##### Wiktionary

**formal system**

n. (context logic English) The grouping of a formal language and a set of inference rules and/or axioms.

##### Wikipedia

**Formal system**

A **formal system** is broadly defined as any well-defined system of abstract thought based on the model of mathematics. The entailment of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory. A formal system need not be mathematical as such; for example, Spinoza's * Ethics* imitates the form of Euclid's *Elements*.

Each formal system has a formal language, which is composed by primitive symbols. These symbols act on certain rules of formation and are developed by inference from a set of axioms. The system thus consists of any number of formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.

Formal systems in mathematics consist of the following elements:

- A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite strings of symbols).
- A grammar, which tells how well-formed formulas (abbreviated
*wff*) are constructed out of the symbols in the alphabet. It is usually required that there be a decision procedure for deciding whether a formula is well formed or not. - A set of axioms or axiom schemata: each axiom must be a wff.
- A set of inference rules.

A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, respectively.

Some theorists use the term *formalism* as a rough synonym for *formal system*, but the term is also used to refer to a particular style of *notation*, for example, Paul Dirac's bra–ket notation.

#### Usage examples of "formal system".

Using Omar Narayama's universal syntax, the holists of the Order constructed a __formal system__, or science, that treated the individual mind as sub-programs of a universal algorithm.

There has to be a __formal system__ of rules, according to which the numbers are combined.

They were almost too logical to make good analysts (as far as Avram knew, no Gorm in recorded history had ever played a hunch), and their lack of any __formal system__ of permanent naval or military ranks sometimes confused their imperial partners .

They were almost too logical to make good analysts (as far as Avram knew, no Gorm in recorded history had ever played a hunch), and their lack of any __formal system__ of permanent naval or military ranks sometimes confused their imperial partners.

But each __formal system__, logically consistent internally, describes a possible universe, which therefore exists.

The universe, this universe, is described-umm, that's the wrong word-by a __formal system__.

It was a peace enforced by William's own decrees, not by a __formal system__ of law and justice.

Perhaps heand Troi could devise a more __formal system__ for the othersand she could then take charge of it.

When ve'd first met Radiya in the Mines, Yatima had asked ver why some non-sentient program couldn't just take each __formal system__ used by the miners and crank out all its theorems automatically sparing citizens the effort.

Though no more eerie than many of the other oddities of this place, the lack of a __formal system__ of beliefs gave Collins goose bumps.

His own country had no more __formal system__ for succession than did Germany.