Return to BS5

Sometimes, you return to something that you worked hard on in the past and an obvious aspect leaps out at you that wasn't obvious at the time.  In The Fans of Plantinga's BS5, I was arguing against the claim:

This was primarily because this is used to claim that, since we cannot prove that (a maximally great) god does not exist, it is possible that (a maximally great) god does exist, furthermore since we cannot prove that (a maximally great) god is does not exist necessarily it is possible that (a maximally great) god does exist necessarily and it therefore follows that (a maximally great) god does exist (necessarily).  Sounds like BS, right?

Anyway, I also argued against a more complex version of the argument (raised by cpdavey):

1. ◊□A (Assumption for Conditional Proof (CP))

2. ~□A (Assumption for Reductio Ad Absurdum (RAA))

3. ~~◊~A (2 E2)

4. ◊~A (3 Double Negation (DN))

5. □| ◊~A (4 S5-reit)

6. □◊~A (5 nec intro)

7. □~□~~A (6 E1)

8. □~□A (7 DN)

9. ~◊~~□A (8 E2)

10. ~◊□A (9 DN)

11. ◊□A & ~◊□A (1,10 Conjunction)

12. ~~□A (1-11 RAA)

13. □A (12 DN)

14. ◊□A -> □A Q.E.D. (1-13 CP)

I concluded that the Reductio Ad Absurdum wasn't actually necessary, because you can do it like this:

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)

But here we can see that the step between (6) and (7) is not valid (because it's reversed) - so there's a problem which is being disguised by the process of going through the Reductio Ad Absurdum.

I pretty much left it at that, inviting the reader to point out where the problem is, if they could see it.  That was five years ago.

So, what was the obvious thing that recently leaped out at me?  The proof put together, or at least presented by cpdavey relies rather heavily on the introduction of double negatives and swapping from ~□~ to ◊ and from ~◊~ to □.  Both of these latter two moves can be reversed (see Classical Modal Logic).  What it doesn't rely on, is the value of A.  Which means we can do this:

1. ◊~G (Initial weakish atheism position)
2. ~□~~G (1 classical modal logic)

3. ~~□~□~~G (2 n2, right way around)

4. □~□G (3 DN removed)

5. ~◊□G (4 classic modal logic)

This means we arrive at the same conclusion whether we assume that it is possible that there is no god (◊~G) or that it is not necessary that there is a god (~□G), namely that it is not possible that there is necessarily a god (~◊□G), which is precisely what we would expect.

Effectively all cpdavey was arguing was that if his presumption was correct, then assuming the truth of the opposite of his presumption would lead to a conclusion that his presumption was incorrect.  Which it did and which we would expect.  That's why there's no great problem within his "Reductio Ad Absurdum"  but there is when we run his proof with no RAA.  His whole argument came down to a "heads I'm right, tails you're wrong" sort of con.

The error is quite clear in Step 12 above, where cpdavey writes "12. ~~□A (1-11 RAA)".  Clearly the RAA started at Step 2, not Step 1.

Sometimes we just don't see things that are staring us in the face.  Perhaps I was trying too hard to be clever.  But then again, so was cpdavey, since the result that he needed to get his RAA to work was precisely the one he was aiming at, via use of the RAA, ie the fallacious claim that, given ◊□A, □A.

God as the Greatest Conceivable Being

It has been claimed that "God is the greatest conceivable being".  I usually interpret this to be a specific claim about the specific god held to exist by a specific claimant, but I note that it also applies more generally to the god of any WLC-affiliated theist, because WLC makes a similar claim in his variation on Plantinga's version of the ontological argument.

I put it to you though that I can conceive of a being that is greater than the god of any specific theist.  This being, for the sake of the argument, is the Grand Pixie, which has not only all the characteristics of the god of the theist, but also purpleness.  In fact, the Grand Pixie is purpleness so the Grand Pixie has "being purpleness" as one of his characteristics.

Now, I claim that "being purpleness" is a feature that improves on anything that has all the other characteristics, even if "being purpleness" is entirely neutral, because it's one more characteristic thus contributing to a greater grade of numerical greatness.

An argument that might be raised against my argument here is that no-one believes in the Grand Pixie, but I respond that the number of believers is not only irrelevant but also damaging to the "my god is the greatest conceivable" argument, since a conceivably greater god than anyone's god would be one that is believed in more people - since I don't believe in any god, then a conceivably greater one is the one that I too would believe in, being one more believer.

Another is that I myself am the person making claims about the Grand Pixie and I can't refer to someone back in history who made the claims.  True, but this is just context.  The same applies if a theist goes back to the original claimant with respect to the maximal conceivable greatness of her god, even if we don't know who that original claimant was.  The only way out of this regression is to arrive at the maximally great god and have it tell someone that it is maximally great, but this is a sort of thing that non-maximally great things are also capable of (demons for example, in the worldview of some theists, and the figments of insane people in the worldview of some atheists).  The point here being that the claim to maximal greatness made by a human on the part of her god is precisely that, a claim, and nothing more.

I can make a counterclaim that there is a conceivably greater being than anyone's god, the Grand Pixie where Grand Pixie =your god + "being purpleness"/some other feature absent from your god, say "grooviness" or "being paisleyness".  The theist can try to address that, by claiming that her god actually does have "being purpleness", "grooviness", "being paisleyness " or anything else that I might come up with, but then the theist’s conception of god is revealed to be ad hoc.

While some might accuse me here of being insulting to their conception of god, and there might be some truth to that although my intent is more towards light-heartedness (and I know that some people are insulted by people not talking about their god with anything other than utter seriousness), the central point remains.  No matter how great you make your god, someone else can come along, add even a neutral feature and conceive of something greater.

So how can anyone truly claim that their god is the greatest conceivable being?

This is a Necessary Post

Well, it's not really.  What we can say is that it is an existent post and we can then ponder on whether it is possible, necessary or contingent.

We could certainly equivocate in order to claim that it is a necessary post, because the title of it tells us that it is "a Necessary Post" - necessity is in its self-described nature.  But that would not make it necessary, would it?  Or you could rely on my word for it, telling you that it's necessary, but some readers would not be inclined to take my word for it.

And that's only when we know that the post exists.  What about if this was one of those special posts that no-one else can read, because I haven't the right privileges for general access?  Would a philosopher be able to work from the notion that there is a asserted necessary post (or the possibility of a necessary post) to the conclusion that there is an existent necessary post?

It seems to me that she couldn't, at least not without cheating, and that discussions about necessity (or mere contingency) follow existence, rather than the other way around.

Perhaps someone can explain why the attempt to logic something into existence via the presumption of necessity is something more than theatrics?  (see also here)

Being The Grand Pixie

We tend to think of The Grand Pixie as a big man with a beard, or some sort of powerful "person" like a human being, although one who can do amazing things. This is just the childish version, it is conditioned in our thinking by a pedestrian approach to religion.

There are religions that don't have a "Grand Pixie" per se, such as Buddhism. Essentially, there is no reason to think of The Grand Pixie as a person, certainly not one with a corporeal body. That image, which is hinted at in the Pixie Manual, is merely metaphor. Depending upon the religious tradition, however, one can have very abstract views of The Grand Pixie which have nothing to do with a father figure or a mother figure.

There is a more abstract way to think about The Grand Pixie: that is "Transcendental Signifier;" the notion of a metaphysical first principle that organizes everything into a metaphysical hierarchy. This is the more sophisticated view of The Grand Pixie, and most of the works of the great Faerian philosophers hint at notions of The Grand Pixie in these abstract terms.

Some ancient Faerian, who developed a nonsensical “ontological argument” that no-one really takes seriously anymore, defined The Grand Pixie as "that which nothing greater than can be conceived." If he could have been bothered going through the charade, this old windbag would have ended all of his arguments by saying "this thing we call The Grand Pixie," as a means of keeping the exact nature of The Grand Pixie open ended. This is because The Grand Pixie is beyond our understanding, as the Pixie Manual says, but we can leave a "place marker" for the concept of The Grand Pixie by understanding that the ultimate logical function of the The Grand Pixie concept is that of the transcendental signifier.

Ground of Being

One of the sophisticated concepts used by great Faerian theologians is that of "The Ground of Being." This concept indicates, not that The Grand Pixie is the fact of things existing, but that The Grand Pixie is the basis for the existence of all things. The Grand Pixie is more fundamental to existing things than anything else. So fundamental to the existence of all things is The Grand Pixie – that The Grand Pixie can be thought of as the basis upon which things exist – the ground of their being. To say that The Grand Pixie is The ground of being or being itself, is to say that there is something we can sense that is so special about the nature of being that it hints at this fundamental reality upon which all else is based.

The phrases "Ground of Being" and "Being itself" are basically the same concept. Some use both at different times, and other pixiologians prefer "Being Itself," but they really speak to the same concept. Now Sceptics are always asking "how can The Grand Pixie be being?" I think this question comes from the fact that the term is misleading. The term "Being itself" gives one the impression that The Grand Pixie is the actual fact of "my existence," or the existence of my flowerbed, or any object one might care to name. Some random guy, on the other hand, said explicitly (in some book, so it must be true, right?) that this does not refer to an existential fact but to an ontological status. What is being said is not that The Grand Pixie is the fact of the being of some particular object, but, that he is the basis upon which being proceeds and upon which objects participate in being. In other words, since The Grand Pixie exists forever, nothing else can come to be without The Grand Pixie's will or thought, and since there can't even be a potential for any being without The Grand Pixie's thought, all potentialities for being arise in the "mind of The Grand Pixie" than in that sense The Grand Pixie is actually "Being Itself." I think "Ground of Being" is a less confusing term. The Grand Pixie is the ground upon which all being is based and from which all being proceeds.

How Can "a Being" be Being Itself?

Part of the confusion stems from a misunderstanding of what is being said. I say that The Grand Pixie is 'necessary being' not "a necessary being," not because I forgot the "a" but because The Grand Pixie is not "a being." He is above the level of any particular being that participates in being, but exists on the level of the Being, the thing itself, apart from any particular beings. There is Being, and there are "the beings." This is a crucial distinction, but it leaves one wondering what it means and how it could be. I think the answer lies in the fact that The Grand Pixie is ultimate reality. The Grand Pixie is the first, and highest and only necessary thing that exists, and thus, had The Grand Pixie not created, The Grand Pixie would be the only thing that exists. Could one somehow ponder a universe in which The Grand Pixie had not created, in which The Grand Pixie was all that was, one might well ask "what is it to be in this universe where there is only The Grand Pixie?" In such a universe the only conceivable answer is "to be is to be The Grand Pixie." In that sense The Grand Pixie is Being Itself.

(Original article being satirised is here.)

Removing BS5 and the Ontological Argument from All Possible Worlds

I’ve touched on the Ontological Argument a few times in the past (here for example).  In its original form, it’s an argument that because a non-existent great being is less great than an existent great being, then if we can conceive of a maximally great being, then it must exist.  William Lane Craig uses a modal logic form of the argument that was devised by Alvin Plantinga.  As I argued here, this argument hinges on the use of a ruse that I have labelled BS5.  BS5 in turn relies on the use of “possible world semantics” which is a model used to express modal claims – which means statements that express knowledge and belief – in the analysis of linguistics.

I’ve not stressed this before, and perhaps I should have.  Modal logic is quite different to propositional or syllogistic logic in which propositions are either true or false.  In modal logic, the operators “necessary” and “possible” are introduced.  These operators allow us to analyse statements of belief and potentiality and thus are useful in nutting out what philosophers are really saying.  Unfortunately, to most people, the expressions and symbology of modal logic quickly become unintelligible, so modal logic can also be used to confuse the unwary.

This is from page 3 of Modal Logic for Philosophers, by James W Garson:

It’s no surprise, therefore, to see that modal logic is used by William Lane Craig (per Plantinga) to attempt to prove the existence of his god, or rather to prove the existence of a “maximally great being” (which is then assumed to be his god).  Due to increasing levels of frustration, I had to retire from a heated discussion over at Craig-Land as to the validity/soundness of an argument based on a modified axiom in this logic system.  I wouldn’t mind having a green car, in fact I have had one, but when I am being told that an argument that could easily prove that my blue car is green is completely valid when proving that a god exists, I am apt to get a little hot under the collar.

As a very quick reminder, S5 is a system within modal logic which incorporates the axiom known as (5), namely ◊A->□◊A or “if A is possible, then it is necessary that A is possible”.  This can be worded in possible world semantics as “if A is true in some possible world, then in all possible worlds it is true that A is true in some possible world”.

BS5, however, relies on a reworking of (5) to arrive at the derived statement ◊□A->□A which means that “if in some possible world it is true that A is true in all possible worlds, then A is true in all possible worlds” which in standard logic (ie not in possible world semantics), could be worded as “if it is possible that A is necessary, then A is necessary”. 

The axiom that makes this possible is known as (B) (Garson, page 39):

There is a problem with what I call BS5, as James Garson points out (page 43):

Sadly, I don’t feel that Professor Garson has pointed out the problem with the use of BS5 sufficiently clearly or in sufficiently strong terms.  In his last sentence, however, he seems to be saying (to me at least) that while we can and should respond positively to □(A->◊A), we should not respond as positively to (B) (A->□◊A). 

So, while I’ve argued that there’s something wrong with the key BS5 statement (◊□A->□A), particularly if it is used to try to argue something into existence, Garson seems to be going further and saying that we should be suspicious of (B) itself – specifically because it results in BS5.

I acknowledge that there is something inherently wrong about using a system created to consider knowledge and belief in this way, but I’ve also wondered whether there is something wrong about grouping up the □ or ◊ symbols in possible world semantics and it’s this that I’d like to address here.

As can be seen in the extract from page 43 of Modal Logic for Philosophers, S5 is a strengthening of the simplification principles of S4.  In S4, the principles □□A->□A and ◊◊A->◊A can be iterated so that infinitely long strings of either □ or ◊ can be contracted down to one symbol.  However, when we word these iterated principles in possible world semantics we hit upon a vagary associated with the string of “possible”:

·         if in some possible world it is true that A is true in some possible world then it is true that A is true in some possible world

Well, yes, but we aren’t necessarily talking about the same “some possible world”.  Let’s use Earth and Mars as possible worlds and use “abundant life exists” as A.  On Mars it is true that in some possible world there exists abundant life (it exists on Earth), but that specific possible world isn’t Mars.  However, if we considered a slightly different universe, we could even say that it is possible that there could be abundant life on Mars (because we have no evidence that allows us to eliminate that possibility).

This leads me to wonder whether it would not be better to think about longer strings of □ or ◊ symbols as referring to different layers of necessity and possibility within a hierarchy of phase spaces.

Such an approach would have no effect on S4.  For example, where a “universe” phase space contains subordinate “world” phase spaces, □□A would mean that “in all possible universes it is true that A is true in all possible worlds” and this would resolve down to “A is true in all possible worlds (across all universes)”.  Similarly, ◊◊A would mean that “in some possible universe it is true that A is true in some possible world” and this would resolve down to “A is true in some possible world (across all universes)”.  Conversely, when a single symbol is presented, this would be equivalent to a (notionally) infinite string of that symbol.

BS5, on the other hand, would be affected.  ◊□A would mean that “in some possible universe it is true that A is true in all possible worlds (within that universe)” and could not be resolved down to □A.  The more common representation of S5, ◊A->□◊A, would not be affected since it would follow that if “A is true in some possible world (across all universes)” then “in all possible universes it is true that A is true in some possible world (in at least one of the possible universes)”.

The problem with (B), A->□◊A, would become clear because we don’t have a firm understanding of what we are saying with an unqualified A in such a scheme.  To resolve this, an unqualified statement A in possible world semantics would mean that A is true in some possible world in some possible universe, or that A is not impossible.  In other words: A<->◊A and it would be valid to say that ◊□A->A, since this is equivalent to ◊□A->◊A (if something is possibly necessary, then it’s possible).

What we could not say, however, is that □A->A, but this is an existent problem in the fundamental system K (see the Stanford Encyclopedia of Philosophy entry on Modal Logic) and introducing BS5 doesn’t solve this problem.

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Note that “world” and “universe” are used for convenience here, but in reality they refer to hierarchical epistemic alternatives.  A “world” could refer to the range of possibilities available to a one player within a single game of cards, while a “universe” could refer to the range of possibilities available to that player over many different games of cards.  The hierarchy would not necessarily be limited to “world” and “universe”.  While the terminology might need some further thought, we could have layers equivalent to town, state, country, world, universe and multiverse (something that was true in all towns in all states, in all countries, in all worlds, in all universes, in all multiverses is apparently pretty necessary).

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Because I like visual things, here are a couple of explanatory images:

What the last image is saying is that to be “necessary”, without qualification, A must be necessary across all universes and all worlds (and any other layers, if they exist) and to be possible, without qualification, A need only be possible in one world (where “world” is considered to be the lowest tier in the hierarchy).  An example of a “possible necessary” is shown with B, which is necessary in Universe 1, possible in Universes 2 and 3 and not possible in Universe 4.  Thus it would be accurate to say ◊□B, that “in some possible universe it is true that B is true in all possible worlds (in that universe)”, but not accurate to say □B that “B is true in all possible worlds” since this unqualified statement tacitly implies “across all universes”.

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Some people might argue that my proposal would make possible world semantics unwieldy and complicated.  Those people might want to browse through Handbook of Modal Logic (Blackburn et al.) or Modal Logic for Philosophers (Garson).  They’ve already got unwieldy and complicated completely covered.

Others will argue that my proposal isn’t sufficiently developed.  These people are right.  I’m not the right person to develop the proposal into yet another variant of modal logic, I’m not a logician.  It’s entirely possible that my proposal won’t work for some arcane reason, but it just seems obvious to me that there’s a problem with BS5 and that my proposal might address that problem.

If there are logicians out there who can explain why my proposal won’t work, I’d appreciate it.  What won’t be appreciated is an arrogant, uninformative dismissal along the lines of “that’s just not how we do things, and by the way you have to have a doctorate before you can be involved in the discussion”.

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):

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.

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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 ◊□A->A is provable from (B). So ◊□A->A should be acceptable if (B) is. However,◊□A->A 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 □(A->B). 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:◊□A->A, 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 (1 double DN)

3. ~□~□A (2 E2)

4. □A (3 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.