# Modern Logic: The Boolean Period: De Morgan

# MODERN LOGIC: THE BOOLEAN PERIOD: DE MORGAN

The above criticisms of Hamilton's system are primarily due to Augustus De Morgan (1806–1871), whom Hamilton, in 1846, had misguidedly accused of plagiarizing his quantification. In the famous and protracted controversy that ensued, De Morgan was led into a thorough dissection of the whole system, and subsequent critics, from Mill, Peirce, and Venn onward, have taken most of their ammunition from him.

Though greatly superior as to insight and technical ability, the logic of De Morgan has affinities with that of his rival in that it, too, lays stress on the autonomy of logic and on the extensional point of view. It equally shares Hamilton's interest in reforming and enlarging the traditional syllogistic, an enterprise now outdated, which has caused it to fall into unmerited neglect. Apart from his early *Formal Logic* (London, 1847; 2nd ed., Chicago, 1926), the bulk of De Morgan's logical writings are to be found in five memoirs (plus a sixth, still unpublished) contributed to the *Cambridge Philosophical Transactions* between 1846 and 1862. The *Syllabus of a Proposed System of Logic* (London, 1860) gives a cursory account of his scheme, as does his article "Logic" in the *English Cyclopaedia* (*Arts and Science Division*, V, London, 1860, pp. 340–354).

The basis of common logic, for De Morgan, consists in relations of partial or total inclusion, or exclusion, among classes. Where information about a majority of class members is available or where, as in the "numerically definite" syllogism, precise numbers are given, it is possible, as he shows, to draw valid conclusions of a non-Aristotelian type. But these conditions are seldom realized. A more radical departure is the admission into ordinary propositions of negative terms and class names (symbolized by lower-case letters), such that a term *X* and its "contrary" *x* between them exhaust the "universe of discourse" (a useful device that has since been generally adopted). Assuming these classes to have at least notional members, it follows that two classes and their contraries can be related in eight possible ways:

- All
*X*'s are*Y*'s. - All
*x*'s are*y*'s. - All
*X*'s are*y*'s. - All
*x*'s are*Y*'s. - Some
*X*'s are*Y*'s. - Some
*x*'s are*y*'s. - Some
*X*'s are*y*'s. - Some
*x*'s are*Y*'s.

These can be rewritten without negative symbols as:

- All
*X*'s are*Y*'s. - All
*Y*'s are*X*'s. - No
*X*'s are*Y*'s. - Everything is either
*X*or*Y*. - Some
*X*'s are*Y*'s. - Some things are neither
*X*'s nor*Y*'s. - Some
*X*'s are not*Y*'s. - Some
*Y*'s are not*X*'s.

Of these the contradictory pairs are (1) and (7), (2) and (8), (3) and (5), and (4) and (6). Since the distribution of terms is given or implied throughout, these forms are simply convertible by reading them in reverse. "Contraversion" (or obversion) is obtained by altering the distribution of a term, replacing it by its contrary, and denying the result. "All *X* 's are *Y* 's" becomes successively "No *X* 's are *y* 's," "All *y* 's are *x* 's," and "Everything is either *x* or *Y*." The procedure is the same for the other seven forms, making 32 possibilities in all.

De Morgan's rule of syllogism is either that both premises should be universal or, when only one is, that the middle term should have different quantities in each. Inference takes place by erasing the middle term and its quantities. Since, including the syllogisms of weakened conclusion, there are 4 basic patterns, and since 3 terms and their contraries can be paired off, in premises and conclusion, in 8 different ways, there are 32 valid syllogisms, of which half have two universal premises and 8 a universal conclusion.

To remedy the "terminal ambiguity" whereby the undistributed term in the universal "All *X* 's are *Y* 's" may refer indifferently to some or all of the *Y* 's, De Morgan investigated the complex propositions produced by combining pairs of elementary forms. It is in this connection that he gives the well-known rules for negation of conjunctions which have since received his name—though he did not, in fact, invent them.

In endeavoring to patch up Hamilton's quantified system De Morgan made further distinctions between "cumular" (collective) and "exemplar" (distributive) forms of predication; struggled, unavailingly, to bring the intensional interpretation of terms (as attributes) into line with the extensional and to subsume both under a pure logic of terms (the "onymatic" system); and explored in passing such nontraditional forms of inference as the syllogisms of "undecided assertion" and "transposed quantity." More important is his recognition that the copula performs its function in inference, not as a sign of identity, but only through its role as a transitive and convertible relation.

De Morgan's generalization of the copula leads on, in his fourth Cambridge memoir, to a pioneer investigation of relations in general, which is the foundation of all subsequent work in the field. He there distinguishes a relation (say, "lover of") from its denial, its contrary, and its converse ("loved by"); proceeds to compound relations, or relative products ("*L* of *M* of"), and to quantified versions of these ("*L* of every *M*," "of none but *M* 's," etc.); and discusses a variety of equivalences that hold between these different sorts of relations and the rules for their discovery and manipulation. The purpose of this, typically enough, was to exhibit the syllogism in its most general form, as a series of combinations of relations. Despite the ingenuity and resource with which he treated it, this devotion to the syllogism was something of a weakness in De Morgan's work. It tethered him too closely to tradition, so that it was not until others exploited them that his own most fruitful discoveries were seen for what they were.

** See also ** De Morgan, Augustus; Hamilton, William; Mill, John Stuart; Peirce, Charles Sanders; Venn, John.

*P. L. Heath (1967)*

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# Modern Logic: The Boolean Period: De Morgan

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