The Poverty of Empathy

Does empathy enable effective charitable dispositions? I think not.

Consider the toy model illustrating the problem.

Glossary:
x = Jack
y = Jill
Hx = Jack is Human.
Hy = Jill is Human.
Sx = Jack thinks his suffering matters in guiding his actions so as to prevent personal suffering.
Sy = Jill thinks her suffering matters in guiding her actions so as to prevent personal suffering.
Sxy = Jack thinks Jill’s suffering matters in guiding his actions so as to prevent her suffering.
Syx = Jill thinks Jack’s suffering matters in guiding her actions so as to prevent his suffering.
Kx = Jack knows what can be done to prevent personal suffering.
Ky = Jill knows what can be done to prevent personal suffering.
Kxy = Jack knows what can be done to prevent Jill’s suffering.
Kyx = Jill knows what can be done to prevent Jack’s suffering.

I’ll show that ∀y∀x((x≠y)∧(((Hx∧Sx)→Kx)∧((Hy∧Sy)→Ky))) → ∀y∀x(((Hx∧Hy)∧(Sxy∧Syx))→(¬Kxy∧¬Kyx))

[Informal English: For distinct humans Jack and Jill, if they think their own and the other’s suffering matters as a guide for their own actions so as to prevent their own and the other’s suffering then it’s not the case that either of them knows what can be done to prevent the other’s suffering.]

Argument: ∀y∀x((x≠y)∧(((Hx∧Sx)→Kx)∧((Hy∧Sy)→Ky))) → ∀y∀x(((Hx∧Hy)∧(Sxy∧Syx))→(¬Kxy∧¬Kyx))
PROOF:
0. ∀y ∀x ((x≠y) ∧ (((Hx ∧ Sx) → Kx) ∧ ((Hy ∧ Sy) → Ky)))
1. ¬∀y∀x(((Hx∧Hy)∧(Sxy∧Syx))→(¬Kxy∧¬Kyx))
2. ((Ha∧Sa)→Ka)∧((Ha∧Sa)→Ka)
3. (a≠a)
4. (a≠a)∧(((Ha∧Sa)→Ka)∧((Ha∧Sa)→Ka))
5. ¬(((Hb∧Ha)∧(Sba∧Sab))→(¬Kba∧¬Kab))
6. ∀x((x≠a)∧(((Hx∧Sx)→Kx)∧((Ha∧Sa)→Ka)))
7. ∃x¬(((Hx∧Ha)∧(Sxa∧Sax))→(¬Kxa∧¬Kax))
8. ¬∀x(((Hx∧Ha)∧(Sxa∧Sax))→(¬Kxa∧¬Kax))
9. ∃y¬∀x(((Hx∧Hy)∧(Sxy∧Syx))→(¬Kxy∧¬Kyx))
10. ∀y ∀x (((Hx ∧ Hy) ∧ (Sxy ∧ Syx)) → (¬Kxy ∧ ¬Kyx))▄

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