Whether there is a "logic-sense" is a question about consciousness so fundamental and yet so hard that it's scarcely even recognized by science-friendly philosophy of mind. Phenomenologists have something to say about it because they are just trying to characterize experience, without concern for whether or how their descriptions are compatible with a particular scientific theory of nature. But if you look at "naturalist" philosophers (naturalism = physicalism = materialism = an intent that one's philosophy should be consistent with natural science), the discussion scarcely gets beyond the existence of colors and other "five-sense" qualities.

The usual approach is to talk as if a conscious state is a heap of elementary sense-qualia, somehow in the same way that a physical object could be a pile of atoms. But experience is about the perception of form as well, and this is related to the idea of a logic-sense, because logic is about concepts and abstract properties, and the properties of a "form" have an abstractness about them, compared to the "stuff" that the form is made from.

In the centuries before Kant and Husserl, there was a long-running philosophical "problem of universals", which is just this question of how substance and property are related. How is the greenness in one blade of grass, related to the greenness in another blade of grass? Suppose it were the exact same shade of green. Is it the same thing, mysteriously "exemplified" in two different places? If you say yes, then what is "exemplification" or "instantiation"? Is it a new primitive ontological relation? If you say no, and say that these are separate "color-instances", you still need to explain their sameness or similarity.

With the rise of consciousness itself as a theme of human thought, the problem has assumed a new character, because now the greenness is in the observer rather than in the blade of grass. We can still raise the classic questions, about the greenness in one experience and the greenness in another experience, but the deeper problem is whether we are even conceiving of the basic facts correctly. Experience isn't just "green stuff happening" or "round stuff happening", it's "green round stuff happening to my hand, eyes, and mouth" (if I'm eating an apple), it's "happening to me " (whoever and whatever "I" am), it's "green round stuff being experienced as green and round" - and that little word "as" sums up a whole dimension of consciousness that the focus on sense-qualia obscures; the aspect of consciousness known as its "intentionality", the fact that an experience is an experience "of" something or "about" something.

Names can be useful. The sense that "stuff is happening to me" has been called apperception. (That's jargon that you won't see on LW. Jargon that you will see, that comes at the same phenomenon from a different angle, is indexicality, the me-here-now component of an experience. One also needs to distinguish between me-here-now experienced or conceptualized in terms of difference to other "me-here-now"s, and me-here-now as simply another component of an experience, even if you're not thinking about other people at the time. Apperception is more about this second aspect, whereas discussions of indexicality tend to puzzle over what it is that distinguishes one person, as a locus of experience, from another - they're both "me" to themselves, but ontologically they are two entities, not one.) If there is a logic-sense, then it is presumably at work both in intentionality and in apperception; in fact the latter appears to contain a sort of indexical intentionality, the logic-sense applied to the perceiving self.

Two other very different perspectives: First, in Objectivism, you will see "concept formation" discussed as "measurement omission". The idea being that a concept is a perception with something removed - the sensory and indexical particularities. It doesn't quite deal with the ontological problem of what "instantiation of a property" is in the first place, but it highlights a psychological and cognitive/computational aspect.

Second, for the five senses, there are sense organs. If there is a logic sense, one should ask whether there's a logic organ too. Here the neurocomputational answer is going to be that it's a structure in the brain which has the outputs of sense organs as its inputs. This answer doesn't do away with the miasma of dualism that hangs over all functionalist explanations of experience, but it does plausibly mimic the causal dependence of higher-order experience on raw experience.

Finally, I'll point out that the nature of logic and a logic-sense is tied up with the nature of being and our awareness of it. We have a sense that reality exists, that individual things exist, and that they are a certain way. If you can stand to read something like Heidegger's historical phenomenology of Being, you'll see that grammar and logic have roots in a certain experience of being and a certain analysis of that experience, e.g. into "thatness" and "whatness", existence and essence: that a thing is, and what a thing is. These perceptions and distinctions were originally profound insights, but they were codified in language and became the everyday tools of the thinking, wilful mind. Heidegger's work was partly about recovering a perception of being prior to its resolution into existence and essence, out of a conviction that that is not the end of the story. The problem with trying to think "beyond" or "before" subject-predicate thinking is that it just turns into not thinking at all. Is there intellectual progress to be had beyond the raw observation that "Something is there", if you don't "apply concepts", or is the latter simply an essential condition of understanding? Et cetera, ad infinitum.