What is direct and indirect denial in mathematical linguistics? And how do logic proofs work?
To my understanding, mathematical linguistics may include logical proofs as a part of its coverage, but the two are distinct entities and do not necessarily overlap academically or epistemologically. Of course, when you get into anything as multifaceted as logic and philosophy, it's possible that some terms may be applied generously or somewhat abstractly. However, based on the way this question is phrased, I'm assuming that you're using the term "mathematical linguistics" to refer more to formal reasoning and the use of mathematical terms to do so, than a mathematical analysis of the rules of language, which is a closer approximation of what mathematic linguistics is.
Similarly to the way that science uses different definitions of words like "theory" than we do in everyday conversation, logic uses the word "proof" in a different way than we commonly do. "Proof" in everyday terms is different from logical proof; for example, "proving" that 2+2=4, by writing the equation, is different from actually demonstrating that quantities called 2 and 4 exist, and that they are related. Logical proofs require the construction of a sort of network, within which realities and relationships are defined, and using this network to establish the validity or error of a statement.
There is no set of rules that establish the validity of a proof; they are not "on rails" so to speak. Proofs must be constructed meticulously in order to ensure that their reasoning is fully visible. This is where phrases like "If P then Q" apply. You may have seen reasoning like this before, usually phrased in a manner similar to, "All ducks quack, X quacks, therefore X is a duck." This statement would not be logically sound, because we have not defined that nothing else quacks. Rephrase this statement to see its incompleteness; "All fires are hot, X is hot, therefore X is a fire". You can see how proofs would quickly become very complex.
Denial is considered the logical opposite of assertion; it is a statement of negation. Directness and indirectness have to do with the way in which the denial is articulated. For example, referring again to the hot fire; direct denial would say "the fire is not hot because it is less than Y degrees". Indirect denial would say "The definition of heat is arbitrary and abstract, therefore fire cannot be defined as hot". An indirect denial doesn't really provide a response to the assertion that has been made, but addresses or attacks the structure that the assertion has been built upon; it is a "reflection upon the definition".