I was amused to see this video of two mathematicians trying for an hour to figure out from scratch how to play Lewis Carroll's Game of Logic:
They struggle a lot, in part because they are pretty clearly rusty with syllogisms, although they begin slowly to make serious headway once they realize that Carroll's literal diagrams are square Venn diagrams. This is an important aspect of them; one reason why Carroll developed literal diagrams is that he wanted diagrams that would do what Venn diagrams but scale better to larger problems, leading to the famous Octoliteral Diagram in Symbolic Logic:
That's for a Sorites with eight distinct terms.
However, the Game of Logic is played only with Triliteral Diagrams (for premises) and a Biliteral Diagram (for the conclusion); the goal of the game is to take a pair of premises on the Triliteral Diagram and reduce it to a conclusion on the Biliteral Diagram. In other words, you are doing basic syllogisms; you'll have three terms (x, y, and m), which can have either positive or negative forms (Carroll prefers to designate negative forms with an apostrophe, like x', y', and m'). Your premises will share a middle term (m) and in effect you get the Biliteral Diagram by pulling out the middle term from the Triliteral Diagram.
The Triliteral Diagram looks like this (if we mark what terms each compartment covers):
The whole Diagram is the universe of discourse, and we are just dividing the universe of discourse according to the positive and negative versions of the terms. If the Universe is "Dogs", then in the upper left outer corner we have X Y Not-M Dogs; in the upper left inner corner we have X Y M Dogs, in the lower right outer corner we have Not-X Not-Y Not-M Dogs, and so forth, whatever our terms X Y and M may be.
To actually play, we use two counters. One counter is a DOESN'T EXIST counter (Grey, in the Game); the other counter is a DEFINITELY EXISTS counter (Red, in the Game). Those are my terms, rather than Carroll's, but they are accurate, because one counter something definitely exists in the universe, while other says something definitely doesn't exist in the universe, given your premises.
Suppose you have a premise, No X are M. If no X are M, we have to put a DOESN'T EXIST counter in any box that has both X and M. That would be the upper inside boxes in the center. If you had a different premise, Some X are M, however, that tells you that there is definitely an X that is M in this universe, BUT it doesn't tell you whether this is Y or Not-Y. So the DEFINITELY EXISTS counters get put on lines -- if Some X are M, then it has to go on the line between the two upper inside boxes, because we know there is something in the X M boxes, but we don't know yet whether it should go in the Y or the Not-Y box.
This is where the dynamics of the game come in, because the DEFINITELY EXISTS counters like sitting on the fence, but the DOESN'T EXIST counters are bullies, and are always knocking the DEFINITELY EXISTS off the fence into a box. If I have
No X are M
then, as said, before, that will put DOESN'T EXIST counters in the upper inside boxes. If I add to this the premise
All Y are M,
this will do two things: it will put DOESN'T EXIST counters in all Y Not-M boxes, but, according to the Game rules, it will be a DEFINITELY EXISTS counter on a line between Y M boxes (because it doesn't tell us anything about X or Not-X). The Y Not-M boxes are the left outside boxes, and we'll put DOESN'T EXIST counters in those. The Y M boxes are the left inside boxes; we put a DEFINITELY EXISTS counter on the fence between the two.
But the first premise told us that the upper left inside box had a DOESN'T EXIST counter in it. DOESN'T EXIST counters are bullies; they knock DEFINITELY EXISTS counters off the fence, so we move our DEFINITELY EXIST counter from the line between the two Y M boxes into the Not-X Y M box (i.e., the lower left inside box).
To get our conclusion, we turn this Triliteral Diagram into a Biliteral Diagram. A Biliteral Diagram looks like the Triliteral Diagram with the center boxes (M and M') taken out, leaving only the big quadrants. The rules for reducing a Triliteral Diagram to a Biliteral Diagram are simple:
(1) If both parts of a quadrant in the Triliteral Diagram have DOESN'T EXIST counters, put a DOESN'T EXIST counter in the same quadrant of the Biliteral Diagram.
(2) If a DEFINITELY EXISTS counter is definitely in one of the quadrants (not on the fence between two quadrants), put a DEFINITELY EXISTS counter in the same quadrant of the Biliteral Diagram.
(3) All other counters (DOESN'T EXIST counters that only cover part of a quadrant, DEFINITELY EXISTS counters that are on a fence between two quadrants) disappear.
Thus the whole Game of Logic is just ordinary syllogisms, where the Game rules have the same effect that rules of syllogisms do, although Carroll has his own particular interpretation of categorical propositions. In Carroll's interpretation, in the Triliteral Diagram:
"All S are P" gives you two DOESN'T EXIST counters and one DEFINITELY EXISTS counter;
"No S are P" gives you two DOESN'T EXIST counters;
"Some S are P" gives you one DEFINITELY EXISTS counter;
"Some S are not P" gives you one DEFINITELY EXISTS counter.
You can also get completely coherent games if you change these interpretations so that "All S are P" only gives you two DOESN'T EXIST counters (this makes the Game work a bit more like how syllogisms are treated in predicate calculus, by removing affirmative subalternation), or so that "No S are P" gives you two DOESN'T EXIST counters and one DEFINITELY EXISTS counter (this makes the Game work a bit more like traditional Aristotelian syllogisms, by adding negative subalternation).
You can learn more about the Game of Logic from Carroll's own book on it (the one that the mathematicians were having trouble with in the video above), The Game of Logic.
