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)
~F1 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 => <= == ~ 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)].