"No philosophers are conceited; Some conceited persons are not-gamblers. .'. Some not-gamblers are not philosophers" This can be proved by reduction to _Ferio_, thus:-- "No conceited persons are philosophers; Some not-gamblers are conceited. .'. Some not-gamblers are not philosophers". The validity of _Ferio_ follows directly from the Axiom '_De Omni et Nullo_'. (2) _Symbolic Representation._ Before proceeding to discuss other Methods of Solution, it is necessary to translate our Syllogism into an _abstract_ form. Let us take "persons" as our 'Universe of Discourse'; and let x = "philosophers", m = "conceited", and y = "gamblers." Then the Syllogism may be written thus:-- "No x are m; Some m are y'. .'. Some y' are x'." (3) _Solution by Euler's Method of Diagrams._ The Major Premiss requires only _one_ Diagram, viz. 1 _____ _____ _/ \_ _/ \_ / \ / \ | x | | m | \_ _/ \_ _/ \_____/ \_____/ pg181 The Minor requires _three_, viz. 2 _____ _____ _/ \_ _/ \_ / \ / \ | y | | m | \_ _/ \_ _/ \_____/ \_____/ 3 _____ _____ _/ \_/ \_ / / \ \ | y | | m | \_ \_/ _/ \_____/ \_____/ 4 _____ _/ ___ \_ / / y \ \ | \___/ | \_ m _/ \_____/ The combination of Major and Minor, in every possible way requires _nine_, viz. Figs. 1 and 2 give 5 _____ _____ _____ _/ \_ _/ \_ _/ \_ / \ / \ / \ | x | | y | | m | \_ _/ \_ _/ \_ _/ \_____/ \_____/ \_____/ 6 _____ _____ _____ _/ \_/ \_ _/ \_ / / \ \ / \ | x | | y | | m | \_ \_/ _/ \_ _/ \_____/ \_____/ \_____/ 7 _____ _____ _/ \_ _/ \_ / \ / \ | xy | | m | \_ _/ \_ _/ \_____/ \_____/ 8 _____ _____ _/ ___ \_ _/ \_ / / x \ \ / \ | \___/ | | m | \_ y _/ \_ _/ \_____/ \_____/ 9 _____ _____ _/ ___ \_ _/ \_ / / y \ \ / \ | \___/ | | m | \_ x _/ \_ _/ \_____/ \_____/ Figs. 1 and 3 give