Languages, natural like English or French, or subject to specification like the mathematical language or formal logic itself, do not make any assumption, and this for the obvious reason that assumptions are made by logical beings which have a language, not by the languages themselves.
Someone said that natural language doesn't require or imply classical logic, but I thought this example might point to the contrary.
I was thinking about the law of identity. This states that "A is A" - i.e., anything is identical to itself. In grammar, this is reflected in the use of definite articles (the) and pronouns to refer to specific entities.
Example: "The cat is on the mat." Here, "the cat" refers specifically to a particular cat.
So how am I wrong? Could you explain?