Ontological Argument Improved Again

Let,

E!x ≝ x exists in re
Ix ≝ x exists in intellectu
Gx ≝ x admits of more greatness
G<Px,~Px> ≝ x having P is greater than x not having P
Gxy ≝ x is greater than y
©… ≝ it is conceivable that…

g ≝ (ɿx)(~©Gx ∧ ~©(∃y)Gyx)

1. (∀x)[(Ix ∧ ~E!x) ⊃ ©E!x] (premise)
2. (∀x)G<E!x, (~E!x ∧ Ix)> (premise)
3. (∀x){[[~E!x ∧ G<E!x, (~E!x ∧ Ix)>] ∧ ©E!x] ⊃ ©Gx}(premise)
4. Ig (premise)
5. ~E!g (IP)
6. Ig ∧ ~E!g (4,5 Conj)
7. (Ig ∧ ~E!g) ⊃ ©E!g (1 UI)
8. ©E!g (6,7 MP)
9. G<E!g, (~E!g ∧ Ig)> (2 UI)
10. ~E!g ∧ G<E!g, (~E!g ∧ Ig)> (5,9 Conj)
11. [~E!g ∧ G<E!g, (~E!g ∧ Ig)>] ∧ ©E!g (8,10 Conj)
12. [[~E!g ∧ G<E!g, (~E!g ∧ Ig)>] ∧ ©E!g] ⊃ ©Gg (3 UI)
13. ©Gg (11,12 MP)
14. (∃x){{[~©Gx ∧ ~©(∃y)Gyx] ∧ (∀z){[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = x)]}} ∧ ©Gx} (13 theory of descriptions)
15. {[~©Gμ ∧ ~©(∃y)Gyμ] ∧ (∀z){[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]}} ∧ ©Gμ (14 EI)
16. {(∀z){[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]} ∧ [~©Gμ ∧ ~©(∃y)Gyμ]} ∧ ©Gμ (15 Comm)
17. {(∀z){[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]} ∧ [~©(∃y)Gyμ ∧ ~©Gμ]} ∧ ©Gμ (16 Comm)
18. {(∀z){[[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]} ∧ ~©(∃y)Gyμ] ∧ ~©Gμ} ∧ ©Gμ (17 Assoc)
19. (∀z){[[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]} ∧ ~©(∃y)Gyμ] ∧ {~©Gμ ∧ ©Gμ} (18 Assoc)
20. ~©Gμ ∧ ©Gμ (19 Simp
21. E!g (5-20 IP)

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A Possible Interpretation of Proslogion 2

One of my struggles in trying to understand Proslogion 2 is how Anselm gets to the actual existence of God rather than what he arrives at in Proslogion 3, namely the inconceivability of God’s non-existence.  I’ve also struggled with the notion of using a two-place predicate like “greater than”, since Anselm tells us that if God exists in the mind alone, a greater could be conceived, i.e. to think of God as existing in reality.  Here, we are saying that we could conceive of one and the same concept in greater ways rather than conducting a comparison of the God concept to other items in the world.  The following interpretation approximates what Anselm seems to be arguing, and I would say that it is a sound argument for God’s existence.

D1. God is defined as that which cannot be conceived to admit of more greatness.
P1. For all x, if x exists in intelletu and not in re, then it can be conceived that x exists in intellectu and not in re.
P2. For all x, if it can be conceived that x exists in intellectu and not in re, then it can be conceived that x exists in intellectu and in re.
P3. For all x, if it can be conceived that x exists in intellectu and not in re and it can be conceived that x exists in intellectu and in re, then it is conceivable that x admits of more greatness.
P4. God exists in intellectu.
C. Therefore, God exists in re.

Let,

E!x ≝ x exists in re
Ix ≝ x exists in intellectu
Gx ≝ x admits of more greatness
©… ≝ it is conceivable that…

g ≝ (ɿx)~©Gx

1. (∀x)[(Ix ∧ ~E!x) ⊃ ©(Ix ∧~E!x)] (premise)
2. (∀x)[©(Ix ∧ ~E!x) ⊃ ©(Ix ∧ E!x)] (premise)
3. (∀x){[©(Ix ∧ ~E!x) ∧ ©(Ix ∧ E!x)] ⊃ ©Gx} (premise)
4. Ig (premise)
5. ~E!g (IP)
6. Ig ∧ ~E!g (4,5 Conj)
7. (Ig ∧ ~E!g) ⊃ ©(Ig ∧~E!g) (1 UI)
8. ©(Ig ∧~E!g) (6,7 MP)
9. ©(Ig ∧ ~E!g) ⊃ ©(Ig ∧ E!g) (2 UI)
10. ©(Ig ∧ E!g) (8,9 MP)
11. ©(Ig ∧~E!g) ∧ ©(Ig ∧ E!g) (8,10 Conj)
12. ©(Ig ∧ ~E!g) ∧ ©(Ig ∧ E!g)] ⊃ ©Gg (3 UI)
13. ©Gg (11,12 MP)
14. (∃x){{~©Gx ∧ (∀y)[~©Gy ⊃ (y = x)]} ∧ ©Gx} (13 theory of descriptions)
15. {~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} ∧ ©Gμ (14 EI)
16. {(∀y)[~©Gy ⊃ (y = μ)] ∧ ~©Gμ} ∧ ©Gμ (15 Comm)
17. (∀y)[~©Gy ⊃ (y = μ)] ∧ {~©Gμ ∧ ©Gμ} (16 Assoc)
18. ~©Gμ ∧ ©Gμ (17 Simp)
19. E!g (5-18 IP)

QED

[Edit: My friend, Matt, thinks my argument may be susceptible to parody.  Here is my response]

Generally, I think parodies fail because such supposed objects, like islands of which none greater can be conceived, do not really exist in the intellect for the very same reason round squares are not abstract objects in the mind.  The phrase is nonesense, and so does not pick out any object of the understanding.

Islands just are the sorts of things that admit of degrees of greatness, so are other objects used in parody. For example, islands are present in a specified location that is surrounded by water, but it is unclear how big an island should be when considering its greatness.  It certainly cannot be omnipresent and be an island.  How many trees, island beauties, or sandy beaches ought there to be on the island which cannot be conceivably greater?  

My argument can motivate this response by proving that the greatest conceivable island is not an object that exists in the intellect.  This is because specifying that there is an island than which none greater can be conceived leads to the conclusion that God is an island, and that seems like a good reductio of the idea such a concept can be conceived.

So, if we grant the parody, I could prove that island can be predicated of God, or a being than which a greater cannot be conceived. But since islands are essentially contingent and admit of degrees of greatness, island cannot be a predicate of God, who is the being than which none greater can be conceived. So, we must reject the assumption that a greatest conceivable island exists in intellectu and we can base it on the somewhat reasonable premise that God is not an island. I would argue as follows:

Let,

Lx ≝ x is an island

i ≝ (ɿx)(~©Gx ∧ Lx)

20. ~Lg (premise)
21. (∃x){{~©Gx ∧ (∀y)[~©Gy ⊃ (y = x)]} ∧ E!x} (19 theory of descriptions)
22. Ii (IP)
23. (∃x){{(~©Gx ∧ Lx) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = x)]} ∧ Ix} (22 theory of descriptions)
24. {~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} ∧ E!μ (21 EI)
25. {(~©Gν ∧ Lν) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = ν)]} ∧ Iν (23 ΕΙ)
26. ~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)] (24 Simp)
27. (∀y)[~©Gy ⊃ (y = μ)] (26 Simp)
28. (~©Gν ∧ Lν) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = ν)] (25 Simp)
29. ~©Gν ∧ Lν (28 Simp)
30. ~©Gν (29 Simp)
31. ~©Gν ⊃ (ν = μ) (27 UI)
32. ν = μ (30,31 MP)
33. ~©Gμ ∧ Lμ (29,32 ID)
34. (~©Gμ ∧ Lμ) ∧ (∀y)[~©Gy ⊃ (y = μ)] (27,33 Conj)
35. ~©Gμ ∧ {Lμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} (34 Assoc)
36. ~©Gμ ∧ {(∀y)[~©Gy ⊃ (y = μ)] ∧ Lμ} (35 Comm)
37. {~©Gμ ∧ {(∀y)[~©Gy ⊃ (y = μ)]} ∧ Lμ (36 Assoc)
38. (∃x){{~©Gx ∧ {(∀y)[~©Gy ⊃ (y = x)]} ∧ Lx} (37 EG)
39. Lg (38 theory of descriptions)
40. ~Lg ∧ Lg
41. ~Ii (22-40 IP)

So as long as you can provide the premise that God is not an island, not a pizza, etc. the proof works to show that such objects really are not in the intellect.

Absolute and Relative Identity

I have seen some argue that any relative identity claim can be reduce to an absolute identity claim in the following manner:

1) x and y are the same F  ≝ is an Fis an F, and x = y.

However, I don’t think this works.  Part of the motivation for relative identity is that there may be circumstances like:

2) x and y are the same F, but x and y are not the same G.

But (1) and (2) are not compatible, since we would have to affirm and deny absolute identity between x and y.  So the relative identity theorist should reject (1) given his commitment to (2).

Relative identity is not just absolute identity, plus the idea that x and y fall under the same sortal.  Moreover, this would be to suggest that relative identity is derivative, and absolute identity is the more primitive notion.  I would argue that is it the other way around.  So I would define absolute identity in terms of relative identity in the following manner:

4) x = y ≝ for any sortal, S, if x is an S or y is an S, then x and are the same S.

In other words, the absolute identity between x and y is derived from the fact that for any sortal which belongs to either x or y, it is the case that x and y count as the same S.  I say “either x is an S, or y is an S” as opposed to “both x is an S and y is an S” to avoid situations where x can be counted as an S and some y cannot, but they are the same S for any sortal underwhich both can be counted.  For there to be absolute identity, it must be the case that all sortals that belong to x must also belong to y.  I believe (4) captures this.

So to say x and y are absolutely identical is to say that for any sortal underwhich x or y can be counted, x and y are the same sortal.

 

A More Iconic Trio?

Merry Christmas!  Happy New Year!  Here is a meme for you!

An Argument Against Subjective Morality

P1. If morality is relative to the subject, then morality is a domain that is a  matter of personal opinion.

P2. All domains that are matters of personal opinion, are domains where facts and evidence cannot determine correct belief.

P3.  All domains where facts and evidence cannot determine correct belief are domains that lack propositions for which it is worth dying before giving assent under coercion.

P4.  Morality is a domain with propositions for which it is worth dying before giving assent under coercion.

C. Morality is not relative to the subject.

P1 and P2 are not too objectionable.  That is just what we mean when we say that morality is subjective.  I think that if you are going to object to the argument, you will object to P3 or P4.

A Remix of Anselm’s Conceptual Ontological Argument

st-20anselm20weninger

D1. God is defined as the x such that there is not something, y, where y is conceivably greater than x.
P1. For all x, if x is conceivable, then there is something, y, such that either y is identical to x and y exists or there is something, z, such that z is identical to x, z does not exist, and y is conceivably greater than z.
P2. There is some x such that x is conceivable and it is not the case that there is some y such that y is conceivably greater than x.
P3. For all x and y, either x is conceivably greater than y or y is conceivably greater than x, or if it is not the case that either x is conceivably greater than y or that y is conceivably greater than x, there is some z such that z is the mereological sum of x and y, and either z is conceivably greater than x or z is conceivably greater than y.
C. God exists.1

E!x ≝ x exists
Cx ≝ x is conceivable
Gxy ≝ x is conceivably greater than y
σ<x,y> ≝ the mereological sum of x and y
g ≝ (ɿx)~(∃y)Gyx

1. (∀x){Cx ⊃ (∃y){[(y = x) ∧ E!y] ∨ (∃z)[(z = x) ∧ (~E!z ∧ Gyz)]}} (premise)
2. (∃x)(Cx ∧ ~(∃y)Gyx) (premise)
3. (∀x)(∀y){[Gxy ∨ Gyx] ∨ {~(Gxy ∨ Gyx) ⊃ (∃z)[(z = σ<x,y>) ∧ (Gzx ∨ Gzy)]}} (premise)
4. Cμ ∧ ~(∃y)Gyμ (2 EI)
5. ~(∃y)Gyμ (4 Simp)
6. (∃z)[~(∃z1)Gz1z ∧ ~(z = μ)] (IP)
7. ~(∃z1)Gz1ν ∧ ~(ν = μ) (6 EI)
8. (∀y){[Gνy ∨ Gyν] ∨ {~(Gνy ∨ Gyν) ⊃ (∃z)[(z = σ<ν,y>) ∧ (Gzν ∨ Gzy)]}} (3 UI)
9. [Gνμ ∨ Gμν] ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]} (8 UI)
10. (∀y)~Gyμ (5 QN)
11. ~Gνμ (10 UI)
12. ~(∃z1)Gz1ν (7 Simp)
13. (∀z1)~Gz1ν (12 QN)
14. ~Gμν (13 UI)
15. Gνμ ∨ [Gμν ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]}] (9 Assoc)
16. Gμν ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]} (11,15 DS)
17. ~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)] (14,16 DS)
18. ~Gνμ ∧ ~Gμν (11,14 Conj)
19. ~(Gνμ ∨ Gμν) (18 DeM)
20. (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)] (17,19 MP)
21. (ζ = σ<ν,μ>) ∧ (Gζν ∨ Gζμ) (20 EI)
22. Gζν ∨ Gζμ (21 Simp)
23. ~Gζμ (10 UI)
24. Gζν (22,23 DS)
25. ~Gζν (13 UI)
26. Gζν ∧ ~Gζν (24,25 Conj)
24. ~(∃z)[~(∃z1)Gz1z ∧ ~(z = μ)] (6-23 IP)
25. (∀z)~[~(∃z1)Gz1z ∧ ~(z = μ)] (24 QN)
26. (∀z)[~~(∃z1)Gz1z ∨ ~~(z = μ)] (25 DeM)
27. (∀z)[~(∃z1)Gz1z ⊃ ~~(z = μ)] (26 Impl)
28. (∀z)[~(∃z1)Gz1z ⊃ (z = μ)] (27 DN)
29. {Cμ ∧ ~(∃y)Gyμ} ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = μ)] (4,28 Conj)
30. Cμ ∧ {~(∃y)Gyμ ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = μ)]} (29 Assoc)
31. {~(∃y)Gyμ ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = μ)]} ∧ Cμ (30 Comm)
32. (∃x){~(∃y)Gyx ∧ (∀z)[~(∃z1)Gz1z ⊃ (z =x)]} ∧ Cx} (31 EG)
33. Cg (32 theory of descriptions)
34. Cg ⊃ (∃y){[(y = g) ∧ E!y] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gyz)]} (1 UI)
35. (∃y){[(y = g) ∧ E!y] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gyz)]} (33,34 MP)
36. [(ξ = g) ∧ E!ξ] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (35 EI)
37. (∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (IP)
38. (ν = g) ∧ (~E!ν ∧ Gξν) (37 EI)
39. ~E!ν ∧ Gξν (38 Simp)
40. Gξν (39 Simp)
41. (ν = g) (38 Simp)
42. Gξg (40,41 ID)
43. (∃x){~(∃y)Gyx ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = x)]} ∧ Gξx} (42 theory of descriptions)
44. {~(∃y)Gyζ ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = ζ)]} ∧ Gξζ (43 EI)
45. ~(∃y)Gyζ ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = ζ)](44 Simp)
46. ~(∃y)Gyζ (45 Simp)
47. (∀y)~Gyζ (46 QN)
48. ~Gξζ (47 UI)
49. Gξζ (44 Simp)
50. Gξζ ∧ ~Gξζ (48,49 Conj)
51. ~(∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (37-50 IP)
52. (ξ = g) ∧ E!ξ (36,51 DS)
53. (ξ = g) (52 Simp)
54. E!ξ (52 Simp)
55. E!g (53,54 ID)

QED

1 Some aspects of this argument are influenced by Oppenheimer & Zalta (1991), i.e. the existential quantifier carries no existential import and is analogous to Anselm’s existence in intellectu whereas E! is a predicate that indicates existence in re. One weakness of Oppenheimer & Zalta’s argument is that it depends on a non-logical axiom regarding Gxy such that it is connected. In other words, either Gxy or Gyx or (x = y). This requires all individuals to stand in a greater than relationship. It is plausible, though, that two non-identical individuals could share equal greatness. I am able to derive the uniqueness of the being than which none greater can be conceived by appealing to the notion that the merelogical composite of two equally great individuals is at least greater than one of its proper parts, which I take to be a modest premise. The interesting thing about my formulation is the first premise, which distinguishes in intellectu from in re existence, and captures Anselm’s claim that a greater could be conceived than a being that exists in the understanding alone without begging the question that this greater thing actually exists—it is merely conceptually greater.  See P.E Oppenheimer & E.N. Zalta. (1991). “On the Logic of the Ontological Argument.” In Philosophical Perspectives. Vol. 5. 509-529.

Calvin’s basket is full

Domino Logic

Here are some really interesting videos in which dominos simulate logical reasoning and computing:

 

Novena Prayer Challenge

rosary

(Photo Credit: Sign.org)

There are many intentions for which I have been asked to pray. While I do offer a moment of prayer for these requests, I think I can do better. So I am would like to pray the rosary for the next nine days (starting September 19) for all of these intentions.

The Rosary is the most excellent form of prayer and the most efficacious means of attaining eternal life. It is the remedy for all our evils, the root of all our blessings. There is no more excellent way of praying (Pope Leo XIII).

I found a beautiful recitation of the rosary sung in Latin, so this will be what I will be listening to as I pray:

Sign of the Cross:

In nomine Patris, et Filii, et Spiritus Sancti. Amen

Apostles’ Creed:

Credo in Deum Patrem omnipotentem, Creatorem caeli et terrae. Et in Iesum Christum, Filium eius unicum, Dominum nostrum, qui conceptus est de Spiritu Sancto, natus ex Maria Virgine, passus sub Pontio Pilato, crucifixus, mortuus, et sepultus, descendit ad infernos, tertia die resurrexit a mortuis, ascendit ad caelos, sedet ad dexteram Dei Patris omnipotentis, inde venturus est iudicare vivos et mortuos. Credo in Spiritum Sanctum, sanctam Ecclesiam catholicam, sanctorum communionem, remissionem peccatorum, carnis resurrectionem, vitam aeternam. Amen.

The Lord’s Prayer:

PATER NOSTER, qui es in caelis, sanctificetur nomen tuum. Adveniat regnum tuum. Fiat voluntas tua, sicut in caelo et in terra. Panem nostrum quotidianum da nobis hodie, et dimitte nobis debita nostra sicut et nos dimittimus debitoribus nostris. Et ne nos inducas in tentationem, sed libera nos a malo. Amen.

The Hail Mary:

AVE MARIA, gratia plena, Dominus tecum. Benedicta tu in mulieribus, et benedictus fructus ventris tui, Iesus. Sancta Maria, Mater Dei, ora pro nobis peccatoribus, nunc, et in hora mortis nostrae. Amen.

Glory Be:

GLORIA PATRI, et Filio, et Spiritui Sancto. Sicut erat in principio, et nunc, et semper, et in saecula saeculorum. Amen.

Oratio Fatimae (The Fatima Prayer)

Domine Iesu, dimitte nobis debita nostra, salva nos ab igne inferiori, perduc in caelum omnes animas, praesertim eas, quae misericordiae tuae maxime indigent.

Hail, Holy Queen:

SALVE REGINA, Mater misericordiae. Vita, dulcedo, et spes nostra, salve. Ad te clamamus exsules filii Hevae. Ad te Suspiramus, gementes et flentes in hac lacrimarum valle. Eia ergo, Advocata nostra, illos tuos misericordes oculos ad nos converte. Et Iesum, benedictum fructum ventris tui, nobis post hoc exsilium ostende. O clemens, o pia, o dulcis Virgo Maria.

V. Ora pro nobis, Sancta Dei Genitrix.
R. Ut digni efficiamur promissionibus Christi.

caelum omnes animas, praesertim eas, quae misericordiae tuae maxime indigent.

Hail, Holy Queen:

SALVE REGINA, Mater misericordiae. Vita, dulcedo, et spes nostra, salve. Ad te clamamus exsules filii Hevae. Ad te Suspiramus, gementes et flentes in hac lacrimarum valle. Eia ergo, Advocata nostra, illos tuos misericordes oculos ad nos converte. Et Iesum, benedictum fructum ventris tui, nobis post hoc exsilium ostende. O clemens, o pia, o dulcis Virgo Maria.

V. Ora pro nobis, Sancta Dei Genitrix.
R. Ut digni efficiamur promissionibus Christi.

(EWTN, Latin Prayers).

My Intentions:

1) The continued health of my daughter, my wife, and myself.
2) Full employment and financial stability for my family.
3) The health of my Aunt Laura.
4) The health of my grandmother, Clare.
5) The speedy recovery of my cousin’s husband, Mark.
6) For the recovery of my compadre, Armando.
7) For my sister and her family.
8) The requests of Michael Shenigo.
9) The requests of Helen Marple-Horavt.
10) For those who doubt God.
11) For the victims of recent violent attacks.
12) For defenseless children absused by their parents and adults.
13) For the unborn and the victims of abortion.
14) For the United States in the election of candidates who truly seek to bring justice and peace to the world.
15) For the souls of my mother, Matthew, Lupe, Tia Socorro, and Carol who recently passed.

If you would like to participate in this challenge, post your intentions in the comments below and I will include them in my intentions as I pray over the next nine days. You don’t have to be a Catholic or believer to pray or join in this challenge, in fact, it might make sense to pray even if you don’t believe.

The Outsider’s Test Passed

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