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Could anyone explain, or provide a good source to, the language used in theoretical economic papers?
For example, the difference between an assumption, proposition, postulation, theorem, lemma, axiom, corollary, conjecture, and so on.
I think that there are also different conventions across fields so I'm not sure if what I'm currently reading in mathematics/physics (e.g., this and this) is applicable to economics.
The language you refer to is the same in economics and in mathematics, for the simple reason that economics uses this language borrowed from mathematics, as it uses mathematics. So, you have to look at the language of mathematics. The question of mathStackExchange you linked gives you important explanations.
I can just add something to what is said in an answer in mathStackExchange, a paragraph of which I quote:
A theorem is a logical consequence of the axioms. In Geometry, the "propositions" are all theorems: they are derived using the axioms and the valid rules. A "Corollary" is a theorem that is usually considered an "easy consequence" of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a 'corollary' is deemed more important than the corresponding theorem. (The same goes for "Lemma"s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).
I want specify something about propositions and theorems. It is true that propositions and theorems are conceptually the same thing: they are derived from the axioms through the inference rules, they are 'proved'.
But when you read a text of mathematics, some statements are called 'Theorem' and other 'Proposition'. This is a convention.
Usually, the name 'Theorem' is reserved to important theorems, often, but not necessarily, they have a name: 'The Hahn-Banach Theorem, the 'Theorem of contractions', and so on.
The name 'Proposition' denotes a less important theorem, which doesn't deserve the name 'Theorem'.
A Lemma usually is a theorem that is important to prove another theorem, and not very important on its own, but it is possible to find Lemmas considered important, for example in differential equations theory there is the 'Gronwall's Lemma', or 'Fatou's Lemma' in real analysis, which are considered very important theorems on its own.
A conjecture is just a hypothesis that a statement is true, but it hasn't been proved (or disproved): the Riemann conjecture, the Fermat conjecture (this last conjecture has been proved).
Returning to this topic, I realized that I made a serious omission, I forgot the very important mathematical term of definition. I make up for this omission in a separate answer, otherwise the previous answer becomes cumbersome.
Often mathematicians say that the most important things of mathematics are not theorems, but definitions.
A definition is a statement that describes a mathematical entity using terms that have a known meaning, as previously defined or assumed as primitive.
Definitions are at the heart of mathematics: new mathematical objects can be introduced through definitions, and definitions can generate new mathematics. The creative function of definitions is even more evident when it is defined not a particular object inside a theory, but the scope of a theory itself: for instance, the definition of group in algebra leads to a whole field of algebra, group theory.
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As an advice that can be useful from a practical point of view, when reading texts of mathematics: it should be noticed that which statements are definitions and which are theorems can vary depending on the author, so it mustn’t generate wonder if definitions differ among texts.
This can happen sometimes going from books of mathematics more aimed at economics to ‘purer’ mathematics books.
A common example can be the definition of a rank of a matrix in linear algebra. Often, more applicative texts define the rank of a matrix as the number of rows (or columns) linearly independent.
Other texts, usually more theoretical, define the rank of a matrix $A$ as the dimension of the image of the linear application generated by $A$, $Ax$, where $x$ is a vector. The statement about rows is then given as a theorem. The two definitions are equivalent.
