TITLE: What's the difference between a logic, an internal logic (language) of a category, an internal logic of a topos and a type theory?
QUESTION [20 upvotes]: maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). 
However, vaguely speaking by a logic I mean a pair $(\Sigma, \vdash)$ where $\Sigma$ is a signature (it has the types, functionals and relational symbols) and a consequence operator $\vdash$. 
By an internal logic in a category I mean viewing each object as a type and each morphism as a term. 
So what's the difference between a logic, an internal logic (language) of a category, an internal logic (language) of a topos and a type theory? Furthermore why the interchanging in the literature between the word "logic" and "language" when dealing with the internal properties of a category? Moreover, how higher order logic, modal logic and fuzzy logic suit in these concepts stated above?
Thanks in advance.

REPLY [8 votes]: Your definition of logic is pretty much correct. A logic contains both the language which the signature $\Sigma$ generates and the deductive system defined by $\vdash$.
A type theory is a logic with different sorts of individuals (called "types") and constructions that generate new types from existing ones, like product and arrow types.
An internal logic is a type theory derived from a category and the internal language is the language part of that logic. Specifically, the atomic sorts of the internal language are the objects of the category. Since a topos is a specific category of categories, the internal logic of a topos is the derived type theory.
The modalities of modal logic can sometimes be related to operators on subobjects in a category, but only if they preserve logical equivalence: $\alpha\iff\beta$ should imply $\Box\alpha \iff \Box \beta$. Otherwise, the induced function of subobjects is not well-defined.
Fuzzy logic is misnamed. It is more like a model theory than a logic.