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)].