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REVIEW 3 major objections 6 minor 36 references

Possibility of a real-world event is defined as the Łukasiewicz evaluation of its prerequisites and constraints.

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2026-08-04 12:37 UTC pith:BCYKZ7LJ

load-bearing objection A clearly-written consolidation of Schwartz's earlier event-possibility work; formal core is sound, but the 'objective' claim is overstated because Poss(E) depends on a subjective choice of complete context. the 3 major comments →

arxiv 2510.02655 v3 pith:BCYKZ7LJ submitted 2025-10-03 cs.AI

A Concept of Possibility for Real-World Events

Daniel G. Schwartz This is my paper
classification cs.AI
keywords possibility theoryreal-world eventsprerequisites and constraintsŁukasiewicz logicplanningfeasibilityvehicle route planningdecision analysis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes replacing the subjective, fuzzy-set-based notion of possibility with a probability-driven degree for real-world events. The core idea is that every event has prerequisites that enable it and constraints that impede it; the event's possibility equals the Łukasiewicz-logic value of their conjunction, computed as the minimum of the probabilities of prerequisites and of negated constraints, with maximum for alternatives. The author argues this yields a measure of the 'ease' of an event that can be grounded in statistical data, making it more applicable than earlier subjective notions. A theorem shows that two ways of writing the same context give the same possibility degree when they are strongly equivalent in a restricted logical sense, and a vehicle route-planning example shows how the degree could drive dynamic plan choice.

Core claim

Given an event E, choose a complete list of prerequisites p_i and constraints c_j, form a contextual construct C by combining them with conjunction and disjunction, assign each atomic item its probability, and interpret negation, conjunction, and disjunction as 1−, min, and max. Then the possibility of E is v(C). The paper claims this provides an objective, computational measure of the ease of E: the probabilities can come from statistical sampling, and the min/max combination captures how the least available prerequisite or least mitigated constraint limits the event. It also proves that the degree is invariant under reordering and reassociating a context after conversion to disjunctive nor

What carries the argument

The central object is the complete contextual construct, a Boolean expression built from prerequisites and negated constraints and treated as a proposition. The mechanism is the Łukasiewicz possibility valuation v: atomic constructs map to their probabilities, conjunction to min, disjunction to max, and negation to 1−, so Poss(E) = v(C). The strong-equivalence theorem ensures that any two complete contexts whose disjunctive normal forms differ only by commutative and associative rearrangements yield the same possibility degree, giving the construction a form of well-definedness.

Load-bearing premise

The computation assumes the atomic prerequisite and constraint probabilities are statistically independent, and it takes the human-chosen 'complete' context as genuinely complete; if either fails, Poss(E) = v(C) may not correspond to the actual ease of the event.

What would settle it

Take one leg of the waypoint example where rain and traffic accidents are positively correlated. Compute v(C) with the min formula and compare it with a model that uses their joint distribution; if the two values diverge enough to flip which route is judged most possible, the independence premise is load-bearing. A reader could also compare route rankings from real traffic data against observed traversal ease.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Planners can rank competing plans by a min/max score computed from probability data rather than from subjective possibility assignments.
  • Route or project choices become dynamically reassessable: when a constraint probability changes, such as a traffic accident report jumping to 1, the possibility degree re-evaluates immediately and the chosen path can be updated.
  • The same structure transfers to any activity/event pair with prerequisites and constraints, so robot action planning, project networks, and communication routing can each inherit the same feasibility measure.
  • The min aggregation stays high even when several low-probability constraints exist, unlike a probabilistic product, matching the intuition that a plan is only as possible as its weakest element.
  • Classical logical equivalence is not enough for two contexts to give the same degree; the paper identifies strong equivalence as the needed equivalence notion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural stress test not pursued in the paper: if constraints are correlated, say rain makes accidents more likely, the min/max formula ignores the joint distribution. A testable extension would replace atomic probabilities with joint or conditional probabilities while keeping the Łukasiewicz connectives, and check whether route rankings survive.
  • The 'complete context' depends on a human choice, so the possibility degree is arguably relative to an explicitly stated context rather than a property of the event alone. One could measure how sensitive the resulting rankings are when additional prerequisites or constraints are added.
  • The paper speculates that this min/max reasoning matches ordinary human planning. That is directly testable: present humans with the same probabilities and see whether their judged 'ease' of routes follows min/max rather than product-probability reasoning.
  • The strong-equivalence theorem relies specifically on min and max; replacing them with other t-norms would break the invariance. This suggests a boundary, worth exploring, for how far the logical connectives can be varied before plan rankings become order-dependent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a new notion of possibility for real-world events, distinct from Zadeh's fuzzy-set-based possibility theory. An event is described by a "contextual construct" built from prerequisites (p), constraints (c) represented as negated constraints (¬c), and combined with ∧ and ∨. The possibility valuation v is defined by assigning probabilities to atomic constructs and aggregating with min, max, and 1− (Łukasiewicz logic). If C is a complete contextual construct for event E, then Poss(E)=v(C). The paper proves that strongly equivalent constructs yield equal possibility degrees, illustrates the framework on a vehicle route-planning example, compares with probability theory, and suggests applications in robotics, project planning, computer networks, and games.

Significance. If the central definition were well-defined, the approach would offer a simple, computational, probability-based alternative to existing possibility theory, with potential value in planning and decision analysis. The formal theorem in Section 5, once repaired, would provide a precise invariance condition for the proposed measure. The paper is self-contained, gives an explicit algorithm and worked example, and correctly identifies that classical logical equivalence is insufficient for invariance. However, the claimed objectivity is not supported, because the notion of a "complete" contextual construct is left informal and the measure is not a function of the event alone. The paper is a promising starting point, not yet a finished theory.

major comments (3)
  1. [§3, §5 (definition of completeness and Scott's example)] Poss(E) is not well-defined as a function of E. §3 states that completeness is "fundamentally intuitive and inherently ambiguous." Since no formal criterion is given, two users may choose complete contexts for the same event that are classically equivalent but not strongly equivalent; Scott's example in §5, (p∨¬p)∧q vs (p∧¬p)∨q with v(p)=0.5, v(q)=0, gives 0 vs 0.5. The strong-equivalence theorem does not constrain this choice, so Poss(E) depends on an arbitrary syntactic selection. This contradicts the claimed objectivity in the Introduction and Conclusion. The fix is either to relativize Poss to a chosen context, e.g., Poss(E,C), or to impose a canonical completeness criterion that guarantees strong equivalence of all legitimate complete contexts.
  2. [§5 (conv algorithm and Theorem proof)] The conversion algorithm conv is incomplete. It includes distribution rules for p∨(q∧r) and (p∧q)∨r but not the dual rule p∧(q∨r) ⇒ (p∧q)∨(p∧r). The example in §5 applies exactly this missing rule when rewriting ¬c1∧(¬c2∨p3) as (¬c1∧¬c2)∨(¬c1∧p3). Thus the proof's claim that "only items 5, and 6 will be employed" is false, and the Theorem is not proved as written. The theorem is probably salvageable by adding the missing distribution rule, but the formal argument needs correction. Also, item 2 is labeled "Idempotence Law" but is actually double negation.
  3. [§3 (independence assumption)] The assumption that "the atomic contextual constructs are statistically independent" is not used anywhere: v(C) is computed via min/max, not products, and the theorem does not rely on independence. In the waypoint example (§7), constraints such as bad weather (c2) and traffic accidents (c3) are plausibly correlated, so the assumption is unrealistic. If the goal is to provide objective probabilities, independence is neither necessary nor justified. The paper should either remove the assumption or explicitly show how it is used in the derivation.
minor comments (6)
  1. [§4, Eq. (4)] Equation (4) is missing a closing parenthesis after "environmental_issue"; it should read min[Prob(capital), Prob(employees), 1−Prob(environmental_issue), max[...]].
  2. [§5 (proof text)] In the proof of the Theorem, "conceptual constructs" should be "contextual constructs."
  3. [§8] The text refers to "the case of Jane in Section 2," but Section 2 contains no Jane; the example uses ABC Corporation. Either rename or correct the reference.
  4. [References [33]] Reference [33] is cited as the source of the DNF conversion algorithm, but the listed Wikipedia page is "Probability axioms," not a page about disjunctive normal form. The citation appears to be incorrect.
  5. [Abstract and Introduction] The claim that the approach "has nothing in common" with Zadeh's theory besides Łukasiewicz connectives is overstated: the operations min, max, and 1− are also used in standard possibility theory. The paper should acknowledge this overlap more carefully.
  6. [§8] The argument that probability theory would give "close to 0" for a conjunction of several probable prerequisites is not compelling; the product of many high probabilities can still be reasonably large. This comparison is not essential to the paper's main proposal.

Circularity Check

0 steps flagged

No significant circularity: Poss(E) is explicitly defined from input probabilities, and the paper's equivalence theorem is a proved property of that definition, not a fitted prediction.

full rationale

The paper's central definition, "if C is a complete contextual construct for E, set Poss(E) = v(C)" (Section 3), directly constructs the possibility degree from user-supplied prerequisite and constraint probabilities using Łukasiewicz min/max/1− valuations. There is no parameter fitted to a target quantity and no prediction that is statistically forced by the inputs. The strong-equivalence theorem in Section 5 is a genuine derived property of the definition, and the proof is given in the paper; the citation to the author's [29] is explicitly noted as a generalization and is not load-bearing. The acknowledged subjectivity of "complete contextual construct" is a well-definedness limitation rather than a circular reduction. Similarly, the complex-event propagation rule "Poss(E) = Poss(E′)" in Section 6 is an explicit modeling choice, not an attempt to derive a fitted result. The waypoint example applies the definition to illustrative probability values without using the outputs to determine the inputs. There is no self-citation chain used to force the central claim, no ansatz smuggled in via citation, and no renaming of a known result as a new derivation.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The theory's inputs are user-chosen probabilities and a user-chosen context; no new entities are introduced. The main assumptions are standard probability theory, statistical independence, and the subjective choice of context, which together limit the objectivity of the resulting possibility measure.

free parameters (3)
  • Selection of complete contextual construct
    The set of prerequisites and constraints chosen for an event is not determined by the theory; the paper acknowledges it 'depends on a human user’s perception'. This choice directly determines Poss(E), so it is a free parameter of the model.
  • Probability values for atomic constructs
    Probabilities Prob(p) and Prob(c) are inputs, provided subjectively or by sampling. They are not fitted by the theory, but they are free in the sense that the formalism imposes no constraints on them beyond [0,1].
  • Choice of inference operator for complex events
    In Section 6, the paper allows various operators for conditional possibility (e.g., Łukasiewicz, nine others from [6], ten from [26]). The choice affects how possibility propagates and is left to the user.
axioms (4)
  • standard math Kolmogorov probability axioms
    The paper explicitly uses a standard probability measure satisfying the Kolmogorov axioms (Section 3).
  • domain assumption Atomic contextual constructs are statistically independent
    Stated in Section 3: 'it is assumed that the atomic contextual constructs are statistically independent'. This is load-bearing for the min/max aggregation to be a meaningful combination.
  • domain assumption Completeness of a context is a well-defined, agreed-upon notion
    The paper defines a complete contextual construct as a 'full description' of the relevant context, but admits this is 'fundamentally intuitive and inherently ambiguous' (Section 3). The theory requires that users agree on this choice.
  • ad hoc to paper Łukasiewicz logic (min/max/1−) is a plausible model for human reasoning about plans
    The paper adopts Łukasiewicz connectives and speculates that this 'correctly captures normal human reasoning' (Section 8), but provides no empirical evidence.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of A Concept of Possibility for Real-World Events." pith.science (2026). https://pith.science/paper/BCYKZ7LJ

@misc{pith2026251002655,
  author       = {Pith},
  title        = {Pith review of: A Concept of Possibility for Real-World Events},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BCYKZ7LJ}},
  note         = {Machine review of arXiv:2510.02655}
}
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read the original abstract

This paper offers a new concept of {\it possibility} as an alternative to the now-a-days standard concept originally introduced by L.A. Zadeh in 1978. This new version was inspired by the original but, formally, has nothing in common with it other than that they both adopt the {\L}ukasiewicz multivalent interpretation of the logical connectives. Moreover, rather than seeking to provide a general notion of possibility, this focuses specifically on the possibility of a real-world event. An event is viewed as having prerequisites that enable its occurrence and constraints that may impede its occurrence, and the possibility of the event is computed as a function of the probabilities that the prerequisites hold and the constraints do not. This version of possibility might appropriately be applied to problems of planning. When there are multiple plans available for achieving a goal, this theory can be used to determine which plan is most possible, i.e., easiest or most feasible to complete. It is speculated that this model of reasoning correctly captures normal human reasoning about plans. The theory is elaborated and an illustrative example for vehicle route planning is provided. There is also a suggestion of potential future applications.

Figures

Figures reproduced from arXiv: 2510.02655 by Daniel G. Schwartz.

Figure 1
Figure 1. Figure 1: Example street network. 7. An Example: Vehicle Waypoint Navigation These ideas may be illustrated with a hypothetical real-world application. Consider the task of navigating a vehicle (with or without a driver) through a network of city streets. Suppose that, as depicted in [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

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Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.