and so on.
I propose to call any such series of combinations the Logical Alphabet. It holds in logical science a position the importance of which cannot be exaggerated, and as we proceed from logical to mathematical considerations, it will become apparent that there is a close connection between these combinations and the fundamental theorems of mathematical science. For the convenience of the reader who may wish to employ the Alphabet in logical questions, I have had printed on the next page a complete series of the combinations up to those of six terms. At the very commencement, in the first column, is placed a single letter X, which might seem to be superfluous. This letter serves to denote that it is always some higher class which is divided up. Thus the combination AB really means ABX, or that part of some larger class, say X, which has the qualities of A and B present. The letter X is omitted in the greater part of the table merely for the sake of brevity and clearness. In a later chapter on Combinations it will become apparent that the introduction of this unit class is requisite in order to complete the analogy with the Arithmetical Triangle there described.
The reader ought to bear in mind that though the Logical Alphabet seems to give mere lists of combinations, these combinations are intended in every case to constitute the development of a term of a proposition. Thus the four combinations AB, Ab, aB, ab really mean that any class X is described by the following proposition,
X = XAB ꖌ XAb ꖌ XaB ꖌ Xab.
If we select the A’s, we obtain the following proposition
AX = XAB ꖌ XAb.
Thus whatever group of combinations we treat must be conceived as part of a higher class, summum genus or universe symbolised in the term X; but, bearing this in mind, it is needless to complicate our formulæ by always introducing the letter. All inference consists in passing from propositions to propositions, and combinations per se have no meaning. They are consequently to be regarded in all cases as forming parts of propositions.
The Logical Alphabet.
| I. | II. | III. | IV. | V. | VI. | VII. |
| X | AX | AB | ABC | ABCD | ABCDE | ABCDEF |
| aX | Ab | ABc | ABCd | ABCDe | ABCDEf | |
| aB | AbC | ABcD | ABCdE | ABCDeF | ||
| ab | Abc | ABcd | ABCde | ABCDef | ||
| aBC | AbCD | ABcDE | ABCdEF | |||
| aBc | AbCd | ABcDe | ABCdEf | |||
| abC | AbcD | ABcdE | ABCdeF | |||
| abc | Abcd | ABcde | ABCdef | |||
| aBCD | AbCDE | ABcDEF | ||||
| aBCd | AbCDe | ABcDEf | ||||
| aBcD | AbCdE | ABcDeF | ||||
| aBcd | AbCde | ABcDef | ||||
| abCD | AbcDE | ABcdEF | ||||
| abCd | AbcDe | ABcdEf | ||||
| abcD | AbcdE | ABcdeF | ||||
| abcd | Abcde | ABcdef | ||||
| aBCDE | AbCDEF | |||||
| aBCDe | AbCDEf | |||||
| aBCdE | AbCDeF | |||||
| aBCde | AbCDef | |||||
| aBcDE | AbCdEF | |||||
| aBcDe | AbCdEf | |||||
| aBcdE | AbCdeF | |||||
| aBcde | AbCdef | |||||
| abCDE | AbcDEF | |||||
| abCDe | AbcDEf | |||||
| abCdE | AbcDeF | |||||
| abCde | AbcDef | |||||
| abcDE | AbcdEF | |||||
| abcDe | AbcdEf | |||||
| abcdE | AbcdeF | |||||
| abcde | Abcdef | |||||
| aBCDEF | ||||||
| aBCDEf | ||||||
| aBCDeF | ||||||
| aBCDef | ||||||
| aBCdEF | ||||||
| aBCdEf | ||||||
| aBCdeF | ||||||
| aBCdef | ||||||
| aBcDEF | ||||||
| aBcDEf | ||||||
| aBcDeF | ||||||
| aBcDef | ||||||
| aBcdEF | ||||||
| aBcdEf | ||||||
| aBcdeF | ||||||
| aBcdef | ||||||
| abCDEF | ||||||
| abCDEf | ||||||
| abCDeF | ||||||
| abCDef | ||||||
| abCdEF | ||||||
| abCdEf | ||||||
| abCdeF | ||||||
| abCdef | ||||||
| abcDEF | ||||||
| abcDEf | ||||||
| abcDeF | ||||||
| abcDef | ||||||
| abcdEF | ||||||
| abcdEf | ||||||
| abcdeF | ||||||
| abcdef |
In a theoretical point of view we may conceive that the Logical Alphabet is infinitely extended. Every new quality or circumstance which can belong to an object, subdivides each combination or class, so that the number of such combinations, when unrestricted by logical conditions, is represented by an infinitely high power of two. The extremely rapid increase in the number of subdivisions obliges us to confine our attention to a few qualities at a time.