Friday, 10 February 2012

“Modal Ontological Arguments” and meta-modality

In which I find that the Modal Ontological Argument "works" because Modal Logic languages cannot cope with modalities about modalities.  I fix that.

Like its non-modal counterpart, Anslem’s plain old “Ontological Argument”, the Modal Onotological Argument can be maddening.  Whilst both seem obviously fallacious it is not at all obvious where the fallacy lies.  Is it a parlour trick?  If so, where is the sleight?

The argument seems to commence with the assumption that God possibly exists.  Then with no further assumptions than conventional modal logic, the argument concludes that God actually exists.  The assumption that God possibly exists seems innocuous enough, the claim that God actually exists is a little more contentious.  As the only bridge between the two is modal logic it suggests that the fault lies there.

That there is a fault can be shown by running the argument for other entities we are quite sure do not exist but will accept that they could possibly exist; such as  The Golden Mountain, the Flying Spaghetti Monster or the Decent Pint of Mass Produced Lager. 

A version by William Lane Craig, itself a slightly reworded version of Alvin Plantinga’s “victorious” argument, goes like this:

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.

(Craig, 2010)  (Plantinga, 1974, p.214-217)

Possible Worlds

Modal logic is expressed both formally in symbolic language and less formally in everyday language.  In everyday language the modal statements are expressed by means of statements about “possible worlds”, complete and discrete descriptions of the way the world could be.  If p is possible there will be a complete and discrete description of the way the world could be that contains p: p will “exist in” a “possible world”.

If p is necessary then every description of the way the world could be will include p: p will exist in all possible worlds.  “Impossible” can be expressed by absence from all possible worlds.  “Actual” involves existence in one, special, possible world: the actual world. (As the actual world is such a special world it deserves its own name and we shall follow Plantinga in calling it “Kronos”).

“It is possible that a maximally great being exists”

A “maximally great being” takes the place of God in the argument.  “Maximal greatness” is “maximal excellence” in every possible world.  “Maximal excellence” is the collection of those God-like properties of omniscience, omnipotence and moral perfection (“OOM”). 

The “being” is mentioned partly because this is an argument about a being, God, but mostly to avoid issues of trans-world identity.  OOM are attributes of beings or nothing at all.  If OOM exists in a possible world then a being exists in that world to possess them.  The existence of OOM entails a being but the existence of OOM in more than one world does not entail the same being in each one.  The disadvantage in talking about OOM rather than a specific OOM being is that you might prove that a god is in each possible world without showing that God is in each possible world.  The advantage of discussing OOM alone is that it makes the discussion that follows a lot simpler.  So long as we undertake not to introduce a trans-world-identity sleight, no one should mind us simplifying matters by talking directly of OOM.

There is a claim of OOM qualified by the modality, necessity, conferred by OOM being subsumed in “maximally greatness”.  There is also a qualification by possibility. 

What does the “possible” qualify, what is its range?  The range of the necessity is clear: OOM.  But does “possible” range over OOM or over “necessary OOM”?

If “possible” ranges over OOM there is a claim that OOM is possible.  There is also, clearly, a claim of necessary OOM.  Two claims about OOM are made and the modal claims of the first premise can be re-phrased:

1a. OOM is possible and necessary

Expressing this in terms of possible worlds is simple: OOM is in all of them. 

The alternate reading is that the “possible” qualifies the necessity of OOM (call that “OOM*”):

            1b. OOM* is possible

It is much more difficult to express just what the modal claim is here.  The modality of OOM* is not the modality of OOM; a claim that OOM* is possible is not a claim that OOM is possible.  For example, let p be a plainly contingent entity; such as the piece of paper or the screen you are reading this on.  Let p* be the claim that p is necessary. p* is plainly impossible, p is a contingent entity and contingent entities are not necessary.  p, though, is plainly possible: if it exists, it exists in Kronos, which is a possible world, so it's possible.

Extending the language

Expressing these modalities of modalities within the language of possible worlds sketched out above is more than tricky, it may require an extension of that language.  The possible worlds we are used to for speaking of modalities are populated by actuals.    A possible p is actually in a possible world.  To talk about the modality not of an actual but of a modality, to talk meta-modally, somewhere populated by modalities is required. 

Modalities populate the set of possible worlds.

Now the language of possible worlds requires, in a way, complete description of all possible worlds.  We need the full set of possible worlds to “look into” to check whether p is present in all, some or none.  So, when asked whether p is possible we rustle up the full set of possible worlds, “S1”, and have a look.

But S1 may be wrong, it may not be the set of possible worlds, just a set of worlds that could be the set of possible worlds.  We have imagined a possible set of possible worlds, a set in which the modality of p resides but only possibly. 

The full set of sets of possible worlds is where modalities of modalities reside.

- To say that X is possibly [a particular modality] is to say that in at least one set of possible worlds X has that modality.
- To say that X is necessarily [a particular modality] is to say that in all sets of possible worlds X has that modality
- To say that X is impossibly [a particular modality] is to say that X has that modality in no sets of possible worlds
- To say that X is actually [a particular modality] is to say that X has that modality in the actual set of possible worlds (meta-Kronos).

Extending symbolic modal systems

To express this formally we need to distinguish modalities and meta-modalities.  We shall add subscripts to the modal operators, □ (necessity) and ◊ (possibility). An operator that is not subscripted is taken to be subscripted with zero.

We need to make clear the range of the meta-modalities.  This we shall do by restricting the permissible expressions in the extended language, "well formed formulae" or "wff".  We shall introduce two rules:

1. A wff in whichever “standard” modal logic we are extending is a wff.
2. A wff enclosed in brackets with a modal operator to the left with a subscript one higher than the highest subscripted operator within the brackets is a wff.

Finally we need to be able to work on a modal statement unencumbered by the meta-modality, to figure out what we are qualifying before applying the meta-logic.  We shall add a new argumentation rule, the "detachment/re-attachment rule":

- On a line in an argument the operators with the highest subscript may be removed ("detachment")
- If the operators are detached they must be reintroduced on a later line to form a wff ("attachement")
- When attached the operators are given the lowest subscript then available.

The confusion in the argument

We can now clearly distinguish three different readings of the first premise.  We can express them both formally and in the more everyday language of possible worlds. Our opening position clear we can proceed to argue modally.

The first reading, 1a: OOM is possible and necessary, is expressible in the non-extended language.  OOM is in all possible worlds and every possible world.  As it is in all possible worlds, it is in Kronos.  Therefore OOM exists.


1.      ◊OOM & □OOM (Premise)
2.      □x → x (Axiom)
3.      □OOM (1 Elimination)
4.      OOM (2,3)

The second reading, 1b: possibly OOM*, requires the extensions to the languages.  OOM*, the necessity of OOM, entails the presence of OOM in each world in a set of possible worlds.  In that set of possible worlds OOM will be in Kronos and will, therefore, exist.  OOM is actually exisiting in at least one possible set of possible worlds and is, thus, possibly existing.


1.      1(□OOM) (Premise)
2.      □x → x (Axiom)
3.      □OOM (Detachment)
4.      OOM (2,3)
5.      ◊OOM (Attachment)

A third reading, not mentioned above, is the meaning someone probably “hears” on first encountering the argument:

1c  God is possible

In the language of possible worlds, God is in at least one possible world.  So God is possible. 


1.      ◊G (Premise and conclusion)

Notice how 1b reaches the same conclusion as the premise that is “heard” and which will be readily agreed.  The unextended languages, though, are unable to adequately express 1b.  As a result any formalisation, “full” or “everyday language”,   is forced onto 1a, or similar.  1a naturally reaches the conclusion of the Modal Ontological Argument but, if made clear would not be assented to by atheists, agnostics and, even, many theists.

There's a nice little Evil-God Modal Ontological Argument here.


Craig, W. L. (2010, February 4). Five Arguments for God. Retrieved February 8, 2012, from The Gospel Coalition:
Plantinga, A. (1974). The Nature of Necessity. Oxford: Oxford University Press.


Rick Taylor said...

I recommend you study an introductory book on modal logic. One is able to deal with the modality of the modalities as it were by introducing an "accessibility relation" between possible worlds. So C might be a possible world for A, while not being a possible world for B. By varying the assumptions of the accessibility relation, one is able to come up with different modal logics.

The modal logic Plantinga uses is S5, which assumes every possible world is possible for every possible world. But this is certainly not the only such system.

Tony Lloyd said...

Hi Rick

Do you have a suggestion? There are a few around. Including this one:

Which comes in at $121.87!!

BTW I'm sceptical about accessability relations between "worlds" to be able to replicate relations between modalities. Are we not still going to get down to the question of whether or not x is actually accessible from y: a question we might be right or wrong about?

Rick Taylor said...

I haven't read the book you linked myself, but I've heard lots of people recommend it, so that's probably a good one to read. You don't have to get the hardcover version; there's both a paperback and a kindle edition that are far cheaper.

I read "First order Modal Logic" by Fitting and Mendelsohn which is excellent, but I'd only recommend it to some who has a strong background in classical predicate logic, as used day to day by a working mathematician.

Rick Taylor said...

This webpage recommends a number of books on modal logic.

The site also has a page with recommended books for classical logic.

Tony Lloyd said...

That sites's pretty good. I've gone for the Graham Priest book. The site highly recommends it, I can get it secondhand and (so long as Priest doesn't try and sell me on dialethism too much) it should be a good read.


Rick Taylor said...

Interesting. I might take a look at that book myself.

Rick Taylor said...

Just to add, the system S4 addresses the problem you describe in modal logic. Everything valid in S4 is valid in S5, but S4 does not contain the assumption that if something is possibly necessarily true, it must be necessarily true. In S4, a proposition might be necessarily true in one possible world, contingently true in a second, contingently false in a third, and impossible in a fourth.

S4 still has the properties that something necessarily true must necessarily be necessarily true, something that's possibly possible must be possible, and something impossible cannot possibly be possible. These all seem reasonable to me (and are equivalent to one another under the usual assumptions about possibly and necessarily).

Before I learned about the access relation between worlds, I thought one would have to have a multi-tiered system of worlds to reason about modality. But the access relation does the same thing more simply and elegantly.