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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)

~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 of M.

["The Realm of First-Order Logic", Jon Barwise, Handbook of

Mathematical Logic (Barwise, ed., North Holland, NYC, 1977)].