UoD: Everything.

Gx: x is God-like

Ex: x has essential properties.

Ax: x is an essence of A.

Bx: x is a property of B.

Px: property x is positive.

Nx: x is a General property.

Xx: x is Positive existence.

Cx: x is consistent.

The final argument by my interpretation is presented below in 4 parts:

1.

1 │ □Ex ≡ □Px ≡ □Gx A (A is Assumption)

2 │ □Ex A

3 │ ◊Px ≡ □Px A

4 │ ◊Px A

------------------

5 │ □Px ≡ □Gx 1, 2 ≡E

6 │ □Px 3, 4 ≡E

------------------

7 │ □Gx 5, 6 ≡E

Alt. 1, 1st.

1 │ □Ex ≡ □Px ≡ □Gx A (A is Assumption)

2 │ □Ex A

3 │ ◊Px ⊃ □Px A

4 │ ◊Px A

------------------

5 │ □Px ≡ □Gx 1, 2 ≡E

6 │ □Px 3, 4 ⊃E

------------------

7 │ □Gx 5, 6 ≡E

Alt. 1, 2nd.

1 │ □Ex ≡ □Px ≡ □Gx A (A is Assumption)

2 │ (□Px ⊃ □Nx) ⊃ □Px A

3 │ □Px ⊃ □Nx A

4 │ □Ex A

------------------

5 │ □Px 2, 3 ⊃E

6 │ □Px ≡ □Gx 1, 2 ≡E

------------------

7 │ □Gx 6, 5 ≡E

This alternative, nr. 2, takes care of the former line ”6 │ (□Px ⊃ □Nx) ⊃ □Px A” and adds overall description by this!

2.

1 │ □Px ≡ □Gx A (A is Assumption)

2 │ □Xx ⊃ □Px A

3 │ □Xx A

------------------

4 │ □Px 2, 3 ⊃E

------------------

5 │ □Gx 1, 4 ≡E

3.

1 │ ◊Cx ≡ □Gx A (A is Assumption)

2 │ □Px ∨ ~□Px A

3 │ □Px ⊃ ◊Cx A

------------------

4 ││ □Px A

0 ││-----------------

5 ││ □Px 6 R

6 ││ ~□Px A

0 ││-----------------

7 ││ □Px 6 R

8 │ □Px 4, 6-9 ∨E

9 │ ◊Cx 8, 3 ⊃E

------------------

10│ □Gx 9, 1 ≡E

4.

1 │ □Bx ≡ □Gx A (A is Assumption)

2 │ ◊Ax ≡ □Bx ≡ (◊Ax ⊃ □Bx) A

3 │ ◊Ax A

------------------

4 │ □Bx 3, 2 ≡E

------------------

5 │ □Gx 4, 1 ≡E

Note for the 4th part: Consider (◊Ax ⊃ □Bx) as “added explanation”!

Also, line 2 of the 4th part is Definition 2 from the original argument of Gödel.

Note2: The following lines are taken out for having no use in this interpretation of the argument.

8 │ □Gx ⊃ □Px A

16│ □Gx ⊃ □Cx A

17│ □Gx ⊃ □Ax A

No need to put any emphasis to the line numbers 8, 16 and 17 above.

Note3: A forgotten line 6 and its own alternative has been added now, 16:28, 13.03.2012 CET.

Note4: Small corrections. Adding the direct relations to Gödel's Ontological Argument. Added now, 23:40, 13.03.2012 CET.

Note5: The above time stamps relate to publishing on Facebook as note! The whole presentation has therefore been imported from the original place on Facebook to this blog. By Leonardo F. Olsnes-Lea on Facebook on Saturday, 14 January 2012 at 02:43 CET(?).

Else, development has been:

(relating to Gödel's argument directly, put after the connector),

1 │ □Ex ≡ □Px ≡ □Gx A (A is Assumption) – Def. 1

2 │ ◊ Ax ≡ □ Bx ≡ (◊Ax ⊃ □Bx) A - Def. 2

3 │ □Ex A – Def. 3

4 │ □Px A

5 │ □Px ∨ ~□Px A – Axiom 1

6 │ (□Px ⊃ □Nx) ⊃ □Px A – Axiom 2

7 │ ◊ Ax A

8 │ □Gx ⊃ □Px A – Axiom 3

9 │ ◊Px ≡ □Px A – Axiom 4

10│ ◊Px A

11│ □Xx ⊃ □Px A – Axiom 5

12│ □Xx A

13│ ◊ Px ⊃ □Px A – Axiom 6

14│ □Px ⊃ ◊Cx A – Theorem 1

15│ ◊Px ≡ □Px A

16│ □Gx ⊃ □Cx A – Corollary 1

17│ □Gx ⊃ □Ax A – Theorem 2

------------------------------

18│ □Ex R (R is Reiteration)

19│ □Px ≡ □Gx 1, 13 ≡E (≡E is Biconditional Elimination)

20│ □Px 4 R

------------------------------

21│ □Gx 14, 15 ≡E – Theorem 3

Various notes to the development have also been written under the argument as note on Facebook!

Time stamp above is correct again, CET-wise!

ReplyDeleteAlso:

3.

Development, but in several bits:

1. bla, bla, bla, bla...

Note1: Temporary writing that needs a good deal of mending yet to come through.

Note2: The rest of the argument can be considered description and not (really) part of the deduction as you can see from above! Alright? So how do we make the deductive "walk" to include all of it? Yes, one simple way is to insert what is missing and then make sure all of it follows necessarily... Well, well, I'll wait with it until tomorrow! You can finish it if you want!

Note3: I've published this now in several versions (due to clipboard problems) for about an hour ago and up to now, 03:52 CET.

So on 1 │ □Ex ≡ □Px ≡ □Gx ≡ □ Bx A (A is Assumption)

one adds □Px and □Ex ≡ □Gx ≡ □ Bx until the conclusion remains: □Gx on line 21 !

Kurt Gödel himself. The picture is from his Wikipedia page.

[Picture.]

Leonardo F. Olsnes-Lea For line 13, or what Gödel writes as "Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing." must be □Gx ⊃ □Ax A (What have I been thinking? Out of the blue...!)

19 January at 22:45

Leonardo F. Olsnes-Lea

The whole line of his: Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive

Definition 2: A is an essence of x if and only if for every property B, x has B necessarily i...See more

19 January at 22:45

Leonardo F. Olsnes-Lea I'll think I'

23 January at 04:31

Leonardo F. Olsnes-Lea I'll think I'll just set-up a long line of bi-conditionals that "play down" the other arguments, everything connected or set-up multiple arguments to say one thing several time, □Gx!

23 January at 04:32

Leonardo F. Olsnes-Lea ...thing several times, □Gx! Well, well, 3 stages set-up, only 2 necessary. This is what I bother to do for now!

23 January at 04:41

Leonardo F. Olsnes-Lea ...first in the World??? As Fitch compliant and modern, consistent logical language!!!

7 February at 08:29

Leonardo F. Olsnes-Lea Suggestion for 1. line again: 1 │ □Ex ≡ □Px ≡ □Gx ≡ □ Bx ≡ □Px A (A is Assumption)

27 February at 05:06

Leonardo F. Olsnes-Lea This may seem brutal and I think it is, but logics are only that complex. From this top line it should now be possible to pull down what's necessary. Not only that, but the set-up gives you explanatory force on how Gödel thinks how God relates to the World! Cheers!

27 February at 05:09

Leonardo F. Olsnes-Lea □Px has been added twice above . New: Suggestion for 1. line again: 1 │ □Ex ≡ □Px ≡ □Gx ≡ □ Bx A (A is Assumption) !

27 February at 05:19

[Not important, but can be nice to read.]

Theorem 2 has been listed wrongly above. The correct Definition 2 is now put there instead!

ReplyDeleteGödel's Ontological Argument in its completeness, taken from Wikipedia:

ReplyDeleteDefinition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive

Definition 2: A is an essence of x if and only if for every property B, x has B necessarily if and only if A entails B

Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified

Axiom 1: If a property is positive, then its negation is not positive.

Axiom 2: Any property entailed by—i.e., strictly implied by—a positive property is positive

Axiom 3: The property of being God-like is positive

Axiom 4: If a property is positive, then it is necessarily positive

Axiom 5: Necessary existence is positive

Axiom 6: For any property P, if P is positive, then being necessarily P is positive.

Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.

Corollary 1: The property of being God-like is consistent.

Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.

Theorem 3: Necessarily, the property of being God-like is exemplified.