The Free On-line Dictionary of Computing (30 December 2018):

first-order 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)

    ~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)].

   (2005-12-27)