To solve problems using the 'Game of Logic' the premises given in the problem are plotted onto a diagram (the triliteral diagram) from which data may be transferred to a second diagram (the biliteral diagram). If a conclusion can be reached from the two premises, then it will be evident in the biliteral diagram from which it may be read as a statement in the same form as the original premises.

The Triliteral Diagram

This diagram is used to represent the attributes: x, y, m.

The top half represents x; the bottom represents x';

The left half represents y; the right half represents y';

The square in the middle represents m;

The area outside the inner square represents m'.

Combinations of attributes such as: x and m, x and m', y and m, y' and m, etc are represented on the diagram by those cells which are within both of the areas representing the two individual attributes. Click on the grey button on the right to explore how this works.

Plotting the Premises on the Diagram

The premises are plotted on the diagram by placing grey and pink counters in various cells according to the premises. Grey counters signify that a cell is empty and pink counters signify that there ares one or more things in a particular cell.

For a premiss of the type: "No x are m" ... locate the cells which correspond to x and m and place a grey counter in both of them, in order to show that those cells are empty.

For "Some x are m" we need to place a pink counter in the relevant cells. Although there are two cells representing X and m, we cannot say which of the two cells contains anything - just that at least one of them contains at least one thing. To signify this, we place the pink counter on the border between the two cells.

For "All x are m" we must keep in mind that Dodgson regarded such premises as being equivalent to two propositions: No x are not-m" and "Some X are m" hence we put two grey counters on the diagram (in the cells representing x and m') and one pink counter on the border between the cells representing x and m.

 

Drawing the Conclusion

Once the two premises have been plotted on the diagram, we are ready to draw the conclusion. In doing this we check to see if there are any pink counters which are sitting on the border lines and examine the cells each side of the border to see if either contains a grey cell. If so, then the pink counter can be moved from the border into the empty cell next to it.

The conclusion is drawn by transferring data from the triliteral diagram to a “biliteral diagram”. The biliteral diagram is equiavalent to the triliteral diagram without the central square, thus it is used to show relationships between the X and Y attributes.

Each of the four squares on the biliteral diagram corresponds with a square in the same location on the triliteral diagram.

We place a grey counter in a cell on the biliteral diagram if both cells contain grey counters in the corresponding square on the triliteral diagram. We place a pink counter in a cell on the biliteral diagram if either cell contains a pink counter in the corresponding square on the triliteral diagram.

If no counters can be placed according to these rules, then no conclusion can be drawn, otherwise the conclusion is simply 'read' off the biliteral diagram.

In the example on the right, only the pink counter can be transferred to the biliteral diagram (for grey counters to be transferred, both cells in the equivalent quarter of the triliteral diagram require a grey counter).

Examples