CHAPTER XXIV.

THE MEASURE OF LOGICAL FORCE.

I. THE combinational analysis of the meaning of assertions enables us to establish an almost mathematical system of measurement of the comparative force of assertions. Given the number of independent terms involved, that form of proposition has the least possible force which negatives only a single combination. Thus with three terms, a proposition of the form AB = ABC negatives only the single combination ABc; but A = ABC negatives three, and A BC as many as four combinations. These latter propositions may be said to have three and four times the logical force of the first given.

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Thus

2. I have not yet been able to discover any general laws regarding this subject of logical force, but many curious and perhaps important observations may be made. a great many forms of assertion agree in having the logical force one-half, that is to say, they negative half the com binations. Such is the case, the terms being three in number, with the propositions A = BC; A B | BC; A = Bc bC. Indeed, it is very frequently true that any proposition having no term common to both sides of the equation negatives half the combinations. This is true of all propositions of the types A = B, A = BC, and generally A BCD... Y. But it is not true of the type AB = CD. The appearance of the same term in both members of an

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equation always weakens its force; thus A = ABC has the force only of, whereas ABC has the force. Again, BC has the force, but A B = B | C only the

A =

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force.

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3. The best ostensive instance of logical power is found in a form of proposition which embraces the greatest intension in one member with the greatest extension in the other. This kind of assertion has the general form ABC... P · Q · R ·| .; and as the terms increase the logical force approaches indefinitely to unity. Thus while A BC has the value, AB = CD has that of 10 out of 16, and A - B = CDE that of 22 out of 32. A few other observations on this subject are thrown into the form of the following questions :

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4. Show that the logical force of equations of the

form A

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5. Prove that a single proposition of the type ABC..

P ·· Q ·|· R ·· . . . ., there being in all n independent

letter terms, and no term common to both sides, has the

6. Can you discover any equation between a single term and any expression not involving that term which has a logical force other than one-half ?

7. What form of proposition involving only A and B in one member, and C, D, in the other, has the lowest possible logical force ?

8. What is the utmost number of combinations of n terms which can be negatived without producing contradiction?

9. What is the utmost number of combinations of four

terms which can be negatived by a proposition involving only three of them?

10. What two propositions involving five terms negative the utmost possible number of combinations, without selfcontradiction?

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II. Show that m successive propositions of the type A AB, B BC... ., that is to say, in the form of the Sorites, leave m + 2 combinations unnegatived, so that the

12. Prove that the amount of surplus assertion, or overlapping of the propositions, in a Sorites as treated in the last question, increases indefinitely. Investigate the law of the surplusage.

CHAPTER XXV.

INDUCTIVE OR INVERSE LOGICAL PROBLEMS.

I. THE direct or deductive process of logical analysis consists in determining the combinations which are, under the Laws of Thought, consistent with assumed conditions. The Inverse Problem is—given certain combinations inconsistent with conditions, to determine those conditions. explained in the Principles of Science (chapter vii.) the inverse problem is always tentative, and consists in inventing laws, and trying whether their results agree with those before us. An American correspondent, Mr. M. H. Doolittle, points out that in making trials we should always pay attention to combinations in proportion to their infrequency, or solitariness, infrequency being the mark of deep correlation. The infrequency may be that either of presence or of absence.

2. The following inductive problems consist of series of combinations of three terms and their negatives which are supposed to remain uncontradicted under the condition of a certain proposition or group of propositions. The student

is requested to discover such propositions, express them equationally, and then assign them to the proper type in the table on p. 222. If in any problem the conditions are self-contradictory the student is to detect the fact.

3. Assuming each of the following series of combinations to consist of those excluded or contradicted by certain propositions, assign the propositions which are just sufficient to exclude them in each problem, express these propositions equationally, and refer them as in the last question to the proper type in the table :