This entry traces the historical development of the Square of Opposition, a collection of logical relationships traditionally embodied in a square diagram. This body of doctrine provided a foundation for work in logic for over two millenia. For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms. Certain additional principles ("contraposition" (...) and "obversion") were sometimes adopted along with the Square, and they genuinely yielded inconsistency. By the nineteenth century an inconsistent set of doctrines was widely adopted. Strawson's 1952 attempt to rehabilitate the Square does not apply to the traditional doctrine; it does salvage the nineteenth century version but at the cost of yielding inferences that lead from truth to falsity when strung together. (shrink)

I introduce a new method for investigating those original textswhich illustrate a theory which is aimed at solving a general problem. Themethod comprises two investigations. The first investigation is aimed atrecognising, within the original texts, all the occurrences of double negationsand ad absurdum arguments. These occurrences mean that the author argueswithin non-classical logic, his logical thread develops through a chain of doublenegated sentences which compose units of argument, each ending with an adabsurdum argument. More generally, the theory is organised in (...) an alternativeway to the classical deductive organisation. The second investigation takes intoaccount the recent formalization of a hierarchy of degrees of infinity, through ahierarchy of types of mathematics. The mathematics of almost the least degreeof actual infinity relies upon the use of one only quantifier. Hence, the secondinvestigation within the original text is aimed at recognising the occurrences ofquantifiers, as an evidence of the introduction of this degree of actual infinity.I apply this method to the first chapter of De Docta Ignorantia byCusanus, a scholar belonging to the school of “negative thinking”. I obtainevidence showing an exceptional capability of innovating his method ofarguing, through a clever use of both double negated sentences and adabsurdum arguments. But he was unsuccessful in organizing the entire theoryin this way, because he mixed logical arguments with theological suggestions.Moreover, from the direct reading of his text one may surmise that his appeal tothe intellectualis visus refers to geometrical intuition, which is the basis of amathematics involving the lower degree of actual infinity. Instead, Cusanuscorrectly introduced the basic notion, i.e. the infinitesimal, of an higher degreeof mathematics; which, two centuries after him, was intuitively introduced inmathematics and in the last century was eventually defined in a formal way.As a general appraisal, Cusanus’ effort to obtain knowledge of divinesubjects appears much more consistent in its use of non-classical logic than inits use of mathematical objects, although, in the past, it was the latter thatbrought him celebrity. However, although anticipating methods and notionswhich were to be formalised by subsequent science, his method, to transferre mathematical properties to divine properties, is largely an allusive method.In conclusion, the application of the above-mentioned method of analysisto a first part of Cusanus’ De Docta Ignorantia was successfully inasmuch as itwas capable to circumscribe both the logical content and the mathematicalcontent of this text. (shrink)