The 'Game of Logic' is a method of solving logic problems called syllogisms, in which two premises are given from which a conclusion may or may not be drawn. By plotting the premises on a diagram it will be seen whether or not there is a conclusion to be drawn from the premises and what that conclusion is.

The premises take the form of propositions about classes of things and their attributes.

Basic Forms of Premises

A premise generally takes one of three forms of which these are representative examples:

Some new cakes are nice

No new cakes are nice

All new cakes are nice

Formally the construction of a premiss is:

<quantifier><subject><copula><predicate>

The quantifier may be one of "some", "no" or "all". The copula is the word "is" or "are". The subject and predicate describe things and their attributes.

More precisely, the three example premises, above, have the meaning

Some new cakes are nice cakes

No new cakes are nice cakes

All new cakes are nice cakes

Clearly these premises refer to "cakes" and their attributes "new" and "nice".

We represent these attributes by letters (such as x, y and m) and substitute the letters for the words thus:

Some x are y

No x are y

All x are y

Negative Attributes

Sometimes the negative form of an attribute is used in a premiss such as "No new cakes are not nice" in which "not nice" might be regarded as the negative of "nice". Where a letter such as x or y is used to represent an attribute, x' or y' is used to represent the negative of the attribute. Thus "No new cakes are not nice" would be represented by " No x are y' ".

This is an instance of where Dodgson has adopted a slightly different approach to the traditional textbooks on Aristotles logic. Traditionally the "not" is connected to the word "are" (the "copula") rather than the second attribute (the "predicate") which in the example above is "nice". Thus, where Carroll is content to allow a premiss of the form "No new cakes are not-nice", traditional logicians would write it: "No new cakes are-not nice". In essence, the meaning is the same, but traditional methods for solving syllogism do not allow for negative predicates and by doing so reduce the number of pairs of premises that yield a conclusion when compared with Dodgson's method.

Dodgson's Definition of Premises Beginning "All"

Dodgson had a particular interpretation of premises beginning "All", which makes his approach to symbolic logic far more complicated than the traditional methods described in contemporary logic text books.

Whereas the proposition "All new cakes are nice" infers that "No new cakes are not-nice", the traditional view is that it does not necessarily infer the existence of any new cakes at all. Dodgson, however, took the view that when we use the word "All" we are, in choosing that word, asserting the existence of some "new cakes". Hence, for Dodgson, "All new cakes are nice" is equivalent to two propositions: "No new cakes are not-nice" and "Some new cakes are nice". This distinction is important when it comes to plotting the premises on the diagram that is used for drawing conclusions.

Dodgson's approach is far more in keeping with everyday use of the language, but is contrary to definitions of these forms of proposition adopted by his contemporary logicians and when they are used in the study of logic today. For further analysis of the forms of proposition used as premises and how Dodgson’s interpretations differ from other forms, use the "Square of Opposition" demonstrator, which can be found by clicking on the "Demonstrators" menu-choice, above.

The 16 possible forms of the premises

From the three forms of quantifier, four combinations of positive and negative attributes and two possible orders for the attributes, there are 24 different forms for propostions with attributes x and y. Because "No x are y" is essentially the same as "No y are x" and a similar relationship exists for premises starting "some", there are only 16 forms of premises with unique meanings.

Solving Syllogisms

In its simplest form, a syllogism comprises two propositions (each from the set of 16 possible forms), each including two attributes, but with one attribute in common. The common attribute is referred to as the "middle term", which Dodgson later describes as the "eliminand". The other two attributes are called "retinends" and, if there is a valid conclusion to be drawn it will be a proposition which is constructed using the two retinends.

In the 'Game of Logic' the solution is found by plotting the premises on a diagram as shown in the 'Demonstrator'.