Abstract
A failed prediction is taken to refute the theory. This paper shows that refutation falls on the whole conjunction and logic does not say which assumption failed—with five assumptions there are $\mathbf{31}$ candidates, and only $\mathbf{4}$ separators make the blame unique. No new theorem or law is claimed.
First, refuting a conjunction of $k$ assumptions leaves $\mathbf{2^{k}-1}$ subsets that could bear the blame (Section 2).
Second, that is $\mathbf{31}$ at $k=5$ and $\mathbf{63}$ at $k=6$ (Section 2).
Third, this is the core of the paper. Settling $j$ assumptions leaves $2^{k-j}-1$, reaching $\mathbf{1}$ only at $j=k-1$ (Section 3).
Fourth, at $j=0$ nothing is reduced and $31$ remain (Section 3).
Fifth, the thirteen instances the corpus found of “one turned out to be two” are the work of writing those separators (Section 4).
Sixth, the separator is whether refutation falls on the conjunction or on one assumption (Section 4).
Keywords: Duhem-Quine thesis, falsifiability, auxiliary hypotheses, philosophy of science, Lakatos.
Scope (what is not claimed)
No new theorem or law is claimed—that refutation falls on the whole conjunction has long been discussed in the philosophy of science. No general philosophy is written—the merits of positions, their history and the course of the debates are not the subject; only a count and the corpus’s own instances. The corpus is not defended—whether its scope notes form a protective belt or a hard core is returned to the author. This paper goes as far as the numbers. Assumptions are treated as independent—the count $2^{k}-1$ idealises them as independent and equally suspect. In practice they carry weights and an order. Probability is excluded—prior credences over which assumption failed are not the subject; only the number of combinations. Relation to earlier papers: Paper 241 said to ask whether there is a binary operation before asking whether it can be made into addition—writing separators has exactly that shape. Paper 300 showed that a common root can be decided—“one turned out to be two” is the discovery of a distinct root. Paper 480 showed that the five criteria disagree—there too, a verdict failed to be unique. Paper 479 split the seven appearances of $1/e$ into three—three roots beneath one number. What is added is counting the candidates as $2^{k}-1$, tabulating $k=1$ to $6$, giving the effect of separators as $31\to15\to7\to3\to1$, showing that $k-1$ are needed for uniqueness, and placing the separator at conjunction versus single assumption.
1. Notation and the Question
Setting: Let a prediction be drawn from assumptions $A_{1},\dots,A_{k}$. The prediction $P$ follows from their conjunction.
$$A_{1}\wedge A_{2}\wedge\cdots\wedge A_{k}\ \Rightarrow\ P$$If observation returns $\lnot P$, the contrapositive yields only
$$\lnot(A_{1}\wedge\cdots\wedge A_{k})$$that is, at least one is false, with no indication which. Count the subsets that could bear the blame.
Observation 1.1 A failed prediction is taken to refute the theory. Refutation lands on the theory—one arrow striking one target.
Observation 1.2 But a prediction does not follow from the theory alone. The apparatus working, auxiliary laws, approximations, initial conditions—any one of them failing makes the prediction fail.
Question: How many combinations could bear the blame.
2. Five Assumptions Leave Thirty-One
Computation 2.1: Count the subsets that could bear the blame when a conjunction of $k$ assumptions is refuted.
| Assumptions $k$ | Candidate subsets |
|---|---|
| $1$ | $\mathbf{1}$ |
| $2$ | $3$ |
| $3$ | $7$ |
| $4$ | $15$ |
| $5$ | $\mathbf{31}$ |
| $6$ | $\mathbf{63}$ |
Observation 2.2 The closed form is $2^{k}-1$. The number of non-empty subsets—$31$ at $k=5$ and $63$ at $k=6$.
Observation 2.3 Enumeration agrees with the closed form. Listing every subset of size $1$ to $5$ gives $31$—an independent check.
Observation 2.4 With a single assumption there is no ambiguity. $2^{1}-1=1$—a boundary: without a conjunction, refutation has somewhere to land.
Observation 2.5 But an experiment with one assumption is rare. Calibration of instruments, statistical assumptions, estimates of systematic error—start counting and five is soon passed.
Observation 2.6 And each extra assumption nearly doubles the ambiguity. From $31$ to $63$—adding an assumption doubles the places blame can go.
3. Only Four Separators Make It Unique
Computation 3.1: With five assumptions, count what remains when separators settle $j$ of them as unbroken.
| Settled $j$ | Remaining | Fraction of the original |
|---|---|---|
| $0$ | $\mathbf{31}$ | $1.0000$ |
| $1$ | $15$ | $0.4839$ |
| $2$ | $7$ | $0.2258$ |
| $3$ | $3$ | $0.0968$ |
| $4$ | $\mathbf{1}$ | $0.0323$ |
Observation 3.2 Each one settled nearly halves it. $31\to15\to7\to3\to1$—the closed form is $2^{k-j}-1$.
Observation 3.3 Uniqueness arrives at $j=4$, that is $k-1$. Four of five removed, leaving one—to say which assumption failed, every other must be defended.
Observation 3.4 At $j=0$ nothing is reduced. Still $31$—write no separator and the refutation carries no information.
Observation 3.5 So “the theory has been refuted” states a $j=4$ conclusion from a $j=0$ position. Four short—write the four that were skipped and the statement is supported.
Observation 3.6 The decline is exponential. Two more settled takes $31$ to $7$, a factor of $4.43$—each separator buys the same again.
4. The Separator
The separator: does refutation fall on the conjunction, or on one assumption.
Observation 4.1 On the conjunction, and there are $2^{k}-1$ destinations. Logic cannot choose—which to doubt becomes a judgement outside logic.
Observation 4.2 On one assumption, and it can be repaired. A unique destination—requiring the other $k-1$ to be settled first.
Observation 4.3 These are distinct roots by the criterion of Paper 300. Both are called “refuted”—one the negation of a conjunction, one of a named assumption.
Observation 4.4 What the corpus has been doing is writing those $k-1$. The shape “what was thought one turned out to be two” arose independently in thirteen fields—Moore and Dennard, melting point and glass transition, phase and group velocity, fit and predictive power. In each, one word named two quantities.
Observation 4.5 Then the refutation acquires a destination. Not “the law broke” but “two things were being called one”—one of the $31$ is named.
Observation 4.6 Paper 241 has the same shape. Ask whether there is a binary operation before asking whether it can be made into addition—presume the operation and two quantities fold into one, and the refutation loses its destination.
Observation 4.7 Papers 479 and 480 did that work inside the corpus. 479 found three roots beneath one number, 480 counted the disagreements among five criteria—both have the shape of separating what looked like one.
Observation 4.8 To be honest—the count $2^{k}-1$ is an idealisation. Assumptions were treated as independent and equally suspect—in practice they carry weights and an order, and “calibration of the instrument” does not stand equal with “the core of the theory”. What is counted is logical possibility, not actual suspicion.
5. Roll Call and Closing
- $2^{k}-1$ candidates (Observation 2.2): non-empty subsets.
- $31$ at $k=5$ (Computation 2.1): matching enumeration.
- No ambiguity at $k=1$ (Observation 2.4): the boundary.
- Nearly doubling with $k$ (Observation 2.6): $31$ to $63$.
- $2^{k-j}-1$ after $j$ separators (Computation 3.1): the core.
- Unique only at $j=4$ (Observation 3.3): defend all the others.
- $j=0$ reduces nothing (Observation 3.4): still $31$.
- “Refuted” is four short (Observation 3.5): write the skipped four.
- The separator is conjunction or one (Observation 4.3): distinct roots by Paper 300.
- Logical possibility, not suspicion (Observation 4.8): what was counted.
Closing: A failed prediction is taken to refute the theory—one arrow striking one target. But a prediction does not follow from the theory alone: the apparatus working, auxiliary laws, approximations and initial conditions all enter, and any one of them failing makes the prediction fail. The destinations are surprisingly many—refuting a conjunction of $k$ assumptions leaves $\mathbf{2^{k}-1}$ subsets that could bear the blame: $\mathbf{31}$ at $k=5$ and $\mathbf{63}$ at $k=6$. Listing every subset of size $1$ to $5$ also gives $31$, so enumeration and closed form agree. With a single assumption there is no ambiguity, which is the boundary: without a conjunction, refutation has somewhere to land. But an experiment with one assumption is rare—calibration, statistical assumptions, systematic error, and five is soon passed. Each extra assumption nearly doubles the ambiguity: adding one doubles the places blame can go. Only four separators make it unique—settling $j$ of five leaves $\mathbf{31\to15\to7\to3\to1}$, nearly halving each time, with closed form $2^{k-j}-1$. Uniqueness arrives at $j=4$, that is $k-1$: to say which assumption failed, every other must be defended. At $j=0$ nothing is reduced and $31$ remain—write no separator and the refutation carries no information. So “the theory has been refuted” states a $j=4$ conclusion from a $j=0$ position. It is four short, and writing those four supports it. One thing separates them—does refutation fall on the conjunction, or on one assumption. On the conjunction there are $2^{k}-1$ destinations and logic cannot choose; which to doubt becomes a judgement outside logic. On one assumption the destination is unique, which requires the other $k-1$ to be settled. They are distinct roots by the criterion of Paper 300, one the negation of a conjunction and one of a named assumption. And what the corpus has been doing is writing those $k-1$: the shape “what was thought one turned out to be two” arose independently in thirteen fields—Moore and Dennard, melting point and glass transition, phase and group velocity, fit and predictive power. In each, one word named two quantities. Then the refutation acquires a destination: not “the law broke” but “two things were being called one”, and one of the $31$ is named. Paper 241 has the same shape—ask whether there is a binary operation before asking whether it can be made into addition, since presuming the operation folds two quantities into one and the refutation loses its destination. Papers 479 and 480 did that work inside the corpus, one finding three roots beneath one number and the other counting disagreements among five criteria. To be honest—the count $2^{k}-1$ is an idealisation treating assumptions as independent and equally suspect. In practice they carry weights and an order, and “calibration of the instrument” does not stand equal with “the core of the theory”. What is counted is logical possibility, not actual suspicion.
References
- Duhem, P. (1906). La Théorie Physique: Son Objet et sa Structure. Chevalier & Rivière.
- Quine, W. V. O. (1951). Two Dogmas of Empiricism. The Philosophical Review.
- Lakatos, I. (1970). Falsification and the Methodology of Scientific Research Programmes. In Criticism and the Growth of Knowledge, Cambridge University Press.
- Yamagishi, Y. (2026). Before Asking Whether It Can Be Made into Addition, Ask Whether There Is a Binary Operation at All [Paper 241]. Zenodo. 10.5281/zenodo.22130189
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Same Number Appears Seven Times and Splits into Two Groups and One [Paper 479]. Zenodo. 10.5281/zenodo.22733844
- Yamagishi, Y. (2026). Putting the Criteria Themselves to the Test [Paper 480]. Zenodo. 10.5281/zenodo.22733871
Reference
Yamagishi, Y. (2026). A Broken Law Can Be Blamed on Any of the Assumptions [Paper 481]. YOI: https://www.yamagishiyuuki.com/yoi/481
DOI: 10.5281/zenodo.22733927
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 481. This paper is a preprint and has not been peer reviewed.