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MatPlus.Net  Forum General Problemas - July 2015, issue n. 11 |
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| | Page: [Previous] [Next] 1 2 3 4 | | (21) Posted by Geir Sune Tallaksen Østmoe [Tuesday, Jul 21, 2015 22:27] |
And I think it was Agdestein and Shirov who both had severe time trouble, and after move 40 they found themselves in a position where Black had two black-squared bishops. It was obvious from the position that something had gone wrong, and nobody could tell how. Did they stop playing chess after the illegal move was made?
Who knows how often an illegal move has been made in time trouble, neither player notices, and nobody knows in aftermath that anything illegal had occured? I would never claim that the players didn't play chess after the illegal move was made.
What implications this has for problem chess, and whether this means that Andrew's problem are orthodox or fairy... I don't know. I can understand both opinions. What I can say is that I like the problems anyway.
| | | (22) Posted by Nikola Predrag [Tuesday, Jul 21, 2015 23:19]; edited by Nikola Predrag [15-07-21] |
Joost, you seem to be confused about the relevance.
My opinion about natural could be relevant or irrelevant concerning the Andrew's proposals for CHANGING the rules.
Concerning the mentioned Andrew's problems, there is no "my opinion", there are EXCLUSIVELY the rules which define chess. To help you to come to a CLEARLY VALID conclusion about these problems, I suggest you to answer the question:
Is the statement "the side in check has the move" CLEARLY VALID ALWAYS in chess, or not?
Andrew has written "...In any case, your "hidden axiom" is not an axiom, but already an implication of the other rules..."
This should actually mean that he's aware that in chess, the checking side can't have the move. And that implicates that his problems are NOT CHESS.
You are building the case on some absurd premisses. This is legitimate, if you can correctly handle an absurd premiss. Forgetting it, and reasoning as if the premiss was not absurd, might be deceiving. When you introduce an irregularity, the wholeness of the rules is instantly disqualified and the definition of the system is automatically changed, but what is that new definition? And you try to carry on as though nothing was changed.
Yes, after the illegal castling, the players were not playing the same game anymore. If the respective position was legal, they were playing a part of a new game from that position. It was certainly not the same chess game which they had begun from the initial position. Or, are you trying to say that the illegal move (castling) was "playing chess"?
What if the whole game was a series of "illegal moves", unnoticed until the game was over? Was this "playing chess"?
I repeat, there's by far greater absurd in Andrew's problems than the legality of position. If you're convinced that the solution of Andrew's #2 is "playing chess", that's indeed not my trouble.
Anyway, I've contributed with something or nothing. It doesn't seem that you find that as being relevant for the issue so, I wish you good luck in further speculations.
| | | (23) Posted by Andrew Buchanan [Wednesday, Jul 22, 2015 05:38]; edited by Andrew Buchanan [15-07-22] |
Dear Nikola,
Thanks very much for your thoughtful replies.
Firstly, I now think that you and I were using the word "axiom" in different senses. I was using it in the sense of "premise": the initial things we assume about a world, from which other things are deduced. E.g. Euclid's Axioms, Peano Axioms of Arithmetic etc. You only the other hand took it to an essential property which is deduced *from* the rules; and if a certain axiom cannot be deduced, then the rules must be wrong. Hilarity ensued from this misunderstanding between us. :-)
So to be clear in the Lese Majeste #2 above I think that Black is in check but not checkmated, because the latter would require it to be his turn. And I think the checking does not matter, except to indicate that the diagram is illegal.
For you the notion of being checked only when it's one turn to play appears quintessential to what is chess in the larger sense. But not all people would agree: (e.g. me for one). People (including me) have taken it for granted historically, but that doesn't mean it's essential. For most of history, the checking player would just capture the king immediately, so such positions were dull. But now that this capture is illegal with the wonderful unnecessary clause in Article A1.2, we find that such illegal positions can be interesting.
So to my mind, this is an example of the famous philosophical "Problem of Induction" (see https://en.wikipedia.org/wiki/Problem_of_induction). Another example of this is the inference that "all swans we have seen are white, and therefore all swans are white", before the discovery of black swans.
Now I understand that your real objection is to me "changing" the rules. Chess13^64 absolutely does not do this. It merely extends the set of legal starting positions. If one happened to start playing Chess13^64 from the regular game array, then the rules would apply just as in Chess1. And also Chess13^64 is consistent with Chess960. Naturally if one expands a game, some different behaviours would be observed. But the subjective experience of encountering those different behaviours could be as positive for some as they are apparently negative for you, Nikola. And the overall sense to me would remain sufficiently chesslike. I would not feel that I am now playing Scrabble or cricket. :-)
All the best,
Andrew.
| | | (24) Posted by Andrew Buchanan [Wednesday, Jul 22, 2015 05:58]; edited by Andrew Buchanan [15-07-22] |
Is Lese Majeste a fairy concept? Well initially I felt something so alien must be, but Dirk Borst of Probleemblad convinced me otherwise. The decisive point for me is that the conventions appear to regard illegality and fairiness as separable. Similarly, Blathy's wonderful illegal #30 is not fairy.
Is Lese Majeste a joke concept? I guess it depends what you think of as a chess problem joke. For me, a joke problem is one where the solver has to disregard the exact stipulation or diagram and think "outside the box". On the contrary, with Lese Majeste everything is spelled out and you must interpret the rules very literally.
Finally, I want to report that I have myself carried only playing in an illegal game. This was in the Edinburgh inter-schools league when I was a boy. As White, reaching for the clock, my sleeve accidentally swept queen's rook and a-pawn from their starting squares onto the floor. Embarrassed, I picked up the rook and put it back on a1, but the pawn had rolled further under a desk and it did not register for me or my opponent that this piece too was missing! So we played on. Indeed developing my queen's rook directly up the a-file contributed to a fast kingside attack that won me the game. It was only a few days later, looking through the game score, that I realized something very odd had happened. But we had still been playing chess!
| | | (25) Posted by Hauke Reddmann [Wednesday, Jul 22, 2015 09:36] |
My 1 cent: Lese Majeste is the chess equivalent of a division
by zero. Crazyness is bound to happen.
Now, keeping the analogy, all Analysis (the math area) is
based on division by zero. The trick is to formalize the
crazyness until the result is meaningful.
Our first task thus shouldn't be heated discussions whether
this condition is to be classified as fairy (I couldn't care
less) but whether the condition is meaningful. (I read the
article and then solved the problem in five seconds, which
is circumstantial evidence it is, but then, I am a
professional troll :-)
Surplus Snippet: If you answer in rapid chess on an illegal
move hanging the king by capturing it, YOU lose.
So let's start the trolling:
 (= 2+3 )
Lese Majeste - Stalemate?!
| | | (26) Posted by Joost de Heer [Wednesday, Jul 22, 2015 10:31]; edited by Joost de Heer [15-07-22] |
> Is the statement "the side in check has the move" CLEARLY VALID ALWAYS in chess, or not?
No, because illegal moves can be played. Usually, when such an illegality is noticed, the position is/should be reverted to the position before the illegality (7.5a), but such positions -can- arise.
> (Lese Majeste) stalemate?
Yes. Although one could argue that it's lost for white, since his only move results in immediate loss of the game :).
| | | (27) Posted by Nikola Predrag [Wednesday, Jul 22, 2015 14:28] |
Joost, what is relevant critical point in your post 26?
A point where the things change character?
Why the "play" after illegal position is cancelled and the game is reverted to the position before the illegality, if an illegal move can be played?
You are speculating about an absurdity without recognizing it.
You insist on treating the rules mathematically but mathematics requires mathematical definitions. Firstly, the RELEVANT elements have to be abstracted from a worded formulation. Mathematics requires the terms with an exact meaning of the words.
Chess rules define clearly what is chess. The very mentioning and treatment of illegalities ADDITIONALLY CONFIRMS that illegality is NOT chess and the chess game can continue only from the moment before it ceased to be a chess game.
How could you miss this very clear meaning of the rules?
You abstract an irrelevant meaning and what's worse, an absurd one - "illegal moves can be played". You have "dropped" the essential relevant meaning-"...played in a chess game".
A 7 says that a game will be continued as BEFORE the irregularity and NOT AFTER.
What do you expect, that the rule should explicitely say that "game may not continue after an irregularity because it would not be a chess game anymore"?
The rules expect a bit of common sense from a reader.
And to deal with the destructive absurd insisting of some individual, there's an arbiter who will interprete the rules.
Why should I even listen about the "competition rules", they are written only for practical reasons because there will always be some "wise guy" who'll try to spoil the purity of chess. And the most relevant factor in the competition rules is an ARBITER.
There is nothing in A7 defining that an irregularity is chess. There are "basic rules" to tell you clearly what is chess and that should be relevant for an arbiter.
The play after an irregularity is cancelled! The Andrew's solutions are instantly cancelled including the diagram, because the irregularity has been instantly noticed! And the author himself has obviously noticed the irregularities before explicitely mentioning the illegal position!
That is RELEVANT, how could you missed it?
Andrew, do you comprehend what is an axiom and why it might be accepted as self-evident?
The induction of white Swans is certain as much as the completeness of the relevant system is certain. When the system is changed(widened), the certainty of induction is changed.
Chess is a completely defined system with the self-evident axioms. You should first define a wider system (e.g. "the curved spacetime" or whatever) to affect the certainity of the "self-evident".
And that would not be the chess as it is defined now.
If you should apply the present definition, the case is clear and there's nothing to be added.
I don't have time to ask repeatedly for pointing out the rellevant critical points, and after all, you pay no intention to it.
With such a manner of speculating, it would be foolish from me to expect you to care about the differences between chess as related to mathematics and chess as related to the human concepts of the world we're living in. That's related to the suggested changes of the rules.
| | | (28) Posted by Joost de Heer [Wednesday, Jul 22, 2015 18:03] |
Let's explain with an example which you hopefully will understand.
In football, playing the ball with your hands is illegal. Yet sometimes it happens, and sometimes even a goal is made by playing the ball with the hand. Is this allowed? No. Is the goal valid? No. Does it occasionally count as a goal, and the game continues? Yes, because a hands ball needs to be seen by the referee. Are the players stopping playing football after a hands goal? Of course not. The only difference with the chess rules is that even a long time after the illegality, the illegality can/should be reversed, whereas in football the irregularity, once play continues, is accepted.
Rule 7.5a states: "If during a game it is found that an illegal move has been completed, the position immediately before the irregularity shall be reinstated. If the position immediately before the irregularity cannot be determined, the game shall continue from the last identifiable position prior to the irregularity. Articles 4.3 and 4.7 apply to the move replacing the illegal move. The game shall then continue from this reinstated position.". For a composition, no previous positions are available, so there is no last identifiable position prior to the irregularity. One could decide two ways: "This is rubbish!" (Nikola's interpretation), or "Well, let's continue anyway" (Andrew's interpretation). I follow the latter interpretation.
| | | (29) Posted by Sarah Hornecker [Wednesday, Jul 22, 2015 18:42] |
So I collect from your information that technically playing a ball with the hand is allowed as long as no referee sees it?
| | | (30) Posted by Geir Sune Tallaksen Østmoe [Wednesday, Jul 22, 2015 19:01] |
Regarding stalemate, I believe we are really dividing by zero...
5.2b: "The game is drawn when the player to move has no legal move and his king is not in check. The game is said to end in ‘stalemate’. This immediately ends the game, provided that the move producing the stalemate position was in accordance with Article 3 and Articles 4.2 – 4.7."
3.9 states that you cannot leave your king in check, so Black's last move in Hauke's position cannot have been in accordance with Article 3. Therefore, this is not stalemate according to 5.2b. But what is it then, considering that we don't have any known position before the irregularity occured?
| | | (31) Posted by Nikola Predrag [Wednesday, Jul 22, 2015 19:57] |
When you understand the whole concept you might instantly notice an alien interpretation. The wholeness of a concept might help you to see that some formally legal interpretation was not intended in the system and that a legal interpretation which appears as intended should prevail.
But that could and should be discussed!
I won't discuss the clear situations. The game-array position is legal so, A7.5a actually might require playing a new game, add here A7.2a.
About an interpretation of Codex, first decide whether you'll use or misuse the flexibility of definitions. I don't think that "conditional" acceptance of illegal positions includes making a mockery of chess.
Excessive number of Pawn-captures is one thing and the "checking side having the move" is essentially different because the legality of the very first move of the future play is disqualified.
It's becoming entertaining, you see a difference between football and chess, although only one so far :-)
| | | (32) Posted by Andrew Buchanan [Thursday, Jul 23, 2015 04:35] |
(1) Discounting Article 7
The domain of chess problems is a "film of a book". Adaptation of the rules is required. This broadly requires three elements:
(1) "framing": elimination of rules & clauses which only have bearing on OTB play
(2) addition of conventions concerning a default history of the game
(3) addition of conventions concerning player decisions
Task (1) is performed (inexpertly) by Footnote 12, which specifically excludes Article 7 covering irregularities. But to be diligent, I also itemized other parts of the Codex referring to illegal positions. These indicate that illegal positions are deprecated but may be permitted. Lese Majeste aligns itself with this long tradition in the problem world of tolerating worthwhile illegal positions and just playing chess in them. Blathy's #30 shown earlier is a great example.
And just because a stipulation acknowledges that the diagram is illegal (to avoid confusing the solvers) does not mean that somehow imaginary players in the game are "aware" that an irregularity has happened, and somehow Article 7 is resuscitated, and sneaks in again by the back door. Article 7 is out.
Therefore I discount all arguments based on Article 7. And I am not being original or controversial in so doing.
(2) A real "divide by zero"
Geir (sorry I don't know if this is the name you are called by) raises a fascinating point in pointing out the clause at the end of 5.2b, which says that the last move must be legal. How do I take this? The morale I have always taken of this kind of clause is how deeply the basic rules and competitive rules are enmeshed, and hence the technical difficulties in making "the film of the book".
This is a real "divide by zero" and can be handled by proper framing: taking all the irregularity clauses out of the "Basic Laws of Chess".
(3) Axioms and theorems
I am asked, what is an axiom? I take it to be one of the propositions on which a logical structure is based. Axioms do not require proof: indeed if one axiom could be proved from the others it should be discarded as redundant.
For chess problems the axioms are the "Basic Laws of Chess" (Articles 1,2,3,5 & 9, having removed OTB aspects) together with certain conventions, especially covering tasks (2) & (3) above. We must overlook that they are all a bit loosely worded.
That a checked player has the move is a *theorem* (let's call it CM), rather than an axiom. It is provable from the axioms/rules. While the axioms/rules are "self-evident", theorems (including CM) are not self-evident: they derive their validity from the axioms.
...
I need to head out now into the torrential tropical rain to take a boat to the little island of Cheung Chau to see a friend. In my next post I will talk about white swans.
| | | (33) Posted by Nikola Predrag [Thursday, Jul 23, 2015 10:10] |
Andrew, to comprehend the concept of axiom the meaning of "self-evident" is crucial. Any nonsense could be claimed as being an axiom.
| | | (34) Posted by Joost de Heer [Thursday, Jul 23, 2015 10:37] |
> Andrew, to comprehend the concept of axiom the meaning of "self-evident" is crucial. Any nonsense could be claimed as being an axiom.
Nonsense. You can build perfectly valid systems on non-selfevident axioms. Ever heard of p-adic numbers for instance?
The only property that an axiom should have is that it's not in contradiction with the other used axioms, and that it can't be derived from the other used axioms. Self-evident, logical, natural, whatever, are irrelevant properties for an axiom.
| | | (35) Posted by Nikola Predrag [Thursday, Jul 23, 2015 13:35]; edited by Nikola Predrag [15-07-23] |
Joost, it appears as an improvement in perceiving the relevance. However, when you say "Nonsense.", I'm not convinced that the relevance is understood.
The wording is not relevant, while the meaning of "self-evident" is. You have replaced the "self-evident" with the "not in contradiction with the other used axioms".
Now it's even less clear what might be the essence of an axiom. How do you know that some axiom is not in contradiction...?
The question is, how and why a formulation of an axiom comes to existence.
Was there some "first" axiom which was neither in contradiction with nor derived from the other axioms, since no axioms were there?
Do the axioms just pop-up randomly by themselves out of nothing?
| | | (36) Posted by Hauke Reddmann [Thursday, Jul 23, 2015 13:59] |
Hey, I just *began* trolling...
 (= 1+1 )
Lese Majeste - White is checkmated :-)
In my secret identity as Eric-Half-A-Mathematician, I'm completely
on the formalist side - axioms just have to be self-consistent.
And Lese Majeste looks self-consistent for the moment. (But I
just began trolling :-)
@Football: Not a good example, hand is a foul. (Like, what do I
know, smoking OTB :-) A foul leads to a punishment but the game
goes on. An illegality would be something that forces the referee
to break off the game permanently.
Maybe it helps to generalize the condition to an explicitely
fairy one: Doom Chess. Black has one Invulnerability Pill
(no permament iddqd cheatcode please :-) He can swallow it on
a White check instead of making a move and his king can't be
captured. After Whites next move, the effect is off and the play
continues normally.
| | | (37) Posted by Andrew Buchanan [Friday, Jul 24, 2015 02:42]; edited by Andrew Buchanan [15-07-24] |
Dear Nikola,
I am happy that my previous post "compiled" almost to the end before it encountered an objection from you. :-) I view this as progress!
So let's deal with "self-evident".
I am not familiar with the term "self-evident" as a mathematical term. I don't think it is one. It appears in wikipedia in an article without references or sources, and seems to be a term from epistomology (philosophy). Mathematically, I take the position that any set of propositions can constitute a set of axioms. (Yes, "any old nonsense"!) If the axioms are inconsistent or redundant then that may or may not be discovered or discoverable. In this viewpoint, I described the axioms as "self-evident", and all theorems are not "self-evident", but that's really saying that "self-evident" is a synonym for "is an axiom", which is not I think what you mean.
If there is a mathematical (or other) definition for "self-evident" in your mind, please simply tell us. Then we can be on common, solid ground. However, I have more to say in this post anyway :-)
Clearly, one can critique a set of axioms not just in terms of possible consistency or completeness, but also in terms of how accurately they represent some underlying world. One might invent a term that says, with respect to that underlying world, certain axioms or even theorems may be more or less intuitive than others. Maybe that's what you mean by "self-evident". But finally, for a world as small as Chess1, one can hope to capture with one's axioms all legal games and only all legal games. That seems very doable, since the original basic rules are constructive in form.
I want to extend chess in new, interesting and humorous directions. As we go further, certain properties we took for granted are no longer going to apply. I don't really understand why this is a problem for you. Specifically, I don't understand why you would form a particular attachment for the idea that only the person with the move can be in check. It turns out that there is an exception for starting positions, but normally in Chess1 we don't notice it because in the game array position, neither player is in check. (By the way, in the current Laws there is a insignificant bug which implies that White does not have the move in the starting position! This is because "having the move" is defined in terms of who moved last.)
Nikola, we seem to have different tastes. To my mind, one of the joys of mathematics is how familiar intuitions are no longer applicable when one pushes ideas further and further. For example, the progression of normed division algebras from reals to complex to quaternions and finally to octonions. People reel when they face the square root of -1 for the first time in the complex numbers. But that's just the first hurdle. Beyond that, quaternions are not commutative, and the octonions are not even associative! To me, there is is an "aesthetic dissonance" here which is extremely attractive.
So that a theorem ceases to be true as we extend our scope can be very much a plus factor. It means we are dealing with a more interesting, richer set of axioms.
In a much humbler way, I find it nice in our little world of chess problems, that as we go from Chess1 through Chess960 to Chess13^64, we will find that weird things are happening. Pawns may be on the first or eighth ranks. We may have zero or many kings on each side. Castling rules may have to be adapted. And once we have decided what kind of multi-rex rules to adopt, we may confront the challenge of how to deal with the player to move already checking his opponent.
However, there is a more practical side to this. I have argued that the rules of chess must also apply to illegal positions, since otherwise there is a requirement to provide a proof game for every diagram. Chess13^64 is maybe too big. The question is where to draw the line. It is easy to define the smaller subset of positions in which each side has exactly one king, and there are no pawns on the end ranks. Call this ChessN. This excludes most of the weird behaviours in a very simple way. And in fact N could be calculated very easily. But going further to exclude the positions in which "the wrong player is in check" is much more complicated to specify, and arguably we should not exclude this set of illegal positions which the Laws already make a specific point of handling! So to be faithful to the Laws, wherever we precisely draw the line we should ensure that the illegal check positions are inside our scope!
| | | (38) Posted by Nikola Predrag [Friday, Jul 24, 2015 14:55] |
Dear Andrew,
the very first and the deepest ground-stone for the abstraction is the RELEVANCE.
You try to answer my questions by complicating the reasoning. You apparently think that I'm trying to complicate by introducing NEW premisses so, you're polite enough to bother with the supposed complications. Thank you sincerely for your efforts but this actually only wastes a lot of time, both yours and mine.
I'm not interesed in computing or logical processes, at least not before the INPUT was determined. I ask what is "INSIDE" some point, what makes it a critical point and where/when that might be relevant.
The one and only relevant feature of an axiom is about how it can be used as a fact (truth, valid premiss or, whatever term indicating such an essece of that feature).
A proven theorem has the same relevant feature and if I can use something as a fact, I don't care about the terminology. The relevance of "self-evident" or "derived from..." is "being considered as a valid premiss".
I'm heavily discoureged about disscusing since it seems that you don't care about the relevance right from the first and basic steps of your reasoning. How much should I write in hope to explain the very essential meaning of "relevance" (being relevant/irrelevant) for an abstraction.
The whole post comes out of your statement from the article:
--------------------------
Article A1.2 of the FIDE Laws of Chess concerns checkmate. In 2000, it was amended, to add that:
"Leaving one’s own king under attack, exposing one’s own king to attack and also ‘capturing’ the
opponent’s king are not allowed."
It’s the last part I want to focus on is this article: that ‘capturing’ the opponent's king is not allowed.
It’s a strange kind of rule, because it should never trigger if the players are otherwise following the
Laws. But whatever the impact on over-the-board players, it’s a cute gift to problemists, as I hope to demonstrate in this article.
---------------------------
Now, you "focus" on the statement (S) = "capturing’ the opponent’s king is not allowed".
And (somewhere ), you said: "(S) is redundant". That's the first alarm that you should have heard.
How can it make a difference if it's redundant?
Before (S) was added you knew that a King could never be captured in chess. What a difference could make the addition of the statement which you already knew?
The very origin of your enthusiastic fantasy is a delusion/phantom, at least as concerning the problems shown in your article.
A complete neglection of the relevance.
Answer:
What is the relevance of (S) for your reasoning about these problems? Or, what would be different without (S) (or, before (S) was added)?
Please, answer shortly, just mentioning the crucial point(s).
You're actually changing chess-system, apparently without even having a "sense for relevance". It's not about you personally, that "lack" appears as a dominant characteristic of the new millennium due to the fascinating power of computing. I understand, computing is much easier than thinking.
Nevertheless, the relevance comes first, before any simple or complicated reasoning.
| | | (39) Posted by Andrew Buchanan [Friday, Jul 24, 2015 20:00] |
QUOTE
What is the relevance of (S) for your reasoning about these problems? Or, what would be different without (S) (or, before (S) was added)?
This is a good question, but I don't think you will like my answer :-)
In the QQKqrrnk diagram above, without (S), White could simply capture BK. ((S) is the only rule preventing this.) This would not mean that Black has lost: on the contrary, after losing his/her king, Black can never be checkmated, and so can never lose! As White would rather not do that, he will avoid playing that move, and we don't need A1.2 for the problem to be sound. That's even better!
Is this what you were getting at?
| | | (40) Posted by Nikola Predrag [Friday, Jul 24, 2015 21:54]; edited by Nikola Predrag [15-07-24] |
Yes, after 1.QxK Qf8# or e.g. 1...Rh6+ Black would win. Actually, without a King Black would only have to care about avoiding stalemate, in order to win.
And instead of saying "That's even better", you should be alarmed about your recognition of what is relevant. You have "recognized" that the addition of (S) allows something that was not allowed for centuries. But it was a completely inverted notion of the relevance.
And instead of rushing to some "new conclusions", you should first detect what went wrong, why exactly you have thought that the official formulation of (S) was relevant for the "historical" change.
The rules were written for the COMMON SENSE and not for practicing the mathematical rigorousness which would require the exact meaning of every word, in the first place.
You would follow the rules relying on the centuries of common sense, as long as it suits your purpose and then abruptly, you would apply the mathematical rigorousness which, BTW, would still have to rely on the same common sense to interpret the wording?!
This is exactly a point of a JOKE PROBLEM. It could be logically interesting or artistic, but the common sense knows that it's a joke and not chess, for many long centuries.
I'll not waste my time on proving, what the common sense sees as obvious, that "the side in check has the move" and that you may play chess only if the side in check has the move. Any different interpretation INSTANTLY indicates a different game.
Before "pushing ideas further and further", one should know the critical points and then decide about their relevance.
Then it might be worth discussing.
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