Colbert on Faith, Logic, Humor and Gratitude

In the video below, Stephen Colbert talks about faith, logic, and humor.  Even though Colbert says that the ontological argument is “logically perfect”, like Pascal, he does not think logic can lead to faith in God.  There must be a movement in the heart, which Colbert connects to gratitude, and which he lives out in his work as a comedian.  But it isn’t as though logic and emotion as opposed forces.  The feeling of gratitude makes sense within a worldview where there is a being than which none greater can be conceived.

When we reflect on our existence, the love we share, the struggles, the joys, the busy days, and the quiet nights, we feel we ought to give thanks.  This gratitude is not conditioned by the kind of life we have.  For we see that gratitude is often freely expressed by the most lowly among us, and we are irked when the richest and most powerful lack gratitude.  Such a duty to feel gratitude seems to exist for us all and it doesn’t matter who we are or the sort of life we have.

Now, if we ought to express an unconditioned gratitude, then we can do so.  But if we can express such gratitude, there must be at least possible that there is an object worthy of such gratitude.  It is, after all, impossible to express gratitude if there cannot be anyone to whom the gratitude is due.  So, we might say that our ability to express unconditioned gratitude is at least predicated on the possibility of there being someone worthy of such gratitude.  So, I think only a perfect being is worthy of unconditioned gratitude, and if is possible that there is such a being, such a being exists.  That is, for me, one way in which gratitude and logic connect to bolster faith.

Anyways, here is the Colbert video.  I love a comedian who can name drop Anselm and Aquinas!

Everyday Apologist: What is Scientism?

Tim Hull interviewed me on his YouTube show the Everyday Apologist.  The interview is divided into three parts:

1. What is Scientism?

2. Why Should Christians Care About Scientism?

3.  How to Respond to Scientism:

Please subscribe to the Everyday Apologist for more interviews and great content.

Vexing Links (6/13/2015)

Some recent links of note:

Short Riddle

15 + 12 = 3
4 – 7 = 21
24 = 0

Explain the context in which these statements are true.

The Modesty of Maydole’s Temporal Contingency Argument

In a recent discussion that I had, my interlocutor claimed that “contingency” was an outdated scholastic concept. Really it is just a modal property. Sometimes it is called “two-way” possibility, i.e. x is contingent iff possibly and possibly not x. Temporal contingency the possibility of existing at some point in time and not existing at some point in time. We experience temporal contingency all the time. Anyways, I promised to explain how contingency is still relevant today in the philosophy of religion. In fact, I think it is relevant in one of the most powerful arguments for God’s existence. I can’t really imagine a good reason to deny any of the premises, and it is of course logically valid. So I am compelled to conclude that it is a sound argument for the existence of a supreme being, which I call “God”.

In a sense, The argument originates with Thomas Aquinas’s third way, but is developed by Robert Maydole, who fuses it with a modal ontological argument to devise an ingenious new argument.

Maydole defines a supreme being as follows:

D1. A supreme being is such that it is not possible that there exists anything greater than it and it is not possible that it is not greater than anything else that is non-identical to it.

He then proves the following, which we will call T1:

T1. If possibly a supreme being exists, then a supreme being exists.

Maydole does this by making use of a few theorems, like Barcan Formula, and other theorems in modal logic (I will reproduce the argument below, for those who are interested, see the conditional proof on lines 4-19 for the exact proof). Then Maydole constructs an argument for the possibility of a supreme being. He lists the following premises (but don’t attack them straight off, something interesting happens):

P1. Something presently exists.
P2. Only a finitely many things have existed to date.
P3. Every temporally contingent being begins to exist at some time and ceases to exist at some time.
P4. Everything that begins to exist at some time and ceases to exist at some time exists for a finite period of time.
P5. If everything exists only for a finite period of time, and there have been only a finitely many things to date, then there was a time when nothing existed.
P6. If there was a time when nothing existed, then nothing presently exists.
P7. A being is temporally necessary if and only if it is not temporally contingent.
P8. Everything has a sufficient reason for its existence.
P9. Anything that has a sufficient reason for its existence also has a sufficient reason for its existence that is a sufficient reason for its own existence.
P10. No temporally contingent being is a sufficient reason for its own existence.
P11. Every temporally necessary being that is a sufficient reason for its own existence is a being without limitations.
P12. A being without any limitations is necessarily greater than any other being.
P13. It is not possible for anything to be greater than itself.
P14. It is necessarily the case that “greater than” is asymmetric.

From P1-P14 one can prove C1:

C1. A supreme being exists.

The proof from P1-P14 to C1 is a bit long, and I believe Maydole even made a few typographical mistakes along the way. Here is my adaptation of this part of the argument, if you are interested.

Next consider what was said, before, that if it is possible that a supreme being exists, then a supreme being exists, i.e. T1. Maydole’s argument is surprisingly modest. What he does is argue that POSSIBLY (P1-P14) is true. Since C1 is provable from (P1-P14), we can say POSSIBLY C1 is true, which is to say that possibly a supreme being exists. Given T1 and the possibility that a supreme being exists, we can conclude that a supreme being exists (which is rightly called God)!

Now, the argument is very strong, because it is plausible that P1-P14 are actually true. However, Maydole only requires that the premises be possibly true rather than actually true, which is to say that they are not logically or metaphysically incoherent, or that they are true in some metaphysically possible world (as contemporary modal logicians would say). The deduction is valid, and it is very hard for me to think any of the premises are false. So I am compelled to think that this is, indeed, a sound argument for God’s existence.
So the proof looks something like this:


Gxy ≝ x is greater than y
Sx ≝ (~◊(∃y)Gyx & ~◊(∃y)(x≠y & ~Gxy))

1. ◊(P1-P14) (premise)
2. (P1-P14) ⊢ C1 (premise that C1 is provable from P1-P14)
3. {◊(P1-P14) & [(P1-P14) ⊢ C1]} ⊃ ◊C1 (premise)
4. ◊(∃x)Sx (Assump CP)
5. ◊(∃x)Sx ⊃ (∃x)◊Sx (BF theorem)
6. (∃x)◊Sx (4,5 MP)
7. ◊Su (6 EI)
8. ◊(~◊(∃y)Gyu & ~◊(∃y)(u≠y & ~Guy)) (7, df “Sx”)
9. ◊(~◊(∃y)Gyu & ~◊(∃y)(u≠y & ~Guy)) ⊃ (◊~◊(∃y)Gyu & ◊~◊(∃y)(u≠y & ~Guy)) (theorem)
10. ◊~◊(∃y)Gyu & ◊~◊(∃y)(u≠y & ~Guy) (8,9 MP)
11. ◊~◊(∃y)Gyu (10 Simp)
12. ◊~◊(∃y)(u≠y & ~Guy) (10 Simp)
13. ◊~◊(∃y)Gyu ⊃ ~◊(∃y)Gyu (theorem, by “S5”)
14. ◊~◊(∃y)(u≠y & ~Guy) ⊃ ~◊(∃y)(u≠y & ~Guy) (theorem, by “S5”)
15. ~◊(∃y)Gyu (11,13 MP)
16. ~◊(∃y)(u≠y & ~Guy) (12,14 MP)
17. ~◊(∃y)Gyu & ~◊(∃y)(u≠y & ~Guy) (15,16 Conj)
18. Su (17, df “Sx”)
19. (∃x)Sx (18 EG)
20. ◊(∃x)Sx ⊃ (∃x)Sx (4-19 CP, which proves T1)
21. {◊(P1-P14) & [(P1-P14) ⊢ C1] (1,2 Conj)
22. ◊C1 (3,22 MP)
23. ◊(∃x)Sx (22, def “C1”)
24. (∃x)Sx (20,23 MP)


To me, it is P11 that needs more explanation. It certainly seems right that a temporally necessary being who is the sufficient reason for its own existence has the sort of existence that is not limited by time nor by the existence of any other thing. But to say that the existence of x is not limited by time nor any thing seems a bit different from saying thag such a being is essentially without limitations. I believe the idea is that if there is no time nor state of affairs in which such a being would cease to exist or lack a reason for existing, then it is not limited by anything at all, and must be greater than every other thing.

Another person noted that P5 did not make sense to him because time is something that exists, so there could never be a time when nothing exists. Maydole, however, is quantifying over things in a way that is distinct from moments (in his “Modal Third Way” you see a more careful distinction between moments and things). With the right qualifications, and stipulations, this worry can be alleviated, e.g. one might say “no concrete things” or “no subsitent things” rather than “nothing”.

Maydole, R. 2012. “The Ontological Argument”. In The Blackwell Companion to Natural Theology. Ed. W.L. Craig & J.P. Moreland. Malden, MA: Blackwell Publishing, pp. 580-586.

Vexing Links (5/25/2015)

Some recent links of note:

  • Robin Smith has recently updated the SEP article on Aristotle’s Logic
  • Tuomas Tahko updates an entry at the SEP on Ontological Dependence originally authored by the late great E.J. Lowe
  • Peter Adamson’s History of Philosophy without any Gaps has a new podcast episode  on 13th century Logic
  • Massimo Pigliucci took the New Atheists to the woodshed (almost feel sorry for them)
  • Jeffery Jay Lowder notes that David Wood took John Loftus to the woodshed on the question “Did Jesus Rise from the Dead?” (I agree with Lowder and couldn’t help but get the impression that Loftus knew he had been whipped by the end of the debate—granting that he failed to address 1 Cor 15)
  • Messianic Drew constructs a similar argument for God from Fitch’s paradox as I did previously on this blog.  One difference is that I use the BCF (Big Conjunctive Fact) to explicitly argue for an omniscient mind (which isn’t a big slice of God, but certainly troubling for naturalism)
  • Alex Pruss as a nice neat argument for God from life (I list biogenesis as evidence that supports theism, though that is always subject to new discoveries)
  • Speaking of which, a new theory of abiogenesis is being lauded by internet atheists as putting God on the ropes (Should theists be sweating? It might be worth noting that the scientist who has devised the theory, Dr. England, is an observant Jew who prays to God three times a day.  Classical theists don’t require that the creation of life to be a miraculous intervention, but the general order of nature points to a living source of creation)
  • I recently found an interesting clip of evolutionary biologist, Ken Miller (who testified against ID in the Dover case) defend Aquinas’s fifth way (though the fifth way is a teleological argument, it is not the same as the sorts of arguments ID theorists put forward, as Ed Feser likes to point out)
  • Inspiring Philosophy has a great video response to the question of whether the Trinity is a pagan concept
  • Bill Vallicella and Dale Tuggy are discussing God’s relationship to being (this is the latest from Vallicella, but it all started from this interview on Tuggy’s superb Trinities podcast)
  • Lastly, and most importantly, if you are wondering which superhero would win in a one-on-one battle, wonder no more

π = 3?

Occasionally you will hear an anti-theist mock the doctrine of the inspiration of scripture by arguing that the Bible says that π = 3. They cite 1 Kings 7:23:

Now he made the sea of cast metal ten cubits from brim to brim, circular in form, and its height was five cubits, and thirty cubits in circumference.

or 2 Chronicles 4:2:

Also he made the cast metal sea, ten cubits from brim to brim, circular in form, and its height was five cubits and its circumference thirty cubits.

Now let’s leave aside that this is a) a description of a real physical bowl and not a treatise on abstract Euclidean circles (so perhaps the object wasn’t a perfect circle), and b) the point of these passages is to describe physical objects in terms that the people of the time would have understood. So if they didn’t know π or that every circle has the same ratio of circumference to diameter, they would have had an incomplete description without being informed of rough approximations of both. I think the best response to this “Bible contradiction” is this:


I don’t really take such an objection seriously. I think it betrays some basic ignorance about what the doctrine of inspiration means and what we should expect an inspired text to look like. For the anti-theist who cites this, the expectation is that God should have handed down a math treatise and a few books on general relativity and quantum mechanics (assuming those theories are not overturned by some new paradigm in physics). But why would God do that? Why would God spend pages explaining geometric and arithmetic relations when he gave us the intelligence to do these things ourselves? This only reinforces the pet-hamster view of humanity’s relationship to God. His role is just to satisfy out every need so that we don’t have to stretch ourselves in any way. I’m sorry, but I disbelieve in that sort of God too.

File “π = 3 in the Bible” under really really bad arguments against Biblical inspiration.

Oh, and in case you don’t get the joke, Lawrence Krauss once tried to refute William Lane Craig in a debate by arguing that 2 + 2 can equal 5: here.

Some Proposed Corrections to Maydole’s Temporal Contingency Argument

Robert Maydole presents an interesting argument for a supreme being, called the temporal contingency argument.  The argument is a long deduction, and so is seen as somewhat difficult to comprehend. The version that I am critiquing appears in the Blackwell Companion to Natural Theology and appears as follows (with highlighted lines that I believe are problematic)[1]:


These errors are not fatal to the argument, however.  Here is a quick workaround that I think preserves the spirit of Maydole’s deduction (using nested conditional proofs and the identity rule, for example).  I’ve simplified some of the lexicon, but if pretty much follows Maydole’s definitions.  A revised deduction is as follows:

Bx ≝ x begins to exist at some time and ceases to exist at some time
Tx ≝ x is temporally necessary
Cx ≝ x is temporally-contingent
Fx ≝ x exists for a finite period of time
≝ Only finitely many things have existed to date
≝ Something presently exists
≝ There was a time when nothing existed
Sxy ≝ x is a sufficient reason for the existence of y
Wx ≝ x is without any limitations
Gxy ≝ x is greater than y
Sx ≝ (~◊(∃y)Gyx & ~◊(∃y)(x≠y & ~Gxy))

1. P (premise)
2. M (premise)
3. (
∀x)(Cx ⊃ Bx) (premise)
4. (∀x)(Bx ⊃ Fx) (premise)
5. ((∀x)Fx & M) ⊃ N (premise)
6. N ⊃ ~P (premise)
7. (
∀x)(Tx ≡ ~Cx) (premise)
8. (∀x)Cx (IP)
9. Cμ ⊃ Bμ (3 UI)
10. Cμ (8 UI)
11. B
μ (9,10 MP)
12. B
μ ⊃ Fμ (4 UI)
13. Fμ (11,12 MP)
14. (∀x)Fx (13 UG)
15. (∀x)Fx & M (2,14 Conj)
16. N (5,15 MP)
17 ~P (6,16 MP)
18. P & ~P (1,17 Conj)
19. ~(
∀x)Cx (8–18 IP)
20. (∃x)~Cx (19 QN)
21. ~Cν (20 EI)
22. Tν ≡ ~Cν (7 UI)
23. (T
ν ⊃ ~Cν) & (~Cν ⊃ Tν) (22 Equiv)
24. (~C
ν ⊃ Tν) (23 Simp)
25. Tν (21,24 MP)
26. (∃x)Tx (25 EG)
27. (
∀x)(∃y)Syx (premise)
28. (∀x)[(∃y)Syx ⊃ (∃z)(Szx & Szz)] (premise)
29. (∀x)(∀y)[(Tx & Syx) ⊃ ~Cy] (premise)
30. (∀y)[(Ty & Syy) ⊃ Wy] (premise)
31. (∀y)[Wy ⊃ ☐(∀z)(z≠y ⊃ Gyz)] (premise)
32. ~◊(∃y)Gyy (premise)
☐(∀x)(∀y)(Gxy ⊃ ~Gyx) (premise)
34. (∃y)Syν (27 UI)
35. (∃y)Syν ⊃ (∃z)(Szν & Szz) (28 UI)
36. (∃z)(Szν & Szz) (34,35 MP)
37. Suν & Suu (36 EI)
38. (∀y)[(Tν & Syν) ⊃ ~Cy] (29 UI)
39. (Tν & Suν) ⊃ ~Cu (38 UI)
40. Suν (37 Simp)
41. Tν & Suν (25,40 Conj)
42. ~Cu (39,41 MP)
43. Tu ≡ ~Cu
 (7 UI)
44. (Tu ⊃ ~Cu) & (~Cu ⊃ Tu) (43 Equiv)
45. ~Cu ⊃ Tu (44 Simp)
46. Tu (42,45 MP)
47. Suu (37 Simp)
48. Tu & Suu (46,47 Conj)
49. (Tu & Suu) ⊃ Wu (30 UI)
50. Wu ⊃ 
☐(∀z)(z≠u ⊃ Guz) (31 UI)
51. Wu (48,49 MP)
52. ☐(∀z)(z≠u ⊃ Guz) (50,51 MP)
53. ☐(∀z)(~z≠u ∨ Guz) (52 Impl)
54. ☐(∀z)(~z≠u ∨ ~~Guz) (53 DN)
55. ☐(∀z)~(z≠u & ~Guz) (54 DeM)
56. ☐~(∃z)(z≠u & ~Guz) (55 QN)
57. ~◊(∃z)(z≠u & ~Guz) (56 MN)
☐~(∃y)Gyy (32 MN)
59. ☐(∀y)~Gyy (58 QN)
60. (∀y)~Gyy (CP)
61. μ=ν (CP)
62. ~Gμμ (60 UI)
63. ~Gμν (61,62 IR)
64. μ=ν ⊃ ~Gμν (61-63 CP)
65. (∀y)~Gyy ⊃ (μ=ν ⊃ ~Gμν) (60-64 CP)
66. ☐[(∀y)~Gyy ⊃ (μ=ν ⊃ ~Gμν)] (65 NI)
67. ☐(μ=ν ⊃ ~Gμν) (59,66 MMP)
68. ☐(∀x)(∀y)(Gxy ⊃ ~Gyx) & ☐(∀z)(z≠ν ⊃ Gνz) (33,52 Conj)
69. [☐(∀x)(∀y)(Gxy ⊃ ~Gyx) & ☐(∀z)(z≠ν ⊃ Gνz)] ⊃ ☐[(∀x)(∀y)(Gxy ⊃ ~Gyx) & (∀z)(z≠ν ⊃ Gνz)] (theorem)
70. ☐[(∀x)(∀y)(Gxy ⊃ ~Gyx) & (∀z)(z≠ν ⊃ Gνz)] (68,69 MP)
71. {[(∀x)(∀y)(Gxy ⊃ ~Gyx) & (∀z)(z≠ν ⊃ Gvz)] ⊃ (μ≠ν ⊃ ~Gμν)} (theorem)
72. ☐(μ≠ν ⊃ ~Gμν) (70,71 MMP)
73. [☐(μ=ν ⊃ ~Gμν) & ☐(μ≠ν ⊃ ~Gμν)] ⊃ ☐[(μ=ν ∨ μ≠ν) ⊃ (~Gμν ∨ ~Gμν)] (theorem)
74. ☐(μ=ν ⊃ ~Gμν) & ☐(μ≠ν ⊃ ~Gμν) (67,72 Conj)
75. ☐[(μ=ν ∨ μ≠ν) ⊃ (~Gμν ∨ ~Gμν)] (73,74 MP)
76. ☐(μ=ν ∨ μ≠ν) (theorem)
77. ☐(~Gμν ∨ ~Gμν) (75,76 MMP)
78. ☐(~Gμν ∨ ~Gμν) ⊃ ☐~Gμν (theorem)
79. ☐~Gμν (77,78 MP)
80. (∀z)☐~Gzν (79 UG)
81. (∀z)☐~Gzν ⊃ ☐(∀z)~Gzν (theorem)
82. ☐(∀z)~Gzν (80,81 MP)
83. ☐~(∃z)Gzν (82 QN)
84. ~◊(∃z)Gzν (83 MN)
85. ~◊(∃z)Gzν & ~◊(∃z)(z≠ν & ~Gνz) (57,84 Conj)
86. Sν (85 def “S”)
87. (∃x)Sx (86 EG)

[1]R. Maydole. 2012. “The Ontological Argument”. The Blackwell Companion to Natural Theology. Ed. W.L. Craig & J.P. Moreland. Malden, MA: Blackwell Publishing. Document image retrieved from <>.

Bernstein/Ahmed debate on Unbelievable?

Unbelievable? hosted a great debate between C’Zar Bernstein and Arif Ahmed on the Argument from Consciousness for God’s existence: listen here.

A rough outline for Bernstein’s argument was something like:
1. There are non-physical minds.
2. The explanation for (1) is either personal or natural.
3. The explanation is not natural.
4. Therefore, the explanation is personal.

Fleshed out, Bernstein defended an evidential argument, where consciousness doesn’t logically entail the God of classical theism, but that consciousness provides evidential support for classical theism. Most of the debate came down to the first premise, which Bernstein defended by way of the modal argument for the soul.

Ahmed focuses on an eliminativist/Humean response and basically just denied there were persons, and fell back on the claim that we should really only admit into our ontology whatever is strictly needed for science (so no need to talk about conscious persons or moral properties).  A good deal of the discussion focused on whether we have good reason to think persons exist, and I think Bernstein got the better of Ahmed in the end (pointing out how Ahmed couldn’t even really talk about pain without referencing his own awareness of it).  However, this meant that little time was focused on showing why consciousness is good evidence in support of classical theism.  Indeed, I agree with Bernstein that it is good evidence.  However, I think more needs to be said for why this is so.

It’s worth a listen, that is for sure.

An Ontological Argument from Actuality

Here is a refinement on my ontological argument from actuality:

1. Something is an Anselmian God if and only if it is conceivable, nothing can be conceived of which is more actual, and it necessarily exists (definition Θ).

2. There is something conceivable such that nothing can be conceived of which is more actual (premise).

3. For all x, if the possibility of failing to conceive of x implies the possibility that x doesn’t exist, x is mentally dependent (premise).

4. For all x, if x is mentally dependent, there is something conceivable that is more actual than x (premise).


5. An Anselmian God exists.

I start out with a definition of an Anselmian God, which is a stipulation, but is rooted in the idea that a Being of Pure Actuality is arguably perfect and possesses a good number of divine attributes.  

As I noted in a previous post, the traditional argument uses a “greater than” relation, which some find suspect.  “Greatness” would have been understood by Anselm as something that can be evaluated objectively on a scale, as in the Neo-Platonic notion of the Great Chain of Being.  To the contemporary ear, “greatness” seems subjective and vague.  I think “actual” in the Thomistic-Aristotelian sense is a fair approximation of greatness, but we can have a better sense of what “actual” means.  Thomas is able to derive the divine attributes from a being of Pure Actuality, so “most actual” is plausibly a divinely loaded superlative.  Moreover, it seems to me that the act-potency distinction is not something the contemporary ear would take to be dependent on subjective opinions.  So, I think (2) is fairly impeccable.  

I think (3) is a bit clunky, but it basically means that if something is merely a concept, then it is mentally dependent.  So, in the case of God, if God is merely a concept in the mind, then the possibility that God could fail to be conceived by all minds that exist implies that God, as a mere concept, could fail to exist, and so depends upon minds to continue to exist.  Put another, if God is merely a concept, then there was no God in the Jurassic period, as William Lane Craig once suggested to John Dominic Crossan.

Finally, (4) says that if something is mentally dependent, then something is conceivable that is more actual than it.  Some people think, for instance, that moral values are mind-dependent.  So, for instance, the actuality of the value of human life, VHL, depends on there being an actual community of minds that actually conceive of human life as valuable.  Were such a community to cease to exist, the VHL would only potentially exist, even if humans existed.  If the VHL were an objective fact grounded in human nature, then the actuality of the VHL would obtain whenever humans actual exist.  There is a certain assymetry that suggests that grounding the VHL in human nature is to view VHL as more actual than grounding VHL in the subjective opinions of a community of minds.  For the VHL to be actual in one case, there need only be actual humans exemplifying human nature, where as in the latter, there needs to be actual humans exemplifying human nature and an actual community of minds that actually is of the opinion that human life is valuable.  For, without the humans, a community of minds that endorses the VHL would really just be saying that VHL potentially exists and would be actual upon the occassion of human life.  We could say, then, that x is more actual than y iff the existence of x depends upon the actualization of fewer potentials than y depends upon.  VHL grounded in the actuality of human nature depends upon the actualization of fewer potentials than VHL grounded in subjective opinions about humans. So (4) just tells us that for any x that depends upon the mental for its actuality, it is conceivable that there is something that is more actual (and less dependent) than x, e.g. to conceive that x can actually exist independent of mentally conceiving of x.


Cx – x is conceived
Mx – x is mentally dependent
Axy – x is more actual than y
Θx- x is an Anselmian God, 

that is: 

1. (∀x){Θx ≝ ([♢Cx & ~(∃y)(Ayx & ♢Cy)] & ☐(∃z)(z=x))} (Def Θ)
2. (∃x)[♢Cx & ~(∃y)(Ayx & ♢Cy)] (premise)
3. (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx} (premise)
4. (∀x){Mx ⊃ [(∃y)(Ayx & ♢Cy)]} (premise)
5. (∀x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (IP)
6. ♢Cu & ~(∃y)(Ayu & ♢Cy) (2 EI)
7. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ Mu (3 UI)
8. Mu ⊃ [(∃y)(Ayu & ♢Cy)] (4 UI)
9. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ [(∃y)(Ayu & ♢Cy)] (7,8 HS)
10. [♢Cu & ~(∃y)(Ayu & ♢Cy)] ⊃ [♢~Cu ⊃ ♢~(∃z)(z=u)] (5 UI)
11. ♢~Cu ⊃ ♢~(∃z)(z=u) (6,10 MP)
12. (∃y)(Ayu & ♢Cy) (9,11 MP)
13. Avu & ♢Cv (12 EI)
14. ~(∃y)(Ayu & ♢Cy) (6 Simp)
15. (∀y)~(Ayu & ♢Cy) (14 QN)
16. ~(Avu & ♢Cv) (15 UI)
17. (Avu & ♢Cv) & ~(Avu & ♢Cv) (13,16 Conj)
18. ~(∀x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (5-17 IP)
19. (∃x)~{[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (18 QN)
20. (∃x) ~{~[♢Cx & ~(∃y)(Ayx & ♢Cy)] ∨ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (19 Impl)
21. (∃x){~~[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (20 DeM)
22. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (21 DN)
23. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[~♢~Cx ∨ ♢~(∃z)(z=x)]} (22 Impl)
24. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[☐Cx ∨ ♢~(∃z)(z=x)]} (23 ME)
25. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & [~☐Cx & ~♢~(∃z)(z=x)]} (24 DeM)
26. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & [~☐Cx & ☐(∃z)(z=x)]} (25 ME)
27. [♢Cu & ~(∃y)(Ayu & ♢Cy)] & [~☐Cu & ☐(∃z)(z=u)] (26 EI)
28. ~☐Cu & ☐(∃z)(z=u) (27 Simp)
29. ☐(∃z)(z=u) (28 Simp)
30. [♢Cu & ~(∃y)(Ayu & ♢Cy)] (27 Simp)
31. [♢Cu & ~(∃y)(Ayu & ♢Cy)] & ☐(∃z)(z=u) (29,30 Conj)
32. Θu (1,31 “Def Θ”)
33. (∃x)Θx (32 EG)


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