For Sommers notation we go back to the parts of the categorical proposition.
(1) universe of discourse
(2) judgment (which can be affirmative or negative)
(3) quantity of subject (which can be universal or particular)
(4) subject term (which can be positive, like ‘punished’, or negative, like ‘unpunished’)
(5) quality of predication (which can be affirmative or negative)
(6) predicate term (which can be positive or negative)
We’ll leave the universe of discourse in the background as assumed. We’ll assign letters to our subject term and predicate term, just as we do with literal diagrams, e.g., S and P. These can be positive or negative, so we’ll indicate this with a plus or minus, for instance:
(+S) (+P)
We want to add in our quality; we’ll give affirmative quality a plus and a negative quality a minus:
(+S)+(+P)
We need to add quantity. We’ll assign a plus sign to the particular and a minus sign to the universal:
+(+S)+(+P)
And we’ll add judgment, by putting it outside:
+(+(+S)+(+P))
This is the expanded form. To make it more handy, we can also take the reduced form, which works just as with algebra. So the reduced form of the above example would be:
+S+P
We can do the same with any categorical proposition. For instance, "No P is R" would be, in expanded form:
+(-(+P)-(+R))
And in reduced form:
-P-R
All A propositions will have the following (reduced) format:
-S+P
All E propositions will have the following format:
-S-P
All I propositions will have the following format:
+S+P
All O propositions will have the following format:
+S-P
We can then use our notation to solve logic problems. To do this we need to understand the basic principle of validity for Sommers notation:
An argument is valid whenever the premises add up to the conclusion and the whole argument is regular.
An argument may be regular in one of two ways: either it has all universal propositions (in both premises and conclusions) or it has one (and only one) particular premise and one particular conclusion.
