k The next theorem shows that membership of a given maximal consistent theory has the same properties as a truth assignment just replace membership in the theory with truth in each of th
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The next theorem shows that membership of a given maximal consistent theory has the same properties as a truth assignment (just replace membership in the theory with truth in each of the clauses below.)
Theorem 71 (Maximal consistent theory 3) For every maximal D-consistent theory Γ
and formulae A, B, the following hold:
1 ¬ A ∈ Γ iff A / ∈ Γ.
2 A ∧ B ∈ Γ iff A ∈ Γ and B ∈ Γ.
3 A ∨ B ∈ Γ iff A ∈ Γ or B ∈ Γ.
4 A → B ∈ Γ iff A ∈ Γ implies B ∈ Γ (i.e., A / ∈ Γ or B ∈ Γ).
The proof is specific to each deductive system as it uses its specific deductive machinery
Each proof is left as an exercise
Given a theory Γ, consider the following truth assignment:
SΓ(p) :=
T, if p ∈ Γ;
F, otherwise.for every propositional variablep.
The truth assignmentSΓextends to a truth valuation of every formula by applying a recursive definition according to the truth tables (see Section 1.4.5.2) The truth valuation
is denotedSΓ
Lemma 72 (Truth Lemma) If Γ is a maximal D-consistent theory, then for every
for-mula A, SΓ(A ) = T iff A ∈ Γ.
Proof Exercise Use Theorem 71.
Corollary 73 Every maximal D-consistent theory is satisfiable.
Lemma 74 (Lindenbaum’s Lemma) Every D-consistent theory Γ can be extended to
a maximal D-consistent theory.
Proof Let A0, A1, be a list of all propositional formulae (NB: they are countably many,
so we can list them in a sequence.) I will define a chain by inclusion of theories Γ0 ⊆ Γ1 ⊆
defined by recursion onnas follows:
• Γ0:= Γ;
• Γ n+1:=
Γn ∪ { A n } , if Γn ∪ { A n } is D-consistent;
Γn ∪ {¬ A n } , otherwise
Note that every Γn is an D-consistent theory Prove this by induction onn, using the properties of the deductive consequence from Proposition 63
Now, we define
Γ∗:=
n∈N
Γn
Clearly, Γ⊆ Γ ∗ Γ∗is a maximal D-consistent theory Indeed: