Section 1.2: Truth Assignments
A set of truth values:
(falsity),
(truth).
A truth assignment for a set
of sentence symbols is a function
.
We further consider the extension
defined on the set
of all wffs built up from
such that
a) for a sentence symbol
,
, and
b) for wffs
,
 for any on a set , there is a unique extension on .
For a formula
containing sentence symbols from
only,
satisfies
iff
. A wff or a set of wffs is called satisfiable if there is a truth assignment that satisfies it.
(Compactness Theorem: to be proved in Section 1.7) If every finite subset of an infinite set
is satisfiable, then
is satisfiable.
A set
of wffs (hypotheses) tautologically implies a wff
(conclusion), written as
, if every truth assignment for the sentence symbols used in
and
that satisfies all wffs in
also satisfies
.
 If , the we write .
 If and , then and are tautologically equivalent.
 is said to be a tautology, written as , if .

Two sets of wffs are equivalent if one tautologically implies any formula iff the other does so too. And a set of wffs is called independent iff no member of it is tautologically implied by the remaining members of the set.
 Every finite set of wffs has an equivalent independent subset, but this is not true, in general, for an infinite set of wffs.
Some laws
 Associative and commutative laws for , , and .
 Distributive laws for and , and vice versa.
 De Morgan’s laws for the negation of and (and, in more general, the duality principle).
 The substitution principle: if a wff is a tautology, then by substituting wffs for sentence symbols we still get a tautology.

Tautologies including the conditional symbol:
 Contraposition: .
 Exportation: .
 .

Tautological implications:
 iff .