10/17/2014 – Think of the worst blunder in the opening that does not lead to immediate checkmate. Say, giving away your queen on the second move with 1.e4 d6 2.Qg4. It is a losing move, as every beginner will tell you. But do we really know it, or are we just making an educated guess that is impossible to demonstrate? This is the philosophical problem of chess knowledge, explained by Pedro Múrias.

Bust this! A philosophical opening

By Pedro Múrias

Think of the worst blunder in the opening that does not lead to immediate checkmate. Say, giving away your queen on the second move. It is a losing move, every beginner knows that. But do we really know it, or are we just making an educated guess that is impossible to demonstrate? This is the philosophical problem of chess knowledge.

Fischer famously claimed to have busted the King’s Gambit, but not everyone was convinced. Again Vasil Rajlich claimed to have done it, this time for sure, with the help of 3000 processor cores, but at the end of the day it was an April Fools’ hoax. Perhaps we should start with simpler openings. What about busting your author’s attack against 1. e4 d6? Take a look.

1.e4 d6 2.Qg4 ...

Some people would place a few question marks after 2. Qg4, but you may want to save the marks for the following question: How do you know that 2. Qg4 is a bad move? True, Black can take the queen with 2. ... Bxg4, but the point of the game is checkmate and there is no checkmate to be seen here. What is so bad about losing the queen on the second move? And whatever it is, how do you know that is really bad?

We have to draw a distinction — not clear-cut but useful — between practical chess and “ideal” or “perfect chess”. That is the distinction between chess as an empirical matter involving human or computer players in concrete situations, and chess as a mathematical object constituted only by its formal rules, each game being a series of values, called “moves”, in accordance with the rules. If you see chess as an empirical matter, then you can rely on experience and argue that, as a matter of fact, whoever loses their queen without some sort of “compensation” will be expedien­t­ly defeated by any average opponent. It has always been so. That is regular justification by in­duc­tion, not unlike a scientist’s explanation of some other aspect of the physical world. For evaluating less silly moves, psychology may help.

If, however, you are not interested in what players actually do, but in ideal chess, you will most likely want to know which moves are objectively best and to know the outcome of perfect play after some position. Here you will not be satisfied with induction from past experience. You are aiming at the kind of abstraction mathema­ticians deal with, and you want a demonstration, a mathematical proof. As in most of mathematics, deduction would seem the way to go. Notice that even so called “mathematical induction” is, despite its name, pure deduction.

Deduction does help in chess matters in one of two ways. If you are solving an endgame with a small number of pieces or a chess problem for a short term specific result — say, a mate in n moves, when n is small — you are able and expected to check all possible valid moves, according to the rules of the game. If you follow the rules and check all the moves, it is logically impossible for your solution to be wrong. That is deduction in its clearest form, and computers are perfect at it, endgame tablebases being their most remarkable output.

Alas, it is not possible to evaluate most opening and middlegame moves this way. There is no purely mathematical obstacle to it, as Ernst Zermelo argued for the first time in a 1913 article (in German). But there are zillions of possible moves until checkmate or draw, and even the fastest computer could not check them all in the next thousands of years. The King’s Gambit is not busted, as no respected opening is. To be sure, there may be a forced mate for Black after 1.e4 d6 2.Qg4 in so few moves that a supercomputer would be able to find it. But no computer has done it yet, and our question is whether we know now that 2.Qg4 is a losing move. Ernst Friedrich Ferdinand Zermelo, German logician and mathematician, known for developing the Zermelo– Fraenkel axiomatic set theory and his proof of the well-ordering theorem. [Photo Konrad Jacobs/Wiki]

Nevertheless, computers, with their deductive powers, are also good in openings and the middle­game. Since it is impossible to calculate most openings and middlegames until checkmate or draw, computers use heuristics as guidance. A heuristic is an added piece of information. Heuris­tics evaluate positions as better or worse, can conflict with each other, and are duly ignored when the computer can calculate a position until forced checkmate or draw. Rules for checkmate and draw are the only absolute criteria for evaluating positions. But applying a heuristic is still a matter of deduction: a computer would evaluate 2. Qg4, after 1. e4 d6, as a terrible move in simple application of the heuristic that being down a queen is terrible, unless either there is a forced mate or draw, or other heuristics apply that find enough compensation for the queen.

Do we have, after all, a fully deductive justification for believing that 2.Qg4 is a losing move? Certainly not. Heuristics themselves are not deductively justified from the rules of chess. In fact heuristics are generally seen as mere practical devices for dealing with problems. They usually work, as a matter of fact, but they are not “the truth” about their subject matter. If we are interested in perfect play chess, heuristics cannot help.

So it seems that we do not really know whether White will lose after 1.e4 d6 2.Qg4, assuming perfect play.... But that, of course, is preposterous. After 2. ... Bxg4, Black will use the extra queen to increase their advantage in the opening and middlegame until it becomes (even more) crushing and mate comes up. There is no way to avoid it. Since it is preposterous to think other­wise, a reasonable philosophical inquiry about the knowledge of perfect chess — the epistemo­logy of perfect play chess — should not start by asking whether we have such knowledge, but rather how it is justified. And since deduction is not the answer, the question is: how can induction justify some knowledge about the mathematical object called chess?

The author Pedro Múrias is a Portuguese jurist with an inclination for chess and for philosophy.

In the above picture he is arguing about the law of obligations.

One thing in favour of chess is that it is not infinite. Its game-tree complexity is over 10^120, which is quite a number, but is almost nothing compared to infinity. This is important because using induction to justify beliefs over mathematical infinite domains may be asking too much (see this Stanford Encyclopedia of Philosophy article on the subject). Since chess is not infinite, we can give it a try. It is still peculiar to use induction and its generalizations over finite samples to get to understand an abstract entity like perfect chess, and not the physical universe, but there is no principled objection against it. And a clear difference remains between practical chess and perfect chess. Practical chess knowledge depends, at least partly, on past experience from real world games and positions, a bit like scientific theories depend on past observations of the real world. For perfect chess, the basic information comes from any possible games and positions, it comes from games in the mathematical abstract sense.

So here is how we do it. We start with problems that can be deductively solved. Deduction still has some role, of course. We start with some endgames, middlegame issues to be settled during the middlegame, short term goals in openings, regular chess problems, and the like. Then we observe with “the eyes of thought” some regularities. Say, in most positions, it is better for the player’s goal to have a queen than a knight or a rook; checkmate can be reached faster if one has a queen; in middlegames, it is usually better to have more space and material; and so on. Some regularities are very similar to some exceptionless truths deduced from the rules of the game. For instance, on an empty board, a queen can directly move to more squares than a rook, and, in most positions with several pieces, a queen can directly move to more squares than a rook.

Generalizations involved in discovering regularities are tantamount to theories about the perfect chess universe — theories which, you may notice, bear some resemblance to computer heuris­tics. With enough confirmation from more solvable problems and after checking theories against each other, we come to formulate descriptive laws of perfect play chess (not to be confused with the constitutive rules of the game). These laws are “ceteris paribus”, meaning roughly that they allow for exceptions: e.g., one may have to sac the queen for a smothered mate, or to sac a lesser piece for a big space advantage. Some laws of nature are ceteris paribus too. To know ceteris paribus laws is to know something, in spite of the exceptions allowed. And the good thing is that we may look for and formulate as many ceteris paribus laws as we feel necessary. You may e.g. find laws concerning moves after the 15th move in the variation of your choice.

Galileo Galilei, 1564 – 1642, Italian physicist, mathematician, engineer, astronomer, and philosopher

who played a major role in the scientific revolution. [Portrait of Galileo by Justus Sustermans, 1636]

Now a sceptic might argue: how do we know that we are observing the relevant positions and that we are making the generalizations that matter? We do not have independent criteria for such assessment. But that is not a problem specific to chess knowledge. How did Galileo know he was making the relevant observations of the skies? Well, he did not, not when he started. What any scientist or indeed anybody can do is make more observations that will confirm or disprove (falsify) his/her hypotheses and theories. And every scientist has to check theories against each other for consistency. At some stage, further checking becomes pointless: one came to know some natural or chess laws, even if allowing for exceptions. The sceptic could go on to point out that some laws of chess remain unknown and some of those may be important. But again that is not a problem for chess only, and it does not prevent us from having some knowledge of perfect chess or the physical world. Moreover, as the laws fit together to form a larger picture, one can even know, at least for some kinds of problems, which relevant theories are still to be confirmed or falsified. You would have to be a very radical sceptic to deny that we know something.

Armed with such knowledge, there are a few positions and moves for which we can give an eva­luation beyond any reasonable doubt. What happens is that those moves are bad (or good) moves according to some ceteris paribus laws, and they are not good (bad) moves according to any law, of all that we can figure. And so we may confidently assert that 2. Qg4 is a blunder, and that 2. ... Bxg4 is a perfect move, as confidently as we assert that shooting at one’s own foot is an unfortunate idea. We know something about perfect chess, and it is enough to recognize that 2. Qg4 loses the game. We do not have a mathematical proof, but we know it anyway. Please do not try your author’s attack against perfect players. It’s busted.

To conclude, it looks interesting that the study and knowledge of perfect chess has many simila­rities to the study and knowledge of the physical world, although chess is an abstract entity and the product of arbitrary rules. Chess is a universe of its own and, as the saying goes, it mirrors life, at least in the way we know it.

About the author Pedro Múrias

Born in 1970, I got my first law degree in 1994, and my mestrado in 1999. I worked as a lecturer for the school of law of the Lisbon University from 1994 to 2007. During the last few years I have been working for the Portuguese government as a consultant. I have published in Portuguese, mostly on the law of obligations and on legal philosophy. You can find details and a complete cv on my page, but it is all in Portuguese. I play at Playchess.com. When I was writing this article, my Elo rose from the usual 1700 to 1897, but when I finished it swiftly went down to 1563. I'm sure that was a coincidence. Chessbase.com is a great news site which I check several times a day – and many more times on April 1st. I would like to thank Teresa Marques, Alan Baker, Luís Duarte d'Almeida, Margarida Lima Rego and Gonçalo Almeida Ribeiro for very helpful comments and suggestions.

Luke McShane: I agree with the author

Luke McShane [photo by Ray Morris-Hill]

Luke James McShane (born 7 January 1984) is an English chess player, former World Youth Champion and chess prodigy. Luke has become one of England's leading players and a member of the Olympiad team. He attended Westminster Under School and then City of London School, before entering Oxford University in 2003, where he read philosophy and mathematics at University College. In the summer of 2006 he interned at Goldman Sachs in FICC. Since then he has been a trader in London's financial sector.

We knew that Luke had discussed precisely this subject of logical and inductive truth with Carlsen second GM Peter Heine Nielsen some years ago, so we asked him what he thought of the article by Pedro Múrias. He replied within the hour: