I'd like to present an example of logical analysis and interpretation because the arguments can sometimes be reformulated and perhaps made differently too as you like:
Aetixintro wrote: |
Gödel's ontological proof can be questioned, however I've contributed with a version that makes it stand out as splendid and at the same time being accepted without a question outside the logical soundness objections to a "necessary God".
Here is:
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
...
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.
From:
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 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: Any property entailed by—i.e., strictly implied by—a positive property is positive
Axiom 2: If a property is positive, then its negation is not 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.
http://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof
http://en.wikipedia.org/wiki/Universe_of_Discourse - UoD from above.
http://en.wikipedia.org/wiki/Logical_consequence - "Entailment"/"entails".
Some of the text is from http://whatiswritten777.blogspot.no/2012/03/presentation-of-my-interpretation-of.html. |
Aetixintro wrote: |
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. |
So there it is. Enjoy logics! |