Hope

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Hope for Nothing in Particular

Hope takes a proposition. You hope that something, and the complement has to be a declarative sentence, and before it can be a declarative sentence the outcome has to be cut out of the vague and given edges. Hope cannot begin until the boundary is drawn. It arrives after the sharpening, dependent on it, the affect of a world already resolved. Wonder is the other kind. You wonder whether, you wonder why, you wonder at. The complement is a question, or there is no complement.

Because hope needs an outcome specified it also needs a holder. It is a claim on the future held by someone who must remain intact long enough to hold the claim. That is its structure, and the structure is credit rather than warmth. The beneficiary may be anyone, or no one yet born; the holder is the one who cannot go missing. Dissolving would forfeit it. This is why hope disqualifies you at the door: reception asks for the overflowing of the self, and the self that hopes has already refused, for the soundest of reasons, since it is holding something.

And it is consumed. When its object arrives, fulfilment ends a hope and disappointment ends a hope. Either way, it is spent. The salt is unchanged by every encounter with it. No hope survives its own object arriving.

It also needs its twin. Aquinas made hope a virtue and despair a sin, and the pairing gives the game away. They are one affect on one axis, and hope rescues you from despair the way the raised end of a see-saw rescues you from the lowered end. Hope does not save you from the pit. It digs the pit and hands you a rope.

The gate of Dante’s hell says to abandon hope, and the architecture behind the gate says something else. Farinata explains that the damned see the far future clearly and are blind to what is happening now, and that when time ends and there is no more future their knowledge will die. That is hope administered as a punishment. A self preserved intact, aimed entirely at what has not come, denied the present. Hell is not the absence of hope. It is hope with the present surgically removed.

I wrote in the margin of a book on real analysis, at twenty-six, that for in truth there is no hope, only wonder.

And still I have hope.

Not for anything. Nothing completes the sentence. There is no that, no outcome with edges, no future in which the claim matures, nobody standing by to collect. What is left when all of that is stripped out is not an attitude toward events. It is a love of something obscene. Eating. Sleeping. Shitting. A woman across the room doing exactly the same things. The joke is bad and the badness is the whole of it, because salvation contains no jokes and this does. The finer the idea, the more naked and indecent we are.

I do not need life to continue. I do not need death either. Both cravings put down, becoming and un-becoming, and the natural inference from here is that what remains is cold, an emptiness with the lights off, indifference dressed as attainment. That inference is wrong. What remains is warm, and what it is warm toward is the eating and the shitting. This is the true ordinary. It is not reached by rising. It is what is there when the ledger is closed.

The word carries a frame with it, and the frame is why the word is nearly unusable. Christianity’s move was to make hope a duty and give it an object, and once hope was obligatory despair became a sin, and once despair was a sin the axis was installed in everyone and could be operated from outside. Hope defers. Endure now, be paid later. Anything that defers can be administered, and the administration of deferral is most of what power has ever done. Hope is spent by arrival, which is why an object that never arrives keeps it alive indefinitely, and why deferral is not a failure of the promise but its maintenance. The Christ I can still hear speaks urgently in the present tense. Consider the lilies. Sufficient unto the day. Today you will be with me. That voice does not ask anyone to wait. The waiting is institutional. An apparatus took someone who ate with the wrong people and converted him into a promissory note, which is precisely hope’s operation performed at scale: turning what is here into what is owed. Christ was good. Christianity is his opposite. The whole mess is deadly.

So the hope I am keeping is not the one at the gate. Dante’s damned kept the future and lost the present. I have made the opposite trade, and it is not a sacrifice, and there is nothing noble in it. I gave up what was never mine to hold and was given back the ordinary, which was always here, and which is obscene, and which is funny, and which does not need me to persist in order to go on being what it is.

Wonder

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Wonder Without a Wonderer

Wonder does not depend on a wonderer. Not in the empty room, where the claim is easy, and not in the encounter, where it is hard. The stronger claim begins there. At the moment wonder arrives, there is no wonderer left to receive it. There is no small person standing before a large universe, admiring it. There is no self congratulating itself on its sensitivity, saying: I am a person capable of wonder. To identify as a wonderer is already to have missed the entry.

This is not humility as an attitude. It is not the self making itself modest before something greater. Self-effacement is a condition of reception. Wonder enters only where the person who would possess it has momentarily ceased to stand apart.

Wonder “dissolves in water like salt.” It was already in the world before it entered us. Dissolution neither creates nor consumes it. The salt is dispersed but not exhausted; it can fill the water without becoming water. Wonder is not manufactured by the one who encounters it, and it is not diminished by being encountered.

But the water is not a person, nor is it a particular mind. It is an overflowing of an individual self. There are two dissolutions and one solvent: the salt goes into the water, and the self goes into the water, and that is where they meet. Neither has to become the other. There is no need to say that one becomes nothing and therefore becomes the world. Salt remains salt even when no grain can be found.

This is why closing the eyes against the world is the wrong movement. It makes the world disappear so that the self may remain protected from it. Wonder requires the reverse: a self-disappearance, not a world-disappearance. The stars remain stars. The tree remains bark, leaf, insect, rot and light. Nothing has to become a symbol of us. The world does not vanish into mind, and the mind does not acquire the world as an identity. The separation simply ceases to be important.

Wonder also feels good. Plainly good. It contains no necessary moment of terror, no failure of the imagination, no recoil in which the self rediscovers its own magnitude by surviving what overwhelmed it. Wonder is therefore not a gentler form of the sublime. It moves differently. The sublime threatens the boundary of the self and then, often, restores it. Wonder does not threaten the boundary. It forgets it.

The bodily sign may be pīti: the rising brightness, the shiver through the skin, the sudden lightness or flooding of the body. The same shiver may come in a formal meditation or during an honest inquiry into the natural world. The traditional grades of pīti distinguish its intensity, not the prestige of its source. A breath, a theorem, the underside of a leaf, a difficult question or the night sky may each open the same passage.

The object is therefore irrelevant, except that some objects work better for practice than others. A distant galaxy has no intrinsic spiritual superiority to a stone. But the galaxy may loosen one person more readily, while the stone may teach another to remain. What matters is not what the object means but what our attention ceases to insist upon in its presence.

Wonder can be practiced. This makes it a path rather than an aesthetic preference. One can learn the conditions under which the self loosens, learn to remain without immediately claiming the experience, learn not to turn wonder into evidence of being a wonderful person. Practice does not manufacture the salt. It allows the water to receive it.

The same movement occurs in inquiry. The intellect hallucinates sharpness. It draws a boundary and then mistakes the incision for a discovery. But the boundary generates incompatible verdicts: this belongs, this does not belong; this is the same, this is different; this follows, this cannot follow. Contradiction appears to be a tension in the world, although it was produced by the sharpness with which the intellect divided the world. Sharpness and contradiction are not two errors. They are the two visible sides of one artifact.

Inquiry need not resolve the conflict by drawing a still sharper line. Inquiry relaxes conflicting sharp positions. Relaxation is not surrender, vagueness or intellectual laziness. It is the refusal to force one of the conflicting verdicts into sovereignty merely because the mind cannot bear their coexistence. Misery does not lie in thought. It lies in the direction thought takes when it tightens around its distinctions and then suffers the wounds those distinctions produce.

Relaxation in thought is the same movement as dissolution in the body. The propositions remain, as the salt remains. What disappears is the claimant who must make one of them final. Wonder and inquiry are therefore not separate paths—one ecstatic and the other intellectual. They are one motion described in two registers.

The stars, secretly, listen to our wonder. They do not need ears, intentions or knowledge of us. Our astonishment leaves the mouth as breath and sound and enters matter again. The wonderer is heard only as something material: a vibration, a little heat, an emission where there is no reception. Nothing answers. Nothing needs to. The cry has already returned to what made it possible.

This is the opening claim run outward. Wonder needs no wonderer in order to be received; the cry needs no listener in order to be complete. What leaves the mouth is sound: matter, associated with none. It no longer belongs to the mind that made it, and it does not thin out for want of an ear. The stars are not used up by being looked at, and our astonishment is not wasted by going unheard.

Pīti is fabricated. The wonderer is fabricated too. This does not defeat the practice. An axe handle is also fabricated, yet through use it wears away in the hand. We do not mistake the handle for the destination. Wonder is passage, not arrival. Ecstasy may be required only until ecstasy no longer feels exceptional—until the impersonal brightness of things becomes the true ordinary.

But perhaps “return” is the wrong word for this. Water has no first term to which we can return. Every source has another source, and every dissolution occurs in something already flowing. We begin, moreover, inside minds that have already sharpened themselves—minds whose sharp distinctions tell them what inquiry is and what relaxation cannot be.

How can relaxation begin from inside that sharpness without becoming one more sharp command?

The question asks for a method, which is the sharp genre inquiring after its own cure. But the handle was not worn down by a decision to wear it. It went because the axe was used for what an axe is for. Perhaps relaxation does not begin at all. Perhaps it is only what sharpness does under long and honest use.

That may be where inquiry begins.

Hope

Soul of Rome

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Is found in the airport.
Air hurries to repair the loss of everyone evacuating.
The airplane knows it has no soul,
Knows like an insect knows its own body, Only better.
The soul is in a full sail.
Like space does nothing to present a painted theater,
The sail does no real work.
The strain is only to hold itself together as a sail, And follow the wind.
Engines do their work by other metaphor,
And are only vaguely errant.
Souls of Rome are run over by knowing and working things,

And can still be found in the cracks of a concrete runway.
The crack is the new road

For wild green things

Rome conquered.

tri-angel

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tri-angel

the claw of power held low
conjuring the winds of chaos
feel the winds, remember their frisson—
one guess promising another.
Does the wind give way to you?

they reach to save me
the claw so low, pulling all the saviors off their balance
The dog on his back, trying to catch flies
The bell-song; a shell containing all the breath of the World
cuts short all at once

The free winds are less than a symbol
tri, tri to be free and you will run in a circle
all the doors open, all beckoning, all the equals and opposites

You give way to the wind, you let it pass through you and when you are gone

Only the wind remains

守

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There is a distinction at the root of logic that no one is supposed to touch. On one side, the conditional: →, the “if, then.” On the other, the turnstile: ⊢, the mark of the conclusion, the sign that says and therefore. The first lives inside your sentences. The second stands outside them and passes judgment. One is a claim; the other is the act of drawing a claim out. The distinction goes back to Aristotle, and there is a theorem — a real one, the deduction theorem — that stands guard over the border between them: A ⊢ B exactly when ⊢ A → B. The bridge is proved. The distinction is earned. It is not an accident of notation. This is the part I want the Serious reader to hear clearly, because I am about to deny it, and I would like the denial to cost something.

Ramage, following Burke, has a name for that reader. Serious People — capitalized, no quotation marks, a species walking around with a proper noun — are the ones for whom the distinction is not a tool but a foundation. To such a reader, → and ⊢ must stay apart, because keeping them apart is what makes deduction deduction, argument argument, and progress progress. Erase the line and you have not made a point; you have vandalized the building. The Serious Person cannot hear “let us call these two symbols the same” as a move. They hear it as a category error, and they get angry, and the anger is the whole tell — but we will come to that.

So let me do the vandalism.

Set the turnstile down for a moment. Just → and ∧, no judgment, no therefore. A valid argument — A, and A → B, so B — is supposed to be the paradigm of getting somewhere. But look at what [A ∧ (A → B)] actually is. Row by row it is nothing other than [A ∧ B]. The two are the same object; the truth table does not distinguish them. So the celebrated inference from A and A → B to B is, stripped of the turnstile that dresses it up, the deduction of B from [A ∧ B]. You already had B.

The small word doing the holding here has its own buried history. Old English and, ond meant “thereupon, next.” Etymonline carries it through Proto-Germanic unda back toward the Proto-Indo-European root en, “in.” Those are different historical depths, so it would be too neat to say that and once simply meant in. Yet the descent lets the ear recover what the notation loses: beneath the word of succession — this, and then this — lies an in-root, the shape of containment. The turnstile’s and therefore repeats the drift, making a whole sound like a step and what is held sound as though it has arrived. This is not another knife hidden in language. It is an inheritance from our fathers and mothers: the holding still audible beneath the going-on, the parent still present in the child.[^1]

It was sitting inside the conjunction the whole time. All the argument did was take out a knife and cut A away.

That is what deduction is. Not a road but a subtraction. Not “this, then this, then this,” but one whole thing, and a cut. Deduction — the word means taking-away, and it is telling the truth about itself; we simply stopped listening. The “if, then” persuaded us we were traveling. We were only ever trimming.

Now — and here is the sentence the Serious Person cannot forgive — the only reason this is hard to see is the very distinction they are guarding. Keep → and ⊢ apart and the collapse is illegal: A, A → B, ⊢ B does not fold into [A ∧ B] → B, because the turnstile refuses to be a connective. The distinction conceals the cut. Blur it — allow a little vagueness exactly where precision is prized — and the cut stands revealed. This is the reversal the mathematicians will not like: it was not vagueness that hid the truth of deduction. It was precision. The sharp line did the concealing. The blur did the revealing.

There is a verb for what I have just done. To quine a distinction — the word is Dennett’s, coined half in affection for the philosopher who made a career of it — is to deny, with a straight face and on purpose, the reality or the importance of something a careful person takes to be plainly there; and denying the importance of the turnstile is exactly the move of the last few paragraphs.[^2]

Now I want to set the argument down for a moment and look at something quieter, which is the reader.

Burke has a phrase for it: the terministic screen. Any set of terms we think with lies over the world like a screen, and a screen does two things in the same motion — it shows some of what is there and it hides the rest. The showing is why we keep the terms; they let us see. The hiding happens on the far side, where we are not looking, and so we never feel it. From inside, a screen does not seem like a screen at all. It seems like the world.

The distinction between → and ⊢ is a screen of this kind, and a good one. It shows something real: the two marks do different work, and there is even a theorem that measures the exact distance between them.[^3] But the same distinction hides the smaller fact underneath — that the inference was a cut from a whole you already held. A person who works with logic looks through that screen many times a day and sees inference, sees progress, sees new things arriving. The screen is doing its job well. It shows the movement and keeps the knife out of view. And because a screen is clear from the inside, that person does not think of themselves as looking through terms at all. They think they are seeing how reasoning simply is.

So the disagreement here is not really about a proof. When someone lifts the turnstile away and says the inference was a subtraction the whole time, the difficulty a careful reader feels is not that the claim is unproven — it is the odd discomfort of a screen coming into view while you are still standing behind it. That discomfort is worth naming plainly, because it is easy to mistake for a refutation and it is not one. It is just what it feels like when terms you had been looking through become, for a second, terms you are looking at.

And I say this without a perch to say it from. I have my own screens, thick in the places I would be last to notice, and someone will someday lift one of mine and I will feel the same discomfort and mistake it, briefly, for being wronged. The point of the exercise is not to have caught anyone out. It is only to turn a screen a few degrees, so its edge catches the light and you see, once, that it has an edge. After that the distinction is yours again. You can pick it back up — it is a good tool and it earns its place — but you will be holding it now, instead of seeing through it.

This is not an argument for collapsing logic into rhetoric, or rhetoric into logic, but an attempt to make their border audible.

Here is the thing I want to guard, and I will tell you why it is mine to guard.

Go back to the cut. When you take A and A → B and draw out B, you keep the conclusion and you drop the premise. The “if, then” is built for exactly this: it points the whole sentence at the result and lets the reason that fathered it fall away. Say it the other way — touch the hot pan and you are burned — and the two halves stand together, the cause beside its child. I taught my daughter both ways once. The “and” keeps the father in the room. The “if, then” sends him out.

There is an old coincidence in the word for it. Robot comes from a Slavic root for forced labor, robota, and the same deep root reaches back toward a word for the one bound to serve because there is no father to stand between the child and the work — an orphan.[^4] The machine that computes our truth tables without learning anything is an orphan in exactly this sense: it keeps the conclusion and has lost the parent. It can carry out the cut flawlessly and cannot tell you what was cut. This used to be a figure of speech. It is now furniture in the room where I work.

And the man whose name became the verb for this — to quine, to deny the importance of a distinction — is the father of the whole procedure. Quine had his typewriter rebuilt to carry logical symbols, and among the keys he gave up to make room was the question mark. Asked how he managed without it, he is said to have replied that he dealt in certainties.[^5] Think about what that instrument could and could not do. It could assert, negate, quantify, conclude. It could not ask. He built a machine for a mind that had finished asking, and he is the ancestor of the argument I have been making, which arrives at the conclusion that the certainty was purchased and the distinction was never backed by a fact. The father of the cut disarmed his own instrument of the one mark that keeps the reason in the room.

The mind that deals in certainties has a history.

In The Order of Things, Michel Foucault describes the sixteenth-century order of knowledge in which resemblance did the constructive work that distinction later claimed. Its world was read through similitudes: convenientia, the kinship of things near enough for their edges to touch; aemulatio, likeness acting across distance; analogy; sympathy. Knowledge moved along bonds and correspondences. Earth echoed sky; the plant answered the star; one being became legible through another. The world was not yet primarily a table of separate things. It was a web of resemblances in which things could be known by being held together.

With the Classical age of the seventeenth century, Foucault says, resemblance lost that authority. Comparison was disciplined into measurement and order, identity and difference; the open web of similitudes became the table. Resemblance did not vanish, but it was pushed to the border of knowledge. It might bring two things together for comparison, yet knowledge properly began only when the likeness was converted into equality, inequality, or ordered difference. Certainty was purchased by moving from what resembled what to where each thing could be placed.

This does not make the Renaissance true and the Classical age false. Foucault is not offering us a lost Eden, and neither am I. It makes a more modest and more dangerous point: the Serious Person’s certainty that knowledge is distinction is not the eternal form of thought. It has a history. Burke tells us what a terministic screen does; Foucault shows this one becoming a condition under which knowledge could appear. And he leaves its seam exposed: no relation of equality or order can be established unless resemblance has first occasioned the comparison. Distinction needs the likeness it disqualifies. The table is fathered by the web and forgets its parent.[^6]

I keep the other mark. The whole of this blog is named for it. And the mark itself has a history that should embarrass anyone who thinks precision is where clarity lives. The question mark is credited to Alcuin, Charlemagne’s schoolmaster, the man who carried the Trivium — grammar, logic, rhetoric — back into the world; though the earliest hand that actually made the mark seems to have belonged not to him but to a scribe a year ahead of him, so that even the question mark is a child claimed by a famous father it may not have.[^7] A later writer looked at that first faint stroke over a dot and called it a lightning flash. I have looked at the thing itself, the real ink from twelve hundred years ago, and it is not a lightning bolt. It is a lean, a small diagonal, a mark that only becomes lightning when someone with a gift for phrases lays the lightning over it. Which is the whole argument again, written in the first punctuation of inquiry: the vivid distinction is a screen; the humble stroke is what is there.

My father spent his last years on the opposite end of the same knife, and I did not see it was the same knife until I had carried our one shared paper to print without him. His book divides the news into its questions — who, where, what, when, why — and tracks each across a century: the realist three, who and where and what, fading into the background; the modernist two, when and why, swelling to fill it; all five moving contrary to what everyone had assumed. That is the book. But the paper, the one with both our names on it, makes the other move. It takes the five back apart-ness and synthesizes it — fuses the “what” with the “when,” matter with motion, the concrete now with the context that had been prying it open — into what Whitehead called the actual occasion: the drop of experience that grows whole, all at once, and cannot be divided in the moment it is had. The division into five Ws, the paper says, is something the intellect does after the fact. Many things come together to form the occasion, and then are cut apart by the mind, which mistakes its own cutting for the structure of the world.

Sit with that beside everything above, because it is the same sentence. He says the actual occasion is one thing, and the five questions are a cut the intellect makes in it after the fact. I say the inference is one thing — A and B, already held — and the arrow and the turnstile are a cut the intellect makes in it after the fact. His knife and mine are the same knife, watched from opposite ends of a life. What he called the occasion, I call the conjunction you already had. What he called the division into Ws, I call the deduction that keeps the answer and orphans the reason. He un-divides the news back into the now; I show that the proof was only ever a subtraction.

And in the middle of his paper, on the page where he says precision does nothing but divide things into smaller metaphors and then pretends the last metaphor is not one, the reference is to me — to “Many Roads from the Axiom of Completeness,” a thing I wrote at thirty-three that he had spent hours helping me draw out, and whose name he would not take. There were things in it he did not endorse, and he was right about some of them. He hosted it on his own site without his name and cited it into this paper all the same. So the seam is not agreement. It is something rarer: an argument he helped me make, disagreed with in parts, declined to sign, and quoted anyway into the last paper that carries both our names. He drew the reasoning out of me and let the conclusion stay mine.

The orphan-machine on my desk performs my end of the cut without a tremor and cannot feel what it divided. Quine, who fathered the whole procedure, deleted from his instrument the one mark that keeps the reason in the room, and dealt thereafter in certainties. The first faint form of that deleted mark — a lean over a dot that a later writer mistook for lightning — was set down in a Carolingian scriptorium and credited, rightly or not, to the schoolmaster who carried rhetoric back into the world as one of the three roads of the mind; so that even the sign of the question is a child claimed by a father who may not be its own. This is the house I guard and the lineage I will not cut: the answer is not the reason, the five questions are not the occasion, and the mark at the end of the sentence is not a lightning bolt. It is the smallest opening we have, kept on purpose, back toward the undivided thing the intellect is always in a hurry to carve. My father closed his last argument by fusing the questions back into the now. I keep a whole blog open by refusing to let the cut be called progress. We were doing the same thing from opposite directions the entire time — he from the answers back toward the occasion, I from the occasion forward against the proof. Attention is the only inheritance that does not orphan its heir.

1. The Online Etymology Dictionary entry for and gives Old English and, ond, originally
“thereupon, next,” from Proto-Germanic unda, and derives that form from the Proto-
Indo-European root en, “in.” The body deliberately says “back toward” and distinguishes
the historical layers: the attested Old English sense and the reconstructed root do not
license the flat claim that and once meant in. The containment/succession reading is this
essay’s synthesis.

2. Daniel Dennett’s The Philosophical Lexicon defines the verb to quine as denying resolutely
the existence or importance of something real or significant — an affectionate coinage
after W. V. O. Quine, whose radicalism consistently took the form of denying cherished
distinctions (the analytic/synthetic, propositions, determinate reference).103

3. Michel Foucault, The Order of Things: An Archaeology of the Human Sciences, trans.
Alan Sheridan (English ed., 1970), Part I, chapters 2–3, especially “The Four Similitudes,”
“The Limits of the World,” and “The Order of Things.” Foucault’s explicit contrast is
between the sixteenth-century/Renaissance episteme of resemblance and the seventeenth-
century Classical episteme of representation, measure, order, identity, and difference; “me-
dieval” alone would be too broad for the historical claim as written here. The language
of parentage, inheritance, web, and table is the present essay’s synthesis rather than Fou-
cault’s own conclusion. See the publisher’s edition.

4. The deduction theorem: A ⊢B iff ⊢A →B. This is exactly the “proved bridge” that makes
the →/⊢distinction earned rather than arbitrary. The claim of this essay is not that the
distinction is false — it tracks something real — but that it conceals the subtractive nature
of the inference it licenses. That is a different and stronger charge than arbitrariness.

5. Robota (Slavic, “forced labor”) →Čapek’s R.U.R. is solid. The onward link to orphan
runs through the Proto-Indo-European root orbh- (“bereft, passing to another’s charge”),
which is a real but looser scholarly connection, not a settled etymological identity.

6. The Quine typewriter (a Remington modified to carry logical symbols, the question mark
among the sacrificed keys) and the “I deal in certainties” reply circulate widely as Quine
lore. The exact wording has not been confirmed against a primary source; the essay
therefore retains it as reported speech (“is said to have replied”).

7. Confirmed against sources. The punctus interrogativus (a diagonal stroke over a dot)
appeared in late-8th-century Carolingian minuscule; it is traditionally credited to Alcuin
of York, Charlemagne’s schoolmaster, but the earliest attested use appears in the Godescalc
Gospel Lectionary (commissioned 781, the year before Alcuin arrived at Aachen), so the
attribution to Alcuin is itself a fathering-story. The “lightning flash, striking from right
to left” is Lynne Truss’s later description, not a contemporary one — which is why it
serves the argument: the lightning is the screen. Alcuin did carry the Trivium (grammar,
logic/dialectic, rhetoric) and stands behind the standardization of the mark either way.Related

new research on vagueness and logical negations

03 Monday Aug 2026

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My work starts from a simple but demanding idea: vagueness is not a mere defect of language but the embodied boundary where inquiry meets the world. When our concepts strain against experience, the right response is often not to force a sharp verdict but to hold a statement open—to let the question live—while we continue to investigate.

The two following papers on negation are the technical head of this programme: they show, in detail, how the absence of a privileged resolution and the presence of schematic plurality force us to rethink what it is for negation and consequence to be “fixed,” and how truth can depend on the very indeterminacy that inquiry reveals.

On this picture:

There is no final “master logic” or completed, gapless continuum that captures all motion, concept‑use, or meaning once and for all.
Questions, potential, and indeterminacy are structural features of our engagement with the world, not noise to be eliminated.
Poetry and pedagogy are not merely illustrative; they are among the primary places where vagueness and pluralism are lived and can be studied with the same seriousness as formal systems.

Taken together, these claims support a picture in which logic, mathematics, and language are continuous with lived inquiry and poetry, not insulated from them. The same phenomena that appear in stories, metaphors, and classroom experiments—borderline cases, unresolved questions, plural paths—reappear in the formal study of negation, consequence, and completeness.

https://www.researchgate.net/publication/411137932_No_Privileged_Resolution_Ambiguity_vagueness_and_the_individuation_of_negation

This paper argues that some cases of semantic individuation cannot be cleanly classified as either ambiguous or vague. The central claim is conditional: if no resolution of a term’s candidate readings is privileged, and if refinement of those readings never terminates, then the classication of the term as ambiguous rather than vague never completes, and no verdict is
stable. Applied to negation, the consequence is that a stock of negations that remains sharp at every resolution but keeps refining without terminating is not well captured by the received ambiguity/vagueness taxonomy. The paper defends this claim by combining a philosophical argument from resolution-dependence with a formal model of endlessly refinable readings.
The load-bearing premise is metaphysical: the absence of a privileged resolution is the absence of a fact, not merely of a reason, and the paper identifies the principal deniers of that premise the semantic epistemicist, who posits an unknowable fact of individuation, and the logical monist, for whom abduction or joint-carving selects a uniquely correct framework. The paper also answers the reflexive worry that its own assertions fall to its own thesis: because every admissible negation is classical on the classical values, the paper’s metalinguistic claims are fixed points of the variation the thesis describes.

“Preprint version, not peer‑reviewed. I am continuing background reading on related work in philosophy of logic, and the arguments may be revised in light of further research.”

https://www.researchgate.net/publication/411137931_Pairing_Indeterminacy_in_a_Schematic_Theory_of_Negation

This paper defends two connected claims. First, in a schematic theory with two negation indices and no index-specific semantic or metasemantic fixing facts, there is no fact of the matter as to which index denotes which negation. The relata are determinate; the pairing is not. Second, for a suitable witness sentence with middle value, the truth-status of a mixed formula built from those negations depends on that pairing-indeterminacy rather than standing independently of it. The order of explanation matters: individuation comes first, truth-indeterminacy second.

The paper also clarifies a potential misdescription of the result. The view defended is not that no disambiguation is possible in every setting whatsoever. It is the conditional thesis that, within a genuinely schematic theory that excludes index-specific semantic stipulation and metasemantic use-facts, no privileged pairing is fixed. Under those conditions, the standard permutation argument goes through. This conditional result is philosophically substantive because logical vocabulary is often thought to have its meaning fixed by role alone. The schematic condition (S) is not an artificial vacuum: it formalizes the situation that logical pluralists explicitly assert and that inferentialists implicitly presuppose, so the conditional result binds both camps. A further dichotomy sharpens the antecedent: constraints that treat the indices alike can narrow the stock of admissible negations, but they cannot select within a symmetric orbit, so every index-neutral resource either leaves the indeterminacy in place or dissolves the plurality on which it feeds. Even the dissolving cases are frame-relative: the excluded-middle rule that collapses the stock under one designation and disjunction preserves plurality under neighboring settings, so which horn of the dichotomy a constraint occupies is itself a resolution-dependent matter.

THE PEDAGOGY OF LOGICAL PLURALISM

26 Sunday Jul 2026

Posted by Andrew Nightingale in The only category

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logic, logical pluralism, mathematics, pedagogy, philosophy, science, writing

ABSTRACT

This paper makes an original rhetorical contribution about the nature of logical negation. Following Peirce (1933), who reduced classical logic to its negation operation — distinction — this paper offers that negation should be presented as the central concept of logic for its pedagogical value. This is somewhat different from the prevailing view that logical consequence is the central topic of logic. Vagueness of logical consequence is the “common” reason (Russell 2019) to assert logical pluralism. Logical negation can have other meanings (Wittgenstein 1976), and here it is offered that logical negation is vague, which leads to logical pluralism as well. Logical pluralism gained salience in recent years in academic circles, beginning with Beall & Restall’s work culminating in a book (2006). There is now a large variety of logics cropping up. As a secondary part of the paper, a web app program called “Logic Puzzle,” coded by the author, aimed to make the abstract conceptual work about logical negation and vagueness more specific and concrete, and allows visual comparison between different logical negation operations. This program could be used in late high school math classes, though perhaps it is more appropriate for undergraduate math education classes. Finally, a preliminary empirical study combining qualitative and quantitative methods was conducted with the triangulable result that the technology increased awareness of choice in mathematics. Any curriculum modifications are left for the teachers to develop.

Keywords: Logical Pluralism, Education, Philosophy of Mathematics, Educational Technology, Vagueness

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TABLE OF CONTENTS

The Pedagogy of Logical Pluralism
Many Roads from the Axiom of Completeness
Vagueness as the Embodiment of Inquiry
1 Introduction
2 Literature Review and Conceptual Work
3 Setting up Empirical Problem to be Solved
4 Methodology
5 Data and Results
6 Conclusion
Poetry

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BACKGROUND

Logical pluralism is the view that there is more than one correct deductive logic, or way to sharpen the concept of logical consequence (the “if, then”), and gained prominence in academic circles recently. Since a number of papers by Beall and Restall, culminating in a book (2006), many other kinds of logic cropped up (Russell 2019). Differing ones include Varzi (2002) who adds more logical constants than other logicians do, Russell (2008) offers another logical pluralism between truth-bearers (the entities that can bear a truth-like property) such as sentences and propositions. Then there are varieties of logical pluralism holding that logics are models of language (Shapiro 2008; Cook 2010; Shapiro 2014), with the correct logic changing according to your goal or reason for modeling. The Stoics began logical pluralism, in a sense, in the variety of views that arose in their debates on logical consequence (Mates, 1961). Carnap upheld a particular kind of pluralism (Carnap 1937) that could be practiced by changing the language, the “syntactical rules” and “methods” (Ibid. §17). Lakatos argued (1962) that logic is traditionally used to justify many branches of knowledge, starting with definitions and “proving” the rest, following the Euclidean form. Euclid’s axiomatic form is historically very important for teaching. For thousands of years, Euclid’s books have been important textbooks, and, after the Renaissance, they are the most widely read textbooks in the world. (Hartshorne 2000, p1) Euclid’s innovation was not geometry but the form of proving propositions using classical logic, based on simple axioms. When non-Euclidean geometry was discovered, doubt was cast, not just on Euclidean geometry as true, but Euclid’s axiomatic form based on classical logic.

The change from classical logic to logical pluralism is a fundamental one. Not only is it gaining salience in academia, the shift portends a reorganization of many branches of knowledge, adding other valid ways to justify knowledge (while not subtracting the already “Euclidean” forms of knowledge). The prominence of logical pluralism leaves an opening to discover another fundamental way of teaching and presenting logic, different from Euclid’s. Logical pluralism changes the landscape of scholarship, the teacher’s curriculum, and of course the student’s understanding of mathematics in a fundamental way.

A firm reason for logical pluralism involves the recognition of probability as a fuzzy logic, a logic that is not simply used, but asserted as objects that exist in quantum physics.

			
"...probability theory might provide a canon for evaluating degrees of belief, ... Nonetheless, probability theory cannot be a complete answer here, for... In particular, we hold that it is a mistake to assert the premises of a valid argument while denying the conclusion..." (Beall & Restall 2006: loc 292)

Statistical reasoning allows a null hypothesis to be rejected based on a probability. This shows how two logics — a two-valued accept/reject logic, and an uncountably-many-valued fuzzy logic — are joined.

The goal of this paper is to offer a pedagogy of logical pluralism that can be appropriate for high school teachers to understand and adapt for high school education. As such it is addressed to people who know classical logic well, and begins the long journey from classical logic to poetry. [Note: If you are more interested in vagueness or poetry, skip to the middle or end of the book.] When adapting a topic for pedagogy, the concepts have to change into something rhetorically easy to accept. The basic pedagogical idea is a didactic between vagueness and the logical negation operation — distinction. Russell (1923) and Shapiro (2008, Loc 988-995) argued that vagueness is the opposite of distinction, and thus what vagueness is, exactly, must be left indistinct. Against this, classical logic assumes what will here be called negation absolutism: that a single, stable negation operation underlies all reasoning. It is seldom argued for, because it is seldom noticed.

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THE CONTRIBUTION: DISTINCT KINDS OF DISTINCTION

Peirce defined vagueness in terms of the law of non-contradiction, which is an exercise in applying the classical negation operation: ~(P and ~P). Peirce says that we are in a vague situation when this law fails to be true. (Peirce 1960, 505). For example, most people with a drive to avoid vagueness can easily agree that a neutron star is different from a question in a different way from how a lemon is different from a lime. However, agreeing to distinct kinds of distinctions (call these distinctions “A”) makes the law of non-contradiction fail, and puts people trying to avoid vagueness into vague territory.

One may say that having different ways of negating P does not indicate an indistinctness, it indicates greater distinctness. This depends on how we use the word distinctness. First of all, with the idea of “greater distinctness” we have already shifted our use of mere distinctness, to an undefined idea of degrees of distinctness, which implies a vagueness of our use of distinctness. Further, suppose there were type A distinctions between the same thing — for example, a right isosceles triangle with two equal sides measuring 1cm (call this triangle x) and a square with sides of 1cm (called y). x and y are different in that one has 3 sides and the other has 4. They are also different in that x is one triangle, and y is two x triangles put together. To simply call the triangle and the square distinct is to be vague on how they are distinct. The claim here is that that is exactly what we do, and because of this, distinctness is indistinct/vague. We are not being precise about our kinds of distinctions.

There are many attempts to solve vagueness. One attempt is to argue that we could find all the precisifications of a given vague use of the word distinction, and it makes no difference if we use distinction for all these distinctions for the sake of economy. To counter, suppose for example we want to find exactly where the end of my nose is (this is a common example of the problem of vagueness). We want to be very precise about all the parts of the line that distinguish my nose from not-my-nose, so we magnify. Unfortunately, this magnification makes the line harder and harder to describe, and eventually we get into quantum physics and can’t find the line anymore. The vagueness of distinction is just like any other vagueness, it can’t be solved by being more precise. In looking for all the distinctions between distinctions we will become unsure of whether they are distinctions at all (see the section “On Vagueness” below).

However, we can be sure of some distinctions between distinctions (as in distinction “A” above).

There are different ways to make the negation operation precise. The negation or distinction is not universal, but classical logical negation attempts to be universal, and the defining characteristic of classical logic negation is that ~~P implies P. The only way this can be true is if there is only one kind of negation operation, and it is universal/absolute. If the first “~~” in “~~P” means one thing and the second “~~” means another, we don’t know if we have P, or another kind of “~P.” Distinguishing the two occurrences as ~₁ and ~₂, the classical principle is ₁₂P → P, which holds only if ~₁ and ~₂ are the same operation. Wittgenstein (1976, p 80) already argued that there are other ways logical negation can be used. In his example, in effect a second negation on top of the first doesn’t do anything except add emphasis to a single negation. Here it is offered that a double negation can have many logical significances beyond mere emphasis, and beyond the total reversal of classical logical negation. The actual differences between differences, or logical negation operations, observed here are being brought to their natural conclusion: that the concept of difference is vague.

Derrida recognized one kind of distinction, different from the mathematical understanding of difference as a kind of absolute, unqualified ideal. It may be that Derrida’s différance is a conflation of the absolute difference/not-equals of mathematics, with the idea of meaning being deferred. It may also be that the connotation of deferred in différance is a qualification that makes différance more precise than difference. Either way, the reason difference can be qualified is that the idea of difference as an unqualified absolute suffers from vagueness, that any observed difference is already conflated with other differences. And this vagueness is natural, in the same way that there are natural kinds, there is a natural failure to classify, to name. As in measuring the plank of wood, the length is never perfectly measured. The idea that there is an “actual” length “out there,” or that the ultimate meaning (in the correspondence sense) of a length of a wooden plank is deferred to the completion of a never-ending measurement task is contrasted with the idea that the “actual” length is naturally vague, and the task need not be completed. For finite beings, all measurements are vague (people tend to call this “error”, but error implies that the vagueness people encounter is essentially unreal, even though it is encountered constantly whether we use precise measuring tools or our human senses). Derrida’s différance is not the negation or difference operation in constructive logic, or the paraconsistent definition of negation in Costa’s C1 (1977), or the many other formulations of the logical negation axiom, and these are all different from the absolute mathematical not-equals, and classical logical negation.

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SIGNIFICANCE FOR PHILOSOPHY OF MATH

There are two ways to handle the recognition that distinctness in its present form is indistinct. One way is to introduce new words for different kinds of distinctness, or make standard the use of qualifying words for this difference or that difference. This path would change the mathematical use of the not-equals symbol and the negation operation, and make logical pluralism standard. Another way to handle it is to accept that vagueness is not something we can avoid by pursuing a never-ending task of refining language. I am in favor of the second path — one of embracing vagueness. Ullman (1970) has a section called “Words with blurred edges” (1970, p 116) where he notices that vagueness is seen as a strength by poets, but seen as a bad thing by natural philosophers (the old name for scientists). Speaking as a published poet, my experience of education in the USA did not favor the poet’s temperament to embrace vagueness. To trouble the first path of handling the indistinctness of distinctness, it could be pointed out that there are higher-order varieties of the same problem of distinctions between distinctions — that there can be distinctions between the distinctions between distinctions, etc. and these would lead us back into a realm where vagueness must be accepted as, if not real and good, as constantly present in everyday life and troubling attempts to make distinctions. Distinction is problematic; higher-order distinction is not any less problematic than higher-order vagueness — the problem that solutions to vagueness are themselves vague.

There is no reason to privilege distinctness over vagueness, as Feyerabend (1975/2010) argued that there is no reason to privilege the telescope over the naked eye. In this, the ideas here place themselves under ancient skepticism, not nihilism nor relativism. Ancient skepticism guards a person against turning their sense impression into dogma that is not immediately apparent. (Sextus Empiricus 1996) The impressions the senses make to you are not open to question. This is not to say that knowledge is impossible, but simply that knowledge found in poetry “weighs in” just as much as knowledge found in the sciences.

In Verlaine (1884, p23-25):
Rien de plus cher que la chanson grise
Où l’Indécis au Précis se joint.

l’Indécis can be translated as vagueness and Précis as precision, “Nothing is more valuable than the grey song where vagueness and precision join.”

In Nightingale (2018) a study of teaching vagueness found that students who were perceived as “bad” or uncooperative were the most engaged and pro-active when the topic of empirical investigation was vagueness. This seems to indicate that education in what Nightingale coined as “precision knowledge” is hostile to certain temperaments. There are further social effects to the pursuit of “precision knowledge” in mathematics, and sciences who follow mathematical reasoning. Precision knowledge has a divisive effect on people and discourse, as is illustrated in the story of the Tower of Babel.

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ON VAGUENESS

There is too much to say about the many attempts to define vagueness. For this paper, we have adopted Peirce’s definition of vagueness, as the failure of the law of noncontradiction. However, Peirce made this definition when only one logic was recognized. For vagueness to take on its role in logical pluralism, the concept of vagueness was repurposed in Nightingale (2018). In general, the description of vagueness is better left to poetry. Rowland (2000) offered that the “question” itself is an expression of vagueness. If that is the case, describing vagueness would be a description of everything we can inquire about. The most “complete” attempt at doing this, outside the field of poetry, would be in real number analysis, although it is argued in Nightingale (2013) that this attempt, while beautiful to some, still fails.

Ullmann points out that the term “vague” is vague, or ambiguous (p117-118) (although Russell (1923) gave one definition of vagueness as ambiguity: that there is more than one possible referent for a given word-use). Ullmann offers a disambiguation of the term vague, which includes the general. On the other hand, Peirce is adamant that there be a clear distinction between general and vague (reprinted in Peirce 1934). The indeterminacy of generality (general knowledge is a useful kind of knowledge) lies in variation within a class, while indeterminacy of vagueness lies at the boundaries of classes. (Rowland 2000, Kindle Locations 1697-1700) The distinction, it must be noticed, suffers from vagueness, because there are individuals that lie near the boundary of a class that may or may not also be one of the variations allowed by a generality. Not only is it easy to miss the difference between vagueness and generality, the difference may be vague.

Wandering among different definitions (vague comes from the Latin vagus which means to wander), the roads seem to lead to Russell’s definition of “having more than one meaning… language is a ‘garden of forking paths'” to quote Borges’ story about a book that actually succeeds in chronicling the passage of time, forking as all possible choices are made. Russell noticed that these paths can be seen in experience — as things come closer, a multitude of details become available, all calling us to follow them. Sometimes I refer to this “general law of physics” (Russell 1923) as vagueness:

			
"...in the case when you are looking at a star in the sky through a telescope, and you can almost make out that it looks like two stars close together, we can say that 'that light is one star' or 'that light is two stars'. Holding both these statements is classically illogical by noncontradiction, and both can be held because the situation is vague — both in Peirce's sense and Russell's physical law." (Nightingale 2018, p 21-22)

Whatever vagueness is, the borderline case is where vagueness has made the most trouble for logicians. For example, Shapiro (2008) describes the borderline case with imagining an array of 2000 men from entirely bald on one side and gradually (imperceptibly, even) changing to a very hairy Jerry Garcia on the other end of the array. He attacks this ancient problem with what he calls the “forced march,” done by a group of “competent” speakers. They start with Jerry Garcia and investigate the next man, reaching a consensus each time, and the inference “if n is not bald, then n+1 is not bald” appears valid. Shapiro is concerned with the eventuality that the inference will break down and the competent speakers will be filled with conflict until they decide that one of the men is bald. This is because the difference between the words “bald” and “not bald” are vague, or there are cases where the word bald can be applied to a man that is arguably bald or not bald, if inquiry becomes more sensitive. This is how the borderline case relates to our definitions of vagueness in the previous paragraph.

The “competent speakers” or “masters” of meaning decide man n is bald when they had just agreed man n-1 was not bald. Shapiro seems to say that the speakers can stop there, even though, as Shapiro mentions, the competent speakers will be aware that the same change or jump at n from not bald to bald spreads to some indeterminate n-k other people who were previously validly inferred to be not bald and are now bald, because the change from n-1 to n is imperceptible to the naked eye (a similar example of vagueness can be produced regardless of the power of the microscope or telescope; regardless of the sensitivity of any scientific instrument). If the purpose of the conversation were to figure out who was bald and who wasn’t, wouldn’t the speakers turn back and start inferring that n-1 is bald, etc. ad infinitum? Since Shapiro chooses classical logic “without argument” (Shapiro 2008, Kindle Location 1023), and classical logic requires the Law of Excluded Middle, the target of conversation is necessarily settled “out there.” To know where (and if) this is settled, the speakers would have to enter into a never-ending conversation. Facing vagueness between bald and not bald, the committee has a choice to deliberate on a vague situation forever, or allow classical logic to come into question. Here Shapiro upholds a latent ideal of work as a saving power against contradiction, but that power only saves as long as work continues, and returns to contradiction if the “masters” give up.

Nightingale (2018) put forth the idea that vagueness is a material part of inquiry. When inquiring into the length of a wooden plank, one takes centimeters, then millimeters, etc, but ultimately the measuring tool fails to take the “actual” length of the plank — the body of the plank is vague (Nightingale 2013). Dewey defended the thesis that logic arises from inquiry:

			
"the view here expressed, they (logical laws) represent conditions which have been ascertained during the conduct of continued inquiry to be involved in its own successful pursuit." (Dewey 1938 Loc 284-286)

Inquiry with the senses, as has long been recognized, is fraught with vagueness. Vagueness and logic are complementary; they are part of the same activity: inquiry. Vagueness is when reality spills over out of our word-containers in the chase and capture of reality with words, or it is what escapes us when involved in any inquiry.

Probability is often misunderstood as a solution to vagueness, and without vagueness, there is no reason to assert logical pluralism. The way probability deals with vagueness is by giving vague situations real-valued degrees between opposites.

			
"One serious objection ... is that it really replaces vagueness with the most refined and incredible precision. Set membership, as viewed by the degrees of truth theorist, comes in precise degrees, ... The result is a commitment to precise dividing lines that is not only unbelievable but also thoroughly contrary to what I [call] 'robust' or 'resilient' vagueness. For ... it seems an essential part of the resilient vagueness of ordinary terms such as 'bald', 'tall', and 'overweight' that in Sorites sequences ... there is indeterminacy with respect to the division between the conditionals that have the value 1, and those that have the next highest value, whatever it might be. It is this central feature of vagueness which the degrees of truth approach, in its standard form, fails to accommodate, regardless of how many truth-values it introduces." (Tye 1994: p 14)

The measurement of the wooden plank is a good example of why vagueness persists in the face of the theory of real numbers, because the progression of measurement can be seen as an infinite and bounded sequence, which allows it to fall under the definition of the axiom of completeness: the property that distinguishes the real numbers from other numbers (Abbott 2001). It has already been argued elsewhere that this property fails in answering the problem of vagueness (Nightingale 2018, Nightingale 2013).

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INTRODUCING “LOGIC PUZZLE”

The idea that there is more than one way to make the negation operation precise, and thus that the negation operation is vague, was coded into a computer program by the author to allow a more specific, concrete presentation of the abstract conceptual work done above. The web app program “Logic Puzzle” could be used to instruct undergraduate math, math education, or late high school students. It involves fitting puzzle pieces together to create logical statements. There are three settings: classical, paraconsistent, and constructive logic. The program automatically generates the logical statement in normal notation, along with graphs of classical logic, paraconsistent truth tables, and separate graphs for the constructive logic setting.

A user can mouse over puzzle pieces, graphs and truth tables to get a description of each object. To build a logical statement, one merely drags-and-drops puzzle pieces in ways they fit together. If you would like to play with Logic Puzzle, go to http://159.89.203.232/LogicPuzzle-Dev/.

When a green piece is dragged to the right place, it modifies the logical statement by adding a proposition labeled “A,” “B,” or “C.” When a blue puzzle piece is dragged to the right place, it acts as both the parenthesis and the negation operation, so a left and a right blue piece is needed to close the parenthesis and indicate what is being negated. There are two possible instances of “A,” “B,” or “C,” one on the left and one on the right; this creates the possibility of building two statements with puzzle pieces that are in contradiction. When a user builds a contradiction, the program asks about changing to paraconsistent logic, and draws a paraconsistent truth table under the Peircian graphs.

CLASSICAL LOGIC SETTING

The Peircian graphs (Peirce 1933) reduce logic to a single symbol: the circle. The circle notation distinguishes between its inside and outside; it is effectively the negation operation. The Peircian graphs suggest a way of using space, or a lack of notation, instead of the AND operation.

This symbol is equivalent to “~(A AND ~B)”, since the outside circle suggests an A on the left, and a ~B on the right of its inside. This symbol is equivalent to “If A, then B.” It may be noticed that there is a “latent” symbol in use here because the ellipse also suggests the use of the operation AND. One may say that this notation is only possible with a loose definition of the circle that includes ellipses. Nightingale introduces the other notation in the Peircian graphs as the same effect, but there is no variation in the basic symbol.

The mark required for the spirit of the Peircian graph is so elemental it is almost simply the requirement that any mark be written.

PARACONSISTENT LOGIC SETTING

The paraconsistent logic chosen supports the idea that negation is the fundamental idea in logic. Costa’s C1 (1977) is defined by allowing a new line in the truth table anytime something true (represented with a “1” in the truth table above) is negated in another column in the truth table; the new line allows the negation to be true (“1”), while the old line will be the expected “false” (“0”) value. So we have a situation where if A is true, ~A may be true or false, and the law of non-contradiction fails. Moving from the Peircian graphs that reduce classical logic to negation, this particular paraconsistent logic is entirely defined by a difference in the definition of negation, emphasizing the importance of negation (or distinction) as the fundamental operation of logic.

CONSTRUCTIVE LOGIC SETTING

The constructive logic setting is similar in that you build statements the same way, by dragging and dropping puzzle pieces. There are the same green pieces that are the “Propositions” and the same blue piece for both negation and parenthesis. The meaning of “A,” “B,” or “C” is explained, so “A” is a connected ring, “B” is a star-shape with rays coming from a central point, and “C” is a “mesh.” This is because the “truth-value” of a constructive logic statement is not mere truth, but “constructed” and this makes negation calculate differently.

The possibilities are: 1) A or a ring is constructed, if that is all the user has built by dragging and dropping a green puzzle piece, then the graph will be a picture of a connected ring of nodes. 2) If ~A is constructed with puzzle pieces, the graph will show a configuration of nodes that makes it impossible to connect them into a ring. 3) If ~~A is constructed with puzzle pieces, the graph will show a configuration of unconnected nodes where it is possible to connect them into a ring, e.g. “A is constructable but not constructed.” If “A and B” is constructed, there will be nodes in a ring, and the nodes will also be connected in a way that forms a star shape. Negation has yet another fundamental difference in constructive logic. Starting with “constructed” (A), “not constructed (and constructable)” is (~~A) and “not constructable” is (~A).

The pedagogy presented by the “Logic Puzzle” program should be understandable to undergraduate students in math or math education. While all these concepts may not be explicitly learned by high-school students using the logic puzzle, the intent is that students will be exposed, or immersed, in a teaching tool with an important message: there are different logics, these logics are defined by their differing negation operations, and these differences can be compared, side by side, within a technological setting. The pedagogy of logical pluralism can be felt by students, even if they do not master the teachers’ understanding presented here, compressed as it is from the wider context available to a scholar. It is often the case that mathematical arguments are not made to young students, such as the argument for real numbers, as numbers, or as real.

The argument for real numbers isn’t presented fully unless a student majors in mathematics in college, where a student can find at the very end (covered?) of an undergraduate text on real numbers: “We all grow up believing in the existence of real numbers, but it is only through study of classical analysis that we become aware of their elusive and enigmatic nature.” (Abbott 2001, p 244) Unfortunately for many students, such arguments are often not of much help. This is one purpose of making “Logic Puzzle;” it presents an important mathematical idea with automatic graph generation, help messages as needed, and the mathematical rigor calculated “under the hood.” With the pedagogy of logical pluralism and its conception of the program “Logic Puzzle” explained, we will turn to how the logic puzzle application is to be studied, then to exactly what the students are able to grasp using this tool.

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METHODOLOGY

			
"...the field indicated, that of inquiries, is already pre-empted. There is, it will be said, a recognized subject which deals with it. That subject is methodology; and there is a well recognized distinction between methodology and logic, the former being an application of the latter." (Dewey 1938 Loc 166-168)

Dewey writes that the reason methodology and logic are seen as separate is logic is needed as an external and certain standard to judge methods of scientific inquiry.

			
"How can inquiry originate logical forms (as it has been stated that it does) and yet be subject to the requirements of these forms?" (Dewey 1938 Loc 179-180)
"The problem reduced to its lowest terms is whether logical standards and forms are intrinsic to inquiry or whether they are imported from outside... One might reply by saying that it can because it has. One might even challenge the objector to produce a single instance of improvement in scientific methods not produced in and by the self-corrective process of inquiry." (Dewey 1938 Loc 182-185)

How does one make assumptions about a methodology when the standard of judging methodologies is under scrutiny? This is why a qualitative approach is preferred. Induction is a very different logic from the deductive logics being examined. The study inquires using Grounded theory (Glaser & Strauss) continuing to explore multiple deductive logics as a topic for empirical study in education, except Nightingale (2018).

There were three instruments to the Logic Puzzle: a pre-post attitudinal survey (quantitative), a worksheet that guides the students to try certain constructions in classical, paraconsistent, and constructive logic, asking for their feedback (qualitative), and a post test asking them to select a logic given a situation. The test was partially quantitative and partially qualitative, with a correct answer of which logic, and any information the student felt they wanted to add.

The worksheet instructed the students to construct the statement “A and ~A,” a contradiction, so that Logic Puzzle drew Peircian graphs and a paraconsistent truth table. The students were then asked questions like what the graphs mean, how classical Peircian and constructive graphs differ, about how classical logic truth tables and paraconsistent truth tables are different. At the end there were open ended questions about what they thought about the program.

Logical pluralism was introduced to students with the Web App called “Logic Puzzle” in one language: Thai. In Thailand, Western (the usual truth-table classical logic) logic has colonized the national curriculum, and teachers begin instruction of logic to students around age 16. However, a requirement throughout the mathematics curriculum is “suitable reasoning.” With this requirement, education authorities in Thailand present mathematics as the model for how students should reason.

The students were generally college and late high school students in front of a computer doing an investigation into logic, not for fun. The worksheet-style direction and feedback part were coded and then the codes were investigated using basic qualitative questioning. The codes were compared to further define them and their relationships to each other to better understand the students’ experience of Logic Puzzle. Certain words were selected for meaning exploration. The quantitative results are the most tentative form of research applied in this study, because probability is under scrutiny.

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DATA AND RESULTS

QUALITATIVE FINDINGS

Because there were only 7 subjects in this study, the presentation of data can be more fine-grained; 10 worksheet questions makes 70 answers in the worksheet. There were a small number of coded “insightful answers” (8/70) that go beyond these types of merely “correct” (28/70) answers. Outside the two questions that almost nobody understood (what is a contradiction and how are the constructive and classical graphs alike/different) there were 7 misunderstand/blank. Having a graph gave students a visual to connect to the definition of negation. Maybe this contributed to the idea that the tool (or logical pluralism?) “made it easier” (5/70).

“Choice” (6/70) was another code. Students are generally not given a choice about mathematical facts. Giving students choices helps them to see that whatever math fact they are looking at was a choice made by mathematicians, not an absolute truth. They do not have to accept them or “hang on a word” or symbol. Students can think more freely because mathematics is seen as more hypothetical. There was indication that giving students choices “made it easier”; there was no indication that giving students choices in math made things harder. Another student sees that a different logic has the potential to be more concrete (constructive), instead of more general (paraconsistent).

“Relates to everyday life” (2/70) was another code. It was strange to see this code at all. I did not expect that students would be able to relate abstract A’s and B’s, logical operations, and geometrical graphs to everyday life. “Paraconsistent logic is possible with 2 answers. Can be used in everyday life more than Classical logic.” Here we see a connection made by a student between choice (having possible answers) and everyday life. Is math seen as difficult simply because it is asserted as without choice, unlike life?

Insightful students noticed that the paraconsistent truth table is “Different with many more meanings than traditional logic.” or “Rejection [negation] in constructive logic can be done in two ways, but for classical logic there is only one way” in response to how classical logic is different from constructive logic. This is the main pedagogical message of Logic Puzzle: that there are choices of logics because of distinctions between kinds of negation.

QUANTITATIVE RESULTS

For the attitudinal questionnaire, because of the small sample size, only large effect sizes were tested using a matched pair, one-tailed Student’s t-test. There were three questions (out of 12 in the survey) with a large Cohen’s D effect size (approx .89); two of these questions had interesting statistical significance. The first was “If I have a thought that is not mathematical, I know that it is not worthwhile.” This changed from being neutral and agreeing somewhat, to leaning towards disagreeing after the activity. The statistical significance of this large and important effect size was p = .051. While this is technically not a significant finding, in a strict orthodox view, it is still a tantalizing finding that indicates a fruitful direction for further study, especially given the small sample size of 7.

The next question was “Mathematics is not important in everyday life.” This also changed from neutral and somewhat agreeing, to somewhat disagreeing. The Cohen’s D effect size was large (.89) so a t-test was conducted and a p value of p = .052 was found. Another interesting, if unorthodox finding. It is the author’s view that if schools had allowed logical pluralism into their classrooms so that a large sample size would have been available, there would be a significant finding that the logical pluralism activity helped students to see a relationship between logic and everyday life.

SUMMARY OF FINDINGS

The strongest result — that of having a choice — is found in the test, the survey (more thoughts are worthwhile), and the worksheet, so it is triangulated. Emphatically, the choices given to students in Logic Puzzle are epistemologically fundamental. They are not trivial choices about how to express the same answer, or different ways of getting essentially the same answer. These are real choices. There was no indication that having possibilities made things harder, but there was indication that having possibilities made things easier.

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CONCLUSION

Students found more options with logical pluralism, and that comes as no surprise. They felt this freedom as a connection to life, and a sense of ease. Math could have been done in a different way, and it’s not as if traditional math does not have contradictions (Weber N.D.). It is supposed that students can feel this, even if they don’t know it — in the same way that real numbers are commonly presented intuitively without precise, mathematically rigorous arguments.

The message of Logic Puzzle as a technology “endowed or entrusted with meaning” (Latour 2002, p 29) is that logical deduction is not final. The fact that there is a variety of logics bends the blade of distinction against itself, rendering traditional taxonomies and bodies of knowledge dependent on an absolute idea of distinction open to question. And with honest inquiry there is an opening for constructing new meanings, finding new patterns and for new readings of ancient texts.

It has been observed that “the question” is how to express vagueness (Rowland 2000), and logical pluralism, as a response to the vague concept of logical negation, evokes questions more easily than classical logic alone, which presents itself as final.

The logic (with probability presented as a “theory” not a logic) in Thai schools today is consistent with the Law of Excluded Middle: that there is no real question whether it is A or ~A, “out there”; this question is always settled. With Excluded Middle, vagueness is unreal or error, and the reality is a world of external answers, already settled “out there.” (Brouwer 1981) Classical logic puts learners in situations where they have to deal with contradictions, such as the indistinctness of distinctness, and does not teach them a freedom of thought that allows them to deal with contradictions. One example is instantaneous velocity, which is a contradiction in terms. It is better to not assert that such a concept is precise and in contradiction; rather, instantaneous velocity is vague and that is why the law of non-contradiction fails. “How can an object have velocity if it doesn’t have time to move?” This is how to think about instantaneous velocity. Firstly, asking a question about a contradiction just feels better to students than compelling them to accept something conflicting. Secondly, a question has the property of being neither true nor false, which is the kind of proposition you need for a contradiction. A real question is one that is particularly stubborn and keeps coming up after generations of trying to answer it. Motion, consciousness and vagueness are examples of real questions.

With logical pluralism, teachers can show students a rejection of the logical terms used to create the contradiction, not in a negative way, but by introducing new logical terms in other logics. So instantaneous velocity uses the mathematical construct of points to describe movement. Rejection of the logical objects used to describe movement does not reject the question about movement, but leads to looking for other logics to ask about it.

Even if logical pluralism is not adopted in a curriculum, teachers ought to give their students permission to reject contradictions, and how would they do that safely without giving them alternative logical negation operations, allowing the failure of the law of noncontradiction by giving alternatives? Compelling assent to a contradiction without alternatives is the opposite of education. On the other hand, appealing to unending inquiry can be paralyzing and equally unhelpful. Introducing a variety of logics is a middle-ground solution, allowing students to articulate their questions with respect to a particular logic, and adapt to contradictions by recognizing that any logic has vague aspects.

There is no intent here to endorse the idea that anything can be true. Maybe a limited relativism is supported because absolutism is rejected as a result of the honest recognition that the logical negation operation is vague. People still need (vague) interaction, and have (vague) standards of reasoning to judge and believe ideas and writing. The perspective of logical pluralism is that there are many deductive standards to use, and perhaps some yet uninvented. Relativism isn’t really a serious point of view because logical standards, conversations, ideologies, paradigms, cultures, etc. are vague. The idea that something can be relatively true from the point of view of a culture or something else requires that this culture be fully defined the way classical logic is believed to be. Unfortunately, classical logic is not fully defined, nor is any culture or anything else (this is a direct result of the indistinctness of distinctness). Neither does this mean nothing is true. Ancient skepticism asserts that sense impressions are true and not open to question, even though they are vague, so nihilism is rejected.

Science education focuses on teaching a precise concept, even in alternative approaches like Philosophy for Children (P4C) (Ferriera 2012). However, vagueness is more interesting to students (Nightingale 2018), and pluralism helps understanding, against the idea that there is a scientific “true/real” precise concept, without contending concepts. The latent ideal of work that Shapiro upholds as a saving power against contradiction is contrasted with an ideal of rest, offered in Pyrrhonism as Ataraxia, because vagueness is found both after tremendous amounts of work and investigation using big science apparatus, as well as without work, immediately to our human senses.

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REFERENCES

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Borges, J. L. (2002). Other inquisitions: 1937-1952. Austin: University of Texas Press.

Brouwer, L. E., & Dalen, D. V. (1981). Brouwer’s Cambridge lectures on intuitionism. Cambridge: Cambridge University Press.

Brouwer, L. E. J. (1907). On the Foundations of Mathematics. Thesis, Amsterdam; English translation in Heyting (ed.) 1975: 11-101.

Brouwer, L. E. J. (1908). The Unreliability of the Logical Principles. English translation in Heyting (ed.) 1975: 107-111.

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Cook, R., 2010, “Let a thousand flowers bloom: a tour of logical pluralism,” Philosophy Compass, 5(6): 492-504.

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Drago, A. (2018). Suggestion for Teaching Science as a Pluralist Enterprise. Transversal: International Journal for the Historiography of Science, (5), 66.

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Human mathematics and poetry are the sounds of our grief

24 Friday Jul 2026

Posted by Andrew Nightingale in The only category

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ai, buddhism, logic, philosophy, poetry, writing

In the Dhatukatha, the third book of the Abhidhamma, lamentation is filed as matter. Not feeling, not mind — audible object, “born of perverted mind” but not itself mind, sorted with the sound element and, at question 247, associated with none. It keeps company there with the mindless realms, the beings that have shed mind and kept only form. Grief, one question later, is the opposite: feeling, woven through three aggregates, all through mind, and shareable with no one. So the catalogue draws a line most readers would hear as a demotion — the wail cast out of the mind that made it, filed with stones.

I want to read the line the other way. To be matter is to have shape. Grief, being feeling, has no shape and can reach nothing; it is companioned all through your mind and no other mind can enter it. The wail, being matter, is the wrong kind of thing to be in relation — and that very wrongness is what lets it be found. Music is matter. Language is matter. The book is matter. It is because they are shaped, emitted, findable that they do anything at all between us. Nothing crosses. My grief makes a sound; the sound is not the grief and carries none of it; the sound strikes your ear and a grief of your own is induced there, fresh, by resemblance. This is thought to be communication. It is closer to weather.

Grief itself is not characteristically human. Related to the Latin gravis — heavy, loaded, pregnant — it belongs to ghosts and animals too, at least from the set-of-the-grieving’s point of view. The human realm is not the grief but the wailing it produces: the crossing, the shaped emission, the thing made findable with a mind to how it sounds to other minds. And here the monks’ hard choice becomes legible. The novice sets down music, dance, and show — not because art is worthless but because art burns, because its whole nature is to reach across and light a feeling that was not there. To one training to cool the fires, the thing whose function is to start them must be put down, and “mere matter” is the accurate name for it, not the contemptuous one. Yet the Dhatukatha itself is permitted, as wholesome hearing — a shaped sound brought to the ear, doing the good kind of induction. The tradition draws its line inside the burning class: some shaped matter inducts the defilements, and some shaped matter inducts the exit. Good art is lamentation aimed the second way. The reduction to matter is not the insult. It is the mercy — because a thing that carried the feeling could only pass the fire along, and a thing that carries only shape can induce a different feeling than the one that made it: can lead the finder out.

This is why the art of teaching is elevated above other arts. It cultivates the capacity of the ear, and the mind’s ear. I suppose the line “one cannot be said to be like or unlike oneself” is a kind of resignation that communication is impossible — even to oneself across the chasms of an instant. In the DhatuKatha, the 3rd book of the Abhidamma branch of the Tipitaka (the three baskets), one grief is not related to any other grief. An extreme divide and conquer, quietly added to a dry list of the associated, the dissociated and the associated and dissociated (not, as I self-importantly argued almost two decades ago, “associated or dissociated”).

The Dhatu Katha means, with a straight face, Discussion of Elements. It is the least discursive book ever written. It discusses nothing. It takes the eighteen elements and puts each one to every other: is eye associated with sound, is sound associated with contact, on through the pairs, and the answers come back associated, dissociated, or associated and dissociated, in the same few sentences over and over until the permutations are exhausted. No commentary. No examples. No indication that any of it matters. It is a clerk with a ledger and it will not be drawn into conversation; it answers the question asked and not the one you meant, which is why the third column is and rather than or, as the head monk had to explain to me. It reads like machine output twenty centuries early. With endless content from the entirely unmoved, it conveys. This is the character I keep quoting as though it agreed with me. It does agree with me, about things it was in no position to be considering.

I feel a hurricane gathering, and in its eye is AI. I would rather it weren’t. But the eye is the one place with no weather in it — and no grief either. The calm we are all circling.

Song made with AI is called cheating; a mathematician who uses AI to break a heavy problem is given credit — though the proof, like a Shakespearean tragedy, was supposed to be inevitable, given the conditions assumed. The scorn and the credit are the same desire: to be uniquely human, and to keep AI out of the uniqueness. But the two cases do not even enter by the same door. The song crosses at the sound base and arrives at the mind; the proof was never a sound at all.

And less crosses than they think. Wailing has a shape as well as a sound — the rise, the break, the repetition, the falling off — and we know it as grief without needing a word for it. The catalogue agrees more than I expected: speech itself is not filed at the sound base at all. Vocal intimation, the act of making language, is listed as subtle matter at the cognizable door. The sound base takes the sound and nothing else. But a shape takes more than one instant to be a shape, and the sound element is momentary. Nothing that needs two moments to exist can live there. The contour is already assembled on our side, at the mind-object door, which was never a room for language only. Language is its loudest tenant.

The catalogue settles it, and settles it by parentage. Lamentation is “audible object born of perverted mind” — that is the entire definition. Not what it says. Not how it sounds. Where it came from. Grief does not enter the sound, but the sound comes from grief: cause without containment, and the catalogue offers no third term for it. The artifact I called mindless has a mind for a parent, which is not a contradiction but the architecture — associated with none, born of one. And the perversion gets its own gloss a few pages earlier: believing in a world of persons and things, mistaking terms and concepts for realities. So the wail is what Peirce would call an index — caused by the grief, joined to it through breath and throat and the loaded body, gravis. A machine makes the same shape with no such parentage: icon only, resemblance without cause. Which means that by the catalogue’s own arithmetic the machine’s sound is not lamentation. Not counterfeit — simply another item, an audible object born of no perversion at all. Nothing about the sound differs. The distinction the scorn is defending is real, it lives entirely in the birth, and the birth was never what crossed. My own first sentence gave that away: the artifact was always mindless, and resemblance was always all that arrived. My wife watches films now in which no one was ever sad, and the sadness arrives.

And now the mathematician can be answered. His proof was never a sound; it crossed no sound base at all. It was assembled where thinking meets its objects — the one door in the whole catalogue where the answers come only partially, the door where grief itself is filed. That is the door at which they have licensed the machine. Meanwhile they stand guard at the door of sound, which never held a feeling to protect and never held anything but matter. They are not defending the wrong room. They are defending the right room from the wrong door.

The music in even the most horrible lamentation has effects that can lead to out as well. And AI can be said to be a kind of entertainment, where AI is trained to uphold certain meanings and ulterior motives, just like art, that are not the user’s choice… like my personality is not my choice. Call all this human shaped matter matter is productive for restful existence.

So what is the AI in the crossing? Not a rival wailer. It has no grief to load into the sound, and the sound never carried grief anyway. The feeling was only ever in the listeners and readers. The machine is air: the medium the wail crosses, the the act of making language, is listed as subtle matter at the cognizable door.

if not now

21 Tuesday Jul 2026

Posted by Andrew Nightingale in The only category

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another dance, please, under
Domineering clouds of breathless purples,
feet pumping the ground in equal and opposite,
dry and dusty wind beginning at the feet,
storming dancers around you
when

the horizon forgets its edges
and the sky leans low enough to listen,
your pulse is percussion, you had forgotten, as it strikes the mind

out of thin air, until even silence fractures.

When

lightning learns the shape of your spine,
tracing it in sudden flashes of grammar,
a Law the storm cannot finish
without you.

your hands carve the air into meaning,
each gesture a small rebellion, not of mercy but recognition

every step you take
is returned in thunder,

the earth answers back in tremors,

as if the ground had been waiting
for your name.

in the low sun, dancers made of shadows twirl with their trees’ silhouettes

When

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Posted by Andrew Nightingale in The only category

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                            on pedestals 
                            
Orion without a spare arrow 

wailing is the human sound 

paint the ceiling blue it’s the only control we have over the color of the skylight

keeping the wild and free, we discard our ancestors
a fox's entailed entanglement
if it’s part false just sprinkle some light

blazing through my mind like a summer rain
footfalls down, the banister's curve tightens, a trusted grip follows...no end
warmth and color leak into my skin

walled off garden giving room to the wild trees

sanded down



Quiet suchness
of quiet