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A Theory of Partial Truth for Scientific Statements

This paper proposes a restricted formal framework for quantifying the partial factual truth of scientific statements by evaluating the normalized discrepancies of their components against empirical evidence, thereby allowing for the assessment of models that contain both accurate and inaccurate elements without conflating factual adequacy with confidence or convergence to complete truth.

Original author: Gustavo E. Romero

Published 2026-11-06✓ Author reviewed ⓘ

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CC BY 4.0

Original author: Gustavo E. Romero

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Science often relies on descriptions that are not perfectly true, yet they work remarkably well. Think of a map that leaves out small side streets to show the main highways clearly; it is technically inaccurate because it omits details, but it is incredibly useful for navigation. In physics and astronomy, scientists use similar "maps" of reality. They describe stars as points, surfaces as perfectly smooth, or populations as infinite numbers. These descriptions are strictly speaking false because the real world is messier, but they capture the essential behavior of systems with enough accuracy to make predictions and drive progress. For a long time, the standard way to judge these scientific statements has been a simple "yes or no" question: is the statement true or is it false? This binary approach, however, hides the nuance of how science actually works. It treats a model that is 90% accurate the same as one that is 10% accurate, simply because neither is perfect. It fails to explain how a theory can be false yet still lead to genuine discovery.

A new paper by Gustavo E. Romero, a researcher at the Argentine Institute of Radio Astronomy, proposes a way to measure the "partial truth" of scientific statements. Instead of asking if a claim is simply right or wrong, the paper offers a method to estimate how well a claim fits the facts, even when it contains errors. The core idea is that scientific progress is not about finding a single, perfect description of the universe, but about building representations that become increasingly adequate for the specific problems we are trying to solve. This approach allows scientists to say that a statement is "mostly right" in a precise, numerical way, distinguishing between a small, harmless error and a major flaw, without confusing this measurement with the probability that the statement is true.

The paper begins by clarifying what exactly is being measured. It distinguishes between the physical statement itself—the words written on a page or spoken aloud—and the proposition, which is the actual meaning or claim being made. The goal is to evaluate the factual adequacy of that meaning against a body of evidence. This evidence consists of controlled observations and experiments, such as measurements of a star's brightness or the resistance of an electrical wire. The author argues that we must separate the question of "how confident are we in this data?" from "how well does this claim match the data?" A claim can be very close to the measured facts even if the measurement itself has some uncertainty, and the new framework is designed to capture that closeness.

To do this, the paper introduces a system for assigning a score between zero and one to any scientific claim. A score of one represents the maximum possible adequacy given the current evidence, while zero represents a complete mismatch. For claims involving numbers, such as "the mass of this star is 100 times the mass of the sun," the score is calculated by looking at the difference between the claimed number and the number supported by observation. If the claimed number is very close to the observed number, the score is high. If it is far away, the score drops. The paper specifies that this difference is measured relative to a scale appropriate for the problem, ensuring that a small error in a large number is treated fairly compared to a small error in a tiny number.

A crucial part of the new method is how it handles complex statements that are made up of several smaller claims. Imagine a scientific model that makes three predictions: one about temperature, one about pressure, and one about the shape of a container. If the model gets the temperature and pressure right but gets the shape wrong, a simple "true or false" system would label the entire model as false. The new framework, however, allows the model to retain a high score because the accurate parts still contribute value. The paper shows that by averaging the scores of the individual parts, weighted by how important each part is to the specific question being asked, we can get a fair assessment of the whole. For instance, if a model is being tested for its ability to predict electrical resistance, claims about the color of the wire or the temperature of the lab are irrelevant. The new system ignores these irrelevant details, ensuring that a model isn't unfairly penalized for getting unimportant things right or wrong.

The author demonstrates this method with concrete examples, including a hypothetical assessment of electrical resistance and a real-world application involving a massive star called Zeta Puppis. In the star example, researchers had previously estimated the rate at which the star loses mass based on X-ray observations. The paper takes a specific claim about this rate and compares it to the observed data. Even though the claimed rate was slightly different from the observed rate, the new method assigns it a high score of about 0.86, indicating that it is a very good approximation of reality. This score is not a guess about the likelihood of the claim being true; it is a direct measure of how well the claim aligns with the evidence, taking into account the known limitations of the measurements.

The paper also addresses potential pitfalls in this kind of measurement. It warns that the score depends heavily on how the problem is defined and which parts of a model are considered important. If a scientist changes the scale of measurement or decides that a different part of the model is more critical, the score can change. The author is careful to state that this does not mean the facts have changed, but rather that the assessment criteria have shifted. The method is designed to be transparent about these choices, requiring scientists to explicitly state what they are measuring and how they are weighing the different pieces of evidence. This prevents the scores from being misleading or arbitrary.

Ultimately, the paper offers a tool for a more honest and nuanced view of scientific progress. It suggests that we should stop thinking of science as a journey toward a single, perfect truth that we will eventually reach. Instead, science is a process of refining our descriptions, making them more adequate for the specific questions we need to answer. A theory can be "partially true" in a way that is mathematically precise, allowing us to see exactly where it succeeds and where it fails. This approach respects the fact that scientific models are always approximations, but it also gives us a way to measure how good those approximations are. By distinguishing between the accuracy of a claim and the confidence we have in it, and by weighing the importance of different parts of a theory, the new framework helps us understand how science moves forward, even when it is not yet perfect.

Technical Summary: A Theory of Partial Truth for Scientific Statements

Problem Statement
Scientific practice frequently relies on representations—models, theories, and statements—that achieve empirical success despite being strictly false due to idealizations, approximations, or specific errors. Classical bivalent logic, which assigns a single truth value (true or false) to a proposition, fails to distinguish between representations that are "more" or "less" factually adequate. It cannot account for the heterogeneous structure of complex scientific claims, where precise predictions may coexist with mistaken assumptions. Existing approaches, such as probabilistic confirmation (Bayesianism), fuzzy logic, and theories of verisimilitude (truthlikeness), address related issues but conflate factual adequacy with probability, vagueness, or distance from a complete truth. The paper addresses the specific question of whether factual adequacy can be represented in degrees without confusing it with confidence, probability, or subjective belief, while avoiding the formal inconsistencies found in earlier axiomatic attempts.

Methodology
The paper develops a restricted formal framework for estimating partial factual truth ( $V_E$ ) based on the following structural components:

Truth Bearers and Domain: The theory distinguishes between statements (physical occurrences) and propositions (equivalence classes of equisignificant statements). The valuation function $V_E$ applies only to a domain $D_E$ of meaningful factual propositions open to empirical assessment relative to a body of evidence $E$ .
Quantitative Valuation: For quantitative propositions asserting a value $\mu$ $μ$ against an empirical estimate $e \pm \delta$ $e \pm δ$ , the framework introduces a normalized relative discrepancy ( $\beta_E$ $β_{E}$ ).
- $\beta_E(p) = \min\left(1, \frac{|\mu - e|}{s_E(M)}\right)$ , where $s_E(M)$ is a characteristic scale appropriate to the problem.
- The factual adequacy is defined linearly: $V_E(p) = 1 - \beta_E(p)$ . This maps perfect agreement to 1 and discrepancies exceeding the characteristic scale to 0.
Aggregation and Composition:
- Elementary Basis: Complex propositions are decomposed into a fixed basis of distinct, independent, and jointly consistent elementary components ( $S(P)$ ).
- Logical Equivalence: To resolve the "parenthesization problem" identified in prior work (where recursive binary averaging yields different values for logically equivalent expressions), the framework evaluates the set of elementary components rather than applying recursive binary operators.
- Relevance Weighting: A relevance function $Rel_F(p)$ assigns weights to components based on their importance to a specific scientific problem $F$ .
- Weighted Mean: The contextual valuation is calculated as a weighted arithmetic mean:
  $V_{E,F}(P) = \frac{\sum r_i V_E(p_i)}{\sum r_i}$
  where $r_i$ is the relevance of component $i$ . Components with zero relevance do not alter the valuation, while components with positive relevance allow for compensation (an accurate component can offset an inaccurate one).
Axiomatic Structure: The theory is governed by four axioms: Boundedness ( $0 \le V \le 1$ ), Normalization (1 for maximal adequacy, 0 for minimal), Composition (arithmetic or weighted mean over elementary components), and Quantitative Valuation (linear complement of normalized discrepancy).

Key Contributions

Resolution of the Parenthesization Paradox: By abandoning recursive binary averaging in favor of aggregation over a fixed set of elementary components, the framework ensures that logically equivalent expressions (e.g., $(p \land q) \land r$ vs. $p \land (q \land r)$ ) receive identical valuations, provided the elementary basis remains constant.
Distinction between Truth and Relevance: The framework explicitly separates the factual adequacy of a claim from its relevance to a specific problem. This prevents irrelevant but accurate claims from artificially inflating the adequacy score of a representation regarding a specific scientific question.
Separation of Adequacy from Probability: The valuation $V_E$ is interpreted strictly as an estimate of representational adequacy relative to evidence, distinct from the probability of a proposition being true or the degree of confidence in it.
Handling of Measurement Indeterminacy: The framework distinguishes between discrepancy (the mismatch between assertion and evidence) and measurement indeterminacy (the uncertainty of the evidence itself). The valuation is relative to the evidence, but the target is the evidence-independent adequacy to the facts.

Results

Formal Properties: The derived properties include bounds preservation (the aggregate value lies between the minimum and maximum component values), grouping invariance (results are independent of how components are grouped, provided total weights are retained), and monotonicity.
Compensation vs. Thresholds: The arithmetic mean allows for compensation; a high adequacy in one component can offset a low adequacy in another. The paper notes that this differs from a "minimum" rule, which would require all components to meet a threshold.
Sensitivity Analysis: Worked examples (electrical resistance and astrophysical mass-loss rates) demonstrate that aggregate scores are sensitive to:
- The choice of normalization scale (e.g., relative to magnitude vs. operating range).
- The assignment of relevance weights.
- The granularity of the elementary basis (e.g., treating an inequality as a single claim vs. two bounds).
Astrophysical Application: The framework is applied to a published observation of the mass-loss rate of the star $\zeta$ Puppis. A proposition asserting a rate of $4.0 \times 10^{-6} M_\odot \text{yr}^{-1}$ is evaluated against an empirical estimate of $3.5 \times 10^{-6} M_\odot \text{yr}^{-1}$ , yielding a partial truth value of $\approx 0.857$ , illustrating how a non-zero discrepancy can still support substantial factual adequacy.

Significance and Claims
The paper claims to offer a restricted reformulation of partial truth theories rather than a new mathematical theory of averaging. Its primary significance lies in providing a tool for conditional comparisons of factual adequacy among scientific models and theories.

It does not provide a unique method for choosing scales or weights, nor does it offer a numerical calculus for arbitrary formulas (negation and disjunction are not recursively defined).
It clarifies that scientific progress need not be viewed as convergence toward a complete, final truth. Instead, progress can be understood as the construction of representations that become increasingly adequate in the respects most relevant to the problems science seeks to solve.
The framework distinguishes "partial factual truth" (graded adequacy under explicit criteria) from Elgin's concept of being "true enough" (epistemic value of idealizations), focusing strictly on the quantitative assessment of empirical claims.

The author emphasizes that the valuation is an estimate of adequacy relative to available evidence and methodological warrants; it does not guarantee correspondence with reality, nor does it replace the need for rigorous empirical testing and the identification of systematic errors.

Technical Summary: A Theory of Partial Truth for Scientific Statements

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