This is the second time I see a Stack Exchange question on the usage of special quotation marks in philosophical literature, the first time being this question about p- and d-shaped quotation marks in a work by Carnap. During my research for this question I found another philosophical work, which uses ⌜-, ⌝-, m-, M-shaped quotation marks itself and intensively talks about the use of quotation marks in other philosophical works¹.
The latter explains among others:
I use Quine’s quasi-quotation marks, ‘⌜’ and ‘⌝’ in combination with ‘α’. In quasi-quotation, all internal expressions are quoted, i.e., mentioned (designated), except for metalinguistic variables, whose values are mentioned. I use single quotation marks for direct (expression) quotation. Following Kaplan, I use superscripted occurrences of ‘m’ as indirect-quotation marks, and superscripted occurrences of ‘M’ as indirect-quasi-quotation marks.²
Given this and my small understanding of what is going on, I make the somewhat informed guess that such quotation marks serve to distinguish different forms of quotation or reference on a meta level such as (to give simple examples):
- Citing another author verbatim.
- Referring to the word as such (and not what it means).
- Scare quotes and indicating metaphoric use of a word (such as for lack of a better word).
- Referring to the meaning of a word.
- Marking a word or term that is being defined at that very point.
To sum it up, such quotation marks are used because the author ran out of other types of emphasis and are a special notation that should either be explained somewhere in the book or be commonplace in the respective scientific community.
¹ Nathan Salmon – Metaphysics, Mathematics, and Meaning
² And because somebody is bound to ask: The author also uses boldface to encode something and italics for emphasis and other purposes.