HRG

IMHO, people confuse the mathematical statement that 1+1=2 (or S0+S0=SS0 in Peano arguments) with the physical statement that the natural numbers with addition are a good model for the behavior of some classes of physical objects under juxtaposition. Apples, oranges and rocks belong to this class, while water drops aren't. 1 drop + 1 drop will make 1 (bigger) drops, not two drops.

This is meant in the same sense that the mathematical formalism of Newtonian mechanics is a good approximate model for the behavior of point masses under gravitational attraction,

or that plane Euclidean geometry is a good approximation for what we measure with rods and light rays.

The mathematical statement that 1+1=2 is a deduction within a formal system, and as such unassailable. The physical statement that some objects behave according to Peano arithmetics is empirical - like the corresponding statements for Newtonian mechanics and Euclidean geometry.

At least, that's how I see it.

Regards,
HRG.

"The task of philosophy is not the establishment of propositions, but their clarification" (L. Wittgenstein)

bd-from-kg

Huh?

We have (by stipulation):

P1: Y

P2: ~(Y & ~X)

From these we obtain:

C1: ~Y v X (from P2)

C2: X (from P1 and C1 )

In other words, if Y and ~(Y & ~X) are tautologies, so is X.

Of course, if you're assuming that either Y itself or ~(Y & ~X) is a necessary truth but not a tautology, you're assuming what you set out to prove.

In general, it seems to me that all necessary truths must be tautologies and vice-versa, by the following argument. If a statement is not a tautology, the notion of its being false is not self-contradictory. If the notion of its being false is not self-contradictory, one can conceive without self-contradiction of a world in which it is false. But a world that one can conceive of without self-contradiction is by definition a possible world. Thus a statement that is not a tautology is not a necessary truth. But of course a statement that is a tautology is a necessary truth. So the notion of a tautology is identical to the notion of a necessary truth - or at least the two notions have the same extension.

Finally, the notion that "X exists" can ever be a necessary truth runs up against the problem that one can clearly conceive without self-contradiction of a world in which nothing whatsoever exists. In this world "X exists" is false for all X. So no such statement can be a necessary truth.

Automaton

Kenny, I think we'll have to agree to disagree on this issue... Any further debate would only lead back around in circles.

Thanks for the discussion, it has been a lot of fun.

Kenny

You're right. That was a bad argument on my part. I don't know what I was thinking; please forgive the lapse. I still disagree with the idea that the only necessary truths are logical tautologies, but the argument I provided was question begging.

God Bless,
Kenny

Kenny

I'll second that! Thanks as well!

Kenny

Hello, bd-from-kg.

Sorry for the delay in response. My posting activity is touch an go at the moment so I don’t know how often or how extensively I may be able to respond in the future.

I agree that this argument is informally question begging (though, not formally so), but I do not think that your reasons for labeling it as such hit the mark. The fact that the conclusion follows in an “obvious, transparent way” would not seem to me to make an argument question begging. Consider the classic example of a syllogism:

P1. All men are mortal.
P2. Socrates is a man.
C1. Socrates is mortal.

I don’t think that the conclusion of this argument follows any less transparently from the premises than the conclusion of the argument above, but I think we both would be hard pressed to call this argument question begging.

I think the real issue is one of warrant transfer. I would label an argument as informally question begging if the any sort of potential or claimed warrant for the premises depends entirely and directly upon the potential or claimed warrant one has for the conclusion prior to evaluation of said argument. In the case of the argument you gave for infinity of primes, I do not see how there could be any warrant for the first premise that does not all ready depend on the warrant one has for the conclusion. Of course, the question facing us concerning the OA is just this question – does the warrant for the possibility premise of the OA depend on the warrant for the conclusion – and the whole discussion of rational intuition as a potential source of warrant revolves around this issue.

Huh? If, at a sporting game, all the bleachers are filled, the one of the bleachers is filled so long as one is taken to mean at least one and not only one. Perhaps your objection here boils down to an objection to the notion of transworld properties, which Plantinga is already assuming in his phrasing of this argument (because he has already argued for the coherence and reality of transworld properties elsewhere). If so, this presents another issue to be addressed, but not one that has been dealt with up to this point.

Perhaps a weakness in how this version of the OA is being presented is the fact that, in terms of warrant conduction, at least, the definitions serve much the same function as premises. In other words, the definitions themselves are making appeals to our intuitions concerning notions such as “greatness” and “maximal greatness,” and with sufficient rewording or expansion, the definitions themselves might serve as premises. This makes it appear that all the warrant for the argument (if any) is derived from the first premise, when, in fact, the definitions play a crucial role in this argument as well. Consider this rephrasing:

P1: If it is logically possible for there to be something which manifests greatness to a maximal degree, something which manifests greatness to a maximal degree exists necessarily.

P2: It is possible for there to be something which manifests greatness to a maximal degree.

C1: Something which manifests greatness to a maximal degree exists necessarily.

Essentially, the definitions in Plantinga’s argument serve the function of P1 in the argument above and P1 in Plantinga’s argument is identical with P2 in the argument above. P1 and P2 in the above argument are clearly distinct. In fact, one can affirm P1 and remain an agnostic concerning P2 (or even deny P2) since P1 can be argued for independently of P2. Consider the following argument as quoted <a href="http://www.utm.edu/research/iep/o/ont-arg.htm" target="_blank">here</a>which attempts to show that the existence of an unlimited being (which is closely associated with the idea of maximal greatness) is either logically necessary or logically impossible.

Either an unlimited being exists at world W or it doesn't exist at world W; there are no other possibilities. If an unlimited being does not exist in W, then its nonexistence cannot be explained by reference to any causally contingent feature of W; accordingly, there is no contingent feature of W that explains why that being doesn't exist. Now suppose, per reductio, an unlimited being exists in some other world W'. If so, then it must be some contingent feature f of W' that explains why that being exists in that world. But this entails that the nonexistence of an unlimited being in W can be explained by the absence of f in W; and this contradicts the claim that its nonexistence in W can't be explained by reference to any causally contingent feature. Thus, if God doesn't exist at W, then God doesn't exist in any logically possible world.

Put in reverse, this argument can be used to argue that the notion of an unlimited being, if logically coherent, implies the necessary existence of such a being. If successful, this argument stands regardless of whether or not the notion of an unlimited being is, in fact, logically coherent.

I think that, at best, this is an aesthetic objection to layout of Plantinga’s argument, not an objection to the argument’s actual content. With sufficient rephrasing, such as discussed above, this objection can be easily removed. Hartshorne, for instance, has demonstrated that []P follows from the premises P -&gt; []P and &lt;&gt;P in modal logic S5. So if it can be argued that the concept of God, if coherent, implies the necessary existence of God, and this is combined with a possibility premise, then we obtain the same results without ever having a premise of the form “possibly necessary.”

I agree that there are possible worlds where senses and memories cannot be trusted by the sentient beings within them, but how does this illustrate that we have no intuitions about how things are in other possible worlds? To me this claim seems obviously false. For instance, the proposition, “In no possible world do contradictory states of affairs obtain” is basic intuition we have concerning all possible worlds. Mathematical truths are often regarded as necessary truths (though there is some controversy about this), which would mean that our intuitions concerning mathematical propositions are intuitions concerning all possible worlds. In fact, to deny that we have any intuitions concerning all possible worlds is to deny that we have any intuitions concerning any sort of necessary truths.

I think the assertion that it is not legitimate to affirm something by intuition which can possibly be proven through other means, is clearly false. For instance, suppose that there is a successful argument for the reliability of induction which has thus far eluded the philosophical community (or if you think someone has produced such an argument already, imagine a community of persons for which it is unknown). If that were the case, and it were also the case that in order to rationally affirm something by intuition it must be impossible to provide a proof of it, then all beliefs based on induction by persons unaware of this argument would be irrational (for the community described). But, this does not seem right.

Furthermore, “Necessarily X” is not the type of premise being affirmed by intuition in the OA. “Possibly X” is being affirmed, which then leads, by deduction, to “Necessarily X.”


I’m not convinced that the only sorts of necessary truths are logical tautologies, at least not the logical tautologies which are found in first order predicate logic. They may be tautologies of a sort, but I must admit that my mind is not totally clear on this yet. It is true that the basic tautologies don’t include existence claims, but the tautologies of first order predict logic also do not contain claims about the modal status of propositions, claims about numbers, claims about knowledge, etc., yet most are prepared to concede that propositions such as “Necessarily, P-&gt;P,” “2+2=4” and “If S knows that p, it is true that p” are all necessary truths. In other words, the field of necessary propositions seems to be richer than the sort of tautologies that we encounter in first order predicate logic.

Now, I realize that one might hold that all of the above statements are themselves tautologies, but even if that is the case, this is not devastating for the OA. If there are tautological statements which express truths (often which are far from immediately obvious) about things such as numbers and knowledge, then why not tautological statements about the nature of being and greatness which express necessary truths which are not immediately obvious, and why might these not imply existential claims? In fact, that is precisely what the OA aims to show. If there is a concept of some existing thing g which implies []g, and it is also true that &lt;&gt;g, then it is indeed contradictory to deny the existence of g. So, if it is even logically possible that g, it follows that g.

God Bless,
Kenny

[ September 26, 2002: Message edited by: Kenny ]</p>

Thomas Metcalf

Originally posted by Kenny:

"Either an unlimited being exists at world W or it doesn't exist at world W; there are no other possibilities. If an unlimited being does not exist in W, then its nonexistence cannot be explained by reference to any causally contingent feature of W; accordingly, there is no contingent feature of W that explains why that being doesn't exist. Now suppose, per reductio, an unlimited being exists in some other world W'. If so, then it must be some contingent feature f of W' that explains why that being exists in that world. But this entails that the nonexistence of an unlimited being in W can be explained by the absence of f in W; and this contradicts the claim that its nonexistence in W can't be explained by reference to any causally contingent feature. Thus, if God doesn't exist at W, then God doesn't exist in any logically possible world." (Emphasis original.)

I have two major objections to this argument and its relatives, and one minor. The first has already been mentioned in this thread; it simply makes no sense to speak of a necessary being possibly or necessarily existing as one does when one asserts &lt;&gt;G or []G. I do not believe this is at heart an aesthetic objection; we are not in a position to accept (or even to understand) how God's necessary existence could be necessary in a possible or necessary way. You are correct that []P follows from P--&gt;[]P and &lt;&gt;P in S5, but suppose we take "G'" to be "a being with all of God's attributes except that He may or may not exist necessarily" so "[]G'" is equivalent to "God exists." The ontological argument we have now is:
(A) &lt;&gt;[]G'
(B) []G'--&gt;[][]G'
(C) [][]G'
We are stuck with nested modal operators. I don't think we are entitled to use S5 that way when P itself is a proposition with a modal operator, as it is implicitly when P is equivalent to "A necessary being with attributes as follows... exists." To assert &lt;&gt;G is undeniably to assert that it is possible that something necessary exists, which is unintelligible. Try to make sense of "There is a possible world in which something exists in every possible world," if you can. And to apply to your above argument, your first clause is equivalent to "Either a being Who exists in all possible worlds and is unlimited in all other ways exists at world W or it doesn't exist at world W..." See the problem?

The second objection is related. As I do not believe existence is a determining predicate, I do not believe modal status is a determining predicate, because a modal statement is a kind of existential statement -- specifically, it refers to existence in a number of possible worlds. Think of Plantinga's eunicorns, and compare them to "nunicorns," necessarily existing unicorns. We do not define something as necessarily or possibly existent, because if we did, it would either be obvious a priori that it existed and there would be no reason to search for it (and no way to deny it, like nunicorns), or it would be obvious a priori that it did not exist. When we try to decide whether something exists, there are two stages -- we first define what we are looking for, and then we search the universe and our body of accumulated knowledge to see if anything that follows from our knowledge or observations matches the description we have. To describe something as necessary antecedently is to make this process useless. Are you really ready to believe in nunicorns? To put in context of the argument above, to define God as an unlimited being is to define Him as existing in every possible world, and this is an illicit definition.

My third objection is simply that the verbal formulation of this argument is misleading. It seems intuitively problematic to deny that God is possibly existent, especially because God's possible existence is contained in His definition, but also because it just seems that God is not defined as a self-contradictory being and is therefore not obviously logically impossible. I must warn that logical impossibility should not be confused with modal impossibility here. To assert that God is impossible is only to assert that all the God-resembling things in nearby possible worlds are not in existence in every possible world. Perhaps out of 100 possible worlds, 99 contain a God-resembling thing, but to say that a being Who, if He existed, would exist necessarily, is impossible, is simply to say that none of the beings in any of the possible worlds exist in all of them. This is not so intuitively problematic as "God is impossible." The logical contradiction in asserting God's existence would simply be that it implicitly says "A being Who exists in every possible world does not exist in every possible world," and this is what would make Him impossible. To relate to the above argument, to say God doesn't exist in any logically possible world is simply to say that all the God-resembling things in all their possible worlds aren't existent in every possible world, and this is not particularly counter-intuitive.

In sum: To assert that a necessary proposition (such as "A being exactly like God except that it may or may not exist necessarily, exists necessarily") is possibly or necessarily true is nonsensical. Modal status is not a determining predicate. And it is not abundantly clear that God is possible.

BolshyFaker

So umm 12 pages too late (and having read none of the discussion pages 2-11) could the OP be re-stated as:

1)Argument from Arrogance:

1- I can't think of any sound arguments for Theism
2- Clearly I am the current pinnacle of human evolution.
3- So obviously no one can think of any sound arguments for Theism.
4- Therefore, God does not exist.

Just a (free)thought...

[edited to correct a blatantly stupid spelling of arrogance ]

[ September 30, 2002: Message edited by: BolshyFaker ]</p>

leonarde

Is that anywhere near the Argument from Pamplonance????

Bill Snedden

I don't know if you're still around to see this, but...

I don't see how the existence of grue-like predicates would necessitate the "failure of induction." If the universe is ordered (a necessary condition for this objection to be even considered a problem), then the existence of grue-like predicates simply means that no particular inductive determination can ever be considered final. But that's nothing new; inductive reasoning only leads to higher probabilities of truth, not absolute determinations.

The "justification" of induction rests on the principle of order. If order exists, then induction can be justified, regardless of how "disorderly" that order may appear. The key is that there are no "arbitrary" states arising in the future; all are ordered. If this is true, then in principle, induction is still valid.

To return to the example to which you linked, if there were a property of an emerald that caused its color to change based upon the time of observance, then that property could be discovered by empirical observation and would then become part of the body of knowledge concerning emeralds. We would no longer consider emeralds to be "green", but of two or possibly more colors (as is actually the case with other gemstones like sapphires & diamonds). In other words, empirical observation would still allow us to detect & catalogue the existence of grue-like predicates and refine our knowledge accordingly. Far from being a "failure of induction", this is actually what induction is.

Regards,

Bill Snedden