J u m p t o c o n t e n t
M a i n m e n u
M a i n m e n u
N a v i g a t i o n
● M a i n p a g e
● C o n t e n t s
● C u r r e n t e v e n t s
● R a n d o m a r t i c l e
● A b o u t W i k i p e d i a
● C o n t a c t u s
● D o n a t e
C o n t r i b u t e
● H e l p
● L e a r n t o e d i t
● C o m m u n i t y p o r t a l
● R e c e n t c h a n g e s
● U p l o a d f i l e
S e a r c h
Search
A p p e a r a n c e
● C r e a t e a c c o u n t
● L o g i n
P e r s o n a l t o o l s
● C r e a t e a c c o u n t
● L o g i n
P a g e s f o r l o g g e d o u t e d i t o r s l e a r n m o r e
● C o n t r i b u t i o n s
● T a l k
( T o p )
1
E x a m p l e s
2
F o r m a l d e f i n i t i o n s
T o g g l e F o r m a l d e f i n i t i o n s s u b s e c t i o n
2 . 1
G r o u n d t e r m
2 . 2
G r o u n d a t o m
2 . 3
G r o u n d f o r m u l a
3
S e e a l s o
4
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
G r o u n d e x p r e s s i o n
4 l a n g u a g e s
● ف ا ر س ی
● I t a l i a n o
● M a g y a r
● У к р а ї н с ь к а
E d i t l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
( R e d i r e c t e d f r o m G r o u n d f o r m u l a )
Term that does not contain any variables
Syntax
Semantics
Semantics (logic)
Formal grammar
Formation rule
Well-formed formula
Automata theory
Regular expression
Production
Ground expression
Atomic formula
Predicate logic
Mathematical notation
Natural language processing
Programming language theory
Computational linguistics
Syntax analysis
Formal verification
In mathematical logic , a ground term of a formal system is a term that does not contain any variables . Similarly, a ground formula is a formula that does not contain any variables.
In first-order logic with identity with constant symbols
a
{\displaystyle a}
and
b
{\displaystyle b}
, the sentence
Q
(
a
)
∨
P
(
b
)
{\displaystyle Q(a )\lor P(b )}
is a ground formula. A ground expression is a ground term or ground formula.
Examples [ edit ]
Consider the following expressions in first order logic over a signature containing the constant symbols
0
{\displaystyle 0}
and
1
{\displaystyle 1}
for the numbers 0 and 1, respectively, a unary function symbol
s
{\displaystyle s}
for the successor function and a binary function symbol
+
{\displaystyle +}
for addition.
s
(
0
)
,
s
(
s
(
0
)
)
,
s
(
s
(
s
(
0
)
)
)
,
…
{\displaystyle s(0),s(s(0)),s(s(s(0))),\ldots }
are ground terms;
0
+
1
,
0
+
1
+
1
,
…
{\displaystyle 0+1,\;0+1+1,\ldots }
are ground terms;
0
+
s
(
0
)
,
s
(
0
)
+
s
(
0
)
,
s
(
0
)
+
s
(
s
(
0
)
)
+
0
{\displaystyle 0+s(0),\;s(0)+s(0),\;s(0)+s(s(0))+0}
are ground terms;
x
+
s
(
1
)
{\displaystyle x+s(1 )}
and
s
(
x
)
{\displaystyle s(x )}
are terms, but not ground terms;
s
(
0
)
=
1
{\displaystyle s(0)=1}
and
0
+
0
=
0
{\displaystyle 0+0=0}
are ground formulae.
Formal definitions [ edit ]
What follows is a formal definition for first-order languages . Let a first-order language be given, with
C
{\displaystyle C}
the set of constant symbols,
F
{\displaystyle F}
the set of functional operators, and
P
{\displaystyle P}
the set of predicate symbols .
Ground term [ edit ]
A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):
Elements of
C
{\displaystyle C}
are ground terms;
If
f
∈
F
{\displaystyle f\in F}
is an
n
{\displaystyle n}
-ary function symbol and
α
1
,
α
2
,
…
,
α
n
{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}
are ground terms, then
f
(
α
1
,
α
2
,
…
,
α
n
)
{\displaystyle f\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)}
is a ground term.
Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms).
Roughly speaking, the Herbrand universe is the set of all ground terms.
Ground atom [ edit ]
A ground predicate , ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.
If
p
∈
P
{\displaystyle p\in P}
is an
n
{\displaystyle n}
-ary predicate symbol and
α
1
,
α
2
,
…
,
α
n
{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}
are ground terms, then
p
(
α
1
,
α
2
,
…
,
α
n
)
{\displaystyle p\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)}
is a ground predicate or ground atom.
Roughly speaking, the Herbrand base is the set of all ground atoms,[1] while a Herbrand interpretation assigns a truth value to each ground atom in the base.
Ground formula [ edit ]
A ground formula or ground clause is a formula without variables.
Ground formulas may be defined by syntactic recursion as follows:
A ground atom is a ground formula.
If
φ
{\displaystyle \varphi }
and
ψ
{\displaystyle \psi }
are ground formulas, then
¬
φ
{\displaystyle \lnot \varphi }
,
φ
∨
ψ
{\displaystyle \varphi \lor \psi }
, and
φ
∧
ψ
{\displaystyle \varphi \land \psi }
are ground formulas.
Ground formulas are a particular kind of closed formulas .
See also [ edit ]
Open formula – formula that contains at least one free variablePages displaying wikidata descriptions as a fallback
Sentence (mathematical logic) – in mathematical logic, a well-formed formula with no free variablesPages displaying wikidata descriptions as a fallback
References [ edit ]
Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G. (eds.), Handbook of discrete and combinatorial mathematics , p. 68
Hodges, Wilfrid (1997), A shorter model theory , Cambridge University Press , ISBN 978-0-521-58713-6
First-Order Logic: Syntax and Semantics
Cardinality
First-order logic
Formal proof
Formal semantics
Foundations of mathematics
Information theory
Lemma
Logical consequence
Model
Theorem
Theory
Type theory
Tarski's undefinability
Banach–Tarski paradox
Cantor's theorem, paradox and diagonal argument
Compactness
Halting problem
Lindström's
Löwenheim–Skolem
Russell's paradox
Proposition
Inference
Logical equivalence
Consistency
Argument
Soundness
Validity
Syllogism
Square of opposition
Venn diagram
Boolean functions
Logical connectives
Propositional calculus
Propositional formula
Truth tables
Many-valued logic
Second-order
Higher-order
Fixed-point
Free
Quantifiers
Predicate
Monadic predicate calculus
(Ur- )Element
Ordinal number
Extensionality
Forcing
Relation
Set operations:
Uncountable
Empty
Inhabited
Singleton
Finite
Infinite
Transitive
Ultrafilter
Recursive
Fuzzy
Universal
Universe
codomain
image
In /Sur /Bi -jection
Schröder–Bernstein theorem
Isomorphism
Gödel numbering
Enumeration
Large cardinal
Aleph number
Operation
continuum hypothesis
General
Kripke–Platek
Morse–Kelley
Naive
New Foundations
Tarski–Grothendieck
Von Neumann–Bernays–Gödel
Ackermann
Constructive
Expression
Extension
Relation
Formation rule
Grammar
Formula
Free/bound variable
Language
Metalanguage
Logical connective
Predicate
Proof
Quantifier
Sentence
Signature
String
Substitution
Symbol
Term
Theory
second-order
elementary function
primitive recursive
Robinson
Skolem
of the real numbers
of Boolean algebras
of geometry :
Natural deduction
Logical consequence
Rule of inference
Sequent calculus
Theorem
Systems
Complete theory
Independence (from ZFC )
Proof of impossibility
Ordinal analysis
Reverse mathematics
Self-verifying theories
of models
Model
Non-standard model
Diagram
Categorical theory
Model complete theory
Satisfiability
Semantics of logic
Strength
Theories of truth
T-schema
Transfer principle
Truth predicate
Truth value
Type
Ultraproduct
Validity
Church–Turing thesis
Computably enumerable
Computable function
Computable set
Decision problem
Kolmogorov complexity
Lambda calculus
Primitive recursive function
Recursion
Recursive set
Turing machine
Type theory
Algebraic logic
Automated theorem proving
Category theory
Concrete /Abstract category
Category of sets
History of logic
History of mathematical logic
Logicism
Mathematical object
Philosophy of mathematics
Supertask
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Ground_expression&oldid=1215169678#Ground_formula "
C a t e g o r i e s :
● L o g i c a l e x p r e s s i o n s
● M a t h e m a t i c a l l o g i c
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n i s d i f f e r e n t f r o m W i k i d a t a
● P a g e s d i s p l a y i n g w i k i d a t a d e s c r i p t i o n s a s a f a l l b a c k v i a M o d u l e : A n n o t a t e d l i n k
● P a g e s t h a t u s e a d e p r e c a t e d f o r m a t o f t h e m a t h t a g s
● T h i s p a g e w a s l a s t e d i t e d o n 2 3 M a r c h 2 0 2 4 , a t 1 5 : 1 1 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w