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Suppose, now, we combine any class with its negative, what result do we obtain? If b is taken to mean birds, what does 6(146) mean?
Every one immediately recognizes that it is impossible for any being at the same time to possess a quality ayd not to possess it. One or the other must be true; both cannot be true at the same time. So, whatever quality really defines the class “ birds,” there cannot be in the universe a single individual which at the same time really possesses it and does not possess it. There is nothing which is at the same time a bird and not a bird. To represent “nothing” we have the very convenient symbol 0, nought. So we are driven to the conclusion that b(146)=0, and so of any class.
We proceed to the consideration of logical addition and subtraction. To express the aggregate conception of a group of objects consisting of partial groups, we use the conjunctions 66 and," " or,” etc. -e.g., “ladies and gentlemen," " men or women." In popular language these terms, " and," " or," etc., are often ambiguous. As ordinarily used, we cannot always tell whether they are meant to connect terms mutually exclusive or not. If we say a thing is either x or z, this mere statement does not explicitly inform any one whether we mean 6 if the one, then not the other," or only “if not the oue, then the other;" whether we mean "x or z, but not both," or only “X or 2, or, it may be, both.” In our symbolic language we are able to avoid this vagueness perfectly. We supply the place of these words by the sign +, plus, which shall always mean that the classes which it connects are quite separate, entirely distinct, so that no member of one is found in another. Thus, - either a or 6” in the first or exclusive sense is represented by a (146)+6(1–a), and in the second or non-exclusive sense by a+b (1-a). Here, again, as in the case of logical combination or multiplication, we see that we obtain the same aggregate group in whatever order the terms are taken — e. g., if “a” represents chickens and "]" stands for ducks, then a+b=b+a, with the implication that no chickens are ducks.
The inverse operation to logical addition, or collecting parts into a whole, is logical subtraction, or separating a part from a whole. This, as we have seen, is expressed in common language by the word “ except,” which we represent by the minus sign (-). Here, again, it is indifferent whether we express the excepted case first or last, or in what order we write any series of terms, some of which are affected by the minus sign — e.g., “ all plane figures, except triangles,'' means the same as, "excepting triangles, all plane figures.” That is, x—=—z+x. Moreover, if we make a selection by combining the adjective equilateral with the class “plane figures, excepting triangles," we reach the same result as if we combined it first with the class “ plane figures,” then with the class “ triangles," and then took their difference. That is, y(x-2)=yx-yz; which shows that multiplication is distributive as well for a difference as for a sum. Applying the foregoing results to logical equations, we arrive immediately at the three general axioms (in which we use the word “ equal ” in its broadest sense, as signifying identity of individuals, coexistence of qualities, or equality of numbers) :
1. If equal things are added to equal things, the wholes are equal.
2. If equal things are taken from equal things, the remainders are equal.
3. If equal things are multiplied by equivalents, the results are equal."
Hence we may add, subtract, or multiply logical equations, and transpose terms exactly as in the common algebra of number.
We are now ready to return to the equation given when we combine any class with its contrary or negative. In every such case we get x (1-x)=0, because it is impossible for anything to possess a quality and not to possess it at the same time. But from X (1-x) =0, since multiplication in logic is always distributive, we get, inevitably, x-x?=0; and transposing, x=x?. As x was entirely unrestricted, this law must hold for every logical term ; and here, at last, we have something without a parallel in ordinary algebra. Instead of every number fulfilling this requirement, it is true for only two, namely, 1 and 0.
This peculiar law interferes essentially with division' in our notational algebra, which, as a logical operation, is identical with what is commonly called Abstraction.
But to return. If we always have x=x2 XX, what is its interpretation in this form ? Simply that if we combine a logical class with itself, or from a class select those members which it has in common with itself, the result is the class itself unchanged.
In passing from terms and their combination to the expression of propositions, we premise that if a proposition is negative we attach the negative particle to the predicate, and we denote “ Some” by the indefinite symbol v. It will be convenient to apply the epithets of logical quantity, “universal” and “ particular,” and of logical quality, - affirmative” and “negative,” to the terms of propositions, and not to the propositions themselves. There are, then, four classes of terms, namely: the universal-affirmative, “ all a's:” the particular-affirmative, “ some x's,” or “a's;” the universal-negative, “ all non-x's ;'' the particular-negative, “ some non-a's.”
The expression “no x's" is not properly a term of a proposition, for the meaning of the proposition “no a's are y's" is wall x's are non-y's.” The subject of that proposition is, therefore, universal affirmative ; the predicate, particular-negative.
By the various combinations of the four classes of terms, each with all, retaining the distinction always made in ordinary logics between subject and predicate, sixteen propositions will result. For it will be seen that we have four possible distinct subjects, in treating apart the term and its negative or complementary, x and 1–x, which latter, for the sake of exhibiting symmetry, we will represent by č.
Our four distinct subjects, then, are w, ł, vx, vī; and we have as many distinct predicates, namely, y, y, vy, vy. Com
1 The extraordinary difficulties connected with a rigid and general exposition of algorithmic division, its limitations here, and the true reasons for them, can only be appreciated by one who has worked on that subject. It will require a separate paper, to which this may be taken as introductory.
bining these as above indicated, the sixteen propositions which result are as follows:
1. x =vy. All x's are y's.
Some non-x's are non-y's.
Some x's are all the non-y's.
All non-a's are all the y's. 14. x= y. All x's are all the non-y's. 15. x= y. All x's are all the y's. 16. ž=ý. All non-x's are all the non-y's.
Ordinary Syllogism is inference from two propositions called the premises, having it common term called the middle term.
By the various combinations of the sixteen propositions, each with all, 256 pairs of premises will result. For brevity, these are not given, but, if needed, can be written out from the sixteen propositions already enumerated.
Now, as a first exercise for our Algorithmic Logic, as so far developed, let us apply it to the reduction or solution of these 256 Categorical Syllogisms.
In some sense, it was the perception of some parts of this problem of Syllogism, and of the need for solution, explanation, or reduction, which probably called logic into being. As Professor Bain says in Mind, January, 1878: “ The meaning of Syllogism, then, is the formal relation between the premises and the conclusion, whatever the matter be. If all syllogisms — all cases of argument or inference — were of the type Barbara, I doubt whether Syllogism would ever have been invented. Not that in Barbara there is not an element of form ; but that being so easy, we need not even be conscious of it. But the inventor of the Syllogism was awakened to the fact that in many kinds of reasoning, not unfrequent in their occurrence, the formal relation of premises to conclusion was puzzling and uncertain, not to say misleading.” Aristotle saw the need and value of a solution, and actually solved a considerable part accurately.
i The 256 are given in tabular form, though in a somewhat different notation, on page 89 of Volume VIII of this journal, in a short article “On Logic,” to which our attention was kindly called by the Editor. On page 90, also, are well stated, empirically, some conclusions which here we will apodeictically demonstrate.
The general form in which we have stated the problem, in which every possible case is taken account of, gives us 256 pairs of premise-propositions ; which would seem to make the complete discussion of Categorical Syllogism a matter of dreadful complication. In truth, without the application of mathematical ideas, it must have remained annoyingly intricate. But the result of an analytic solution, however tedious, may often be given synthetically in a very compendious form, and such is the nature of the Reduction of Syllogism which we now present.
Naturally we take it first in its elements — the propositions.
Now, by the simple consideration that as perfectly expressed in our notation every proposition is convertible — may be read backwards as truly as forwards — we see that six of the sixteen propositions (8-13) disappear, since in them no new relation between classes is given.
Against the last three propositions, against any universal substitutive judgment, it has been often and strongly urged that such are not logically simple propositions. The latest and best brief statement on this point is given in the January number of Mind, by the editor, in a short criticism of Professor Jevons. For those who still consider 14, 15, 16 as simple propositions, we can finish with them in a word.
In any pair of premises one of which is a universal substitutive, since this declares that one class or letter is exactly another, neither more nor less, read this other in place of the middle term and you have the conclusion.
We now have left only the seven simple logical propositions