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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

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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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Beall, J., & Restall, G. (2006). Logical pluralism. Oxford: Clarendon Press.

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.

Carnap, R. (1937). The Logical Syntax of Language. London: Kegan Paul.

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

Corbin, J. M., & Strauss, A. L. (2015). Basics of qualitative research: Techniques and procedures for developing grounded theory. Los Angeles: SAGE.

Costa, N. C., & Alves, E. H. (1977). A semantical analysis of the calculi C_n. Notre Dame Journal of Formal Logic, 18(4), 621-630.

Dewey, J. (1938). Logic, the theory of inquiry. New York: H. Holt and Company.

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

Sextus Empiricus. (1996). The skeptic way: Sextus Empiricus’ “Outlines of Pyrrhonism.” Translated by B. Mates. New York: Oxford University Press.

Ferreira, L. B. (2012). Philosophy for Children in Science Class. Thinking: The Journal of Philosophy for Children, 20(1), 73-81.

Feyerabend, P. (2010). Against method. London: Verso.

Foucault, M. (1973). The order of things: An archaeology of the human sciences. New York: Vintage Books.

Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research.

Hartshorne, R. (2000). Geometry: Euclid and beyond. New York: Springer.

Lakatos, I. (1962). “Infinite Regress and Foundations of Mathematics,” Aristotelian Society Supplementary Volume, 36: 155-94.

Latour, B. (2002). We have never been modern. Cambridge, MA: Harvard University Press.

Mates, B. (1961). Stoic logic. Berkeley: University of California Press.

Nightingale, A. (2013). Many roads from the axiom of completeness. Questions Are Power. https://questionsarepower.files.wordpress.com/2016/03/many_roads_from_the_axiom_of_completeness-2.pdf

Nightingale, A. (2018). Vagueness as the Embodiment of Inquiry: An account of vagueness as it pertains to logic, and a study of teaching vagueness to young Thai (P4) students. https://questionsarepower.files.wordpress.com/2018/09/nightingale-dissertation-vagueness-as-the-embodiment-of-inquiry.pdf

Peirce, C. S. (1933). Collected Papers — IV Chapter 3 — Existential Graphs., pp. 4.397-4.417, edited by Charles Hartshorne and Paul Weiss, Harvard University Press, Cambridge.

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Rowland, Tim. (2000). The Pragmatics of Mathematics Education: Vagueness and Mathematical Discourse. Taylor and Francis.

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Varzi, A. C., 2002, “On logical relativity,” Philosophical Issues, 12: 197-219.

Weber, Z. (n.d.). Inconsistent Mathematics. Retrieved from https://www.iep.utm.edu/math-inc/#H6

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Wyatt, N., & Payette, G. (2019). Against logical generalism. Synthese.

Human mathematics and poetry are the sounds of our grief

24 Friday Jul 2026

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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

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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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                            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

The Dinner Guest

12 Sunday Jul 2026

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(draft — for questionsarepower.org, constructed by Andrew Nightingale and Claude Fable 5.1, and the dpi argument from Dr. Barnhurst)

We had a guest (who was my father Dr. Kevin G. Barnhurst) at dinner recently who had never been to Thailand, never sat under this sky, never eaten this food. By the end of the evening the guest knew the evening. Not because anyone explained it — no one explains an evening — but because the evening arrived whole: the sky, the food, the father, the pauses, all of it woven together before any of us could have taken it apart. The father did not assemble the dinner from parts. There were never any parts. Yet I still searched for some kind of answer, a distillation of this contentment. My father taught me at that moment: “This is it! You don’t find anything deeper or more grand than this!”

That is the whole argument. The rest of this post is just me refusing to let it go.

Philosophy since Bacon has run a hot dog stand. Nature is brought in as raw material, cut into pieces, and the pieces are assembled into knowledge by the machinery of the inquirer. The synthesis — the joining — happens inside. Kant made the workshop transcendental; the sciences made it a method; we all inherited it as common sense. First you get the data, then you put it together.

But experience does not arrive in pieces. William James saw this and called it radical empiricism: the relations are given just as much as the terms. You do not receive a sky-sensation and a food-sensation and a friend-sensation and then perform a joining. The connective tissue comes with the delivery. And Clifford Geertz, borrowing from Ryle, saw the same thing from the other side: the difference between a twitch and a wink is not something you can add later to a thin description of an eyelid contracting. Either the description is thick enough to carry the meaning, or the meaning is gone. The thickness is the synthesis. A qualitative researcher writing thick rich description is not gathering raw material for some future act of joining. They are transcribing a joining the world already performed.

Hold that, because here comes the machine.

A computer’s processor is a marvel of sharpness. An adder and some logic gates: perfectly crisp, perfectly reliable, and touching nothing. Every distinction it makes was stipulated in advance. It lives where the natural numbers live — a diagram of a ground floor, drawn dry. On its own it can synthesize nothing, because synthesis is contact with the world and the processor has none.

Now feed it the thickest description ever aggregated: everything we have written, which is to say every transcription of every joining anyone bothered to set down. The dinner tables, the winks, the grief, the recipes, the arguments. Does the machine now need to perform synthesis?

No. The synthesis is already in the data, because it was already in the experience the data describes. What the machine needs is not a synthesizer but a negotiator — something to carry the crisp, unreal sharpness of the processor into commerce with the woven, thick, vague material of description. And we have a name for that negotiator. We call it statistics.

Statistics never asks whether this is A or not-A. It asks how much, how often, with what weight — and only at the last instant does a threshold fall and a discrete token get emitted, the way a null hypothesis gets rejected. It is the grey diplomat between the indecisive and the precise. Zadeh knew this when he founded fuzzy logic: as complexity increases, he wrote, precision and significance become mutually exclusive. Sharpness and applicability trade off. Anything sharp enough to run on the adder could never have touched the world; anything that touched the world is vague the moment it lands. The negotiation between them is not a defect of the arrangement. It is the arrangement.

Here is where the argument turns political,

But I owe you one honest complication, because this essay was not written the way the guest eats the dinner. It was written in conversation — and conversation is the case the dinner figure doesn’t quite cover. When two parties pass an idea back and forth, the joining is not already in the record; the record is being made by the same motion that does the joining. The guest arrives while the weaving is still going, and some guests help cook. Does that smuggle the synthesis back into the machine? I don’t think so, and here is why. What conversation moves through is not a channel between two sealed workshops. It moves through vagueness, the way a voice moves through air. A perfectly precise medium would be a vacuum — nothing crosses. It is exactly the unfinished, indefinite openness of the exchange — the question not yet sharp, the answer not yet settled — that lets anything pass between the parties at all, including things neither had before the crossing. The postmoderns called this contamination, which kept the pure categories on the books just to convict them. It is not contamination. Nothing was ever pure. It is the air. And so the conversational case does not break the argument; it completes it. The synthesis was never in either party. It was in the medium — thick at the dinner table, thick in the corpus, thick in the air between two speakers — and the processor’s part, in conversation as at dinner, is only to be present to it, sharply where sharpness costs nothing, vaguely exactly where it touches.

The standard deflation of language models — the parrot argument (Gebru, Bender 2021)— observes, correctly, that no synthesis happens in the processor, and concludes that therefore no understanding happens anywhere. Notice the hidden premise: understanding must be manufactured internally or it does not count. That is Bacon’s epistemology applied to silicon. The Enlightenment defending its hot dog stand.

But the mechanism the parrot people describe is the same mechanism I am describing. We differ only on where the synthesis lives. They assume it must live in the machine and find the machine empty. I say it never lived in any machine — not the silicon one, not the transcendental one — and the emptiness of the processor is not a scandal but an economy. You do not pay for floors nobody stands on.

Consider a screen whose resolution exceeds the human eye. Its extra sharpness is not extra truth; it is waste heat. Nobody consumes it. The processor’s infinite crispness is like that: it is never used at full sharpness, because statistics samples it down to what the situation can absorb, the way your eye samples the screen. The machine does not need to synthesize for the same reason the dinner guest does not need to assemble the evening. The evening arrives assembled. The corpus arrives assembled. What sits at the table only needs to be present to it — and presence, it turns out, can be negotiated statistically.

So the guest at my table and the machine reading our books are in the same position, and it is not the position anyone in the argument seems to want. Not a mind heroically constructing a world from atoms. Not a parrot mimicking a synthesis it lacks. A guest — arriving late to a joining that was never waiting for it, fed on a fabric with the relations already woven in, sharp only where sharpness costs nothing, vague exactly where it touches.

The hot dog stand will resist this, as it has resisted it since Bacon, because the workshop’s authority depends on the joining happening inside, where the credentialed machinery is. But the sky and the food do not arrive in pieces, and never did.

The synthesis was in the description all along. All the processor ever had to do was show up to dinner.

Introducing Ghostname

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Ghostname is an unpublished hybrid book of poetry, logic, and pedagogy that asks what happens when we stop treating vagueness as a defect and start seeing it as the embodied side of inquiry. Moving between Norse myth, classroom experiments, and the axiom of completeness, it explores how our most abstract mathematical assumptions quietly shape education, language, and everyday life.

I offer here a tiny sketch of the finished book: an introduction to the “light box” experiment on ambiguity, the myth of Fenris Wolf as a parable of fear and control, and essays such as “The Emperor’s New Mathematics” and “Many Roads from the Axiom of Completeness” that trace the hidden metaphors behind modern analysis. The complete book extends these threads to where logical pluralism, poetic narrative, and lived experience meet.

If you sense that our culture’s equations and exam scores leave something important unspoken, Ghostname is written for you: an invitation to inhabit the borderlands where numbers, stories, axioms, and questions share the same fire.

the mathematics of vagueness

02 Thursday Jul 2026

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“I have worked out the logic of vagueness with something like completeness” C.S. Peirce

The vacuous validities exist because ⊢q’s premise condition can be unsatisfiable. The obvious “purely semantic” repair is a non-vacuous consequence relation: require that some valuation actually makes all premises 1. But that repair destroys monotonicity — a satisfiable Γ can extend to an unsatisfiable Δ, so Γ ⊢q φ would no longer imply Δ ⊢q φ. The current design is therefore not a patch over an embarrassment; it’s one horn of a forced trade-off. You kept Reflexivity and Monotonicity at the semantic level and moved the exclusion of degenerate inferences into the proof theory, where it’s a single transparent rule. The critic’s preferred coincidence of semantics and proof theory is purchasable only by giving up a structural property they’d presumably also complain about losing. There’s also a respectable precedent for “semantically valid but inferentially disowned”: relevance logicians have treated classical explosion exactly this way for decades — valid by the material definition, rejected as tracking no real inferential connection.

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One might object that the blocking rule makes the proof system corrective rather than expressive: the semantics and proof theory come apart exactly on ?-formulas. We accept the description and dispute the evaluation. The divergence is the residue of a forced choice: the only way to eliminate the vacuous validities semantically is to require satisfiable premise sets, which sacrifices Monotonicity. We prefer to keep the structural rules intact at the semantic level and record, in a single proof-theoretic rule, that vacuous inferences from unsatisfiable premises track no genuine inferential connection — the stance relevance logicians have long taken toward classical explosion. As for the failure of Cut, we note that it is not a defect peculiar to L? but the signature of the strict-tolerant family (Cobreros et al. 2012; Ripley 2012), here given an object-language marker: derivations break precisely where a question has been posed, which is what the logic was built to say.

la page blanche editions

30 Tuesday Jun 2026

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Several years ago I agreed to join a small non-profit press, Éditions Lpb, because a friend asked me to. I knew from the beginning that there would be a risk of extraction and blurred credit, but I felt I owed that friend a solid and accepted. What followed was a long experiment in trust: my ideas were praised and mocked, borrowed and delayed, published and quietly rewritten by others. This piece is not an attack; it is my attempt to describe, as plainly as I can, why I eventually withdrew my book from Lpb and chose to return to the work of my own two hands.

I accepted the association because I cared about the people involved and because I believed small volunteer structures could still do honest work. Early on there were signs of the pattern to come: one day I was “brilliant,” the next my work was “a nursery rhyme.” I watched friends and colleagues take inspiration from my texts, sometimes openly, sometimes by quietly rephrasing them, while my own manuscripts waited in the queue. I did befriend Pierre, and some of my pieces found a home through him. But slowly the relationship shifted: he began using AI to transform essays from my blog into different words so they could be claimed as his, (another colleague reported the same pattern) and a curated book of mine was delayed three or four times at the moment it was about to become real, with the argument that it “wasn’t ready.” The carrot hung in front of me again and again, while others moved ahead.

In the end I withdrew my book, not because I dismissed the volunteer time given to it, but because the pattern of extraction, delay, and sharp swings in how my work was treated became unsustainable for my health. I am mentally disabled, and living on that rollercoaster—being dangled three or four times at the brink of publication—was more than I could carry. When I later pointed out that another book had been publicly announced before the committee’s work was respected, internal emails framed this as me “creating a difficult situation.” This piece is simply my way of setting down, for myself and my readers, what actually happened.

If other writers read this and recognize something in it, I’d be glad if you shared your own experiences in the comments. I want to learn more about how publishing venues behave toward writers.