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Detaining Criminals

The following is a formal demonstration that under some circumstances it is morally permissible to detain criminals. The question of moral optimality is then discussed informally.

Notation

  • Note: For notation not defined here see the rigorous theory of ethics.
  • Agents
    • d = Whomever does the Action to detain the criminal
    • c = Criminal
    • v = Victim of the original crime
    • Hf = the set of Agents whom might potentially be harmed by c in the future
  • Times
    • tm = The time of the original crime
    • td = The time that detention starts
    • tf = The time that detention ends
  • Actions
    • H = <c,M,tm,tm,v> – The Act by c of committing crime M against victim v at time tm
    • R = <d,D,td,tf,c> – The Act by d of detaining (D) the criminal c from td to tf
    • SH = {H}
    • SR = {R}

Situational Assumptions

  1. c≠d
  2. D∈Act
  3. <d,td,tf,c>∈D
  4. M∈Act
  5. <c,tm,tm,v>∈M
  6. tm < td
  7. Hf≠Ø
  8. Gain(SH) = {c}
  9. Gain(SR) = Hf
  10. Vctm(SH) = Hf ∪ {v}   — The original crime harms the victim and possible future victims
  11. Vctm(SR) = {c} — The only Agent harmed by detention is the criminal. This is a questionable assumption if detention of the criminal harms those who depend on him for any reason whatsoever!
  12. SH∉BAS ] — The crime was not part of a BAS. This could probably be demonstrated from more basic assumptions, but for simplicity I will just assume it outright.
  13. ∀v{ v∈Hf) ⇒ dNPWB(v,u0-min(SR)) | SR ≤ [ NPWB(v,u0-min(SH)) | SH(0) – NPWB(v,u0-min(SH)) ] }  — The benefit due to detention for anyone does not exceed the harm caused by the original crime.
  14. ¬∃SS[ R∈SS & SS∈RAS(SH,td) ]  — Detention of the criminal is not part of any other Restitution Action Set
  15. ¬∃SP[ SP∉BAS & Wrong(SH)⊆Gain(SP) & Gain(SH)⊆Vctm(SP) ]  — SH was not itself for restitution

Proof of Lemmas

  • (A) Proof that SH⊆Exec
    1. M∈Act  — Situational Assumption #4
    2. <c,tm,tm,v>∈M  — Situational Assumption #4
    3. <c,M,tm,tm,v>∈Exec  — (1), (2) & Def. of Exec
    4. H∈Exec  — (3) & Def. of H
    5. SH={H}  — Def. of SH
    6. ∀X{ X∈SH ⇒ X∈Exec }  — (4) & (5)
    7. SH⊆Exec  — (6) & Def. of ⊆
    8. Q.E.D.
  • (B) Proof that SR⊆Exec
    1. D∈Act  — Situational Assumption #2
    2. <d,td,tf,c>∈D  — Situational Assumption #3
    3. <d,D,td,tf,c>∈Exec  — (1), (2) & Def. of Exec
    4. R∈Exec  — (3) & Def. of R
    5. SR={R}  — Def. of SR
    6. ∀X{ X∈SR ⇒ X∈Exec }  — (4) & (5)
    7. SH⊆Exec  — (6) & Def. of ⊆
    8. Q.E.D.
  • (C) Proof that SH∈HAS
    1. SH⊆Exec  — Lemma (A)
    2. SH∉BAS  — Situational Assumption #12
    3. ∃e( e∈Vctm(SH) ) — Situational Assumption #10
    4. ∃e( e∈Wrong(SH) ) — (4) & Def. of Wrong
    5. Let X∈SH  — Assumption
    6. X = H  — (4) & Def. of SH
    7. Wrong(H) = Wrong(SH)  — Def. of SH
    8. Wrong(X) = Wrong(SH)  — (6) & (7)
    9. u0-min(H) = u0-min(SH)  — Def. of SH
    10. u0-min(X) = u0-min(SH)  — (6) & (9)
    11. u1-min(H) = u1-min(SH)  — Def. of SH
    12. u1-min(X) = u1-min(SH)  — (6) & (11)
    13. ∀X[ X∈SH ⇒ Wrong(X)=Wrong(SH) & u0-min(X)=u0-min(SH) & u1-min(X)=u1-min(SH) ]  — (5), (8), (10), & (12)
    14. ¬∃SP[ SP∉BAS & Wrong(SH)⊆Gain(SP) & Gain(SH)⊆Vctm(SP) ] — Situational Assumption #15
    15. SH∈HAS — (1), (2), (4), (13), (14) & Def. of HAS
    16. Q.E.D.
  • (D) Proof that SR∈RAS(SH)
    1. SH∈HAS  — Lemma (C)
    2. SR⊆Exec  — Lemma (B)
    3. Gain(SR) = Hf  — Situational Assumption #9
    4. Vctm(SH) = Hf ∪ {v}  — Situational Assumption #10
    5. ∀a[ a∈Gain(SR) ⇒ a∈Vctm(SH) ]  — (3), (4) & Def. of ∪
    6. Gain(SR)⊆Vctm(SH)  — (5) & Def. of ⊆
    7. Hf≠Ø  — Situational Assumption #7
    8. Gain(SR)≠Ø  — (3) & (7)
    9. Vctm(SR) = {c}  — Situational Assumption #11
    10. SR = {R}  — Def. of SR
    11. R = <d,D,td,tf,c>  — Def. of R
    12. {a: ∃A∃u0∃u1∃x1∃x2…∃xN[ <a,A,u0,u1,x1,x2…xN>∈SR ] } = {d}  — (10), (11) & Logic
    13. Actr(SR) = {d} — (12) & Def. of Actr
    14. c≠d  — Situational Assumption #1
    15. c∉Actr(SR) — (13) & (14)
    16. Wrong(SR) = {c}  — (9), (15) & Def. of Wrong
    17. SH = {H}  — Def. of SH
    18. H = <c,M,tm,tm,v>  — Def. of H
    19. {a: ∃A∃u0∃u1∃x1∃x2…∃xN[ <a,A,u0,u1,x1,x2…xN>∈SH ] } = {d}  — (17), (18) & Logic
    20. Actr(SH) = {c} — (19) & Def. of Actr
    21. ∀x[ x∈Wrong(SR) ⇒ x∈Actr(SH) ]  — (16) & (20)
    22. Wrong(SR)⊆Actr(SH)  — (21) & Def. of ⊆
    23. ∀v{ v∈Hf) ⇒ dNPWB(v,u0-min(SR)) | SR ≤ [ NPWB(v,u0-min(SH)) | SH(0) – NPWB(v,u0-min(SH)) ] }  — Situational Assumption #13
    24. ∀v{ v∈Gain(SR) ⇒ dNPWB(v,u0-min(SR)) | SR ≤ [ NPWB(v,u0-min(SH)) | SH(0) – NPWB(v,u0-min(SH)) ] }  — (3) & (23)
    25. SR∈RAS(SH)  — (1), (2), (6), (8), (22), (24), & Def. of RAS
    26. Q.E.D.
  • (E) Proof that R∉Ex(SR,td)
    1. tm < td — Situational Assumption #6
    2. R = <d,D,td,tf,c>  — Def. of R
    3. {u0: ∃a∃A∃u1∃x1∃x2…∃xN[ <a,A,u0,u1,x1,x2…xN> ∈ {R} ]} = {td}  — (2) & Logic
    4. min{u0: ∃a∃A∃u1∃x1∃x2…∃xN[ <a,A,u0,u1,x1,x2…xN> ∈ {R} ]} = td  — (3) & Def. of min
    5. u0-min({R}) = td — (4) & Def. of u0-min
    6. tm < u0-min({R})  — (1) & (5)
    7. ¬(u0-min({R}) < tm)  — (6) & Irreflexivity of <
    8. R∉Ex(SR,td) — (7) & Def. of Ex
    9. Q.E.D.
  • (F) Proof that SR∈RASO(SH,td)
    1. SR∈RAS(SH) — Lemma (D)
    2. R∈SR  — Def. of SR
    3. R∉Ex(SR,td)  — Lemma (E)
    4. ∃X[ X∈SR & X∉Ex(SR,td) ]  — (2) & (3)
    5. ¬∀X[ X∈SR ⇒ X∈Ex(SR,td) ] (4) & Logic
    6. SR∉CAS(td)  — (5) & Def. of CAS
    7. SR∈OAS(td)  — (6) & Def. of OAS
    8. SR∈RASO(SH,td) — (1), (7) & Def. of RASO
    9. Q.E.D.   
  • (G) Proof that SR∈RASA(SH,td)
    1. SR∈RASO(SH,td) — Lemma (F)
    2. Let X∈SR  — Assumption
    3. X = R  — (2) & Def. of SR
    4. ¬∃SS[ R∈SS & SS∈RAS(SH,td) ]  — Situational Assumption #14
    5. ∀SS[ R∉SS | SS∉RAS(SH,td) ]  — (4) & Logic
    6. ∀SS[ SS∈RASC(SH,td) ⇒ SS∈RAS(SH,td) ]  — Def. of RASC
    7. ∀SS[ SS∉RAS(SH,td) ⇒ SS∉RASC(SH,td) ]  — (6) & Logic
    8. ∀SS[ R∉SS | SS∉RASC(SH,td) ]  — (5) & (7)
    9. ¬∃SS[ R∈SS & SS∈RASC(SH,td) ]  — (8) & Logic
    10. ¬∃SS[ X∈SS & SS∈RASC(SH,td) ]  — (3) & (9)
    11. ∀X{ X∈SR ⇒ ¬∃SS[ X∈SS & SS∈RASC(SH,td) ] }  — (2) & (10)
    12. SR∈RASO(SH,td) & ∀X{ X∈SR ⇒ ¬∃SS[ X∈SS & SS∈RASC(SH,td) ] }  — (1) & (11)
    13. SR∈RASA(SH,td)  — (12) & Def. of RASA
    14. Q.E.D.

Proof that R∈MRLprm(d,td)

  1. c=Actr(H)  — Def. of H
  2. R∈SR  — Def. of SR
  3. SR∈RASA(SH,td)  — Lemma (G)
  4. ∃d∃H∃SR[ d=Actr(SH) & R∈SR & SR∈RASA(SH,td)  — (1-3)
  5. R∉ExecP(d,td)  — (4) & Def. of ExecP
  6. R∈MRLprm(d,td)  — (5) & Def. of MRLprm
  7. Q.E.D.

Moral Optimality
The Moral Objective Function (FMRLobj) for X is dNPWBT(Wrong(SH) ∪ Actor(SH),t)|{X} – dNPWBT(Wrong(SH),u0-min(SH))|SH, because SH occurred in the past.  In general, detention (for any period of time) will hurt c and help Hf.  If this is a linear function of t (which is likely — the relationship should be approximately the same for all t unless the criminal is rehabilitated), then the following holds true:

If dNPWBT(Wrong(SH) ∪ Actor(SH),t)|{X} < dNPWBT(Wrong(SH),u0-min(SH))|SH, then the FMRLobj > 0 by executing X.
Otherwise, FMRLobj ≤ 0 by executing X.

Since dNPWBT(Wrong(SH))|{X} > 0, dNPWBT(Actor(SH))|{X} < 0, and dNPWBT(Wrong(SH),u0-min(SH))|SH < 0, if dNPWBT(Wrong(SH) ∪ Actor(SH),t)|{X} = dNPWBT(Wrong(SH))|{X} + dNPWBT(Actor(SH))|{X}, this becomes:

If -dNPWBT(Actor(SH))|{X} < dNPWBT(Wrong(SH))|{X} – dNPWBT(Wrong(SH),u0-min(SH))|SH, then the FMRLobj > 0 by executing X.
Otherwise, FMRLobj ≤ 0 by executing X.

In other words, we ought to detain the criminal if the harm to the him is less than the gain to the potential victims of future crimes plus the harm of the original crime.  Assuming that the the criminal cannot be rehabilitated, we ought to either detain the criminal for life, or release him immediately.  If the criminal can be rehabilitated to the point where the risk of future crime drops below the threshold defined by the inequality above, then at that point he should be released.
 
December, 2013


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