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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.
Formal language, which is a set of well-formed formulas, which are strings of symbols from an alphabet, formed by a formal grammar (consisting of production rules or formation rules). Deductive system, deductive apparatus, or proof system, which has rules of inference that take axioms and infers theorems, both of which are part of the formal ...
In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. These rules are formalized with a ranking of the operations.
As a writer, I value grammar, spelling, syntax, diction—all that jazz. But as a millennial, I’m programmed to understand that if I use a gratuitous apostrophe within a possessive “its,” I ...
A formal grammar describes which strings from an alphabet of a formal language are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. A formal grammar is defined as a set of production rules for such strings in a formal language.
The rule is allowed if does not appear on the right side of any rule. The languages described by these grammars are exactly all languages that can be recognized by a linear bounded automaton (a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.)
Using this grammar, the sequence of symbols (((p → q) ∧ (r → s)) ∨ (¬q ∧ ¬s)) is a formula, because it is grammatically correct. The sequence of symbols ((p → q)→(qq))p)) is not a formula, because it does not conform to the grammar. A complex formula may be difficult to read, owing to, for example, the proliferation of parentheses.
A formal language is an organized set of symbols the essential feature being that it can be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference to any meanings of any of its expressions; it can exist before any interpretation is assigned to it—that is, before it has any meaning.
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