Hi Martin,

let's focus on the first sentence:

Paul: believes that he and Elmer agree on three things - and he believes that Elmer believes something that he, Paul doesn't believe. – Ok, that’s two (agreed) things that Paul believes.

I would not exclude that the Paul and Elmer agree on three things and do not agree on an addition fourth thing

Cheers

First of all, are you sure that page 197 is correct? i could not find your example on that page in my copy of W&O, which is also 1960, so presumably the same edition.

Also, I read your informal description as follows. "There are A, B, C and D such that for each of A, B, C , Paul believes it iff Elmer does. Moreover, Elmer believes D whereas Paul does not."

It is not clear whether Paul believes ~D or merely fails to believe D. It would be nice to find out exactly what Quine did say, but first I need to know the page number! Or, you can quote his exact words.

Dear Rohit,

unfortunately I’m not at home and cannot quote from my (German) Quine W+O. What I found was on Google Books and I got a screenshot of p. 197 where the problem of P. and E. is mentioned. I know that this is not the only place in the book where the problem is discussed, but I could not see that other place on Google Books. As far as I can see on my cmputer the German translation was “Paul und Elmar sind sich über genau drei Dinge einig, Paul glaubt nur eine Sache, die Elmar nicht glaubt“. And that was on p 365 (German). So I mixed up the two persons. Paul is the one who believes more. Here I quote my excerpt:

I 364

Strang: Intensionale Abstraktion von propositionalen Einstellungen: Undurchsichtigkeit muss irgendwo angesiedelt sein: hier in der intensionalen Abstraktion (Klassen wären durchsichtig)

I 365

Strang: Identitätsbedingungen stark/schwach: Bsp Paul und Elmar

I ~365

Strang: Glaubenssätze und Zitate können als unendlich viele verschiedene Dinge verstanden werden (Unbestimmtheit)

I 365

Es gibt einige recht spezielle Fälle, wo die Identitätsbedingungen noch über die Erfordernisse der propositionalen Einstellungen hinausgehen:

Bsp "Paul und Elmar sind sich über genau drei Dinge einig" "Paul glaubt nur eine Sache die Elmar nicht glaubt",

Sind die Identitätsbedingungen übermäßig stark, so könnte dadurch eine Revision bewirkt werden, bei der die Zahlen Drei und Eins in diesen Beispielen durch immer höhere ersetzt werden müssen, bis unendlich.

Solche Fälle sind seltsam und wir sind uns daher unsicher im Hinblick auf die hinreichenden Identitätsbedingungen.

Indirekte Rede: sie wird durch zu starke Identitätsbedingungen in Mitleidenschaft gezogen. Bsp "p" und "q" seien zeitlose Sätze, die intuitiv mehr oder weniger äquivalent sind. Wir nehmen an w äußere "p" zum Zeitpunkt t. Man ist versucht zu glauben, dass eine Identitätsbedingung, die so stark ist, dass sie zw. "p" bei t und "q" bei t unterscheidet, den Satz "w äußert "q" zum Zeitpunkt t" falsch werden lässt.

Man ist versucht zu glauben, dies gelte nur für die indirekte Rede, nicht aber für den Glauben, da w zum Zeitpunkt t alle möglichen verschiedenen Dinge glauben, aber nur einen einzigen Satz äußern kann.

Das ist jedoch falsch! Denn was w mit der Äußerung seines einen Satzes (im Sinne der indirekten Rede) "sagt" kann als so viele verschiedene "Dinge" (im Sinne von Gegenständen der propositionalen Einstellung der indirekten Rede) verstanden werden, wie wir wollen. (End of excerpt).

Now my question was not about strong and weak identity conditions for beliefs but about the game of inserting sentences. I just wanted to play that game of inserting, ignoring the problem for a moment, that a proposition might change its truth value.

I'm aware that this is not the problem Quine discussed, but I found it interesting to see what comes out if we try to take sentences.

I am still reading on the subject, sorry.

Please consider this post as a pointer to the question.. Would post an answer soon.

Dear Anup,

Thank you for this little reminder of the question. There were not many fellows attracted until now. Hopefully it is interesting for some others.

The whole problem is kind of artificial because it is not contained in the indicated source. My fault was that I used an excerpt of the text where a mistake was included: between the two assertions - Paul and Elmer agree on three things; Paul believes only one thing that Elmer does not believe - there is a semicolon that was missing in my excerpt. This changes the whole problem. But anyway, as I wrote in my original question, let’s take it as an interesting game and a challenge that shows some interesting insights about the relation between sentences, utterances and propositions. And the riddle functions as well, without being “blessed by Quine”.

What Quine showed with his example was that we are not entitled to assume a thing or an object when someone says “Paul and Elmer believe the same thing”. We should not say “There is a sentence, Paul and Elmer agree upon” when we do not have more information than the sentence “P. and E. believe the same sentence”. What Quine was criticizing was the mentalism that assumes mental or “inner” entities, the mind is in a relation to. There are no such entities.

This is not touched nor questioned by our little game.

An extreme form of mentalism would be: I dreamed of a winged horse > There is a winged horse, I dreamed of.

I am quoting two sentences from your first post and the one from the last one.

he  and Elmer agree on three things - and he believes that Elmer believes something that he, Paul doesn't believe.

Paul agree on three things - and he believes that Paul doesn't believe something that he, Elmer, believes

Paul and Elmer agree on three things; Paul believes only one thing that Elmer does not believe

Other than  semicolon there is another difference.that are also indicated above.

Which combination should be considered?

Dear Anup,

I edited my original question on top according to my best information. Unfortunately I do not own the 1976 edition of Word and Object which is a longer text than the 1960 book. My German translation of 1980 is based on this 1976 English version.

Paul is the one who believes more. We should stay with this version. The other question was, whether “only one thing” and “something” should be understood differently. We should understand both in the sense of a complete sentence that can be true or false. No fragments of sentences, no couples of sentences.

One word about the remark at the end of my original question: “Paul and Elmer are sons of colleagues”: can be read as a hint that the two don’t have to be friends because they do not have to be brothers. They can just be sons of colleagues who even do not know each other.

When they don’t know each other the whole situation is changed. They don’t have to refer to the sentences or propositions of the other.

“Quine – in his Quinean way –…” refers to the style of this author who liked to give indirect hints for his hopefully logically gifted readers. 

After fixing the problem statement as the corrected version of the question on top, let us try to understand what do we mean by agreement. From the sentence that "Paul and Elmer agree on three things" what information we get. Can we define a predicate agree(P,E,x) to mean Paul and Elmer agree on x as

agree(P,E,x) = believe(P,x)  and believe(E,x)

If we want to keep the meaning of x unknown we can possibly define an agreement by a predicate expression

agreement(P,E) = E x : agree(P,E,x)

Therefore the sentence "Paul and Elmer agree on three things" can be represented as

E x,y,z : agree(P,E,x) and agree(P,E,y)  and agree(P,E,z)

We use E to denote 'there exist'

I think we can write it that way. Quine would say that there is nothing like an “agreement” that is an object in the world. For Quine this is also true for e.g. “aspects” or “respects”: when someone says “…in respect to…” there is no such thing as a "respect". But I think for our problem here we can use this notation as long as we do not count agreements. We should not say that there are “three agreements between P. and E.” – and I think we don’t need them anyway.

What is not clear to me is the meaning of the box you are using. I read your notation as “There are three sentences…” That seems unproblematic.

I have edited my previous post. You will not find any box there. I understand what you mean by We should not say that there are “three agreements between P. and E.”  in that case the notation needs to be changed as follows. To denote "P and E agree on three things we should write

E x,y,z : agree(P,E,(x, y , z))

I put coma such that order of the logic does not increase.

I would write it this way (somewhat traditional notation):

(Ex)(Ey)(Ez)(x sent & y sent & z sent & P bel (x&y&z) & E bel (x&y&z))

That is “There is at least one sentence x….

In order to make it become exactly one sentence x we normally would have to write

(Ex)(w)(x sent & (w sent > w = x))

“There is a least one sentence x and all other sentences w are identical with sentence x”

This works for normal objects like tables: it says that there is only one table x, all other tables w are identical with the table x (maybe in a particular room). For sentences it would work even better because we can say that the sentence “Snow is white” is identical with the sentence “Snow is white” which is less counterintuitive than for tables.

What we cannot do is to make it become something like "Any sentence w  "Grass is green" is identical with sentence x "Snow is white" ". 

The first problem is that we could not do it for propositions or utterances, because a later utterance of “Snow is white” has a different time stamp.

The second problem is that we don’t want to exclude sentences y and z. But I think there could be a technical solution for this second problem. If not - will it tell us something about the impossibility of counting sentences?

(Ex)(Ey)(Ez)(x sent & y sent & z sent & P bel (x&y&z) & E bel (x&y&z))

I have one problem with this representation. Since x, y and z are sentences and P bel and E bel are  predicates the operator & used in two different instances should have two different semantics. I think if we do not fix these semantics now we will have problem to retain the sentences y and z in our representation.

As you said we can not count agreement, we can consider the meaning of the proposition "Paul and Elmer agree on three things" as follows. We define a collection C of three things x,y and z and assert that both P and E believe this collection C.

I agree that the second part that is

(Ex)(w)(x sent & (w sent > w = x))

is suitable to represent "Paul believes only one thing that Elmer does not believe."

Anup,

Three repsonses to your two last posts:

1. “&” is no operator but a connective or particle – “bel” is an operator. I see no problem with a changing semantics (or meaning) for different occurrences of “&”. The meaning remains “and”.

Oh, I see what you mean! Can we exclude that Paul believes “& to be “and” while Elmer believes “&” to be “or”? – Well that is not likely to be so. The reason why not is explained by Quine, when he speaks about the other name for these particles (connectives): they are sometimes called “logical constants” – here is why they are called constants:

Assume an alien saying “pling plong buff” – after finding out that “buff” is the logical constant (particle), it will turn out whether it means “and” or “or” or “if then..” for the alien, “pling” and “plong” being whatever object or sentence. An “and” as logical constant can never be translated with an equivalent to “or” in whatever language. It will turn out in the context or in the behavior. It is the only thing that is sure when the language is totally different from ours and totally unknown to us. What we will need to know is whether the alien assents or dissents. Then we will see a difference between “pling or plong” and “pling and plong” (for us, when the rest remains untranslated and also for the aborigine). We will share the situation as far as the constants are involved.

I’ll take some more time for the response to your other post.

A reply to Anup’s last post:

2. No problem for a collection C of three sentences, especially no problem because beliefs are at stake. I.e. the collection could be inconsistent but we can believe inconsistent collections of sentences, that often happens.

3. Notation:

“(Ex)(w)…”

is to be translated: “For at least one object x and for all objects w…”

“(x sent & (w sent > w =x)”

Is to be translated:

“x is a sentence and (if w is a sentence, then this sentence w is identical with the sentence x)”.

The classical example in logic books goes with gods (“any other God w is identical with God  x”)

Henry Ford could be called an “early logician”: “My customers can have any color, provided this color is (identical with) black”.

Type of x is  sentence and I think bel(x) is a logical entity. What are the legal value sets for these two types? Are they same?

Ok, when bel(x) is a logical entity, then we might say something like “It is true that E believes the false sentence x”.

Then we can have

False x or true y = true

False x and true y = false      and so on. But:

Bel false x or bel true y = true

Bel false x and bel true y = true (!)

Let’s see whether this will change the situation. I don’t know at the moment. I guess it will make a difference in relation to the counting.

Does False x denotes a proposition that the sentence x is false? If so then this should have a value true if the sentence x is false and true otherwise. Then there is a problem. Who is deciding the value of the proposition False x. If the believer himself is doing this then there is a contradiction else it is not necessary because  ~bel operator will do the job. Another thing I do not understand is why you have used true and true in two different occasions?  .

"Agree" and "believe" are not comparable. The first is a truth, the second is a belief, or desire. In your "Another way" propositions it seems you've turned the s, that is namely a belief, into an agrees-true different from a believes-true type. Thus it seems not true that P/\E agree-true on s, while P believes-true on s /\ P believes-true (E ~believes-true on s) /\ E ~believes-true on s > the agreement seems to remain still desideratum and lost in a Quinean way.

—g

@Anup

My notation was not good. It could then also be true to write ~x.

Maybe it is better to write

~x /\ x or y: true (because it is sufficient, when one side, y, is true for “or”).

~x: what is the case - : x: what is claimed.

~x and x and y: false (because of the meaning of “and”)

This is for the sentences, not for beliefs.

The difference for beliefs is then, that Paul and Elmer can agree on a false belief with no problems. They can also disagree on a false or true belief, as far as I see it.

@Giuseppe,

I agree that there is a decisive difference between agreeing and believing. Now I modified my question again: we will have “Another way (I)” and “(II)” (I didn’t change anything in the formulas).

The “Another way (I)”-expample is the least problematic: I intended it to be the case, when Paul and Elmer don’t know each other. They could believe sentences like “Grass is green” and “Snow is white” quite independently from each other, and in addition to that Paul could believe that the maiden name of his mother was Smith while Elmer has no opinion about it. We could say there is no inserting at all. That was the simple solution I suggested, at the same time believing that it is too simple.

“Another way (II)”: in this case both know each other and their beliefs are related to each other. What I had in mind was something like this:

P and E agree on the color of snow, grass and strawberries but they do not agree on the maiden name of Paul’s mother. (Elmer believes it was Miller). So I wrote

E: … E /\ P ~ agree on s

P: … E /\ P ~ agree on s

s being something like

The maiden name of Paul’s mother was Smith.

Why should we not say “They agree that they do not agree”? And E and P say “We agree that we do not agree”. My point was, that this (together with the agreements on the color of snow, grass and strawberries) is the fourth point, they agree upon.

Ok, now I see what you mean: agreement as kind of meta-belief. P” Snow is white” is a simple sentence, “We agree that we don’t agree that Paul’s mothers name was Smith” is a complex sentence. In order to keep the same level we would then have to say “We agree that we agree that Snow is white”?. But I think this is not so: we already have:

P /\ E agree on p …~ agree on s.

The next level is to simply write

P:…

E:…

At the beginning. (s being “The maiden name of Paul’s mother was Smith.”)

BTW: the reason why I wrote “P /\ E” instead of “We” was that I wanted to avoid propositions.

-Martin.

@ Giuseppe:

I read your post again. Agreeing/believing:

#1 As far as I see it, agreeing is to believe the same sentence – it could be explicit, implicit or kept secret. In our case it is only important that the sentences can be counted.

#2 s is not a belief but a sentence (such as p, q, r). This seems the decisive point we are disagreeing about.

Inserting, embedding, counting: the problems come into play when we have to insert names or sentences or relations: Example “I believe that Paul believes that I believe the same sentence but I don’t” – E.g. the number of sentences can be enlarged or reduced by writing “the other” instead of a name or by “first sentence” instead of “same sentence”..

-Martin

Martin,

(el1) E bel p


(el2) E bel q


(el3) E bel r

(pa1) P bel p


(pa2) P bel q


(pa3) P bel r


[1](pa4) P bel s



> e(el1;3) agr P (pa1;3) ................................................ [AGR1]


[1] >> E bel (E ~bels /\ P bel s) ≣ s(E ~bel ~P bel)


≁


[1] >> P bel (E ~bel s) ≣ s(P bel E ~bel)



> (-s (E bel P bel))  /  (s (P bel E ~bel)) = Δ ................. [~AGR1]


I don't find a way to consider the agreeing via s for [1].


Maybe the way is via (E):(P) as daughters of C'(E):C''(P) ≣ Clim(E;P)


That is to say the opposite of absorbing / inserting, i.e. "scattering".


But I've to muse.

—g

Giuseppe,

I am curious to see your “scattering”. I did not understand “ s(E ~bel ~P bel)” (right hand side of [1].

In the meantime I got some ideas that have not much to do with what Quine was talking about, but anyway:

Can one believe a sentence without exactly knowing what it means? I think Paul can believe that the majority came to the meeting, believing that the majority is 8 people while Elmer, he too, believes that the majority came but he takes the majority to be 9 people. Is that a problem?

Another example: Can one believe ”P hits E” without believing “E is hit by P”? Or “P sprayed paint on the wall” – “P sprayed the wall with paint”?

When we decide to have only sentences, it seems to be a problem; we already banned propositions. But I think we should say that there is only one sentence “P hits E” and the passive form is simply not relevant or if, then we have to count it as one more sentence (= chain of symbols).

Here I invented an even meaner example:

(1) The nice members helped the members.

Some limiting cases for this:

(2) The two guys had a busy afternoon.

(3) The two guys had a lazy afternoon.

(4) Everybody helped himself.

(5) Nobody knew what to do.

Can Paul and Elmer agree on (1) while Paul believes (2) and Elmer believes (3)? I think this is possible! It is even possible to believe (2) and (3) at the same time when there are only 4 members (Special limiting case).

Let there be 4 sentences a, b, c and d. I have a confusion. In order to agree on a say, is it necessary that both P and E should believe a?  If so let us assume that  both of them believe a but E does not know that P believes a and P does not know that E believes a. In that case can we assert that they agree on a?

Good question. I think we should say that they can agree, but I am not a native speaker of English. I think of an opinion poll where it is said that e.g. the majority of the people agree that a project should be stopped. These people don’t know each other as well. Or would you choose another word for that in English?

The fact is that agreement [AGR] ⊧ belief [BEL],

i.e. P/\E agr (a) = (P bel a) /\ E( bel a) ≣ [AGR]a  ⊢ P[BEL]E

while it is not valid the contrary, because belief does not infer agreement by itself.

BEL belongs to the class of on-way subjectivity [in part. desiderata], while AGR assumes at least two-ways, whose end-edges are at minimum 2 [I could agree with myself, who is 2nd to my 1st subj "me" — thus the sentence stops to imply the subjective desideratum, to imply a surjective relation].

AGR needs relation [ethym. Fr. a gré — to give grace, pleasure (surjective)].

BEL implies on-way gradational objectification [of reality: where every object has to be treated as "item"; this is the reason why I could reverse the right hand side of my [1], since in the BEL domain P is any item].

Now, maybe the question of the scattering is to be focused in terms of set theory. As Martin guessed, "if P and E would not know each other…" we would not have any chances that their relation steps from the subjective state [because of objectification] into relationship, determining the space for AGR. As set-doughters, the boundaries of their parents [C]'' do not restrict them, but do inject objective certainty of relation.

Mh?

—g

As to agreement: I hope you will be able to read the copy I took from Quine’s Word and Object, 1960 p. 197, (from Google books) where he uses the expression “agree”. (Attached)

Anyway, I think it would be perfect to say they “believe the same thing”. They could do it without knowing each other. But that would be the trivial case: we would have to count p,q,r for Elmer and p,q,r,s for Paul and that’s all.

Giuseppe, I will have to think over your proposal. I think I understand 80%, e.g. the asymmetry of BEL and AGR you are talking about and I agree (or should I say I believe to believe the same thing?). I still don’t understand “s(E ~bel ~P bel)”.

Mh – (Medium Heavy? Mostly harmless?)

-Martin.