Handbook of the History of Logic: - Fordham University Faculty

Relational Logic of Juan Caramuel 647
presentation of the logic of binary relational statements on the levels of proposition
(Part I) and argument (Part II). We shall take up the matter below in rather more
detail.
Herculis Logici Labores Tres (“Three Labors of Hercules, the Logician”) appears
to be an interesting collection of three studies devoted to three logical problems. 8
The first Labor called Nil-negans (“Negating nothing”) reduces the system of valid
modes of the syllogism based on the reduction of the negative copula to the negation
of the predicate, i.e. term negation. Caramuel appears to have defended this
so-called infinite logic (negated terms are called “infinite”) as part of his bachelor
exams in philosophy at Alcalá, early on in his career. 9 The second Labor entitled
Non-omnis (“Not every”) deals with the issue that in natural language — Latin
— the compound expression non omnis (not every) is equivalent to “some... not...
and some...,” for instance, “not all students smoke” is equivalent to “some students
do not smoke and some do”, rather than to “some...not...” only, as is the purely
logical meaning. It seems that the meaning of the natural language quantifier
includes something over and above the logical one. Caramuel explores this idea
and shows the logical properties of statements including Latin non omnis: equivalences
and oppositions of these statements as well as valid modes of syllogism.
Finally, the third Labor under the name Contingens (“Contingent”) is a systematic
treatment of modal logic. The basic idea is that contingens (contingent) is
not equivalent solely to “possible”, but “possible and possible not”. 10
The second part of Rational Theology, Metalogica, or — as the full title reads
— Metalogica: Disputationes de Logicae Essentia, Proprietatibus et Operationibus
Continens (“Metalogic Containing Disputations on the Essence of Logic, its Properties
and Operations”), rather than being a work in formal logic, is a set of
treatises, formally books, in the fields of the philosophy of logic (Book I), philosophical
logic (Books VII, IX), metaphysics (Books II, III, IV, V, VI, VIII, X)
and informal logic (Book X). The topics covered range from the nature of logic,
the theory of proposition and semantic paradoxes to issues of a metaphysical kind
(beings of reason, universals, individuation, self-evident principles etc.). The last
book, tenth in number, called De Severa Argumentandi Methodo (“On a Rigorous
Method of Argumentation”), is the most extensive of all. 11 The content ranges
from informal logic to logic applied to specific problems in philosophy and theology.
Why is Metalogic needed, according to Caramuel? He draws a parallel between
logic and music, both directing sound, but for a different purpose. As
the knowledge of the production of musical sound, music, needs musical theory
(“metamusic”), 12 so does the knowledge of the production of language sound —
logic — need metalogic. One might object that the knowledge of the production
of language sound is grammar, rather than logic. According to Caramuel, the lat-
8Sometimes listed as published independently in 1651, which is doubtful, however.
9Belazzi [1982], p. 17.
10This was by no means obvious at the time; cf. [Roncaglia, 2003]. For a detailed treatment
in Caramuel cf. [Dvoˇrák, 2005].
11It appeared as an independent work in 1644.
12Caramuel refers to A. Kircher’s Musurgia universalis and calls it “metamusic”.