➘ #35 Source Article

Symbolic Logic Formulation

Language and Domain

Annotation: The entities we reason about are penalties (P), agents (A), durations (T), values (V), and dimensions (D).

Annotation: These predicates describe relationships such as “penalty p requires duration t,” “agent a has value v,” “agent a serves penalty p for time t,” “agent a satisfies penalty p,” “two dimensions are incommensurable,” and so on.

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Axioms

Annotation: If a penalty is defined by a time requirement, then anyone who satisfies it must serve that time.

Annotation: Duration and value are incommensurable kinds — they can’t be measured on a common scale.

Annotation: If two dimensions are incommensurable, there exists no conversion function from one to the other — you can’t trade time for value in any legitimate way.

Annotation: Any time-defined penalty is non-fungible; it can’t be replaced or commuted by other forms of payment or value.

Annotation: For justice to hold, each just penalty must be satisfied by someone.

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Cross-Dimensional Block Principles

Annotation: Having any degree of value, even infinite, does not by itself satisfy a time-based penalty. Value cannot function as temporal payment.

Annotation: If the penalty is tied to a specific offender’s identity, then having someone else serve it changes the nature of the act — it’s no longer the same kind of fulfillment.

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Theorem: Value Cannot Satisfy Duration

Annotation: Even if an agent has great or infinite value, if the penalty requires time, then serving less than the required duration fails to satisfy it. Value cannot substitute for duration.

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Substitution Impossibility Applied to PSA

Let

and assume

Then from D1:

But this contradicts .

Therefore:

Annotation:
If the penalty’s essence is eternal duration, and Christ served only three days, the conditions for satisfaction are not met — regardless of Christ’s infinite worth or divinity. The substitution fails by categorical mismatch rather than by mere magnitude.

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A Fitch-Style Proof

1. 
   Annotation: If a penalty is defined by a time requirement, then anyone who satisfies it must serve exactly that required time.
2. 
   Annotation: The penalty e (eternal punishment) is defined by infinite duration.
3. 
   Annotation: The substitute agent j (Jesus) has infinite value.
4. 
   Annotation: The duration of Jesus’s suffering is three days, a finite temporal span.
5. 
   Annotation: Three days is not equivalent to eternity.
6. 
   Annotation: If two properties are incommensurable (like time and value), there is no legitimate conversion function between them.
7. 
   Annotation: Duration and value are incommensurable dimensions; they cannot be exchanged on a single scale.
8. 
   Annotation: Any penalty whose essence is time served cannot be replaced by another currency of satisfaction.
9. 
   Annotation: Justice requires that each penalty is satisfied by some agent, not necessarily that substitution is allowed.
10. 
    Annotation: Even infinite value does not satisfy a time-defined penalty, since the currencies are non-convertible.
11. 
    Annotation: When a penalty is tied to a specific person, it cannot be fulfilled by another; the act type changes.
12. 
    Annotation: Serving a penalty for a specific duration excludes having served it for any different duration.
13. 
    Annotation: If the penalty requires eternity, any agent who satisfies it must have served for eternity.
14. 
    Annotation: Suppose, for argument’s sake, that Jesus’s act does satisfy the eternal penalty.
15. 
    Annotation: From the assumption and the requirement, it follows that Jesus must have served eternally.
16. 
    Annotation: We note both the finite service and the difference between three days and eternity.
17. 
    Annotation: By the rule of exact service, if Jesus served three days, he did not serve infinity.
18. 
    Annotation: Therefore, Jesus did not serve eternity.
19. 
    Annotation: A contradiction arises — Jesus both did and did not serve eternity.
20. 
    Annotation: Discharging the assumption, we conclude that Jesus does not satisfy the eternal penalty.
21. 
    Annotation: For any agent, serving a finite duration cannot satisfy an eternal penalty.
22. 
    Annotation: Generally, if the required duration differs from the one actually served, satisfaction fails — regardless of value.
23. 
    Annotation: The penal substitution claim fails logically; a finite event cannot satisfy an essentially eternal, durational penalty. The incommensurability of time and value makes substitution impossible.

◉ A plain English walkthrough of the Master Proof above.

1. The proof starts with a basic axiom about duration-defined penalties:
   if a penalty’s essence is the serving of time, then any agent who truly satisfies it must serve exactly that time. This is our foundation (line 1).
2. The specific case being tested is the Christian claim that Jesus satisfied the penalty of “eternal punishment.”
   So, we set up three core facts:
   • The penalty e is defined as eternal (line 2).
   • Jesus has infinite value (line 3).
   • Jesus’s actual suffering lasted only three days (line 4).
3. Arithmetic tells us that three days is not the same as eternity (line 5).
   That may sound trivial, but it’s formally necessary for the contradiction later.
4. The next few axioms establish the dimensional limits of substitution:
   • Time and value are incommensurable; there’s no legitimate function that converts one into the other (lines 6–7).
   • Therefore, any penalty defined by time cannot be replaced by any quantity of value (line 8).
   • Justice, however, always requires some real satisfaction of the penalty, not merely symbolic replacement (line 9).
   • This is codified by the rule that “value cannot substitute for duration” (line 10) and the “identity rule,” meaning one person cannot literally serve another’s penalty because the act type itself changes (line 11).
5. A precision rule is introduced: if an agent serves some duration t₁, they can’t be said to have served a different t₂ for the same penalty (line 12).
   This captures what the paper calls the “exactness” of temporal service.
6. Applying these general principles to the eternal-punishment case, we reason as follows:
   Since e is eternal, anyone who satisfies it must have served eternity (line 13).
7. We then assume, for the sake of contradiction, that Jesus’s three days do satisfy the eternal penalty (line 14).
   If that were true, then Jesus must have served eternity (line 15).
8. But earlier we established that Jesus served only three days and that three ≠ eternity (lines 4–5 combined → line 16).
   From the “exactness” rule (line 12), it follows that anyone who served three days did not serve eternity (line 17).
   Therefore, Jesus did not serve eternity (line 18).
9. This produces a direct contradiction: Jesus both did and did not serve eternity (line 19).
   The contradiction forces us to reject the assumption that he satisfied the penalty (line 20).
10. Once the assumption is discharged, the general conclusion follows:
    for any agent a, if the penalty requires eternal duration and the agent serves a finite duration, then the agent does not satisfy the penalty (line 21).
    In abstract form, if a penalty requires a specific time t, and someone serves a different t′, satisfaction fails (line 22).
11. The final line (line 23) expresses the formal outcome: no finite act can satisfy an essentially eternal penalty, even if the act’s agent is of infinite worth. The two relevant properties—time and value—belong to different categories and cannot substitute for one another.

Summary Insight

This proof dismantles penal substitution not by denying divine power or worth, but by exposing a category error: the attempt to pay a duration-defined debt with a value-defined payment. Time and value do not exist on a shared metric, so infinite value cannot bridge the dimensional gap. Hence, even if Jesus possesses infinite worth, that does not (and cannot) make three days of suffering equivalent to an eternal punishment.

◉ Narrative Summary

The argument begins by taking seriously what the doctrine of penal substitutionary atonement (PSA) itself asserts: that Jesus’s death and suffering “satisfy” the penalty that humans supposedly deserve for sin — a penalty described as eternal punishment. The paper’s goal is not to deny divine capacity or theological claims in general, but to examine whether this particular transaction makes logical sense if one treats the penalty as a definable category of justice.

The proof proceeds by distinguishing two dimensions of penalty: duration and value. Duration-based penalties—like imprisonment or eternal torment—are defined by time served, not by value exchanged. Value-based penalties—like fines—can be satisfied by substitution, because money is fungible. This distinction matters because time and value, as the proof formalizes, belong to different logical dimensions. There is no legitimate conversion function that turns a measure of value into an amount of time; they are incommensurable.

Given this, the paper assigns symbolic predicates to express what must be true for any penalty satisfaction claim to be valid. If a penalty requires a certain duration, then anyone who satisfies it must have actually served that duration. This rule captures the intuitive and institutional meaning of “time served.” When applied to the doctrine of PSA, this yields three facts: (1) the penalty is eternal; (2) Christ’s suffering lasted three days; and (3) three days is not eternity. From this, the proof derives that if Jesus satisfies the eternal penalty, he must have served eternity—yet the record shows that he served only three days.

At that point, the logic hits a contradiction: Jesus both did and did not serve eternity. The contradiction is fatal to the satisfaction claim. It means the substitution cannot count as genuine satisfaction of a duration-defined penalty. The problem is not quantitative—it’s not that three days is too short—but categorical: value and duration do not trade on the same axis.

The broader implication is that the familiar Christian apologetic appeal to Christ’s “infinite worth” or “infinite suffering intensity” does not solve the mismatch. Increasing the intensity of suffering does not change its duration, and assigning infinite worth does not transform three days into eternity. To claim otherwise commits what the author calls the Valuation Fallacy—the false assumption that a high-value act can replace a time-defined requirement.

The proof ends with a general schema showing that this flaw would undermine any justice theory that allows value-substitution for time-defined penalties. The logical outcome is clear: if the penalty is essentially durational, only in-kind duration can satisfy it. Atonement theories that rely on cross-dimensional exchange—paying time with value—collapse under their own formal definitions.

The narrative consequence is that the atonement, as traditionally cast in penal-substitutionary terms, fails by internal logic. It confuses satisfaction (meeting the penalty’s essence) with commutation (mercifully waiving it). A coherent alternative, the paper suggests, would have to abandon the transactional model altogether or redefine the penalty into a non-durational form—effectively conceding that the original doctrine cannot stand as stated.