Reductionism and Supervenience: A Proof in Modal Logic

Below, I prove that if reductionism is true then the supervenience thesis is true.
(See Simon Blackburn, “Supervenience Revisited” in his Essays in Quasi-Realism, 1993.)

Reductionism: N (x)(G*x ⊃ Fx)

Necessarily, for all x, if x is correctly described by a natural property description represented by G*, then x has the moral property represented by F.

The Supervenience Thesis: N (∃x)(Fx & G*x & (G*x U Fx)) ⊃ (y)(G*y ⊃ Fy))

Necessarily, if something exists that has moral property F and is correctly described by G*, and its being F supervenes on its being G* [or, conversely, it’s being G* underlies its being F], then for all y, if y is G*, then y is F.


All modal operators here are to be understood as involving analytic or conceptual necessity/possibility. (As opposed to, say, metaphysical or natural necessity/possibility).

Supposedly G. E. Moore’s open question argument gives good reason to reject reductionism. Reductionism is equivalent to what Moore called the “naturalistic fallacy”. Blackburn argues that anti-reductionists lack a good explanation for their acceptance of the supervenience thesis.


1. N (x)(G*x ⊃ Fx)
// ∴ N (∃x)(Fx & G*x & (G*x U Fx)) ⊃ (y)(G*y ⊃ Fy))

2. ASM: ~ N ((∃x)(Fx & G*x & (G*x U Fx)) ⊃ (y)(G*y ⊃ Fy))       AIP
3. P ~((∃x)(Fx & G*x & (G*x U Fx)) ⊃ (y)(G*y ⊃ Fy))          from 2
4. W1 ~((∃x)(Fx & G*x & (G*x U Fx)) ⊃ (y)(G*y ⊃ Fy))       from 3
5. W1 ~(~(∃x)(Fx & G*x & (G*x U Fx)) ∨ (y)(G*y ⊃ Fy))       from 4
6. W1 ~~(∃x)(Fx & G*x & (G*x U Fx)) & ~(y)(G*y ⊃ Fy)       from 5
7. W1 ~(y)(G*y ⊃ Fy) & ~~(∃x)(Fx & G*x & (G*x U Fx))       from 6
8. W1 ~(y)(G*y ⊃ Fy)                     from 7
9. W1 (∃y)~(G*y ⊃ Fy)                     from 8
10. W1 (∃y)~(~G*y ∨ Fy)                     from 9
11. W1 (∃y)(~~G*y & ~Fy)                     from 10
12. W1 ~~G*a & ~Fa                     from 11
13. W1 G*a & ~Fa                     from 12
14. W1 G*a                     from 13
15. W1 (x)(G*x ⊃ Fx)                     from 1
16. W1 G*a ⊃ Fa                     from 15
17. W1 Fa                     from 14, 16
18. W1 ~Fa & G*a                     from 13
19. W1 ~Fa                     from 18
20. W1 Fa & ~Fa                     from 17, 19

21. N (∃x)(Fx & G*x & (G*x U Fx)) ⊃ (y)(G*y ⊃ Fy))                     from 2-20, IP
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"Make me a channel of Your Peace."

-St. Francis

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Logic Symbols in HTML

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I learned some new HTML codes today for symbolic logic:

∼ ∼
· ·
∨ ∨
⊃ ⊃
≡ ≡

∴ ∴

′ ′
″ ″

∀ ∀
∃ ∃

↔ ↔
⇔ ⇔

≠ ≠

∞ ∞

You can also make any Greek letter, capital or lower-case this way:
φ φ
Φ Φ

I still haven't found really great symbols for modal logic, but I guess we could use these for "possible":
◊ ◊
or, ♦ ♦

I haven't found a satisfactory one for "necessary", although I suppose any symbol that would generate a square when the browser cannot interpret it might work! :)

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"He Himself is our Peace." (Eph 2)

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