The linguistic turn must either escape logic or adopt logic. If it adopts logic, it adopts some kind of semantics. If it doesn't adopt logic, then some role must be found for irrationalism.
The linguistic turn is a metaphor. Language takes up space, and within the space ideas are found. Ideas take up space if they have value, and the aspect of their having reality is called Art.
With this general idea of Art, we have a concept as strong as semantics without any necessary semantic rules. Now consider how rigid this becomes, when, in the broader context of these looser assumptions, we learn that hard logic still applies.
The result is much more rigorous. It is, in some sense, the universal format for all earlier forms of logic. It is as primal as syllogisms or mathematics.
My proof is that coherent knowledge can be found using opposites.
The exclusion of one value leaves another value, but in any context in which value is found, the combination is not found. In some sense, conventionally, neither of two opposites exists. Because, in some sense, it is not measured.
To measure an opposite requires a context of opposition. And, if a context is defined in terms of opposites, then it is fair to say that the context is exclusive in some sense. Through exclusivity, we grant coherency to an idea.
Through multiple ideas---multiple opposites of any complexity---we second-guess the value of reductionism.
There is no need to adopt a single category by itself, without adopting multiple frames of reference. It is then an obvious move to formalize the frame of reference, carrying over the assumption of a kind of correpondence theory to frame information.
However, correspondence is no more than a tool.
Even framing knowledge is only a tool.
But, for now it deserves the profound name of systemics and formulitics.