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A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules , which may be interpreted as valid rules of inference, or a set of axioms , or have both.
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.
In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules can be applied to a nonterminal symbol regardless of its context. In particular, in a context-free grammar, each production rule is of the form
Rudolph Carnap defined the meaning of the adjective formal in 1934 as follows: "A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning of the symbols (for example, the words) or to the sense of the expressions (e.g. the sentences), but simply and solely to the kinds and order of the symbols from which the expressions are ...
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms by a set of inference rules. [1] [non-tertiary source needed] [2] In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics. [3]
Let us notate a formal grammar as = (,,,), with a set of nonterminal symbols, a set of terminal symbols, a set of production rules, and the start symbol.. A string () directly yields, or directly derives to, a string (), denoted as , if v can be obtained from u by an application of some production rule in P, that is, if = and =, where () is a production rule, and , is the unaffected left and ...
For example, in the definition "An elephant is a large gray animal native to Asia and Africa", the word "elephant" is the definiendum, and everything after the word "is" is the definiens. [ 7 ] The definiens is not the meaning of the word defined, but is instead something that conveys the same meaning as that word.
Formal system, an abstract means of generating inferences in a formal language; Formal language, comprising the symbolic "words" or "sentences" of a formal system; Formal grammar, a grammar describing a formal language; Colloquialism, the linguistic style used for informal communication