We may pass on to consider in somewhat more detail a special equational or semi-equational system—open also to special criticisms—by which Hamilton and others sought to revolutionise ordinary logical doctrine.
195 140. The eight propositional forms resulting from the explicit Quantification of the Predicate.—We have seen that in the ordinary fourfold schedule of propositions, the quantity of the predicate is determined by the quality of the proposition, negatives distributing their predicates, while affirmatives do not. It seems a plausible view, however, that by explicit quantification the quantity of the predicate may be made independent of the quality of the proposition, and Sir William Hamilton was thus led to recognise eight distinct propositional forms instead of the customary four:—
| All S is all P, | U. |
| All S is some P, | A. |
| Some S is all P, | Y. |
| Some S is some P, | I. |
| No S is any P, | E. |
| No S is some P, | η. |
| Some S is not any P, | O. |
| Some S is not some P, | ω. |
The symbols attached to the different propositions in the above schedule are those employed by Archbishop Thomson,[191] and they are those now commonly adopted so far as the quantification of the predicate is recognised in modern text-books.
[191] Thomson himself, however, ultimately rejects the forms η and ω.
The symbols used by Hamilton were Afa, Afi, Ifa, Ifi, Ana, Ani, Ina, Ini. Here f indicates an affirmative proposition, n a negative; a means that the corresponding term is distributed, i that it is undistributed.
For the new forms we might also use the symbols SuP, SyP, SηP, SωP, on the principle explained in section [62].
141. Sir William Hamilton’s fundamental Postulate of Logic.—The fundamental postulate of logic, according to Sir William Hamilton, is “that we be allowed to state explicitly in language all that is implicitly contained in thought”; and we may briefly consider the meaning to be attached to this postulate before going on to discuss the use that is made of it in connexion with the doctrine of the quantification of the predicate.
196 Giving the natural interpretation to the phrase “implicitly contained in thought,” the postulate might at first sight appear to be a broad statement of the general principle underlying the logician’s treatment of formal inferences. In all such inferences the conclusion is implicitly contained in the premisses; and since logic has to determine what inferences follow legitimately from given premisses, it may in this sense be said to be part of the function of logic to make explicit in language what is implicitly contained in thought.
It seems clear, however, from the use made of the postulate by Hamilton and his school that he is not thinking of this, and indeed that he is not intending any reference to discursive thought at all. His meaning rather is that we should make explicit in language not what is implicit in thought, but what is explicit in thought, or, as it may be otherwise expressed, that we should make explicit in language all that is really present in thought in the act of judgment.