Yesterday this article by Keith Burgun showed up and sparked a bunch of discussion: a way to better games. Mostly fairly hostile discussion. And naturally so, for the article is written in a very provocative tone (intentionally or not). But there is something of value in there that may be being missed. Not something "new" that will pull us out of some ludic "dark ages" into "enlightenment", but something that might be of practical use to some people designing games.
(There's also a lot that I disagree with, but others are generously criticising most of that already.)
Going to backtrack from games for a bit, start at a high level and work my way down.
Logic allows us to deduce things that are absolutely true. This may be counterintuitive if you haven't studied it, since most of the time in real life we can't know anything for certain, but in logic we can. It's a conditional truth though - we only obtain statements of the form "if A is true, then B is also true", which may not be so useful if we can't confirm A. All mathematics is of this form; everything is logically deduced from a small set of initial axioms which cannot themselves be proved. Logic can only make true statements about abstract systems that don't necessarily correspond to anything in the real world. But as Wigner exclaims in The Unreasonable Effectiveness of Mathematics in the Natural Sciences, surprisingly often these systems can approximate reality well enough to be extremely useful.
Formal definitions are a powerful tool in logic. I posted more succinctly about this last time this topic rose up: definitions. They let us specify a class of objects in a way that's amenable to logical reasoning, allowing us to deduce new facts about it.
How useful this is depends on how structured the objects you're dealing with are. In general, the more restrictive the definition, the more structure there will be, the more truths you will be able to deduce about the objects so defined.
However, if you have too much structure, this can complicate things. Even if you're only interested in studying a single specific object, it often helps to take a step back and abstract it out a bit. Figure out what are the salient features of this object and reason just with those. Very often it is actually easier to prove a general statement that answers more than your original question than to answer it directly on its own. So there's a tension when picking a definition in making it both restrictive enough and broad enough to be a useful abstraction.
You can give a word any definition you want. As when writing a computer program you can define variable names however you choose, when writing a text you can define terms however suits you. But obviously going against common usage makes it harder for others to read, so there's value in trying to construct definitions that are formally useful and also capture the intuitive meaning of the term. It's usually infeasible to define how words are used in a completely precise way, but approximations can still be useful.
A definition is not a value judgement. Specifying a category for consideration does imply that you feel it's worthy of study, but it does not automatically follow that anything outside of it is less worthwhile. (But it's not uncommon for people to treat such a classification as though it contains an implicit valuation. Please don't do this.)
A restrictive definition need not restrict creativity. Sometimes choosing to work under a constraint can be a helpful creative tool; going deep into one area and understanding it thoroughly can generate new ideas. Other times, while you might not have initially chosen to work under a particular constraint, you end up satisfying it anyway; and then you can usefully apply results about it to your work. And if a proposed definition fails to capture what you're interested in, looking outside the intersection of the two categories is often a guide towards good examples.
There are two meanings of the word "game" in common usage. One is very broad; as Wittgenstein points out, the word is used to cover a vast range of activities with little in common between them. But the word is often used in a quite specific way as well, to describe a particular kind of structured play, as when one says "it's not really a game" in an attempt to describe a work of interactive art such as Proteus or Dear Esther.
I use both meanings all the time and expect people to understand from context. This expectation is usually justified, but it sometimes leads to confusion - for example, a few days ago I tweeted this provocative statement: "if a game's not worth playing a hundred times, it's not worth playing once", and I immediately received replies suggesting certain puzzles as counterexamples. I agree that once you know the solution to a puzzle there's usually not much value to interacting with it further, but puzzles are not the type of object I was considering; however there was insufficient context to make my meaning clear. (Note that I don't necessarily believe my initial statement, I was just trying it out.)
The first of these two meanings of "game" refers to a very broad unstructured class of objects. There's not much that can be usefully said about them in general! It includes a lot of interesting stuff, but it's not a category where we can expect to logically reason out much of value. That's okay, there are other ways to explore it. Anna Anthropy vocally encourages inclusiveness in the game-development community in order to get as many different people with different viewpoints as possible making different things, and I heartily agree.
The second meaning, however, refers to a tightly restricted class of objects. There's a lot of structure, and logic is useful for navigating it. Numerous people have studied this class over the years. There is not a consensus definition, but various have been proposed. Some authors seem keen to search for 'one true definition' (Burgun among them) but having multiple competing definitions is not a problem. There are at least six different definitions of "matroid" in common use - all equivalent, but not obviously so - and researchers will use whichever gives most insight into the problem at hand. The case with games is similar (although in general our definitions will probably be closely overlapping but not equivalent), and we can similarly pick and choose to think in terms of whichever characteristics suit our present goals.
Crawford defines a game as an interactive activity in which active agents (including players) compete with each other and can use attacks to interfere with each other. This does a fairly good job of capturing the intuitive common-use meaning, but the word "attacks" is problematic; it fails to describe positive mutually beneficial interactions like resource trades in Settlers of Catan or action selection in Puerto Rico and Race for the Galaxy. The concept of "active agents" is somewhat subjective as well, as he himself admits, but it can be a useful model.
Costikyan builds on this in I Have No Words & I Must Design, he declares decision-making to be the key characteristic of games; defining them as "a form of art in which participants make decisions in order to manage resources through game tokens in the pursuit of a goal."
Burgun's attempt owes much to the previous two, so I take issue with his claim that his definition is "new"; it's an attempt to codify a folk definition that's been in use for decades, and while his exact wording may be different he exalts decision-making just as Costikyan does. The claim of novelty suggests unfamiliarity with the work of prior authors (as do the claims that the craft of game design has not matured - we've seen incredible progress in board games in particular across the last couple of decades). However, it's a worthwhile attempt; the phrasing of "ambiguous decisions" is clear and evocative, and more flexible than Costikyan's discussion of "resources".
I will not attempt to provide a definition of my own, but I will suggest strong replayability as another characteristic of games with which a definition could be crafted (possibly rendering my earlier tweet tautological).
This class of "games" is by no means the only one worth looking at. Let's explore the entire world of play in all its variety! Kanaga presents one interesting framework for examining playspaces in general: Played Meaning (Concerning the Spiritual in Games).
But when the systems we create do fit into an established category, games or otherwise, it's useful to draw upon the body of knowledge accumulated about that category. And if we're not certain how to proceed with a design, whether it fits into an established category or not, it can be helpful to abstract out its salient features and reason logically about them - possibly inventing a new category as we go. As in Science, we must confirm our ideas through experiment, but a sound theoretical framework helps to point out interesting directions to experiment in.
So this may not be an approach that you find personally useful - whether because of how your mind works, or because you're not working within any conveniently structured class of objects - but it is not without value at all. I find it useful, and some of my work I'm most proud of has been inspired by this type of approach.