The second-cycle discussion lands on the light box activity’s essential question: when we “see” something in an experiment, what exactly are we seeing—an objective feature of the world, or a community-stabilized interpretation shaped by instruments, scale, and expectation? The classroom transcript shows that this question did not remain abstract. It became embodied in the children’s insistence—urgent, almost physical—on re-checking the mirrors, re-running the looking, and pressing the teacher for a verdict: “Is that white real?” That moment is the philosophical core of the activity. The experiment is not only about color-mixing; it is about the status of an appearance that emerges from a setup that “shouldn’t” produce it (no white light was shone). The children are effectively asking: Is an emergent phenomenon legitimate evidence, or merely an illusion produced by imperfections?
This is where the Wittgenstein and Frege quotations become more than ornament: they describe two rival ideals of inquiry that the students oscillate between.
Wittgenstein: the virtue of the indistinct
Wittgenstein’s line—“Is it even always an advantage to replace an indistinct picture by a sharp one? Isn’t the indistinct one often exactly what we need?”—names what the light box demonstrates materially: vagueness isn’t always epistemic failure; sometimes it is the very condition of a stable phenomenon. In the books earlier framing, the “truth” that a white surface reports (e.g., under red light, it looks red) depends on diffuse reflection—on micro-roughness that mixes incident light rather than preserving perfectly separated rays. In that sense, whiteness (and “white-looking”) is not the absence of structure but the presence of extremely fine structure whose epistemic role is precisely to blur.
So the light box invites a reversal of the usual moral: instead of “imperfection contaminates truth,” we get “imperfection produces the phenomenon we rely on.” Under this lens, white is not a cheat; it is a real outcome of a real interaction between RGB light and scattering surfaces. The children’s question “is it real?” becomes: Do we count a phenomenon as real when it depends on vagueness? Wittgenstein’s answer, pedagogically enacted, is: often yes—because the indistinct is exactly what a form of life needs to perceive and coordinate around.
Frege: the demand for sharpness (and its hidden cost)
Frege’s microscope analogy pulls hard in the other direction: scientific goals demand “sharpness of resolution,” because ordinary seeing hides imperfections. In the excerpt of Frege represents the methodological impulse: replace everyday language with ideography; replace ordinary seeing with instrumented seeing; replace blurred boundaries with crisp ones. Shapiro’s framing (Frege as realist) matters here because it makes the sharpness-demand sound not merely practical but ontological: to be realistic is to be ever more resolved.
But the classroom data complicates that ideal. The students already believe—prior to what their eyes show—that magnification yields more “information,” more color, more reality. Their tacit epistemology is “zooming in = gaining truth.” This is, in miniature, the Fregean faith. Yet in the light box setup, increased resolution does not enrich the phenomenon; it destroys it. Magnification separates what vagueness had mixed; with only three source lights, separation yields fewer visible colors, not more. The teacher’s students’ confusion is philosophically productive: it exposes the dogma that sharper is always better, even when the phenomenon itself is scale-dependent and exists only as a product of blur.
So the tension isn’t simply “vagueness vs logic.” It’s sharper:
Vagueness is not merely a defect to be eliminated; it is sometimes the mechanism by which a phenomenon becomes perceivable and communicable.
Sharpness is not purely gain; it is a transformation that can erase exactly what matters at a human observational scale.
Community interpretation: the social life of “real”
This is why the opening claim—scientific theory as community interpretation—fits the transcript so well. The students do not just observe; they negotiate. They test each other’s claims, try tools (teacher glasses), and when tools fail they shift to imagination. The classroom becomes a micro-scientific community trying to decide what counts as evidence and what counts as “real.” The question “is that white real?” is not answered by the world alone; it is answered by the community deciding what sorts of dependence (on roughness, on scale, on instrument) disqualify reality and what sorts do not.
And here is the deeper synthesis between the quotes: Wittgenstein and Frege do not merely disagree about clarity. They disagree about the aim of inquiry.
Wittgenstein warns that sharpening can be a loss when the “indistinct picture” is the one that organizes our successful practices.
Frege insists that sharpening is a gain when scientific aims demand resolution beyond ordinary limits.
The light box episode shows that both are right, but about different goods—and this pedagogy makes the difference visible.
The dissertation’s implicit conclusion
So when the teacher asks, “why is demanding ever-more sharp resolutions realistic?”, The material supplies a strong answer:
Realism cannot simply mean “maximize resolution,” because reality is not accessed by a single monotonic ladder of magnification. Reality is structured across scales, and what is real at one scale (white as stable diffuse appearance) can be decomposed at another scale (separated colored reflections) without thereby being “exposed as fake.” What changes is not merely what we see, but what we are able to count as the phenomenon. The demand for sharpness is realistic only relative to a chosen aim—measurement, control, prediction at a certain scale—not as a universal virtue.
The light box activity therefore becomes a pedagogical argument: scientific seeing is not the elimination of vagueness, but the disciplined management of it—including knowing when blur is epistemically enabling and when it is epistemically obstructing. The students’ resistance—denying their senses in favor of the “magnification gives more information” belief—shows how deeply the sharpness-ideal is culturally installed. And the classroom struggle shows why teaching science is not just teaching facts; it is teaching criteria for what counts as a good representation.
I will be compiling a new-and-old collection of works here. I do not have a publisher yet, and when I find one, this post will come down for copyright reasons.
Original above, chatgpt assisted version of Starspin and the Missing Observer:
Tycho Brahe believed he was correcting ancient astronomy with unprecedented precision. He trusted his instruments and his eyes. But the deeper instrument he relied on was neither quadrant nor sextant: it was an aura of faithful observation—a rhetorical position masquerading as scientific fact. We often forget that “objective observation” is not something science has ever proved possible. It is an attitude we adopt, a posture of trust in our measurements. Under Aristotle’s unmoving Earth, this attitude was coherent. But once the Earth is recognized as spinning and hurtling through space, a basic problem emerges: we cannot describe the motion of anything without first knowing the motion of everything else. To know how the stars move, we must record their motion from the moving Earth. To know how the Earth moves, we must record its motion relative to the stars. But without knowing either in advance, we begin only with guesses—guesses shaped by our rhetorical stance toward what counts as “good” observation. Thus every astronomical model, ancient or modern, is built upon a circularity that no amount of mathematical refinement can dispel. We cannot observe from a non-moving point; no such point has ever been found. And until such an absolute vantage is discovered—if it exists—every claim to objective cosmology rests on persuasion, posture, and rhetorical performance. This is not an attack on science. It is an acknowledgment of what Einstein already implied: all measurement is relational, and relation always presupposes a stance.
Scientists normally respond by making the mathematics more complex, as though complexity could wash away the rhetorical underpinnings of knowledge. But complexity persuades; it does not neutralize. A formula can intimidate us into assent just as effectively as a sermon. This is one reason scientific rhetoric so often spills into politics. When Bernie Sanders says that extreme poverty in the richest nation is a contradiction, he is pointing not to a logical error but to a failure of rhetorical thinking—a failure to see contradiction as a lived reality, not a technical one. Americans are no longer trained to think rhetorically. They believe one persuades by making “logical claims,” as though logic were free of stance, persuasion, and communal orientation. The result is a political culture held hostage by certification, authority, and shame. If a person’s belief is not scientifically approved, institutionally validated, or expert-endorsed, it is dismissed automatically. This is why the spectrum from anti-vaxxers to flat-earthers sparks such animosity: the conflict is not about science but about who is allowed to speak.
Jorge Luis Borges once contrasted Argentine and American attitudes toward literature. In Argentina, he said, there is always the possibility that a book which wins a major prize might nonetheless be good. In the United States, this attitude is nearly inconceivable. Here, value must be certified. A book, an idea, even a person must bear official approval before being granted attention. This difference is not minor. It is the difference between a society oriented by rhetoric—where persuasion is communal, flexible, and shared—and a society oriented by credentialed “logic,” where only approved speakers may speak, and where dissent is pathologized. If science rests on rhetoric, then so does politics. If observation rests on stance, then so does community. If the universe lacks a fixed center, then so must our systems of knowledge. The task is not to eliminate rhetoric but to recognize it— and to build a society that can think with it rather than fear it.
Abstract This paper advances an original rhetorical and pedagogical proposal concerning the nature of logical negation. Following Peirce (1933), who reduced classical logic to a single primitive—negation understood as distinction—I argue that negation should be presented as the central concept of logic, especially for educational purposes. This contrasts with the prevailing view that the primary subject of logic is logical consequence. The vagueness of logical consequence (Gillian 2019; Tarski 1956) is the standard argument for logical pluralism. Yet logical negation itself admits different interpretations (Wittgenstein 1976). I propose that negation is also vague, and that this vagueness independently supports logical pluralism. As a secondary contribution, I describe a web-based educational program, Logic Puzzle, designed to make abstract discussions of negation and vagueness more concrete. The program enables visual comparison of logical negation operations. It may be useful in late-high-school settings or, more fittingly, in undergraduate mathematics-education courses. A preliminary mixed-methods study suggests that the tool increases learners’ awareness of choice in mathematics. Curriculum recommendations are left to teachers.
Background Logical pluralism is the view that there is more than one correct deductive logic—more than one legitimate way to sharpen logical consequence. Since the influential work of Beall and Restall (2006), varieties of logic have proliferated (Gillian 2019). Examples include: Varzi’s (2002) logics with expanded sets of logical constants, Russell’s (2008) pluralism concerning truth-bearers, model-theoretic pluralisms based on purposes or contexts (Shapiro 2006; Cook 2010; Shapiro 2014). Historically, debates among the Stoics (Mates 1961) already embodied early forms of pluralism. Carnap (1937) endorsed a version in which pluralism arises by altering the syntactical rules of a language. Lakatos (1962) observed that Euclid’s axiomatic form—definitions followed by proofs from simple axioms—served for centuries as the paradigm of rigorous knowledge. Indeed, Euclid’s Elements became the most widely read textbook in history after the Renaissance (Hartshorne 2000). But the discovery of non-Euclidean geometries undermined not only Euclid’s geometry but the classical logical framework supporting it. The contemporary turn toward logical pluralism is therefore not superficial: it suggests a reorganization of the ways knowledge is justified. Pluralism does not eliminate classical (Euclidean) modes of argument, but adds other valid methods. One compelling reason for pluralism comes from probability theory. Probability functions as a many-valued (fuzzy) logic in scientific practice, including quantum physics. Beall and Restall note that although probability cannot be the whole story of logical consequence, it shows how different logics coexist: a two-valued “accept/reject” structure interacts with an uncountably-valued probability calculus. This paper aims to sketch a pedagogy of logical pluralism accessible to high-school teachers already familiar with classical logic. The journey, I suggest, moves from classical logic toward something closer to poetry. (Readers chiefly interested in vagueness or poetry may skip ahead.) The central pedagogical idea is a didactic interplay between vagueness and the negation operation—distinction. Russell (1923) and Shapiro (2008) noted that vagueness is the contrary of distinction. Thus the very attempt to define vagueness sharply conflicts with its nature.
Vagueness and Logical Consequence Beall and Restall argue that pluralism arises because the intuitive concept of logical consequence is “not sharp” (2006). Shapiro similarly notes that multiple legitimate precisifications of consequence exist. Tarski (1956) explicitly declared the concept vague. Gillian (2019) identifies this vagueness as the common foundation for pluralism. In this paper, I treat vagueness as a central problem in logic—perhaps the central problem—which creates the possibility of logical pluralism. Yet I depart from standard pluralism by placing negation, rather than consequence, at the center of logic. Peirce attempted to reduce logic to a single consequence symbol but succeeded instead in reducing it to negation (Peirce 1933). This simplification motivates my claim: recognizing multiple logics amounts to recognizing multiple kinds of distinction.
Distinct Kinds of Distinction Peirce defined vagueness by the failure of the classical Law of Non-Contradiction (1960). The modern form—¬P → P—assumes a universal negation operation. Pluralism becomes possible when the double negation ¬¬P does not reduce to P, but instead yields a different form of “not-P.” That is: different negations model different kinds of distinctions. For example: A neutron star differs from a question in a different way than a lemon differs from a lime. A right isosceles triangle of side length 1 cm and a square of side length 1 cm differ both in number of sides and in compositional geometry. To simply say they are “distinct” obscures the kind of distinction involved. Our ordinary practice collapses many distinctions under a single term, and this collapse is itself vague. Attempts to cure vagueness via precisification face the standard problem: greater precision eventually dissolves distinctions rather than clarifying them. As with the canonical “find the boundary of my nose” example, increasing resolution leads to quantum uncertainty, not crystalline precision. Classical negation requires universality—one single “¬.” If the two negations in ¬¬P differ, the classical inference to P breaks. Wittgenstein (1976) already noted that negation may have multiple meanings. Here I argue that double negation may have many logical significances—far beyond emphasis or reversal. Recognizing this pushes us to an unavoidable conclusion: difference itself is vague.
Derrida and Logical Negation Derrida’s différance identifies a type of distinction unlike the absolute, unqualified difference of mathematics. Whether différance conflates mathematical difference with semantic deferral, or whether “deferral” qualifies difference more precisely, is secondary. What matters is that difference is qualifiable because absolute difference is inherently vague. We never encounter unmediated difference; distinctions are always entangled with other distinctions. Vagueness here is not an error introduced by limited tools but a structural feature of finite beings. Measurement—even with perfect instruments—reveals indeterminacy. Calling this “error” obscures the reality that vagueness is encountered constantly. Derrida’s différance differs from intuitionistic negation, paraconsistent negation (da Costa 1977), and the classical “≠” symbol. These negations model different kinds of distinction.
Implications for Philosophy of Mathematics Two options arise from recognizing that distinctness is itself indistinct: Proliferate terms for different kinds of difference, revising “≠” and negation to reflect pluralism. Embrace vagueness as unavoidable. I advocate the second. Ullmann (1970) observes that vagueness is prized in poetry but shunned by the sciences. As a poet, I note that U.S. education rarely cultivates a temperament comfortable with vagueness. Higher-order distinctions—distinctions between distinctions between distinctions—merely replicate higher-order vagueness. There is no reason to privilege distinction over vagueness, just as Feyerabend (1975) rejected the privileging of the telescope over the naked eye. This stance aligns with ancient skepticism (Sextus Empiricus), which warns against turning sense impressions into dogma. Poetry and science both inform knowledge; neither stands as the sole authority. Verlaine captured this well: “Nothing is more precious than the grey song where vagueness and precision join.” Empirical work (Nightingale 2018) suggests that students labeled “difficult” or “non-compliant” often excel when investigating vagueness—a sign that curricula built around “precision knowledge” alienate certain temperaments. Precision-centric epistemology can divide communities, as allegorized by the Tower of Babel.
On Vagueness Peirce’s definition of vagueness as failure of non-contradiction must be reconsidered in a pluralist context. In Nightingale (2018), vagueness was repurposed: if a concept leads inquiry into branching interpretations—as in Russell’s “garden of forking paths”—it is vague. The borderline case has always troubled logicians. Shapiro’s (2008) “forced march” example arranges 2000 men from bald to very hairy. Competent speakers progress along the line applying: “If n is not bald, then n+1 is not bald.” But imperceptible differences undermine classical reasoning. At some point the group must switch from “not bald” to “bald,” and because classical logic requires the Law of Excluded Middle, this transition produces conflict. In reality, as sensitivity increases, borderline cases multiply. This entire dynamic reflects vagueness in the sense developed earlier: the failure of a universal, crisp distinction.
Conclusion Logical pluralism stems not only from the vagueness of consequence but from the deeper vagueness of negation—the operation of distinction itself. Negation should therefore be re-centered in logic pedagogy. Embracing vagueness rather than eliminating it widens the conceptual landscape available to teachers, students, and mathematicians. The Logic Puzzle tool demonstrates that students benefit from encountering multiple forms of negation visually. Preliminary empirical results suggest increased awareness of choice in mathematical reasoning. Pedagogical implementations should be developed collaboratively by teachers who understand their own students’ needs.
Numbers Are Metaphors Mathematics proposes numbers to measure real things. The notches on a measuring tape correspond to numbers, and those numbers correspond—imperfectly—to positions on an object. Even when the notches succeed in pointing to something real, they remain signs of reality, not reality itself. And between the notches, the tape contains silent intervals: unmarked gaps, neither measured nor named. The real number system was invented to fill these gaps. It removes the need for metaphor by supplying a number—even infinitely many numbers—for every missing interval. As Giuliani writes, “Metaphorical language is language proper to the extent that it makes up for the gaps of language” (1972). Real numbers attempt to do just that: cover the gaps, smooth the voids, and complete the line. How is this magic accomplished? Through the Axiom of Completeness: Every bounded increasing sequence has a least upper bound. This axiom is the quiet engine behind the real numbers’ apparent perfection. Imagine measuring a plank with a straight-looking edge. You measure the whole meters. Some plank remains. You measure the decimeters. More remains. You continue until the tape’s precision runs out, or your eyes fail, or you simply lose interest. You must fail in measuring the exact length of the plank. The process always stops prematurely; inquiry always encounters fatigue. And yet the Axiom of Completeness assures us that there exists a real number for the “actual” length—even though the process could never reach it. This is the paradox: The axiom gives us confidence in the existence of an exact length, while simultaneously proving that our measurements can never reach it. We are still in the poetic world of metaphor, “a process, not a definitive act; it is an inquiry, a thinking on” (Hejinian, 2000). We began wanting to talk about something real— something as simple as the length of a plank. We arrived with tools, control, and attention. But we still fail. All measurements remain metaphors. The length remains an object just out of reach, falling through the gaps of language. Perhaps the task is not to awaken from metaphor, but to recognize that metaphor is already a kind of contact. And contact itself is ambiguous. When we touch a table, or reach for a loved one, there is no true penetration, no merging of substances— only a nearness where pressures meet and recognize each other. Light behaves the same way. The sun never touches the Earth. Its finger-rays stop at the surface, knowing precisely where to yield. They illuminate without entering, offering warmth without grasping. A form of intimacy without ownership. This is what our inquiries do. They reach toward the real, stop at the threshold, and illuminate what can be illuminated. What is lit has always been vague: a revealed nearness, a direct experience of a world that cannot be fully measured, and does not need to be.
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The Role of Rhetoric (Aphoristic Revision) The parallel postulate was never necessary, only asserted. Euclid held it back as long as he could, building a Neutral Geometry true everywhere. But in the end he chose—persuaded by coherence, by tradition—that only one line runs parallel through a given point. This was less proof than persuasion. That choice shows how we think. The crystallization of intuition into formal mathematics is a rhetorical process. The vague becomes exact, the fluid hardens into proof. We pretend logic alone dictates the result, but behind every axiom stands persuasion. Non-Euclidean geometry proved the point. More than one parallel line can exist. Triangles on a sphere can have three right angles. The Earth itself gave evidence, but we preferred the frame that made space measurable, quantifiable, ours. Even science, in claiming territory, leans on rhetoric. Rhetoric is not an ornament of thought but its orientation. It decides which truths are admitted, which distances are counted, which realities are named. To prove is also to persuade. To formalize is to crystallize choice into necessity. Aristotle called rhetoric an art of persuasion; I call it the art of orientation. It stands at the border where mathematics becomes possible, where science borrows from poetry. Rhetoric guards poetry, translating its openness into form. It shows us that truth, even when written in symbols, is always chosen—always persuaded into being.
Magical thinking in mathematics Andrew Nightingale Today’s goal is to show that magical thinking is omnipresent in mathematics. Definition of magical thinking Let’s call magical thinking a metaphorical thought that seeks to become real. Saying “my heart is the sun” is not just an image: it is a statement that acts as if it could actually warm my heart. The metaphor is not a simple comparison, but an attempt at realization. How mathematics works Mathematics relies on this same mechanism. Let’s take the concept of difference. In everyday life, differences abound: the difference between a crow and a desk is not the same as that between the North Star and a philosophical question; the difference between 3 and 5 is not experienced as equivalent to that between 3 and 9: one is “closer,” the other “further away.” However, in mathematical language, all these differences are reduced to a single operator: subtraction. The symbol “-” metaphorically affirms that all differences are alike, that they belong to the same abstract essence. Thus, 3 – 5 is treated in the same way as 3 – 9: the nature of the difference is overshadowed by its belonging to a single category, “numerical difference.” This gesture is deeply metaphorical—and magical in the sense that I mean. For mathematics does not merely say, “Let’s pretend that all differences are identical”; it asserts that they are in fact identical, at least in the universe of mathematical reasoning. Universalism of difference From this derives an ultimate concept of universal difference, applicable to any measurable situation. Whether we are talking about objects, distances, probabilities, or abstract structures, it is always the same operation that applies. The heterogeneity of real differences is flattened, absorbed, and metaphorized into a single form. It is in this sense that mathematics practices magical thinking. Philosophical counterpoint My approach is the opposite of Derrida’s. With his différance, Derrida enriches the word “difference” with layers of meaning: temporality, language play, deferring and deferring. I propose, on the contrary, to divide difference into a plurality of differences, irreducible to one another. Mathematics, on the other hand, chooses the other path: it reduces and unifies. Consequences for science and poetry Since science expresses itself in the language of mathematics, this magical thinking extends to all scientific disciplines. My thesis advisor in mathematics once told me, “Mathematics is the poetry of science.” But it is a very particular kind of poetry: a poetry that uses and reuses endlessly the metaphor of Difference — and who believes in its power of truth.
Delving into a New Operator: the Questioning Operator (“?”) Andrew Nightingale The “?” operator marks a sentence not as true, nor false, nor contradictory, but as non-final—as something intentionally held open. Instead of asserting, the agent adopts an attitude of inquiry. In this way, the operator functions as a formal expression of suspended acceptance. Classical logic asks whether a proposition is true or false. Paraconsistent logic allows that a proposition may be both. Paracomplete logic allows that it may be neither. The Questioning Operator adds something distinct: ?φ = the agent treats φ as unresolved, declining to fix its truth. This is not ignorance. It is a decision about how to relate to a proposition—a refusal to make the move from thinking to believing.
Syntax Extend a base language LLL: If φφφ is a formula, then ?φ?φ?φ is a formula. Optionally, a higher-order form ??φ??φ??φ can be introduced, meaning “I question the status of ?φ?φ?φ as a question.” The present essay focuses on the unary operator “?”.
Semantics: a Three-Valued Approach A simple and calculable semantics arises from the familiar three-valued set: T (true) F (false) U (undetermined / open) For the standard connectives, we can employ Kleene’s K3 evaluation. The important clause is the truth condition for the new operator: v(?φ)={Fif v(φ)=T,Totherwise.v(?φ) = \begin{cases} F & \text{if } v(φ) = T, \ T & \text{otherwise.} \end{cases}v(?φ)={FTif v(φ)=T,otherwise. Thus: If φφφ is false, we may question it → ?φ=T? φ = T?φ=T. If φφφ is undetermined, we may also question it → ?φ=T? φ = T?φ=T. Only if φφφ is true is it inappropriate to question → ?φ=F? φ = F?φ=F. In classical terms, the operator asserts: “φ is anything but true.” It is emphatically not negation. For instance, ¬φ¬φ¬φ maps U to U, but ?φ?φ φ maps U to T. In short, questioning is not refutation.
Consequences and Examples A few consequences follow immediately: If φ is a tautology, then ?φ?φ?φ is false in every valuation. Questioning destroys logical necessity. The operator is truth-functional, so its behavior in complex formulas is compositional. Consider the Liar sentence L:=“L is not true”L := “L \text{ is not true}”L:=“L is not true”. Under Kripke-K3, v(L)=Uv(L) = Uv(L)=U. Then v(?L)=Tv(?L) = Tv(?L)=T. The paradox becomes a case where the correct attitude is inquiry, not assertion. This captures the spirit of your earlier work on paradox: treat the liar not as a contradiction to be resolved, but as a node of inquiry.
Toward a Proof-Theoretic Account A natural deduction system can be built around the principle: From φ φ φ, infer ¬(?φ)¬(?φ¬(?φ). But from ?φ?φ?φ, one may not infer ¬φ¬φ¬φ. These rules reflect that questioning is an epistemic act, not an evaluative one. A useful metatheorem: If φ is a classical theorem, then ?φ?φ?φ is false in every three-valued valuation. Thus tautologies are exempt from inquiry. This confirms the operator’s philosophical purpose: it marks the stance of suspended assent.
Naming the Operator Several naming conventions are available: The Questioning Operator (Q) The Openness Operator (O) The Inquiry Operator (I) The Suspensive Operator (S) The Non-Truth Operator (N) The best compromise remains simply: the Questioning Operator (“?”).
A Meta-Questioning Form The higher-order form: ??φ:=?(?φ)?? φ := ?(?φ)??φ:=?(?φ) allows self-scrutiny: If questioning φ was inappropriate, ??φ=T?? φ = T??φ=T. If questioning was appropriate (φ false or open), ??φ=F?? φ = F??φ=F. This gives a formal mechanism for examining the stance of inquiry itself.
Conclusion The Questioning Operator formalizes an intentional act often overlooked in logic: the choice to withhold judgment rather than negate or assert. It offers an elegant way to model epistemic humility and preserves the philosophical intuition that some propositions should not be forced into truth values prematurely. In paradox, in vagueness, and in epistemic pause, the operator “?” makes space for inquiry where assertion would mislead.
The essay proposes treating some problematic sentences not by assigning them the classical values true or false, nor by declaring them outright contradictions, but by inserting a question operator ? — an operator that marks a sentence as “anything but true” (i.e. open, interrogative, non-final). The rhetorical intent is to preserve openness, time/flow, and uncertainty instead of forcing a binary assignment.
What these systems do. Paracomplete systems treat some sentences (notably Liar sentences) as neither true nor false — there is a third “gap” or undefined value. Kripke’s 1975 fixed-point theory constructs a minimal fixed language in which paradoxicals remain ungrounded (i.e., lack a truth value) rather than producing contradiction. The Strong Kleene tables are often used to evaluate such gaps. impan.pl+1
Similarity to ?. If ?φ is read as “φ is not true / indeterminate / open,” then ? behaves very much like marking a gap. Viewed semantically, ? could be treated as a predicate that holds of sentences that take the gap value in a K3/Kripke fixed-point semantics.
Key difference. Paracomplete accounts typically define the truth-value directly (a semantic gap), whereas the essay treats ? as an intentional question/attitude that preserves possibilities and invites change. To formalize ? as a gap you must give it truth-conditions and rules for how it composes with other connectives (something the essay doesn’t yet do). impan.pl
2) Supervaluationism —? as “not super-true” / indeterminate across precisifications
What supervaluationism does. Supervaluationism treats borderline sentences as true on all admissible precisifications (super-true), false on all (super-false), and otherwise indeterminate. It thereby preserves classical tautologies for super-true sentences while allowing gaps. Academia
How ? could fit.?φ might be read as “φ is not super-true” or “φ is not true on all precisifications” — a higher-level diagnostic operator saying the sentence lacks a robust classical truth. That would let you keep many classical inference patterns when sentences are super-true, while marking paradoxical sentences as ? (indeterminate).
Issues to watch. Supervaluationists face revenge problems (one can formulate sentences that say “this sentence is not super-true”), so you’d need to show how ? avoids or resolves the same technical pitfalls. Academia+1
3) Paraconsistent / Dialetheist approaches (Priest) —? is very different
What dialetheism does. Dialetheists accept that some sentences are both true and false (true contradictions, dialetheias), and use paraconsistent logics to block explosion (i.e., to avoid triviality when contradictions occur). Stanford Encyclopedia of Philosophy
Contrast with ?. The essay’s ? rejects asserting truth rather than accepting a sentence as both true and false. So while dialetheism embraces contradiction, ? seeks to sidestep it by withholding the affirmation of truth. These are epistemically and metaphysically distinct moves. If you formalize ? as a gap, it aligns with paracomplete, not paraconsistent, strategies. Stanford Encyclopedia of Philosophy
4) Tarski’s hierarchical solution —meta-levels vs. a questioning operator
What Tarski suggested. Ban self-reference by splitting object-language and meta-language levels to avoid a global truth predicate.
Where ? stands.? is not ontologically banning self-reference; it’s a device for marking problematic self-referential sentences as open/indeterminate rather than pushing them into a higher metalanguage. So ? is more permissive than Tarski’s stratification — but if you want to keep a single language with ?, you’ll need to show how ? avoids the contradictions Tarski tried to prevent.
What they do. Some approaches model truth as time-indexed or as evolving under revision: sentences can change truth-value over time or under increasing stages of evaluation. Kripke’s construction also has a staged, monotone build-up which resonates with dynamic viewpoints. impan.pl
? and time. The essay’s stress on change and flow makes it natural to read ? as a temporal/modal operator: ?φ ≈ “φ is not now fixed as true” or “it is currently open whether φ.” This would place the operator in the family of truth-revision or temporal logics and could be formalized with staged evaluation or a modal semantics (possible-stages quantification). That choice would help the essay keep its time-sensitive intuitions while moving to a clean formal semantics.
6) Epistemic / probabilistic approaches —? as an epistemic withholding
What they do. Instead of altering truth-conditions, these accounts treat paradox as a feature of our epistemic state (we should suspend judgment or assign subjective probabilities).
? as epistemic. If ? is interpreted as an epistemic operator—“we currently withhold judgment about φ” — then the operator doesn’t alter the language’s truth-conditions but signals a stance of suspension. That is easier to implement but weaker logically (it doesn’t solve the paradox semantically; it only prescribes behavior about assertion).
Advantages and potential payoffs of the ? idea
Preserves an intuitive, human practice. People ask rather than immediately assert in many philosophical contexts — ? captures that.
Flexible: multiple formalizations possible.? can be implemented as (a) a gap operator (K3/Kripke), (b) a supervaluation diagnostic, (c) a temporal/modal operator, or (d) an epistemic operator. Which one you choose depends on the philosophical commitments you want to preserve (e.g., preserving classical logic where possible vs. admitting true contradictions). impan.pl+1
Pedagogical appeal. As the essay stresses, teaching a logic that foregrounds questioning may be more faithful to real reasoning and less dogmatic about truth.
Main technical challenges you must address (concrete list)
Give formal semantics.
Option A: Define ?φ truth-conditions in a three-valued (or many-valued) semantics (e.g., K3). Prove consistency properties, and show how ? composes with ¬, ∧, ∨, →. Stanford Encyclopedia of Philosophy+1
Option B: Give ? a supervaluational semantics: ?φ true iff φ is not true on all precisifications (or vice versa). This requires addressing revenge sentences. Academia
Specify proof rules / entailment.
Which inference rules preserve ?? Does ?φ block contraposition, conditional proof, or classical explosion? You must specify a proof theory (sequent calculus or natural deduction) and prove soundness/completeness relative to your semantics.
Handle revenge paradoxes.
Any novel semantic device invites new paradoxes (sentences that self-refer to ? itself). Show how ? resists or locates such revenge sentences (this is where Kripke-style fixed points or restricted expressivity may help). ILLC Preprints and Publications+1
Decide metaphysical attitude.
Is ? a semantic device (changes what “is true” means) or an epistemic/intentional device (changes what agents assert)? Be explicit: conflating the two is a common slippery move.
Compare with dialetheism and justify rejection or adoption.
If you reject dialetheism, explain why accepting contradictions is worse than leaving sentences open. If you accept some contradictions, explain whether ? still has a role.
Concrete suggestions to formalize ? (a roadmap)
Start with Kripke/K3 implementation. Treat the object language with a truth predicate T, adopt Kleene/Kripke evaluation rules, and define:
Good — this is a rich move. Here’s a toy model + worked example (handout style) for a supervaluational “?” operator + how to incorporate the nuance from Many Roads from the Axiom of Completeness (and your dissertation) about “Is this a question?” as part of the question-intention. I’ll lay it out in parts: (1) some philosophical framing, (2) formal toy semantics, (3) worked examples (including a “revenge-style” construction), (4) reflection / implications for teaching or your dissertation.
1) Philosophical framing (drawing on Many Roads…)
Based on Many Roads from the Axiom of Completeness (Nightingale), especially as discussed in the PDF, a few important themes guide how we should treat ?:
The essay treats questions not simply as linguistic forms, but as intentional acts — “thinking of ‘the question’ in itself.” Questions Are Power
There is explicit attention to “Is this a question?” — i.e., the question-intention itself can be questioned. As Nightingale writes, exploring what the Axiom hides “releases a vertigo … an inquiry into thinking of ‘the question’ in itself.” Questions Are Power
In your dissertation (as you indicate), you also treat ? (or question-intention) as layered: not only do we mark a sentence as “open / not super-true,” but we must also reflect on whether our very act of questioning is stable or “complete.” This meta-questioning (Is this a question? What kind of question?) is not peripheral — for Nightingale, it’s essential.
So: the ? operator should not just mark indeterminacy of truth, but also bear a second-order character: it signals questioning not only of the statement’s truth, but of the act of questioning itself.
2) Formal toy semantics for ? (supervaluation + meta-questioning)
Here’s a minimal formal setup to capture both indeterminacy and a reflexive question-intention. This is meant for a handout or a “classroom toy model”; it’s not a full dissertation-length system, but enough to illustrate and test the concept.
2.1 Language
Let L be a propositional / first-order language with a truth-predicate T(x) (or just a propositional “sentences as objects” style).
Introduce a unary operator? applied to well-formed formulas φ: ?φ.
Introduce a second-order operator?? (optional) to mark question-reflection: ??φ reads, “I question whether φ is the kind of question it claims to be” (i.e., question-about-question).
2.2 Precisifications (supervaluationist setup)
Define a set of admissible precisifications Π. Each precisification π∈Π is a classical interpretation of L (assigning truth-values in {T, F}) for each atomic sentence, including sentences with T.
But not all self-referential / higher-order question-reflection sentences need to be allowed in all precisifications. We can restrict Π so that for any π∈Π:
If a sentence involves ??, its interpretation in π must respect a stability clause: π must assign truth in such a way that question-intentions do not collapse trivially into “true / false only.” (This mimics restricting precisifications to avoid very pathological self-referencing “I am not a question” loops.)
Alternatively: allow all, but track second-order indeterminacy (see below).
2.3 Semantic clauses
Define the supervaluation semantics for ? and ?? as follows:
A sentence φ is super-true if it is true in all π∈Π.
φ is super-false if it is false in all π∈Π.
Otherwise, φ is indeterminate.
For the question operators:
?φ (first-order question) is super-true iff φ is not super-true.
Intuitively: ?φ = “It is not the case that φ is unambiguously (in all precisifications) true.”
?φ is super-false iff φ is super-true.
In other cases (if φ is indeterminate), ?φ may itself be indeterminate (depending on exactly how you set up composition).
??φ (meta-question) is super-true iff there is at least one precisification π such that in that precisification, the act of questioning φ (i.e., interpreting ?φ) does not correspond to a “stable question”. Formally:
Let’s say in each π, there’s a predicate or evaluation criterion Qπ(φ) that determines whether in π, ?φ is treated as a legitimate question (i.e., nontrivial questioning, not just “φ false / true”).
Then: ??φ is super-true if Qπ(φ) fails in at least one π.
??φ is super-false if Qπ(φ) holds in all π.
Otherwise, ??φ is indeterminate.
(This is a toy clause — in a full system you would need to define exactly what “legitimate questionhood” means in each precisification.)
3) Worked Examples (Toy Handout)
Here are some example sentences + how they might be evaluated under this toy semantics.
Example A: Simple non-paradoxical sentence
Let p = “It rains.” Consider ?p.
In each π∈Π, p is either T or F (classical).
Suppose in all precisifications, p is sometimes true, sometimes false (i.e., it’s not fixed). Then p is not super-true.
So ?p is super-true (because the semantic clause says: ?φ is super-true if φ is not super-true).
This matches our intuition: “It is not unambiguously true that it rains” — a genuine question-intention.
Example B: Liar-style sentence (revenge-type)
Let L = “L is not super-true.” (the canonical super-laier). Now consider:
?L — “I question whether L is (super-)true.”
??L — “I question whether ‘L is not super-true’ is itself a stable question.”
Evaluation under toy semantics:
For some π∈Π, you might assign L = T; for others, L = F. That is typical for a revenge-style situation. So L is indeterminate (neither super-true nor super-false).
Then ?L is super-true (because L is not super-true in every precisification).
But what about ??L? That depends on whether in some precisifications, ?L is not “stable” as a question. If in some precisifications, Q_\pi(L) fails (i.e., we interpret ?L not simply as a “this is uncertain” but as a mis-question, or a destabilized questioning act), then ??L is super-true. That means: “Yes, I even question whether my own question ‘L is not super-true’ is a proper question or stable act of questioning.”
This avoids a direct contradiction: you don’t force L to be super-true or super-false, but you allow that the question-intention (?L) is itself non-final; and then you explicitly reflect (??L) on that instability.
4) Reflection / Implications for Teaching or Dissertation
Here’s how you might use this toy model in your dissertation, or in a classroom (or both):
Pedagogical tool:
Present students with L and ?L. Ask them: “Is ?L a better way to handle the Liar than just declaring L undefined?”
Then pose ??L: have them reflect: “What does it mean to question one’s own question? Is there a kind of ‘meta-uncertainty’?”
This helps concretize the idea from Many Roads… that questioning itself is not monolithic — it can be questioned, destabilized, refined.
Dissertation development:
Use the toy semantics to formalize a portion of your argument, showing that the ? operator (plus maybe ??) can be given a precise, supervaluationist semantics.
Use the restriction on precisifications (or the stability clause) to mirror your philosophical argument from Many Roads…: not all “questions” are legitimate — question-intention itself requires reflection.
Then analyze revenge paradoxes (like the Liar) in light of ??: show that some revenge sentences become higher-order indeterminate rather than outright contradictory.
Finally, you can connect this to pedagogy: how teaching ? and even ?? can help students develop a more nuanced understanding of logic, truth, and inquiry — not just false / true, but questioning stability.
Nightingale argues that vagueness is not just a linguistic or logical problem, but a real embodiment of inquiry. Questions Are Power
He links vagueness to logical pluralism: because classical logic and probability theory have instabilities, vagueness makes visible that there can be more than one valid logic. Questions Are Power
Through this lens, vagueness is philosophically valuable — not just a defect to be eliminated.
Develops a theory of vagueness as interdisciplinary — philosophical, scientific, educational. Questions Are Power
Talks about “creating a subjectivity”: how engaging with vagueness shapes students’ intellectual self. Questions Are Power
Questions the notion of strict “difference” or dualities; proposes a new logic that better embodies inquiry. Questions Are Power
Argues for vagueness as “incarnation of inquiry”, not just a problem to be solved. Questions Are Power
Calls for educational innovation (“how innovations speak”) — reflecting on how his teaching experiments can be scaled / integrated. Questions Are Power
Quick Critique (Strengths + Weaknesses)
Strengths:
Innovative Integration: Nightingale successfully blends philosophy, logic, and education. He doesn’t just theorize; he teaches vagueness, giving concrete curriculum interventions (Light Box, puzzles) to make abstract ideas experiential.
Empirical Methodology: Using action research and grounded theory makes the work more than speculative: Nightingale observes real students, collects data, and reflects on how they engage. This gives weight to his philosophical claims.
Philosophical Depth + Practical Relevance: His claim that vagueness is the “embodiment of inquiry” is powerful. It reframes vagueness from being a purely negative problem to a positive epistemic resource. This is philosophically creative and pedagogically meaningful.
Logical Pluralism Advocacy: By showing how students can meaningfully engage with more than one logic, Nightingale challenges the dominance of classical logic in education. This has broad implications, both philosophically and pedagogically.
Weaknesses / Risks:
Complexity for Young Students: The theoretical underpinnings (higher-order vagueness, plural logics) are quite sophisticated. There’s a risk that P4 students may grasp the sensory or puzzle-based activities, but not fully internalize the deeper logical theory. It’s unclear how much of the philosophical nuance they truly understand.
Assessment & Measurement: While there is both qualitative and quantitative data, the dissertation may not fully justify how much conceptual change occurred, how stable it is over time, or how to reliably measure students’ understanding of “vagueness as inquiry.” More robust assessment (pre/post tests, concept inventories) might help.
Scalability & Generalizability: The study is context-specific (Thai P4 students, particular school). It’s not obvious how well the curriculum or interventions would transfer to very different educational, cultural, or age-group settings.
Philosophical Risk: By valorizing vagueness, there’s a danger of undermining the importance of precision in science or mathematics. If vagueness is always positive, when and how should we teach students to seek clarity? Nightingale needs to address this balance more explicitly.
New Logic Proposition: The “idea for a new logic” is very ambitious. Proposing a new logic is one thing; defending it rigorously (metaphysically, semantically, and normatively) is a tall order. The dissertation may propose more vision than fully fleshed-out formal system.
Overall Impression
This is a very promising dissertation: philosophically rich, empirically grounded, and pedagogically adventurous.
It contributes meaningfully to philosophy of logic (via logical pluralism), vagueness theory, and philosophy education.
With some strengthening (especially on assessment, scalability, and formalization of the “new logic”), it could be very influential both in academic philosophy and in practical educational reform.
As a PhD dissertation, it meets a high bar: proposing a novel thesis, grounding it in literature, designing interventions, analyzing data, and sketching future theoretical work.
Once upon a time, there was a monk who wanted to know where Space was.
So he meditated and meditated and meditated, until his mind reached the angels.
He asked the angels, “Oh Angels, where is Space?”
The angels replied, “We don’t know. But if you meditate longer, you will reach even higher angels. They might know.”
So the monk meditated and meditated and meditated, and his beard grew long and grey as he sat still, until he saw the higher angels.
He asked the higher angels, “Oh High Angels, where is Space?”
And the High Angels replied, “We don’t know. But if you meditate longer, you will reach the Highest Angels. Maybe they will know.”
So the monk meditated and meditated, until his beard grew down to his feet and turned white as he sat unmoving, until he saw the Highest Angels.
He asked them, “Oh Highest Angels, where is Space?”
And they replied, “We don’t know. But if you meditate even longer, you will reach Brahma, the Highest of the High, Creator of all the worlds. He will know.”
So again, the monk meditated and meditated, until his hair fell out and his skin sagged from his bones, spotted and pale with age. At last he reached Brahma.
The monk asked, “Oh Brahma, Highest of the High, Creator of all the worlds, where is Space?”
And Brahma replied, “I am Brahma! Highest of the High, Creator of all the worlds!”
For some, this would have been enough. But the monk persisted.
“Yes,” said the monk, “and… where is Space?”
Brahma realized the monk would not go away. He drew him aside, away from his choir of angels, and whispered,
“Look, don’t tell anyone—but I don’t know where Space is. You are asking a dangerous question. If you must know, go ask the Buddha. But go at your own risk, for you go beyond my domain.”
And so the monk rose slowly from his meditation. His body trembled with age, his steps were unsteady, but his will was clear. Luckily for him, the Buddha was living then, residing in a nearby town.
He reached the Living Buddha, sat respectfully to one side, and asked his question:
“Oh Buddha, the Well-Gone, where is Space?”
The Buddha replied simply,
“It is good you came to me, for no one can answer this question except one who has finished the Noble Eightfold Path. Space can only be found in the mind of the Saint — one who has followed the Way and gone to the end of the world with his mind. For he has found Space, and it is in his mind.”
Then the Buddha, saying nothing more, imparted this knowledge in silence. And at that very moment, the monk attained Enlightenment.
From then on, he lived in supreme peace, knowing the bliss of the boundless mind, until his death and beyond.
The parallel postulate was never necessary, only asserted. Euclid held it back as long as he could, building a Neutral Geometry true everywhere. But in the end he chose—persuaded by coherence, by tradition—that only one line runs parallel through a given point. This was less proof than persuasion.
That choice shows how we think. The crystallization of intuition into formal mathematics is a rhetorical process. The vague becomes exact, the fluid hardens into proof. We pretend logic alone dictates the result, but behind every axiom stands persuasion.
Non-Euclidean geometry proved the point. More than one parallel line can exist. Triangles on a sphere can have three right angles. The Earth itself gave evidence, but we preferred the frame that made space measurable, quantifiable, ours. Even science, in claiming territory, leans on rhetoric.
Rhetoric is not an ornament of thought but its orientation. It decides which truths are admitted, which distances are counted, which realities are named. To prove is also to persuade. To formalize is to crystallize choice into necessity.
Aristotle called rhetoric an art of persuasion; I call it the art of orientation. It stands at the border where mathematics becomes possible, where science borrows from poetry. Rhetoric guards poetry, translating its openness into form. It shows us that truth, even when written in symbols, is always chosen—always persuaded into being.
Vagueness is not a matter of semantics. It is a problem that troubles the basic assertion that, as our old friend Berty Russell asserted as an axiom in the Principlia Mathematica, "Everything that is, is." The basics are a very interesting place to stay. We would like to say this is a basic and acceptable assertion, and it turns out to not be basic at all. In fact, no subject is elementary, and also every subject is elementary.
Channell and Rowland argue that vagueness has pragmatic usefulness: "For language to be fully useful, therefore, in the sense of being able to describe all of human beings' experience, it must incorporate built-in flexibility. This flexibility resides, in part, in its capacity for vagueness" (p201 Channell 1994) Dr. Channell outlines various views of where vagueness comes from, from the difference between the "same idea" in different minds (Fodor 1977 in Channell 1994), to language (Peirce 1902 in Channell 1994), to physical reality (Russell 1923). Vagueness is found discussed in logic (Lakoff 1972 in Channell 1994) where it is argued (along with Russell) that "true" and "false" are vague, and so classical logic could be modified..." (p66 Nightingale 2019)
"[i]t is perfectly obvious, since colours form a continuum, that there are shades of colour concerning which we shall be in doubt whether to call them red or not, not because we are ignorant of the meaning of the word "red," but because it is a word the extent of whose application is essentially doubtful." (1923 Russell as quoted in Nightingale 2019, p66).
"The word "red" is vague in this respect because there are borderline cases where it is not clear whether or not we should call the case "red". Russell says "essentially doubtful" because this uncertainty is essential, in the sense of being a part of the nature of red. One deception here is in asserting that the "continuum" is a perfectly precise reality that can be expressed numerically. This renders vagueness a kind of error; without a perfectly known continuum underneath our words, vagueness is not error but has a reality of its own. Does the continuum suffer from vagueness?... Peirce claimed that another way to describe generality is where the Law of Excluded Middle ("A or ~A is always true") does not hold. This makes sense because normally, the LEM decides which of "A or ~A" is true (even if we don't know which is decided, it asserts that "out there" it is decided.) When the LEM does not apply "A or ~A" is left undecided, which allows for a generalization on "A or ~A", you can choose which. However the claim that something can be essentially uncertain is directly against the LEM." (p66-68 Nightingale 2019)
I mean to say that reifying vagueness proves the LEM is false, in general. (The ideas of general and of vague are intimately connected) Russell asserted "everything that is, is" in order to "prove" the LEM. And here I am arguing against the LEM, which would also be against "everything that is, is" What makes red red? In this question i mean to be vague between term red and the actual red. If everything that is depends on other things to be, there is a certain spaciousness to Being, an undefined vagueness between Being and Space.
Questions Are Power 