A Quick Thought on "Began to Exist" in the Kalam Cosmological Argument

The little finicky bits around "begin" and "exist", normally of no import, are crucial for the Kalam Cosmological Argument. A quick reminder of the Kalam Cosmological Argument (“KCA”):

1. Everything that begins to exist has a cause
2. The universe began to exist.
3. Therefore, the universe has a cause.

Why "everything that begins to exist" and not "everything that exists"? Because the role the KCA plays in arguing for God depends on differentiating God from the universe. There needs to be a causal problem with the universe that is not a problem with God so God can be invoked to solve that problem.  “Everything”, though, includes God; the argument would fail if “everything that exists” were the first premise.

What “began”?

Now we could take something "began to exist" as meaning that

Condition 1: that thing exists
Condition 2: there is an earliest time when that thing existed.

If one accepts the arguments William Lane Craig (the main contemporary advocate of the KCA) takes from philosophy and modern physics then one has to accept that, in this sense, the universe did have a beginning. Modern physics theorises that not only did the universe begin, but time itself began.

The philosophical arguments Craig refers to also posit a beginning of time:

- “In other words, the series of past events must be finite and have had a beginning.”
- “an actually infinite series of past events could never elapse; since the series of past events has obviously elapsed, the number of past events must be finite.”

If time began then there is an earliest time. If there is an earliest time then there is an earliest time when the universe existed. The universe exists (condition 1) and there is an earliest time when the universe existed (condition 2): the universe began to exist.

And so, if He exists, did God.

If He exists He fulfils condition 1. As time itself began there is an earliest time when He existed, fulfilling condition 2.

If time itself began everything that exists began to exist. There is no difference between “everything that exists” and “everything that begins to exist” and, so, the argument fails.

An earliest time but no X

If the argument is to be saved another condition, condition “X”, must be added to the concept of “begins to exist”. Condition X must also be applicable to the universe but not applicable to God.

So:

Condition X1: and there must be a prior time when the thing did not exist

This seems fine. Almost everything, except God, fulfils the three new conditions. They exist, there was an earliest time that they existed and there was a time earlier than when they existed. Almost everything, though. There is no time earlier than the earliest time the universe existed. If we accept condition X1 then God did not begin to exist, but neither did the universe.

Craig’s “Solution”

Craig’s “solution” is along the lines of God “existing timelessly”.

Condition X2: and it must not exist timelessly.

WTF, though, is “existing timelessly”? “Existence” seems to require persistence in time. Neanderthals, though they do not exist now, persisted for some quarter of a million years. Exotic particles produced in particle accelerators exist for nanoseconds. There is no time when Hobbits existed. Did Hobbits exist “timelessly” whilst Neanderthals and exotic particles happened to exist timefully?

It seems that Neanderthals and exotic particles do not so much exist in a certain time-related way but exist because they existed in time. Existing “timelessly”, à la Bilbo Baggins, is another way of saying “not existing”.

Let us, though, allow this idea of “existing timelessly” and allow that something that “exists timelessly” does not begin to exist. Why believe that the universe does not exist timelessly? After all, we’ve such little idea of what existing timelessly amounts to, we can hardly rule it out.

Conclusion

So there we have it. The KCA depends on the universe beginning to exist and God not beginning to exist.

But if we say that the universe must have begun, well so did God. If we say that God did not begin to exist, then neither did the universe.

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A Question on the Transcendental Argument for God

I have a question on the Transcendental Argument for God.  Greg Bahnsen boiled it down to:

The transcendental proof for God's existence is that without him, it is impossible to prove anything.

My question is this: what is the argument?  Seriously.  I can accept, in its entirety, the alleged proof and absolutely nothing follows from it.

"Without [God] it is impossible to prove anything" can be rephrased as:

"God exists or it is impossible to prove anything."

With the "or" being an inclusive "or": the statement is false only if both phrases are false.  The statement is true when either:

1. God exists and it is possible to prove something
2. God exists and it is impossible to prove anything
3. God does not exist and it is impossible to prove anything

Now a conclusion follows from its premises if its negation generates a contradiction, if the negated conclusion and the premises cannot be true at the same time.  So what conclusion follows from "God exists or it is impossible to prove anything"?

It's not "God exists".  The negation of "God exists" is "God does not exist", and "God does not exist" can be true at the same time as "God exists or it is impossible to prove anything." (it's situation 3)

It's not "it is possible to prove something".  The negation of "it is possible to prove something" is "it is impossible to prove anything" and that is the second situation.

It's not "it is impossible to prove anything".  The negation of "it is impossible to prove anything" is "it is possible to prove something" and that is the first situation.

Is there a missing premise? Or is the argument just bollocks? Will any presuppositionalist reply to a question rather than just asking them?

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Draft Note on a Presuppositionalist Argument

There is a forthcoming paper that argues that God is a necessary precondition for the Laws of Logic.  The argument, from the paper's conclusion is:

The laws of logic are necessary truths about truths; they are necessarily true propositions. Propositions are real entities, but cannot be physical entities; they are essentially thoughts. So the laws of logic are necessarily true thoughts. Since they are true in every possible world, they must exist in every possible world. But if there are necessarily existent thoughts, there must be a necessarily existent mind; and if there is a necessarily existent mind, there must be a necessarily existent person. A necessarily existent person must be spiritual in nature, because no physical entity exists necessarily. Thus, if there are laws of logic, there must also be a necessarily existent, personal, spiritual being.

I've drafted a note responding to the argument and would be grateful for any feedback.

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Anderson and Welty[1] seek to establish that “(t)he laws of logic imply the existence of God”.  There is, however, an equivocation in a key step in the argument.  Any univocal re-statement of the argument fails.  

The step in question is given in the summary of their argument in the conclusion to the paper:

 “Since [the laws of logic] are true in every possible world, they must exist in every possible world.”

Now, how are we to take the two sub-clauses, “true in every possible world” and “exist in every possible world” as being logically connected?

If and only if («)

If a law of logic exists in a possible world if and only if it is true in that possible world then the step is, obviously true.  However, given Anderson and Welty’s characterisation of the laws of logic as necessarily true; their conclusion, that they entail the existence of God, is contradictory. 

 “The laws of logic entail the existence of God” entails that if there were no God then the laws of logic would not exist.  Given, though, that the laws of logic are necessarily true and that they exist if and only if they are true then the laws of logic necessarily exist. If the laws of logic necessarily exist then they would exist whether God existed or not.[2]

If (®)

So we must abandon the idea that a law of logic “exists” in a possible world if and only if it holds true in that particular world.  Perhaps it is meant that if, (but not “only if”), a law of logic holds true in a world then it exists in a world? This does not remove the contradiction.  The laws of logic are necessarily true, they hold in every possible world whether or not God exists in that possible world.  As their holding true entails their existence the laws of logic exist in every possible world whether or not God exists in that possible world.

Only if (¬)

Anderson and Welty’s conclusion, that the laws of logic imply the existence of God, is not contradicted if we assume that “only if” is meant:  the laws of logic are true in every possible world only if they exist in every possible world.  Indeed, on this reading, Anderson and Welty’s conclusion would be supported. Given Anderson and Welty’s characterisation of what it means for a law of logic to “exist”, were God not to exist in a possible world the laws of logic would not exist in that possible world.   If the non-existence of the laws of logic in a possible world entailed that they failed to hold in that world then an agreement that, say, the law of non-contradiction does hold entails the existence of God. Whilst this may seem like an improvement, the “only if” reading either contradicts other vital premises in the argument or renders the step itself contradictory.  The step claims that the laws of logic hold true in every possible world.  Why is that?  If it is because they exist in every possible world (maybe because God exists in every possible world) then the laws of logic are not necessarily true; their truth is dependent upon their existence and were they do not exist they are not true.  The first premise in Anderson and Welty’s, summarised, argument is that the laws of logic are necessarily true: the argument contradicts itself.   So let us assume that the fact that the laws of logic hold true in every possible world is because they are necessarily true.  This contradicts the assertion that they are true “only if” they exist: the step, itself, is contradictory.

Other logical connectives and conclusion

Other logical connectives are not only more of a stretch in interpretation but fail destroy the step as any link in chain of reasoning.  “And” turns the step into two simple assertions. “Or”, excluding combinations of “or” and “not” that are equivalent to connectives discussed already, simply asserts the truth of one of the sub-clauses.

There may be other logical connectives that haven’t been considered or have not, even, been thought of yet.   No logic is liable to correct the argument, though, as the two sub-phrases of the step simply talk about different things. 

Anderson and Welty must establish that laws of logic are capable of being contingent in order to argue that they are contingent upon God.  Thus the laws of logic are characterised as thoughts.  Thoughts require a mind and, thus, are contingent on minds. “Exist” in the second sub-phrase refers to being thought of by a mind.  Anderson and Welty also need to establish the laws of logic as necessarily true in order to argue that the mind thinking the laws of logic must be necessarily true.  

However being true and being thought are wholly independent properties of propositions.  The one is dependent on a mind, the other dependent on the world outside that mind.  No logical connectives will bridge the gap.   Differing logical connectives may save the argument from saying something contradictory, but at the expense of saying anything at all.

The assertion that God is a necessary precondition for the laws of logic is either necessarily false, or meaningless.

---

[1] Anderson, J. N., & Welty, G. (Forthcoming). The Lord of Non-Contradiction: An Argument for God from Logic. Philosophia Christi .

[2] It may be objected that God is, Himself, necessary and thus there is no possible world where He does not exist.  The objection, however, is not only question begging but proves the counter argument.  God is necessary for the laws of logic if and only if for all possible worlds were God does not exist there are no laws of logic.  The universal statement, “all possible worlds x”, can be recast existentially as “there does not exist a possible world without x”.  God, therefore, is necessary for the laws of logic if and only if there exists a possible world were God does not exist and neither do the laws of logic.  If God is necessary there is no world were God does not exist and, thus, no world were God does not exist and neither do the laws of logic.

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What's so great about evidence?

I’ve just read a paper, Why Do We Value Knowledge? by Ward E.Jones, that poses the question: if we are concerned with truth, why bother with knowledge?  If knowledge is factive then all knowledge is true, but not all truths are known.  Why value a p which is known more than a p’ which is merely true?  Why do we, and it’s pretty evident that we do, think less of the lucky guesser as opposed to the knower?  Both have truth, both believe the proposition in question, the lucky guesser just lacks warrant, or justification, or reasons, or evidence for the proposition.  What’s so great about evidence?

Jones examines the instrumentalist answer to the question; that we value evidence (or warrant, or justification, or reasons...) because it is a means of getting true beliefs. As Jones points out that is a cracking explanation of why we like evidence but no explanation of why we like true beliefs brought about by evidence above those that we just happen upon.  An example from Jones is getting across a river.  We want to be on the other side and nice, safe, reliable method can be depended upon to get us there.  So we will use one. But the state of being on the other side is no sweeter because we got there by a safe and reliable method. 

My own way of putting it is this: we want money, in part, because it lets us buy beer.  But free beer is not valued any less.

Jones’s suggestion is to look at the incidental attributes of knowledge.  In order to get a robust definition of knowledge the concept has been pared down as far as possible, anything non-essential removed.  According to Jones, it may well be in this cloud of non-essential attributes of knowledge that its real value is found.

I think that there is a simpler and more straight-forward explanation.  We value neither truth nor knowledge.  Of course we will say that we value both, but our actions do not bear our statements out.  Rather like my saying that I am fond of a low fat, alcohol-free diet with plenty of exercise is not borne out by my sitting in a pub eating crisps and drinking beer.

Consider four propositions; e₁, e₂, g₃ and g₄. The “e” propositions have as much evidence (or warrant...) as you like whilst the “g” propositions are complete guesses. The odd numbered propositions, e₁ and g₃, are true.  The even numbered propositions, e₂ and g₄, are complete rubbish.  Our attitude to the propositions will be divided by evidence rather than truth.  We will think e₁ and e₂ are true, be inclined to believe them and consider them examples of knowledge.  Our attitude will not change dependent upon whether they are knowledge or whether they are true or false.  Similarly we will be wary of both g₃ and g₄, further investigate both the true and the false, deny both of them the title “knowledge” and be reluctant to believe them to be true.  Whilst we are avowedly all for truth and knowledge, dispositionally our attitudes track evidence.

We do not have access to the truth of propositions beyond the evidence we have for them. And so, the evidence we have for propositions stands as a proxy for truth, it is something we can aim our beliefs at and adjust our attitudes on the basis of. 

Now on the traditional view of knowledge, knowledge is evidenced, or warranted, or justified or reasoned true belief.  Where we have that evidence (or warrant...) we also believe the proposition to be true and, so, believe that all the ingredients of knowledge are there.  Where we do not have that evidence (or warrant...) we do not call our position in relation to the proposition “knowledge”.  Our attitude to propositions tracks our use of the term “knowledge” and, so, we believe that we favour knowledge when, in fact, we favour evidence.

Reference

Jones, W. E. (1997). Why Do We Value Knowledge? American Philosophical Quarterly , 34 (4), 423-439.

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The Efficacy of the Evidential Argument from Evil

After Stephen Law’s debate with William Lane Craig there was a lot of discussion on the web about the efficacy of Law’s argument, the Evidential Problem of Evil, in countering Craig’s arguments.

Many of those who think the Evidential Problem of Evil ineffective argue that it is a probabilistic argument that, at best, shows that God is unlikely. (e.g. Chab123 2011) Others (e.g. Stuart 2011) felt that insufficient attempt was made to show what was wrong with Craig’s arguments for God, especially the Kalam Cosmological Argument.

Craig’s opposing arguments for God are deductive arguments.  If the premises are true and the logic valid the conclusion is conclusively established:  it is simply impossible for the conclusion to be false. 

The best a “probabilistic” argument can do is make it very, very unlikely that the conclusion is false.  And, as Sherlock Holmes pointed out to Dr. Watson, no matter how unlikely a conclusion seems if nothing else is possible then it must be true.

This is Chab123’s complaint. Law claimed that the Evidential Argument from Evil, when plugged into the Modal Ontological Argument, demonstrated the impossibility of God:

“His confusion on this point is breath-taking. His evidential argument from evil, at its very best, shows, at most, that it is probable that God does not exist. The probability is less than 1. To defeat the ontological argument with an argument from evil, his argument would have to entail that God does not exist. The probability that God does not exist would have to be 1. It would have to prove, as he says, that the conclusion of Craig’s argument is false. But Law’s own argument, as a matter of logic alone, cannot achieve this goal. It is a probabilistic argument. As such, it leaves open the possibility that God exists, even if the probability is quite low.”

Nothing probabilistic about modus tolens

Craig’s arguments make use of a rule of logic called modus ponens.  The rule is:

1. If x then y
2. x
3. Thus y

The picture many people have of a “probabilistic” argument is along the lines of:

1. The first swan seen is white
2. The second swan seen is white
3. The third swan seen is white
4. …
5. The nth swan seen is white
6. Thus all swans are white

It can seem that the Evidential Problem of Evil follows this pattern.  Just as the first white swan is not enough to establish that all swans are white the first bit of gratuitous suffering is not enough to cast doubt on God.  The seemingly endless succession of uniformly white swans seems to make it likely that all swans are white and the endless widespread pointless suffering seems to make it likely that there is no God.

In fact not all swans are white, despite how overwhelmingly improbable that seemed before the discovery of Australia.  “y” in the example above, cannot be wrong, “all swans are white” both can be, and is. 

The Evidential Argument from Evil, though, is not a “probabilistic” argument.  It is just as deductive as Craig’s arguments; making use of a rule of logic called modus tolens.

Modus tolens runs:

1. If x then y
2. Not y
3. Thus not x

And the Evidential Problem of Evil runs like this:

1. If God existed then there wouldn’t be so much pointless suffering in the world
2. There is so much suffering in the world
3. Thus God does not exist.

This is just as “logically airtight” as any of Craig’s arguments; if the premises are accepted then the conclusion must be accepted.    We can, of course, dispute the premises (I am, though, persuaded by Law’s “evil god hypothesis” (Law 2009) that rejection of them would be irrational).  As for the comparison arguments from Craig: though in the above example of modus ponens the conclusion was absolutely certain, the premises were just assumed to be true and the logic was kept simple to ensure its validity.  Craig's premises, and the validity of his logic, are both at risk: they are not at all certain and that uncertainty carries through to the conclusion.

The idea that the Evidential Argument from Evil is a member of some poor-relation-species of argument compared to Craig’s is erroneous.

But you haven’t refuted Craig’s arguments!

The complaint that Law did not adequately argue against the Kalam Cosmological Argument is equally erroneous.

When criticising an argument we could “undermine” it: dispute the premises or find fault with the logic.  Or we could show it to be false. Law certainly did not explain why the Kalam Cosmological Argument concluded falsely but he did give a good argument that it does conclude incorrectly.

If the Kalam Cosmological Argument is a sound deductive argument then no evidence will count against its conclusion.  That evidence does count against its conclusion is good evidence that it’s not a sound deductive argument!   

Similarly, with the Modal Ontological Argument.  This, very briefly, assumes the possibility of God and concludes with His existence.  If one accepts the logic of the argument (I don’t) then, far from showing “breath-taking” confusion Law’s claim is perfectly cogent:

1. If God is possible then God exists (Modal Ontological Argument)
2. God does not exist (Evidential Argument from Evil)
3. Thus God is not possible

Of course we would like to know where the argument falls down, knowing where the argument falls down may expand our knowledge and that knowledge may help us avoid error in future arguments.  But look at this, readily criticised, argument:

1. All elephants are pink
2. Socrates is an elephant
3. Socrates is mortal

The “logic” is hardly worthy of the name, the premises are junk but the conclusion is true.  A demonstration of the falsity of the premises or the invalidity of the logic does not establish that the conclusion is false; a demonstration of the falsity of the conclusion, obviously, does.

The Evidential Argument from Evil is a better counter argument than picking holes in Craig's logic and questioning his premises.  Whilst direct criticisms of Craig's arguments may cast doubt on those arguments the Evidential Problem of Evil shows them to be false.

Bibliography

Craig, W. L. (2010, February 4). Five Arguments for God. Retrieved February 8, 2012, from The Gospel Coalition: http://thegospelcoalition.org/publications/cci/five_arguments_for_god/


chab123 (2011, October 24) The Missing Ontological Argument in the Craig vs. Law Debate. Retreived February 13, 2012, from Ratio Christi-at The Ohio State University:
http://chab123.wordpress.com/2011/10/24/the-missing-ontological-argument-in-the-craig-vs-law-debate/

Stuart (2011, October 20). How Stephen Law Failed in His Debate with William Lane Craig. Retrieved February 13, 2012, from THINKINGmatters:

http://thinkingmatters.org.nz/2011/10/ho-stephen-law-failed-in-his-debate-with-william-lane-craig/

Law (2009, November 30) Draft of PART of chpt 3. VSI Humanism. Retrieved February 13, 2012, from Stephen Law (blog):

http://stephenlaw.blogspot.com/2009/11/draft-of-part-of-chpt-3-vsi-humanism.html

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William Lane Craig's Premises and the Kalam Cosmological Argument

William Lane Craig seeks to put forward deductive arguments for the existence of God.  

If Craig gets his logic right he will have produced a valid deductive argument. If an argument is valid then if the premises are true the conclusion is true.  Validity is not enough though,  Pooh's argument here is valid:

"And if anyone knows anything about anything," said Bear to himself "it's Owl who knows something about something," he said, "or my name's not Winnie the Pooh," he said. "Which it is," he added. "So there you are." (Milne, 1977)

We can only be sure of the truth of the conclusion if the argument is valid and we are sure of the truth of the premises.  Now Craig cannot, none of us can, establish the truth of his premises beyond doubt.  What he seeks to do is adopt premises that are “more plausible than their negations” (2010) 


This is problematical.  Accepting that the premise adopted is more plausible than its negation may lead the reader, or debate attendee, to believe that the premise adopted is the most plausible premise.  Were the only options to be that premise and its negation this would be the case.  Take, however the Kalam Cosmological Argument:

1. Everything that begins to exist has a cause.2. The universe began to exist.3. Therefore, the universe has a cause(Craig 2010)

Other premises

There are a number of alternative first premises other than "not everything that begins to exist has a cause", each with good arguments for its adoption:

Alternate A: Not everything has a cause

Argument for: We might consider that "everything has a cause".  However causes happen before effects, meaning that each effect has a predecessor.  As each cause is "something" and requires it's own cause, a cause which also requires a cause further back.  The regress, if everything had a cause, would never terminate.  There are good arguments for this being impossible: meaning that as there is something not everything had a cause. (Note, for later, that if an effect can precede a cause or be simultaneous with its cause there is no obstacle to everything having a cause")

Alternate B: Everything we can understand has a cause

Argument for: Though we speculate about entities we don't understand we shouldn't introduce that speculation into a deductive argument as, well, we don't understand them.  We can confidently say that anything comprehensible has a cause as causality is part of the way that we understand things.

Alternate C: For any X : if a Plank second of time occurred before X's existence then X had a cause.

Argument for: Again, causes happen before effects.  It appears that a Plank second is the shortest period of time any cause can be before its effect.  For any effect, therefore, there has to be at least one Plank second before it for its cause to be in.  

There we have three alternatives to the first premise given by Craig.  

Whilst it is difficult (probably impossible) to quantify plausibility, each of the alternatives appears to be at least as plausible as the premise given by Craig.  It would be difficult to argue that together they are not considerably more plausible than the premise given by Craig.  To argue that they were not more plausible would require arguing that:

"Alternate A or Alternate B or Alternate C" is more plausible than "Craig's first premise" 

Now "Alternate A or Alternate B or Alternate C" fails to establish the conclusion "the universe had a cause", whilst Craig's first premise does.  But  "Alternate A or Alternate B or Alternate C" is more plausible than Craig's first premise.  Limit our choice to these alternatives and the following is true:

The most plausible conclusion is that Craig's conclusion has not been established

What Craig is arguing for.

Supposing we come to the debate holding Alternate C.  What must Craig show in order to establish his contention that the universe had a cause?  Some of our position (Alternate C) and Craig's overlap, its just that that part of Craig's position fails to establish his conclusion.  What Craig needs to convince us of is the difference between his first premise and ours.  What is that difference? We already agree that all stars, all planets, all living things, all atoms and sub atomic particles, all light and, if they exist, all dark matter and dark energy have causes.  We accept that everything arising since the first Plank second of the universe's existence  had a cause.  

Our dispute is about the first Plank second of the universe.

And what is the first Plank second of the universe's existence?  Is it anything more than the beginning of the universe?  Is there anything else to attribute a cause to other than "the universe"?  In order to persuade us to accept his premise over ours does Craig not have to persuade us that the universe had a cause?  In other words, to establish that the universe has a cause he first has to establish that the universe has a cause in order to establish his premise that everything that began to exist had a cause. 

And what does he have to do to establish this?  He has to persuade us that the universe had a cause even though there was no time for the cause to be in.  If he succeeds in doing that he must either allow simultaneous causes (which allows us to curtail the infinite regress mentioned in Alternate A, which allows "everything has a cause") or exempt the universe from this rule.  If we can exempt the universe from the rule to have a prior cause we can just as well exempt the universe from having a cause.  

Conclusion

It is the dichotomy created by the "more plausible than its negation" that allows other, more plausible, alternatives to be ignored and Craig's need to argue for a cause of the universe tout court to remain hidden. It's the dichotomy which allows a startlingly implausible argument to look plausible.

Craig, W. L. (2010, February 4). Five Arguments for God. Retrieved February 8, 2012, from The Gospel Coalition: http://thegospelcoalition.org/publications/cci/five_arguments_for_god/

Milne, A. A. (1977). Winnie the Pooh. New York: Dell Pub. Co.

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“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.

Formally:

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.

Formally:

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. 

Formally:

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.

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

Bibliography

Craig, W. L. (2010, February 4). Five Arguments for God. Retrieved February 8, 2012, from The Gospel Coalition: http://thegospelcoalition.org/publications/cci/five_arguments_for_god/

Plantinga, A. (1974). The Nature of Necessity. Oxford: Oxford University Press.

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