Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Formal language
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== Formal theories, systems, and proofs === {{Main|Theory (mathematical logic)|Formal system}} [[File:Formal languages.svg|thumb|right|This diagram shows the [[Syntax (logic)|syntactic]] divisions within a [[formal system]]. [[string (computer science)|Strings of symbols]] may be broadly divided into nonsense and [[well-formed formula]]s. The set of well-formed formulas is divided into [[theorem]]s and non-theorems.]] In [[mathematical logic]], a ''formal theory'' is a set of [[sentence (mathematical logic)|sentences]] expressed in a formal language. 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 rule]]s, which may be interpreted as valid rules of inference, or a set of [[axiom]]s, or have both. A formal system is used to [[Proof theory|derive]] one expression from one or more other expressions. Although a formal language can be identified with its formulas, a formal system cannot be likewise identified by its theorems. Two formal systems <math>\mathcal{FS}</math> and <math>\mathcal{FS'}</math> may have all the same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntactic consequence of a formula B in one but not another for instance). A ''formal proof'' or ''derivation'' is a finite sequence of well-formed formulas (which may be interpreted as sentences, or [[proposition]]s) each of which is an axiom or follows from the preceding formulas in the sequence by a [[rule of inference]]. The last sentence in the sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions. ====Interpretations and models==== {{main|Formal semantics (logic)||Interpretation (logic)|Model theory}} Formal languages are entirely syntactic in nature, but may be given [[semantics]] that give meaning to the elements of the language. For instance, in mathematical [[logic]], the set of possible formulas of a particular logic is a formal language, and an [[interpretation (logic)|interpretation]] assigns a meaning to each of the formulas—usually, a [[truth value]]. The study of interpretations of formal languages is called [[Formal semantics (logic)|formal semantics]]. In mathematical logic, this is often done in terms of [[model theory]]. In model theory, the terms that occur in a formula are interpreted as objects within [[Structure (mathematical logic)|mathematical structures]], and fixed compositional interpretation rules determine how the truth value of the formula can be derived from the interpretation of its terms; a ''model'' for a formula is an interpretation of terms such that the formula becomes true.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)