Let a proposition be called universal, when the whole universe must be examined to verify it : particular, when the examination of a portion of the universe may verify it. Let a term be called total, where every instance of it must be examined before the proposition can be verified : partial, when such complete examination may not be needed.
There are two forms of predication in use : the cumular, as in "all as are some rs " and some as are not vs (are distinct from all vs); the exemplar, as in every (each, or any) a is some one Y, and some one a (perhaps not any one) is not any one Y. The cumular proposition speaks by collection ; the exemplar, by selection. Grammatical correct ness (representing a usual tendency of thought, though not an absolute law) allows either mode of expression in affirmatives, but demands exemplar expression in negatives; as in all as are not any rs. The cumular and exemplar modes of expression have differences which will presently be further seen.
We first take the cumular proposition and syllogism. Let a term totally used, as a, be denoted by x) or (x ; partially, by a ( or )x. Let affirmation be denoted by two dots, or none ; negation by one dot. Let propositions be distinguished by the quantities of their terms and the quality (affirmative or negative) : this is found to make complete distinction. Thus (.) means a negative proposition in which both terms are particular : and a (.) Y is that proposition enunciated in terms of a and Y. It turns out to be " Everything is either a or Y." There are eight forms of cumular predication : four universal, with four contrary (usually called contradictory) particulars. These are now given, with their symbols and usual readings : and the correctness of the distribution of the words universal and particular, total and partial, may be gathered by the reader from the definitions. Each universal (U) is followed by its contrary particular (r).
X i• of a)•( Y and x (.) Y. Everything either a or v and nothing both : a and r are contraries.
x ( o ) compounded of x (.) I and a ( ) Y. Everything is either a or 1r, and some things are both.
Each kind of proposition is either simple or complex with respect to the other, according as we think of alternation or conjunction. Thus
) o) is the junction of ) ) and ).); but ) ) is the alternative ) o) or Conversion and eontraversion are under the same rules as before.
The preceding forms are completely arithmetical, and are truly par ticular cases of the numerical proposition, of which it will now be convenient to say something, as well as of the corresponding syllogism.
Let m a It signify that m (or more) xs are rs ; then m x y signifies that on or more as are net vs. Let u be the number of instances in the universe; let x, y, z, be the numbers of as, vs, and zs : then u—x, u—y, u—z, are the numbers of as, ys, and zs. A spurious pro position may exist : that is, one which by the constitution of the universe must be true. Thus if x+ y be greater than u, (x + y--u) a Y must be true. Every proposition has two forms : thua When one of the forms is spurious, the other contains a negative quantity, and is * inconceivable.
The contradiction or contrary of WI x 1r is (x + I —m) a y. Hence deduce contradictions of the other forms. From m x Y and n Y Z we deduce (In+ n—y) a z. That is, there is inference when the quantities of the middle term in the two premises together exceed the whole quantity of that term. The following forms of syllogism may easily be investigated.
The proposition x ) ) Y is x a I : the proposition x) °) Y is X X Y joined with (y—x) X Y. And so of the rest.
In the more subjective view of logic, that is, in the mathematical side of it, the one commonly cultivated to the exclusion of the other, the term becomes the word expressive of a class, arid the relation of class to class is that of inclusion or exclusion, total or partial. Each term, x, divides the universe into two classes, x and a. The names may be applied to the relations of class :— Con traversion—changing a name into its contrary—alters the quantity of the term, and the quality of the proposition : thus x ))Y is X ).(y and a ( (y and x (-) Y. And a term and its contrary always enter one pro position with different quantities. Conversion is simply writing the propositional symbol inverted : thus x) ) Y is conversely Y ( ( x. We shall now use the forms ) ) (•( &e. to signify the propositions.