Bibliography: Logic (exact)

Aggregation. The operation of uniting two or more terms or propositions, called aggregants, to produce an aggregate term or proposition which is true of everything of which any aggregant is true, and false of everything of which all the aggregants are false. It is opposed to composition, which is the operation of producing from two or more terms or propositions, called the components, a new term or proposition, called their compound, which is true of all of which all the components are true, and false of all of which any are false.

… the essential office of the copula is to express a relation of a general term or terms to the universe. The universe must be well known and mutually known to be known and agreed to exist, in some sense, between speaker and hearer, between the mind as appealing to its own further consideration and the mind as so appealed to, or there can be no communication, or “common ground,” at all. The universe is, thus, not a mere concept, but is the most real of experiences.

Aggregation. The operation of uniting two or more terms or propositions, called aggregants, to produce an aggregate term or proposition which is true of everything of which any aggregant is true, and false of everything of which all the aggregants are false. It is opposed to composition, which is the operation of producing from two or more terms or propositions, called the components, a new term or proposition, called their compound, which is true of all of which all the components are true, and false of all of which any are false.

Copula is often defined as that which expresses the relation between the subject-term and the predicate-term of a proposition. But this is not sufficiently accurate for the purposes of exact logic. Passing over the objection that it applies only to categorical propositions, as if conditional and copulative propositions had no copula, contrary to logical tradition, it may be admitted that a copula often does fulfill the function mentioned; but it is only an accidental one, and its essential function is quite different. Thus, the proposition, “Some favoured patriarch is translated” is essentially the same as “A translated favoured patriarch is”; and “Every mother is a lover of that of which she is a mother” is the same as “A mother of something not loved by her is not.” In the second and fourth forms, the copula connects no terms; but if it is dropped, we have a mere term instead of a proposition. Thus the essential office of the copula is to express a relation of a general term or terms to the universe. The universe must be well known and mutually known to be known and agreed to exist, in some sense, between speaker and hearer, between the mind as appealing to its own further consideration and the mind as so appealed to, or there can be no communication, or “common ground,” at all. The universe is, thus, not a mere concept, but is the most real of experiences.

Dialogism. A form of reasoning in which from a single premiss a disjunctive, or alternative, proposition is concluded introducing an additional term; opposed to a syllogism, in which from a copulative proposition a proposition is inferred from which a term is eliminated.

Dimension. An element or respect of extension of a logical universe of such a nature that the same term which is individual in one such element of extension is not so in another. Thus, we may consider different persons as individual in one respect, while they may be divisible in respect to time, and in respect to different admissible hypothetical states of things, etc. This is to be widely distinguished from different universes, as, for example, of things and of characters, where any given individual belonging to one cannot belong to another. The conception of a multidimensional logical universe is one of the fecund conceptions which exact logic owes to O. H. Mitchell. Schröder, in his then second volume, where he is far below himself in many respects, pronounces this conception “untenable.” But a doctrine which has, as a matter of fact, been held by Mitchell, Peirce, and others, on apparently cogent grounds, without meeting any attempt at refutation in about twenty years, may be regarded as being, for the present, at any rate, tenable enough to be held.

Dyadic relation. A fact relating to two individuals. Thus, the fact that A is similar to B, and the fact that A is a lover of B, and the fact that A and B are both men, are dyadic relations; while the fact that A gives B to C is a triadic relation. Every relation of one order of relativity may be regarded as a relative of another order of relativity if desired. Thus, man may be regarded as man coexistent with, and so as a relative expressing a dyadic relation, although for most purposes it will be regarded as a monad or non-relative term.

Logic (exact): Ger. exakte Logik; Fr. logique exacte; Ital. logica esatta. The doctrine that the theory of validity and strength of reasoning ought to be made one of the ‘exact sciences,’ that is, that generalisations from ordinary experience ought, at an early point in its exposition, to be stated in a form from which by mathematical, or expository, Reasoning […], the rest of the theory can be strictly deduced; together with the attempt to carry this doctrine into practice.
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There are those, not merely outside the ranks of exact logic, but even within it, who seem to suppose that the aim is to produce a calculus, or semi-mechanical method, for performing all reasoning, or all deductive inquiry; but there is no reason to suppose that such a project, which is much more consonant with the ideas of the opponents of exact logic than with those of its serious students, can ever be realised. The real aim is to find an indisputable theory of reasoning by the aid of mathematics. The first step in the order of logic towards this end (though not necessarily the first in the order of inquiry) is to formulate with mathematical precision, definiteness, and simplicity, the general facts of experience which logic has to take into account.

Dialogism. A form of reasoning in which from a single premiss a disjunctive, or alternative, proposition is concluded introducing an additional term; opposed to a syllogism, in which from a copulative proposition a proposition is inferred from which a term is eliminated.