#32-Formalization

September 29, 2025

➘ #32 Source Article

Symbolic Logic Formalization

Rehabilitation Aim

$\forall x , (Purpose(x,Rehab) \rightarrow [FinitePun(x) \wedge PossibleChange(x)])$
Annotation: For any $x$ , if the punishment of $x$ has the purpose of rehabilitation, then $x$ ’s punishment must be finite and allow the possibility of demonstrating change.

$\forall x , (EternalPun(x) \rightarrow [\neg FinitePun(x) \wedge \neg PossibleChange(x)])$
Annotation: For any $x$ , if $x$ is eternally punished, then $x$ ’s punishment is not finite and $x$ cannot possibly show change.

$\therefore \forall x , (EternalPun(x) \rightarrow \neg Purpose(x,Rehab))$
Annotation: Therefore, eternal punishment cannot serve the purpose of rehabilitation.

Retributive Proportionality

$\forall x , (Purpose(x,Retrib) \rightarrow Prop(x))$
Annotation: For any $x$ , if the punishment aims at retribution, then proportionality must be satisfied.

$\forall x , OffenseFin(x)$
Annotation: For any $x$ , the offense committed is finite.

$\forall x , (EternalPun(x) \rightarrow PunInf(x))$
Annotation: For any $x$ , if $x$ receives eternal punishment, then $x$ ’s punishment is infinite.

$\forall x , ([OffenseFin(x) \wedge PunInf(x)] \rightarrow \neg Prop(x))$
Annotation: For any $x$ , if the offense is finite but the punishment is infinite, proportionality is violated.

$\therefore \forall x , (EternalPun(x) \rightarrow \neg Purpose(x,Retrib))$
Annotation: Therefore, eternal punishment cannot serve the purpose of retribution.

Appeasement of Wrath and Stability

$\forall y , (Stable(y) \rightarrow \forall x , \exists t , [Finite(t) \wedge PunDur(x)=t \wedge Purpose(x,Appease(y)) \rightarrow WrathAppeased(y)])$
Annotation: For any agent $y$ , if $y$ is emotionally stable, then for any $x$ , there exists some finite time $t$ such that punishment of duration $t$ fulfills the purpose of appeasement and $y$ ’s wrath is satisfied.

$\forall x , (EternalPun(x) \rightarrow Infinite(PunDur(x)))$
Annotation: For any $x$ , eternal punishment implies that the duration of $x$ ’s punishment is infinite.

$\forall y \forall x , (Infinite(PunDur(x)) \rightarrow \neg \exists t , [Finite(t) \wedge WrathAppeased(y) , by , time , t])$
Annotation: For any $y$ and $x$ , if the punishment duration is infinite, then there does not exist a finite time at which $y$ ’s wrath is appeased.

$Stable(y)$
Annotation: Assume the offended party $y$ is emotionally stable.

$\therefore \forall x , (EternalPun(x) \rightarrow \neg Purpose(x,Appease(y)))$
Annotation: Therefore, eternal punishment cannot serve the purpose of appeasement if the agent is emotionally stable.

A Fitch-Style Proof.

Fitch 1 — Rehabilitation Aim

$1.\ \forall x\,(Purpose(x,Rehab)\rightarrow (FinitePun(x)\wedge PossibleChange(x))).\quad \text{Premise}$
Annotation: For any $x$ , if the punishment aims at rehabilitation, then it must be finite and allow the offender a chance to change.

$2.\ \forall x\,(EternalPun(x)\rightarrow (\neg FinitePun(x)\wedge \neg PossibleChange(x))).\quad \text{Premise}$
Annotation: For any $x$ , if the punishment is eternal, then it is not finite and change is not possible.

$3.\ EternalPun(a).\quad \text{Assumption}$
Annotation: Assume, for an arbitrary $a$ , that eternal punishment applies.

$4.\ EternalPun(a)\rightarrow (\neg FinitePun(a)\wedge \neg PossibleChange(a)).\quad \forall\text{-elim on (2)}$
Annotation: Specialize line (2) to the case of $a$ .

$5.\ \neg FinitePun(a)\wedge \neg PossibleChange(a).\quad \text{MP from (3),(4)}$
Annotation: From (3) and (4), it follows that $a$ has neither finite punishment nor possible change.

$6.\ \neg PossibleChange(a).\quad \wedge\text{-elim on (5)}$
Annotation: Extract the second conjunct from (5).

$7.\ Purpose(a,Rehab)\rightarrow (FinitePun(a)\wedge PossibleChange(a)).\quad \forall\text{-elim on (1)}$
Annotation: Specialize line (1) to $a$ .

$8.\ Purpose(a,Rehab).\quad \text{Assumption}$
Annotation: Suppose rehabilitation is the purpose of punishing $a$ .

$9.\ FinitePun(a)\wedge PossibleChange(a).\quad \text{MP from (7),(8)}$
Annotation: If rehabilitation is the purpose, both finitude and change must be present.

$10.\ PossibleChange(a).\quad \wedge\text{-elim on (9)}$
Annotation: From (9), extract the possibility of change for $a$ .

$11.\ \bot.\quad \text{from (6),(10)}$
Annotation: Contradiction: $PossibleChange(a)$ and $\neg PossibleChange(a)$ both appear.

$12.\ \neg Purpose(a,Rehab).\quad \text{neg-intro from (8)-(11)}$
Annotation: Therefore, rehabilitation cannot be the purpose of eternal punishment.

$13.\ EternalPun(a)\rightarrow \neg Purpose(a,Rehab).\quad \text{cond-intro from (3)-(12)}$
Annotation: Discharge assumption (3) to derive the conditional.

$14.\ \forall x\,(EternalPun(x)\rightarrow \neg Purpose(x,Rehab)).\quad \forall\text{-intro from (13)}$
Annotation: Generalize to all $x$ : eternal punishment cannot serve rehabilitation.

Fitch 2 — Retributive Proportionality

$15.\ \forall x\,(Purpose(x,Retrib)\rightarrow Prop(x)).\quad \text{Premise}$
Annotation: For any $x$ , if punishment aims at retribution, proportionality must hold.

$16.\ \forall x\,OffenseFin(x).\quad \text{Premise}$
Annotation: All offenses are finite in scope.

$17.\ \forall x\,(EternalPun(x)\rightarrow PunInf(x)).\quad \text{Premise}$
Annotation: For any $x$ , eternal punishment entails an infinite penalty.

$18.\ \forall x\,((OffenseFin(x)\wedge PunInf(x))\rightarrow \neg Prop(x)).\quad \text{Premise}$
Annotation: For any $x$ , a finite offense combined with an infinite punishment breaks proportionality.

$19.\ EternalPun(a).\quad \text{Assumption}$
Annotation: Assume eternal punishment for arbitrary $a$ .

$20.\ EternalPun(a)\rightarrow PunInf(a).\quad \forall\text{-elim on (17)}$
Annotation: Specialize (17) to $a$ .

$21.\ PunInf(a).\quad \text{MP from (19),(20)}$
Annotation: From eternal punishment for $a$ , its penalty is infinite.

$22.\ OffenseFin(a).\quad \forall\text{-elim on (16)}$
Annotation: From (16), $a$ ’s offense is finite.

$23.\ (OffenseFin(a)\wedge PunInf(a))\rightarrow \neg Prop(a).\quad \forall\text{-elim on (18)}$
Annotation: Specialize (18) to $a$ .

$24.\ OffenseFin(a)\wedge PunInf(a).\quad \wedge\text{-intro from (21),(22)}$
Annotation: Combine (21) and (22).

$25.\ \neg Prop(a).\quad \text{MP from (23),(24)}$
Annotation: Proportionality fails for $a$ .

$26.\ Purpose(a,Retrib)\rightarrow Prop(a).\quad \forall\text{-elim on (15)}$
Annotation: If retribution is the purpose for $a$ , proportionality must hold.

$27.\ Purpose(a,Retrib).\quad \text{Assumption}$
Annotation: Suppose retribution is the purpose of punishing $a$ .

$28.\ Prop(a).\quad \text{MP from (26),(27)}$
Annotation: From (26) and (27), proportionality must hold.

$29.\ \bot.\quad \text{from (25),(28)}$
Annotation: Contradiction: $Prop(a)$ and $\neg Prop(a)$ .

$30.\ \neg Purpose(a,Retrib).\quad \text{neg-intro from (27)-(29)}$
Annotation: Therefore, retribution cannot be the purpose of eternal punishment.

$31.\ EternalPun(a)\rightarrow \neg Purpose(a,Retrib).\quad \text{cond-intro from (19)-(30)}$
Annotation: Discharge assumption (19) to form the conditional.

$32.\ \forall x\,(EternalPun(x)\rightarrow \neg Purpose(x,Retrib)).\quad \forall\text{-intro from (31)}$
Annotation: Generalize to all $x$ : eternal punishment cannot serve retribution.

Fitch 3 — Appeasement and Emotional Stability

$33.\ Stable(y).\quad \text{Assumption}$
Annotation: Assume an arbitrary offended agent $y$ is emotionally stable.

$34.\ Stable(y)\rightarrow \forall x\,\exists t\,(Finite(t)\wedge (Purpose(x,Appease(y))\rightarrow AppeasedBy(y,t))).\quad$ $\text{Premise}$
Annotation: If $y$ is stable, then for each $x$ there is a finite time that would suffice for appeasement if appeasement is the aim.

$35.\ \forall x\,(EternalPun(x)\rightarrow Infinite(PunDur(x))).\quad \text{Premise}$
Annotation: Eternal punishment implies infinite duration.

$36.\ \forall x\,(Infinite(PunDur(x))\rightarrow \neg \exists t\,(Finite(t)\wedge AppeasedBy(y,t))).\quad \text{Premise}$
Annotation: Infinite duration prevents any finite time from appeasing $y$ .

$37.\ EternalPun(a).\quad \text{Assumption}$
Annotation: Assume eternal punishment for arbitrary $a$ .

$38.\ Infinite(PunDur(a)).\quad \text{from (35),(37)}$
Annotation: From eternal punishment, the duration for $a$ is infinite.

$39.\ \neg \exists t\,(Finite(t)\wedge AppeasedBy(y,t)).\quad \text{from (36),(38)}$
Annotation: From infinite duration, there can be no finite appeasement time.

$40.\ \forall x\,\exists t\,(Finite(t)\wedge (Purpose(x,Appease(y))\rightarrow AppeasedBy(y,t))).\quad$ $\text{from (33),(34)}$
Annotation: From stability, there exists a finite time for each $x$ if appeasement is the aim.

$41.\ \exists t\,(Finite(t)\wedge (Purpose(a,Appease(y))\rightarrow AppeasedBy(y,t))).\quad \forall\text{-elim on (40)}$
Annotation: Specialize (40) to $a$ .

$42.\ Finite(c)\wedge (Purpose(a,Appease(y))\rightarrow AppeasedBy(y,c)).\quad \exists\text{-elim on (41)}$
Annotation: Choose a fresh witness $c$ for the finite appeasement time.

$43.\ Purpose(a,Appease(y)).\quad \text{Assumption}$
Annotation: Suppose appeasement is the purpose of punishing $a$ .

$44.\ Purpose(a,Appease(y))\rightarrow AppeasedBy(y,c).\quad \wedge\text{-elim on (42)}$
Annotation: From (42), if appeasement is the purpose, then $y$ is appeased at $c$ .

$45.\ AppeasedBy(y,c).\quad \text{MP from (43),(44)}$
Annotation: From assumption, $y$ is appeased at $c$ .

$46.\ Finite(c).\quad \wedge\text{-elim on (42)}$
Annotation: The chosen witness $c$ is finite.

$47.\ \exists t\,(Finite(t)\wedge AppeasedBy(y,t)).\quad \exists\text{-intro from (45),(46)}$
Annotation: Therefore, there exists a finite time at which $y$ is appeased.

$48.\ \bot.\quad \text{from (39),(47)}$
Annotation: Contradiction: both existence and non-existence of finite appeasement.

$49.\ \neg Purpose(a,Appease(y)).\quad \text{neg-intro from (43)-(48)}$
Annotation: Therefore, appeasement is not the purpose of eternal punishment.

$50.\ EternalPun(a)\rightarrow \neg Purpose(a,Appease(y)).\quad \text{cond-intro from (37)-(49)}$
Annotation: Discharge assumption (37) to form the conditional.

$51.\ \forall x\,(EternalPun(x)\rightarrow \neg Purpose(x,Appease(y))).\quad \forall\text{-intro from (50)}$
Annotation: Generalize to all $x$ : eternal punishment cannot serve appeasement.

Consolidated Master Result

$52.\ \forall x\,(EternalPun(x)\rightarrow (\neg Purpose(x,Rehab)\wedge \neg Purpose(x,Retrib)\wedge \neg Purpose(x,Appease(y)))).\quad \text{from (14),(32),(51)}$
Annotation: Consolidated result: for any $x$ , eternal punishment fails rehabilitation, retribution, and appeasement.

◉ A plain English walkthrough of the Master Proof above.

Rehabilitation Aim (Equations 1–14)

We start by defining what rehabilitation requires: if punishment aims at rehabilitation, then it must be finite and provide the offender a chance to change (1). But eternal punishment, by definition, is not finite and allows no opportunity for change (2).

We assume eternal punishment for an arbitrary individual $a$ (3). From the premises, we infer that $a$ has neither finitude nor the possibility of change (5–7). We then assume for contradiction that the punishment of $a$ had the purpose of rehabilitation (8). That assumption forces us to conclude that $a$ must have the possibility of change (9–10). But this contradicts the earlier result that $a$ cannot change (11). From this contradiction, we conclude that rehabilitation is not the purpose (12).

We discharge the assumption and generalize: for any $x$ , if punishment is eternal, then its purpose cannot be rehabilitation (13–14).

Retributive Proportionality (Equations 15–32)

Next, we analyze whether eternal punishment could serve retribution. Retribution requires proportionality between offense and penalty (15). All offenses are finite (16), but eternal punishment entails an infinite penalty (17). And if a finite offense receives an infinite penalty, proportionality fails (18).

Again, take an arbitrary $a$ and assume eternal punishment (19). This forces us to conclude $a$ has an infinite penalty (21) but a finite offense (22). Combining these yields disproportionality (25). But if retribution were the purpose, proportionality would have to hold (26). Assuming for contradiction that retribution is the purpose (27), we are forced into a direct clash: $a$ both has and does not have proportionality (28–29).

We conclude that retribution cannot be the purpose of eternal punishment (30). By conditional proof and generalization, we state that for any $x$ , eternal punishment cannot serve retribution (31–32).

Appeasement and Stability (Equations 33–51)

The third analysis examines whether eternal punishment could serve appeasement. Suppose the offended party $y$ is emotionally stable (33). Emotional stability implies satiability: if appeasement is the purpose, there must exist some finite time that suffices to appease $y$ (34). But eternal punishment entails infinite duration (35), and infinite duration ensures that there can be no finite appeasing time (36).

Again, take an arbitrary $a$ and assume eternal punishment (37). From this, we derive that $a$ ’s punishment is infinite in duration (38), and thus there is no finite time of appeasement (39). But stability also implies there is such a finite time if appeasement is the purpose (40–41). From this, we introduce a specific finite time $c$ (42). Assuming appeasement is the purpose (43), we conclude $y$ is appeased at that finite time $c$ (45–46). This directly contradicts the earlier result that no such finite time exists (47–48).

Therefore, appeasement cannot be the purpose of eternal punishment (49). By conditional proof and generalization, for any $x$ , eternal punishment cannot serve appeasement (50–51).

Consolidated Conclusion (Equation 52)

Finally, we conjoin the three results: eternal punishment cannot serve rehabilitation (14), cannot serve retribution (32), and cannot serve appeasement (51).

Thus, for any $x$ , eternal punishment has no coherent purpose under any of the three major aims of punishment (52).

◉ Narrative Summary

The argument examines whether eternal punishment can serve any of the three classic aims of punishment—rehabilitation, retribution, or appeasement.

First, consider rehabilitation. Rehabilitation presupposes that punishment is finite and that the offender has the opportunity to change. Eternal punishment, by definition, has no endpoint and denies any possibility of reform. When we test the hypothesis that eternal punishment could be rehabilitative, it collapses into contradiction: eternal punishment simultaneously requires and denies the possibility of change. Thus, eternal punishment cannot serve the purpose of rehabilitation.

Next, consider retribution. Retribution requires proportionality between the severity of the offense and the severity of the punishment. Offenses committed by human beings are finite in nature. Eternal punishment, however, is infinite in scope. Combining a finite offense with an infinite punishment generates disproportionality by definition. Attempting to maintain that eternal punishment could be retributive leads directly into contradiction: it must be proportional, and yet it cannot be proportional. Therefore, eternal punishment cannot serve the purpose of retribution.

Finally, consider appeasement. An emotionally stable being can, in principle, be appeased within a finite period of punishment. If eternal punishment were to serve the purpose of appeasement, there would have to be some finite point at which appeasement is achieved. But eternal punishment extends infinitely, and infinite duration rules out the possibility of appeasement at any finite time. This creates another contradiction: appeasement both must occur at a finite time and cannot occur at any finite time. Consequently, eternal punishment cannot serve the purpose of appeasement either.

Taken together, the three analyses converge on the same conclusion: eternal punishment cannot rationally fulfill any legitimate purpose of punishment. It is neither rehabilitative, nor retributive, nor appeasing. As such, eternal punishment stands outside the domain of coherent or justifiable penal aims.

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