I am trying to figure out the meaning of prefix "Meta-" in English.
Quoted from Wikipedia
Meta- (from Greek: μετά = "after", "beyond", "with", "adjacent", "self"), is a prefix used in English (and other Greek-owing languages) to indicate a concept which is an abstraction from another concept, used to complete or add to the latter.
In epistemology, the prefix meta is used to mean about (its own category). For example, metadata are data about data (who has produced them, when, what format the data are in and so on). Also, metamemory in psychology means an individual's knowledge about whether or not they would remember something if they concentrated on recalling it. Furthermore, metaemotion in psychology means an individual's emotion about his/her own basic emotion, "or somebody else's basic emotion".
Another, slightly different interpretation of this term is "about" but not "on" (exactly its own category). For example, in linguistics a grammar is considered as being expressed in a metalanguage, or a sort of language for describing another language (and not itself). Any subject can be said to have a meta-theory which is the theoretical consideration of its meta-properties, such as its foundations, methods, form and utility.
I was wondering what is the slightly difference between "about (its own category)" and "'about' but not 'on' (exactly its own category)"?
My original problem is to understand the meaning of "metalogic" and "logic". Quoted from Wikipedia:
Metalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves.1 According to Geoffrey Hunter, while logic concerns itself with the "truths of logic," metalogic concerns itself with the theory of "sentences used to express truths of logic."2
The basic objects of study in metalogic are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic known as model theory, while the study of deductive apparatus is the branch known as proof theory.
In this case, is it correct to understand "meta-X" as some study that treat something (here logical system) as a whole without going inside it, whereas "X" study the same thing by going inside it?