Identical truth
logical truth, tautology
A property of formulas in the language of predicate calculus, meaning that the formulas are true in all interpretations and for all admissible values of their free variables. For example, for a formula containing only one -place predicate symbol
and variables of one sort (that is, variables which are interpreted in the same domain of variation), any pair
, where
is an arbitrary non-empty set and
is an arbitrary binary relation on
, is an interpretation. Arbitrary elements of
are admissible values for the free variables. Truth of a formula
at values
(
) of the variables
, respectively, is defined by induction on the structure of the formula, as follows. (Here the free variables run through the set
and the predicate symbol
denotes the relation
.)
Suppose that a formula is given, as well as a finite sequence
of variables containing all the free variables of
; let
denote the set of all finite sequences
of elements of
at which
is true in
. A set of the form
can be constructed inductively as follows (here it is assumed that the logical symbols in
are
,
,
):
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if has the form
;
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where ,
,
denote, respectively, intersection, difference and projection along the
-st coordinate (that is, the image with respect to the mapping
) of sets.
Identical truth for a formula with free variables
then means that for any interpretation
, every sequence
of elements of
belongs to the set
. For
the set
is either empty or a singleton. For example, the formula
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is an identical truth. The converse implication is not an identically-true formula.
In the case where an interpretation is fixed, a formula is sometimes called identically true if it is true in the given interpretation for any values of its free variables.
References
[1] | S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951) |
[2] | J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967) |
Identical truth. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Identical_truth&oldid=16212