**The 17th century**

The Logica Hamburgensis (1638) of Joachim Jung (also called Jungius or Junge) was one replacement for the “Protestant” logic of Melanchthon. Its chief virtue was the care with which late medieval theories and techniques were gathered and presented. Jung devoted considerable attention to valid arguments that do not fit into simpler, standard conceptions of the syllogism and immediate inference. Of special interest is his treatment of quantified relational arguments, then called “oblique” syllogisms because of the oblique (non-nominative) case that is used to express them in Latin. An example is: “The square of an even number is even; 6 is even; therefore, the square of 6 is even.” The technique of dealing with such inferences involved rewriting a premise so that the term in the oblique case (for example, “of an even number”) would occur in the subject position and thus be amenable to standard syllogistic manipulation. Such arguments had in fact been noticed by Aristotle and were also treated in late medieval logic.

An especially widely used text of the 17th century is usually termed simply the Port-Royal Logic after the seat of the anticlerical Jansenist movement outside Paris. It was written by Antoine Arnauld and Pierre Nicole, possibly with others, and was published in French in 1662 with the title La Logique ou l’art de penser “Logic or the Art of Thinking”. It was promptly translated into Latin and English and underwent many reprintings in the late 17th and 18th centuries. In its outline, it followed Ramus’ outline of concept, judgment, argument, and method; it also briefly mentioned oblique syllogisms. The Port-Royal Logic followed the general Reform program of simplifying syllogistic theory, reducing the number of syllogistic figures from four, and minimizing distinctions thought to be useless. In addition, the work contained an important contribution to semantics in the form of the distinction between comprehension and extension. Although medieval semantic theory had used similar notions, the Port-Royal notions found their way into numerous 18th- and 19th-century discussions of the meanings and reference of terms; they appeared, for example, in John Stuart Mill’s influential text A System of Logic (1843). The “comprehension” of a term consisted of all the essential attributes in it (those that cannot be removed without “destroying” the concept), and the extension consisted of all those objects to which the concept applies. Thus the comprehension of the term “triangle” might include the attributes of being a polygon, three-sided, three-angled, and so on. Its extension would include all kinds of triangles. The Port-Royal Logic also contained an influential discussion of definitions that was inspired by the work of the French mathematician and philosopher Blaise Pascal. According to this discussion, some terms could not be defined (“primitive” terms), and definitions were divided between nominal and real ones. Real definitions were descriptive and stated the essential properties in a concept, while nominal definitions were creative and stipulated the conventions by which a linguistic term was to be used.

Discussions of “nominal” and “real” definitions go back at least to the nominalist/realist debates of the 14th century; Pascal’s application of the distinction is interesting for the emphasis that it laid on mathematical definitions being nominal and on the usefulness of nominal definitions. Although the Port-Royal logic itself contained no symbolism, the philosophical foundation for using symbols by nominal definitions was nevertheless laid.

One intriguing 17th-century treatment of logic in terms of demonstrations, postulates, and definitions in a Euclidean fashion occurs in the otherwise quite traditional Logica Demonstrativa (1697; “Demonstrative Logic”) of the Italian Jesuit Gerolamo Saccheri. Saccheri is better known for his suggestion of the possibility of a non-Euclidean geometry in Euclides ab Omni Naevo Vindicatus (1733; “Euclid Cleared of Every Flaw”). Another incisive traditional logic was that of the Dutch philosopher Arnold Geulincx, Logica fundamentis suis restituta (1662; “Logic Restored to its Fundamentals”). This work attempted to resurrect the rich detail of scholastic logic, including the theory of suppositio and issues of existential import.