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Philosophy and Logic of Games, Notre Dame, Fall 2026

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This is a rough outline syllabus for the class I am teaching this semester at the University of Notre Dame on the philosophy and logic of games.

University of Notre Dame                                                                              Fall 2026

Philosophy and Logic of Games
Phil 20615

Joel David Hamkins, O’Hara Professor of Logic

TR 3:30—4:45am 110 O’Shaughnessy Hall

Description. Shall we have a game? The course will explore the philosophy and logic of games, exploring all things games. We shall begin with some elementary game theory and decision theory, investigating game-theoretic ideas in the context of numerous actual games, moving eventually to the fundamental theorem of finite games, game trees, the hypergame paradox, and more. Students will demonstrate playing mastery over a variety of familiar games, including chess, draughts, Nim, Hex, Othello, Connect 4, and others. The course will include philosophical readings on the nature of human game playing. The latter part of the course will delve into infinitary issues, including determinacy and nondeterminacy, connections with the philosophy of mathematics, Conway games, and the surreal numbers as games. It will be a highlight of the class to cover the analysis of various infinite games, including infinite chess, infinite draughts, infinite Hex, infinite Wordle, infinite Sudoku, and more. 

Fulfills University 2nd Philosophy Requirement. The University of Notre Dame is unique to my knowledge in requiring all undergraduate students, regardless of major, to take two philosophy classes. (And this situation is surely part of the explanation for both the large size and high quality of the Notre Dame philosophy department.) At many other universities, including those at which I previously worked, this would entail that there was a contentiously argued but finally standard Phil 1 and Phil 2 class syllabus, which all students took exactly alike. But the Notre Dame style is rather to have a free market in intro and 2nd philosophy classes, and students can choose the ones they like. This games class happens to fulfill the 2nd philosophy requirement. The class is open to all students, with no prerequisites, and the class is full of students with diverse majors, including engineering, computer science, neuroscience, finance, and others. But most of the students happen to have a STEM or math angle in their major, since this is the population attracted to the class.

Readings will be based on selections from:

  • Joel David Hamkins, Infinite Games: Frivolities of the Gods, selected readings available for students on https://infinitelymore.xyz.
  • Bernard Suits, “What is a Game?” Philosophy of Science , Jun., 1967, Vol. 34, No. 2 (Jun., 1967), pp. 148-156. Available at: https://www.jstor.org/stable/pdf/186102.pdf 
  • Thi Nguyen, Games: Agency as Art, 2020, Oxford University Press.
  • Thi Nguyen, The Score, 2026, Penguin Random House.

Each lecture will be on a topic assigned from the reading, with several homework questions and topic assigned. A detailed schedule of topics and readings will be made available.

In-class writing quizzes. We will have a number of in-class quizzes, especially on the readings as well as the more theoretical material. Students are advised to stay on top of the reading and to be prepared to explain the finer points of the ideas we discuss and read about.

Class participation. Students should expect to participate in the class discussion. Every student should aim to participate in every lecture with comments or questions. I will be calling on students in class for contributions.

Game challenges. Students will demonstrate playing mastery over a variety of familiar games, including chess, draughts, Nim, the Gold Coin game, Hex, Othello, Go, Connect 4, Rubik’s cube, and others, undertaking scheduled game play with the professor. Students should plan to practice thoroughly before undertaking the challenge.  Assessment will be based on proficiency and strategic reasoning, as well as the ability to speak sensibly with the professor about relevant general strategic considerations and principles while playing the game and afterward. Games challenges can be scheduled using the Google scheduler form. I will make additional slots available as the semester progresses if there is need. 

Philosophy Games Club. Students are encouraged to participate in the Philosophy Games Club, which I have organized in the interest of this course. The club meets in the philosophy lounge (first floor of Malloy Hall) every Thursday afternoon 1:30-3, with members of the broader philosophical community invited, including especially all students in this class. Free snacks and refreshments will be provided. Students will have a chance to learn some of the games and play each other, as well as the professor.

Use of AI. The various AI and generative text tools are certainly powerful, and some students may find this to be a useful way to learn interactively about a topic, as with a substitute artificial study partner (although it is better to have an actual human study partner). My policy is that students must not submit AI-generated work as their own, including work resulting from AI editing tools. 

Grading Assessment. The final course grade will be based on the in-class quizzes, covering the abstract theoretical and philosophical material, the games challenges, class participation, and a final exam, which will cover the entire course. The class will be highly interactive, and students should come prepared to contribute meaningfully.

  • In-class writing quizzes, 5 points each
  • Games challenges, 5 points each.
  • Final exam, 30 points
  • Class participation and discussion contributions, 10 points

Grading is not based on a rigid translation of scores to letter grades, but rather will take account of the difficulty of the challenges and materials. “On the curve”

Mathematical prerequisite? There is no official mathematical prerequisite, but students should expect to get involved with some detailed or technical arguments in the theory of games in a manner that is similar to mathematical argument. In this sense, openness to a mathematical outlook will be invaluable.

Lecture topics, tentative

Checkmate is not the same as a forced capture of the enemy king in simplified chess

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In my imagination, and perhaps also in historical reality, our current standard rules of chess evolved from a simpler era with a simpler set of rules for the game. Let us call it simplified chess. In simplified chess there was the same 8×8 board with the same pieces as now moving under the same movement rules. But the winning aim was different, and simpler. Namely, the winning goal of simplified chess was simply to capture the enemy king. You won the game by capturing the opposing king, just as you would capture any other piece.

There was therefore no need to mention check or checkmate in the rules. Rather, these words described important situations that might arise during game play. Specifically, you have placed your opponent in check, if you were in a position to capture their king on the next move, unless they did something to prevent it. You placed your opponent in checkmate, if there was indeed nothing that they could do to prevent it.

In particular, in simplified chess there was no rule against leaving your king in check or even moving your king into check. Rather, this was simply an inadvisable move, because your opponent could take it and thereby win. Checkmate was merely the situation that occurred when all your moves were like this.

It is interesting to notice that it is common practice in blitz chess and bullet chess to play with something closer to simplified chess rules—it is often considered winning simply to capture the opposing king, even if there was no checkmate. This is also how the chess variant bughouse is usually played, even in official tournaments. To my way of thinking, there is a certain attractive simplicity to the rules of simplified chess. The modern chess rules might seem to be ridiculous for needlessly codifying into the rules a matter that could simply be played out on the board by actually capturing the king.

Part of what I imagine is that our contemporary rules could easily have evolved from simplified chess from a practice of polite game play. In order to avoid the humiliation of actually capturing and removing the opponent’s king and replacing it with one’s own piece, which indeed might even have been a lowly pawn, a custom developed to declare the game over when this was a foregone conclusion. In other words, I find it very reasonable to suppose that the winning checkmate rule simply arose from a simplified rule set by common practice of respectful game play.

I am not a chess historian, and I don’t really know if chess did indeed evolve from a simpler version of the game like this, but it strikes me as very likely that something like this must have happened. I await comment from the chess historians. Let me add though that I would also find it reasonable to expect that simplified chess might also have had no provision for opening pawns moving two squares instead of just one. Such a rule could arise naturally as an agreed upon compromise to quicken the game and get to the more interesting positions more quickly. But once that rule was adopted, then the en passant rule is a natural corrective to prevent abuse. I speculate that castling may have arisen similarly—perhaps many players in a community customarily took several moves, perhaps in a standard manuver sequence, to accomplish the effect of hiding their kings away toward the corner and also bringing their rooks to the center files; the practice could have simply been codified into a one-move practice.

My main point in this post, however, does not concern these other rules, but rather only the checkmate winning condition rule and to a certain logic-of-chess issue it brings to light.

When teaching chess to beginners, it is common to describe the checkmate winning situation in terms of the necessary possibility of capturing the king. One might say that a checkmate situation means the king is attacked, and there is nothing the opponent can do to prevent the king from being captured.

This explanation suggests a general claim in the logic of chess: a position is in checkmate (in the contemporary rules) if and only if the winning player can necessarily capture the opposing king on the next move (in simplified chess).

This claim is mostly true. In most instances, when you have placed your opponent in checkmate, then you would be able to capture their king on your next move in simplified chess, since all their moves would leave the king in check, and so you could take it straight away.

But I would like to point out something I found interesting about this checkmate logic claim. Namely, it isn’t true in all cases. There is a position, I claim, that is checkmate in the modern rules, but in simplified chess, the winning player would not be able to capture the enemy king on the next move.

My example proceeds from the following (rather silly) position, with black to move. (White pawns move upward.)

Of course, Black should play the winning move: knight to C7, as shown here:

This places the white king in check, and there is nothing to be done about it, and so it is checkmate. Black has won, according to the usual chess rules of today.

But let us consider the position under the rules of simplified chess. Will Black be able to capture the white king? Well, after Black’s nC7, it is now White’s turn. Remember that in simplified chess, it is allowed (though inadvisable) to leave one’s king in check at the end of turn or even to move the king into check. But the trouble with this position is not that White can somehow move to avoid check, but rather simply that White has no moves at all. There are no White moves, not even moves that leave White in check. But therefore, in simplified chess, this position is a stalemate, rather than a Black win. In particular, Black will not actually be able to capture the White king, since White has no moves, and so the game will not proceed to that juncture.

Conclusion: checkmate in contemporary chess is not always the same thing as being necessarily able to capture the opposing king on the next move in simplified chess.

Of course, perhaps in simplified chess, one wouldn’t regard stalemate as a draw, but as a win for the player who placed the opponent in stalemate. That would be fine by me (and it would align with the rules currently used in draughts, where one loses when there is no move available), but it doesn’t negate my point. The position above would still be a Black win under that rule, sure, but still Black wouldn’t be able to capture the White king. That is, my claim would stand that checkmate (in modern rules) is not the same thing as the necessary possibility to capture the opposing king.

I had made a Tweet earlier today about this idea (below), but opted for a fuller explanation in this blog post to explain the idea more carefully.

On Twitter, user Gro-Tsen pointed out a similar situation arises with stalemates. Namely, consider the following position, with White to play:

Perhaps Black had just played qB5, which perhaps was a blunder, since now the White king sits in stalemate, with no move. So in the usual chess rules, this is a drawn position.

But in simplified chess, according to the rules I have described, the White king is not yet explicitly captured, and so he is free still to move around, to A6, A8, B6, or B7 (moving into check), or also capturing the black rook at B8. But in any case, whichever move White makes, Black will be able to capture the White king on the next move. So in the simplified chess rules, this position is White-to-win-in-1, and not a draw.

Furthermore, this position therefore shows once again that checkmate (in normal rules) is not the same as the necessary possibility to capture the king (in simplified rules), since this is a position where Black has a necessary possibility to capture the White king on the next move in simplified chess, but it is not checkmate in ordinary chess, being rather stalemate.

;TLDR Ultimately, my view is that our current rules of chess likely evolved from simplified rules and the idea that checkmate is what occurs when you have a necessary possibility of capturing the enemy king on the next move in those rules. But nevertheless, example chess positions show that these two notions are not quite exactly the same.