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A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values : as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
The consequence of these features is that a mathematical text is generally not understandable without some prerequisite knowledge. For example, the sentence "a free module is a module that has a basis" is perfectly correct, although it appears only as a grammatically correct nonsense, when one does not know the definitions of basis, module, and free module.
To do so technically would require a more sophisticated grammar, like a Chomsky Type 1 grammar, also termed a context-sensitive grammar. However, parser generators for context-free grammars often support the ability for user-written code to introduce limited amounts of context-sensitivity.
Formal semantics is the study of grammatical meaning in natural languages using formal concepts from logic, mathematics and theoretical computer science.It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language.
For example, a grammar for a context-free language is left-recursive if there exists a non-terminal symbol A that can be put through the production rules to produce a string with A as the leftmost symbol. [15] An example of recursive grammar is a clause within a sentence separated by two commas. [16]
At the most basic conceptual level, a parser, sometimes generated by a parser generator like yacc, attempts to decide if the source program is syntactically valid, that is if it is well formed with respect to the programming language grammar for which the compiler was built.
Generative grammar proposes models of language consisting of explicit rule systems, which make testable falsifiable predictions. This is different from traditional grammar where grammatical patterns are often described more loosely. [9] [10] These models are intended to be parsimonious, capturing generalizations in the data with as few rules as ...
Traditional grammar (also known as classical grammar) is a framework for the description of the structure of a language or group of languages. [1] The roots of traditional grammar are in the work of classical Greek and Latin philologists. [2] The formal study of grammar based on these models became popular during the Renaissance. [3]