# Propositional Logic Game

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zmobie
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Joined: 11/19/2008

Hi all. I am a college student at U of M, and I am the Student Instructor for a Logic class. I also love playing and writing games, so I thought it was a no brainer that I should make a game to help people learn how to do propositional logic proofs. The problem is that I am having a hard time translating the rules of propositional logic into a fun and easy to learn game. Every idea I come up with assumes way too high of a skill level going into the game.

I would like to support high level play at later stages in the game, maybe with modular difficulty levels so when you first start, the game is easy, but as you progress it gets more challenging.

I am also having trouble with theme. I thought it would be fun to play the part of various philosophers throughout time, each with their own specialties in the game, but this might interfere with the goal of actually teaching people how to do logic proofs. If anyone here has experience with symbolic logic, or has any ideas that could help me out, i'd love to hear them.

InvisibleJon
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Joined: 07/27/2008
An interesting challenge...

Wow. That's a toughie.

I'm working up a game that uses (and therefore teaches) very basic boolean logic (AND, NAND, OR, NOR, XOR, etc...).

It's a neat design challenge, though... I need to think on it (and refresh my knowledge of propositional logic).

Kjev
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Joined: 03/03/2009
WTB primer on subject

zmobie wrote:
Hi all. I am a college student at U of M, and I am the Student Instructor for a Logic class. I also love playing and writing games, so I thought it was a no brainer that I should make a game to help people learn how to do propositional logic proofs. The problem is that I am having a hard time translating the rules of propositional logic into a fun and easy to learn game. Every idea I come up with assumes way too high of a skill level going into the game.

I would like to support high level play at later stages in the game, maybe with modular difficulty levels so when you first start, the game is easy, but as you progress it gets more challenging.

I am also having trouble with theme. I thought it would be fun to play the part of various philosophers throughout time, each with their own specialties in the game, but this might interfere with the goal of actually teaching people how to do logic proofs. If anyone here has experience with symbolic logic, or has any ideas that could help me out, i'd love to hear them.

Being a serious games designer in my professional life (digital games, in this case), this sounds like an interesting challenge. Sadly enough I have no experience (that I know of) concerning 'symbolic logic' / 'propositional logic proofs'.

Is there a way to give me and others at this forum that are interested a primer on the subject? E.g. websites with information that you, as a Student Instructor (and most likely the 'expert' on the subject here), believe to capture the essence of the subject?

zmobie
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Joined: 11/19/2008
Well, propositional logic is

Well, propositional logic is used to capture a propositional argument in symbolic form in order to determine its validity. Lets use a simple example

If I jump off the building, I am dead.

morbid i know, but a good example nonetheless.

This is a conditional statement. If - then. Thats how you know its a conditional. To symbolize a conditional statement we use the arrow.

-->

Now, propositions are statements about the world with truth value. In any conditional statement, there are two propositions, the antecedent and the consequent. The antecedent is the first one, and the consequent is the second one.

Lets let J be the first one and stand for "I jump off the building"
Lets let D be the second one and stand for "I am dead"

To represent the previous statement we would write this

J --> D

If J, then D. If I jump off the building, I'm dead. Pretty simple so far. The conditional is known as a logical connective. There are other logical connectives.

& - And - a conjunction
v - or - a disjunction
= - if and only if - The Biconditional

Then there is the ~ which means NOT.

So I could write ~J v D which would mean "Either I didn't jump off the building or I am dead"
I could write ~( J & ~D) which would mean "It is not the case that I Jumped off the building and didn't die"

OK, now to the fun part. Every connective has certain rules of inference that you can use to operate on them. Its easy to illustrate that with an example.

say we are given 2 statements, and it is asserted that they are true. one looks like this

J --> D (if i jump off the building, I am dead)

and the other looks like this

J (I jump off the building)

What we know from these 2 statements is:

J --> D
J
therefore: D

This is one of our rules of inference. It is called affirming the antecedent, or Modus Ponens. Modus Ponens is a valid rule of inference because it is a truth preserving rule. If the first two statements are true, then we know the last one is.

Other rules of inference are like this

J --> D (If i jump of the building, I am dead)
therefore: ~J (I didn't jump off the building)

That one is called denying the consequent or Modus Tollens.

There is also some obvious ones like

J
D
Therefore: J & D

Which is called conjunction

There is

J v D
~J
therefore: D

Which is disjunctive syllogism.

I think you are starting to get the picture. Now, to do a logical proof, you use these rules to prove why your argument is valid. For instance

1. J-->~D (This is a premise. All premises are assumed to be true)
2. J (This is also a premise)
3. D v P / P (This is a premise, and after the slash is the conclusion)

We want to use the premises to come up with the conclusion. So here is how its done

4. ~D Modus Ponens lines 1 and 2
5. P Disjunctive Syllogism lines 3 and 4

Then we are done! We've proved that this is a valid argument. Obviously there are a lot more rules, and it can get pretty difficult to figure out how to prove these things, but I think you sort of understand the basics of how it is done...

Now, how to make a game out of it.

seo
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Joined: 07/21/2008
Now I want to make a game

Excellent explanation. And inspiring.

Now I want to create a game based on this. My first idea is two decks of cards: one with Letters and symbols, for the players to draw to their hands and play to the board in sets that form valid statements. Players can also form sets of statements to form inferences. Maybe each card has a scoring value, and inferences work as multipliers or something (so a three-lines inference will score three times the card values of their statements).

The other deck has statements, and is used by all the players, to draw and immediately play to the table. These cards determine what propositions and statements and are true or false, and are used to set the variable "rules" that will make the player's statements valid or not. We could call this deck the Axioms deck, perhaps?

So, if the Current Axioms are: P (which means P is true), ~D (so D is false), J --> D cannot be played.

On their turn, players will draw cards (2 or 3, I think) to their hands from the players reserve, and update the Axioms: draw a card, add it to the current axioms, remove whichever axioms required to leave a coherent set of axioms (here they will try to leave a set of axioms that favor their game while canceling the validity of some of his opponents current statements).
Then all the existing player statements that are no longer valid are removed from play (cards are used to form a new reserve pile and added to the bottom of the current one).
Then the player can play a new valid statement (or more than one). These can be independent from the player's current statements, or form an inference. Or he can just pass if he wants to keep the cards for future turns.

So, that's more or less a possible game. Feel free to use it as is or develop it further. I like the idea and will probably work on it, but I think there's still room for more than one propositional logic game in the market. ;)

zmobie
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Joined: 11/19/2008
I was thinking of something

I was thinking of something sort of similar, where the 'axioms' as you call them are drawn and played (or asserted as it were) or you can keep it in your hand and try to prove it for points. All played axioms can be used by all players. Once you prove a statement, you get a certain amount of points for it depending on how difficult it is to prove; and proven statements can be eliminated from a players point base by proving its contradiction. The main problem I'm having is how to implement the inference rules in a way that makes it easy to learn. Sure, I could easily make a game for myself because I already know all the rules. It would be in depth and fun and easy (for me or anyone who already knew all this stuff) but for someone just learning it would be impossible. Got any ideas about that? Have you ever developed a learning game?

Kjev
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Joined: 03/03/2009
zmobie wrote:The main problem

zmobie wrote:
The main problem I'm having is how to implement the inference rules in a way that makes it easy to learn. Sure, I could easily make a game for myself because I already know all the rules. It would be in depth and fun and easy (for me or anyone who already knew all this stuff) but for someone just learning it would be impossible. Got any ideas about that? Have you ever developed a learning game?

Well, how would you normally teach someone about the inference rules?

Also, have you considered looking at a system based on Flux.

seo wrote:
Excellent explanation. And inspiring.

Now I want to create a game based on this.

Let's make a secret GDS out of it! ^_^

seo
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Joined: 07/21/2008
zmobie wrote:The main problem

zmobie wrote:
The main problem I'm having is how to implement the inference rules in a way that makes it easy to learn. Sure, I could easily make a game for myself because I already know all the rules. It would be in depth and fun and easy (for me or anyone who already knew all this stuff) but for someone just learning it would be impossible. Got any ideas about that? Have you ever developed a learning game?

I developed a learning game about street addressing for the World Bank, so propositional logic sounds like a fun and light family game theme, in comparison. ;)

I thought about making it easy to learn, but forgot to mention on my earlier post. My idea is to have the cards on both decks divided into levels. A first level could comprise just some of the propositions (with true and false values), and simple statements using just the conditional. This level might be based just on separate statements, with no inferences.

A second level could add more propositions, and inferences.

Next level could add AND and OR, etc.

So, just like your post led us from the simple P --> D to more complex concepts like Modus Ponens and Modus Tollens, the game, by adding cards as certain scores are reached or a certain number of turns is played or whatever you see fit, would take the players/students from simple, easy to understand statements to more complex constructs and syllogisms.
Advanced players could just start the game at their complexity lever of choice (so when playing with fellow teachers you can have fun from the start).

seo
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Joined: 07/21/2008
Kjev wrote:seo

Kjev wrote:
seo wrote:
Excellent explanation. And inspiring.

Now I want to create a game based on this.

Let's make a secret GDS out of it! ^_^

That was exactly the first thing that came to my mind. :)
It wouldn't be fair, though, as some of us would have a head start. But "design a game to teach propositional logic" is 100% GDS fodder.

zmobie
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Joined: 11/19/2008
Kjev wrote:zmobie wrote:The

Kjev wrote:
zmobie wrote:
The main problem I'm having is how to implement the inference rules in a way that makes it easy to learn. Sure, I could easily make a game for myself because I already know all the rules. It would be in depth and fun and easy (for me or anyone who already knew all this stuff) but for someone just learning it would be impossible. Got any ideas about that? Have you ever developed a learning game?

Well, how would you normally teach someone about the inference rules?

Also, have you considered looking at a system based on Flux.

seo wrote:
Excellent explanation. And inspiring.

Now I want to create a game based on this.

Let's make a secret GDS out of it! ^_^

I vote against that idea.

Kjev
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Joined: 03/03/2009
zmobie wrote:I vote against

zmobie wrote:
I vote against that idea.

I wasn't serious, but why, if I may ask, would you vote against it? It might bring up some pretty good ideas / concepts for the game you wish to make.

zmobie
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Joined: 11/19/2008
I'm sure it would, i'm just

I'm sure it would, i'm just being paranoid.

Jpwoo
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Joined: 03/26/2009
I'm not thinking entirely

I'm not thinking entirely straight right now. so these may be awful.

A friend of mine has a game called mathmaticians dice, where you roll X number of polyhedral dice and you have to write a balanced equation given those numbers. Perhaps the same kind of thing could be used here. You could be given a set of 'facts' A, B, if A than ~B, C only if A. and then you are randomly assigned a group of arguments, nor xor cor, therefore etc... and you have to use what you are given to create a true statement.

The game would mostly be coming up with interesting sets of givens.

clearclaw
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Joined: 07/21/2008
Wff'n'Proof?

While not an exact match, have you looked into Wff'n'Proof?

http://wffnproof.com/

peat
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Joined: 08/03/2008
Propositional Logic Game

I would think a spy game or mystery would set a tone. The clues are provided via stringing logic together to come to a conclusion. I do like your idea of famous people in the mix.

JB
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Joined: 02/06/2009
I took decutive systems back

I took decutive systems back in my college days.

Let me riff out some ideas.

So I'm thinking a two player card game. You have two decks. One with statments such as "A*~D". And the other with several copies of the rules of Inferance. (Including the rules of deduction and all the equivalents.)

To set up the game you deal both players a Stament card. This is the statment they are trying to prove. It goes face up in front of them. You also deal the players a small hand (maybe 3) inferance rules. Finially, you put one statement face up in the center. This is a Shared Assumption.

Now each turn, a player can do one of four things:

1. Draw a new Inferance card into her hand.
2. Draw a new Statement card as a Shared Assumption.
3. Redectio ad absurdum: Use her inferance cards and two or more Shared Assumtions to prove a contradiction. This gives her the right to remove one of the Assumtions.
4. Prove his point: Use her inferance cards and the Shared Assumtions to prove her own conclusion and win the game.

For more advanced play, you can break down the rules of inferance into catagorised decks so there is more skill in choosing which deck to draw from. But I think just knowing when you have all the cards you need in a timely manner will take enough skill- you may want a timer.

Anyway, that's an idea.

Which U of M do you go to, Maryland?

Oh and I've had the idea before of having a battle of the philosophers game, but right now I'm working on a tribal politics game. Besides, I'm not sure how I balanced such a game could be. I mean the text for the Hume card has to be something like: "Skeptical Solution: if you hold this card when you lose, you instead win." and then Descartes power would be like: "It's all a dream: remove all cards in play, take the whole deck and sit in an oven until everyone else goes home." It just gets riddiculus fast.

zmobie
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Joined: 11/19/2008
I've thrown together a little

I've thrown together a little playtest deck already and a lot of your suggestions have sort of come out of the playtesting, which is cool.

The way it works so far is.

1. Draw 3 cards
2. Play one to the shared assumptions area ( you can't skip this step ). Any inferences you can make from your placed assumption may be used immediately. So if you play a P, and notice that you can do both a Modus Ponens and a Disjunctive Syllogism, you get to do both of these. Whenever you make an inference, you put that inference rule-card down in front of you. The more variety of inferences you use, the more points you get. Conclusions you get from your inference are put in play in front of you and are your own conclusions that the other player cannot use.
3. Step 3 is take an action. Actions are as follows.

Draw a conclusion from the shared inferences. (it happens automatically when you play a card, but if you want a conclusion your opponent got, you can use an action to get it)

Play a card as a conclusion you want to prove. ( You play a card face down, if you can recreate that proposition with your inferences, you get the points listed on the card)

or Use your own conclusions to form new conclusions. (You can use an action to mess with your own personal conclusions you've created from other inferences)

Also, at any time, if you have formed a conclusion that is the contradiction of another statement, both statements disappear. So if P is in your opponents conclusions, and you play ~P, both disappear.

This also applies to conclusions that your opponent has proven. So if he proves (P v Q) and you make ~(P v Q) somehow, you've erased some of his points.

Thats the game so far. Its pretty simple rules wise because the actual inference rules can be hard to wrap your mind around at first. I'm building it up one step at a time. I'm starting with just inference rules, and no equivalence rules. There are going to be 3 levels of difficulty, but each level will be pretty much the same i think.