Unifying Game Theory, Information Theory and Formal Logic

A unifying theory

Victor Morgante
7 min readDec 22, 2024
Game Theory underpinning Formal Logic and Information/Communication Theory. Image by author.

It seems clear to me that Game Theory underpins Formal Logic and Information/Communication Theory, with people or automata exchanging tokens (what I call symbols) with a payoff function that measures the utility of the tokens/symbols expressed within a medium. This includes the utility of the structures or models formed within the mind or processing unit of an automaton, as in the workings of an artificial intelligence.

Classical diagrams of Information/Communication Theory see messages past via a channel/medium from a Sender to a Receiver, as below:

Communication / Information Theory. Image by author.

If we discount noise within the actual Message (of the Sender), introduced within the Medium/Channel or as introduced by the Receiver, in interpretation of the Message, then we have the aim of the communication, being either that the Receiver interpret the Message as intended by the Sender, or the Sender deliberately and intentionally deceiving or misleading the receiver as to the purpose of the communication message. Utility, therefore, is either a function of the value of the message to the receiver, or the utility of the communication as a whole to the sender, both sender and receiver, or to an outside observer analysing the communication as and under game theory with both sender and receiver being players in a game that is either cooperative or non-cooperative, and allowing for two way communication or even multi-player communication.

Formal Logic under Game Theory

If you would like to jump ahead, I have written an article that reflects a paper I have written on the Formalism for Game-Theoretic Approach to Formal Logic, available [here]. And another, further outlining the rational for formal logic to be under game theory, [here].

Extending the unification to communication theory we will get to later in this article, however the rational for game theory underpinning formal logic, or to say that formal logic is under game theory, is quite simple.

If we take people, or sufficiently intelligent automata as in AI, as sender and receiver, the message as theorems of a theory, and interpretations as functional with a payoff function, we have game theory and Information/Communication Theory combined and subsuming formal logic:

Formal Logic subsumed by Game Theory and Information/Communication Theory. Image by author. Brain images royalty free from Pixabay.

If we take Model Theory of formal logic, and look at it closely, we understand that the subjective or shared model of the structure/s of a theory and rules of inference provide a payoff function in being able to prove this-or-that under the accepted model of the theory. That payoff function is, I say, a payoff function under game theory.

Theorems of a theory may be shared between players or with one’s self. And we are well aware of games of logic, such as Ehrenfeucht Fraisse Games (EFGs) which acknowledge that games may be played with and between competing models.

Formal Logic and Information/Communication Theory

Theorems of formal logic, written in the symbols of a theory, are a message that may be passed between players (interpreters) acting as sender and receiver in classical Information/Communication Theory. If working through a problem by oneself, on paper or a computer screen or even in one’s own mind, an interpreter/player becomes both sender and receiver.

There is not more one can really say or add to that, other than the paradigm shift to perceive of a set of theorems as a message, and that message to be interpreted.

The corollary is also true. Natural language, if not gibberish, has the aim of forming within one or other’s mind structures of combined concepts that make what, if intents pure, we call logical sense.

Our focus, however, here, is to unify Game Theory, Information/Communication Theory and Formal Logic, and so we focus on formal logic which has equally an aim of achieving logical sense over a domain, or a Universe of Discourse (UoD).

That debates rage, banter ensue, rivalries take place, or new theories emerge when in discussion over theories of formal logic or a set of theorems, is part and parcel of there being room for various interpretations (as in an Ehrenfeucht Fraisse Game) and with room for agreement on an interpretation or continued disagreement on an interpretation.

I have argued that even this aspect of formal logic can be positioned under a game, a Coherent Cooperative Game (with aim of win-win) which allows for different interpretations (in deed or thought) under what I call a Differential Interpretation Game, or DIG.

Mostly formal logic, in its playing, aims to be under a Coherent Cooperative Game, with players aiming for win-win in the sharing of theorems and proofs therein, however we cannot discount that much of the discussion around formal logic comes about by deliberate or unintentional differentiated interpretation of those theorems (as in those learning a new theory, for instance).

Wrapping it up

Formal Logic, in action, has all the elements of being under game theory:

  1. Players: Interpreters, either people or automata are players in the game of Formal Logic;
  2. Actions/Strategies: The choices available to each player are to pick from the structures of a theory, and to use their rules of inference, when forming theorems and proofs under the chosen theory;
  3. Payoffs: A subjective or shared interpretation of a theory has payoff in an interpretation making sense, or theorems and proofs making sense subjectively or collectively under the game;
  4. Information: Sets of theorems, and in proofs, are information that are shared or shared with oneself when playing the game;
  5. Nash Equilibrium: A shared interpretation forms a Nash Equilibrium, and under what I call a Coherent Cooperative Game. Alternate strategies exist under a Differential Interpretation Game (DIG);
  6. Dominant Strategy: Unless playing a DIG, the dominant strategy of a game of Formal Logic is to have and work under a shared interpretation of a theory.

The types of game that can be played under formal logic are as follows:

  1. Zero-Sum vs Non-Zero-Sum: If two players cannot agree on an interpretation then we have a zero-sum game, else we have a non-zero-sum game;
  2. Cooperative vs Non-Cooperative: When played as a Coherent Cooperative Game, formal logic is cooperative. When a DIG exists we have a Non-Cooperative Game which may manifest as a new Cooperative Game (as in a new interpretation);
  3. Simultaneous vs Sequential:
    a) Two philosophers simultaneously presenting competing proofs for the same theorem;
    b) Multiple mathematicians working independently on solving a conjecture;
    c) Parallel hypothesis formation in scientific reasoning.
  4. One-shot vs Repeated:
    a) One-shot: A single formal proof of a theorem (like Gödel’s Incompleteness Theorem);
    b) Repeated: The iterative process of mathematical induction, where each step builds on previous ones;

In all instances, formal logic may be viewed as a game between verifier vs falsifier. “Does this make logical sense, or is there a different interpretation?”. One player trying to establish truth, the other seeking counterexamples.

Formal Logic, in action, has all the elements of being under Information/Communication Theory:

If we limit Information Theory to classical Communication Theory, formal logic as a Sender and Receiver, a Message and a Medium (paper or computer screen, or the machinations of one’s own mind), Noise (deliberate or unintentional mal-construction of theorems), and Interpretation. Interpretation being central to the very notion of formal logic’s Model Theory.

Afterthought

There may be those who would argue that a unified theory of Game Theory, Information/Communication Theory and Formal Logic would require a tomb of a work to produce. E.g. A book hundreds or over 1000 pages long. However Isaac Newton stated as Rule 1: “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.”

That people, and now AI, are natural and normal as interpreters we can take for granted. That people wish for logical truth and certainty we can take for granted. That people wish for clear and logical sense of what they interpret we can take for granted. That Formal Logic is a construct of the human mind, we can take for granted. That people play games and communicate is all natural. All these things are very natural, and so too is the unification of Game Theory, Information/Communication Theory and Formal Logic.

Thank you for reading. As time permits, I will write more on the unification of Game Theory, Information/Game Theory and Formal Logic.

======Noise in the Communication\Message

Please refer to this article which extends the thesis, and which speaks about noise in the message (which may be as a result of deliberate or accidental construction/delivery of the message by the Sender, introduced in the Medium/Channel or as misinterpretation by the Receiver): Communication Theory and Formal Logic | by Victor Morgante | Dec, 2024 | Medium

Many thanks to Modeller, who recognised the need for acknowledging noise, which is a fundamental component of modern-day Communication Theory (10th Jan 2025)

=====Further Reading=====

  1. Coherent Cooperative Games — As I describe them;
  2. The Genesis of Coherent Cooperative Games;
  3. Coherent Cooperative Games — Described;
  4. Coherent Cooperative Games and the Law;
  5. Formal Logic — And Coherent Cooperative Games;

….and…

  1. All of Logic is a Game;
  2. What is Formal Logic;
  3. Applied Use of Ehrenfeucht Fraisse Games;
  4. What is a Graph Database;

…and…

  1. The Atoms of Knowledge;
  2. The Richmond Architecture;

…and where it all started:

  1. Morphing Conceptual Models.

===============End==============

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Victor Morgante
Victor Morgante

Written by Victor Morgante

@FactEngine_AI. Manager, Architect, Data Scientist, Researcher at www.factengine.ai and www.perceptible.ai

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