So there we have it.
@tygxc doesn't know what "deductive logic" means and doesn't know what "relevant degree" means either.
It is more difficult to believe that someone with a relevant degree would forget what deduction is. A bit like a literature graduate forgetting what a noun is.
Perhaps I was wrong to think he would be one of those who understood what the value of a position is. For clarity: V(given position) is defined in three simple stages:
Let W be a white strategy and B be a black strategy (always assumed to be deterministic)
V(W, B) is defined as the (deterministic) result when these strategies are played against each other.
V(W) is defined as minimum over all black strategies B of V(W, B),
V(given position) is defined as maximum over all white strategies W of V(W)
This is a good example of verbal reasoning being far better than the attempt to depict it algebraically, which we see here. I could portray it verbally. It would be much clearer and without need for further explanation. One thing I don't understand is this: why is the interplay of strategies deterministic? What's the reasoning behind that, because I think it's incorrect?
Happy to clarify that by adding a definition that was left implicit (I shouldn't have assumed it was obvious).
A deterministic strategy is one which always plays the same move in any specified position.
If you play two specific deterministic strategies against each other, you always get the exact same game and the same result.
@6482
After this formalistic intermezzo, can we now agree in your lingo that
the value to white of the position after 1. Nh3 <= the value to white of the position after 1. Nf3
and
the value to white of the position after 1. e4 e5 2 Ba6 <= the value to white of the position after 1. e4 e5 2 Nf3
It's a definition that is used in the definition of the value of a position.
@6482
After this formalistic intermezzo, can we now agree in your lingo that
the value to white of the position after 1. Nh3 <= the value to white of the position after 1. Nf3
and
the value to white of the position after 1. e4 e5 2 Ba6 <= the value to white of the position after 1. e4 e5 2 Nf3
As chess players willing to take a good bet we could.
But game theorists trying to solve chess definitely cannot. These results are unproven, just like it is unproven that chess is a draw, and just like until recent history it was unproven that checkers was a draw (until it was proved).
You are the former.
@6488
Let us go back to your own previous post @6445:
"Alpha–beta pruning is a search algorithm that seeks to decrease the number of nodes
that are evaluated by the minimax algorithm in its search tree."
"It stops evaluating a move when at least one possibility has been found
that proves the move to be no better than a previously examined move."
++ 1 Nh3 is no better than 1 Nf3. 1 e4 e5 2 Ba6? is no better than 1 e4 e5 2 Nf3
"Such moves need not be evaluated further."
++ Thus 1 Nh3 and 1 e4 e5 2 Ba6? need not be evaluated further.
"When applied to a standard minimax tree, it returns the same move as minimax would,
but prunes away branches that cannot possibly influence the final decision."
++ The branches 1 Nh3 and 1 e4 e5 2 Ba6? cannot possibly influence the final decision.
Yes, I can see. I suppose it makes the logic easier but I wonder if it's accurate.
I'm sorry, but your thinking doesn't make sense. A definition only needs to be valid, there is no notion of "accuracy".
Given the set S of legal states s in chess where white is to move, each s of which has a non-empty set of legal moves M(s), a deterministic strategy for white is a mapping f from S where f(s) is always a member of M(s).
That definition is valid because it determines whether something is a deterministic strategy for white or not.
I'm sorry, but your thinking doesn't make sense. A definition only needs to be valid, there is no notion of "accuracy".>>
Probably in the same way that it's ok to define 1. e4 e5 2. Ba6 as losing for white. Glad that's sorted, anyway.
No, that is a good example of what is not ok.
Asserting an unresolved proposition as an axiom (which is what you are suggesting - nothing to do with definition, which is about labelling, not truth) may cause inconsistency. To be safe, you need to prove relative consistency. In this case this requires proving the result that you (extremely eccentrically) wish to use as an axiom!
Understand the difference between an axiom, a proposition and a definition?
[Please note that your own posts demonstrate why it is necessary to be precise about these things, rather than what you suggest].
Yes, I can see. I suppose it makes the logic easier but I wonder if it's accurate.
I'm sorry, but your thinking doesn't make sense. A definition only needs to be valid, there is no notion of "accuracy".
Given the set S of legal states s in chess where white is to move, each s of which has a non-empty set of legal moves M(s), a deterministic strategy for white is a mapping f from S where f(s) is always a member of M(s).
That definition is valid because it determines whether something is a deterministic strategy for white or not.
I think I know what you're saying. It's a way of defining determinism itself. However, determinism isn't shown to define chess and so there seem to be double-standards at play here. You're saying that you can define something into existence. This may be acceptable to mathematicians but then that existence only holds for the specific paradigm and only works if it can be cancelled out at the other end. But here you're making a statement, regarding the nature of chess, which is similar in type to those you condemn from tygxc.
It's far from convincing. You would have to do much better than that.
No, you don't understand.
The definitions of deterministic strategy was made in order to be able to define the value of strategies and positions.
The definition of value of a position was made in order to be able to state a proposition like:
"The value of the position after 1. Nh3 is less than or equal to the value of the position after 1. Nf3".
This is the routine way in which mathematics (and very closely related subjects) are done. You can't shortcut it except if everyone is so familiar with the topics that it is obvious without being said. That is not true in this forum!
It's almost the same game if the purpose is to capture the king. The reason it's not is that stalemate would become a win.
@6488
"As chess players willing to take a good bet we could." ++ You do not understand the difference between probabilistic and deterministic. You can bet on probabilistic subjects, like if player A can win positions 1 Nh3 or 1 e4 e5 2 Ba6? against player B or not. As for 1 Nh3 or 1 e4 e5 2 Ba6? drawing, winning, or losing it is deterministic, either or, there is neither probability nor betting.
"game theorists trying to solve chess definitely cannot" ++ They definitely can and should. Incorporating game knowledge is beneficial in solving a game - van den Herik.
"These results are unproven" ++ It is proven that 1 Nh3 cannot be better than 1 Nf3, see above. It is proven that 1 e4 e5 2 Ba6? loses for white: a checkmate in 82, see above.
"it is unproven that chess is a draw"
++ It is proven that chess is a draw: there is evidence compelling the mind to accept the truth or fact. There is a deductive argument that a single tempo is not enough to win as well as inductive evidence from millions of human and engine games, especially from ICCF.
"it was unproven that checkers was a draw (until it was proved)."
++ The weak solution of Checkers uses only 19 of the 300 openings. That solution prunes away irrelevant lines, so a weak solution of Chess can prune away 1 Nh3, 1 a4, and 1 e4 e5 2 Ba6?
On one side you say 1 e4 e5 2 Ba6? wins for white despite being a whole bishop down.
On the other side you say the initial position wins for white because of a single tempo.
That makes no sense. In the initial position a bishop or a pawn is enough to win, a tempo is not.
It's almost the same game if the purpose is to capture the king. The reason it's not is that stalemate would become a win.
Not always
(You cannot queen a tempo, but an extra pawn is enough to win. - Confucius.)
Yes, I was going to draw attention to that distinction, but edited it out.
But it was unforgiveable not to be precise in saying "most stalemates would become losses" or "all stalemates where the player has a move which would otherwise be legal but moves into check would become losses". So kudos for taking me to task for that. 
It's almost the same game if the purpose is to capture the king. The reason it's not is that stalemate would become a win.
I never could understand this supposed difference. The game ends when the king is attacked and has no means of escape. Probably for historical reasons, the king is never removed from the board. The game ends when either king is attacked and has no means of escape.
Stalemate cannot be a win for either side because the rules state that each side makes alternate moves. In a stalemate, that's impossible so the game ends inconclusively because neither king is attacked and the game cannot progress. It really is quite simple. Like my mind. Clear and simple.
Yes. The presumption is to remove the rule that you are not allowed to be in check after a move (this is necessary to make it possible for the king to be captured) and to permit the capture of the king (not actually a new rule since the possibility cannot arise in normal chess).
After this all stalemates where the side to move has a move which would be legal but moves into check become losses (because the move can be made and the opponent then captures the king) but, as MARattigan correctly pointed out, stalemates where the side to move has no legal move, even one that would move into check, remain draws unless you add a further rule (that not having a legal moves loses. Or that this wins!).
@6478
There is no contradiction of any kind.
Your formalistic descriptions simply mean 'the outcome if all participants play optimally.
So V = min [V(W)] is indeed no higher for 1 Nh3 than for 1 Nf3. q.e.d.