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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.
https://questionsarepower.org/2020/03/22/the-fools-song/
Related: https://questionsarepower.org/2014/09/08/the-valid-logical-argument/
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.
https://questionsarepower.org/2022/10/07/the-naive-and-the-mature/
https://questionsarepower.org/2024/06/15/ancient-cave-paintings/
Pedagogy of logical pluralism (Original in link is followed by chatgpt assisted version)
https://questionsarepower.org/2025/11/22/the-pedagogy-of-logical-pluralism-review-and-critique/
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.
—
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.
(original link followed by chatgpt version)
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.
—
https://questionsarepower.org/2023/05/01/questions-and-definitions/
Questions Are Power 