Friday, 7 June 2013

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.

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.


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).


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”.


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”.

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