h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hi Bx4,

Ok, I look forward to it.

ttfn

Apologies for continuing delay had to fly to Scotland to see an old friend who is terminally ill.

bs

Really sorry to hear that.

hi psi

Apologies for the delay

>>In the meantime, let's try a different tack. Do you agree that if the argument is circular then this is structural rather than dependent on the semantics or meaning of the variables?<<
Indeed, I have so stated on several occasions.

>> In other words, you think the following is circular:

P1: p iff q.r.s

[additional premisses and deductions as required here]

Conclusion: r entails s

...because the circularity is contained in P1? (premiss 1)<<

Well not precisely. Merricks' conclusion 'Warrant entails truth' is not equivalent to your 'r entails s' but rather 'r entails truth' giving

P1: p iff q.r.s
P1a: q.r.s->p (Strict implication equivalent of P1)

[additional premisses and deductions as required here]

Conclusion r

Now for any deductive structure to have a valid conclusion the premisses that lead to that conclusion must be true. Now P1/P1a can only be true if the individual atomic propositions,|q|r|s, are all true. Consequently, r must be true in P1/P1a as well as in the conclusion. which would satisfy the criterion for circularity:

'Begging the Question is a fallacy in which the premises include the claim that the conclusion is true or (directly or indirectly) assume that the conclusion is true. This sort of "reasoning" typically has the following form.

Premises in which the truth of the conclusion is claimed or the truth of the conclusion is assumed (either directly or indirectly).

Claim C (the conclusion) is true.

This sort of "reasoning" is fallacious because simply assuming that the conclusion is true (directly or indirectly) in the premises does not constitute evidence for that conclusion. Obviously, simply assuming a claim is true does not serve as evidence for that claim. This is especially clear in particularly blatant cases: "X is true. The evidence for this claim is that X is true."'

http://www.nizkor.org/features/fallacies/begging-the-question.html

>>I don't think there is a circularity there. Of course a circularity could be introduced in the square bracket section, but this isn't the source you have identified in your arguments so far is it? <<

I suppose technically if I were to take the modus ponens structure then the square bracket section would include the proposition

P2: q.r.s

which would highlight the circularity of r more explicitly but I don't think it necessary.


bs

Hi Bx4,

>>Well not precisely. Merricks' conclusion 'Warrant entails truth' is not equivalent to your 'r entails s' but rather 'r entails truth' giving

P1: p iff q.r.s
P1a: q.r.s->p (Strict implication equivalent of P1)

[additional premisses and deductions as required here]

Conclusion r <<

I think you are mistaken here and that this might be the source of our disagreement.

Consider:

K(p) iff p.B(p).W(p)

I had introduced a generic proposition with the same logical structure, namely:

p iff q.r.s

but I can see that the particular letters I used might lead to confusion if I am to explain where I think you've gone wrong. This is because I need to compare the general proposition with the particular one taken from Merricks. So let's use instead:

P1: q iff r.s.t

and compare it with

Pm: K(p) iff p.B(p).W(p)

Merricks' conclusion is:

Cm: W(p) entails p

Notice, crucially that it is not the truth of the proposition W(p) that is implied by 'Warrant entails truth', rather it is the truth of the proposition p itself. This is the distinction you have overlooked in saying that the conclusion is not 'r entails s' but 'r entails truth', hence just r. You have confused the truth of W(p) and the truth of p in the original.

So using P1 above,

the conclusion is t entails r

which is generalisation of W(p) entails p.

Since, for the reasons I mentioned involving the truth conditions of the premise, this is not circular in either the particular or the general case, your argument is incorrect.

I hope that makes sense.

ttfn

hi psi

I don't think the issue is whether one uses your generic proposition or the one specific to Merricks' paper but I think where we disagree is over Merricks' conclusion. You say that:

Merricks' conclusion is:

Cm: W(p) entails p

is in some sense equivalent to 'warrant entails truth' but the latter makes no mention of p.

So I think your assertion that:

>>Notice, crucially that it is not the truth of the proposition W(p) that is implied by 'Warrant entails truth', rather it is the truth of the proposition p itself. <<

is mistaken.

However even if I accept your version this does not defeat my point about circularity. If we take your:

P1: q iff r.s.t

Then this simply says

the consequent q is true if and only if the antecedent, the conjunction r.s.t is true. As such it is equivalent to the Lewisian strict implication

P2: r.s.t->q

that is to say if the antecedent, the conjunction r.s.t, is true then the consequent q cannot be false.

The problem with your argument is that r.s.t can only be true if all the individual atomic proposition are true.

So the conclusion the 'conclusion'

>>t entails r<< is present in the antecedent of the premise and hence the argument is circular.

Note also that t entails r is not unique since we can equally say that r entails t, r entails s, s entails r and so on. As you say the structure of the specific BcP.WP.P precisely mirrors that of r.s.t so all the possible conclusions, not just WP entails P, are also available and all suffer the same problem that if they are true in the antecedent of the premise then asserting they are true in the conclusion is a circular argument.

>>Since, for the reasons I mentioned involving the truth conditions of the premise, this is not circular in either the particular or the general case, your argument is incorrect.<<

Not so since you have yet to show the structure of the argument which shows which does not presuppose the truth of r.s.t (or of BcP.WP.P)in the antecedent.

One obviously difficulty you have to get round is that Merricks'

KcP iff BcP.WP.P

is not as he claims a definition of WP but of KcP. That is it is (pace Quine) 'true by definition' and if KcP is true then the antecedent is also true.

bs

Hi Bx4,

Only time for a quick reply now, I'll reply more fully tomorrow, firstly Merricks clearly formulates his examples in terms of the truth and falsity of p, eg:

"Suppose that S has a warranted belief that p, and p is false"

So my characterization of his conclusion is valid.

Secondly, you have made the same point again about the conjunction, so you don't seem to have understood that q iff r.s.t can be true if one or more of the atomic propositions r,s and t is false. I can only suggest you look at the truth table for this. So this argument you have repeatedly made for circularity fails for this reason.

ttfn

Hi Bx4,

So in more detail, point by point:

>>I don't think the issue is whether one uses your generic proposition or the one specific to Merricks' paper but I think where we disagree is over Merricks' conclusion. You say that:

Merricks' conclusion is:

Cm: W(p) entails p

is in some sense equivalent to 'warrant entails truth' but the latter makes no mention of p.

So I think your assertion that:

>>Notice, crucially that it is not the truth of the proposition W(p) that is implied by 'Warrant entails truth', rather it is the truth of the proposition p itself. <<

is mistaken.<<

It is clear from Merricks' paper that 'Warrant entails truth' means that if a belief in p is warranted then this entails that p is true. For example, he sets up a reductio fairly early on by supposing that warrant doesn't imply truth and deriving a contradiction. In the preamble to this he says:
"The denial that warrant entails truth leads either to contradiction (which is surely undesirable), or to a set of claims that is far less plausible than the claim that a belief’s being warranted entails that it is true. We should conclude, therefore, that it is not logically possible that a warranted belief be false. This is, of course, equivalent to the claim that warrant entails truth."

Here Merricks clearly equates 'warrant entails truth' with the longer form 'a belief's being warranted entails it is true'.

Then, in the reductio itself he makes it clear that what he means by saying that a belief is false is simply to believe p when p is false. By the same logic, a true belief is to believe p and for p to be true.

Hence my formulation of his conclusion, W(p) -> p is correct.

>>However even if I accept your version this does not defeat my point about circularity. If we take your:

P1: q iff r.s.t

Then this simply says

the consequent q is true if and only if the antecedent, the conjunction r.s.t is true. As such it is equivalent to the Lewisian strict implication

P2: r.s.t->q

that is to say if the antecedent, the conjunction r.s.t, is true then the consequent q cannot be false.<<
P2 is not equivalent to a biconditional even given the Lewisian interpretation because P2 does not say that if q is true then r.s.t is true. However, this is not relevant to the main point, which is:

>>The problem with your argument is that r.s.t can only be true if all the individual atomic proposition are true.

So the conclusion the 'conclusion'

>>t entails r<< is present in the antecedent of the premise and hence the argument is circular.<<

This is wrong. You would be right if assuming the truth of the premise P1: q iff r.s.t thereby assumed the truth of r,s and t. You seem to have become convinced that the law of conjunction has this consequence in this case. It does not. A simple counter example is the case q is false, t and s are true but r is false. Hence t entails r is not contained in the premise P1 because P1 can be true while t -> r is false.

Remember, iff is the biconditional <->, and look up the truth table for p <-> q and you will see that I am correct.

>>Not so since you have yet to show the structure of the argument which shows which does not presuppose the truth of r.s.t (or of BcP.WP.P)in the antecedent.<<

See above.

>>One obviously difficulty you have to get round is that Merricks'

KcP iff BcP.WP.P

is not as he claims a definition of WP but of KcP. That is it is (pace Quine) 'true by definition' and if KcP is true then the antecedent is also true.<<

As I've pointed out a few times, KcP can be false and the iff premise true.

I hope I have been clear in my explanation, and if I have appeared blunt in my assertions, this was for brevity and clarity, I hope it doesn't come across as arrogant or hectoring.

ttfn

i psi

>>It is clear from Merricks' paper that 'Warrant entails truth' means that if a belief in p is warranted then this entails that p is true<<

Perhaps Merricks should reflect on his lack of precision in his paper's title .


>>t is clear from Merricks' paper that 'Warrant entails truth' means that if a belief in p is warranted then this entails that p is true. For example, he sets up a reductio fairly early on by supposing that warrant doesn't imply truth and deriving a contradiction. In the preamble to this he says:

"The denial that warrant entails truth leads either to contradiction (which is surely undesirable), or to a set of claims that is far less plausible than the claim that a belief’s being warranted entails that it is true. We should conclude, therefore, that it is not logically possible that a warranted belief be false. This is, of course, equivalent to the claim that warrant entails truth."

This seems to be a different argument from your assertion that:

Cm: W(p) entails p

since his conclusion seems to be that

Cm: W(p) entails BcP

>>A simple counter example is the case q is false, t and s are true but r is false. Hence t entails r is not contained in the premise P1 because P1 can be true while t -> r is false.

I don't think your counter-example works since if q is false then the conjunction r.s.t is also false even if any one, any two or all more of the atomic propositions is false. Contrariwise q can /only/ be true if all the atomic propositions and hence the conjunction, are true and in P1: A single false atomic proposition in the antecedent means the antecedent is necessarily false and hence the consequent is necessarily false.

P1 q iff r.s.t
P2 ¬r
|= ¬r.s.t
|= ¬q

>>As I've pointed out a few times, KcP can be false and the iff premise true<<

You have so asserted but you have not shown how this can be so. Surely:

P1: KcP iff BcP.WP.P
P2: BcP.WP.P
C: |= KcP

Actually Merricks made a misstep in his assertion that:

'S knows that p, therefore, if and only if S’s belief that p is warranted and p is true.

'is a purely formal characterization of warrant.' when rather it is a purely formal characterisation of knowledge. Surely '' a purely formal characterization of warrant would be:

'P is warranted if and only if S believes that p and S knows that p and p is true '

Formally:

WP iff BcP.KcP.P

hence:

P1: WP iff BcP.KcP.P
P2: BcP.KcP.P
C: WP

However while it eliminates it does not lead to Merricks deductively invalid pseudo-conclusion:

WP entails BcP

>>I hope I have been clear in my explanation, and if I have appeared blunt in my assertions, this was for brevity and clarity, I hope it doesn't come across as arrogant or hectoring.<<

My problem is I think that in seeking brevity you have sacrificed clarity since I cannot see the deductively valid argument that allows you to conclude that:

'KcP can be false and the iff premise true'

bs


bs

Hi Bx4,

>>Perhaps Merricks should reflect on his lack of precision in his paper's title .<<

Indeed.

>>..This seems to be a different argument from your assertion that:

Cm: W(p) entails p

since his conclusion seems to be that

Cm: W(p) entails BcP<<

No I don't think so, perhaps, as with the title, he should have been clearer that the condition of a belief being true, as in: "a belief’s being warranted entails that it is true" means BcP.P. If you look at the reductio, this is clearer. Merricks uses 'true belief' and 'false belief' in this way: 'true belief' = C believes p AND p is true, false belief = C believes p AND p is false.

So the correct conclusion is actually Cm: W(p) entails p, although I can see why Merricks' lack of clarity caused you to think otherwise.

>>I don't think your counter-example works since if q is false then the conjunction r.s.t is also false even if any one, any two or all more of the atomic propositions is false. Contrariwise q can /only/ be true if all the atomic propositions and hence the conjunction, are true and in P1: A single false atomic proposition in the antecedent means the antecedent is necessarily false and hence the consequent is necessarily false.<<

The counter example does work despite my agreement with most of the rest of the above paragraph. This is because you seem to have overlooked one small detail, that's why I suggested you have a look at the truth table for iff (if and only if, <-&gt. It is a /biconditional/. My only other quibble with the above is that 'consequent' and 'antecedent' surely belong to '->' propositions rather than '<->' as we have here.



>>P1 q iff r.s.t
P2 ¬r
|= ¬r.s.t
|= ¬q <<

That seems fine.

>>You have so asserted but you have not shown how this can be so. <<

The most difficult things to show are the ones one finds self evident. I have directed you to the relevant truth table, if you have read it then I honestly can't see how you would avoid seeing that P1 below can be true if KcP is false.

>>Surely:

P1: KcP iff BcP.WP.P
P2: BcP.WP.P
C: |= KcP <<

How is the above valid argument relevant?

>>Actually Merricks made a misstep in his assertion that:

'S knows that p, therefore, if and only if S’s belief that p is warranted and p is true.

'is a purely formal characterization of warrant.' when rather it is a purely formal characterisation of knowledge. Surely '' a purely formal characterization of warrant would be:

'P is warranted if and only if S believes that p and S knows that p and p is true '

Formally:

WP iff BcP.KcP.P

hence:

P1: WP iff BcP.KcP.P
P2: BcP.KcP.P
C: WP <<

Interesting. I'm not sure about this, so I'll have a think.


>>My problem is I think that in seeking brevity you have sacrificed clarity since I cannot see the deductively valid argument that allows you to conclude that:

'KcP can be false and the iff premise true'<<

I'll set out the argument again as clearly as I can in my next post.

ttfn

Hi Bx4,

So you are unconvinced by my assertion that, given:

P1: q iff r.s.t

t -> r can be false.

It seems to me that the simplest way to demonstrate that this can be so is to assign truth values to q, r, s and t such that P1 is true and t -> r is false.

So I assign the following values:

q=F
t=T
s=T
r=F

With these values, t -> r is false. Now consider P1, and for clarity we replace 'iff' with the logical symbol '<->' (This was the cause of the accidental smiley in my last post).

P1 is then:

q <-> r.s.t

q is false, and by the law of conjucntion r.s.t is false. So P1 is true. This is because any statement of the form u <-> v yields T if both u and v have the truth value F. (Check the truth table, it is equivalent to XNOR).

QED

ttfn

hi psi

V. busy at the moment. Will reply at weekend
bs

Hi Bx4,

Looking forward to your reply.

ttfn

hi psi

>>No I don't think so, perhaps, as with the title, he should have been clearer that the condition of a belief being true, as in: "a belief’s being warranted entails that it is true" means BcP.P. If you look at the reductio, this is clearer. Merricks uses 'true belief' and 'false belief' in this way: 'true belief' = C believes p AND p is true, false belief = C believes p AND p is false.

So the correct conclusion is actually Cm: W(p) entails p, although I can see why Merricks' lack of clarity caused you to think otherwise. <<

But Merricks original claim after Plantinga is 'Warrant is “that, whatever precisely it is, which makes the difference between knowledge and mere true belief.' So a discussion of 'false belief would seem irrelevant since BcP and P can both be true making BcP.P true without the need for warrant (whatever that is).

>>The counter example does work despite my agreement with most of the rest of the above paragraph. This is because you seem to have overlooked one small detail, that's why I suggested you have a look at the truth table for iff...It is a /biconditional/. <<

I don't think its that simple since 'iff' seems to be ambiguous:

In material implication we have P implies Q iff (P^Q)or(~P^Q)or(~P^~Q).

In strict implication we have P implies Q iff (P^~Q) is not possible. This is a modal logic which deals with the concepts of possibility and necessity.

http://www.physicsforums.com


If therefore the biconditional is one of strict implication:

(KcP->BcP.WP.P).(BcP.WP.P->KcP)

rather than one of material implication:

(KcP=>BcP.WP.P).(BcP.WP.P=>KcP)

Then truth tables are immaterial:

'As a proposition, a strict conditional is either true or false.[20] A strict conditional is true if and only if the conclusion is true in every case that the hypothesis is true[21]

http://strictconditional.wikispaces.com/

The biconditional of a strict conditional is the proposition produced out of the conjunction of the strict conditional and its converse. When written in its standard English form, the hypothesis and conclusion are joined by the words "if and only if." The biconditional of a strict conditional is equivalent to the conjunction of the strict conditional and its converse.
(ibid)

I think I am warranted in assuming the strict biconditional given Merricks' claim that 'warrant entails truth' since:

'The word "entailment" is used instead of "implication" to describe this connective. Entailment is not truth-functional, and we will not deal with it in this course. (If you're interested, see Alan Ross Anderson and Nuel D. Belnap, Jr., Entailment: The Logic of Relevance and Necessity, Princeton University Press, Vol. 1, 1975, Vol. 2 (with J. Michael Dunn), 1993.) '

http://legacy.earlham.edu/~peters/courses/log/mat-imp.htm

>>The most difficult things to show are the ones one finds self evident. I have directed you to the relevant truth table, if you have read it then I honestly can't see how you would avoid seeing that P1 below can be true if KcP is false.<<
Because I am less persuaded than you that, for the reasons given above, truth tables are of any relevance.

>>How is the above valid argument relevant?<<

It shows that 'WP entails 'is not a valid conclusion from one 'horn' of the biconditional. Nor it would appear is it a valid conclusion from the other.

KcP->BcP.WP.P
KCP
|=BcP.WP.P

>>Interesting. I'm not sure about this, so I'll have a think. <<
OK

>>QUED<<

If the biconditional is the biconditional of a strict conditional then surely NQED

bs

Hi Bx4,

In my view your latest reply has two main flaws:

1) The assertion that the biconditional is best seen as strict is not well supported by any plausible interpretation of Merricks' paper.

2) Even if we view P1 as a strict biconditional, this does not answer the substantive objection to your argument for circularity, namely that you have the logic wrong.

In support of 1) I need only point out that Merricks is explicit in his use of 'entailment': p entails q means it is logically impossible for p to be true and q false. This is logical entailment for which truth tables are a recognised and valid method of checking. Nothing in Merricks' paper suggests he is at all wedded to Lewisian strict implication or any other relevance logics, and his examples suggest the opposite, eg p or Brown is in Barcelona.

Turning to 2), suppose we grant for the sake of argument that the biconditional:

KcP <-> BcP.W(P).P

is interpreted as strict.
Then if, as according to your quote:
"'As a proposition, a strict conditional is either true or false.[20] A strict conditional is true if and only if the conclusion is true in every case that the hypothesis is true[21]" and we interpret the biconditional as you suggest, we have:

P3: KcP -> BcP.W(P).P
P4: BcP.W(P).P -> KcP

or we could write the conjunction (KcP -> BcP.W(P).P).(BcP.W(P).P -> KcP).

Yet still the strict conditional W(P) -> P is not contained in these premises. All you have changed is that we are now saying that in every case that C knows p, C believes p, p is warranted and p is true, and vice versa. This does not meet my counter example, namely this can be true, in every case that KcP and yet there can still be cases where C does not know P (this does not falsify statements about cases in which C does know P). So the truth of P3 and P4 still does not guarantee the truth of W -> p, since in the cases where KcP is false the strict part of the conditional does not apply.

In summary, the invocation of strict conditionals is unsupported and even if accepted it doesn't affect my objection to your argument for circularity.

ttfn

Hi Bx4,

I thought I'd also reply to some specific points.

>>So a discussion of 'false belief would seem irrelevant since BcP and P can both be true making BcP.P true without the need for warrant (whatever that is).<<
The discussion of false belief is relevant in making the distinction between:
B1: It is false that C believes P
B2: C believes P AND P is false,

and this seemed useful due to an apparent confusing by you of the two notions.

>>Then truth tables are immaterial:<<
This is not a valid conclusion. You are using equivalent methods to conclude circularity, for example citing the law of conjunction. You cannot have things both ways. You seem to have overlooked that whilst, for example @¬(P^~Q) may not be truth functional, ¬(P^~Q) is.

>>It shows that 'WP entails 'is not a valid conclusion from one 'horn' of the biconditional. Nor it would appear is it a valid conclusion from the other.<<
I assume that by 'valid' you mean non circular? Otherwise you would be arguing against yourself somewhat. It doesn't show that anyway, because yet again you have helped yourself to an additional premise, namely BcP.WP.P. However, in my counterexample this is false. Strict conditionals do not change anything relevant to my objection.

I'm still thinking about the other thing.

ttfn

Hi psi

I'll reply to your latest when I have more time. However your specific claim that

>>As I've pointed out a few times, KcP can be false and the iff premise true.'<<

we have:

'p<=>q "p and q are either both true or both false"

Implication statements (p => q) are sometimes called conditionals, and equivalence statements (p q) are sometimes called biconditionals.'

http://legacy.earlham.edu/~peters/courses/log/terms2.htm

which would seem to contradict your assertion.

(Note:I have substituted '=>' and '<=>' for Suber's non-printing symbols)

bs

Hi Bx4,

>>I'll reply to your latest when I have more time. However your specific claim that

>>As I've pointed out a few times, KcP can be false and the iff premise true.'<<

we have:

'p<=>q "p and q are either both true or both false"

Implication statements (p => q) are sometimes called conditionals, and equivalence statements (p q) are sometimes called biconditionals.'

http://legacy.earlham.edu/~peters/courses/log/terms2.htm

which would seem to contradict your assertion.<<

No, this is the same error on your part. Look at the truth table under the sentence 'The truth table columns that define the basic connectives are as follows:' from the page you linked to above.
Note that the last row and last column contains 'T' for p <=> q when p and q are both false. Hence my assertion that the iff premise can be true if KcP is false. This happens if the conjunction BcP.W(P).(P) is also false, since then we have a situation p <=> q where p and q are false. This returns T as in the truth table I mentioned above. I did mention in might be a good idea to check the truth table for the biconditional relation.

In summary, the iff premise is:

P1: KcP <=> BcP.W(P).P

P1 returns T in the case that KcP is F and BcP.W(P).P is F

I hope that clarifies.

ttfn

hi psi v. busy at moment. Unlikely to be able to reply until weekend
bs

Hi Bx4,

I had a think about this:

>>Actually Merricks made a misstep in his assertion that:

'S knows that p, therefore, if and only if S’s belief that p is warranted and p is true.

'is a purely formal characterization of warrant.' when rather it is a purely formal characterisation of knowledge. Surely '' a purely formal characterization of warrant would be:

'P is warranted if and only if S believes that p and S knows that p and p is true '<<

Why? Just because you have the W(P) term on the left on its own? That doesn't make sense to me. I think also a belief that p might be warranted even if S doesn't actually believe it. So Merricks' formulation does plug W(P) in the correct slot given its relation to knowledge in my view.

ttfn