You're using a different concept of 'actual infinity' than what is here and has traditionally been used in the cosmological argument.

From wiki:

Aristotle also distinguished between actual and potential infinities. An actual infinity is something which is completed and definite and consists of infinitely many elements, and according to Aristotle, a paradoxical idea, both in theory and in nature.

And on the math side:

http://alexzen.by.ru/papers/Cantor/10_mistakes.html

9) The only reason of the paradoxality (2) is the actual infinity of the enumeration (*). It proves that "actual" and "infinite" are algorithmically contradictory notions in the framework of Cantor's proofs and, consequently, the notions "actual" and "finite" are here algorithmically identical ones. Thus, it proves first the Aristotle, Leibniz, Kant, Gauss, Cauchy, Kronecker, Poincare, Brouwer, Wittgenstein, Weil, Luzin, etc. intuitive opinion that the actual infinity notion itself is a self-contradictory notion.

As is well known, according to the ancient mathematical tradition, any rigorously proved mathematical statement is called a Theorem. So, taking into account all said above, we can write:
ARISTOTLE'S THEOREM (III B.C.). INFINITUM ACTU NON DATUR, i.e., the notion "actual infinity" is a self-contradictory one, and all infinite sets in true mathematics are potentially infinite.

I won't have time to look into the details of the video for at least a little while, but I suspect that it's not particularly hard to reject it. Most likely, it's something that you view to be "self-evident" (which does not make it so). If it's not self-evident, then it's "intuitively plausible." But centrally, it's not something to "discuss" because such a rejection is always logically possible. Whether you find this to be a plausible rejection is up to discussion, and that's a different conversation.

Again, it's the nature of the debate. You can reject the "absurdity" and thereby reject the entire argument. The statement that it "seems" absurd to him is the statement that he is taking for the sake of argument that it "IS" absurd. I seriously doubt that the words he spoke were to be taken as some sort of formal argument in favor of the absurdity of an infinite past.

I find your objection to be strange. If we remove the comment about scientists and philosophers, this is your objection:

And now you're in a position where he makes a claim, and your counter-claim is "I disagree." It's not as if you have presented anyone any reason to accept your position over his. Therefore, the entire weight of your position is being measured by your appeal to the scientific and philosophical authority.

The first sentence is false. You can appeal to authority if the authority is authoritative on the topic being presented. As to the second sentence, for statements that are broadly accepted, name-dropping actually adds very little to the conversation.

You seem to have an unusual take on how these debates are structured. And if you were intellectually honest, you would be applying the exact same system of understanding to the people against whom Craig is debating, and I suspect that you'll find the whole thing hangs together in exactly the same way.

Also, if you hold yourself to the same standard that you are holding Craig, you didn't exactly "show" anything. Where have you presented any evidence or dropped names for your claims?

The absurdity doesn't come from imagining an infinite regress, but for an infinite regress to occur there cannot be a definite beginning or starting point. If a beginning is necessitated, then an infinite regress leads to absurdity.

"I think it is commonly accepted that William Craig is one of the most educated and sophisticated Christian apologist."

that he is says a lot about the quality of christian apologists...

In his defense, the answer to that has only been known to above average teenagers for the last 200 years.

Yeah, it is his actual quote. 5:30 in the first link I gave in OP.

Here is his main argument:

You would agree that if the first premise is false, the whole argument is false. How does he support the first premise? By claiming that an actual infinity is impossible. And then...? He goes to his intuition. So, the argument above ultimately rests on Craig's intuition.


I am not but I can, I just didn't wanna make the post even longer than it already is. What I'm criticizing him about is not that he didn't cite enough scientists or philosophers who hold that position, but that even if he did, there are other scientists and philosophers who disagree. Therefore, we can't resolve this issue with name citing, but with actual arguments.



No you can't. At least not if you want to make logically valid arguments.

1. Person A says X is true.
2. Person A is an expert in the field of X.
3. Therefore, X is true.

This is not valid.

By the way, you know why I am talking about valid arguments? Because in the very beginning of his opening statement, Craig accused Shook of presenting arguments for naturalism which are not valid. So, he sets up a standard for the arguments to be used in the discussion to be valid, and then he himself presents arguments which are not valid.

If you examine what Craig says about this I think you will find that he distinguishes between strict logic and broad logic, and also that which is metaphysically possible or absurd.

For instance, he gives the phrase "a married bachelor" as an example of something that isn't self-contradictory on strict logic, but is metaphysically impossible and absurd.

As for mathematicians it's very easy to find famous ones who think an actual infinite is absurd - for instance, Hilbert, who devised the illustration of the hotel, which Craig uses frequently.

No doubt his language at times isn't entirely precise, especially given the circumstances of debate and the need to react in real time, but I think he makes a very strong case for the metaphysical impossibility of the actually infinite.

Let me start with your closing remark:
I'm certainly in Madnak's camp here. I don't consider it meaningful to analyse his public debates based on how logically sound they are, since I doubt he's trying to make them logically sound but rather compelling to the popular audience. Nonetheless, the given quote seems quite wrong to me (and hence careless) - I'm not claiming that a simple slip in logic means he loses, nor even that he is necessarily wrong.
What's the difference? I think a married bachelor is self-contradictory on strict logic (in the sense that no object can be both) - the set of married bachelors is the empty set, what's the difference between that and metaphysically impossible?
I don't think Hilbert's hotel establishes that actual infinities are absurd (assuming by absurd you mean 'don't exist'), merely that, were infinite sets of objects to actually exist, they would have different properties than finite sets. Is there anywhere you know where Hilbert (or some other equally prominent mathematician) claimed that actual infinities are absurd?

Well, apparently some Russian guy is editorializing against Cantor, so I guess there's cherry-picking going on from slightly more recently.

I wouldn't say "Craig's intuition" but rather anyone who finds his basic premise to be intellectually plausible.

This is the whole point! Given the structure, he CANNOT do the things you are suggesting that he does without making the presentation longer than the allotted time (or completely boring the audience).

Maybe you didn't think this through very carefully. Supposing that scientists and philosophers disagree about statement X, what are the changes that an "actual argument" for statement X confined to this format would be convincing? Or one that is simple enough for the audience to follow?

There is only a minor change to this statement that feels like it's quoted directly from wikipedia to make it a valid informal argument.

This is a valid argument. And this is the approopriate structure of these types of debates.

The underlying question is whether Craig is talking about formal logical fallacies (which can never be a good argument) or whether he's talking about informal fallacies (in which case Craig is doing that which he is accusing his opponent of doing). I suspect that he's referring to the former, but I doubt that there's any clarification in the video for this.

Though he doesn't say it's absurd, here's a quote from Hilbert:

http://www.muslimphilosophy.com/ma/w...n_Infinity.doc

Craig has so much on this I don't see any need to repeat it here - his case is strong and based on what other mathematicians and philosophers have said. His basic method is to make a plausible case about something and force you to accept it or accept its negation while showing the negation is more implausible than the premise. So if you're comfortable with an actual infinity I guess his argument is ineffective for you.

Let me just say that to be fair, you need to read the other fallacies that I pointed to. Because now it seems like you found certain fallacies that you think aren't really fallacies (which is possible, that's why I put them up for discussion) but you ignored the obvious fallacies. If they are indeed fallacies, my question about Craig (the one in the OP) still needs an answer. Why does he need to resort to apparent logical fallacies in his arguments?

DOUBLE EDIT:

When you find the time, could you get back to me on this?

I am aware of the limitations coming from the format of the debate and I agree that he can't present a 5 page argument for everything he says. But I think you are missing the point I am trying to make about the argument from authority. I am not accusing Craig of not citing his sources, nor am I accusing him of not reciting their work. He is trying to prove something grand here and he needs to use logically valid arguments. When he makes the soundness of his argument dependent on the truth of certain scientists' views, he is shooting himself in the foot, because if those views happen to be falsified and replaced by other theories (as happens in science all the time), his argument goes down the drain as well. But the existence of God is not something we can replace the way we replace a scientific theory. Do you see my point? If God's existence is dependent on the truth of those scientist's views which he is referring to, then as soon as those views change, Craig has to immediately agree that we no longer have evidence for God's existence (since the evidence he presented was based on views that are no longer held - something he was counting on).


Let him present an actual argument and leave it to the audience to decide if it's convincing or not. That is the purpose of a public debate anyway. When you simply say "Scientist X claims Y", you are achieving nothing, because people who are familiar with what X claims already have an opinion about Y, and people who haven't heard of it can only shrug and either trust or distrust Craig's judgment. Nobody is more convinced one way or another.

And as I said, I disagree with this. When you replace "X is true" with "it is reasonable to accept X is true" you are not making it any more valid, you are just making it sound a little more modest. Let me give you a specific example to show why I don't agree with the validity of this argument:

1. Michael Behe says the bacterial flagellum is irreducibly complex.
2. Michael Behe is an expert in the field of biochemistry.
3. Therefore, it is reasonable to accept that the bacterial flagellum is irreducibly complex.

Well, as it turns out, the notion of irreducible complexity is absolute garbage. Was it reasonable to trust Behe?

Let me explain my view a little further. How can we make this argument better?

1. Michael Behe has conducted studies which suggest that the bacterial flagellum is irreducibly complex.
2. The studies have been analyzed and don't have any apparent flaws.
3. Therefore, it is reasonable to accept that the bacterial flagellum is irreducibly complex.

By taking away the focus from the scientist and putting it on his research, now you are given an opportunity to show that premise 2 above is false and therefore the argument is false. If the focus is on the scientist alone, there is no way to falsify the argument, other than the skepticism card of saying "I don't think Behe has any authority in the field of biochemistry". And notice that with the change in the argument I made, you are no longer appealing to authority, but to empirical evidence (which is a lot easier to analyze whether it is supporting or not supporting the thing you're trying to defend).


He is talking about formal fallacies, but so what? Is he somebody who determines the rules of the debate by deciding which fallacies are not to be made and which fallacies are not a problem? Craig's arguments are full of informal fallacies, how is this acceptable?

"On his website he (Dr. Shook) gives only one argument for naturalism, and to my surprise, that argument was logically invalid." ("I, on the other hand, am making many *informal fallacies*, but that's okay."

EDIT: Not to mention that the false choice fallacy is a formal fallacy which Craig also makes.

I don't think one exists - as hilbert says here. That's very different from him saying it's absurd (which you said was very easy to find). Also here he's talking specifically about a continuously dividable continuum which is different from (for example) a hotel with a countably infinite number of rooms or, perhaps more relevantly, a universe with a countably infinite number of stars.

Any comment on the distinction between strict logic and broad logic? Or is that what you were referring to with the term implausible? If a married bachelor is not self contradictory on strict logical grounds but only in 'broad logic', what is an example of something which is self contradictory wrt the former?

I said they think it's absurd. You don't think the hotel illustration is designed to show an absurdity?


"An unmarried married man" would be strictly self-contradictory. I believe Craig is just trying to be as precise as possible when he distinguishes between strict and broad logic. He says "married bachelor" is logically incoherent but not strictly illogical - a very fine point of distinction. He also distinguishes between logically possible and metaphysically or ontologically possible. For instance, it's logically possible that Peter would not deny Christ, but given the circumstances, it was metaphysically impossible for God to create Peter as He did and Peter not deny. Not all logically possible worlds are metaphysically feasible.

I believe he would say, in a scholarly work, that an actual infinite isn't strictly logically contradictory but that if one existed there would be metaphysical impossibilities or absurdities(illustrated by Hilbert's Hotel), so it's actually impossible for one to exist. Which is what I think Hilbert was saying in the quote.

I addressed a couple things which jumped out to me as I skimmed over your numerous posts. If you're looking for a pat on the back for the things that you've rightly identified, you'll be waiting a long time. I don't claim that Craig is perfect in his presentation, but there are some aspects in your presentation that do not seem to be fair-handed. These are the things that I've addressed.

It probably won't be in the next two weeks.

Again, I don't believe that his presentation was meant to be taken as a formal, deductive proof. I don't think anyone in modern philosophy believes that the question of the existence of God is a formal logical conclusion. So again I'll suggest that you may not understand the actual nature of the discussion at hand.

I don't think you understand. All you're doing is pushing the authority off to a different level. In your secondary presentation, there's no reason to refer to Behe.

And this brings it back to a statement derived from general scientific consensus. You've added more strength to the argument by implicitly appealing to multiple scientists instead of a single one. But this is still just an appeal to authority.

An appeal to authority does not guarantee the truth of a particular claim. This is why it is not a formal logical fallacy (which can never be correct). But the non-guaranteed-ness of the claim does not mean that it's wrong to appeal to authority. Again, it's a matter of the type of argument that is being presented (and making sure that the authority speaks authoritatively.)

Because an informal fallacy may not be wrong. It's simply a class of inductive claims. Inductive reasoning is in its very nature not as iron-clad as deductive reasoning. This is why the appeal to authority may turn out to ultimately be wrong. But again, I don't think any contemporary philosophical mind believes that the question of God can be reached on purely deductive reasoning.

Not at all - I think it's designed to show the differences between infinite sets and finite sets. An infinite set can be defined as a set which can be placed in one to one correspondence with a subset of itself - this is a bizarre thing to try to picture because we try and extend our finite concepts to the infinite whereas in fact they are two quite separate things. Infinity is not a really, really big number (which is part of why Craig's "What's infinity minus infinity?" question is so careless - you subtract numbers, you don't subtract infinities).
I'm trying to understand this distinction - I think it's nonexistent. Your example just seems like linguistics to me - Bachelor means unmarried man. So "Married bachelor" and "Married unmarried man" are identical concepts in my mind. I can't see what criteria is ruling the first "not illogical" but the second is self-contradictory. They both purport to refer to something which necessarily doesn't exist - what distinguishes them other than number of words?
I can understand something being logically possible but not metaphysically feasible - it's similar to what I think Hilbert is getting at, in fact. I think an infinitely divisible continuum is logically possible, but not physically possible in our universe (given quantum discreteness and all that jazz..)
I don't think this is what Hilbert was saying at all - as I mentioned, Hilbert was referring to an infinitely divisible continuum which is not what the hotel is modelling anyhow.

Is it inconceivable for the universe (not just the bits we can see) to extend infinitely? With an actual infinite number of stars? I don't see what's impossible about that.

I didn't open this thread to get a pat on the back, nor would I expect a pat on the back on anything from you in particular. I opened it because I wanted to hear what theists and atheists think about William Craig. Since he is smart and knowledgeable enough to know the nature of logical fallacies, why would he resort* to them in his arguments? I don't feel like writing the detailed question again, it's in the OP. And since I have been accused by many people that I am not being precise enough in my claims, I decided to write a longer post pointing to specific fallacies. While it may be true that some of those are not actual fallacies (something you haven't been able to demonstrate so far), the presence of the other fallacies is a sufficient justification for my question about him.

*While his opponents resort to fallacies between very rarely and never.


Again, you are wrong. Watch the debate in two weeks if you don't want to take my word for it, there is nothing more I can do to persuade you. He specifically said that he is going to present formal deductive arguments for the existence of God.


Yes, you don't need to mention Behe. But Craig was using scientists and philosophers to prove his points and you were saying that this is legitimate. I am saying that it is not correct to mention scientists in your arguments, relying on their authority. You are kind of dodging my criticism.

Also, I am not sure that you are talking about the same "argument from authority" as the commonly accepted meaning of the expression. An argument from authority is regarded as an informal fallacy. You said that it is actually fine to use it if the authority is from the same domain as the argument you're trying to defend. But you have simply asserted this, you didn't explain why it's really okay, while I am pointing to the negative sides of using them.



No, that is not what I am doing. Here's the difference:

A- Scientist X (or many scientists) showed that there are no apparent flaws in Behe's studies using reasonable arguments.
B- Scientist X (or many scientists) asserted that there are no apparent flaws in Behe's studies because their intuition suggested there weren't.

Only in the latter case is there an appeal to authority. In all my posts I am talking about the former case, however.

A formally fallacious argument can also have a correct conclusion:

1. Some birds are black.
2. All ravens are birds.
3. Therefore, all ravens are black.

The point isn't whether Craig's arguments can be correct, despite being informal fallacies. The point is that he is not justified to reach that conclusion using informal fallacies. In both formal and informal fallacies we have a conclusion which doesn't logically follow from the premises. Why can't he make arguments (like his opponents) without invoking fallacies?


Should I take it that you don't consider the ontological argument an argument for the existence of God?

OK I've been looking around his site and I can't find anywhere he argues this, although I'm not a member if it's in some 'private section'. He states often that Hilbert's hotel leads to absurdity but I don't see why - the only justification I've found is in trying to apply subtraction to infinite sets. As I mentioned "What's infinity minus infinity?" is unanswerable in exactly the same way that "What's black minus black?" is unanswerable.

I can't see any problem with an actual infinity, nor any reason to think our finitely derived intuitions should help us understand it. (They do a pretty poor job of understanding light, for example). The only mathematicians I can find who seem to agree are (like Hilbert) saying that an actual infinity doesn't exist - which is not enough to establish it as an absurdity.

If he has some written paper on this, I'd appreciate a link (I don't find podcasts useful). So far I haven't seen any reason to think actual infinities are absurd other than "Craig thinks they are" which isn't particularly compelling. (I also don't find this position particularly damning for the Kalam Cosmological Argument, since Big Bang + Quantum Nature of the Universe can probably be extended to 'there is no actual infinite set of past moments'. It just annoys me when people treat infinity so carelessly).

...

Hmm.

OK, Craig was careless. I forgive him given the sheer amount of product he generates at such a high level. If "impossible" or "non-existent" work better than "absurd", fine.

Non-existent is better - impossible is a synonym so that's no good either. "Not observed" is best of all, imo or "probably non-existent".

This post of yours reads kind of irritated to me, and if so I apologise.

I agree with Craig that God exists, I just think "actual infinities are absurd" is unproven and implausible, so I don't think he should base his arguments on that. I was genuinely surprised that you thought Hilbert considered them absurd and think my interpretation of his quote is much more likely to be correct - so that part was curiousity more than anything else. I wasn't taking a cheap shot or anything.

EDIT: If Craig is claiming an actual infinity is absurd (and he repeatedly does, not just in passing) then he is claiming any theory requiring the existence of an actual infinity is necessarily wrong. I see no reason to grant this - if he retreats to "an actual infinity has never been observed" then he can no longer declare such competing theories to be ruled out on logical grounds. If it's about plausibility, which I don't think is a bad way to go in such things, then one can't form a judgement until one is able to understand infinity - something which takes a reasonable amount of mathematical effort to come to grips with. The error in laypeople's thinking about infinities is nearly always based around thinking of infinity as "a number bigger than any other number" and resting a defence of God on this misunderstanding would be a mistake, in my view.

An infinitely divisible continuum is different from an infinite set of discrete entities. (The set of rationals is countably infinite and doesn't form a continuum, the set of reals is uncountably infinite and does. Hilbert's hotel and the postulated infinite number of stars are not a continuum either.)

http://www.michaelhorner.com/article...ist/index.html
If something doesn't "provide a legitimate basis for rational thought" that seems a virtual synonym for "absurd".

Craig's point isn't that the bare concept is absurd in the same way as, for instance, a married bachelor, but that a real "actual infinite" would produce absurdities.

Also note that Hilbert says "the infinite" without distinguishing types of infinite.

Cheers - I'll get it out.