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

hi psi

>>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.<<

Not to worry I have been pretty much offline myself for a bit though for much less dramatic reasons than a lightning strike. We recently returned to the Western Isles for the summer vac (and the referendum) after a long absence and our house and surrounding land are a bit neglected an 'she who must be obeyed' has decreed a program of refurbishment which is taking up most of my time.

>>Have you caught much of the Edinburgh Festival? <<

No we gave it a miss this year.

158

>>"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.<<

Here I was merely trying to establish whether you accepted that the circularity of an argument could be implicit in the premise and you have answered this in the affirmative.

I did raise the point later as to whether you accepted that in a definition //expressed// as a biconditional then the conditional and its converse must be true. Again you have answered in the affirmative.

>>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 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).<<

The problem with the argument is that by virtue of being a //definition// neither the conditional nor its converse /can/ be false. So setting KcP = F, W(P) = T, P = F simply means that this biconditional is not a definition since the conditional KcP and its converse BcP.WP.P are both false.

I don't see the relevance of your demonstration is relevant in the context where:

KcP<=>BcP.WP.P

is a true biconditional since

'(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)

bs

Hi Bx4,

>>The problem with the argument is that by virtue of being a //definition// neither the conditional nor its converse /can/ be false. So setting KcP = F, W(P) = T, P = F simply means that this biconditional is not a definition since the conditional KcP and its converse BcP.WP.P are both false.<<

The conditional is KcP -> BcP.WP.P,

The converse is BcP.WP.P -> KcP

Both are true when KcP = F, W(P) = T, P = F,if you check the truth tables you'll see that F -> F = T. Since F <-> F is also T, the biconditional is T.

>>I don't see the relevance of your demonstration is relevant in the context where:<<

I don't know what this means. I have shown that the biconditional can be true while W(P) -> P is false, you don't seem to have understood.

ttfn

hi psi

I'm fairly busy at the moment but before I reply to your latest I want for the sake of continuity I want to deal with your responses to my ante-penultimate post. Hopefully in next cuople of days.

bs

Hi Bx4,

Ok.

ttfn

Hi Bx4,

How does the following symbol appear to you?

∃

ttfn

hi psi

159

>>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.<<

But this doesn't seem to make provision for your 'a belief that P is warranted' or the fact ¬BcP.W(P)were

true then BcP.W(P).P would be false and therefore the biconditional

KcP<=>BcP.W(P).P [1]

would not satisfy the requirement that, from Q1 and Q2, when a definition is expressed as a biconditional //both// the conditional and its converse are true.

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

It work as well as [1] since if BcP were false in [1] the the the biconditional could not be true since the

converse BCP.W(P).P would be false. So if

W(P)<=>Kcp.BcP.P [2]

is a definition of warrant expressed as a biconditional then both the conditional and its converse //must//

be true

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

This seems to be one of the significant points of our disagreement but since my [2] is a 'purely formal

characterisation', i.e. a definition, of warrant the situation you describe, because of(Q1) and (Q2), cannot

obtain since both the conditional and its converse must BOTH be true.

Expanding [2] we have:

(W(P)->KcP.BcP.P).(KcP.BcP.P->W(P)) [3]

Then if any one or any two of the three terms in KcP.BcP.P were false then the conjunction itself would be

false so in [3] this would mean that relevant truth value for the conditional W(P)->KcP.BcP.P would be the

following:

W(P) KcP.BcP.P W(P)->KcP.BcP.P
T F F

whereas for the converse, KcP.BcP.P->W(P), the truth value would be:

KcP.BcP.P W(P) KcP.BcP.P->W(P)
F T T

http://www.mathgoodies.com/lessons/vol9/conditional.html

So we have a situation in which while the conditional is false but the converse is true which does not

satisfy the the requirements of (Q1) and (Q2) as far as the characterisation of a definition as a biconditional is concerned.

There is one context in which 'B(P) could be false but W(P) true' which is when all of the atomic

propositions in the conjunction, KcP.BcP.P, are false which would mean that W(P) could be true and BcP false

but it would also require that the atomic propositions KcP and P are also false.

Note that a similar situation obtains in Merricks supposed 'purely formal characterisation of warrant':

KcP<=>BcP.W(P).P [1]

which we can expand as

(KcP->BcP.W(P).P).(BcP.W(P).P->KCP) [4]

Since if follows that if any one or two of the atomic propositions, BcP, and P, in the conjunction are false

and if the atomic proposition W(P) is true then triple conjunction, BcP.W(P).P is also false and hence:


KcP BC.W(P)P.P KcP->BcP.W(P).P
T F F

the conditional is also false. Moreover for the converse, if the triple conjunction is false then:

BcP.W(P).P KcP BcP.W(p).P->W(P)
F T T

Which is inconsistent with the characterisation () of a definition expressed as a biconditional which

requires that //both// the conditional and its converse are true.

In fact their are only two sets of values which would make the triple conjunction true:

BcP W(P) P BcP.W(P).P [5]
F F F T

and

BcP W(P) P BcP.W(P).P [6]
T T T T

Now [5] is incoherent since it requires that KcP is true when P is false which contradicts the epistemic

axiom KcP->P.

So the only case which satisfies the characterisation of [1] as a definition expressed as a biconditional is

the case where [6] obtains. However this means if W(P) is true then P cannot be false but since:

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

the truth of A.'

It follows that in the //premise// [1] W(P) entails P which is precisely the conclusion that Merricks

somewhat circuitously and redundantly seeks to show. The Nizkor definition of circularity is met.

>>Perhaps we have been wrong in considering that Merricks gives a definition then. Maybe a characterization

is a better term?<<

Since you have not shown that Merricks 'purely formal characterisation' of W(P) expressed as a biconditional

differs from a definition of W(P) expressed as biconditional then I am somewhat unclear as to why it 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.[sic]

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

Except that it is not an analogy but has an similar logical which I thought showed the problem with Merricks

assertion that:

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

as 'a purely formal characterization of warrant'. I thought the problem was obvious since it would be

patently absurd to assert that:

'x is a bachelor if and only if x is a man and x is unmarried'

is a purely formal characterisation of 'man'.

>>"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."

This seems something of a red herring as there is a clear distinction between a natural language and a formal

language. Merricks 'a purely formal characterization' would indicate that:

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

is a sentence in a formal rather than a natural language. A formal language is a language:

'a language designed for use in situations in which natural language is unsuitable, as for example in

mathematics, logic, or computer programming. The symbols and formulas of such languages stand in precisely

specified syntactic and semantic relations to one .'

('Formal Language': Collins English Dictionary)

Moreover:

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

Is a single definition in a formal language' and not a 'comprehensive list of lexical definitions' in a natural language so it does not follow that in a single definition there is a 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?<<

In part the position but more significantly as I have pointed out one cannot argue that:

'x is a bachelor if and only if x is a man and x is unmarried'

is a 'formal characterisation' of 'man' or 'unmarried'.

>> 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'.<<

Indeed, but this surely does not mean that the reader would think that the sentence:

'x is a bachelor if and only if x is a man and x is unmarried'

constitutes a definition of 'man' or 'unmarried' so the relevance of your 'hierarchy of concepts' is obscure.

>>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.<<

Some examples which actualise this possibility might be helpful.

>>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)<<

This is not a biconditional so it cannot be an example of 'the possibility of giving a definition where the definiendum is part of an expression on the right hand side of a biconditional'.


bs

Oops! Sometimes when I post from Notepad I get unnecessary line breaks inserted. Not sure why.

Existential quantifier? Odd.

bs

Hi Bx4,

So we can get some symbols to work, splendid.

Before we deal with the characterization/definition issue, let's try once and for all to sort the logic.

I'm finding the logic part of your post very hard to follow, I wonder whether something might be amiss here given the formatting problems you've mentioned? Maybe you could respond again to the following with an edited version?

So, we have:

KcP<=>BcP.W(P).P [1]

which we can expand as

(KcP->BcP.W(P).P).(BcP.W(P).P->KCP) [4]

So for this to be a definition [1] and hence [4] must be true since both the conditional and its converse must be true, there we agree. But consider KcP = F, BcP = T, P = F and W(P) = T

Then both the conditional and its converse /are/ true. This is because both reduce to F -> F which yields T. However, with these values, W(P)->P is false. Hence this cannot be circular.

ttfn

Hi psi

Apologies for delayed reply. Busy undecorating and getting sucked into referendum.

I am replying to your #168 because it may be the nub of our disagreement.

>>Then both the conditional and its converse /are/ true. This is because both reduce to F -> F which yields T<<

However in Q1 and Q2 what is state ind s that the conditional /and/ its converse are BOTH true. These are statements about the truth values of the separate elements, conditional and converse, individually rather than their conjunction is true. Though of course, if the conditional and it converse are both true then the conjunction, the biconditional, is also true.

(Note that I am not saying that your ' F -> F which yields T' is generally wrong but under the constraints of Q1 and Q2 which //requi re// the conditional and converse are both true //not// false.

I have some difficulty with your setting KcP to F.


>> there we agree. But consider KcP = F, BcP = T, P = F and W(P) = T <<

If we consider Merricks putative 'purely formal characterization of warrant':

'S knows that

Hi psi

Apologies for delayed reply. Busy undecorating and getting sucked into referendum.

I am replying to your #168 because it may be the nub of our disagreement.

>>Then both the conditional and its converse /are/ true. This is because both reduce to F -> F which yields T<<

However in Q1 and Q2 what is state ind s that the conditional /and/ its converse are BOTH true. These are statements about the truth values of the separate elements, conditional and converse, individually rather than their conjunction is true. Though of course, if the conditional and it converse are both true then the conjunction, the biconditional, is also true.

(Note that I am not saying that your ' F -> F which yields T' is generally wrong but under the constraints of Q1 and Q2 which //requi re// the conditional and converse are both true //not// false.

I have some difficulty with your setting KcP to F.


>> there we agree. But consider KcP = F, BcP = T, P = F and W(P) = T <<

If we consider Merricks putative 'purely formal characterization of warrant':

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

KcP<=BcP.W(P).P (1)

Then I am unclear how you can set KcP to false since this means that ¬KcP is true but Merricks 'characterization' is of the context where 'S knows that P...' and not 'S does not know that P...
bs

Hi Bx4,

I agree, this is the nub.

>>However in Q1 and Q2 what is state ind s that the conditional /and/ its converse are BOTH true. <<

As I said, with the truth values I have assigned, they /are/ both true.

>>These are statements about the truth values of the separate elements, conditional and converse, individually rather than their conjunction is true. Though of course, if the conditional and it converse are both true then the conjunction, the biconditional, is also true.<<

Exactly. So with these truth values assigned to the individual variables, the conditional and the converse are both true. Hence the conjunction of these is true.

>>(Note that I am not saying that your ' F -> F which yields T' is generally wrong but under the constraints of Q1 and Q2 which //requi re// the conditional and converse are both true //not// false.<<

The conditional and the converse /are/ both true even though in each case both the antecedent and the consequent are false.

>>I have some difficulty with your setting KcP to F.

If we consider Merricks putative 'purely formal characterization of warrant':

'S knows that <<

I am unclear what difficulty you are having. Merricks' characterization might more usefully truncated thus: 'S knows that p iff...' There is no assertion that S knows p entailed by the iff statement.

Referendum result: are you relieved, disappointed, neither or both?

ttfn

Hi Bx4,

I hope you are ok.

I visited Edinburgh at the weekend, my first ever trip to Scotland.
I liked the city a lot.

ttfn

Hi psi

Post referendum went on a serious blue water sail with no access to t'interweb thingy. Just back so I need to catch up with your latest and have a bit of a think before I reply.

Glad you liked Edinburgh. Lived there for 22 years and I'm really fond of it. Still visit often as my daughter now lives in our old house

bs

hi psi

>>The conditional and the converse /are/ both true even though in each case both the antecedent and the consequent are false.

<<

I am somewhat confused here since in:

q<->r (1)

P is the conditional and q is the converse so neither is properly the antecedent and the consequent. Though, of course if one

expands (1) to give:

(q->r).(r->q) (2)

Where p is the antecedent and q the consequent in the first term and vice versa in the second term. Some I am not clear why

you introduce antecedent and consequent here rather than simply saying both the propositions KcP and BcP.W(p).p take the

truth value F. Could you clarify?

I have no problem with your argument that setting:

'KcP = F, BcP = T, P = F and W(P) = T' (a)

yields a biconditional with a truth value of T. However we can achieve the same effect by setting:

(b) KcP = F, BcP = F, P = T and W(P) = T

or

(c) KcP = F, BcP = T, P = T and W(P) = F

making the conjunction BcP.W(P).P take the truth value F in all cases

However as I have already pointed out that if these conditions apply then:

KcP BcP.W(P).P KcP<->BcP.W(P).P (4)
F F T

is formally equivalent to:

¬KcP ¬(BcP.W(P).P) ¬KcP<->¬(BcP.W(P).P) (5)
T T F

So the problem with you argument would seem to be that it is a 'formal characterisation of the conditions under which S does

NOT know P rather than those under which S DOES know P. Merricks' 'characterisation' is after all:

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

which is clearly a definition of the condition where S knows P and not S does not know P. That is KcP has a truth value T

when BcP, W(P) and P individually take the truth value T. So the relevance of your argument base on (a) seems somewhat

problematic.

>>I am unclear what difficulty you are having. Merricks' characterization might more usefully truncated thus: 'S knows that p

iff...' There is no assertion that S knows p entailed by the iff statement.<<

' If sentence X is a logical consequence of a set of sentences K, then we may say that K implies or entails X'

http://www.iep.utm.edu/logcon/

>>Referendum result: are you relieved, disappointed, neither or both?<<

Disappointed

bs

Hi Bx4,

>>I am somewhat confused here since in:

q<->r (1)

P is the conditional and q is the converse so neither is properly the antecedent and the consequent.<<

P doesn't appear in (1) though. In (1) the conditional is q -> r whereas the converse is r -> q

>>Though, of course if one

expands (1) to give:

(q->r).(r->q) (2)

Where p is the antecedent and q the consequent in the first term and vice versa in the second term.<<

I think you might mean 'q is the antecedent...'

>>Some I am not clear why you introduce antecedent and consequent here rather than simply saying both the propositions KcP and BcP.W(p).p take the truth value F. Could you clarify?<<

I was just trying to make clear the distinction between on the one hand the antecedent/consequent and on the other the conditional/converse.

>>I have no problem with your argument that setting:

'KcP = F, BcP = T, P = F and W(P) = T' (a)

yields a biconditional with a truth value of T. However we can achieve the same effect by setting:

(b) KcP = F, BcP = F, P = T and W(P) = T

or

(c) KcP = F, BcP = T, P = T and W(P) = F

making the conjunction BcP.W(P).P take the truth value F in all cases<<

I don't see the relevance of this. The point about the particular values I used was the biconditional being true whilst W(p) -> p is false, showing that the latter is not circularly contained in the former.

>>However as I have already pointed out that if these conditions apply then:

KcP BcP.W(P).P KcP<->BcP.W(P).P (4)
F F T

is formally equivalent to:

¬KcP ¬(BcP.W(P).P) ¬KcP<->¬(BcP.W(P).P) (5)
T T F

So the problem with you argument would seem to be that it is a 'formal characterisation of the conditions under which S does

NOT know P rather than those under which S DOES know P. Merricks' 'characterisation' is after all:

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

which is clearly a definition of the condition where S knows P and not S does not know P. That is KcP has a truth value T

when BcP, W(P) and P individually take the truth value T. So the relevance of your argument base on (a) seems somewhat

problematic.<<

Again I can't see the problem. These are equivalent. To specify the conditions under which something is true is also to specify when it is false. q <-> r or ¬q <-> ¬r, it doesn't matter. The characterization of warrant specifies the conditions under which an agent C knows p. Taking this as a premise does not in any way constitute an assertion that C actually does know p though.I have shown that the premise is true when the agent does not know p, yet W(P)-> p is false. For circularity, W(P) -> p would have to be true for all truth values of the variables in the premise.

You don't seem to have taken on board the idea that if:

"If it is raining then it is wet. (p -> q)"

this does not mean that we should assume it actually is raining in all deductive arguments that take this as a premise.

'>>If sentence X is a logical consequence of a set of sentences K, then we may say that K implies or entails X'

http://www.iep.utm.edu/logcon/<<

I'm not sure how this is relevant either. The sentence expressing the characterisation of warrant can be true whilst W(p)->p is false. Hence there is no circularity.

Sorry the referendum didn't work out the way you wanted.

ttfn

Hi psi

Sorry for delay in replying just returned to hamburg and for some reason could not connect to h2g2. I kept getting a server unavailable message. Have you had same problem? Anyhow seems OK now (albeit on a different machine).

Hopefully will reply to your latest by during weekend.

bs

Hi Bx4,

I got a server unavailable message yesterday but it has been ok since.

By the way, I'm reading Slaughterhouse 5 at the moment.

So it goes.

hi psi

back on though h2g2 seems a tad slow. I'm a bit time poor so may only manage a partial reply.

(Sorry while the truth values in the biconditional truth table //were// correctly aligned (separated by spaces) under the propositions in the original draft the spaces seem to have been eliminated in the 'Preview'. Another 'feature'?


>>Slaughterhouse 5<<

Something of a fan of his. Read all of his novels from 'Player Piano' to 'Breakfast of Champions' though 'Cat's Cradle' was m favourite. 'Slaugherhouse Five' interesting because semi-autobiographical.

>>P doesn't appear in (1)<<

Mea culpa. I meant 'q'

>>I think you might mean 'q is the antecedent...<<

No, I meant q as the converse in the biconditional. I agree if you expand the biconditional p and q alternate as antecedent and consequent but I was thinking in terms of the unexpanded biconditional.

>>I was just trying to make clear the distinction between on the one hand the antecedent/consequent and on the other the conditional/converse.<<

My point was that in the unexpanded biconditional that since q and r are variously antecedent and consequent it doesn't make much sense to refer to q as the antecedent.

>>I don't see the relevance of this. The point about the particular values I used was the biconditional being true whilst W(p) -> p is false, showing that the latter is not circularly contained in the former.<<

My point was that by arbitrarily setting //any// one of the constituent atomic propositions of the converse to false produces the same effect (i.e. the converse is false) but in all cases this only works if you also arbitrarily make the conditional false. However, I am not sure that this is valid if the biconditional is a definition. Given that the truth table for the biconditional is:

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

But for a biconditional to be a definition, from Q1 ad Q2, both the conditional and its converse must be true which is only satisfied by row 1 of the truth table. Eee :

http://www.youtube.com/watch?v=m-hYC5YSj94

I don't really see how you get from the expansion of the biconditional:

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

to your implication W(P)->P or do you mean the supposed entailment W(P)|->P (syntactic consequence)?

It is late and I have an early start tomorrow so this seems a good place to stop but it is probably best to wait until I finish my response to your 175 before you reply (though it would be helpful if you were to clarify what you mean by W(p)->p.

bs

Hi Bx4,

I am interpreting W(p) -> p as 'warrant entails truth', or more precisely, 'if a belief that p is warranted then p is true'.

I hope that helps.

I watched the Youtube video that you linked to, it was most helpful in supplying another example we could use: the definition of a quadrilateral expressed as a biconditional.

Before you continue replying I would urge you to watch this video with a particular focus on what is meant by the 'conditional' and the 'converse', since your latest reply seems still to conflate these notions with the antecedent and consequent.

ttfn

hi psi

Just did a reply to your latest only to have it disappear when I tried to post it. Will recreate in Notebook and try to repost.

bs