Formal system Totally Explained

Everything about Formalization totally explained

In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist of the following elements:

A finite set of symbols (for example the alphabet), that can be used for constructing formulas (for example 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 (for example effective) if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, according to context.
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.

Related subjects

Formal language

A formal language is a set A of strings (finite sequences) on a fixed alphabet α.

Formal grammar

In computer science and linguistics a formal grammar is a precise description of a formal language: a set of strings. The two main categories of formal grammar are that of generative grammars, which are sets of rules for how strings in a language can be generated, and that of analytic grammars, which are sets of rules for how a string can be analyzed to determine whether it's a member of the language. In short, an analytic grammar describes how to recognize when strings are members in the set, whereas a generative grammar describes how to write only those strings in the set.

Formal proofs

Formal proofs are sequences of wffs. For a wff to qualify as part of a proof, it might either be an axiom or be the product of applying an inference rule on previous wffs in the proof sequence. The last wff in the sequence is recognized as a theorem.
The point of view that generating formal proofs is all there's to mathematics is often called formalism. David Hilbert founded metamathematics as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a metalanguage. The metalanguage may be nothing more than ordinary natural language, or it may be partially formalized itself, but it's generally less completely formalized than the formal language component of the formal system under examination, which is then called the object language, that is, the object of the discussion in question.
Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all wffs of which there's a proof for. Thus all axioms are considered theorems. Unlike the grammar for wffs, there's no guarantee that there will be a decision procedure for deciding whether a given wff is a theorem or not. The notion of theorem just defined shouldn't be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems.

Formal interpretations

formal interpretation of a formal system is the assignment of meanings, to the symbols, and truth-values to the sentences of the formal system. The study of formal interpretations is called Formal semantics. Giving an interpretation is synonymous with constructing a model.
   An interpreted formal system is a formal language for which both syntactical rules for deduction, and semantical rules of interpretation are given. An interpreted formal system can be expressed as the ordered quadruple <α, $mathcal$ _i_{_n},p_n) and p₁ and ... and p_n, q.
   For interpreted formal systems there are also alternative, more explicit definitions which include ds, or both ds and D, analogous to those given for interpreted formal languages.

Further Information
Get more info on 'Formalization'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://formal_system.totallyexplained.com">Formal system Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
   As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.