What is Formal Logic?
Unifying Game Theory, Communication/Information Theory and Formal Logic
Formal Logic is a game, the tokens of which are symbols, passed as a communication, the payoff of which is something that makes sense to the individual.
In my 26 year dive into formal logic and as far back as my university days when I was first introduced to Object-Role Modeling and cursory formal logic at university some 29 years ago, it quickly became clear to me that formal logic is a game and all of formal logic needs to be seen under game theory, rather than game theory being an adjunct to formal logic or even a subset of formal logic.
Why?
Because without you, or a computer/AI of some sort, interpreting the symbols and theorems of formal logic, there is no formal logic. The universe doesn’t care if something makes sense or not. You do. I do. We do if we hand over the interpretation to AI.
It also became clear to me very early in my research that I am a meta-logician, interested in formal logic in its entirety rather than endlessly formulating theorems and proofs…studying proof theory holistically and model theory holistically as to have a top-down view of formal logic, its purpose, how to use it and how to bend it to one’s own will if there a desire to achieve something.
If logic wasn’t used to form models in one’s own mind, you just wouldn’t sit down and write out a whole set of theorems in a sequence as to come up with a logical answer or something that at least proves to you that something is real or not.
If we put aside that some things at least appear to be universal, as when mathematics best approximates or has a complete isomorphism with what we witness in reality (e.g. the phyllotaxis arrangement of seed spirals on a sunflower drawing on Fibonacci Sequences), the only reason we would go to the trouble of working out Fibonacci Sequences, mapping (or at least attempting to) mathematics to formal logic (Russell/Whitehead, Principia Mathematica and Russell, Principals of Mathematics), and coming up with an arrangement of theorems and formulae that mimic reality is because we have a desire that the things we see, the way we think, and the conclusions that we reach that seem logical make sense to us. The universe simply is, and if we presume for a moment that the universe applies the rules of a universal mathematics into creating spirals on a sunflower, or more pointedly applies the math stored within the DNA of a sunflower, then we best have a view of my conception of formal logic…without something first having interpreted the message in that DNA, and/or applying it to achieve its goal/end-game (spirals-of-seeds on a sunflower), then we have simply nothing.
So the interpreter is as important as the theorems themselves, and we call it a communication and a game when we consider a payoff being the end result (something making logical sense or the resulting spirals-on-a-sunflower) and the tokens of which game the very information stored within those theorems, and where both the theorems and the data they operate over (e.g. what colour to produce seeds, or variable data in applied logic) being that information.
Where it might be confusing to some to picture theorems of formal logic as data we must consider this, and as best describes the unification of game theory, communication/information theory and formal logic:
If we consider for a second that theorems of formal logic define a computer program stripped down of all control flow statements (GOTO, CASE, SWITCH, IF…THEN…ELSE, function and procedure calling etc), and view theorems of formal logic as defining something akin to a Lambda Calculus in its own right…then theorems of formal logic merely (and I do not say that flippantly or lightly) computer programs that we run in our head, or interpreted/run by a computer system/AI, as to arrive at a result.
That it makes sense to us, or achieves faithfully a desired result is neither here-nor-their…that is what we designed formal logic to do. When we test a proof, we run that program in our heads and if it makes logical sense (holds) then we pat-ourselves-on-the-back, learn-something-new, tap into the universal isomorphism between our system of logic and that of the universe (perhaps), and consider it useful to our purposes and happily share those findings with others as to be good corporate citizens and sharers of knowledge.
Nobody can argue that computer programs are simply data, even if they operate over data.
So we can take a very high level view of formal logic and picture it as a means of communication such as to achieve the/a desired result (the payoff for sharing those tokens) of forming a similar model that makes equal logical sense in the receiver’s mind/computer-brain and if that person works under the same interpretation as us and in all circumstances.
I call that a Coherent Cooperative Game of Formal Logic and as I define Coherent Cooperative Games.
In short, when theorems of a first-order-logic under even Finite Model theory can be interpretted as theorems of higher order logic, given such things as an extended Ehrenfeucth Fraisse Game extended to players being able to choose higher-order interpretations, or including higher-order interpretations when considering the infinite variety of interpretations possible over a suitably verbose first-order logic with an adapted Lowenheim Skolem correspondence that allows of such things…then it becomes irrelevant the intended formalisation of a theory of formal logic, but more about individual interpretation and whether or not there is a shared interpretation within a Coherent Cooperative Game that actually means something, holistically and to the individual player.
It wouldn’t go too far as to say that formal logic serves a purpose to effect the will of those who write its theorems. That will being to share knowledge, formulate in ones own mind a manner of thinking that is logical, learn something new, or to prove a point. All of that being very subjective to the individual who engages in formal logic in the first place.
Which is to say…I pretty much stopped writing theorems as soon as I started and took the view that first I had to ask the question, “Why?”, “What’s this all about?”, “What is formal logic?” and “Why waste my time when I can write a computer program that has far more means of achieving goals with such things as GOTO, CASE, SWITCH, IF…THEN…ELSE statements, function and procedure calling etc?”.
That might sound like a harsh indictment of formal logic, and reduction to triviality, however my deeper sense is I value formal logic and as a means of communication. I simply reserve the right to take the higher view and abstain from being forced into either defining a theory, or working within a theory, when I know that I can otherwise bend a theory to a common good and/or my will if it suits me as a tool-for-the-job-at-hand.
So, what is Formal Logic?
We may never know whether formal logic the reflection of processes in the universe or that we define mathematic/logic rather than simply pull from the ether that which already exists. It is possible, and I say that it is…that Formal Logic is what we make of it, and there would be no such thing as finite-model theory differing from straight first-order-logic differing from higher-order logic unless there a purpose for our will to be effected as to achieve a goal of conveying some semblance of clarity of information/data sharing to achieve what we would call a logical goal or consequence, and when one or more interpreters (players) involved…that becomes a game.
Thank you for reading. As time permits I will write more on information theory, communication theory, formal logic, meta-logic and game theory.
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