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

hi psi

Sorry about the delayed reply. I'm still quite time poor so it will be easier to reply to your posts individually. This is a reply to #136. I perhaps should have replied to #135 first so I might be better to wait until I reply to that one before responding.


>>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.<<

Except I have never made any reference to B1 which would seem irrelevant in any case since Merricks biconditional definition of knowledge

KcP<=>BcP.WP.P

can only be true if the conjunction BcP.WP.P is true and it can only be true if each of the atomic propositions are true. Conversely if KcP is true then the conjunction is true and hence the constituent atomic propositions are true.

Moreover if any of the terms of the biconditional are false then the true biconditional would be:

¬KcP<=>¬(Bcp.Wp.P)

Which would be the outcome whether B1 or B2 or both are true so it is unclear what the actual relevance is of the distinction you are making.

>>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.<<

The confusion is yours since I did not say that the strict conditional or the strict biconditional (consequent on Merricks' use of 'entails') are not truth functional but rather that truth tables while relevant to the material conditional (such as your ¬(P^~Q)) are not so for strict conditionals or biconditionals. A point which your @¬(P^~Q) would seem to accept since it says Lewisian terms 'if P is true the Q cannot be false'.

Your reference to ¬(P^~Q) seems obscure since you have not shown that Merricks KcP<=>BcP.WP.P is a material biconditional rather than a strict biconditional

>>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 am using valid to refer to an argument that does not contain a //formal// logical fallacy such as circularity.


I am somewhat unclear why you think I am relying on an 'additional premise' since it is clearly part of Merricks definition of knowledge and the other premises of my argument are logically derivable from it:

P1 KcP<=>BcP.WP.P

which is equivalent to:

P1a (KcP->BcP.WP.P).(BcP.WP.P->KcP)

p2 BcP.WP.P<=>BcP|WP|P

which is equivalent to

p2a (BcP.WP.P->BcP|WP|P).(BcP|WP|P->BcP.WP.P))

so a conclusion that:

Wp|=P

is invalid because the conclusion is implicit in the premises.

I am not clear why you consider that stict conditionals are not relevant to your argument since your argument seem to be that given p=>q.r.S then p.can be true even if r is false. This only only holds for material conditionals . However Merricks use of 'entails' requires that P1a is comprised of two strict conditionals. See for example 'Strict Implication, Entailment, and Modal Iteration', Arthur Pap (The Philosophical Review Vol. 64, No. 4 (Oct., 1955), pp. 604-613)

bs

Hi Bx4,

I'll wait for your reply to #135 before I respond then.

ttfn

hi psi

#135

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

It is not really a matter of interpretation plausible or otherwise but of the definition of the iff biconditional:


'A biconditional statement can be either true or false. To be true, //BOTH// the conditional statement and its converse must be true. This means that a true biconditional statement is true both "forward" and "backward". //All definitions// can be written as //true// biconditionals' [emphases added]

https://docs.google.com/presentation/d/1JVSuRiX06kpJpdtakIkHjyy0Tj4uGdif-QHglEGv8Iw/edit?pli=1#slide=id.i57

It therefore follows that Merricks:

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

since it is a definition of knowledge and hence from the above must be true. I think therefore I am justified in expressing (a) as:

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

where '->' stands for the Lewisian strict conditional 'fishhook' operator such that the generic 'a->b' means ' if a is true then b cannot be false' and the converse 'b->a' means 'If b is true then a cannot be false'. So for any generic //definition// where 'BOTH the statement and it converse must be true' then:

(c)a<->b

is expressible as

(d)(a->b).(b->a)

Clearly this is therefore not a matter of the semantics of (a) and (b) but rather of the syntax of //any// definition expressed as a true biconditional.

Since Merricks definition of knowledge as a true biconditional requires that the statement, and its converse, BcP.WP.P, are necessaily both true (else it is not a definition) and this is the original premise of Merricks' argument then it follows that since the converse, BcP.WP.P, is true then that the atomic propositions BcP, WP and P cannot be false. Hence given that:

'Entailment relates, by logical implication, two sentences, A and B, such that the truth of B follows from the truth of A'.

http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Entailment.html

Then it follows that Merricks conclusion (as you prefer to express it) 'WP|=P is implicit in his true biconditional definition of KcP. Hence the circularity.

>>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.<<

The point you seem to miss is that

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

is a /definition/ of knowledge where both the statement and its converse are necessarily true hence your appeal to truth tables is irrelevant since the truth table would only have /one/ row; the one in which the statement and it converse are both true yielding a true biconditional. Since a truth table is normally constructed to show how taking different truth values (true or false) of the indiviual statements affect the truth value of the combined statement then clearly if different truth values of the individual statements are not possible (as they are not in a true biconditional) then a truth table cannot be constructed.

Moreover I am somewhat puzzled by your argument that Merricks use of entailment; 'p entails q means it is logically impossible for p to be true and q false.'excludes strict implication since

'To explicate this notion he [Lewis] defined strict implication, according to which the if-then conditional p ->q expressing the strict implication of q by p is equivalent to ~#(p & ~q), and is true just in case it is not possible that p is true and q is false.' [where -> substituted for 'fishhook operator and # for 'possible' operator.]

would seem to be equivalent to Merricks entailment. This is not surprising since the Arthur Pap paper, 'Strict Implication, Entailment, and Modal Iteration', that I cited in 142 makes the point that Lewis considered strict implication and entailment to be equivaelnt.

Moreover, I am unclear why you are citing Merricks; 'p or Brown is in Barcelona' since it does nor seem to be an example of a conditional, strict or otherwise, or of a biconditional or of entailment.

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

Since Merricks uses a //true// biconditional as his definition of KcP then it follows from the earlier description of a definition as a true biconditional:

'A biconditional statement can be either true or false. To be true, //BOTH// the conditional statement and its converse must be true. This means that a true biconditional statement is true both "forward" and "backward". //All definitions// can be written as //true// biconditionals' [emphases added]

the I would suggest that this form follows from theis description.

>>Yet still the strict conditional W(P) -> P is not contained in these premises.<<

Circularity does not require that W(P)->P is contained in thes premises merely that it is //indirectly// contained in them:

'[Circular reasoning] 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.' [Emphases added]

So given Merricks' definition of KcP as the true biconditional:

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

then it follows that:

(e) BcP.WP.P<->BcP|WP|P (where | separates the individual atomic propositions.)

and (e) is (by the definition of conjunction) a true biconditional. Therefore:

(f) KcP<->BcP|WP|P

is also a true biconditional hence in terms of //this// biconditional since BcP|WP|P is necessarily true then it follows that it is logically impossible for WP to be true and P to be false, i.e. WP entails P. so the conclusion is indirectly contained in the premise.

>> 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.<<

This seems to be something of a red herring since I m are only concerned with the case where KcP is true is since Merricks definition of KcP is the true biconditional:

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

which since the biconditional can only be true if the statement and its converse are true and (since the biconditional is true by definition) therefore:

(g) @(KcP->BcP.WP.P).@(BcP.WP.P->KcP) (using the modern modal version of strict implication)

hence

(h) (@KcP->@(BcP.WP.P)).(@(BcP.Wp.P)->@Kcp)

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

Except that I think that I have shown that Merricks definition of knowledge involves a true biconditional which requires that KcP and BcP.WP.P are necessarily true. Moreover as I have shown above that our definition of entailment and Lewis' definition strict implication seem equivalent.

Your objection to my argument on circularity seems to be that premise (a) does not contain an explicit //direct// inclusion of the conclusion in the premise but I think I have shown that the conclusion is implicit in the premise which if I have is sufficient to demonstrate circularity.

BTW. I'm sure I came across a note that said italics can now be used in h2g2 posts but I have not been able to find it again.


bs

Hi Bx4,

There are some interesting points here. The Pap paper does indeed make the point that the distinction between entailment and strict implication cannot be maintained. He argues against C. Lewy's attempt to make the distinction in order to do this. I doubt Merricks had this in mind, all he wanted to show is that p logically follows from W(p).

Consider the following, where '=>' is material implication:

(p => q).p entails q

You might want to say that 'entails' is equivalent to a strict implication, but that doesn't mean we have to rewrite p => q as a strict implication. So I'm not clear why you think the biconditional in Merricks' argument is strict. I had in mind that the motivation behind strict implication is to guarantee relevance, but that doesn't seem to be an issue here.

In any case, let's proceed on the basis that the implications are strict, in my view, your fundamental error remains.

To use your example, I agree that a<->b can be expressed as:
(a->b).(b->a)

From this you have somehow formed the notion that in order for a<->b to be true, a and b must both be true. That's not the case, it is also true if a and b are both false. The truth table for <-> shows this, and the statement, for example that 'if a is true then b cannot be false' does not apply since a is false. And vice versa of course. (The statement and its converse are both true, since a -> b yields T and b -> a yields T if a =F, b=F. Check the truth tables.)

It is late so I'll reply in more detail anon.

ttfn

Hi Bx4,

I'll try to answer some specific points now.

>>Since Merricks definition of knowledge as a true biconditional requires that the statement, and its converse, BcP.WP.P, are necessaily both true (else it is not a definition) and this is the original premise of Merricks' argument then it follows that since the converse, BcP.WP.P, is true then that the atomic propositions BcP, WP and P cannot be false. Hence given that:

'Entailment relates, by logical implication, two sentences, A and B, such that the truth of B follows from the truth of A'.

http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Entailment.html

Then it follows that Merricks conclusion (as you prefer to express it) 'WP|=P is implicit in his true biconditional definition of KcP. Hence the circularity.<<

No. The statement's converse is not BcP.WP.P, I wonder whether you are confusing 'converse' and 'consequent'? The statement could be:

KcP -> BcP.WP.P

then its converse is:

BcP.WP.P -> KcP

You are correct to say that in order for Kcp <-> BcP.WP.P to be true, both of the above statements must be true. You are wrong to infer that therefore the conjunction BcP.WP.P must be true, this is wrong because F -> F yields T.

>>The point you seem to miss is that

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

is a /definition/ of knowledge where both the statement and its converse are necessarily true hence your appeal to truth tables is irrelevant since the truth table would only have /one/ row; the one in which the statement and it converse are both true yielding a true biconditional. Since a truth table is normally constructed to show how taking different truth values (true or false) of the indiviual statements affect the truth value of the combined statement then clearly if different truth values of the individual statements are not possible (as they are not in a true biconditional) then a truth table cannot be constructed.<<

This is incorrect. The truth table for a <-> b has four rows, the fourth is F F <-> T. You are right that a -> b must be true and b -> a must be true, what you have overlooked is that this can be the case if both a and b are false. We can construct the truth table in order to reveal how the truth values of a and b affect the truth of the biconditional. As a definition, we must take Merricks' biconditional to be true. It is not at all the same thing to suppose that the expression on each side of it must be true, as the truth table reveals.

>>Moreover I am somewhat puzzled by your argument that Merricks use of entailment; 'p entails q means it is logically impossible for p to be true and q false.'excludes strict implication since<<

I agree that we can interpret the 'entails' as a strict implication. I am unclear as to how this applies to the biconditional, as I said I can see nothing that would warrant the adoption of a Lewisian logic paradigm for all logicical statements in Merricks' argument.


>>Circularity does not require that W(P)->P is contained in thes premises merely that it is //indirectly// contained in them:<<

I agree. I did not mean W(P)->P is contained explicitly in the premises either. I can see the circularity immediately if it is assumed that for the biconditional to be true then the exressions on each side must be true. I accept that as a definition, the biconditional must be true in Merricks' argument. But you still have not seen that this can be so if KcP is false, W(P) is true and P is false. This is because, for whatever reason, you can't see that a <-> b is true if both a and b are false.

>>Therefore:

(f) KcP<->BcP|WP|P

is also a true biconditional hence in terms of //this// biconditional since BcP|WP|P is necessarily true then it follows that it is logically impossible for WP to be true and P to be false, i.e. WP entails P. so the conclusion is indirectly contained in the premise. <<

Same mistake. BcP|WP|P is not necessarily true, the biconditional can be true if KcP is false and BcP|WP|P is false. In particular this means the biconditional is true and W(P) -> P is false in the case KcP = F, W(P) = T, P = F. This is fatal to the argument for circularity I'm afraid.

>>This seems to be something of a red herring since I m are only concerned with the case where KcP is true is since Merricks definition of KcP is the true biconditional:

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

which since the biconditional can only be true if the statement and its converse are true and (since the biconditional is true by definition)<<

See above, statement KcP ->BcP.WP.P, converse BcP.WP.P -> KcP, look at truth table for '->' both statement and converse are true if KcP = F, W(P) = T, P = F.

>>Your objection to my argument on circularity seems to be that premise (a) does not contain an explicit //direct// inclusion of the conclusion in the premise but I think I have shown that the conclusion is implicit in the premise which if I have is sufficient to demonstrate circularity.<<

No my objection is nothing to do with explicit or direct inclusion. The circularity is obvious if it is assumed KcP must be true. I hope I've helped you to see that this need not be the case.

In summary, you have the logic wrong.

ttfn

Hi Bx4,

Now some points from your #141:

>>Except I have never made any reference to B1 which would seem irrelevant in any case since Merricks biconditional definition of knowledge <<

As I said, B1 was relevant because in your previous post (#129?) you said:

"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 "

Which showed that you had confused the concept of false belief and it being false that an agent believes.

>>Merricks biconditional definition of knowledge

KcP<=>BcP.WP.P

can only be true if the conjunction BcP.WP.P is true <<

That's our disagreement as I regard this as your fundamental error. I have explained why in the previous post.

>>Moreover if any of the terms of the biconditional are false then the true biconditional would be:

¬KcP<=>¬(Bcp.Wp.P) <<

No, the true biconditional would still be:

KcP<=>(Bcp.Wp.P)

Aren't they equivalent anyway?

>>Which would be the outcome whether B1 or B2 or both are true so it is unclear what the actual relevance is of the distinction you are making. <<

They are relevant just in order to clear up any confusion between the two concepts which, as apparently you had indeed confused them, seemed advisable.

>>The confusion is yours since I did not say that the strict conditional or the strict biconditional (consequent on Merricks' use of 'entails') are not truth functional but rather that truth tables while relevant to the material conditional (such as your ¬(P^~Q)) are not so for strict conditionals or biconditionals. A point which your @¬(P^~Q) would seem to accept since it says Lewisian terms 'if P is true the Q cannot be false'. <<

No, truth tables do apply to strict conditionals as well as material ones. In fact, as Pap's paper points out, truth tables can be used to establish the modal status of propositions:

"The same method of analysis of logical ranges, the "truth-table" method, which assures us of the truth of this statement though we may be ignorant of the truth-values of the component statements, also assures us of its necessary truth,..." 'Strict Implication, Entailment, and Modal Iteration', Arthur Pap (The Philosophical Review Vol. 64, No. 4 (Oct., 1955), p612.

The truth table for the strict biconditional is the same as material biconditional, it is just that an extra modal element of necessity is added. So a <-> b has:

TT <-> T
TF <-> F
FT <-> F
FF <-> T

However the condition: if a is true, it is not possible for b to be false and if b is true it is not possible for a to be false /only/ applies if the biconditional itself yields T, ie in rows 1 and 4. So you can't just strike out rows 2 and 3. Moreover, even if you did, that leaves row 4, upon which the statements "if a is true, it is not possible for b to be false and if b is true it is not possible for a to be false" are /silent/ since neither are true.

I hope this clarifies.

ttfn

Hi psi

Apologies. Hors de combat with a recurrence (for 3rd time ) of chest/URT infection. Too feverish to manage a coherent reply for now
bs

Hi Bx4,

Sorry to hear that, get well soon.

ttfn

Hi psi

Recently diagnosed with Type II Diabetes and until blood sugar under control recurrent infections are a side effect (something to do with little beasties snacking on elevated blood sugar -or so I m told) anyhow onwards and upwards.

I have read tour last substantive post and want to have a bit of a think before I reply

bs

Hi Bx4,

I can only hope that the management of your diabetes turns out to be relatively unproblematic and that you are well again soon.

ttfn

hi psi

Type II

Coincidentally before being diagnosed I had started an new diet/exercise regime which tends to reduce blood glucose levels and with medication (metformin) I have reduced my blood glucose level from 80MMmol/mol to 56 mmol/mol. Not quite at target of 48 mmol/mol but getting there. Funnily I have not had any of the symptoms of Type II diabetes.

I have finally begun a reply to your substantive posts but it will take a few days to get there.
bs

Hi Bx4,

I'm glad your diet/exercise regime has had a good result and I'm looking forward to your reply.

ttfn

hi psi

I had rather lost the thread of my argument so I am revisiting ~135 and subsequent posts to reconstruct it.

BTW, I'm sure I came across a comment that h2g2 now supports italics but I'm blowed if I can find it.

bs

Hi Bx4,

If they could just get the basic functionality together then there'd be a viable cake on which italics could be the icing.

ttfn

hi psi

Apologies for much delayed reply. We recently returned to Scotland for the summer vac/referendum after an absence of nearly two years and have been much engaged in domestic reclamation and the 'social whirl'.

I have spent some time reading our conversation but have rather lost sight of the wood or the trees (an unoriginal analogy ).

So rather than replying to all your posts I thought that I might try an establish points of agreement and disagreement.

I had written a post to this end but I missed up the 'Post Message and lost it!

My own fault for composing it in h2g2 rather than notepad.

Anyhow I should have a bit of free time to-morrow so I'll have another go then.

bs

hi psi

I want to try and establish where and, if so, why we disagree on certain key points so here is my starter for ten:

(A) begging the questtion/circularity/petitio principii

In several of your posts you have argued that no circularity can exist because the premise does not 'contain' the conclusion.

Now I agree that if this is the case then my argument falls at the first hurdle. However, the Nizkor Project characterises the fallacy of circularity thuswise:

'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.

1. Premises in which the truth of the conclusion is claimed or the truth of the conclusion is assumed //directly// or //indirectly//).

2. Claim C (the conclusion) is true.' [Emphases added]

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

So circularity can either be when the conclusion is //explicit// in the premise or when the conclusion is //implicit// in the premise.

I am arguing for the latter case in Merricks premise not the former. However, I am unclear whether your use of 'contain' covers both cases. If it does I am unclear how you would demonstrate, rather than simply assert, that the premise does not contain an //implicit// circularity.

(B) '[A] a purely formal characterization of warrant.'

>>Why? Just because you have the W(P) term on the left on its own?<<

No the position is irrelevant. However, what //is// relevant is that Merricks confuses the role of the definiendum and the definiens in a definition:

'A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).The term to be defined is the

definiendum. The term may have many different senses and multiple meanings. For each meaning, a definiens is a cluster of words that defines that term'
http://en.wikipedia.org/wiki/Definition

Merricks 'purely formal characterisation definition:

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

which we have written as the biconditional:

(iia)KcP <=> BcP.W(P).P

would seem to have KcP as the definiendum and W(P) as part of the definiens.

>>That doesn't make sense to me.<<

What doesn't make sense to me is you think Merricks is justified in considering W(P) the definiendum rather than part of the

definiens. Suppose we take a different biconditional definition.

(ii) x is a bachelor if and only if x is a man and x is unmarried married.

(iia) b <=> m.u

Do you consider that that (ii) is a 'purely formal characterisation' of ' man' given that that m is part of the definiens? If not can you explain what is different about (i) given that W(P) would similarly be part of the definiens?

>>I think also a belief that p might be warranted even if S doesn't actually believe it<<

I don't see the relevance of this point. However Merricks (i) does not assert that S believes that p is warranted but that S has a belief that p is true and that belief is warranted so I am a bit unclear who it is that has a 'belief that P might be warranted'. Is this the agent S or another agent?

Morever, if contrary to (1) if 'S does not actually believe it' that p is true then surely S would not know that p is true and hence (i) would not be a definition?

>> So Merricks' formulation does plug W(P) in the correct slot given its relation to knowledge in my view.<<

I have argued that Merricks formulation and your supporting argument are flawed. Perhaps you could refute my arguments?

(C) Expansion of a biconditional
I agree that a bi conditional p<=>q can be expressed as the conjunction of the conditional and its converse (p->q).(q->p) and
I will use it later.

(D)The biconditional characterisation of a definition
In an earlier post I quoted from this account:

(Q1)'A biconditional can be either true or false. To be true BOTH the conditional and its converse /must/ be true. This means that a true biconditional statement is true both 'forward' and 'backward'.[// emphasis added. 'BOTH' in original)

(https://docs.google.com/presentation/d/1JVSuRiX06kpJpdtakIkHjyy0Tj4uGdif-QHglEGv8Iw/edit?pli=1#slide=id.i57)

Moreover:

(Q2)'All //definitions// can be written as /true/ biconditionals.' (ibid) [emphases added]

I am not clear whether you accept this account or not but since it is important to my argument on circularity I don't want to go much further at the moment.

I also want to clarify some of your points about truth tables, strict implication and entailment but again my relies will be somewhat dependent on our replies to the above.

bs

Hi Bx4,

Sorry for the delay in replying. About a week ago there was a lightening strike very close by which set off a house alarm on our road and coincided with our broadband hub malfunctioning. So I've been offline for a week.

I'll try to reply as clearly as I can next post.

Have you caught much of the Edinburgh Festival?

ttfn

Hi Bx4,

I'll try to deal with each point, possibly over several posts.

>>(A) begging the questtion/circularity/petitio principii<<

>>So circularity can either be when the conclusion is //explicit// in the premise or when the conclusion is //implicit// in the premise.

I am arguing for the latter case in Merricks premise not the former. However, I am unclear whether your use of 'contain' covers both cases. <<

I my post #145 I addressed this point as follows:
"I did not mean W(P)->P is contained explicitly in the premises either. I can see the circularity immediately if it is assumed that for the biconditional to be true then the expressions on each side must be true. I accept that as a definition, the biconditional must be true in Merricks' argument. But you still have not seen that this can be so if KcP is false, W(P) is true and P is false. This is because, for whatever reason, you can't see that a <-> b is true if both a and b are false."

>>If it does I am unclear how you would demonstrate, rather than simply assert, that the premise does not contain an //implicit// circularity.<<

I have already demonstrated that it does not contain an implicit circularity. To recap, I showed that W(P)->P is not implicitly contained in the premise KcP<=>(Bcp.Wp.P) by giving truth values for each variable such that KcP<=>(Bcp.Wp.P) is true and W(P)->P is false.

I hope you can see that given a premise P1, a proposition q that shares some variables with P1, and an assignment of truth values, then if P1 can be true whilst q is false it cannot be the case that P1 contains q (implicitly or explicitly). I gave such an example of an assignment of truth values in my #145, KcP = F, W(P) = T, P = F. It does not matter if Bcp is T or F.

With these truth values, KcP<=>(Bcp.Wp.P) is T but W(P)->P is false. Hence W(P)->P is not contained in KcP<=>(Bcp.Wp.P).

I hope that's clear.

ttfn

Hi Bx4,

First, I want to address this, I had said:

"I think also a belief that p might be warranted even if S doesn't actually believe it"

Your reply was:
>>I don't see the relevance of this point. However Merricks (i) does not assert that S believes that p is warranted but that S has a belief that p is true and that belief is warranted so I am a bit unclear who it is that has a 'belief that P might be warranted'. Is this the agent S or another agent? <<

What I meant was that we could imagine a case W(P).P.¬B(P). That is to say, it could be the case that a belief that p is warranted and p is true and S doesn't believe p. The relevance of this is that it was a response to your idea that:

>>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 '<<

Given B(P) could be false but W(P) true, the above doesn't work as a characterization of warrant.

Now to the rest of B:

>>(B) '[A] a purely formal characterization of warrant.' <<

Perhaps we have been wrong in considering that Merricks gives a definition then. Maybe a characterization is a better term? Let's assume that it is a definition for now. Your argument seems to be that Merricks has confused the definiendum and the definiens. But although you say that it isn't the position of words that matters, you haven't said why you think W(P) isn't the former, other than by an intuitive appeal to the following analogy:

>>(ii) x is a bachelor if and only if x is a man and x is unmarried married.[sic]

(iia) b <=> m.u <<

As the wiki article you linked to points out:

"Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions. If every term of every definiens must itself be defined, "where at last should we stop?"[12][13] A dictionary, for instance, insofar as it is a comprehensive list of lexical definitions, must resort to circularity."

In the case of 'x is a bachelor if and only if x is a man and x is unmarried', what is it that tells us that 'bachelor' is the definiendum? Perhaps it is both the position of 'bachelor' on the left and that it is on its own, whereas 'male' and 'unmarried' are part of a conjunction? I think also we have a hierarchy of concepts. We assume that for the reader to understand a definition that talks about a man and marriage, said reader must already possess concepts for 'man' and 'married'.

However, there seems to me to be nothing in principle to exclude the possibility of giving a definition where the definiendum is part of an expression on the right hand side of a biconditional.

For example suppose we already know that the identity element for 'multiply' is 1, and what all the rationals are. Then we could define a rational number's inverse element for multiply like this:

1 = n x (inverse of n)

ttfn

Hi Bx4,

>>(D)The biconditional characterisation of a definition
In an earlier post I quoted from this account:

(Q1)'A biconditional can be either true or false. To be true BOTH the conditional and its converse /must/ be true. This means that a true biconditional statement is true both 'forward' and 'backward'.[// emphasis added. 'BOTH' in original) <<

I don't see anything wrong with this. I think the trouble comes when we have, for example, a true biconditional of the form:

p <=> q.r.s

You seem to conclude that q,r and s must all be true in that case. This is a mistake. If we use your unpacking method above then both

p => q.r.s

and

q.r.s => p

are true. But this is the case when, for example r and p are both false. Again, check the truth table. F => F = T you see.

I doubt I can put it clearer than that.

ttfn