Formal system
A formal system is broadly defined as any well-defined system of abstract thought based on the model of mathematics.
Description
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.
These combinations that are formed from the axioms in accordance with the stated rules.
Mathematics
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.
Recursion
A formal system is said to be recursive (i.e. effective) if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, according to context.
Formalism
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.
See also
External links
- Formal system @ Wikipedia
