A Modest Formulation of the Ontological Argument

In this post, I have formulated Anselm’s argument for the necessary existence of a being than which none greater can be conceived.  However, I have noted that the argument depends upon a two-place “greater than” predicate that functions with something like the Neo-Platonic “Great Chain of Being” in mind.  Some thing, x, is conceived to be greater than y in the sense that x is understood to have more capacities or has an essence that can be actualized to a greater degree. For example, a plant is understood to contingently exists, grows, takes in nutrients, and reproduces. An animal is understood to be greater in the sense that it too contingently exists, grows, takes in nutrients, and reproduces, but it also has capacities like sentience, and can self-move, etc. So the greater something is, the more powers/more capacities it is understood to have. If God exists, then God would be that being which none more powerful could be conceived, which is just to say “none greater”. I find the metaphysics where a two-place “conceivably greater than” predicate can be objectively exemplified to be extremely plausible. There is an objective sense in which I have greater capacities and abilities than a flea.

The argument is as follows:

D1. Some x is an Anselmian God if and only if x is conceivable, it is not the case that there is something that is conceivably greater than x, and x necessarily exists.

P1. There is some x conceivable such that there is nothing conceivably greater than x.

P2. 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).

P3. For all x, if x is mentally dependent, there is some y such that y is conceivably greater than x (premise).

P4. If there is some x such that necessarily there is some z and z is identical to x, and x is an Anselmian God, then necessarily there exists an Anselmian God.


C1. Necessarily, there is an Anselmian God.

That is the argument in ordinary language. To show that it is a formally valid syllogism, I offer the following formal deduction:


Cx ≝ x is conceived
Mx ≝ x is mentally dependent
Gxy ≝ x is conceived to be greater than y
Θx ≝ (∃x){[♢Cx & ~(∃y)♢Gyx]& ☐(∃z)(z=x)} (Def Θx)

1. (∃x)[♢Cx & ~(∃y)♢Gyx] (premise)
2. (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx} (premise)
3. (∀x)[Mx ⊃ (∃y)♢Gyx] (premise)
4. (∃x)[☐(∃z)(z=x)& Θx] ⊃ ☐(∃x)Θx (premise)
5. (∀x){[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (IP)
6. ♢Cμ & ~(∃y)♢Gyμ (1 EI)
7. [♢~Cμ ⊃ ♢~(∃z)(z=μ)] ⊃ Mμ (2 UI)
8. Mμ ⊃ (∃y)(♢Gyμ) (3 UI)
9. [♢~Cμ ⊃ ♢~(∃z)(z=μ)] ⊃ (♢Gyμ)(7,8 HS)
10. ♢Cμ & ~(∃y)♢Gyμ] ⊃ [♢~Cμ ⊃ ♢~(∃z)(z=μ)] (5 UI)
11. ♢~Cμ ⊃ ♢~(∃z)(z=μ) (6,10 MP)
12. (∃y)♢Gyμ (7,9 MP)
13. ♢Gνμ (12 EI)
14. ~(∃y)♢Gyμ (6 Simp)
15. (∀y)~(♢Gyμ) (14 QN)
16. ~♢Gνμ (15 UI)
17. ♢Gνμ & ~♢Gνμ (13,16 Conj)
18. ~(∀x){[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (5-17 IP)
19. (∃x)~{[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (18 QN)
20. (∃x) ~{~[♢Cx & ~(∃y)♢Gyx] ∨ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (19 Impl)
21. (∃x){~~[♢Cx & ~(∃y)♢Gyx] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (20 DeM)
22. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (21 DN)
23. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[~♢~Cx ∨ ♢~(∃z)(z=x)]} (22 Impl)
24. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[☐Cx ∨ ♢~(∃z)(z=x)]} (23 ME)
25. (∃x){[♢Cx & ~(∃y)♢Gyx] & [~☐Cx & ~♢~(∃z)(z=x)]} (24 DeM)
26. (∃x){[♢Cx & ~(∃y)♢Gyx] & [~☐Cx & ☐(∃z)(z=x)]} (25 ME)
27. [♢Cμ & ~(∃y)♢Gyμ] & [~☐Cμ & ☐(∃z)(z=μ)] (26 EI)
28. ~☐Cμ & ☐(∃z)(z=μ) (27 Simp)
29. ☐(∃z)(z=μ) (28 Simp)
30. [♢Cμ & ~(∃y)♢Gyμ] (27 Simp)
31. [♢Cμ & ~(∃y)♢Gyμ] & ☐(∃z)(z=μ) (29,30 Conj)
32. Θμ (31 Def “Θx”)
33. ☐(∃z)(z=μ) & Θμ (29,32 Conj)
34 (∃x)[☐(∃z)(z=x) & Θx] (33 EG)
35. ☐(∃x)Θx (4,34 MP)


Indeed, I find the above argument very persuasive. However, there may be some who are resistant to the notion that the two-place “conceivably greater-than” predicate can actually and objectively be exemplified. For such a person, I propose a more modest version of the argument. The more modest version is that, since C1, i.e. “☐(∃x)Θx”, is provable given P1-P4,one can argue that if P1-P4 are jointly possible, C1 is possible, and so an Anselmian God necessarily exists. This follows given S5 in modal logic, which says that ◊☐P entails ☐P. The argument can be formally proved as follows:

Let, also:

P1 ≝ (∃x)[♢Cx & ~(∃y)♢Gyx]
P2 ≝ (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx}
P3 ≝ (∀x)[Mx ⊃ (∃y)♢Gyx]
P4 ≝ (∃x)[☐(∃z)(z=x) & Θx] ⊃ ☐(∃x)Θx
C1 ≝ ☐(∃x)Θx

36. ◊[(P1 & P2) & (P3 & P4)] (premise)
37. [(P1 & P2) & (P3 & P4)] ⊢ C1 (premise; proved by 1-35)
38. [◊[(P1 & P2) & (P3 & P4)]& {[(P1 & P2) & (P3 & P4)]⊢ C1}] ⊃ ◊C1 (premise)
39. ◊[(P1 & P2) & (P3 & P4)] & {[(P1 & P2) & (P3 & P4)] ⊢ C1} (36,37 Conj)
40. ◊C1 (38,39 MP)
41. ◊☐(∃x)Θx (40 Def “C1”)
42. ☐(∃x)Θx (41 by “S5”)

QED (again)

Since (37) is established, and (38) merely argues that if premises are jointly possible, and those premises prove some conclusion, then the conclusion is possible, (38) is relatively uncontroversial.  So, if one objects that P1-P4 are not actually true, but admits that they are at least broadly logically, or metaphysically compossible, then one ought to agree that, necessarily, an Anselmian God exists.

Non-physical thought processes

Image from the American Heart Association Blog

An argument for the non-physical intellect and the possibility that it can survive the death of the body (based on a recent Facebook discussion and also roughly on James F. Ross’s Immaterial Aspects of Thought)1:

D1) For all x, (x is a semantically determinate process ≝ there exists a y such that x contains y, and y is a set of operations that have a fixed and well-defined syntax and are semantically unique in their referents).
P1) For all x, (if x is a physical process, it is not the case that x is a semantically determinate process).
P2) There exists an x and there exists a y, such that {x is a formal thought process in my intellect, [x contains y, and (y = Modus Ponens)]}
P3) For all y, [ if (y = Modus Ponens), y is a set of operations that have a fixed and well-defined syntax and is semantically unique in its referents].
C1) There exists an x such that (x is a formal thought process in my intellect and it is not the case that x is a physical process). [From D1 and P1-P3]
P4) For all x, [if (x is a formal thought process in my intellect, and the mode of being of my intellect is physical), then x is a physical process].
P5) For all x, (if it is not the case that the mode of being of x is physical, then x is non-physical).
C2) My intellect is non-physical. [From C1, P4 and P5]
P6) For all x, if x is non-physical, then x cannot be physically destroyed.
P7) For all x and all y, if x cannot be physically destroyed and y can be physically destroyed, x can survive the physical destruction of y.
P8) My body can be physically destroyed.
C3) My intellect can survive the physical destruction of my body. [From C2 and P6-P8]

The point of the argument is essentially this: A physical process can be mapped onto a language, as we have computers do. But that physical process is only simulating the use of language and the way it computes symbols is only insofar as we tether symbols to physical states undergoing various processes. But the physical process itself does not fix the semantic content or the syntax, we do. And so we say that a computer might fail to “add” properly because of a hardware malfunction. But there is no telos intrinsic to the physical process that distinguishes functioning from malfunctioning, so it is merely our attempt to simulate adding that can, at times, be frustrated by a computer functioning in ways we did not anticipate or intend.

This is why no physical process can be semantically determinate. You can have a physical process that is given semantic content by a mind, and then it will be semantic, in a sense, but indeterminate in that the process doesn’t have to fix upon the syntax or semantics assigned to it.

However, a mental process like reasoning according to Modus Ponens is a syntactically well-defined operation that a mind can do. When the mind is doing this operation, it is preserving truth values. A mind cannot “do Modus Ponens” and “not do Modus Ponens” at the same time and in the same way. But a physical process “programmed” to track “Modus Ponens-like inferences” can run a program that makes “Modus Ponens-like inferences” while never actually doing Modus Ponens. It might be doing some other operation all together that is indistinguishable from Modus Ponens up to any given point in time, but in the next run of the program, the hardware catches on fire and it spits out on its display “if p, q/ p// not-q”. You can’t say that catching on fire and displaying an invalid argument on a screen was not part of the process, since the process just is however the hardware happens to function.

Given this, and given that the thing known is in the knower according to the mode of the knower, the rest follows from relatively uncontroversial premises.

Deduction: Let,
Px ≝ x is a physical process
Cxy ≝ x contains y
Ox ≝ x is a set of operations
Tx ≝ x has a well-defined syntax
Sx ≝ x is semantically unique in its referents
Fxy ≝ x is a formal thought process in y
Mx ≝ x has a mode of being that is physical
Nx ≝ x is non-physical
Rx ≝ x is physically destroyed
Vxy ≝ x survives the destruction of y
Dx ≝ (∃y){Cxy & [Oy & (Ty & Sy)]}
m ≝ Modus Ponens
i ≝ my intellect
b ≝ my body

1. (∀x)(Px ⊃ ~Dx) (premise)
2. (∃x)(∃y){Fxi & [Cxy & (y = m)]} (premise)
3. (∀y){(y = m) ⊃ [Oy & (Ty & Sy)]} (premise)
4. (∀x)[(Fxi & Mi) ⊃ Px] (premise)
5. (∀x)(~Mx ⊃ Nx) (premise)
6. (∀x)(Nx ⊃ ~◊Rx) (premise)
7. (∀x)(∀y)[(~◊Rx & ◊Ry) ⊃ ◊Vxy] (premise)
8. ◊Rb (premise)
9. (∃y){Fμi & [Cμy & (y = m)]} (2 EI)
10. Fμi & [Cμν & (ν = m)] (9 EI)
11. (ν = m) ⊃ [Oν & (Tν & Sν)] (3 UI)
12. Cμν & (ν = m) (10 Simp)
13.(ν = m) (12 Simp)
14. Oν & (Tν & Sν) (11,13 MP)
15. Cμν (12 Simp)
16. Cμν & [Oν & (Tν & Sν)] (14,15 Conj)
17. (∃y){Cμy & [Oy & (Ty & Sy)]} (16 EG)
18. Dμ (17 Def “Dx”)
19. ~~Dμ (18 DN)
20. Pμ ⊃ ~Dμ (1 UI)
21. ~Pμ (19,20 MT)
22. (Fμi & Mi) ⊃ Pμ (4 UI)
23. ~(Fμi & Mi) (21,22 MT)
24. ~Fμi ∨ ~Mi (23 DeM)
25. Fμi (10 Simp)
26. ~~Fμi (25 DN)
27. ~Mi (24,26 DS)
28. ~Mi ⊃ Ni (5 UI)
29. Ni (27,28 MP)
30. Ni ⊃ ~◊Ri (6 UI)
31. ~◊Ri (29,30 MP)
32. (∀y)[(~◊Ri & ◊Ry) ⊃ ◊Viy] (7 UI)
33. (~◊Ri & ◊Rb) ⊃ ◊Vib (32 UI)
34. ~◊Ri & ◊Rb (8,31 Conj)
35. ◊Vib (33,34 MP)
36. Fμi & ~Pμ (21,25 Conj)
37. (∃x)(Fxi & ~Px) (36 EG)
38. (∃x)(Fxi & ~Px) & Ni (29,37 Conj)
39.[(∃x)(Fxi & ~Px) & Ni] & ◊Vib (35,38 Conj, which is C1-C3)

J.F. Ross. 1992. “Immaterial Aspects of Thought.” In The Journal of Philosophy. Vol. 89. No. 3. 136-150
I. Niiniluto. 1987. “Verisimilitude with Indefinite Truth.” What is Closer-to-the-truth: A Parade of Approaches to Truthlikeness. Ed. T.A.F. Kuipers. Amsterdam: Rodopi. pp. 187-188
(P4) is based upon the principle that a thing known is in the knower according to the mode of the knower. See, for example, Thomas Aquinas Summa Theologiae I.14.1.

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 <http://commonsenseatheism.com/wp-content/uploads/2009/05/irrefutable.png>.


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