Regarding logic, the answer is a mixed one.
The development of medieval logic is wide area of study; see at least SEP's entries :
and more ..., as well as :
The Reanaissance rediscovery of ancient Greek mathematics and philosophy induced a general "devaluation" of scholastic logic, seen as cumbersome and futile.
The "champions" of scientific revolution, like Bacon, Galileo and Descates was highly critical or quite silent about it; see :
but also :
Among modern "pioneer" of logic, like Leibniz, Bolzano and Frege (all quite unrecognized as such during their time), only Leibniz has a clear knowledge of medieval logic.
The Algebra of Logic Tradition was "motivated" more by the development of algebra than by traditional logic.
See on this :
But there are at least two considerations to be done :
First, the humanists devaluation of scholastic logic is contemporary to the rediscovery of ancient Greek philosophy, included Aristotle's one and its logic.
Thus, we have a "underground" continuity in "aristotelian" logic also during the Renaissance and Early Modern Era; see for example :
Hobbes is generally considered by the scholars as one of the first great early modern
thinkers to break with tradition and direct his work instead towards the new philosophical
and scientific developments. Not infrequently, passages have been considered out
of context, with scholars stating that Hobbes abandoned Aristotelian philosophy and
logic, because of his vitriolic attack upon Scholastic philosophy and theology, even
though a large part of his thought, and especially of his logic, had been decisively
in fluenced by the Aristotelian tradition.
It is well known that Hobbes never wrote a textbook of logic, nor taught logic in
the university. However, the general introduction to his Elementa philosophiae
presents a dense treatment of logic, the result of 10 years’ thought on the topic; this
can be considered as a work in its own right. [...] Hobbes’ knowledge of the Aristotelians must have increased with his travel in Italy (1610–1613) [...] where he knew Fulgenzio Micanzio, a friend of Paolo Sarpi and Galileo. In this period Hobbes began to read Galileo and Euclid, and hatched the plan to establish a rigorous mechanical science of reality as a whole. Probably under the impulse of Galilei’s philosophy, which was full of Aristotelian ideas, Hobbes focused his interests on Paduan logic, whose legacy is quite evident in his works.
From the incipit of his Logica, which could be considered as a work in its own right, Hobbes shares with his contemporaries the intention of establishing a scientific method. Such a method should take as its model the advancements and developments of geometry and should proceed with the same rigour.
- Russell Wahl, Port Royal : The Stirrings of Modernity, in Handbook of the History of Logic. Vol 3, page 667-on :
Logic or the Art of Thinking, popularly known as the Port-Royal Logic, was probably
the most important logic text book from the time after the mediaeval period until the middle of the nineteenth century. [...]
Like other authors of the seventeenth century, Arnauld and Nicole were very critical of the old logic, which for them included not only the “scholastic” work on syllogisms, but also the humanistic logic of Ramus. They saw their logic as new and they were particularly influenced by Descartes, who had also been critical of syllogistic logic, and they incorporated many of his doctrines into their work. This new seventeenth-century logic, with its rejection of much traditional logic and its concern with clarifying ideas and determining the truth of simple propositions, has often been criticized by more recent logicians of mixing psychology and epistemology with logic. Often it has been compared unfavorably with mediaeval logic. Despite the fact that most of its topics are closer to the mediaeval period, the Port-Royal Logic has a very modern feel to it, and covers several topics now included in more informal introductory logic courses, such as discussions of clarification of concepts, informal fallacies, causal reasoning, and probability, as well as more traditional accounts of propositions and syllogisms.
Second, what "complicates" a reconstruction of the status of logic during the Early Modern Era are :
the difficulty of separating the influence of "old" medieval logic from that of "new" (rediscovered) aristotelian one
the characteristic "mixture" in the logical discussions of this period of formal logic, analysis of language (both present in medieval tradition) and search for a method for acquiring knowledge, that was clearly a "modern" issue, but again linked to the (rediscovered) aristotelian doctrines of the Posterior Analytics.