Thursday, March 08, 2007

Arguments and Evidence

The following principle seems plausible:

Arguments and Evidence (AE): Necessarily, for all x and y, if x is a valid argument and y has evidence for each of x's premises, then y has evidence for x's conclusion.

After all, we are concerned with valid arguments because we are concerned with evaluating evidence for and against different claims. Using valid arguments helps us to determine whether we have evidence for their conclusions, and the reason for this seems to be that valid arguments are evidence-preserving: If we have evidence for the premises of a valid argument, then we have evidence for its conclusion.

However, (AE) also seems to have implausible consequences. For instance, suppose that, at time t1, Alan's total evidence supports each of the premises of a modus ponens argument:
1. P
2. If P, then Q.
3. Therefore, Q.
Then, at t2, Alan gains some additional evidence without losing any of his previous evidence and that the evidence he has gained makes it the case that, at t2, his total evidence, while still supporting that if P, then Q, supports the negation of P. In addition, the evidence Alan gains at t2 does not support the negation of Q. Surely such a case is possible. But if (AE) is true, then it follows that, at t2, Alan has evidence in favor of Q, since he still has evidence in favor of P and if P, then Q, since he has not lost any of the evidence he previously had. Making certain other plausible assumptions, it also follows that Alan has just as much evidence in favor of Q at t2 as he had at t1. And this seems implausible. Are we then to reject (AE)?

One suggestion would be that we should reject (AE) and replace it with:

Arguments and Total Evidence (ATE): Necessarily, for all x and y, if x is a valid argument and y's total evidence supports each of x's premises, then y has evidence for x's conclusion.

However, (ATE) cannot do all of the explanatory work that (AE) did. For instance, consider the following argument:
1. The fact that the physical constants allow for the existence of life is more probable given theism than given atheism.
2. If (1), then theism is true.
3. Therefore, theism is true.
I take it that I have some evidence in favor of each of the premises of this argument and that because of this, I have some reason to believe its conclusion. However, I also believe that my total evidence supports the negation of premise (2). So, (ATE) does not help to explain why I have evidence in favor of the conclusion of this argument, whereas (AE) does. Thus, (ATE) cannot do all of the explanatory work (AE) did.

Let me end with some questions: Is (AE) false, as I have suggested? Are the reasons I have given to think that (AE) is false convincing? If (AE) is false, can we replace it with a principle that does not have its implausible consequences and yet still explains my situation with respect to the fine-tuning argument mentioned above and other such situations? More generally, what is the relationship between valid arguments and evidence?

15 Comments:

Blogger Joshua said...

I guess I have an inclination to believe that there is some principle that connects up an individuals evidence for the premises of an argument with that individuals evidence for the conclusion. However, I guess I worry that both (AE) and (ATE) are false. One reason to worry that they are false might have to do with valid arguments that are too complicated for an individual to understand. But, I know that is not a conclusive reason sicne the individual might still have the evidence and fail to be able to ground his belief in the conclusion in the appropriate way.

In any case, those kinds of worries are not really connected to the puzzle that you bring up since, presumably, one might be able to generate the puzzle even with an appropriately restriction on (AE).

Let's call the person who has evidence for (1) but whose total evidence supports the denial of (1) "George". As I mentioned before, you might think that George does have evidence for (3) even though his total evidence doesn't support (1). That might be a bit strange, but, one might argue, it is better to preserve the original principle than to deny that he has evidence for (3).

But, we might strengthen the puzzle as follows. Suppose that George has no evidence for or against (3) other than the evidence that comes by way of the argument (1)-(3). Then it seems that the following might be true.

A. George has some evidence for (1) and for (2)
B. George's total evidence supports the denial of (1).
C. By way of (AE), George has evidence for (3)

And, since he has no other evidence for or against (3),

D. George's total evidence supports (3).


(A)-(D) seem kind of puzzling. I think that the thing to deny is that (D) is true. We can deny that for any S and any P, if S has evidence for P and S has no other evidence for or against P, then S's total evidence supports P. It seems like the case presented might be a viable counterexample to this principle.

12:40 PM  
Blogger Chris Tillman said...

Also the principles would need to be amended to fix for cases in which our evidence/total evidence supports each of the premises, but not their conjunction. Lottery arguments are like this, but there are less exotic examples.

Also, is your question intended to be different from the question of whether there are true closure principles for being justified in believing something (where we're construing this evidentially)?

5:46 PM  
Blogger Joshua said...

I think reg's question is over a principle that is weaker than closer of justification. He is invoking some kind of closer of having some evidence for.

11:11 AM  
Blogger jon said...

I don't know if I'm allowed to comment here, but I will anyway.

I agree with Joshua that you would want to add something to the principles that includes the individual grasping that the argument is valid.

What I don't see is why one should think that Q is supported by one's total body of evidence. It might be true that the extra evidence that comes against P does not count against Q, but it does defeat a piece of the only evidence one has in favor of Q. If one's reasons for P are undercut in such a way, it seems to me that one's total evidence no longer supports Q.

9:29 AM  
Blogger jon said...

Better put: it seems to me that one's total body of evidence supports Q only if one's total body of evidence supports the conjunction of P and if P then Q. Thus in the case described one's total body of evidence does not support Q. This requires endorsing the possibility that one's total body of evidence can fail to support a proposition even though one has some evidence for and no evidence against that proposition, but I think cases like these give some plausibility to that

4:50 AM  
Blogger Joshua said...

Jon,

I take it that you are responding to my revised version of Greg's puzzle (since Greg's original puzzle did not rely on the claim that someone's total evidence supports Q)

You said "it seems to me that one's total body of evidence supports Q only if one's total body of evidence supports the conjunction of P and if P then Q." But this claim cannot be correct (or at least it cannot be a necessary truth about evidential support). This is because one can have evidence that supports Q yet is not of the form 'P and if P, then Q'.

You also said something about defeat. But I am not sure how defeat plays into the picture. Are you suggesting that Greg's original principle is false because it does not take into account the fact that evidence against P is also evidence against Q (or, to put it in defeat terms, evidence against P is a defeater for Q). But this suggests a general principle that is surely false. Here is the general principle:

If (i) S's only evidence for Q is that P and P entails Q is true and (ii) S comes to have evidence against P, then S comes to have evidence against Q.

This priniple is surely false. We often warn our students against disbelieving a conclusion when we have only shown that an argument for a conclusion is no good.

So, in the end, I guess I am just not sure what you are saying in response to the new puzzle. It might be that I am confused because the puzzle was't clearly spelled out. So, let me try to give a particular example that (I hope) will make things more clear.

Here is Greg's principle:
Arguments and Evidence (AE): Necessarily, for all x and y, if x is a valid argument and y has evidence for each of x's premises, then y has evidence for x's conclusion.

Suppose S has evidence for P and for P entails Q. So, by Greg's pinciple, S has evidence for Q. Suppose further that S has no other evidence for or against Q.

Now, S's professor, whom S takes to be an authority, tells him that P is false. S's total evidence does not support that P. But, S still has all the other evidence he once had for P and P entails Q (he just now has more evidence against P). By Greg's principle, S still has evidence for Q. By the supposition that S has no other evidence for or against Q, it follows that S's total evidence supports Q. But now we have generated the puzzle.

You might think that the following claim is false: "By the supposition that S has no other evidence for or against Q, it follows that S's total evidence supports Q.". But, how can that be so? The professor only said that P is false, not that Q is false. Moreover, we might add to the case that the professor says explcitly that he doesn't have any opinion about Q. Then, it seems, that the professor's testimony doesn't change S's evidence for Q.

11:34 AM  
Blogger jon said...

Hi Joshua,

time for me to clarify. With some clarifications about the individual understanding that the deduction is valid, I do want to endorse:
(AE*) Necessarily for all x and y, if x is a valid argument and y has evidence for the conjunction of x's premises, then y has evidence for x's conclusion.

I would also endorse:
(ATE*) Necessarily for all x and y if x is a valid argument and y's total evidence supports the conjuction of x's premises, then y's total evidence supports x's conclusion.

The reason that I do not see a problem with consequence Greg points out regarding (AE). I reject that if x has evidence for p and no evidence against p, then x's total body of evidence supports p. From the case described by Greg it seems clear that one's total body of evidence can fail to support a proposition that only has (direct) evidence that supports it since there is also indirect or defeating evidence possessed by x that changes the way p is supported by x's evidence without there being new evidence that counts directly against p.

6:24 AM  
Blogger rock* said...

I e-mailed Rich yesterday concerning my puzzle (or, rather, Joshua's strengthened version of it) and, after an e-mail exchange, I believe that I'm able to accurately summarize his response. Regarding the (Spencerian modification of the) particular case presented concerning Alan, Rich believes that (i) at t1, Alan has some evidence for Q and no evidence against Q, (ii) at t1, Alan's total evidence supports Q, (iii) at t2, Alan has some evidence for Q and no evidence against Q, and (iv) at t2, Alan's total evidence does not support Q.

More generally, Rich accepts (AE). (Actually, he accepts a modification of (AE) that accommodates the worries concerning complex arguments that Joshua mentions. However, so far as the puzzle goes this modification is irrelevant.) Thus, he believes that Alan has some evidence for Q at t2, since Alan has some evidence for P at t2 and Alan has some evidence for If P, then Q at t2. And he believes that, as specified in the case, Alan has no evidence against Q at t2. So, Rich believes that at t2, Alan has some evidence for Q and no evidence against Q. However, he denies that at t2, Alan's total evidence supports Q. This is because he denies the following principle:

Total Evidence (TE): Necessarily, for all x and propositions p, if x has evidence for p and x has no evidence against p, then x's total evidence supports p.

(I take it that Joshua was recommending denying this principle in the final paragraph of his first comment on the post, although he may have changed his mind by the time he wrote the last paragraph of his most recent comment on the post.)

This is an interesting situation. At this point, I'm not sure what to say about (TE). I'm pretty sure that (AE) is true, although I still think it's somewhat odd to think that Alan has some evidence for Q at t2. (In fact, I should mention that I'm willing to endorse (AE) with no modifications. I think that considerations about complex arguments can best be handled by saying something about properly grounding belief, as Joshua mentions at the end of the first paragraph of his first comment.) However, I find it quite a bit stranger to think that at t2, Alan's total evidence supports Q. So, perhaps the think for me to do is to deny (TE). But that seems pretty counterintuitive to me as well. (I said the following to Mary this morning: "Rich thinks that someone can have evidence in favor of something and no evidence against it without their total evidence being in favor of it." And she said: "That's crazy!")

So, I guess I'm in the following situation:
(AE): Pretty intuitive (modulo worries about complex arguments that can be accommodated)
(TE): Even more intuitive, I think
That Alan has some evidence for Q at t2: Somewhat counterintuitive
That Alan's total evidence supports Q at t2: Very counterintuitive

6:34 AM  
Blogger Joshua said...

This comment has been removed by the author.

7:47 AM  
Blogger Joshua said...

Rock*,

It is true that I think the best thing to do is to reject TE. But I think it is really wierd. I intended to highlight the weirdness of rejecting it by giving a case where the evidence against P is testimonial evidence where the testifier explicitly says that he has no opinion about Q. It really seems that in the case given in my last post, the individual S's total evidence still supports Q (given AE). So, although I am inclined to think rejecting TE might be the best thing to do, I am worried that I am mistaken.

Jon,

You seem to be giving us a theory on which TE fails; a theory that involves defeat in some way. But I don't understand the view that you are proposing.

You say:
"From the case described by Greg it seems clear that one's total body of evidence can fail to support a proposition that only has (direct) evidence that supports it since there is also indirect or defeating evidence possessed by x that changes the way p is supported by x's evidence without there being new evidence that counts directly against p."

In this excerpt you invoke the notions of (i) direct evidence and (ii) indirect evidence. You also invoke the notion of (iii) defeat and the notion of (iv) a proposition being supported in a way. I don't understand these notions. I have some vague ideas about them, but insofar as I understand them, I fail to see how they have bearing on this case.

Here is one example of how I fail to understand the import of these notions. I have been told that defeat comes in two varieties, the rebutting variety and the undercutting variety. The rebutting variety just seems to be evidence against a proposition. But I don't see how we have any evidence agaisnt Q. The undercutting variety is supposed to give reason to beleive some supported proposition is false even though the supporting propositions are all true. But we normally don't come across this kind of defeat when the reasoning involved is valid and we had no such defeater in the case described. So, how does defeat play a role in explaining why TE is false?

I may be mistaken about defeat. it has been several years since I read and talked about it and I am trying to remember what I supposedly learned about this topic.

Everyone,

I am beginning to wonder about the connection between this case and cases of justified true belief without knowledge (sometimes called Gettier cases).

If we add the following (perhaps dubious) principle, then we will be able to generate a case of justified true belief without knowledge by slightly modifying the example:

(JTE) S is justified in believing P iff S's total evidence supports P and believes P on the basis of that evidence.

Suppose that our notorious S, (AKA George and Alan) believes Q on the basis of P and if P, then Q. suppose also that Q is true. Then, it seems, we have a case where S's total evidence supports Q and he believes Q on the basis of that evidence. So, by JTE, S is justified in believing Q. But Q is also true. So, S has a justified true belief. But, there is somethign wrong with S's rationality. Although he believes Q partially on the basis of P, his total evidence does not support P. (we might even add that P is false, but I am not sure if that will add or detract fromt he case). So, it seems that S does not know Q. So, it seems that we have a case of justified true belief without knowledge. We might take this to show that JTE is false. But, we might instead take it to bolster the case against TE.

I am not sure what to say. I am sure that there is something epistemically wrong in this case and that this mistake prevents S's knowling Q. But I am not sure what is wrong. Maybe JTE is false and he is not justified in his belief. or maybe TE is false and he is not justified in his belief or maybe some other principle is false and although he is justified, he does not know. What I am interested in is how each of these principles interact and what various combinations of views will say about the natures of knowledge, justification and evidential support.

7:53 AM  
Blogger rock* said...

Everyone,

I should mention that in my e-mail correspondence with Rich he too appealed to the claim that at t2, Alan has an undercutting defeater for Q (as some have been inclined to do in the comments here). Unfortunately, that claim doesn't really help me understand the situation here, since I'm not sure I understand the notion of a defeater nor of an undercutting defeater. (Thus, I sympathize with what Joshua has to say to Jon on this score.) What I would like is a way to fill in the blank in the following principles:

(D) Necessarily, for all x and y, x is a defeater for y iff ____________.

(Or perhaps:
(D*) Necessarily, for all x, y and z, x is an undercutting defeater of y for z iff __________.
Or maybe something even more complicated.)

(D*) Necessarily, for all x and y, x is an undercutting defeater for y iff __________.

(Or, perhaps, a similar but more complicated principle.)

It is worth noting that the way in which Rich responds to the puzzle commits him to denying the claim that two people cannot differ with respect to whether their total evidence supports a proposition without differing with respect to the evidence they have for and against that proposition. (Which claim could probably be formulated as a more precise supervenience thesis.) Again, the denial of this claim seems puzzling to me, but perhaps denying it affords the best response to the puzzle.

9:52 AM  
Blogger jon said...

Joshua,
I think that those notions that I invoke in that paragraph give us some tools to think that rejecting (TE) is not so implausible after all. Let's say the following:

p is direct evidence for q (for S) iff p evidentially supports q all by itself.

p is indirect evidence for q (for S) iff (i)p is not direct evidence for q,(ii) p is direct evidence for r, (iii) r is direct evidence for q.

It makes sense to me that there is indirect evidence. Defeaters are examples of indirect evidence.

Necessarily, (ignoring the individual) for all propositions x, y and z, x is an undercutting defeater of y regarding z iff y is evidence for z, x is not evidence for or against z, and (y&x) is not evidence for z.

Necessarily, (ignoring the individual) for all propositions x, y and z, x is a rebutting defeater of y regarding z iff y is evidence for z, x is evidence for not-z, and (x&y) is evidence for not-z.

If these notions make sense, and they seem to, then it makes sense that one's total body of evidence can fail to support a proposition which has some evidence in favor of it and no evidence against it. I would say that this is because indirect evidence affects what one's total body of evidence supports. Put differently, considerations that affect the evidential standing of a piece of evidence used to support another proposition, affect whether the total body of evidence supports the conclusion even though they do not concern the conclusion.

Greg,
I do not see how Rich's response denies the supervenience claim you cite.

1:54 PM  
Blogger Joshua said...

Jon,

I have some worries about your biconditionals. I will voice just a couple, though, because my main concern is with the claims involving defeaters. Here is a worry that I have concerning the first two biconditions. I am not sure I understand what it means for one proposition to evidentially support another all by itself. In particular, I am not sure I undersntand how it can be that P support Q all by itself and Q supports R all by itself but it is not the case that P supports R all by itself. However, this kind of situation seems to be necessary for any proposition to be indirectly supported.

Now for my real concerns. You have given two notions of defeat. The notion of an undercutting defeater is the only one that is of concern though. I take it that the suggested response to the original puzzle is as follows:

S has evidence for P and for P entials Q and thus has evidence for Q. However, although S has no other evidence for or against Q, her total evidence does not support Q because the professor's testimony is an undercutting defeater of Q regarding P (or is it the conjunction P and P entails Q? I'm really not sure).

So, something like that is the proposal. I have two questions. First, what reason do we have to believe that there are undercutting defeaters? I have one word of warning about concerning answers to this question. I have been given some cases in conversation, but it is not clear to me that these cases aren't really cases of rebutting defeaters. So, my warning is that I might we worried about the cases that are brought to my attention.

My second question is this: Why believe that the situation involving S and her professor is a situation that involves an undercutting defeater? Here is one initial reason to believe that it is not. The putative cases of undercutting defeat that have been drawn to my attention are cases where the defeating proposition provides reason to believe that the evidence for the defeated proposition is not good evidence. But nothing like that is involved in this case. The professor did not say that Modus Ponens is invalid or that the conditional (if P, then Q) is false. So it looks like the situation does not fit putative pardigm cases of undercutting defeat.

8:35 PM  
Blogger jon said...

Joshua,

By 'evidentially support all by itself' I mean that p has that relation to q iff: if S's total body of evidence included p and p alone then S's total body of evidence would support q.

I don't have any additional reasons to believe in undercutting defeaters, but the traditional cases are very compelling to me. What do you not like about them? I would just add that most people find them plausible, and if so then most should find a rejection of TE is plausible.

With regard to the case at hand, some might distinguish between having one's evidence undercut and having the connection between one's evidence and what it supports undercut. I take the evidence to be the conjunction of the data and its connection to what it supports. In the prof. case this gets undercut -- G no longer has reason to believe that conjunction since he no longer has reason to believe P. Nothing the prof says bears directly on Q, so I can't see how it could be considered a rebutting defeater.

1:56 PM  
Blogger Joshua said...

Jon,

I will accept that George's evidence is a concjunction (P & (P->Q)) and accept a revision on Greg's original principle according to which if someone has evidence for the conjunction (p & (p->Q)) then he has evidence for Q. Moreover, I will accept taht the professor's testimony makes it such that George's total evidence no longer supports that conjunction. However, George still has evidence for the conjunction. It just isn't true that when your total evidence fails to support something, then you have absolutely no evidence for it. Moreoever, it is part of the case that although George's total evidence fails to support the conjunction, he still has some evidence for the conjunction.

Now you said "G no longer has reason to believe that conjunction since he no longer has reason to believe P". Given what I just said above, it is clear that the clause whcih follows the word 'since' in the above quoted passage is false and hence, the entire statement is false. So, I am still left wondering how it is that the professor's testimony if an undercutting defeater for Q.

I'll respond to the rest of you post later. Right now I have to get back to grading.

Joshua

1:57 PM  

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