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
Sequent calculus
(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!
===Natural deduction systems=== {{Main|Natural deduction}} In natural deduction, judgments have the shape :<math>A_1, A_2, \ldots, A_n \vdash B</math> where the <math>A_i</math>'s and <math>B</math> are again formulas and <math>n\geq 0</math>. In other words, a judgment consists of a ''list'' (possibly empty) of formulas on the left-hand side of a [[Turnstile (symbol)|turnstile]] symbol "<math>\vdash</math>", with a single formula on the right-hand side,<ref>{{harvnb|Curry|1977|pp=184β244}}, compares natural deduction systems, denoted LA, and Gentzen systems, denoted LC. Curry's emphasis is more theoretical than practical.</ref><ref>{{harvnb|Suppes|1999|pp=25β150}}, is an introductory presentation of practical natural deduction of this kind. This became the basis of [[System L]].</ref><ref>{{harvnb|Lemmon|1965}} is an elementary introduction to practical natural deduction based on the convenient abbreviated proof layout style [[System L]] based on {{harvnb|Suppes|1999|pp=25β150}}.</ref> (though permutations of the <math>A_i</math>'s are often immaterial). The theorems are those formulae <math>B</math> such that <math>\vdash B</math> (with an empty left-hand side) is the conclusion of a valid proof. (In some presentations of natural deduction, the <math>A_i</math>s and the turnstile are not written down explicitly; instead a two-dimensional notation from which they can be inferred is used.) The standard semantics of a judgment in natural deduction is that it asserts that whenever<ref>Here, "whenever" is used as an informal abbreviation "for every assignment of values to the free variables in the judgment"</ref> <math>A_1</math>, <math>A_2</math>, etc., are all true, <math>B</math> will also be true. The judgments :<math>A_1, \ldots, A_n \vdash B</math> and :<math>\vdash (A_1 \land \cdots \land A_n) \rightarrow B</math> are equivalent in the strong sense that a proof of either one may be extended to a proof of the other.
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)