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.
Mathematics is not a language but a collection of non-necessarily mutually consistent theories, each theory being based on its own assumptions, only some of them axioms, and couched in broadly the same language of mathematics.
Mathematical work requires mathematicians to make assumptions, but not necessarily the same as each other. Mathematics is more of a commitment to a general method of work than a unified system, method which essentially involves the use of one symbolic scheme and some inordinate rigour in the definition of mathematical concepts. Beyond that, and adherence to logical reasoning, each mathematician is on their own.
Like the mathematical language, a natural language in based on the speakers' assumption that the words used refer. That is, that any word used whatever means something and not nothing. Not even the word "nothing" means nothing. All speakers assume that all words mean something to all speakers of the language, which is perhaps just wishful thinking. All languages are also based on the assumption that the syntax is shared among different speakers of the same language.
However, such assumptions are essentially technical, somewhat like a code of politeness between civilised people whereby you allow the prisoner to make a last wish before being hanged. The form is assumed, but the content is left to your unfettered imagination, and do humans make the best of that!
Languages are also supported by the speakers' unvarying assumption that the audience shares some minimal common perception of reality. Talking would have no utility if not. It would be a waste of time and energy.
So, broadly, languages require common-sense assumptions.
Assumptions about logic as such? No. People used natural languages, and some mathematical language, long before the notion of logic emerged from the travails of the Greek Antiquity. Humans are innately logical. That is, they have some logical capacity. However, this is not an assumption, it is just a fact of human nature. Without it, there would be no language, natural or otherwise, but the same could be said of our linguistic capacity, or even about our capacity to perceive the world.
Formal logic is something else. Like all sciences, formal logic requires logicians to make assumptions about the nature of their object, but this is not language-dependent.
