So this is a rough sentence to parse, but once you know what these words are, it’s a neat concept.
(If anyone out there is familiar with Richard McKeon, I’d appreciate feedback on how accurate my reading is. Philosophy is hard, and the only reason I made it through college was because I could ask professors to explain this stuff to me in plain English. I’m practicing how to read here.)
"Philosophic semantics" is just another way of saying solutions to problems proposed by philosophy—as McKeon puts it, "philosophic semantics is an examination of different solutions of philosophic problems."
"Philosophic inquiries," on the other hand, are those problems proposed by philosophy. Again, McKeon: "philosophic inquiry is an examination of common issues to which different philosophic resolutions may be found."
From here, Richard goes on to discuss the cycle these two create. Long story short, you can compare two different philosophies based on the questions they ask (“different philosophies are significantly related by the common problems they treat, and philosophic inquiry may provide modes by which to relate the stages of different solutions”). By the same token, the solutions used by a certain philosophy can pin down just what that philosophy means (“philosophic semantics may provide schemata by which to make unambiguously clear the meanings that are attributed in a proposed interpretation to statements made in any philosophy”).
In other words, you can compare philosophies with one another by what they ask and how. You can hone in a single philosophy—pin it to one particular stance—by understanding the answers it finds.
This leads him to paint a picture of the cycle. First, “a great philosopher” cleans things up by finding clear, unambiguous ways to state philosophic solutions (“the semantic distinctions have accumulated a mass of ambiguities from which they are rescued periodically by the precisions of a great philosopher engaged in one of the modes of inquiry”). At this point, it looks like we have a perfect understanding of philosophy in general. Next, a “rival school” comes along and restates the problems (“the modes of inquiry […] are reduced to precise repetitions [by the previously mentioned great philosopher] from which they are rescued periodically by the controversial reformulations of common problems in the mode of a rival school”).
So, some hot shot whiz philosopher comes along and says he/she has the perfect answer to everything. Then, someone examines his/her approach; not the answers, but the questions that the hot shot answered. The “rival school” goes on to come up with new problems that are “reformulations” of old ones. This in turn leads to new answers, which leads to new problems.
He sums things up using the sentence I quoted above. “The precisions of philosophic semantics may be preserved by connections established by the modes of inquiry”—in other words, a perfect understanding of problems leads to immutable answers. “The communications among modes of inquiry may be preserved by precisions established by the distinctions of semantics”—in other words, immutable facts beg discussion between modes of inquiry and engender new inquiries.
In short, when you think you’ve found the answer to everything, it highlights what questions you have left to answer.