Inferentialist Game Semantics (Extended Abstract)
This paper establishes a fully abstract correlation between base-extension semantics (B-eS) and Hyland-Ong game semantics to provide an intensional theory of meaning for logical systems, illustrated through the example of 4x4 Sudoku.
Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer
Imagine you are trying to understand how a computer thinks or how a mathematician proves a theorem. For a long time, we've looked at these processes like a map: we check if the final destination (the answer) is "true" based on a static picture of the world. But there's another way to look at it, one that treats logic like a conversation or a game. In this view, a "proof" isn't just a static fact; it's a winning strategy in a dialogue between two players. One player, the "Proponent," tries to defend a claim, while the other, the "Opponent," acts like a skeptical environment, throwing challenges and asking for justifications. If the Proponent can answer every possible challenge the Opponent throws, they have a winning strategy, and that strategy is the proof. This approach, known as game semantics, makes logic feel dynamic and interactive, like a sport rather than a statue.
Now, imagine a different way of defining logic, one that doesn't rely on maps or games at all, but on pure rules of inference. This is called "proof-theoretic semantics." Here, the meaning of a statement comes entirely from how you can build it up from basic rules, like a chef who defines a dish not by its taste, but by the specific recipe steps used to make it. For a long time, these two worlds—the dynamic game of "Proponent vs. Opponent" and the rule-based "recipe" approach—seemed to be speaking different languages. The big question was: Are they actually describing the same thing, just in different ways? Could the rules of the game be built directly from the basic recipe steps, making the game itself a natural consequence of the rules?
This paper says "yes." The authors, Joaquim T. Waddington, Alexander V. Gheorghiu, and David J. Pym, have successfully translated the "game" language into the "recipe" language. They show that the complex interactions of a logical game can be reconstructed entirely from the basic building blocks of proof-theoretic semantics. They didn't just guess this; they proved it mathematically. They created a perfect dictionary where a "base" of rules (the recipe) becomes an "arena" (the game board), a "derivation" (the recipe steps) becomes a "play" (the moves in the game), and a "proof" becomes a "winning strategy."
To make this concrete, they even used a 4x4 Sudoku puzzle as a test case. In their model, the Sudoku board is the "arena." The rules of Sudoku are the "atomic rules." The "Proponent" is the player trying to solve the puzzle, and the "Opponent" is the environment that grants or denies moves based on the rules. They demonstrated that if you can solve the Sudoku (win the game), you have a "winning strategy" that corresponds exactly to a valid logical proof.
The paper goes further to handle the tricky parts of logic, like "OR" statements. In a normal game, if you have to choose between two paths (A or B), you might have to guess which one is right. But in this new framework, a winning strategy for an "OR" statement doesn't mean you have to pick one path immediately. Instead, it means you have a plan that works no matter which path turns out to be the right one. It's like having a backup plan for every possible outcome, ensuring you win regardless of how the game unfolds. This approach avoids the need for "backtracking" (changing your mind later), which is a common trick in other game models.
The authors are very sure about their results. They didn't just simulate this on a computer; they provided rigorous mathematical proofs showing that their "game-extension semantics" is perfectly aligned with standard intuitionistic logic. They proved that if a statement is provable in their game system, it is provable in standard logic, and vice versa. They also explicitly ruled out a simpler, more naive way of handling "OR" statements (where you just pick one winner), showing that such a simple approach fails to capture the full power of logical reasoning.
In short, this paper bridges two major ways of thinking about logic. It shows that the dynamic, interactive world of game semantics isn't an external layer added on top of logic; it can be built from the ground up using the fundamental rules of proof. By doing so, it gives us a deeper, more unified understanding of what it means to "know" something is true: it means you have a strategy that wins the game, no matter how the opponent plays.
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