this bandwidth in the conditions most people now live inside is no mystery, though it is rarely named with this precision. The environments that now occupy the largest share of a typical waking day are engineered, deliberately and with enormous technical sophistication, to reward the opposite of bandwidth. They reward the fastest possible reaction, the binary judgment issued in under a second, the sentence read in isolation from its paragraph, the paragraph read in isolation from the argument it served, the argument encountered without its opponent’s strongest version ever being presented alongside it. None of this is an accident of a neutral technology being misused. The architectures in question were built, tested and refined against the explicit goal of maximizing engagement, and engagement correlates far more reliably with speed, simplicity and heat than with depth.

The heat has been measured. William Brady and his colleagues analyzed more than half a million messages about gun control, same-sex marriage and climate change and found that each additional word expressing a moral emotion, outrage, disgust, contempt, raised the rate at which a message was shared by about twenty per cent, and that the sharing stayed largely inside each political camp. A system tuned for diffusion therefore selects, without anyone needing to intend it, for exactly the sentences that close a question fastest, sentences built on a single structure and charged with a single feeling. The reading surface plays its own part. A meta-analysis of studies comparing reading on paper with reading on screens found that comprehension of informational texts is lower on screens, and that the disadvantage grows under time pressure and has grown over the years rather than shrinking as readers became more familiar with screens, as though the hurried eye of the screen were learning to skim rather than to hold.

The evidence has limits that deserve to be stated with the same care. The popular claim that heavy media multitaskers have measurably damaged attention rests on studies whose effects are small and have often failed to replicate, and no one has yet shown a clean causal line from years on a feed to a shrunken capacity for holding arguments. What the evidence does show is a selection pressure, an environment that rewards the single structure and the hot word every minute of the day, and the body, which becomes what it practices, practices there for hours. A mind trained for years inside such an environment does not become stupid in any crude sense. It becomes narrow in a very specific way, capable of enormous speed within a single structure and increasingly reluctant to hold two structures in relation for the time it takes to feel their friction.

XIV

There is a further dimension to this narrowing that a purely technological explanation misses, because it treats the mind as though it were merely a processor being fed the wrong kind of input. The deeper truth, the one antropismo trágico keeps insisting on against every temptation to flatten the human into an information system, is that reasoning has always been inseparable from feeling. The current impoverishment of reasoning has little to do with a triumph of emotion over logic, the tired framing inherited from a rationalism that never understood what it was dismissing. It is a specific corruption of the relationship between the two.

The word itself remembers what that relationship was. Emotion comes into English from the French émotion, and behind it lies the Latin emovere, from ex, out, and movere, to move, to set in motion, to displace. An emotion, at its root, is a movement out of a place, the moment the body is shifted from where it stood, and in the oldest uses of the word it could name a riot in the streets or a tremor of the earth before it named anything inside a person. Felt fully, in the body, as the incarnate register through which a person first notices that something matters, emotion is the very thing that moves a mind out of its resting place toward a structure worth holding. It is the flicker of unease that says this argument does not add up before the mind has consciously located where the fallacy lies, the flush of recognition that says this claim matters enormously before any syllogism has been constructed to justify why.

Neuroscience arrived at a version of the same thought by its own road. Antonio Damasio proposed in 1994 that bodily signals, which he called somatic markers, tag the options before a mind deliberates and narrow the field in which deliberation happens, and that patients who lose those signals through damage to the frontal lobes keep their logic intact and still make ruinous decisions. The experimental details of his hypothesis have been contested, and the central intuition has survived the contest. Jonathan Haidt went further in 2001 and argued that in moral matters a fast intuition usually delivers the verdict and reasoning arrives afterward to defend it, like a lawyer hired after the sentence has been passed, the emotional dog wagging its rational tail. Both pictures describe the same hinge. A feeling can open an inquiry or close it, and everything depends on which way the hinge turns.

What has corrupted reasoning is the substitution of emotional shortcut for emotional signal, the moment when a flicker of anger or comfort or tribal recognition, instead of moving the mind to look closer, is allowed to stand in for the looking itself, to close the inquiry rather than open it. Fear works this way constantly in public discourse, arriving as a fast, cheap substitute for the slower work of holding a genuinely uncertain claim in its uncertainty, offering instead an immediate two-valued verdict, dangerous or safe, us or them, that spares the listener the exhausting work of finding out where along the many-valued scale the truth actually sits.

Most people, most of the time, are operating with nothing so elaborate as a syllogism. They are operating with a small set of these emotional shortcuts, worn into grooves by repetition, and the grooves have a grammar of their own that deserves to be stated plainly. If it feels unfair, it is wrong, one groove says, collapsing an entire structure of competing claims about fairness into whatever the body registers first as a sting. If experts disagree, nobody actually knows anything, says another, treating the ordinary friction of a living field of inquiry, which always contains disagreement at its edges, as proof that the field contains no knowledge at all.

If someone is caught being inconsistent once, reject everything they have ever said, runs a third, a groove so old that the law gave it a Latin name, falsus in uno, falsus in omnibus, false in one thing, false in everything, and juries were long instructed that they might discard the whole testimony of a witness caught lying in a single part of it. Even the law came to treat the maxim with suspicion, permitting the inference without ever compelling it, and the public square has kept the maxim in its harshest form while forgetting the caution that came to surround it.

A person who reasons exclusively along two or three grooves of this kind is neither lying nor stupid. They are running the only inferential software they were ever given sustained practice installing, and every issue that reaches them, no matter its actual shape, gets forced through whichever groove happens to be nearest, the way water forced downhill takes whatever channel already exists rather than the channel the terrain would ideally suggest. The tragedy, in the specific sense that antropismo trágico gives that word, lies elsewhere than in the fact that human beings feel. Feeling, which ought to be the doorway into the harder work of holding, has been trained by the environments most people inhabit into a substitute for that work, a closing rather than an opening, and a creature that cannot tell the difference between a door and a wall will eventually stop trying doors at all.

XV

The table of the opening scene can now be seen again, with the two people and the sentence that only half arrived. What actually happens in that scene, described with the precision the preceding pages have tried to earn, has little to do with one person being smart and the other not, or with one person arguing in good faith and the other in bad faith, though bad faith certainly exists and deserves its own reckoning. A sentence carrying three claims, perhaps a factual claim, a modal claim about what necessarily follows from that fact, and a normative claim about what ought to be done in light of both, arrived at a body that had bandwidth available for one claim only, and that body did the only thing available to it, which was to seize the first claim, treat it as though it were the whole of what was said, and answer that fragment as though answering the whole. The listener is telling the truth when they say that is not what I meant, because from where they stand, inside their own comparecencia, what actually arrived at them was already a mutilated fraction of the sentence spoken. This is why raising one’s voice never fixes the problem and often deepens it. Volume adds no structure, only pressure, and pressure applied to a channel that is already saturated simply produces static.

The same table also holds a hope that the solitary picture of reasoning hides. Hugo Mercier and Dan Sperber have argued that human reasoning evolved less for the private discovery of truth than for the exchange of arguments between people, which would explain why a lone reasoner so readily builds a case for what they already believe, and also why the same people, set to argue in small groups, solve logical problems that almost all of them fail alone. Two bodies at a table can hold together what neither could hold alone, each one carrying a strand while the other carries another, provided that each lets the other’s sentence arrive whole. The table is where bandwidth fails most visibly, and it is also the oldest place where it was ever enlarged.

XVI

None of this makes bandwidth destiny, fixed at birth and unchangeable afterward, because everything said about how it narrows under certain conditions applies with equal force to how it can be rebuilt under different ones. A short list of habits promising to restore the capacity for reasoning would belong to the same impoverished architecture these pages describe, the binary promise that a few actions will resolve a long and genuinely difficult condition. What can be said, because it has been visible in my own life and in the handful of students and colleagues I have watched grow across years rather than months, is that the rebuilding, when it happens, never happens through information alone. It happens through repeated, uncomfortable exposure to material that refuses to resolve quickly, a text that will not yield its meaning on first reading, an opposing position that turns out, once actually inhabited rather than caricatured, to contain a truth the reader had not expected, a piece of music whose contrapuntal lines refuse to let the ear rest in a single voice and demand instead that the ear learn, slowly, painfully, to hold all the voices at once. The body that does this repeatedly is undergoing something closer to training in the oldest sense, the forging of a capacity through sustained exposure to resistance, and the resistance is the whole point, because a structure held without any friction was never really held at all, it was merely glanced at and set back down.

This rebuilding has a specific and exact history, and it took place far from any discipline called critical thinking, a discipline that barely existed before the last century and that mostly teaches the names of fallacies rather than the felt experience of holding a difficult structure open. It happened inside philosophy, inside law, inside theology, inside mathematics, inside music, inside literature read slowly enough to notice its own architecture, because each of these disciplines, in its own idiom, forces a mind to acquire a specific inferential form it did not already possess and then forces that form to be used under real pressure. A judge’s argument must survive an appeal, a proof must survive a colleague’s scrutiny, a theological distinction must survive centuries of the sharpest minds trying to break it, a fugue must survive the ear of anyone who knows how the subject should return in the bass, and a novel yields its structure only to a reader willing to hold five strands of consequence in view at the same time near the final page. None of these disciplines advertises itself as bandwidth training. They advertise themselves as pursuits of truth, or beauty, or justice, or God, and the widening of what a mind can hold arrives as a side effect of pursuing something else entirely, which may be the only way it can arrive at all, since bandwidth built directly, as an exercise for its own sake, rarely survives contact with a real argument, while bandwidth built in service of something a person actually cared about tends to stay.

There is something almost unbearable in admitting that the restoration of reasoning, at a moment when the need for it has never been more visible, cannot be delivered at the speed the moment itself seems to demand. The very urgency so many people feel about the coarsening of public argument pushes them toward fast interventions, a better algorithm, a media literacy course completed in an afternoon, a set of critical thinking rules memorized like multiplication tables, when what is actually missing cannot be supplied at that pace, because the deficit was never a deficit of rules. It was a deficit of comparecencia, of bodies willing to remain present, uncomfortably present, inside a difficulty long enough for the difficulty to do its slow work on them. A rule can be taught in an afternoon. A capacity to hold two incompatible truths without collapsing either one has never, in the whole history of anyone who has ever acquired it, been built in an afternoon, and that fact sits uncomfortably, and perhaps it should.

Returning to the two people at the table, being right has never by itself moved anyone, and what separates the argument that goes somewhere from the argument that goes nowhere is whether either body in the room is still willing, after the exhaustion of the day, after every incentive around them has trained the opposite reflex, to let the other person’s sentence arrive whole before answering it. This willingness costs something every single time, and there is no device, no course, no page that can make the cost disappear. A page can make the cost visible enough that a reader who feels it, the next time a difficult sentence arrives carrying more than they are ready to hold, might recognize the sensation for what it actually is, the ordinary and recoverable strain of a mind being asked, for one more moment, to hold open a little longer than it has been trained to. Perhaps this is also, quietly, the deepest reason to keep reading pages that resist a first pass, pages whose sentences refuse to resolve on the first attempt and demand a second, and to keep listening to music whose voices refuse to merge into one, since each such page and each such fugue does to a solitary mind exactly what the two people at the table needed to be able to do for each other, holding a structure open, alone, in silence, until the structure finishes forming. A mind that has done this often enough in solitude arrives at the table changed, though changed toward what, and how far, and for how long before the next exhausting day erodes it again, remains open, like the sea battle of tomorrow, whose truth no one at the table can yet decide.

Bibliografía

Aristotle. Prior Analytics, De Interpretatione and Sophistical Refutations, in The Complete Works of Aristotle. The Revised Oxford Translation, edited by Jonathan Barnes, Princeton, Princeton University Press, 1984.

Bayes, Thomas. “An Essay towards Solving a Problem in the Doctrine of Chances”, communicated by Richard Price. Philosophical Transactions of the Royal Society of London 53 (1763), 370-418. https://doi.org/10.1098/rstl.1763.0053

Beall, JC, Michael Glanzberg and David Ripley. “Liar Paradox”, in The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/liar-paradox/

Bobzien, Susanne. “Dialectical School”, in The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/dialectical-school/

Brady, William J., Julian A. Wills, John T. Jost, Joshua A. Tucker and Jay J. Van Bavel. “Emotion shapes the diffusion of moralized content in social networks”. Proceedings of the National Academy of Sciences 114, 28 (2017), 7313-7318. https://doi.org/10.1073/pnas.1618923114

Chase, William G., and Herbert A. Simon. “Perception in chess”. Cognitive Psychology 4, 1 (1973), 55-81. https://doi.org/10.1016/0010-0285(73)90004-2

Damasio, Antonio R. Descartes’ Error. Emotion, Reason, and the Human Brain. New York, G. P. Putnam’s Sons, 1994.

Delgado, Pablo, Cristina Vargas, Rakefet Ackerman and Ladislao Salmerón. “Don’t throw away your printed books. A meta-analysis on the effects of reading media on reading comprehension”. Educational Research Review 25 (2018), 23-38. https://doi.org/10.1016/j.edurev.2018.09.003

Dennett, Daniel C. Intuition Pumps and Other Tools for Thinking. New York, W. W. Norton, 2013.

Dunn, Barnaby D., Tim Dalgleish and Andrew D. Lawrence. “The somatic marker hypothesis. A critical evaluation”. Neuroscience and Biobehavioral Reviews 30, 2 (2006), 239-271. https://doi.org/10.1016/j.neubiorev.2005.07.001

Ecker, Ullrich K. H., Stephan Lewandowsky, John Cook, Philipp Schmid, Lisa K. Fazio, Nadia Brashier, Panayiota Kendeou, Emily K. Vraga and Michelle A. Amazeen. “The psychological drivers of misinformation belief and its resistance to correction”. Nature Reviews Psychology 1 (2022), 13-29. https://doi.org/10.1038/s44159-021-00006-y

Ericsson, K. Anders, and Walter Kintsch. “Long-term working memory”. Psychological Review 102, 2 (1995), 211-245. https://doi.org/10.1037/0033-295x.102.2.211

Frege, Gottlob. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle, Louis Nebert, 1879.

Fux, Johann Joseph. Gradus ad Parnassum, sive manuductio ad compositionem musicae regularem. Vienna, Johann Peter van Ghelen, 1725.

Gentner, Dedre. “Structure-mapping. A theoretical framework for analogy”. Cognitive Science 7, 2 (1983), 155-170. https://doi.org/10.1207/s15516709cog0702_3

Gjerdingen, Robert O. Music in the Galant Style. New York, Oxford University Press, 2007.

Gödel, Kurt. “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”. Monatshefte für Mathematik und Physik 38 (1931), 173-198. https://doi.org/10.1007/BF01700692

Haidt, Jonathan. “The emotional dog and its rational tail. A social intuitionist approach to moral judgment”. Psychological Review 108, 4 (2001), 814-834. https://doi.org/10.1037/0033-295x.108.4.814

Halford, Graeme S., William H. Wilson and Steven Phillips. “Processing capacity defined by relational complexity. Implications for comparative, developmental, and cognitive psychology”. Behavioral and Brain Sciences 21, 6 (1998), 803-831. https://doi.org/10.1017/s0140525x98001769

Hume, David. A Treatise of Human Nature. London, John Noon, 1739-1740.

Huron, David. “Voice denumerability in polyphonic music of homogeneous timbres”. Music Perception 6, 4 (1989), 361-382. https://doi.org/10.2307/40285438

Kahan, Dan M., Ellen Peters, Erica Cantrell Dawson and Paul Slovic. “Motivated numeracy and enlightened self-government”. Behavioural Public Policy 1, 1 (2017), 54-86. https://doi.org/10.1017/bpp.2016.2

Laplace, Pierre-Simon. “Mémoire sur la probabilité des causes par les événements”. Mémoires de mathématique et de physique présentés à l’Académie royale des sciences par divers savans 6 (1774), 621-656.

Łukasiewicz, Jan. “O logice trójwartościowej”. Ruch Filozoficzny 5 (1920), 170-171.

Łukasiewicz, Jan, and Alfred Tarski. “Untersuchungen über den Aussagenkalkül”. Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, classe III, 23 (1930), 30-50.

Mani, Anandi, Sendhil Mullainathan, Eldar Shafir and Jiaying Zhao. “Poverty impedes cognitive function”. Science 341, 6149 (2013), 976-980. https://doi.org/10.1126/science.1238041

Melby-Lervåg, Monica, and Charles Hulme. “Is working memory training effective? A meta-analytic review”. Developmental Psychology 49, 2 (2013), 270-291. https://doi.org/10.1037/a0028228

Mercier, Hugo, and Dan Sperber. “Why do humans reason? Arguments for an argumentative theory”. Behavioral and Brain Sciences 34, 2 (2011), 57-74. https://doi.org/10.1017/s0140525x10000968

Mullainathan, Sendhil, and Eldar Shafir. Scarcity. Why Having Too Little Means So Much. New York, Times Books, 2013.

Nendaz, Mathieu, and Arnaud Perrier. “Diagnostic errors and flaws in clinical reasoning. Mechanisms and prevention in practice”. Swiss Medical Weekly 142 (2012), w13706. https://doi.org/10.4414/smw.2012.13706

Peirce, Charles Sanders. Collected Papers, volume 5, edited by Charles Hartshorne and Paul Weiss. Cambridge (Massachusetts), Harvard University Press, 1934.

Sala, Giovanni, and Fernand Gobet. “Working memory training in typically developing children. A multilevel meta-analysis”. Psychonomic Bulletin and Review 27, 3 (2020), 423-434. https://doi.org/10.3758/s13423-019-01681-y

Sanguinetti, Giorgio. The Art of Partimento. History, Theory, and Practice. New York, Oxford University Press, 2012.

Tversky, Amos, and Daniel Kahneman. “Judgment under Uncertainty. Heuristics and Biases”. Science 185, 4157 (1974), 1124-1131. https://doi.org/10.1126/science.185.4157.1124

Wiradhany, Wisnu, and Mark R. Nieuwenstein. “Cognitive control in media multitaskers. Two replication studies and a meta-analysis”. Attention, Perception, and Psychophysics 79 (2017), 2620-2641. https://doi.org/10.3758/s13414-017-1408-4

Wittgenstein, Ludwig. Philosophische Untersuchungen. Oxford, Blackwell, 1953.

Wright, Georg Henrik von. “Deontic Logic”. Mind 60, 237 (1951), 1-15. https://doi.org/10.1093/mind/LX.237.1