Posted by: Calum Miller | March 12, 2012

Maydole’s Modal Perfection Argument

Here it is, in the same form (largely) as the original paper in Philo, 2003. I will explicate at some point. Enjoy.

P(X) =df it is better to have X than to not

Gxy =df x is greater than y

Sx =df (¬◊(∃y)Gyx & ¬◊(∃y)(x≠y & ¬Gxy))

M1: (X)(P[aX]) → ¬P([a¬X]))

M2: (Y)(P(Y) → (Z)(□(x)(Yx → Zx) → P(Z))

M3: P([aS])

Th 1: ◊(p & q) → (◊p & ◊q)

Th 2: ◊¬◊p → ¬◊p

Th 3: ◊(∃x)Fx → (∃x)◊Fx

‎1. ¬◊(∃x)Sx (AIP)

2. □¬(∃x)Sx (1, ME, DN)

3. □(x)¬Sx (2, QN, DN)

4. (x)¬Sx (3, NE, CP)

5. ¬Sa (4, UI)

6. ¬Sa v ¬Sa (5, Add)

7. Sa → ¬Sa (6, Impl)

8. (x)([aS]x ≡ Sx) (Abs)

9. (x)([a¬S]x ≡ ¬Sx) (Abs)

10. ([aS]a ≡ Sa) (8, UI)

11. ([a¬S]a ≡ ¬Sa) (9, UI)

12. [aS]a → [a¬S]a (7, 10, 11, Equiv, Simp, HS)

13. (x)([aS]x → [a¬S]x) (12, UG)

14. (x)¬Sx → (x)([aS]x → [a¬S]x) (4-13, CP)

15. □[(x)¬Sx → (x)([aS]x → [a¬S]x)] (14, NI)

16. P([aS]) → (Z)[(□(x)([aS]x → Zx)] → P(Z) (M2, UI)

17. P([aS]) (M3)

18. (Z)[(□(x)([aS]x → Zx)] → P(Z) (16, 17, MP)

19. (x)[([aS]x → [a¬S]x) → P([a¬S])] (18, UI)

20. □(x)([aS]x → [a¬S]x) (3, 15, MMP)

21. P([a¬S]) (19, 20, MP)

22. P([aS]) → ¬P([a¬S]) (M1, UI)

23. ¬P([a¬S]) (17, 22, MP)

24. ◊(∃x)Sx (21, 23, IP)

25. ◊(∃x)Sx → (∃x)◊Sx  (Th 3)

26. (∃x)◊Sx (24, 25, MP)

27. ◊Sv (26, EI)

28. ◊[¬◊(∃y)Gyv & ¬◊(∃y)(v≠y & ~Gvy)] (27, def. “S”)

29. ◊¬◊(∃y)Gyv & ◊¬◊(∃y)(v≠y & ~Gvy)] (28, Th 1, MP)

30. ¬◊(∃y)Gyv & ¬◊(∃y)(v≠y & ~Gvy) (29, Simp, Com, Th 2, MP, Conj)

31. Sv (30, def. “S”)

32. (∃x)Sx (32, EG)

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