Automaton

Note, how the author writes "the set of all truths". A logical impossibility, by definition, cannot be a fact nor truth.

bd-from-kg

jpbrooks:

No.

In standard logic, if A is “It’s raining” and B is “John is ten feet tall”, “A -> B is true if it isn’t raining. This isn’t “weird” or a “problem” of any kind; it just meant that the operator “->” is not correctly construed as meaning “implies” as that word is used in ordinary English.

If you want to have an “implication connective” that behaves more like “implies” does in normal discourse, try one of the many modal logics. In these logics there is a connective (I’ll call it “=>”) such that “A => B” (for the A and B above) is false regardless of whether A or B is true, because the truth of A does not necessitate the truth of B.

The choice of logical system is a practical one. Any logical system whose rules of implication preserve truth is perfectly OK theoretically. Of course, one may be more useful to you than another under given conditions, just as one definition of a word may be more useful than another under given conditions (and another may be more useful under others). Logicians generally prefer the standard “->” operator for technical reasons. (A major reason is that statements involving operators like “=>” cannot be evaluated using simple truth tables like the one you displayed.) As Microsoft might say, this is not a “bug”, it’s a “feature”.

Yes, but the point is that we want the statement that God is omniscient to mean, among other things, that God does not believe that the Japanese attacked Pearl Harbor in 1841. If we redefine “true” in such a way that “for every P, P is true if and only if God knows that P” is a tautology, we need to find some other way to say what we originally meant by saying that God is omniscient, which was of course that for every proposition P, God has a belief about P, and that all of these beliefs correspond to reality. However we try to rephrase or reformulate this, it will unavoidably involve quantification across all propositions.

bd-from-kg

Kenny:

Here finally is a response to your second post of Feb. 16. As you noted, this subject is complicated and subtle, and it takes a while to figure out how to explain what I’m trying to say in a way that has some chance of being understood. Don’t worry; I’m not in a hurry.

At this point it seems more productive to try to get at the essence of our disagreement rather than try to give a point-by-point reply. Also, I’m going to try to avoid using terms like “set” and “totality” here as much as possible to try to avoid misunderstandings based on the different meanings we seem to attach to such terms.

The “toy box” example seems as good a place to start as any.

True enough. But suppose that the things in the box are not toys, but the constituent parts of a toy. Now for it even to make sense to call the thing in question a toy (or for that matter for it to make sense to regard it as an entity of any kind) these constituent parts must form a coherent whole: they must interact in ways that can be conceptualized as the operation of a single entity. This is an integral, fundamental part of what it means to say that something is an entity; it has to be (at an absolute minimum) possible to think of all its parts as parts of a single thing. In fact, to be meaningfully considered a single thing, it has to be organized in some sense: a bunch of rocks scattered hither and yon can be conceptualized as a “set”, but it would be absurd to say that they constitute a real, concrete entity. But this organization involves a whole complex of relationships between the parts. And none of the parts can be left out of this network of relationships, or it wouldn’t really be a “part”. But if these parts cannot be thought of coherently as a single entity, none of this makes any sense. How can there be a network of relationships between the parts that enables them to function as a single, coherent entity if the very concept of these parts as a “completed whole” is logically incoherent? If this were true, the toy that they are supposedly parts of could not be thought of coherently as a single entity. But if something cannot even be thought of as a conceptual entity, it surely cannot reasonably be said to exist as an actual, concrete entity. Surely saying that something is a real, concrete entity is a stronger statement than saying that it exists as an idea or concept; the first must logically entail the second. So anything that doesn’t even constitute a conceptual entity – that cannot even be conceived of without self-contradiction – cannot be a real, concrete entity.

It’s true, as you have pointed out repeatedly, that if the parts of God’s mind include beliefs about all propositions, the parts of God’s mind cannot constitute such a “completed whole” or conceptual entity. But as I pointed out before, that’s the problem with the concept of omniscience.

Perhaps this will make the point clearer. We have:

(1) The contents of X’s mind.
(2) All possible true beliefs.

What I am arguing is that, by virtue of the fact that (1) is a concept of the type “the parts of a real, concrete entity”, the concept of (1) as a completed whole must necessarily be logically coherent. It must be possible to think of the contents of any real concrete entity as a “completed whole”, because otherwise we don’t have anything that can reasonably be called a real, concrete entity. On the other hand, we agree that the concept of “all possible true beliefs” as a “completed whole” is self-contradictory. But if X is omniscient, these two concepts are identical, which is impossible. So the concept of an omniscient entity is logically incoherent.

As for your IFSAC story, I have no idea what make of it. Of course the contents of God’s mind (even if we consider only beliefs about propositions) cannot be listed. Nothing larger than the set of integers can be listed. But even if you mean something beyond my understanding by “making a list” I don’t see the point. Obviously you can stipulate that the IFSAC can make a list (whatever that means) of any set, but that it can’t make a list of the contents of God’s mind. So what? How is this different from just saying that the contents of God’s mind don’t constitute a set?

But the last sentence is an excellent illustration of where we part company:

To me, this sentence seems to make no sense. The contents of God’s mind are already “collected together”. It’s pointless to simply refuse to call the resulting collection a “group” or “set” as though the words you apply to it change this reality. But to you, apparently, this sentence does seem to make sense. Apparently you can (or think you can) comprehend the idea that it is logically incoherent to think of a bunch of things as a completed whole, yet that this same bunch of things constitute a real, existing entity. I can’t.

From here on out, you insist on identifying the concept of a group of things with the concept of a list of this group, which as I have said I can make no sense of, so I am unable to follow your thinking. But it doesn’t seem to matter, since it appears that the point can be summarized by saying that you do not consider the contents of God’s mind to constitute a set. Thus if m is a set things which are all contents of God’s mind, T(m) is a meaningful proposition, but w* is not a set, because “all sets consisting of contents of God’s mind” is not a set. This makes perfect sense if the contents of God’s mind do not constitute a set, but I have already explained why I consider this position untenable.

[ February 23, 2002: Message edited by: bd-from-kg ]</p>

jpbrooks

I'm sorry about the late reply, bd. I usually spend most of the weekend perusing other threads and articles on this site.

But if the choice of logical system is subject to human judgment, this seems to introduce an element of uncertainty in the way that reality is represented by a given logical system. And this, in turn, seems to suggest that none of the conclusions that we draw from any of our arguments can be considered deductively certain. In other words, there is, in the end, no significant difference between inductive and deductive logic.

You may be right about this.
Before I read your comment, I was about to propose the definition that God's beliefs (accurately) describe reality (rather than correspond to it), but this too involves quantification over all propositions and is thus subject to the same objection.

[ February 23, 2002: Message edited by: jpbrooks ]</p>

bd-from-kg

jpbrooks:
Not at all. A “logical system” is essentially a set of definitions of the logical terms that occur in it. Thus “If A and A -&gt; B, then B” is (part of) the definition of the connective “-&gt;”. Define it differently and you get different results. Some logics (such as paraconsistent logics) even redefine “true”. (Or rather, the meta- interpretation of “true” [or “verum”] is nonstandard.) None of this “casts doubt” on results obtained from standard logic, any more than the fact that grass is “blue” if we interchange the definitions of “blue” and “green” casts doubt on whether grass is “really” green (using the standard definitions). It just means that with different definitions the same string of words often means something different.

By the way, modal logics (at least the ones I know of) do not redefine “-&gt;”. They introduce a new operator (“=&gt;” in my notation). The “-&gt;” operator is retained with the “old” meaning. Thus all the “standard” results hold. It would be nice if this were always done; it would eliminate a lot of confusion.

But this is getting off-subject. Discussions of nonstandard logics and the like really belong in the Philosophy forum.

jpbrooks

You're right. I'm just not at all comfortable with the idea of there being little or no restriction on the way logic can be defined, (while maintaining logic's applicability to reality). But (as you suggest) this is a subject for a possible future discussion in the philosophy forum.

However, getting back to the subject of this thread, I have been trying to figure out what Kenny might be attempting to point out in his posts above. And, it seems that his point about the "totality" of all truths not being a "set" may have some plausibility. In fact, we have an analogous situation with facts about the universe. I.e., there can be no "completed" "set" of all of the facts about the universe. And yet we know that the universe exists as a "completed" whole.
So, if this can be true in the case of the universe, why would it be incorrect to apply the same line of argumentation to God's knowledge of all truths? Why couldn't God's knowledge of all truths be a "totality" that cannot be represented as a completed "list" or "set'?

I have to leave now, but I will be back later tonight, hopefully.

[ February 28, 2002: Message edited by: jpbrooks ]</p>

bd-from-kg

jpbrooks:

This is not only plausible, it is certain. In fact, the point of the OP was precisely that the totality of all truths cannot be a set.

This is not at all analogous. No one claims that the universe contains all truths about itself, or any truths for that matter. “Truths” are simply not the sort of things that “exist” in the relevant sense. Beliefs, on the other hand, do exist in the relevant sense. The problem with omniscience is that it requires that the beliefs of a specified actual, concrete being have a one-to-one correspondence with “all true propositions”. Or if you prefer (since one-to-one correspondences are ordinarily defined to be sets of a certain kind) the same kind of diagonal argument used to show that there cannot be a set of all truths can be used to show that there cannot be a set of “all true beliefs”. The rest of my argument (to which both you and Kenny seem to be completely oblivious) is devoted to showing that this is logically inconsistent with the concept of an entity which has “all true beliefs”.