## Thursday, 9 May 2013

### The Fans of Plantinga's BS5

Over at Craig-Land, I’ve been struggling to make a point to a couple of fans of William Lane Craig’s Ontological Argument and Plantinga’s attendant BS5, it’s not so much that I can’t make the point (I’ve made it repeatedly – here, here and many times here) but I apparently need to express it in some bizarre blend of complexity and simplicity in order for them to comprehend.  So, I’ll try one last time. (Please note that the first part of this article is directed more at awestruck people like veka@RF rather than the more thoughtful cpdavey24@RF.  I address the latter later in the article.)

For those who have lived lives of bliss and were previously unaware of the argument, it goes like this:

cpdavey24’s formulation

cpdavey24’s formulation in words (BS5)

Assertion:
it is possible that it is necessary that (there exists a being x) [where] x is a Maximally Excellent being

Premise:
(IF) it is possible that it is necessary that (there exists a being x) [where] x is a Maximally Excellent being
(THEN) it is necessary that (there exists a being x) [where] x is a Maximally Excellent being

Conclusion:
Therefore it is necessary that (there exists a being x) [where] x is a Maximally Excellent being

William Lane Craig’s formulation

1.    It is possible that a maximally great being exists.
2.    If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
3.    If a maximally great being exists in some possible world, then it exists in every possible world.
4.    If a maximally great being exists in every possible world, then it exists in the actual world.
5.    If a maximally great being exists in the actual world, then a maximally great being exists.
6.    Therefore, a maximally great being exists.

(For completeness, there is also the version presented in The Blackwell Companion to Natural Theology (which ontologicalme@RF provided, along with the correction at step 11):

Ax =df x is maximally great
Bx =df x is maximally excellent
W(Y) =df Y is a universal property
Ox =df x is omniscient, omnipotent, and morally perfect Deduction

1. ◊(Ǝx)Ax                                                                           pr
2. (x)(Ax ≣ Bx)                                                             pr
3. (x)(Bx ⊃ Ox)                                                                pr
4. (Y)[W(Y) ≣ ((Ǝx)Yx ⋁ (☐∼(Ǝx)Yx)]                             pr
5. (Y)[(ƎZ)(x)(Yx ≣ Zx) ⊃ W(Y)]                                  pr
6. (ƎZ)(x)(Ax ≣ Zx)                                                      2, EG
7. [(ƎZ)(x)(Ax ≣ Zx) ⊃ W(A)]                                       5, UI
8. W(A) ≣ ((Ǝx)Ax ⋁ (☐∼(Ǝx)Ax)                                    4, UI
9. W(A)                                                                                6, 7 MP
10. W(A) ⊃ ((Ǝx)Ax ⋁ (☐∼(Ǝx)Ax)                                  8, Equiv, Simp
11. (Ǝx)Ax ⋁ (☐∼(Ǝx)Ax)                                                9, 10 MP
12. ∼◊∼∼(Ǝx)Ax ⋁ ((Ǝx)Ax)                                           11, Com, ME
13. ◊(Ǝx)Ax ⊃ (Ǝx)Ax                                                      DN, Impl
14. (Ǝx)Ax                                                                       1, 13 MP
15. (x)(Ax ≣ Bx) ⊃ ((Ǝx)Ax ⊃ (Ǝx)Bx)               theorem
16. (Ǝx)Bx                                                                    14, 15 MP (twice)
17. (x)(Bx ⊃ Ox) ⊃ ((Ǝx)Bx ⊃ (Ǝx)Ox)              theorem
18. (Ǝx)Ox                                                                    16, 17 MP (twice)
19. (Ǝx)Ox                                                                        18, NE

I’ve basically not bothered with this last “proof” for a couple of reasons, one being that it’s rather inaccessible to the casual reader [and, to be honest, me], the other will be touched on shortly.)

My objection to the Ontological Argument in general, and Plantinga’s specifically, is that the possibility of a thing, or even the possibility of the necessity of thing, doesn’t by itself make that thing necessary.

It should not be controversial that:

1. a thing may be either possible (◊) or not possible (~◊) but not both, and
2. a thing may be either necessary (□) or not necessary (~□) but not both.

Even the fans of Plantinga’s BS5 seem to agree on these points.

Essentially, the disagreement seems to centre on the meaning of “possible” and “necessary”.  If you look at the freely available (and apparently well regarded) Stanford Encyclopedia of Philosophy (entry on Modal Logic), you will find a discussion of necessity, including the following:

The system K is too weak to provide an adequate account of necessity. The following axiom is not provable in K, but it is clearly desirable.

(M) □A®A

(M) claims that whatever is necessary is the case.

A little further on it continues with:

One could engage in endless argument over the correctness or incorrectness of these and other iteration principles for □ and ◊. The controversy can be partly resolved by recognizing that the words ‘necessarily’ and ‘possibly’, have many different uses. So the acceptability of axioms for modal logic depends on which of these uses we have in mind.

In other words, in K you can’t get from a statement that a thing is necessary to a statement that that thing actually is and the very validity of an axiom rests on what you mean by “necessity” and “possibility” (via the meanings of “necessarily” and “possibly”).

Let’s not focus too deeply, therefore, on what “necessity” and “possibility” mean.  What I’d like to do instead is introduce the following “neopolitonian axioms” with the actual axiom designation following in (round brackets) or [square brackets]:

n1: If A is possible, then it is not possible that A is not possible … A®~~A … [double negation]
n2: If A is necessary, then it is not necessary that A is not necessary … □A®~□~□A … [double negation]
n3: If A is necessary, then A is possible … □A®A … (D)

Because I am a visual sort of person, I present n3 in a Venn Diagram:

In other words, the set of things that are necessary fit into the set of things that are possible.

If we use Transposition we can reach:

n4: If A is not possible, then A is not necessary … ~A®~□A

Here’s a visual n4:

In other words, if something is in the set of things that are not possible they are necessarily not part of the set of things that are necessary (or ~A®~□□A and since □□A®□A therefore ~A®~□A).

Now that I have an excuse to present such things visually, I want to illustrate that in the real world there is an intersection of things that are not necessary but are nevertheless possible (A ~□A).  It’s possible for me to be a dwarf, but it’s not necessary.  It’s possible for my car to be green, but it’s not necessary.  Visually:

Now if Craig (via Plantinga) is arguing that if a thing is possible then it must therefore exist, then they surely have a problem.  There is clearly a category of things which are possible and not necessary and therefore Craig and Plantinga cannot reach the conclusion that they are grasping at.

The best they can do is argue all the way to “necessity” in K (which they might attempt by writing long extremely inaccessible "proofs" in arcane logic schemas which might easily mean absolutely nothing once you decode it and will be a complete waste of time for the poor bastard who attempts that decoding effort), at which point they are stuck because K is too weak to get you from necessity to existence (so that huge decoding effort was a waste of time even before that poor bastard started).  If they want to, eager theists can use a stronger system of logic which can get them from necessity to existence, but then they can’t reach necessity from possibility.  So, what they do is use a weak form like K up until they achieve necessity, then they swap taxi-cabs and pretend that they have been using a stronger system of logic all the time.

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

The SEP entry on modal logic actually addresses the confusion that leads to BS5, so let’s have a quick look at that:

It is interesting to note that S5 can be formulated equivalently by adding (B) to S4. The axiom (B) raises an important point about the interpretation of modal formulas. (B) says that if A is the case, thenA is necessarily possible. One might argue that (B) should always be adopted in any modal logic, for surely if A is the case, then it is necessary that A is possible. However, there is a problem with this claim that can be exposed by noting that ◊□AA is provable from (B). So ◊□AA should be acceptable if (B) is. However,◊□AA says that if A is possibly necessary, then A is the case, and this is far from obvious. Why does (B) seem obvious, while one of the things it entails seems not obvious at all? The answer is that there is a dangerous ambiguity in the English interpretation of A→□◊A. We often use the expression ‘If A then necessarily B’ to express that the conditional ‘if A then B’ is necessary. This interpretation corresponds to □(AB). On other occasions, we mean that if A, then B is necessary: A→□B. In English, ‘necessarily’ is an adverb, and since adverbs are usually placed near verbs, we have no natural way to indicate whether the modal operator applies to the whole conditional, or to its consequent. For these reasons, there is a tendency to confuse (B): A→□◊A with □(A→◊A). But □(A→◊A) is not the same as (B), for □(A→◊A) is already a theorem of M, and (B) is not. One must take special care that our positive reaction to □(A→◊A) does not infect our evaluation of (B). One simple way to protect ourselves is to formulate B in an equivalent way using the axiom:◊□AA, where these ambiguities of scope do not arise.

This makes me wonder if this confusion is fundamentally an American English issue.  It’s been a constant irritation to me that Americans insist on using poetic English literally:

All that glisters is not gold – Merchant of Venice, Shakespeare

This is poetic/archaic, it is not correct modern English.  Think on it for a moment.  Imagine that I suffer from dwarfism and I have seven children, I could say when addressing a session of Little People Anonymous:

·         “Hello, I am neopolitan, I am a dwarf; all of my seven children are not dwarves” or

·         “Hello, I am neopolitan, I am a dwarf; not all of my seven children are dwarves” or

·         “Hello, I am neopolitan, I am a dwarf; all of my seven children are dwarves”

When shown together like this, we can clearly see that I am making three different hypothetical statements.  I have seven children with dwarfism, seven children without dwarfism or seven children, some of whom have dwarfism (but am conveying something by highlighting those who don’t have dwarfism, perhaps I am about to launch into a discussion of how I adopted four of my children in order to have someone in the house who can reach the top shelves and how silly I felt when I realised that I could have just bought a stepladder).

Returning to Shakespeare, some things that glister actually are gold, otherwise the claim makes no sense … so if he was writing in proper English today, and wanted to convey his message clearly rather than poetically, he would have written “Not all that glisters is gold” or, alternatively, “Some things that glister are not gold”.  It does amuse me that I can say “All Americans are not smart”, while pointing to some exception to the hypothetical assertion “All Americans are smart” (for example Bush when talking to non-Republicans and Clinton when talking to non-Democrats), and the vast majority of Americans won’t be offended.  I do know that some Americans are frighteningly intelligent, it’s merely the little grammatical anomaly that I find amusing and that amusement allows me to overcome some of the frustration associated with American cultural imperialism.

Anyhoo, Americans (and increasingly the rest of the English-talkificating world) are thus rendered incapable of placing adverbs correctly and so, as a consequence, when someone says “it is necessary that, if A, then it is possible that A” it should come as no great surprise that there is confusion as to whether this means □(A→◊A) or A→□◊A.

An example of an equivalent confusion is in cpdavey24’s attempt to prove the validity of BS5, the “cpdavey24 proof” (note that he uses p rather than A, so I must make the one additional assertion to bring his “proof” in line with the notation at SEP, that p«A.   I’ll provide clarifications in green and English language versions in gold):

1. ◊□A (Assumption for Conditional Proof (CP))this is just the starting position
it is possible that A is necessary

2. ~□A (Assumption for Reductio Ad Absurdum (RAA))indication of intent to prove A is necessary by showing that the claim that A is not necessary is absurd, given the assumption that A is possibly necessary

3. ~~◊~A (2 E2)grouping brackets have been omitted, should read ~(~◊~A), uses the rule (E2) that if a thing is necessary, then it cannot be impossible
it is not impossible that A is not actual

4. ◊~A (3 Double Negation (DN))if it is not the case that it is not possible that A is not actual, then it follows that …
it is possible that A is not actual

5. □| ◊~A (4 S5-reit)this appears to be a reinterpretation of (5), the valid form of which states that ◊A®□◊A
it is necessary that it is possible that A is not actual

6. □◊~A (5 nec intro)the introduction of necessity, now states that it is necessary that it is possible that A is not actual
it is necessary that it is possible that A is not actual

7. □~□~~A (6 E1)grouping brackets have been omitted, should read □(~□~(~A)), uses the rule (E1) that if a thing is possible, then it cannot be necessarily non-actual
it is necessary that it is unnecessary that A is not not actual

8. □~□A (7 DN)if it is necessary that it not be necessary that A not be not actual then it follows that …
it is necessary that it is unnecessary that A is actual

9. ~◊~~□A (8 E2)grouping brackets have been omitted, should read ~◊~(~□A), uses E2 again
it is impossible that it is not unnecessary that A is actual

10. ~◊□A (9 DN)if it is not possible that it not be not necessary that A is actual then it follows that …
it is impossible that it is necessary that A is actual

11. ◊□A & ~◊□A (1,10 Conjunction)indicates that from 2 we arrive at 10, which conflicts with the starting position, the necessity of A cannot be both possible and impossible, so the claim is that the absurd is achieved

12. ~~□A (1-11 RAA)the negation of 2, since the proof seems to indicate that 2 cannot be true and only other option is not 2
it is not unnecessary that A is actual

13. □A (12 DN)simple double negation
it is necessary that A is actual

14. ◊□A -> □A Q.E.D. (1-13 CP)a “look at me” statement, claiming that it has been proven that …
if something is possibly necessary then it is necessary

cpdavey24 did want me to make comment on this proof, so here here’s my comment in the form of a question: cpdavey24, does it not feel strange that you are obliged to use an RAA approach?

Let’s try it without:

1. ◊□A (Initial position)
2. A <--> ~B (Assumption for Logical Fiddle)
3. ◊□~B (Substitution of 2 into 1)
4. ~~◊~~□~B (3 double DN)
5. ~□~□~B (4 E2)
6. ~□~□A (Substitution of 2 into 5)
7. □A (6 n2)
8. ◊□A -> □A Q.E.D. (1-7)

This is a superior proof since it doesn’t rely on the distracting RAA.  But it’s still wrong.  Note that the substitution steps (A <--> ~B) aren’t really necessary, they are just a distraction.

1. ◊□A (Initial position)
2. ~~◊~~□A (3 double DN)
3. ~□~□A (4 E2)
4. □A (6 n2)
5. ◊□A -> □A Q.E.D. (1-4)

Where’s the error?

Well, for starters, I used n2 in reverse, which isn’t valid.  However, I also think that there might be an issue with using generating a “not” via a double negative in front of a symbol and then using it with the preceding symbol.

However, if I now try to use cpdavey24’s RAA approach:

1. ◊□A (Initial position)
2. ~□A (Assumption for RAA)
3. ~~□~□A (2 n2)
note that this is no longer in reverse, so it is now valid
4. ~~~◊~~□A (3 E2)
note that I am now using E2 in the same direction as cpdavey24 did
5. ~◊□A (4 DN)
note that I am now using DN in the same direction as cpdavey24 did
6. ◊□A & ~◊□A (1,5 Conjunction)
7. ~~□A (2-6 RAA)
8. □A (7 DN)
9. ◊□A -> □A Q.E.D. (1-8)

So what we have is what appears to be a perfectly valid proof, despite the fact that I just showed you there was an error in it.  All I’ve done is hidden it with a Reductio Ad Absurdum.

Suddenly, it doesn’t seem as strange that cpdavey24 used the RAA approach.

I’ll leave it to the reader (perhaps even cpdavey24) to discover whether the error in the “cpdavey24 proof” can be found when it is run forward, rather than in reverse.  If you do find it, feel free to post it in the comments section.

EDIT:  I had a pretty major typo in the last "proof".  Word generated numbered lists seem to cause all sorts of problems and when I rewrote the "proof" ... well, I screwed up.  Typo is now fixed.