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What (who) is first-order - definition
first-order
WIKIMEDIA DISAMBIGUATION PAGE
First order; First-order (disambiguation)
Not higher-order.
(1995-03-06)
first-order
WIKIMEDIA DISAMBIGUATION PAGE
First order; First-order (disambiguation)
¦ adjective
1. most fundamental; primary.
2. denoting mathematical equations involving only the first power of the independent variable or only the first derivative of a function.
first-order logic
COLLECTION OF FORMAL SYSTEMS USED IN MATHEMATICS, PHILOSOPHY, LINGUISTICS, AND COMPUTER SCIENCE
First-order predicate calculus; First-order predicate logic; Predicate logic; Predicate calculus; First order logic; Predicate Calculus; First Order Logic; First order language; First-order language; Quantification theory; First order predicate calculus; Predicate logic (Philosophy); First order logic with equality; 1st order logic; First Order Language; FOPL; First order predicate logic; Polyadic predicate calculus; Predicate logic (philosophy); First-order logic with equality; First-Order Logic; First-order sentence; Quantification calculus; Satisfaction relation; Predicate Logic; Many-sorted first-order logic; First-order Peano arithmetic; FOPC; Lower Predicate Calculus; Tarskian semantics; Classical predicate logic; First-order-logic; Equational first-order logic; Semantics of first-order logic; Deductive systems for first-order logic
<language, logic> The language describing the truth of
mathematical formulas. Formulas describe properties of
terms and have a truth value. The following are atomic
formulas:
True
False
p(t1,..tn) where t1,..,tn are terms and p is a predicate.
If F1, F2 and F3 are formulas and v is a variable then the
following are compound formulas:
F1 ^ F2 conjunction - true if both F1 and F2 are true,
F1 V F2 disjunction - true if either or both are true,
F1 => F2 implication - true if F1 is false or F2 is
true, F1 is the antecedent, F2 is the
consequent (sometimes written with a thin
arrow),
F1 <= F2 true if F1 is true or F2 is false,
F1 == F2 true if F1 and F2 are both true or both false
(normally written with a three line
equivalence symbol)
first-order logicF1 negation - true if f1 is false (normally
written as a dash '-' with a shorter vertical
line hanging from its right hand end).
For all v . F universal quantification - true if F is true
for all values of v (normally written with an
inverted A).
Exists v . F existential quantification - true if there
exists some value of v for which F is true.
(Normally written with a reversed E).
The operators ^ V => <= == first-order logic are called connectives. "For
all" and "Exists" are quantifiers whose scope is F. A
term is a mathematical expression involving numbers,
operators, functions and variables.
The "order" of a logic specifies what entities "For all" and
"Exists" may quantify over. First-order logic can only
quantify over sets of atomic propositions. (E.g. For all p
. p => p). Second-order logic can quantify over functions on
propositions, and higher-order logic can quantify over any
type of entity. The sets over which quantifiers operate are
usually implicit but can be deduced from well-formedness
constraints.
In first-order logic quantifiers always range over ALL the
elements of the domain of discourse. By contrast,
second-order logic allows one to quantify over subsets.
["The Realm of First-Order Logic", Jon Barwise, Handbook of
Mathematical Logic (Barwise, ed., North Holland, NYC, 1977) ].
(2005-12-27)
Wikipedia
First-order
In mathematics and other formal sciences, first-order or first order most often means either:
Examples of use of first-order
1. This is political and diplomatic incompetence of the first order.
2. He‘s a dear old–fashioned chauvinist of the first order.
3. Iraq presents a policy quandary of the first order.
4. The first order of business was stoking a water pipe.
5. "The first order of Bill‘s care has been to stop the bleeding and save his life.