Kenny

First of all, a disclaimer: I am not an expert in set theory. Though I a familiar enough with it to understand the argument bd-from-kg gives, there is much which I still have to learn about the subject in general. Seeing how bd-from-kg is a mathematician by trade (as indicated by his profile), I acknowledge that I am outgunned in terms of expertise. This summer (when I am not taking 20 credit hours in the hopes of graduating in May) I intend (God willing ) to brush up on set theory a little more thoroughly. My coments are thus tentative and subject to revision as my thoughts become more clear on these matters. That being said...


To set the stage, I concede that there is no set of all true statements. This is an interesting philosophical conclusion in its own right with far reaching implications in other areas (such as how to define possible worlds in modal logic -- it appears that something along the lines of a maximal set of true propositions describing a coherent state of affairs won’t do the trick). However, I am not convinced that this creates a problem for omniscience defined as the property of knowing all true propositions and believing no false ones.

There appears to be a difference between the notion of a property and the notion of belonging to a set. There are mathematical objects which have the property of being a set, for instance -- “being a set” is a perfectly coherent property -- however, there is no set of all sets as bd has pointed out. Likewise, some propositions have the property of being true, but (as I have conceded) there is no set of all true statements. Also, it seems to me that all true propositions could very well have the property of being known by God without there being a set or class of all the propositions God knows. Hopefully, this will become clearer as I elaborate.

For the sake of argument, let us assume that God exists.

The notion of “a collection of propositions God knows” is not a concrete thing. It is an attempt to draw an abstraction from concrete aspects of God’s mind. We are taking individual propositions which exist in God’s mind and attempting to group them based on a shared property between them (in this case, their truth-value). In doing so, we are creating another abstract notion in which to place them, but there are no guarantees that the notion we have created is itself coherent. Our intuition betrays us in that it tells us that every time we find several particular things that share common traits it is possible to put them into a group based on those traits. As Russell’s paradox shows, such is not the case. There are individual propositions that God knows, and God knows all true propositions, but there is no set, no group, of all the propositions which God knows. In other words, if (per impossible) God were to set out to create a list of all the propositions He knows, He would find Himself unable to do so, because there is no coherent way to define such a list. Now, that’s counter-intuitive, but I don’t see how saying that God can’t make a list of all the propositions He knows entails that God doesn’t know all true propositions anymore than the fact that there is no set of all sets entails that there are no sets.

Agreed. If it is the case that no proof of GC exists but that GC is true, then God still knows GC is true, but He is unable to produce a proof of it. Likewise, it happens to be the case, IMO, that God knows all true propositions even though He is unable to generated a complete list of what those propositions are.

True, provided that such global propositions are actually coherent. God doesn’t know, for instance, whether or not a particular set belongs to the set of all sets because the question of whether or not something belongs to the set of all sets is literally meaningless. It is equivalent to asking if God knows that “sumesad estnat in da dabu” is either true or false -- there is no meaningful proposition there for God to evaluate.

Ah, but this is the crux of the matter. What your argument really proves, it seems to me, is that there are some apparently (but not actually) meaningful notions of collections regarding propositions which are actually incoherent. God doesn’t know whether propositions belong to such collections because they don’t really describe anything meaningful, but this reflects on the incoherence of the collections themselves, not on the coherence or lack thereof of God’s omniscience.

God Bless,
Kenny

P.S. Please forgive me if I am slow to respond. This is going to be a busy week.

[ February 11, 2002: Message edited by: Kenny ]</p>

jpbrooks

Thanks, Kenny. (I'm not sure that I can agree with everything that you posted above, but) that is essentially what I meant when I pointed out that God's inability to know (or to do, for that matter) what is logically impossible cannot be considered an actual limitation on His omniscience (or omnipotence).

Having taken only one class in set theory, I'm no expert in it either. But your point about the incoherence of collections seems to provide a reason to be cautious about applying set theoretical concepts indiscriminately to theology.

[ February 11, 2002: Message edited by: jpbrooks ]</p>

bd-from-kg

Kenny:

Thanks for your thoughtful response. It clarifies some key issues very well. Naturally I disagree with nearly all of what you have to say.

Before proceeding I want to make it clear that I’m not at all sure that the argument I presented is correct. But I do think that, if there is a flaw, you haven’t located it yet.

Enough preliminaries. Let’s “go to the tape”.

“Being a set” isn’t a good example here, because it isn’t a “property” at all. For example, take this toy box. Do the toys in the box constitute a “set”? Not really. The toys themselves are just toys. But we can think of them as a collection or group. Another word for a collection or group is “set”. The point is that this collection “exists” only as a concept, and the concept, by its nature, is a concept of a collection – that is, of a “set”. So its “existence” consists of its being a concept of the kind we call a “set”. To think of it as a “thing” that happens to have the “property” of “being a set” is exactly backwards.

This is a better example, but it still doesn’t reach the crucial point. The problem here is that “being true or false” is a definitional property of propositions. That is, part (if not all) of the meaning of “proposition” is that it is something that is either true or false. Anything that cannot meaningfully be said to be true or false (like a tree, or “Shut the door!&#8221 is by definition not a proposition. And, of course, if p is a proposition, “Not p” is also a proposition – one that is true iff p is false. So necessarily (by the definition of “proposition” and “not&#8221 some propositions are true.

In fact, in the G/P article you cited, Plantinga keeps harping on the fact that “All propositions are either true or false” as an example of a universal proposition (i.e., a proposition about all propositions). Unlike Grim, I would agree that this is a valid proposition, but it is valid because it is definitional. That is to say, it is not really about propositions, but about the definition of the term “proposition”. It is not properly construed as “For every definition p, either p has the property of being true or p has the property of being false” (which involves an illegitimate quantification over all propositions) but as “Part of what it means to be a ‘proposition’ is to be either true of false”. This is quite different from a (pseudo)proposition such as “All true propositions are known by God”. “Being known by God” is not logically entailed by the meaning of “proposition”, so the only possible way to construe this sentence is “For every proposition [i]p/i], if p has the property of being true, p has the property of being known by God”. But this does inherently involve a quantification over all propositions, which is illegitimate, and is therefore meaningless.

[Actually the last paragraph was aimed more at Grim than at you. I think he blew it by agreeing that statements like “All propositions are either true or false” do not express propositions. This gave Plantinga a huge opening which he proceeded to drive a Sherman tank through.]

And hopefully it will soon become clear why I disagree.

Well, sure. As I pointed out earlier, the notion of the set of toys in a toy box is not a concrete thing. It is an attempt to draw an abstraction from concrete aspects of the real world. A set is an abstraction. And if we aren’t careful we can wind up with a logically incoherent abstraction, like a square circle or the “set of all sets”. But if we start to seriously question whether the lowest possible level of abstraction of this kind is logically coherent, rational thought becomes impossible. And the lowest possible level of abstraction, when it comes to sets, is a set which consists of the elements of a concrete, existing thing. For example, if we have some toys in this box, the concept of all of the toys in the box is clearly a logically coherent concept. So this group constitutes a “set”. This is absolutely foundational. If this isn’t a “set”, nothing is a set. If I cannot form the concept of this particular bunch of toys without logical contradiction, I am literally incapable of rational thought. Rational thought absolutely requires being able to form abstract concepts, and as abstract concepts go, this is as simple as it gets.

Now if the notion of a “mind” as an actual, existing entity means anything at all, the concept of the “contents” of that mind must have some actual concrete meaning. I am unable to form a concept of a mind without any contents.

But if God’s mind is an actual existing entity, and it makes sense to talk about the contents of His mind, the concept of the contents of His mind must be logically coherent. Which is to say that the concept of the contents of God’s mind is a set. (Or, if you prefer, a collection, a totality, or a completed whole; it doesn’t really matter).

This is absolutely crucial, so I want to emphasize how very basic this part of the argument is. If the concept of a “set” means anything at all, the concept of the contents of an actual, existing, concrete entity is a set. This is the same principle that was appealed to implicitly in the case of the toys in the box. There is nothing fundamentally different about the case of the contents of God’s mind. The concept of the contents of God’s mind is a “set” for the exact same reason that the concept of the toys in the box is a “set”.

If you want to argue that there is a difference, in that God’s mind doesn’t exist in quite the same “concrete” sense as the toys in the box, here’s another example: consider the hard drive of the computer I’m working on. Now the “contents” of this drive don’t “really” exist as “concrete” entities. The hard drive is “really” just a bunch of atoms and electrons, etc. arranged in a certain way. This post, for example, doesn’t “really” exist on it. Its “existence”, like the existence of many other files, is an interpretation of the hard drive; it exists only as a concept. Nevertheless, both it and the other contents of the drive do exist in some sense, and it would be absurd to deny that the concept of these “contents” is logically coherent, and therefore constitutes a “set”, merely on the grounds that the “contents” themselves are abstractions.

Yes, of course. If we are going to be able to think in terms of abstract collections of things, we have to be able to think about subcollections defined by a particular property. Thus, if the concept of a “set” means anything at all, the bunch of alphabet blocks in the toy box must constitute a set. (In formal set theory this is called the “axiom of separation”. I know of no one who thinks that the axiom of separation can be dispensed with; it’s a fundamental part of our thinking equipment. And absolutely no one (to my knowledge) thinks that the axiom of separation is capable (in itself) of producing any logical difficulties.)

Not only is this notion very intuitive, but it is an intrinsic part of any meaningful notion of omniscience. It is simply absurd to say that God knows everything there is to know about a particular bunch of things, but doesn’t know everything there is to know about the things in this bunch that have a particular property.

Next, if we are talking about a mind which is capable of knowing things, one of the sorts of things that must form a part of the contents of that mind is “beliefs”. It is impossible to imagine what it would mean to call a mind that contained no beliefs “omniscient”. So God’s mind must have contents, and among those contents are things that have the property of being “beliefs”. Thus, by the “axiom of separation” (or if you prefer, the fundamental intuition on which this axiom is based) the concept of all of the things that God believes is logically coherent – i.e., it is a “set” (or totality, etc.). But if God is omniscient, this is identical to the set (or totality or whatever) of true propositions.

This is really the crux of the argument against omniscience. If there is an omniscient being, the set of true propositions is a subset of the set of the contents of God’s mind, and must therefore itself be a set. As Grim put it, “Were there an omniscient being, what that being would know would constitute a set of all truths.” Unfortunately he failed to defend this claim. You might say that I’m completing his argument by filling in this crucial detail.

Hopefully the exposition above explains why, if the concept of an omniscient being is logically coherent, the notion of all of the beliefs of such a being – i.e., of all true propositions – is guaranteed to be logically coherent.

True. But if they are already elements of a set (or group), it is indeed possible to “put them into” a subset based on those traits. The axiom of separation again.

In a way, this is perfectly correct. The problem is that the existence of an omniscient being logically implies that these notions are coherent – which is to say, meaningful. Since it can be demonstrated that they aren’t, there can be no such thing as an omniscient being.

[ February 11, 2002: Message edited by: bd-from-kg ]</p>

Kenny

bd-from-kg

Thanks as well. This whole discussion, I think, raises some very interesting issues (and not just for theology).

Likewise, I am not sure that my refutation hits the mark or that I completely understand all the points you are trying to make, but I remain unconvinced that there is a genuine problem for omniscience here.

Typically, we do assign properties to abstractions. We say, for instance, that 7 has the property of being prime. A property is simply a mark or characteristic which can be used to distinguish an “object” (whether concrete or abstract) from another. Sets clearly have distinguishing characteristics which distinguish them from other sorts of objects or constructs. 7 is not a set nor is my Grandma, but {1,2,3,...} is. Furthermore, sets have characteristics which distinguish them from one another as well as characteristics which seem to be uniquely shared among sets (such as cardinality). If you don’t like to apply the term “property” to abstractions then substitute a word like “characteristic” or what ever it is that makes you happy -- whatever you wish to call it, sets clearly have distinguishing marks of their own. Regardless, clearly there are certain characteristics (such as being non-self-membered) actually held in someway by certain types of objects (even if they are abstract) for which it is impossible to use as a criterion for grouping or creating a class or a set without involving oneself in a contradiction.

You may be able to escape having to say that “being true or false” is actually a property held by propositions. However, all propositions do have either one of two properties -- they either have the property of being true or the property of being false. Whether a proposition is true or false is not something which follows from its definition alone, but something which follows, in some way, to its relationship with reality (a correspondence of some sort with the way things are). In other words, there is such a property as “being true” (or “being false&#8221 which is held by some propositions which is not merely part of the definitions involved. Yet, (as we both seem to agree) there is no class or group of propositions which share this property.

I’m sure you probably have a response to this, but I think the assertion that “A: quantification over all propositions is meaningless” is self-referentially incoherent in that it renders itself meaningless. It seems to me that A is equivalent to “B: No proposition expresses a meaningful statement about all propositions” which in turn is logically equivalent to “C: For every proposition p, if p is a proposition, p makes no meaningful statements concerning all propositions.” But, B and C are themselves propositions that attempt to make a meaningful statement about all propositions. In other words, if any argument succeeds in demonstrating that there can be no quantification over all propositions, it fails as a result of its success. It follows that there can be no such demonstration.

True, but we're not talking about toys in a toy box here, we're talking about objects of which there is an infinite number as well as notions such as self-reference and the like. That’s hardly the “lowest level” of abstraction.

Agreed, however, that does not demonstrate that certain ways of trying to group those contents are not incoherent.

Okay.

Not necessarily. First, I’m not sure that the notion of a “totality” or a “completed whole” is equivalent to the notions of “a collection” or “a set” (more on this later). Second, I still don’t see why the contents of God’s mind (which are infinitely many) must be such that they can be meaningfully grouped as a set. As we both seem to agree, a “set” is an abstraction based on an attempt to group items with shared characteristics, and not all such attempts can actually succeed.

I disagree. There are only a finite number of toys in the toy box; the contents of God’s mind are infinite. Toys don’t make statements about other toys or statements about themselves; propositions do.

My understanding of the axiom of separation is that for any set A, it must be possible to define subsets of A. This, however, does not say that for any shared property, p, among objects, it must be possible to define a set, A, of all the objects which share p. This later assertion, it would seem to me, is the way right to the door of Russell’s paradox (keep in mind the above discussion of abstract notions like sets still having distinguishing characteristics and the discussion of the fact that there is a property of “being true” even though there is no set of true propositions.

God does precisely know everything there is to know about a particular bunch of things and everything there is to know about the properties which define the things in this bunch. However, this does not entail, as counter-intuitive as it may seem, that for every property held by more than one thing it is possible to group those things into a “bunch” on the basis of that property. As already noted, there is a property of being true that is held by multiple propositions, but there is no collection or “bunch” of all true propositions. There are some abstractions which are properly characterized as “sets,” but there is no abstraction that is properly characterized as the collection of all sets.

Again, this is not what I understand the axiom of separation to say. The axiom of separation tells us something about sets which are known to exist, but it does not inform us if a commonly held property can be used as the basis to form a set of all things which share that property.

As I already alluded to in passing, I fail to see that “all true propositions” is necessarily equivalent to the set or the group of all true propositions. For instance, I can talk about properties (or traits or characteristics if you prefer) shared by “all sets” (“All sets have cardinality,” for instance), but I cannot meaningfully talk about the set of all sets. Clearly, there is some sort of distinction involved between the notion of “all” or a “totality” and a set of all objects.

Right, provided that one knows that such a set exists. However, I am still not convinced that the existence of an omniscient being entails that there be a set of all true propositions in the first place, in which case the argument fails at the very first step.

Anyway, isn't philosophy of religion fun (even if you happen to be an atheist)? I suppose I better get back to my school work for now, however. Furhter replies will have to wait until, at least, after Thursday (a day which I will not be spending on the internet )

God Bless,
Kenny

[ February 11, 2002: Message edited by: Kenny ]

&lt;edited to fix quotation&gt;

[ February 12, 2002: Message edited by: Kenny ]</p>

Kenny

I know I said I wouldn’t post anything further until after Thursday, but I think it may help matters to briefly summarize my position.

Essentially, I deny that the proposition “There exists a totality of X” necessarily implies the proposition “There exists a set (collection, class, etc.) of all X.” It contend that the notion of a set (a group) of things is a different sort of abstract category than a totality of things and that it is possible for there to be a totality of something without a coherent concept of “the group of that something.” This assertion may seem counter-intuitive, but I do not believe it to be without support. One can make meaningful statements about all sets for example (e.g. the axioms of set theory). Statements such as “All sets have cardinality,” in turn, would seem to imply that it is meaningful (at least conceptually) to think about “all sets” -- that is “a totality of sets.” There is however, no set of all sets. Likewise I think it is meaningful to speak of a totality of truths, but I deny that this implies that it must also be true that it is meaningful to talk about a set or group of all truths or to define subsets with respect to the totality of all truths (and there is no violation of the axiom of separation in this assertion since it is being denied that a totality constitutes a set to begin with). In other words, with respect to the issue of omniscience, I affirm that the totality of true propositions exist in God’s mind; I deny that it is possible for God (or anyone else) to construct a meaningful notion of a set from that totality (because of the paradox bd and Grim have (independently, it seems) pointed out) of all true propositions. All this means is that there are certain ways of attempting to group things which are not coherent, not that a totality of such things cannot exist. In other words, the attempt to place all propositions in a group winds up being a misapplication of the concept of “a group” to the reality of all propositions existing, not a proof that the reality itself fails to exist.

God Bless,
Kenny

[ February 13, 2002: Message edited by: Kenny ]</p>

bd-from-kg

Kenny:

On reflection is appears that at this point we are actually discussing two distinct arguments for the incoherence of the notion of omnipotence: Grim’s and mine. I’ll discuss Grim’s first.

1. Grim’s argument

I should admit at the outset that I don’t fully understand Grim’s reasons (apparently given in the paper “Logic and Limits of Knowledge and Truth”) for believing that quantification across “all propositions” is illegitimate. But I think that I can dimly see his reasons for saying this. At any rate, my main concern here will be to show that this is a reasonable position – that it is not (as Plantinga argues) self-defeating or self-evidently false - and that it does imply that omniscience is a logically incoherent concept.

The latter point is pretty clear. If we try to define what it means to say “X is omniscient” it seems to be unavoidably necessary to quantify across all propositions. Thus we might say “X is omniscient if, for every true proposition p, X knows that p is true.” But if quantification across all propositions is illegitimate, and this definition (and any other definition of omniscience, it would seem) is logically incoherent. Now it is generally accepted that there are “irreducibly simple” properties which cannot be defined in any kind of rigorous way (such as the property of being “true”). But it is highly implausible that being “omniscient” is an irreducibly simple property. And for other properties it is pretty much axiomatic that if it is logically coherent it is possible to define it in a logically coherent way. So if omniscience cannot in principle be defined in a logically coherent way, it is not a logically coherent property.

Plantinga offers two arguments against the claim that quantification across all propositions is illegitimate: (1) it is self-refuting because (when expressed rigorously in logical notation) it involves quantification across all properties, and (2) it is self-evidently false, because there are clearly universal propositions – for example “All propositions are either true or false.” As you put it:

But my point was not that being true or false is not a property of propositions, but that “being either true or false” is a definitional property of propositions – that is, it is part of the meaning of “proposition” that a proposition is either true or false. It also follows from the meaning of “proposition” (together with the meaning of “not”) that some propositions are true. The important point here is that statements like “All propositions are true or false” and “Some propositions are true” express propositions that do not intrinsically involve quantifiers. The reason for this is that they are not propositions about propositions, but propositions about the meaning of the word “proposition”.

Actually statement A is phrased a bit carelessly. Let’s consider instead A′: “A statement that involves quantification over all propositions is not well-formed, and therefore does not express a proposition”. In a sense this is a statement about all propositions, but it doesn’t involve quantification over all propositions.

Here’s an example that might help clarify this distinction. Consider C: “All swans have long necks”, and D: “No swans are black”. (We will assume that having a long neck is part of the definition of “swan”, but being non-black isn’t.) C is definitional of swans, and so can be expressed in a way that does not involve quantification: “In order for X to qualify as a swan, X must have a long neck”. There is no such way of translating D into an equivalent expression that does not involve quantification. So while D is unequivocally a statement about “all swans”, C is more properly interpreted as a statement about the word “swan” than as a statement about swans.

This suggests yet another way to express the true statement that A is intended (but fails) to express: A′′: “Only propositions that are definitional of the meaning of ‘proposition’ express meaningful statements about all propositions.” Or in other words, the only meaningful statements about all propositions are statements about what “proposition” means, or that follow tautologically from the meaning of “proposition”. (We include here statements about the meaning of the basic logical connectives and quantifiers that appear in propositions.) Obviously, such propositions can often be expressed in the form “For all propositions...” or “For every proposition p ...”. But they do no intrinsically involve quantification over all propositions. And of course, as I have pointed out, strictly speaking such statements are not really about “all propositions”, but about the meaning of the word “proposition”.

It’s essential to insist on this distinction, since otherwise the status of the “axioms of logic” themselves becomes problematic. For example, consider “If A is true and ‘A implies B’ is true, then B is true.” There is an implied double quantifier here: the full statement of this axiom is “For all propositions A and B,...” And these quantifiers, it would seem, range over all propositions. But according to the analysis above, this is not really a problem, since this axiom (like all of the axioms of logic) is definitional: it defines the meaning of “implies” (or if you want to be pedantic, it follow tautologically from the meaning of “implies”, “if”, and “then”).

2. My argument

(Well, OK, it’s really Russell’s argument.)

The argument I presented in the OP differs significantly from Grim’s. In particular, it does not appeal to the principle that quantification across all propositions is illegitimate. But the notion of a “set” (or collection, or class, or totality) does play an important role. So it’s important to get this notion straight before proceeding.

My point about “being a set” not being a property is similar to the point that existence is not a property. A set is a concept; its “existence” consists entirely of its being a concept of a collection of things. To say that it’s a set is to say nothing more than that it’s a logically coherent concept of this type.

Whether “being a set” is properly regarded as a property isn’t important to the argument in itself. What is important is a proper understanding of what a “set” is, and what it means to say “X is a set”, or that a certain set “exists”.

Now let’s look at some of your statements about sets and why I think they may reflect some confusion about what a “set” is. (I’m taking statements out of order here for the sake of making the argument flow logically.)

This seems to be based on a neo-Platonic conception of a “set” as a thing that “really” exists in a sense other than merely being a concept. Thus, even if we can conceive of a certain collection of things as a “totality” or “completed whole”, it still (according to this view) makes sense to ask whether this collection is “really” a set – i.e., whether it “really” exists, as opposed to our merely having a concept of it. This would make sense if we were talking about unicorns, for example: I can have the concept of unicorns even though unicorns don’t really exist. but it doesn’t make sense for sets, because sets are concepts. If I have the concept of a bunch of things as a collection – i.e., a “completed whole” - this is a concept of the “set type”, which is to say that it is a set.

Let’s examine this further. You say that not all “attempts” to “group” items with shared characteristics can “actually succeed”. In a sense this is true, but it’s important to understand in what sense. What does it mean to say that such an attempt has “failed”? After all, we are not “attempting to group” the items in question in a physical sense; we are “grouping” them only in the sense of thinking of them as a group. The only way that this can “fail” (in the sense that what we are trying to do is impossible in principle) is if the concept of the group in question is logically incoherent.

So we arrive at the following principle: Any concept of a group or bunch of “things” is a set, provided only that the concept is logically coherent.

We are now in a position to consider the following statement:

But as we have seen, the concept “all true propositions” is the set of all true propositions. If the concept exists (i.e., if it is logically coherent) then the set exists, because they are one and the same.

Next, before getting to the main event, let’s clear up a side issue.

Right. The things in question, in addition to having a shared property, must already be elements or members of some set or collection. The property “separates them out” from the other elements to form a subcollection (which is why I used the term “subcollection”). My original argument here was in the form of a logical progression, not a series of disconnected points. First a long argument to the effect that the contents of God’s mind form a set (or collection, or totality, or completed whole), then the point that those contents that have the property of being “beliefs” form a set on the grounds that they are a subcollection of the contents of God’s mind that share a common property.

Since it appears that you do not dispute this point once it is properly understood, there seems to be no need to discuss it further.

Finally we proceed to the crucial question of whether the contents of God’s mind “constitute” a set.

If we consider any real, existent entity, the things that make up or constitute this entity must necessarily also be real, existent entities. So there is simply no question that the concept of them as a group is logically coherent. How could it not be? If something is a real, existent entity, the concept of it is necessarily logically coherent, and so the concept of its contents is necessarily logically coherent. As I pointed out before, this is absolutely foundational for the whole concept of a “set”, or for being able to think rationally at all for that matter. It is simply taken for granted in all of our thinking that the constituent parts of a thing that actually exists can be thought of as a “totality”. Indeed, it seems to me that these constituent parts, along with any relationships that may exist between them, are the thing in question. But even if this be denied, it cannot be denied that they are a part of the thing in question. It is impossible to imagine how the concept of the collection of these constituent parts could be logically incoherent, without making the concept of the thing itself logically incoherent.

Now you objected to applying the principle to God based on the nature of the (supposed) contents of God’s mind. (For example, they are infinite and self-referential.) But that’s the point. The argument, boiled down to its essentials, is this: On the one hand, if God is an actual, existent entity, the principle applies to Him for the same reason that it applies to any other actual existent entity. But on the other hand, if God is omniscient, applying it to Him yields a logical contradiction. It is not a refutation of this argument to point out that if God is omniscient, applying the principle to Him yields a logical contradiction; that’s a part of the argument, not a refutation of it! To refute the argument, you need to offer some independent reason, which is not based on the fact the “the contents of God’s mind” is a logically incoherent concept, why this principle does not apply to Him.

Let’s try looking at this another way. On the one hand we have:

(1) The contents of X’s mind..

On the other we have:

(2) The collection of all true propositions.

If X is an actual, existent entity, the first concept is necessarily logically coherent. But we have seen that the second concept is not logically coherent. But if X were omniscient, the collection of all true propositions would be a part of the contents of X’s mind. This is logically impossible, so we must conclude that there is no such X – i.e., no actual, existent entity is omniscient.
_____________________________________

The above was written before I saw your most recent post. I don’t have time to reply to it in detail right now (I, too, have a life) but my first reaction is that I have no idea what the distinction is (in your mind) between a “totality” and a “set”. For example, why would a “totality” not satisfy the axiom of separation? Why would a diagonal argument not apply to it just as well as to a “set”? In particular, this distinction seems to make no sense if by a “set” you mean (as I do) a concept of a group or collection.

I would remind you of Grim’s statement in the G/P paper:

He later makes some more important points against this whole approach to the problem:

As to the point that the statement “all sets have cardinality” seems to express a valid proposition, I’ll have to defer it to a later time, but it seems clear that it can be dealt with by the distinction I made between “definitional” statements and statements that intrinsically involve a quantification (one that would in this case clearly be illegitimate).

Deputy42

howdy, this is a long thread so im not sure anyone will read this but i'll forge ahead regardless. in response to the original message the author assumes a couple things:
1)gods omniscience needs to be logical
2)logic itself is true
starting with 2, im sure all of you are familiar with the greeks. further i will assume that you are familiar with the fact that their epistemology was based on two things. the only things they took to be self evident without need of proof were logic and geometry. hmmmmm......
and 1 well hey if god were omniscient, wow he certainly would know how to think alogically, almost well....nevermind. i suppose the omnipotence would have something to do with that. thanx, well i liked the proof but i think you should know what you're going in with before you start cuz it might have some bearing on what you get out. latez

Kenny

bd-from-kg,

Part 1: Concerning Grim

I find your responses to Plantinga's refutation of Grim’s argument that there can be no quantification over all propositions to be powerful. Of course, this does not mean that I find Grim’s argument itself compelling (I still do not buy all the premises). I shall have to think about this issue some more. I still have reasons to be skeptical of such a result which I will detail a little later.

Nevertheless, I believe you may have just provided the defender of omniscience another way out of Grim’s version of the argument. You said:

Somewhat earlier you had asserted:

Given the discussion regarding definitional properties which follows, however, a defender of omniscience (as is myself) could maintain that God’s knowledge of all true propositions itself involves a definitional property of true propositions. Elsewhere, you maintain that this is implausible, but, perhaps, not as much as one might think. Consider the question “What does it mean for a proposition to be true?” One answer which is often given is that a proposition is true if and only if it corresponds to reality. Of course, this raises the question as to what it means for something like a proposition to “correspond to reality.” Since, in Christian metaphysics, God is understood to be that in which all reality finds its origin and ultimate definition, a reasonable answer to that question would be (from a Christian point of view) that a proposition corresponds with reality if it corresponds to what God believes about reality (since it is with reference to God, the ground of all being, in which reality finds its definition to begin with). Furthermore, since God’s beliefs cannot fail to be true (by definition) it follows that all of God’s beliefs are warranted as well (i.e. known to be true). This means that, on Christian metaphysics, the notion of “a true proposition” may very well be defined in the following manner: “P is true, if and only if, P is known by God.” The definition of omniscience that you give above, then (substituting “God” for “X&#8221 , plugging in this definition of a true proposition, would read “God is omniscient if, for every (proposition known by God), p, God knows that p.” Since this involves a tautology, it follows that God cannot fail to be omniscient. In other words, even if Grim’s argument that there can be no quantification over all propositions is correct (and I am not convinced that it is), your defense against Plantinga’s objections can be turned around and used in favor of omniscience if it can be plausibly maintained that “being known by God” is somehow wrapped up in the definition of what it is to be a true proposition. Given a correspondence view of truth and Christian metaphysics, this can be plausibly maintained, and so it does not follow that the concept of omniscience necessarily involves quantification over all propositions.

Of course, that takes care of Grim’s argument, but not yours, as it still might be the case that the notion of omniscience entails that there must be a set of all true propositions (which would render the concept of omniscience incoherent). I would point out, however, that there is a close relationship between Grim’s argument that there can be no quantification over all true propositions and your argument that there can be no totality of propositions such that if Grim’s argument fails, so does your own. If meaningful things can be said about “all propositions,” then, conceptually at least, the notion of all propositions is meaningful even if there can be no set of all propositions.

That being said, allow me to present what I believe are a couple of reasons for still remaining skeptical of the idea that there can be no quantification over all propositions:

First, consider universal accidental truths such as “A: There are no black swans.” The fact that there are no black swans (at least, let us suppose) is not a necessary truth but a contingent feature of our world. Now, it seems meaningful, at least at for blush, to assert that “B: For every proposition, p, if p implies that some swans are black, p is false.” In other words, if it is meaningful to say that there are no black swans, then intuitively, at least, it seems that it is also meaningful to say that no true proposition implies the existence of black swans. In fact, it seems that A implies B. Actually, to say that A implies B seems very close to the tautology: (~q -&gt; ((p-&gt;q) -&gt; ~p)). However, B is quantification over all propositions. Furthermore, since B expresses an accidental truth, B is not definitional, in any way, of the meaning of propositions. Now, of course, it is possible for you to deny that B is itself a meaningful proposition or that A implies B, on the grounds of a demonstration that there can be no quantification over all propositions. A demonstration, however, is only as good as the strength of its intuitions. It seems to me that your argument rests on the intuition that for any meaningful notion of a totality, it is possible to view that totality as some sort of set or group or class of which subsets or sub-groups or sub-classes can be defined, and that this is also what leads to Grim’s conclusion that there can be no quantification over all propositions. We can weigh this intuition against the intuition that statements such as B are meaningful. In effect, what your argument (along with Grim’s) demonstrates is that we cannot hold on to both of these intuitions simultaneously and retain coherence. We have to drop one or the other, but which one? That is something which your argument doesn’t really say. Both intuitions seem fairly strong to me, but given the inherent difficulties involved with our intuitions concerning collections of things, demonstrated by such things as Russell’s paradox, my suspicions fall toward the intuition that it is possible to define subsets for every totality before they land on the intuition that statements such as B are meaningful (and I shall elaborate on this in my next post).

[ February 16, 2002: Message edited by: Kenny ]</p>

Kenny

Part 2: Concerning bd-from-kg’s Argument.

If the charge is holding a Platonic notion of abstractions, then I am afraid I must plead guilty, but no matter -- I don’t think the point I am trying to make depends on the holding of such a view.

What I am not convinced of is that a concept of a totality (in the sense that it is meaningful to speak of “all X&#8221 or a completed whole corresponds to or implies the concept of a set, or that for any property shared by more than one thing, it is possible to actually define a group of things which share such a property.

I agree, but if we are going to apply set theory to concrete things, we have to consider how it applies actual concrete things which share various properties. If we want to talk about the set of toys in the toy box, for instance, we have to have some notion of what it actually means to conceive of real physical toys as a collection of things. We also (as you seem to be trying to point out) need to realize that our conception of the toys in the toy box as a group is not the same thing as the toys in the toy box, but the application of an abstraction to them. If it turns out that certain of the ways in which we attempt to abstract about the toys in the toy box are incoherent, all it means is that there is something wrong with the way that we are applying our abstractions to them, not that there really are no toys in the toy box. Likewise, it could be that our difficulties in applying certain of our abstractions concerning groups to the concrete contents of God’s mind reflects a misapplication of our concepts of groups to those contents rather than being an indication that those contents do not exist. For your argument to really succeed, you have to demonstrate that the notion of God’s omniscience necessitates that such abstract concepts must be capable of being applied to God’s mind. You have made some attempts, but I am not yet convinced that you have succeeded in doing so.

To get another sort of perspective on this, consider the following story (never mind its theological feasibility):

Suppose there is an elite circle of angels in Heaven to whom God has decided to give unlimited access to the contents of His mind. One of these angels, Bob, is a collector of sorts -- he likes to group and categorize things, and he decides to do so with the contents of God’s mind so he assigns various categories based on different sorts of properties those contents have. Now Bob doesn’t want to spend forever (literally) sorting through all the contents of God’s mind individually, so Bob whips out the IFSAC (Infinitely Fast Super Angelic Computer) which has infinite storage capacity and is capable of doing an infinite number of operations per second, and programs it to sort through the contents in God’s mind based on various properties those contents have. One such property is “being a proposition.” Bob soon finds himself disappointed, however, because he notices that the program never manages to successfully group all of the propositions in God’s mind together. Oh, it’s finding such propositions alright -- they’re definitely there, but no matter how long Bob runs the program, it never stops. It keeps finding new propositions (even though it is doing an infinite number of operations per second). Frustrated, Bob goes and talks to his fellow angel Jill. Now, Jill’s hobby is mathematics. In fact, she spent a good million years combing both Heaven and Hell for some of the brightest mathematical minds that ever lived, just to glean from their insights (of course, she could have just asked God, but that would have been too easy), and she’s not a bad mathematician herself (having come up with several interesting new theorems over the past couple of eons). Bob wants to know what’s wrong with his program. Jill informs Bob that there is nothing wrong with his program per say; it’s doing exactly what it is supposed to do -- sort through the contents of God’s mind, pick out all the propositions, and then stop when it’s done. The problem, Jill says, is with Bob’s tacit assumption that a complete list of all the propositions in God’s mind (even an infinitely long one) can be constructed. The very idea of such a list, Jill points out, turns out to be incoherent. Of course, this doesn’t mean that all propositions fail to be present in God’s mind -- the computer keeps finding more -- only that it is impossible to collect them all into a complete group.

The purpose of the above story was simply to draw out the idea that there may very well be a distinction between a totality and a set, by putting some sense of concreteness on what such a distinction might involve. In the above story, it was Bob’s attempt to generate a complete list of the propositions in God’s mind that created a problem, not the existence of those propositions in God’s mind. The fact that Bob could not generate such a list was due to the fact that the very idea of such a list turns out to be incoherent. No matter how many propositions such a list contains (even an infinite amount), there will always be more propositions. In other words, there are always more propositions than the amount of propositions on any list.

Indeed, the conclusion that it is impossible to generate any sort of list of the contents of God’s mind has a certain amount of theological appeal. It draws out the Psalmist’s reflection “How numerous are your thoughts, oh God, were I to count them they would be like the grains of sand [i.e a poetic way of saying that it is impossible to count them]” to a whole new level. God’s mind is infinite beyond infinite, such that no list of any sort of the total contents of His mind could be made, and for the believer, this is even further cause of praise. In fact, thinking about such a possibility, left me with a profound sense of awe.

I agree.

No problems here.

I disagree here, if, by a group, what is meant is that the concept of a completed list of those contents must be logically coherent. What I am suggesting is that certain sorts of totalities may involve an infinite number of items in such a way that for any list of items which are a part of such a totality, it is possible to include more items belong to that totality -- (i.e. there are more members of that totality than there can be members of any list).

Hopefully, it is becoming clearer. I shall continue to elaborate.

By a “totality,” I do not mean a collection or a group, but the following: “A totality of X implies that the notion of ’all X’ is conceptually meaningful.” I deny that a totality, in this sense, necessarily implies that the notion of “a set of all X” is conceptually meaningful. As to the question of why the diagonal argument does not apply to the totality of propositions, allow me to take a look at your original argument and point out which premises I find shaky:

I deny that either such class exists if we are to conceptualize such classes as a completed list (which is essentially what the idea of a “set” involves). I hold that any attempt to list all the propositions in God’s mind fails because there are always more propositions that could be added to such a list.

This seems alright provided that m is actually coherent.

This seems okay as well.

I deny that w exists. The existence of w would mean that a list of all T(m)’s could be generated, but it is not at all obvious that such is the case. Again, it may be that there are more T(m)’s than the contents of any list of T(m)’s.

Anyway, that’s all I have time for, for now. The increasing complexity of this discussion is making it very difficult to sustain, so there are no guarantees of that I will have time to generate future responses.

God Bless,
Kenny.

[ February 16, 2002: Message edited by: Kenny ]</p>

jpbrooks

Hi, Kenny.

I'm in general agreement with you. But it seems that you are arguing that infinite "sets", such as "w" above, don't exist simply because all of its elements cannot be put into a list. If that were true, then wouldn't it also apply to any infinite "totality", such as, for example, the set of all integers? If this is the case, then, by extension, set theory cannot be applied to any infinite "totalities" at all!
I always thought that the reason why a "set of all sets" cannot exist is that such a "set" could never contain itself as an "element"; not simply because all of its "elements" cannot be placed sequentially into a list.
Perhaps this is not what you are arguing. But if I am mistaken on this point, can you please clarify your position on this matter.

[ February 17, 2002: Message edited by: jpbrooks ]</p>