Interpretation
المؤلف:
Chang, C.-L. and Lee, R. C.-T
المصدر:
Symbolic Logic and Mechanical Theorem Proving. New York: Academic Press, 1997.
الجزء والصفحة:
...
30-1-2022
1338
Interpretation
An interpretation of first-order logic consists of a non-empty domain 
and mappings for function and predicate symbols. Every 
-place function symbol is mapped to a function from 
to 
, and every 
-place predicate symbol is mapped to a function from 
to the set comprised of two values true and false.
The domain 
is the range of all variables in formulas of first-order logic, and is called the domain of the interpretation.
For a given interpretation, the truth table of any formula is defined by the following rules.
1. The truth tables for propositional connectives apply to evaluate the value of 
(
AND 
), 
(
OR 
), 
(
implies 
), and 
(NOT 
).
2. 
("for all 
, 
") is true if 
is true for any element of 
as value of 
at free occurrences of 
in 
. Otherwise, 
is false.
3. 
("there exists an 
such that 
") is true if 
is true for at least one element of 
as value of 
at free occurrences of 
in 
. Otherwise, 
is false.
Truth tables for infinite domains of interpretation are infinite. The formulas of first-order logic that are tautologies in any interpretation are called valid formulas. A formula is called satisfiable if it takes at least one true value in some interpretation. A formula whose truth table contains only false in any interpretation is called unsatisfiable.
The Löwenheim-Skolem theorem establishes that any satisfiable formula of first-order logic is satisfiable in an 
(aleph-0) domain of interpretation. Hence, aleph-0 domains are sufficient for interpretation of first-order logic.
REFERENCES
Chang, C.-L. and Lee, R. C.-T. Symbolic Logic and Mechanical Theorem Proving. New York: Academic Press, 1997.
Kleene, S. C. Mathematical Logic. New York: Dover, 2002.
Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, pp. 12 and 57, 1997.
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