Monday, December 6, 2010

I just... I mean... and...

While I was at the store buying waffles and french toast (Don't judge me!), I, as one is wont to do, ended up in the checkout lane. "How about $9.98." said the cashier to the lady in front of me, making a statement that seemed less inquisitive and more an assault on the very idea of communication itself. The middle aged lady buying pickles took a $20 out of her purse, looked up at the total, then said "I'll just give you two pennies and you can give me a $10 back." She handed her $20.02 to the cashier, who took the money and plugged the total into the machine. The drawer opened and she began placing the money she had received into it, taking note of the change that should be given. She began putting the pennies into the drawer then paused.

She looked at the pennies, at the lady, at the screen, then repeated this round-robin betwixt those involved several times before saying "I don't know what happened. The total was $9.98, and I put in $20.02." The pickle lady looked at the screen and said "Well, somebodies math was wrong." in an accusatory manner that seemed to suggest the honest belief that she felt the machine was so bad at math that it could not subtract hundredths of a dollar correctly. She left.

I came home, opened my french toast, and noted that there were two separate plastic packages in my box of 6 french toasts. Handy, that will keep the others from getting freezer burn. Then I noticed the back of the box which stated that I was only supposed to eat two of the three toasts, proclaiming through it's little white nutrition box that to do otherwise would be gluttony and, therefor, I would be contributing to the downfall of western civilization.


Plantinga's Ontological Tautology

A thought occurs to me regarding the ontological argument, which, for those not familiar with religious apologetics goes a little like this:

If I can conceive of the greatest possible being, it must exist, because this idea must have come from somewhere.

    1. I can conceive of a perfect being.

    2. To exist is a greater perfection than not existing.

    3. Therefore, God exists.

There is a much longer version known as Plantinga's modal form, which I list below.
I will discuss my problems with both arguments here, not because my thoughts on them are unique, but because it is not easy to come across a good walkthrough of the topics in use. When somebody says that these arguments are begging the question, or that they are a tautology, they seldom explain why.

These objections have been arrived at solely by my own application of Logic. You may find that they match quite closely (or exactly) those of other people, but that is because the roads taken through proper application of logic lead only to one place.

A number of problems with the above mentioned ontological argument are readily apparent. The first to come to mind is simply that you cannot claim that existence is part of a concept. A concept of something is just that: an idea. The reality or existence of a given idea depends solely on our ability to see or experience it. We say that a dog is real, not because the idea of dog involves “existing,” but because we have seen a dog. The idea “dog” does not, in itself, exist, but is only an abstract. A dog exists, “dog” does not.

If all one need do to make something extant was to say “I believe the idea is not complete unless it includes existence” then one could prove anything simply by suggesting that the nature of the idea demands that it be real. Descartes in fact attempts this very thing, such that he is saying “because I cannot conceive God unless as existing, it follows that existence is inseparable from him, and therefore that he really exists.” Essentially, this boils down to “God exists by necessity because I am incapable of imagining god without existence.”

This is meant to imply that a) nobody is capable of conceiving of god without implicit existence, and b) people who claim to be able to think of god without implicit existence are not actually thinking of god. The second of these is what is so frequently adopted by the religious right in America when they say that non-believers don’t understand their beliefs. However, if anyone cannot conceive of a god which must exist, it very quickly degrades the argument that it must exist because it can be conceived.

In these cases God is essentially redefined as “something that exists” or “existence.” When one starts with the premise “God exists,” one cannot logically attempt to prove God exists from said premise.

Plantinga has attempted to weasel out of this by saying that while you cannot claim existence as a standard trait of something, you can claim it out of necessity, e.g. X is such that necessarily A. While making this claim he makes no attempt to show why God’s existence should be necessary, other than once again claiming that it is necessary if one defines God as existing in all possible worlds. This, of course, raises the problem of starting from the assumption that God exists to prove that God exists, which begs the question. X is such that necessarily X, therefore X.

Plantinga’s attempt is as follows:

    1. It is proposed that a being has maximal excellence in a given possible world W if and only if it is omnipotent, omniscient and wholly good in W; and

    2. It is proposed that a being has maximal greatness if it has maximal excellence in every possible world.

    3. Maximal greatness is possibly exemplified. That is, it is possible that there be a being that has maximal greatness. (Premise)

    4. Therefore, possibly it is necessarily true that an omniscient, omnipotent and perfectly good being exists.

    5. Therefore, it is necessarily true that an omniscient, omnipotent and perfectly good being exists. (By S5)

    6. Therefore, an omniscient, omnipotent and perfectly good being exists.

There are several steps of this which are assumed by the S5 axiom in use. To explain S5, I need to explain the terms and uses. First, when Plantinga uses the term “possible” it is used in the form of “logically possible.” For something to be logically possible requires only that it be possible under some circumstances, but not necessarily those which are present. If you take away the Chicxulub impact in possible world Z you get a starting frame in which “Dinosaurs are still alive” is logically possible (yes, I’m aware of birds). Remember that for later.

An attempt is made to suggest that it is “logically possible” for there to be a being of “maximal excellence” in logically possible world W. For this to work, one must first allow for a world W in which “good” is an attribute of something. Good must then either be subjective or objective. If Goodness is subjective, then any being which is “wholly good” in world W must be viewed as maximally good by all minds capable of making such judgments. Given this, it is logically possible for a wholly good being to exist in world W.

However, I am able to imagine a world in which minds do not agree on any maximal goodness, and would in fact say that it is the world we inhabit at this very moment. Without an agreement on subjective goodness, it is not logically possible to be “wholly good.” This leaves us with a logical contradiction. A being cannot be wholly good in all possible worlds if in some worlds it is impossible to be wholly good.

It will then be argued that Goodness is in fact an objective property, independent of minds. This is possible if you consider good to be a moral law. In a world where there is something capable of creating a universal moral law, it is possible for good to be objective. To use Goodness in this way, however, is again begging the question, as for a being to be omnipotent, omniscient, and wholly good in this case, it must be the being which created this universal moral imperative to begin with. The only way to assume an objective good is to assume the existence of a god. *

This means that step 1 in Plantinga’s argument implies:

   □God → □Goodness → ◊God

Such that ◊ is “possible”, □ is “necessary”, and A → B is “if A then B”

   If God necessarily exists, God possibly exists.

Second, a proposition can only be said to be “logically necessary” if it is true in all possible worlds. Again, possible is used in the frame of “logically possible.” If there is no way for a proposition to be false (by way of the negation being true), then the statement can be said to be necessarily true.

“A circle is round.” “A circle is not round.” The latter statement is not logically possible; therefore the former statement is logically necessary. The problem is there is no way to suggest that “an omniscient, omnipotent, and perfectly good being does not exist” is false without saying that it is false because God exists.
Given this, it is not logically necessary that God exists, especially in the case of this argument, where it is uncertain that it can even be shown as possible under the definition provided by proposition 1.

You may find yourself saying “Wait a minute! Plantinga didn’t say it was necessarily true, he said that it was possible that it is necessarily true.” Here’s where the S5 axiom that is in use comes into play.

The S5 axiom is a form of modal logic which is designed to remove excess modal operators. It says that if anything is possibly necessary then it is necessary. This is because being necessary means “true in all worlds,” so if in any possible world something is true in all worlds, it is still true in all worlds.

Likewise, if something is necessarily possible then it is possible. Anything that can be said to be true in all worlds is true in any one. The necessary becomes superfluous in common usage. “In all possible worlds, the sky is green in possible world X.”

In both cases, the doubled modal operators are unnecessary and, for the sake of clarity, the first operator can be removed without changing the meaning of the proposition. Any combination of ◊ and □ can be narrowed down to only the last operator. When Plantinga uses S5 to go from “possible that it is necessary” to “it is necessary,” the use of S5 is correct.

The problem that most people have with his argument is that Plantinga appears, at first glance, to go from “possible” to “possibly necessary” between propositions 3 and 4, and one can logically only go from “possible” to “necessarily possible.”
While it is true that anything necessary is most certainly necessarily possible (if it is true in all worlds, it must be true in any one world), and thereby possible, the reverse which appears to be in use is not correct in any way.

    A → □◊A is correct. (B)

    ◊A → □◊A is correct. (5)

    □A → ◊□A is correct. (5)

The inverse of all of these also follows, such that:

    □◊A → A (B)

    □◊A → ◊A (S5)

    ◊□A → □A (S5)

But ◊A → ◊□A is not a jump that anyone can make.

When he appears to make this jump, Plantinga is actually using linguistic trickery. The trick that is being played here is one of substitution. To demonstrate how it is done, I will show what each of his propositions actually means, logically, once you remove any misleading or confusing wording.

    1. It is proposed that a being God has maximal excellence is real/true in a given possible world W if and only if it is omnipotent, omniscient and wholly good in W; and

    2. It is proposed that a being God has maximal greatness is necessary if it has maximal excellence is real/true in every possible world.

    3. Maximal greatness is possibly exemplified. That is, it is possible that there be a being God has maximal greatness is necessary. (Premise) ◊□God

    4. Therefore, possibly it is necessarily true that an omniscient, omnipotent and perfectly good being God exists. ◊□God

    5. Therefore, it is necessarily true that an omniscient, omnipotent and perfectly good being God exists. (By S5) ◊□God → □God

    6. Therefore, an omniscient, omnipotent and perfectly good being God exists. □God → God

What he has actually done is to define his ideas in such a way that a being of maximal greatness is one which exists in all possible worlds, and stated this as his premise. He is using maximal greatness as a synonym for logically necessary. For clarity, his argument breaks down to:

    1. It is proposed that God exists in possible world W; It is possible that God exists.

    2. It is proposed that God is necessary if it exists in all worlds.

    3. It is possible God is necessarily. (Premise) ◊□God

    4. Therefore, it is possibly necessarily true that this being exists. ◊□God

    5. Therefore it is necessarily true that this being exists. ◊□God → □God

    6. Therefore this being exists. ◊□God → □God → God

That’s all very well and good except for the part where propositions 3 and 4 are identical. When fed through S5 (which is 5 [◊□A → □A] and M [□A → A] combined) you realize that propositions 3 and 6 are logically equivalent. This is a tautology. Plantinga has restated the same sentence 4 different ways, and then claimed “Look, I’ve proven it!”

I respectfully disagree. If proof of existence was really that easy, one could say:

    1. It is proposed that unicorns exist in all worlds. (Premise)

    2. Therefore unicorns exist.

Or, more concisely:

    1. Unicorns exist.

Nobody will ever take that seriously, and they shouldn’t pretend it’s somehow different when drawn out through longwinded obfuscation.

*This argument is valid against any variation of ontological argument, the most frequent being:

    1. I can imagine a perfect being.

    2. It is more perfect to exist than not to exist.

    3. That being exists.

Perfect can only be used objectively if one first assumes that it is not of the mind, but outside of all human judges, i.e. God’s decree.

→ If…then

□ It is necessary that ...

◊ It is possible that …