Here are the steps to solving categorical syllogisms with triliteral diagrams.

If you are drawing conclusions:

(1) Define your universe of discourse. This is the background box.

(2) Assign letters to your terms and divide your background box according to the letters. Put the middle term in the middle to simplify things.

(3) Put the information from your premises onto the diagram, one at a time. Remember, E propositions will result in an X for each of two boxes on a triliteral diagram; I and O propositions will result in an O on a line between two boxes on the triliteral diagram; and A propositions will result in an O on a line between two boxes and an X for each of two boxes.

(4) Transfer any complete information about the quadrant from the triliteral diagram to the corresponding biliteral diagram. This complete information is of two kinds: Two X’s in a quadrant rules out the whole quadrant, and an O in a quadrant (not on a line) rules it in.

(5) Translate your conclusion into English.

If you are checking validity of an argument with a conclusion:

Follow the above steps, then check the conclusion you got with the conclusion provided for the argument. If they match, it is valid.

 

Those who want to practice can use the exercises in Carroll’s Symbolic Logic (online):

Drawing Conclusions: Pages 101-106 (answers on pages 127-129)

Checking Validity: Pages 107-110 (answers on pages 130-131)