Metalanguage in the context of ML (programming language)

Pāṇini (/ˈpɑːnɪni/; Sanskrit: पाणिनि, pāṇini [páːɳin̪i]) was a Sanskrit grammarian, logician, philologist, and revered scholar in ancient India during the mid-1st millennium BCE, dated variously by most scholars between the 6th–5th and 4th centuries BCE.

The historical facts of his life are unknown, except only what can be inferred from his works, and legends recorded long after. His most notable work, the Aṣṭādhyāyī, is conventionally taken to mark the start of Classical Sanskrit. His work formally codified Classical Sanskrit as a refined and standardized language, making use of a technical metalanguage consisting of a syntax, morphology, and lexicon, organised according to a series of meta-rules.

A metatheory or meta-theory is a theory whose subject matter is another theory. Analyses or descriptions of an existing theory would be considered meta-theories. For mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. Meta-theoretical investigations are part of the philosophy of science. The topic of metascience is an attempt to use scientific knowledge to improve the practice of science itself.

The study of metatheory became widespread during the 20th century after its application to various topics, including scientific linguistics and its concept of metalanguage.

In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.

Judgments are used in formalizing deduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments). A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalize the Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simple form. The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of tautologies.

Operator grammar is a mathematical theory of human language that explains how language carries information. This theory is the culmination of the life work of Zellig Harris, with major publications toward the end of the last century. Operator grammar proposes that each human language is a self-organizing system in which both the syntactic and semantic properties of a word are established purely in relation to other words. Thus, no external system (metalanguage) is required to define the rules of a language. Instead, these rules are learned through exposure to usage and through participation, as is the case with most social behavior. The theory is consistent with the idea that language evolved gradually, with each successive generation introducing new complexity and variation.

Operator grammar posits three universal constraints: dependency (certain words depend on the presence of other words to form an utterance), likelihood (some combinations of words and their dependents are more likely than others) and reduction (words in high likelihood combinations can be reduced to shorter forms, and sometimes omitted completely). Together these provide a theory of language information: dependency builds a predicate–argument structure; likelihood creates distinct meanings; reduction allows compact forms for communication.