Thanks for writing about this. I had noticed this, too, and am glad to see someone address this.

What is even more surprising is that Craig has made this sort of daft mistake before *and has had it pointed out to him*.

In his 2004 written debate with Walter Sinnott-Armstrong [available online as an ebook], Sinnott-Armstrong makes the following point:

"Craig says, “both of the premises of the first argument thus seem more

plausible than their denials. Hence, it is plausible that a transcendent

Creator of the universe exists.” This does not follow. Compare

this argument: When I pick a card from a standard deck without looking,

(1) it is not a spade, (2) it is not a heart, (3) it is not a diamond,

(4) it is not a club, so (5) it is not any suit. The conclusion, (5), is obviously

false, even though each premise taken individually has a probability

of 3/4, so each premise is more plausible than its denial, which

has a probability of 1/4. Analogously, Craig’s conclusion might be implausible,

even if each of his premises taken individually is more plausible

than its denial. Small doubts about each premise can accumulate

into large doubts about the conclusion. Anyway, no such subtlety

is needed here, where large doubts about each premise accumulate

into even larger doubts about Craig’s conclusion."

Tim McGrew and Calum Miller have both criticized Craig on this point in peer reviewed papers. https://appearedtoblogly.fi...

AND

https://appearedtoblogly.fi...

The authors incorrectly state that the following is a valid argument:

1. When I next roll this fair, six-sided die, I will roll a 1, 2, 3, or 4.
2. When I next roll this fair, six-sided die, I will roll a 3, 4, 5, or 6.
Therefore, 3. When I next roll this fair, six-sided die, I will roll a 3 or a 4.

The argument is logically invalid since the premises could be true *and* the conclusion could be false. So it is not clear how this could applicable to the point Craig made.

The argument is valid. It works like this:

1. ((1 v 2) v (3 v 4)) ^ ~(5 v 6) (Premise)
2. ((3 v 4) v (5 v 6)) ^ ~(1 v 2) (Premise)
C. (3 v 4)

If premise 1 is true then neither a 5 nor a 6 will be rolled, but one of 1 - 4 will be. If premise 2 is true then neither a 1 nor a 2 will be rolled, but one of 3 - 6 will be. If both premises 1 and 2 are true then only a 3 or a 4 can be rolled.

To show it's valid, use the methods I told you about earlier.

Let's use the following table:

A = 1, B = 2, C = 3, D = 4, E = 5, F = 6

1. ((A ∨ B) ∨ (C ∨ D)) & ~(E ∨ F)
2. ((C ∨ D) ∨ (E ∨ F)) & ~(A ∨ B)
3. ((A ∨ B) ∨ (C ∨ D)) (Conjunction elimination from 1)
4. ~(A ∨ B) (Conjunction elimination from 2)
5. (C ∨ D)

To see that 5 follows, 3 - 5 takes this form:

1. P v Q
2. ~Q
3. P

Thank you for the links to these articles. I will look them over when I get the chance.

In your sample argument, premises 1, 2, 3, and 4 (and the tacit premise that this is a standard deck - which implies a total of four suits) *cannot* all be true. So your sample argument is logically invalid and therefore not analogous to Craig's.

Joseph, you are wrong that the argument (1)-(5) is logically invalid. Validity pertains to whether a conclusion can be derived from the premises using valid rules of inference. While I concede that some (fairly uncontroversial) additional assumptions are needed, which you allude to, it is clear that (5) can be derived from (1)-(4), along with those assumptions, using valid rules of inference.

You note that the premises (1)-(4) are mutually incompatible. Correct, but so what? The purpose of the argument was to refute Craig's implication that all that is needed for an argument to be a good one is that it is logically valid and has premises that are more plausible than their negations. Perhaps you want to add an extra condition to Craig's list. You might add the condition that the premises must be mutually compatible.

In that case, I have another argument for you.

In a standard 1000 ticket lottery, where the result is yet to be determined:

(1) Ticket 1 will not win.
(2) Ticket 2 will not win.
(3) Ticket 3 will not win.
.
.
.
(n) Ticket n will not win. [For every integer value of n between 1 and 999 inclusive]
.
.
.
(999) Ticket 999 will not win.

(1000) Therefore, ticket 1000 WILL win.

Note that this argument is logically valid, that each of the premises (1)-(999) is far more plausible than its negation, that the premises (1)-(999) might all be true, and yet the conclusion is almost certainly false. This seems clearly in violation of what Craig has been telling us about the conditions for an argument to be a good one, even with the extra condition I added, inspired by your post.

Thank you for this helpful counterexample!

Re your 1000 ticket sample argument - Your example seems ambiguous so I'm adding some premises in the interest of clarity (feel free to suggest alternatives if this is not what you were hinting at):
(-2) a ticket will be selected to win
(-1) there are solely 1000 uniquely-numbered tickets
(0) each ticket is assigned one integer between 1 and 1000 inclusive

Then premises (-2), (-1), (0), (1), (2), (3),…, (999) imply the truth of conclusion 1000.
Not sure how your example is related to what Craig said :-)

"Then premises (-2), (-1), (0), (1), (2), (3),…, (999) imply the truth of conclusion 1000."

Yes, I noted that the premises imply the conclusion when I said the argument is logically valid (i.e. it is impossible for the premises to be true and the conclusion false).

Note that the argument (1)-(1000) is logically valid, that each of the premises (1)-(999) is far more plausible than its negation, that the premises (1)-(999) might all be true, and yet the conclusion is almost certainly false. This seems clearly in violation of what Craig has been telling us about the conditions for an argument to be a good one.

Re "and yet the conclusion is almost certainly false"

If premises (-2), (-1), (0), (1), (2), (3),…, (999) are true, then they imply that the conclusion is true. More to the point, how is your example related to what Craig said?

Joseph wrote:
"If premises (-2), (-1), (0), (1), (2), (3),…, (999) are true, then they imply that the conclusion is true."

I have said that if the premises were true the conclusion would have to be true. Why repeat what I have said if you agree with it?

Joseph wrote:
"More to the point, how is your example related to what Craig said"

Craig has implied that the following two conditions are jointly sufficient for an argument to be a good one:

(a) The argument is valid (i.e. if ALL the premises were true TOGETHER then the conclusion would have to be true).
(b) Every premise is individually more likely than its negation.

The lottery ticket argument is a counterexample to that, since it meets both of those conditions but is a very bad argument. The argument is logically valid and every premise is more likely than its negation, yet the argument provides extremely poor support for its conclusion, since the conclusion is very probably false. The winning ticket is probably NOT ticket number 1000, but is much more likely to be a ticket with a different number on it.

Re "The lottery ticket argument is a counterexample to that, since it meets both of those conditions but is a very bad argument" - Why is it a bad argument? The argument seems sound. Recall that I added these premises:

(0) each ticket is assigned one integer between 1 and 1000 inclusive
(-1) there are solely 1000 uniquely-numbered tickets
(-2) a ticket will be selected to win

Joseph wrote:
"Why is it a bad argument?"

Because it is utterly absurd to suppose it is possible to predict the likely winning ticket of a lottery without knowing anything about the lottery except the number of tickets in it. The winning ticket is probably NOT ticket number 1000, but is much more likely to be a ticket with a different number on it.

I have another question for you. Is the following argument a good one (regarding the same lottery)?:

(1) Ticket 1000 will not win.
(2) Ticket 999 will not win.
(3) Ticket 998 will not win.
.
.
.
(n) Ticket 1001-n will not win. [For every integer value of n between 1 and 999 inclusive]
.
.
.
(999) Ticket 2 will not win.
(1000) Therefore, ticket 1 WILL win.

If this second argument is a bad one, why is that?
If you want to tell me both arguments are good ones, then you seem to have committed yourself to the view that ticket 1 will probably win AND ticket 1000 will probably win. They can't BOTH probably be the winning ticket!

Validity can be tricky, if not downright not intuitive when you see how it works in many different logics.

Here is an example to back up Ryan M's point:

1. P
2. not P
Therefore:
3. P AND not P

(3) is clearly implied by (1) and (2).

Or if you prefer a specific argument in English:

1a. Grass is green.
2a. It is not the case that grass is green.
Therefore:
3a. Grass is green AND it is not the case that grass is green.

(3a) is clearly implied by (1a) and (2a).

To affirm both 1 & 2 would violate the law of non-contradiction. Someone who advocates that the law of non-contradiction is false presumes that that their position is true. But if their position is true, then on their view it is false. It is not rational to adhere to a self-refuting position.

My response will be a long one, mostly as a guide to anyone thinking about validity (In propositional logic)

It would be a violation of the LNC if both 1 and 2 are true. However validity is defined in logic in such a way that makes Bradley's argument valid. There are different means of testing for validity in logic. The crucial rule is that an argument is valid just in case there is no truth value assignment where the set of premises are all true but the conclusion is false. Or in another way, the set of premises truth functionally entails the conclusion. A set of sentences truth functionally entails a proposition just in case there is no truth value assignment where all the members of the set are true and the conclusion is false. The symbol for truth functional entailment is often "⊨". So, if a set T truth functionally entails a proposition P then we write that "T ⊨ P". When we test for validity we are testing to see if the set of premises truth functionally entails the conclusion. Here are some means of doing that:

Method 1:

[Create a material conditional out of the premises and the conclusion]

A material conditional is any proposition of the form "P -> Q", which is that P implies Q. In such a statement, P would be called the antecedent and Q is the consequent. Material conditionals are only false when the antecedent is true but the consequent is false.

1. P
2. Q
C. P & Q

We can form a material conditional of the above argument like this: (P & Q) -> (P & Q)

Since material conditionals are only false when the antecedent is true while the consequent is false, it should be clear how this related to truth functional entailment. If the set of premises imply the conclusion then there is no truth value assignment we can give the antecedent that would have the consequent false, so the above argument is logically valid since its set of premises truth functionally entails the conclusion. In the case of Bradley's argument, it would be this:

(P & ~P) -> (P & ~P)

Since the antecedent can never be true there cannot be a truth value assignment where the antecedent is true and the consequent false so the argument is valid.

Method 2:

[Check to see if the conclusion is derivable from the premises]

Example

1. P
2. P -> Q
3. Q -> R
C. R

We can derive Q from the premises in multiple ways. A simple way to derive R is to have inference rules that allow for short cuts, but we can show a long version. One useful inference rule is negation introduction. The rule works by assuming the negation of what you want to prove and show that a contradiction follows from it.

1. P
2. P -> Q
3. Q -> R
4. ~R (Assume)
5. P (Reiteration of 1)
6. P -> Q (Reiteration of 2)
7. Q -> R (Reiteration of 3)
8. Q (Conditional elimination from 5 and 6)
9. R (Conditional elimination from 7 and 8)
10. ~~R (Negation introduction from 4 through 9)
11. R

Since R is derivable from the premises by showing a contradiction follows from the assumption of ~R and the set of premises, the argument is valid. Negation Introduction provides an easy means for showing why arguments with inconsistent premises are always valid.

Example:

1. P
2. ~P
C. R

The argument is valid since R can be derived from 1 and 2 by using negation introduction.

1. P
2. ~P
3. ~R (Assume)
4. P (Reiteration of 1)
5. ~P (Reiteration of 2)
6. ~~R (Negation Introduction from 3 through 5)
7. R

I am assuming that the inference rules used are sound and complete. Any argument where its conclusion is derivable from its set of premises by means of inference rules which are both sound and complete is a valid argument.

Useful links:

https://en.wikipedia.org/wi...
https://en.wikipedia.org/wi...

Actually, since not all the premises can be true, the argument must be valid! Every argument is valid when it is "impossible" for the premises to all be true and the conclusion false. This implies that every argument with either a contradictory premise or inconsistent set of premises must be valid.

I remember reading that long ago. Did Craig happen to reply to Sinnott-Armstrong on this point?

Jeff,

In the following section in response, Craig wrote:

"Since these arguments are logically valid, the only question is whether their premises are more plausibly true than their negations."

So far as I can see, this is all Craig wrote that addressed the issue Sinnott-Armstrong brought up. Craig ignored or missed Sinnott-Armstrong's point and simply repeated the same mistake. So Craig has now made the same very basic mistake at least three times in published works.

If you Google "craig armstrong debate" you can read the whole thing.

John -

Thank you for the BEAUTIFUL counterexample argument. I did not actually give an example argument that is contrary to Craig's general principle. I just described a general category of arguments for which Craig's general principle would fail.

Each of the premises of your counterexample argument are clearly and undeniably more probable than not. In fact, the probability of each premise is clearly .75. The deductive validity of your counterexample argument is also clear and undeniable. Finally, it is clear and undeniable that the conclusion has a probability of less than .5. In fact, the probability of the concusion is clearly and undeniably 0!

Not only have you provided a clear and simple and undeniable counterexample to Craig's principle, but your counterexample shows that there is a third way, besides the two I mention in my post, that Craig's principle can FAIL: the premises of a valid deductive argument can be such that there are dependencies between the truth of those premises. It is because of the dependencies between the premises of your argument that the probability of the conclusion is less than .5.

My abstract description of a category of arguments on which Craig's principle would fail specifies that there are no dependencies between the truth of the premises. Craig's principle fails on the category of arguments that I describe because of the multiplication rule of probability. In most standard valid deductive argument forms all of the premises must be true in order for the argument to work. Thus it is the truth of the CONJUNCTION of the premises that matters and that determines the probability conferred on the conclusion. The multiplication rule thus generally applies even when there is no dependency between the premises.

So, you have helped to show that it should be OBVIOUS that Craig's principle is false, given that there are at least three different ways that it can FAIL.

where do you find the law that says one can only believe that which is given x% probability by your method? is that carved into mount Rushmore? where in the universe do they keep the laws govern what people can believe?
...

The point at which a probability is sufficient to warrant or justify a belief depends on the question at issue, the context, and the values of the person who is examining the question.

William James pointed out that some people may be more interested in obtaining truth and less worrried about believing in falsehoods than others, while others may be more concerned about avoiding false beliefs than in discovering new truths. (Our relationship to truth and falsehood is a bit like our relationship to risks and opportunities; there is room for personal preference and values to play a role here).

So, there is no particular quantity of probability that will be reasonable to use as a threshold for all questions, all contexts, and all people.

But there are good reasons for the use of probability and for specifying a threshold probability for reasonable belief.

First, quantification of probability allows for greater precision and clarity. To say that a claim is "probable" or "very probable" is somewhat vague. To say that the probability of a claim is .7 is more precise and less vague.

Second, once a probability is quantified, we can make use of mathematics and logic to help reason accurately and correctly about the implications of that judgment.

Third, people have a natural tendency to make a variety of mistakes in reasoning about evidence and probability, and the use of mathematics and logic in probability calculations can help people to avoid many common natural tendencies to reason badly.

Fourth, reasoning that makes use of probability estimates and calculations allows for analysis of thinking into clear assumptions and clear logic (as with arguments in general).

My logic might be perfectly acceptable to you, but my initial probability estimates for basic claims/assumptions might not be acceptable to you. By putting forward clear and specific assumptions/premises (including an estimated probability) I make it possible for you to see my thinking more clearly and determine whether our disagreement (if we disagree) is based primarily on my basic assumptions/premises or on my logic/calculations, or if we disagree on both assumptions and on the logic.

That's a good explanation Bradly. Thanks. I do agree that using probability is useful at some junctures, and might be possible in some contexts. But that still leaves the problem of how to make it work. you say:"But there are good reasons for the use of probability and for specifying a threshold probability for reasonable belief." and you give four reasons. I don't dispute them, they are good reasons.
...
That still doesn't tell me, (1) How to avoid bias in setting the prior. (2) where you get new info about God from.
...
To me probability can only be used when you have an empirical issue. We don't have much empirical evidence. One context in which it can be used is in demonstrating the"co-determinate of God, For example in empirical study of mystical experience where probability can be assessed in relation to such control mechanisms as the M scale.
...
http://metacrock.blogspot.c...
;;;

I think the question of what one *should* believe is very hard to pin down, and ultimately there are no hard rules. It will depend on what the person values most. Truth vs falsehood? Consequences of beliefs? Comfort? There is no universal law of belief-justification on these matters.

I think, though, that in order to say a belief is held rationally, it must at the very least be held according to a standard the person applies consistently on all subject. It would be hypocritical to have different standards of belief-justification for different questions and subjects.

For example, both theists and atheists alike often accuse each other of holding inconsistent standards.

Theists typically accuse atheists of having a too high standard with respect to god, that they do not apply consistently in other areas of their life. They say that atheists believe in many other things on much weaker evidence than the evidence for god.

And atheists typically accuse theists of having a too low standard with respect to belief in god, such that they believe on evidence that is much weaker than what they would require to believe in other propositions, such as UFO's, other gods etc. etc.

I don't presume to tell people what they should believe. I tell them why my beliefs are justified.
...
"and atheists typically accuse theists of having a too low standard with respect to belief in god, such that they believe on evidence that is much weaker than what they would require to believe in other propositions, such as UFO's, other gods etc. etc."
...
yes so weak it's only backed 200 empirical studies in academic journals.

I'm sorry that I bothered answering your question. You apparently took this as an attack on theism rather than an example of what I meant.

Regardless, what is it you think is backed by 200 empirical studies?

I* don't mind attacks, (as attacks go had it been one) it would not have been uncivil, I just thought it was an opening to mention my book.

I do not have the slightest clue what you are asking me.

You ask about my "method" and a "law" (neither of which has a clear referent) but I never mentioned a method or a law. I merely noted that Craig had made a similar mistake before and quoted something by Walter Sinnott-Armstrong which illustrates this.

I suggest researching "ordinary Speech. Consult linguistics
Spock. btw Bowen had trouble understanding me, ask him
...
why should we assume that 51% is not sufficient for belief?

Joe, other people here have spoken highly of you before and scolded me for saying I think you are having problems related to comprehension. I've been looking for reasons to doubt my initial thoughts given their response to me but you're not making it easy.

I do think conversations would be much more fruitful if Joe spent more time on polishing the comments he does make even if it meant him posting less comments overall. I often find myself extremely puzzled at what he's responding to and what point he's making. Perhaps that's my own failing, but if so, others seem to have the same failing, so if Joe could be more explicit in his arguments, I think it would benefit us all.

what I said above goes. I've been conditioned over a long time of relating to atheists in a certain way and to expect then to radish a certain attitude., this place is a breath o fresh air really. Si I do apologize for my anger and Let's just agree to move on. Clean slate?

Fair enough.

You aren't alone.

I know I have responded in anger I apologize, I include you in that "let's move on from here."

It depends how you interpret his claim. Suppose the argument in question has the form.
1. A
2. B = "if A then C"
3. Therefore C

Then, the probability of C is (by the law of total probability)
p(C) = p(C|AB) p(AB) + p(C|~(AB)) p(~(AB))

Given the validity of deductive logic, p(C|AB) = 1. Note that this does not imply that p(C|~(AB)) = 0. C could still be true, even if A or B or both are false. So, if p(AB) = 0.51, then p(C) > 0.51. As claimed.

However, if we set p(AB) = p(A|B) p(B), and postulate that p(A|B) = p(B) = 0.51, then p(AB) = 0.2601, and p(C) > 0.2601.

So it depends if "the premises are only 51% probable" means "the conjunction of the premises is 51% probable" [p(AB) = 0.51] or "each premise is 51% probable" [p(A|B) = p(B) = 0.51]. The quotes above seem (to me, at least) ambiguous. It might be worth sending a question of the week to his website.

I can't remember the sources off the top of my head, but I recall seeing Craig make it clear in more than one place that he is only thinks that the premises must individually be more than 50% probable, not the conjunction of the premises.

I think you are right that the "at least 51% probable" claim about the conclusion is based on the fact that there is usually some chance that the conclusion could be true even if the premises of a valid argument for the conclusion were false. In other words, the truth of the premises of a valid argument are usually NOT necessary conditions for the truth of the conclusion.

Your alternative interpretation that Craig was speaking of the probability of the CONJUNCTION of the premises of a valid deductive argument being .51 would get him around my objections. However, this interpretation seems implausible because Craig is talking about belief in specific premises, not about belief in a conjunction of premises: "its premises must be more probable than their opposites." I think it is clear that this criterion for evaluating arguments means this:

EACH premise of the argument must be more probable than not.

The point is to ensure that it is reasonable or rational to believe each of the premises of the argument in question. This criterion is compatible with there being muliple premises in a valid deductive argument argument EACH of which has a probability of .51, in which case it would usually be the case that the CONJUNCTION of those premises would have a probability of LESS THAN .51.

"I think you are right that the "at least 51% probable" claim about the conclusion is based on the fact that there is usually some chance that the conclusion could be true even if the premises of a valid argument for the conclusion were false. In other words, the truth of the premises of a valid argument are usually NOT necessary conditions for the truth of the conclusion."
...
Don't you think atheists and theists will always have different ideas about what constitutes a true premise in relation to God belief?

"Don't you think atheists and theists will always have different ideas
about what constitutes a true premise in relation to God belief?"

Not necessarily. After all, many atheistic arguments depend on propositions about God that theists accept. The purpose of the arguments, more or less, is to show the theist either has an inconsistent set of beliefs or that the theist has a set of beliefs which does not fit well together (i.e. Some beliefs make others very improbable).

of course agreement with some premises don't prove anything, aIl agree with all you should believe the argument, but it[s always a matter of how look at it as to weather or not you accept. I'm talking about "the real arguments" such as Hartshorne's modal. not just BS like creationism or something. Or TAg now there's a crap argument in my opinion.

With respect to those sorts of arguments, there might always be disagreement. Atheists and theists seem to have disagreements about terms such as "Greatness" and "Perfection" that are required for modal arguments for theism. I don't know if the disagreements will necessarily occur though since atheists might doubt premises such as whether God's existence is logically possible of even epistemically possible.

Yes I think a lot just depends upon perspective. There is some truth to the atheist mantra "there's no proof" and also there is a basis for Pascales reasons reason cannot know. I don't want to sound like a flake who is trying to avoid the use of logic. I have to admit my belief is largely grounded in experience but my book proves that;s not as subjective and unverifiable as one might think.
...
In the final analysis I think belief in God will invariably entail some some form of the intuitive weather it be mystical experience or Hartshorne;s deep empiricism.

It looks to me like there are supporting considerations for both sides of the ambiguity. On one hand, Craig was addressing primarily the charge that the probability of his conclusions, (or at one point of the probability of God's existence) being only 51% and also the complaint that a probability of 51% is insufficient justification for a belief. These points seem to support the conjunction-of-premises reading.

On the other hand, though, there is Craig's claim that "in order for a deductive argument to be a good one, it must be logically valid and its premises must be more probable than their opposites." The language here really does seem to imply that each premise must be assessed as being at least 51% more probable than its opposite, and if this turns out to be true, then the argument is a good one (assuming it is deductively valid). And in this case, Bradley's points are fair.

Eric Sotnak said:

On one hand, Craig was addressing primarily the charge that the probability of his conclusions, (or at one point of the probability of God's existence) being only 51% and also the complaint that a probability of 51% is insufficient justification for a belief. These points seem to support the conjunction-of-premises reading.
===========
Response:

I don't see how these points support the conjunction-of-premises interpretation. Can you explain this a bit more?

I was trying to be maximally charitable. However, John's posts showing Craig making the same mistake with Sinnott-Armstrong now leave me inclined to place the burden on Craig to show that a more charitable view is appropriate.

Very interesting and important.

But what about this rejoinder. Say we assess the premises not on a probability standard but on a convincing standard of balance of probabilities. In this case when we are convinced a premise as more probable than not, we assign it the value as "true" or "reasonable to believe as true". We then look at whether the argument is valid and if it is we take it as true, or reasonable to believe true.

I know you will say this is unreasonable. And I think you are correct, but I also think. Y method is how pretty much all legal cases are argued. If they weren't, I doubt many could ever succeed. I practice litigation and this issue has never arisen. I do believe courts have rejected the philosophical approach to probability you outline above. Which I agree is more accurate.

But if we are being this strict, would not it be paralyzing? Would we not have to start with assigning probability to solipsism being false, and then make similar probability assessments with every conclusion and observation, we compound the improbability to the point that I expect none of our conclusions will be more probable than not?