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A Formal Study of Games

Federico Danelli

Abstract

The modal logic was made to be a tool aimed to assess fundamental linguistic concepts,

in order to make evident the underlying assumption of language (any language). This tool

is also very useful to apply the same depth of analysis that we can find in semiotic and

linguistic into ludology and game studies.

This essay will cover the foundational of modal logic applied to games, and the general

conclusions that it allows to derive. Moving from semantics, we can show that any game

inherently define a sheaf of possible game worlds, and because the translation principle

isn’t guaranteed in some game (and, more specifically, is never given in specific

categories of games - like RPG) we are able to built a distinction between “formal games”

and “modal games” (that we will call more exactly procedural and behavioural games

respectively).

Once built, this theory (that we also called “Fictionally Distributed logic” or [fd]logic) runs

across surprisingly conjectures and results, allowing to embrace the infinite variety that

games possess, all that without losing its solid theoretical foundation.

Prefaction to the 4th edition

Originally (2013), this was a little more than a rant (in Italian), towards some game

concepts (specifically about role-playing game). My goals were to show how (thank to the

“flexibility” of games) any theory, be it math or psychology or (case in point) linguistic,

could be used as a cornerstone to assess game .

1

I then produced a second (English) version. Yet the paper resulted too rooted in

philosophy of language and linguistic to be accessible to game designer; and at the same

time yet too generic for an academic journal. This was the reason for third revision, which

included a split that is still the structure of the papers: one (this) specifically related to

linguistic and modal logic, with a foundational goal. Then another one, focused on the

practical applications and examples of said theory (“A Formal Study of Game Design”).

Since then, I submitted only this paper to a dozen of academical journal of philosophy,

linguistic and game studies. Unfortunately, it seems this paper is still either too gamey for

1 Also, do it whilst doing it better than any other supposedly “observation-based” theory.

a philosophy journal, and too philosophical for a game studies journal. That’s why I

worked on a fourth revision, specifically for online self-publishing.

While the theoretical part is almost exactly the same after 6 year, I revised profoundly the

part related to “A Formal Study of Game Design” to make it (at my best) understandable

and practical to use to someone without any particular background knowledges.

However, remember that what you’re reading now is instead the foundational paper. It’s

not exactly easy to comprehend and apply… but nothing worthed ever is!

Summary

Abstract 1

Prefaction to the 4th edition 1

Summary 2

Introduction 3

Overview & methodology 3

Fundamental assumptions 4

Primordial sets 4

Primordial sets: properties 5

Game statements and game elements 5

Possible worlds and game worlds 6

Circumscribing a (single) game world 7

Setting the equivalency 7

Translation principle and translation function t 8

A practical example of translation function applied to games 9

Rigid designators 9

Looking for coherency 10

The perks of narrative 11

Rules and Story 12

The universal game mechanic 13

Rules vs Story 13

What is a game (for)? 14

Cardinality of games 14

[fd]logic 15

Universe, language and games 16

Future developments 16

Addendum: truths 16

Examples of truths 18

Validation in behavioural games 19

A proposal for truths modelization 19

Addendum: closure notes 20

Bibliography 22

Introduction

Games are between the oldest human activity (with music, alcohol and smoking), but they

have usually suffered from a theoretical underdevelopment. This happens even in fields

that are almost directly derived from games (i.e. game theory and probability calculus).

Hopefully, this paper will foster a discussion about the foundational issue hidden within

any theory of games, and it will help progressing in game studies with a more cohesive

and formal point of view (whatever it happens supporting or disproving what’s proposed

here).

Usually any scholars in game studies approach the matter with a taxonomical perspective

like proposing a definition of “game”, or “rules” or “fiction” or whatever. Alternatively, it is

done starting from assumption taken from other field of knowledge (sociology, or

psychology) in order to use the concept proven in other science and brought them into

game studies. In some cases, “pure” observation of games are used instead of derivative

concept.

Overview & methodology

We personally refuse the foundational approach by axiomatization described above, and

firmly believe it’s wrong. No knowledge is created in a bubble, and whilst to define terms

like “rules” of “fiction” beforehand may seems the proper way to proceed, it’s not for

games. Games possess what are called “emergent” properties, and so they cannot be

explained by an axiomatic approach.

So, if we need some foundation for our theory (and we need it indeed!), we have to resort

to the second strategy and take them from somewhere else . And also we have to

2

demonstrate that those definition will work within our new boundaries and landscape.

2We won’t apply the foundational approach based on empirical observation, because, even if theoretically

interesting and valid, in practice it will require the entirety of neurosciences and sociology studies to explain

the simplest of game.

Fundamental assumptions

The idea here is very simple: to use the tools of modal logic (developed primarily by

Frege, Carnap, Hintikka, Quine, Godel, Lewis... and others) to study games.

The reason why we are allowed to apply modal logic to game, is that any “game” is

inherently a specific subset of either a natural or a formal language. If this is somehow

self-evident in those “story game”; nevertheless formal and modal logic are respectively

used to describe the structure of any language, either extensional or intensional (and

games, as we will see, are either one of those).

So, we have to reformulate modal logic to describe games. We will use the “possible

worlds” concept, that is quite fitting. Even without stretching the concept till the modal

realism (which is an impegnative ontological position) this idea of possible worlds fits

3

instinctively and perfectly to games (either if you believe that possible worlds are actual

worlds within a multiverse, or if you instead thinks that those are only concepts that we

use in order to manage our inherently uncertain knowledge).

Primordial sets

Now let’s define the initial sets we will use to build our foundations for a structured theory

of games.

Definition Table 1

I. ∃U | ¬U = Ø

II. ∃ㄙ, ∃x,y ∈ ㄙ | ∀x,y → s(x,y) ∈ 2ㄙ

III. ∃G = {g1, g2, g3, … , gN}

IV. ∃W = {w1, w2, w3, … , wN}

First of all, there is U: the set that includes any logically possible object (define by

opposition: its negation is an empty set, as in I.).

Then we define ㄙas the general linguistic set (or class): any possible symbol (x) and any

composition of symbols s(x) will belong to ㄙ (or one of its subset - II.).

Then there is G, which is the set that includes any game (III.) and W(“world”, a subset of

U - IV.). Please note that provisionally (but for good reason) Gand Ware defined by

enumeration.

3 Lewis, 1973

Primordial sets: properties

There is then a number of relationship we may derive from those very simple sets, and

additional sets that we can establish.

Definition Table 2

V. U ⊃ ㄙ ⊇ G

VI. ∃L = {s1, s2, s3, … , sN}

VII. L ⊆ ㄙ

VIII. ∀s | s ⊆ 2L ⊆ 2ㄙ

We will come back to V. before the end, but let’s state it and simply leave it right (for

now).

One set particularly useful is L(VI.), which is the entire class of natural languages. Since

we don’t need to face yet the translation problem , we don’t need to differentiate within L

4

(as total class of all natural languages) its subset like L1,L2,L3(those are specific

languages like English, French… etcetera).

The VII. lemma states that ㄙas power set of signs combination is wider than natural

language (it includes symbolic languages and casual combination of signs).

VIII. defines that any statement smust be always contained within at least one subset of

natural languages (that serves to differentiate between an unarticulated noise and a

proper statement). Those belonging relationship aren’t written as proper subset (⊂) but

instead as improper subset because, even if there are strings of symbols (thus belonging

to ㄙ) that doesn’t have a correspondence in any language, we cannot exclude at all that

they have a (yet unknown) language that uses them.

Game statements and game elements

Using the elements that we have, we can define the game components (elements those

belongs to the set of games G). Those are splitted in two categories, depending on their

simultaneous belonging - or lack thereof - to both games set Gand natural language set

L.

Definition Table 3

IX. g ⊂ G

X. gS ∈ (g ∩ L)

XI. gE ∈ (g - L)

4 Kripke, 1979

IX. is about a single game g: this is a proper subset of games in general as G.

X. is what we will call a game statement gS, something that belongs both to a (any)

language and to a (any) single game.

XI. is instead a game element gE, something that belongs to a (any) game but that’s not

part of the natural languages (anyway, it will always belong to ㄙ, of which G is a subset) .

5

We can now define the following additional properties about them.

Definition Table 4

XII. gS ∩ gE = Ø

XIII. gS ∪ gE = g

XIV. g = ∑gE(1→n) ∪ ∑gS(1→n)

The intersection and relative complement of a same set is always an empty set, so if XIII.

is true (and it is, by definition) thus XII. is proven true too.

XIV. presents an alternative to define a game g, which is exactly the same as XIII. but here

it is explicitly written that gSand gEmay assume any possible value, and that the game

includes them all.

Those definition may be summarized as follows: in a game we have either game elements

or a game statements, and the sum of them all constitute the game itself.

Possible worlds and game worlds

With our baggage of definition we establish a connection between the concept of

possible world and our emended concept of a game world.

In fact, a game world is indistinguishable from a possible world: they share the same

construction, simply with a little twist on their initial foundation .

6

Definition Table 5

XV. ∀x → ∃x

XVI. ∀P, ∃gS | ∀gS ⇒ gS ∈ Φg

5For example, a game element is a a chess piece: something that has formalized properties (that we could

describe logically). Or, for example a game element may be some specific subset of the code within a video

game.

6We are here using exactly the same solution Quine [Quine, 1960] applies to solve the referential opaque

statement problem: giving them an implicit assertion of existence.

XVII. Φg = ΣnW

XVIII. ∀W | ∃P ⇒ W ∈ Φg

XV. is the assertion of existence as Quine makes it. This, in the possible world logic,

implies that everything logically possible is then possible within at least one possible

world: everything possible is then true (real) at least in one possible world.

Therefore, we can in XVI. redefine existence (∃) using the concept of belonging (∈). We

can use XVI. as: “Player Pbelieves that a game statement gSis compatible with any

possible world W belonging to Φg”.

To be precise, the strictly logical definition (in XVII. and XVIII.) may be paraphrased as:

“There is a player Pand a statement gSso that, given any gS,gSbelong to Φg. Φgis

defined as the sum of npossible worlds W, so that to any Wthere is a Pthat attributes W

to Φg”.

Speaking about game, we can define Φg: that’s the sheaf that define the relationship

between a single game g and all the related possible game world (Wg) that belongs to it.

Circumscribing a (single) game world

Since they share exactly the same construction (in the definition above, substitute “game

world” with “possible world” and you’ll have the classic definition of them), we can

assume that game worlds are “constructed, and not discovered” [Kripke, 1979].

Now, we have to face another problem stated by David Lewis [Lewis, 1973]: any

counterfactual will never be an exhaustive description of a (singular) possible world. That

description will always embrace an unlimited number (sheaf) of game worlds compatible

with the property (or properties) chosen.

The conclusion is that we create our own game worlds depending on how we want to use

them, and in doing so we always create a sheaf ΦGof multiple game worlds that are all

compatible with our requirements.

Setting the equivalency

It seems natural to establish a connection between W(possible worlds) and WG(game

worlds). As stated above, and since their definition are equivalent, it makes perfectly

sense that, given any description of WG, we can create a correspondent W.

Given this fact, maybe should we state that possible world and game world are equal by

definition?

Definition Table 6

XIX. W ≜ WG

That’s exactly what’s written in XIX. This make perfectly sense (even if it is a little

counterintuitive): there are sounded reasons to believe that we may “play” with (within)

any set W, either that is defined by a game or by something else.

Translation principle and translation function

There are other implication in linguistic that descend by what has been written until now,

specifically those related to the so-called “not-extensional statement”. We can now use

them in our game study (motivated by the application of the possible world semantic as

game world semantic).

The inapplicability of the translation principle [Kripke, 1979] is the most relevant for our

purposes. Given its importance, it maybe worth having a brief overview about what it

meant.

If in classical logic (or logic of first order) we can substitute any lemma and statement with

another one with the same truth value, and the composed truth value doesn’t change. So,

if A=B and B=C then A=C (transitive property) .

7

Intensional statement, instead, does not necessarily possess this property. Any

substitution of a game statement with another one does not necessarily save the truth

value, even if we substitute a true lemma with another one, true by itself.

Then, we can try to define the properties of this translation function t(x) in games. It’s

domain and codomain will be of course g1 and g2 (two different game).

Definition Table 7

XX. ∃t | ∀x ∈ g1 ⇒ t(x) ∈ g2

XXI. ∃x,y ∈ g1 | t(x) = t(y) ⇒ x ≠ y

XXII. ∃y ∈ g2 | ∄x ∈ g1 ⇒ y = f(x)

Their properties are simply listed and not explicitly demonstrated due to reason of length:

it’s neither an injective function (XXI.) or a surjective one (XXII.)

If you’re interested in the related proof, here’s the general proceedings: to demonstrate

that at least once a single element of g2is equivalent to more than one element of g1(two

g1elements has the same translation in g2); and to demonstrate that at least in a single

case a function will not exhaust g2(or: there are specific elements in g2that remain

untouched by the translation from g1). Because we need a single case in which properties

7This is an incredibly important point, specifically in game-design, and thus will be also covered by “A

Formal Study of Game Design”

do not apply to deny it, the translation function is games benefit from neither of those

properties.

As we will see, the translation principle will become central later when we will find out that

its one of the distinctive trait that we can use to classify games.

A practical example of translation function applied to games

We’ll use a simple, practical experiment strictly connected to the translation principle (or,

to be precise, lack thereof) that we described above to show how we can “feel” the

difference that a behavioural game possesses in relationship with a procedural one.

1. Imagine a situation in any roleplaying game you know (works with any game, but

it’s easier with roleplaying games). Ideally, think about something that requires “a

test” or any other kind of interaction between player and application of rules and

mechanics. The only requirement, is that it need need to be something not self

evident, like any situation with an undetermined outcome. Now write that situation

down (in natural language, with your own words, without using the specific idiolets

of the game).

2. Pick up the game rules, and assign the numbers / descriptors / bonus-malus /

mechanics (whatever it is) to the situation you written down above, in order to to

solve it (either or a numeric resolution, a poll, or whatever the game use). This

simply means to translate it into games mechanics. Wrote now down the

procedure that “the game” requires.

3. Choose another person. Provide her/him only with the description in natural

language from step 1., and ask her/him to execute the bulletpoint 2.

Are the bullet point 2. and 3. the same? If those are, then the game is (pretty certain) a

procedural one. If the two differ, and none made explicit mistake in interpreting or

applying the rules, then the game is a behavioural one.

Those description does not correspond to a single Playing Characters, but to unlimited

different playing character that share the same description. Yet, they’re each one different

from another. To say it as Kripke will have said that: there are infinite valid translation in

natural language (or fiction) of the same mechanical descriptive element. We never have a

single character, we always have a sheaf of characters compatible one with each other.

Rigid designators

Another important piece of the puzzle was developed by Saul Kripke [Kripke, 1979] as is

called “rigid designator” principle. This is an assumption needed in order to assign a truth

value to any counterfactual. This represent (also) an answer to problem of translation

principle described above.

The translation problem by itself implies that any counterfactual broke the identity

principle, and thus destroy the chain of significance. Literally: once we accepted a single

counterfactual (“this things isn’t itself but it is something else”), we cannot assign any

truth value anymore (because, if something is different, while only that

should be

different? Why something else cannot be? And then, why something else again? And

again? And again…

In order to solve this problem, which is embedded in language and logic itself, Kripke

defines a “rigid designator” by an intuitive way, which however allows to create a very

precise definition. The principle is: “a proper name is always a rigid designator”.

To paraphrase that: if we say “Mark”, we always mean the real Mark here and now, even

if we use him in a counterfactual statement, like: “what if…Mark was 15” taller”. In this

case, Mark is exactly the Mark we know, from every point of view, except his height

,

because that’s what our counterfactual is about.

So, in order to conceive that a counterfactual can be really discussed , we have to

assume that its truth value may change only on what’s the counterfactual is about. The

conclusion of Kripke is straightforward: if something isn’t explicitly stated as different,

than it’s the same.

Note that only proper names are rigid designator: terms that identifies something by a

definition that can change with the pèossible world (like “my friend” or “the 43rd President

of United States”) are called non-rigid designator

, of flaccid designator

. Differently from

the rigid designator, their truth value may change with the counterfactual, not only by the

explicitly cited exception.

Looking for coherency

Now, we have to find out some kind of principle to organize truth values. In modal logic,

this is possible thanks to Kripke’s rigid designator: assuming that an element xof a

possible world WAmaintains all its standard properties p(x); except for those listed in the

Aset that we can explicitly define as “the set of all the differences between WAand W®”.

Note that W® is the “real” world, the one we’re actually in.

Defining it in this way is no more than a simple notation that allows to point out that ®is

the reference systems we use to organize any possible world: in other words, the

reference system is the only world we can experience directly.

Here below a single counterfactual is written as a.

Definition Table 8

XXIII. ∃A = {a1, a2, a3, … , aN}

XXIV. ∀a ∈ WA ⇒ a ∉ W®

The same identical definitions will apply to our game logic. Games in general are not a

freeform babbling and doing stuff, but possesses some kind of inner logical structure.

We can safely assume that simply observing the fact that players are able to

communicate with each other about the game world, even when they discuss about

something (like the effects a wizard can create by her spell) that has not referent in real

world, thus is a counterfactual without a strict meaning.

At this point we need to understand what is (if there is) the difference between a

performative use of language in game compared to the natural language.

In a natural language we have a criteria (the coherence of the truth value) to lock the

significance chain: we have a way to assign truth value to non-extensional statement (by

their similarities with our reference, ®).

In game language, we have to assume that the players create forecast and expectations

about what’s about to happen, and that is exactly as formulating a belief statement

(regulated by modal logic). Then, we need a decisional logic to assign truth value.

It’s interesting to note that the properties of this internal logic has never been explicitly

8

stated in any game. Intuitively, someone call it “narrative coherence” and connect it with

the suspension of disbelief we have in literature. Other relate it to an (implicit or explicit)

agreement between player calling it “creative agenda”. Someone else talk about features

derived from the game world, and use the term “consistency” or “shared imaginary

space” taking for granted that there is a specific reference criteria ® within any game.

None of these ideas is entirely correct. However, there is an underlying feature that those

ideas share. They state the existence of an “ontologic layer” in games, that provide a set

of decisional criteria whenever we cannot directly apply the “standard” criteria we use for

counterfactuals (adherence to the real world).

Thus, a simple disproof of their validity may be the Occam’s razor principle (we should

avoid anywhere’s possible any multiplication of ontological objects to solve a causation

or - or signification - chain).

The perks of narrative

Now, let’s make a small parentesys to face the fact that there are self-evident differences

between various forms of games, and how they applies to the differences between

extensional and intensional statement in games.

9 10

8 The one we will call “Fictionally Distributed Logic” or [fd]logic

9 Obeying classic logic

10 Obeying modal logic

In a chess game we have either intensional statement (“the white player believe he’s

gonna lose

”) and extensional statements (“a pawn is in E2

”). In games, we can determine

if an extensional statement is true or false by applying the game rules and mechanics,

and we have to apply modal logic in order to solve any intensional statements related to

the game. As anticipated, we will call them procedural games.

The most interesting feature of those games is that “what’s matter” into the game are

only the extensional statement. In no procedural games there are any legit intensional

game that can be solved or determined by the rule of the game itself.

There are, conversely, behavioural games. Those, as proven separately , doesn’t have a

11

clear and fixed distinction between extensional and intensional statements, it’s possible

that indistinctly we may need classic logic and game elements, or modal logic and game

statements to assess the truth value of part of the game. In those behavioural games the

truth is “locked” inside a different level than the rules (or to say it in a more accessible

language, “the truth we are interested about regard the story, not the application of

rules”).

This happens only in this category of games: in any other game are the rules themselves

(and the result of their combination) are enough to create a validation structure to

determine any truth value (which should be clear, right now, that is an equivalent for

“playing the game”).

So, if we define the “purpose” of the game as the calculation of truth values for the

game-related statements, there are games that require modal logic (and a specific

behavioural attitude) in order to be used. Those, as said above, are the game where truths

are located within the narrative, into the game world, and not into the game.

From a general standpoint: games are a incredible fast growing and wide ecosystems.

There are literally more games in the world than how many any human being can

experience in a full lifetime. However, we don’t need to list all existent games to provide a

definition and a differentiation strategy to categorize them: it’s enough to observe if the

the game itself includes intensional statements and the procedures necessary to manage

them. If not, it’s a extensional game (and doesn’t matter what are its components, shape,

medium or focus).

Rules and story

We can now give the following logical definition for what’s usually called “story” (XXV.

),

juxtaposed to “rules” (XXVI.

). A “game” generically intended (XXVII.

) is the sum of the two.

11 This is a focal point of “A formal study of game design”. For further explanations and example reference

to that.

Definition Table 9

XXV. Sg = ΣgS ∈ g | Ե(gS) = ⊤

XXVI. Rg = ΣgE ∈ g | Ե(gE) = ⊤

XXVII. g ≣ Φg ≣ Sg ∪ Rg = Σ(gS ∪ gE)

In this definition table we begin to use Եas a univocal symbol for the specific function f(x)

that calculates the truth value of the element x, giving an output which is binary: either ⊤

(true) or ⊥ (false).

Written not in formal language, XXV. predict that the story is the sum of game statements

that belongs to the game, and are true. The rules, similarly, are the sum set of all the true

game elements (XXVI.) A game itself is the union set of those two sets.

Thanks to all the principles enunciated until now we know that we can rewrite a truth

value into a game as a statement of existence, and this statement of existence may be

rewritten as a statement of belonging to a possible world, which really is a sheaf of

compatible possible world differentiated one from another for any counterfactual small as

we want.

The universal game mechanic

We now can create have a more powerful definition that describe what a game (any

game) does, and specifically what a “game mechanics” means.

Definition Table 10

XXVIII. ∃g ∈ G | ∀g ⇒ Ե(g) → [⊤, ⊥]

XXIX. |Ե(g)| = 2

A game mechanics is then any truth value calculation over gin Gthat can be effectively

solved, which values is either “true” or “false” (its codomain has cardinality=2.

Rules vs Story

We have an interesting implication from Definition Table 9 and 10: are either the “story” or

the “rules” enough, by themself, to create a game? The answer, as should be

self-evident, is a sounded “yes”.

There are two ways to prove it: one is that, logically, we can substitute gS=Ø in the

relevant definitions. The other, already cited before, is that procedural games (like chess)

does not have any gS that they legitly can process by their own rules.

What is a game (for)?

Now we have enough elements to provide a preliminary definition of what is a game. Here

is written in natural language (the logical formulation is straightforward from the definitions

used until now).

Definition Table 11

XXX. A game gis a function to assign truth value Եrelated to a specifically

constructed possible world W, called game world Wg.

Using the tool of modal logic that we have created, we can expand from XXX. including

the two different subsets of games that we cited until now.

Definition Table 12

XXXI. A game assign truth values by a specific sub-set of criteria.

XXXII. Those criteria may be explicit (procedural games) or implicitly involved by

a gaming attitude (behavioural game).

Procedural games can be also identified by the greek letter rho (ρ). Behavioural games

will be indicated instead by the greek letter beta

(β).

Cardinality of games

From the description above we can immediately quantify the size of the different sets and

subsets by their cardinality:

Definition Table 13

XXXIII. |Σs| = ℵg

XXXIV. ℵg ⋜ ℵØ → g ∈ ρ

XXXV. ℵg > ℵØ → g ∈ β

XXXVI. ℵρ > ℵβ

Given the cardinality of the sum of all possible statements sdefined on gas in XXXIII., a

game is procedural if this cardinality is at maximum equal to aleph zero (ℵØ). If it’s greater,

than it’s a behavioural game.

Please note that (at least for now) the use of ℵØas measurement of cardinality is purely

arbitrary, and those not implies that cardinality here are equivalent as defined in math.

However, there are sounded reasons to believe that a game like chess - a good example

of procedural game - “contain” exactly the same cardinality of natural numbers (N set),

given the fact that we can explicitly create a chess game that is made of infinite “integer

and positive” moves (the natural number set, which cardinality is exactly ℵØ).

An interesting and still open question is: does behavioural games posses the same

cardinality of the real numbers (R) set? Or, instead, do they have the same cardinality of

the integer (Z) set?

12

Regardless of the exact relationship between the different ℵ, definition XXXVI. is the

pivotal point here: the cardinality of behavioural (β) games is always greater (and never

equal to) than the cardinality of procedural (ρ) games.

[fd]logic

Finally, we are able to start using our Fictionally Distributed logic. The name, now, should

make a lot more sense: it’s the definition of the fact that a game logic cannot be neither

classic logic or modal logic, but have to use them both, respectively in the two different

category of games, and provide with the tool in order to distinguish when should be use

one or the other.

At this point, we can propose some general principles because we have (until now)

created a formal foundation and a definition of applications for this logic.

1. In order to function, [fd]logic presume that what’s written until now is correct. So,

for example, that game may be defined as tool to assess the truth values.

2. The fictional distributed logic is branch of logic applied to games in general, that

allow the description in logical term of the activity commonly defined as “game”.

3. In case of a procedural games, [fd]logic collapse into formal logic; when applied to

behavioural games it collapse in modal logic (with the definition “game world”

instead of “possible world”).

4. [fd]logic allows to identify precisely the component of a game, thus giving

differentiated and aimed procedures applicable to different typology of games .

13

5. [fd]logic is capable of explaining why, for example, procedural game may be

played through any language barrier, by people that only shares the rule of game

themselves. This cannot be true in a behavioural game instead requires a

completely different level of interaction and common basis between players: you

can’t play a narrative game if, at least, all the players doesn’t share a common

language.

12 One of the future development of this study is the idea that is possible to create a Cantor-like diagonal

proof, that can connects the cardinality of a procedural games to those of behavioural games.

13 Even if this point will be better managed into “A Formal Study of Game Design”, it’s worth noting that,

actually, almost all current game design relies only on classical logic, even in behavioural games.

Universe, language and games

We can finally address a simple but profound consequences of what’s written until now. If

it is true that a gaming attitude is necessarily and sufficiently to have a “game”; if also it is

true that the “game world” may be as “big” as the material universe: then the relationship

between U, ㄙ and G (at V. - VIII.), need to be updated as follows.

Definition Table 14

XXXVII. U ⊇ ㄙ ⊇ G

We have to admit that there is actually a terminal possibility (in case of =) that the

universe itself is a symbolic systems, and may, in its entirety,considered a game .

14

Future developments

Those are a number of general conclusions, guidelines, suggestion and topic of further

investigation that derives from what’s written until now. This is only an overview: a

number of them will be discussed and analyzed deeper in “A Formal Study of Games

Design”.

●A diagonal demonstration applied to games with a strong math component (like

chess or domino) should, in theory, be able to prove or disprove the hypothesis

made here about cardinality ℵg = ℵ0.

●There are two kind of games. Some are made up by of procedures, and those

game may be played by a Turing Machine. There are games intrinsically made from

a behavioural attitude (like role playing game). Even if someday, we will be able to

create perfect natural-language syntax machine there are reasonable doubts that

this machine will be able to play a behavioural game. That means that a revised

version of the Turing Test for AI should be the ability to comprehend (thus to play)

a behavioural game. Since behavioural game requires the use of modal logic, this

will be inextricably connected with the recent development in deep learning,

machine learning applied to natural language and the classic Turing test . Our

15

idea, is that a “proper” Turing test for an IA is not based on the capability of a

machine to “travesty” as human, but by their capability of playing a behavioural

game.

●Caillois’ taxonomy of games that defines games by the “experiences” (agon, alea,

16

mimesis, ilinx) also includes the idea that games dispose themself on an axis of

17

between ludus

and paideia

. Caillois thought the two were related to “complexity”

14 This is surprisingly consistent with the holographic theory of the universe.

15 Case in point: today we have machines able to pass a classic Turing test. Yet, we “feel” that they still are

not “true” artificial intelligence.

16 Caillois, 1967

17 There isn’t a direct equivalent of those four area in this paper.

of game rules: an intuitive, but not formalized concept. We believe that those two

extreme correspond to what we discover as procedural (ludus) and behavioural

(paideia) games. Differently from Caillois, however, those are two completely

separated set and there is a precise boundary, not a gradual transition from one to

another .

18

●Playing a game is making expectations about a game world, in order to test them.

Playing means find out what’s right and what’s wrong about our game world. As a

necessary corollary, playing games means improving in both the skill required to

19

validate statements, and knowledge of the prior criteria chosen.

●A β-game is, by definition, a subset of ρ-game. Also, it is without precise

boundaries. There are infinite possible truths that can be verified by the game. A

procedural game, instead, could (theoretically) be exhausted, given enough time

and resources (a.k.a. a Turing-machine can solve it).

●We can define “flaccid-procedures” for behavioural games in order to calculate any

truth values that we need (see “Addendum” below). Those procedures rely on the

game world, not on rules composition. Those procedures are necessarily

incomplete (the game world is incomplete by definition: it’s always a sheaf of game

worlds, exactly like there are infinite possible counterfactual).

●The “practical” game is no sure indication about the ρ or β structure. A game may

seem behavioural because it involves storytelling and offer a great degree of

options to players to contribute to creation of the game world… but being entirely

procedural from the standpoint of validation .

20

●We can define a gamelet: it is the specific subset of game that any different player

uses. Those gamelets may or may not overlap, and may be the results of either

casual of systematic factors .

21

●We can define “story” and “rules” logically, but they do not exist in practice as

separate entities. In a ρ-game, the “story” is equivalent to the rules application,

whilst everything outside that does not belong to the game (do not have a truth

value). In a β-game, instead, the “story” is a rule by itself, since it provide criteria to

assign truth value.

●If games are function applied to sets, it will be interesting to study if those function

are continuous, discrete, neither or both (in the latter cases, it means that different

games can be discrete or continuous). This is a particularly complex thing to prove

or disprove. Actually, we propend towards the idea that the game function operate

18 Intuitively, we can explain the apparent gradual transition due to the fact that many games “seems” more

procedural or behavioural depending on their practical rules. Our distinction is instead theoretical.

19 Either “play many different games” or “play a lot with the same game”

20 A similar game is quite easy to write: “the player to the left determine the truth value of the statements

pronounced by the player on the right”. This is also a “flaccid procedure” as defined above

21 More on them in “A Formal Study of Game Design”

in the continuous, but can be practically managed only in its discrete

approximation.

Addendum: truths

Exactly like there is a number of different truth values and definition in linguistic , so there

22

should be in games. This analysis, even if not needed for foundational goals, allow us to

understand how different statement about the game world (and the game itself) can (or

cannot) be validated... and by what procedures.

This part of the paper will be less formal the the one above, but it’s extremely important if

you want to apply also the notion contained within the other paper “A Formal Study of

Game Design”.

Definition Table 15

XXXVIII. Necessary [NE ] truth includes all those statement that are either true or

false in all possible game world. They are the defining elements of the game, and

can be further divide into:

a. Prior

[Pr ] truth is something self-explaining as true/false. It’s usually

a logical tautology like x=x, or a property that we use to wrap the game

world around it. Usually those are the very cornerstone of the game.

b. Analytical

[An ] truth it’s something true by how it’s presented (by

structure). An analytical truth start from another truth (whatever it is) and

then, demonstrating that there is a connection with the required truth

value. Bug/fallacies within the game belong to this category, as general

statement like “someone will win”.

XXXIX. Contingent [CO ] truth includes those statement either true or false in

some game worlds, while not in others. Any contingent truth need a specific

procedure to determine if it’s true or false for the current game world. Those are

the defining element of the play that we’re having, and not of the game and game

worlds themselves.

a. Posterior

[Ps ] truth is valid in the current game world. It’s true only

because “it happened”. In the case of a narrative game, we can use it as:

“something true because it happened in the game world”. In the case of

games, the most evident truth belonging to this set are those caused by

dice or random chance. It’s a truth valid in our current play, not in the

game in general.

b. Potential

[Pt ] truth, also called as undetermined, is a truth that is

valid in some game world, but not -yet, or eventually never - in the one

22 Lewis, 1973

we’re in. Those are, also, all the question that the game may (and will)

answer while the players are playing.

There are two broad categories that exist in language and we can use directly into games.

Some have “constitutional” truth values, embedded within the entire sheaf of possible

world compatible with the game: we literally cannot change them without changing the

game itself.

The others are truth values specific of a subset of the game worlds: their truth value may

(and often will) change with further explorations of the game world. Their variations,

however, still keep us within “the same” game.

Examples of truths

But let’s see some examples of different truths in case of games. Since we yearn to

create a general game theory, we’ll show how this applies to his original context (a literary

work), to a classical game (chess) and to a video-game (Fallout 3).

Just remember, that in order to analyze the truth value of a statement, we don’t need if to

be necessarily “true”. “False” is a lecit outcome too. A problem will arise if (or when) we

cannot properly assign neither.

Example 1

Truth

“Lord of the Ring”

“Chess”

“Fallout 3”

NE

Pr

Gandalf is Gandalf

Board size is 8x8

Year is 2277

NE

An

Gandalf is an Astari

A single Pawn deploy in D2

It’s been 200 years from war

CO

Ps

Sauron has been defeated

White lose his Queen

Luck stat is < 7

CO

Pt

Sauron wins the war

White win the game

Vault 101 never opened

How this different truth values may be calculated in different scenarios of games isn’t of

capital importance here. The important things is that we can use some of them: we will

call them “validation procedures” or simply “checks”. Also, we can synthetically

demonstrate the difference between procedural (XL.) and behavioural (XLI.) game,

because we have now a way to describe the different possible condition of validation for

both of them.

Definition Table 16

XL. ∃(gS1, gS2) | Ե(gS1) ∈ PS, Ե(gS2) ∈ AN ⇒ Ե(gS1) = Ե(gS2)

XLI. ∃(gS1, gS2) | Ե(gS1) ∈ PS, Ե(gS2) ∈ AN ⇒ Ե(gS1) ≠ Ե(gS2)

This definition differs for the presence of either = or ≠ in the end. It means that we can

pick any statement with a contingent truth value, and analytically derived another

statement from it. Then we have a “procedural” game if the truth value of second

statement changes accordingly with the truth value of original one. If, instead, the derived

statement changes truth value by its own terms, (regardless of the value of the first one),

then we have a behavioural game.

Not surprisingly, this demonstration still operate under the frame of the translation

principle. Indeed, the presence or absence of a translation principle equivalent applicable

to a game is also a necessary and sufficient condition to classify it as either procedural or

behavioural.

Validation in behavioural games

Now we have different notions of truths, and possible procedures to validate them. Let’s

see the most interesting case: how can validate a posterior truth Pt in a behavioural

games.

Definition Table 17

XLII. We can validate the potential truth by abstraction/deduction from other

truth we already know. For example: we can prove that either Pt or ¬Pt is a

tautology (then it’s a Pr truth); we can prove that from a Ps,An or PT) truth we will

derive the Pt (then it’s an An truth); or we can prove that it is part of the definition of

the current Wg (then it is a Ps truth).

XLIII. We can create a validation chain: creating additional Ps or An truth as long

as we are able to connect our Pt truth with one of those, and then apply one of

the procedures above .

23

Note that you can use those strategies also with a procedural game: in this case,

however, any procedure you pick the result will be the same (because the formal logic

that applies to procedural game state that different composition of truth values doesn’t

change if you change the process of calculating it, or translation principle).

Instead in modal logic (and behavioural game) any different procedures we pick up to

calculate the truth value affect the final result.

23 Here basically we develop further characterizations of the game world until we have what we need. We

can do that, since the game world isn’t ever fully and self-sufficiently defined.

There is however a counterintuitive consequence to what we have described right now. It

may seems that any statement may be either true or false depending from how we

describe it, rather than “what it is”.

It seems strange, right? Common sense told us that there is a thing called truth, and this

can’t be both true and false… right? No: that’s wrong. Let’s see why it is.

Addendum: A proposal for truths modelization

The relationship between different truths may also be represented graphically. This kind

of description is a double utility: first, it allow to spatially represent the different validations

procedures we have. Secondly, we can use all to standard tool of cartesian space (for

example standard functions definitions and analysis) to games.

The idea is to draw a cartesian plane. Any point into it may be described as usual as a

couple of real number [x,y]. We use + and - to identify the different quadrants.

There are four possible iteration: [+x,+y], [+x,-y], [-x+y], [-x-y]. Different truth value are

consistently within a single quadrant:

●Analytical truth are [+,+];

●Prior truth [+,-];

●Potential truth [-,-]

●Posterior truths [-,+].

In this representation, necessary truth have a positive component on the x axis [+x], while

contingent truth a negative one [-x].

An interesting hypothesis to make is that the x axis somehow represent the “logical

consistency” of the proposition, while the y axis is connected with a specific moment in

time (to be precise, a certain point in the exploration of the fictional world).

This representation is particularly apt to identify kinds of truth, but we suppose (and save

it for further studies) that it is possible to define functions map the transaction between

different truth defined in the totality of the Wg.

This is also interesting because it shows how games are dynamical system of

relationships, kept together by a description of a game world.

Example 2

It’s still unclear and deserve

further study if a game is an

integer of the boundary

function that discriminate

between True (⊤) and False

(⊥); or if it is instead only the function by itself which result as ⊤and anything outside it is

⊥.

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