Tools and Techniques in Modal Logic Marcus Kracht II ...

1.4. General Logic 21
the relation Σ ⊢Cn ϕ defined by Σ ⊢Cn ϕ iff ϕ ∈ Cn(Σ) is a consequence relation, that
is to say, 〈L, ⊢Cn〉 is a logic. Moreover, two distinct consequence operations determine
distinct consequence relations and distinct consequence relations give rise to
distinct consequence operations.
Finally, call any set of the form Σ ⊢ a theory of ⊢. By (trs.), theories are closed
under consequence, that is, Σ ⊢ not only contains all consequences of Σ but also all
of its own consequences as well. We may therefore say that the theories of ⊢ coincide
with the deductively closed sets. Given ⊢, the following can be said about
⊢–theories.
(top.) L is a ⊢–theory.
(int.) If Ti, i ∈ I, are ⊢–theories, so is � 〈Ti : i ∈ I〉.
(sub.) If T is a ⊢–theory, σ a substitution then
σ −1 [T] = {ϕ : ϕ σ ∈ T} is a ⊢–theory.
(cmp.) If Ti, i ∈ I, is an ascending chain of ⊢–theories then
� 〈Ti : i ∈ I〉 is a ⊢–theory.
Again, the correspondence is exact. Any collection T of subsets of L satisfying
(top.), (int.), (sub.) and (cmp.) defines a consequence operation Σ ↦→ Σ ⊢ by Σ ⊢ =
� 〈T : T ∈ T, T ⊇ Σ〉 which satisfies (ext.), (mon.), (trs.), (sub.) and (cmp.). The
correspondence is biunique. Different consequence operations yield different sets of
theories and different collections of theories yield different consequence operations.
The theories of a logic form a lattice. This lattice is algebraic if the consequence
relation is finitary. The converse does not hold; this has been shown by Burghard
Herrmann and Frank Wolter [103].
Exercise 8. Give a detailed proof of Theorem 1.4.3.
Exercise 9. Let R be a set of axioms or 1–ary rules. Show that ∆ ⊢ R ϕ iff there exists
a δ ∈ ∆ such that δ ⊢ R ϕ.
Exercise 10. Show that every consistent logic is contained in a Post–complete logic.
Hint. You need Zorn’s Lemma here. For readers unfamiliar with it, we will prove
later Tukey’s Lemma, which will give rise to a very short proof for finitary logics.
Exercise 11. Show that in 2–valued logic
ϕ1 ↔ ψ1; ϕ2 ↔ ψ2 ⊢ ϕ1 ∧ ϕ2 ↔ ψ1 ∧ ψ2
ϕ1 ↔ ψ1; ϕ2 ↔ ψ2 ⊢ ϕ1 ∨ ϕ2 ↔ ψ1 ∨ ψ2
ϕ ↔ ψ ⊢ ¬ϕ ↔ ¬ψ
Thus if ϕ ≡ ψ is defined by ⊢ ϕ ↔ ψ, then ≡ is a congruence relation. What is the
cardinality of a congruence class? Hint. We assume that we have ℵ0 many propositional
variables. Show that all congruence classes must have equal cardinality.