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 It equally shares Hamilton's interest in reforming and enlarging the traditional syllogistic, an enterprise now outdated, which has caused it to fall into unmerited neglect.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2nd ed., Chicago, 1926), the bulk of De Morgan's logical writings are to be found in five memoirs (plus a sixth, still unpublished) contributed to the Cambridge Philosophical Transactions between 1846 and 1862.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 nonAristotelian 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 lowercase 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 wellknown 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)
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Modern Logic: The Boolean Period: De Morgan." Encyclopedia of Philosophy. . Encyclopedia.com. 17 Feb. 2019 <https://www.encyclopedia.com>.
"Modern Logic: The Boolean Period: De Morgan." Encyclopedia of Philosophy. . Encyclopedia.com. (February 17, 2019). https://www.encyclopedia.com/humanities/encyclopediasalmanacstranscriptsandmaps/modernlogicbooleanperioddemorgan
"Modern Logic: The Boolean Period: De Morgan." Encyclopedia of Philosophy. . Retrieved February 17, 2019 from Encyclopedia.com: https://www.encyclopedia.com/humanities/encyclopediasalmanacstranscriptsandmaps/modernlogicbooleanperioddemorgan
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.