4. Deduction theorem: Γ ∪ {B } ⊢Σ A if and only if Γ ⊢Σ B →A;
5. Γ ⊢Σ A1 and . . . and Γ ⊢Σ An and A1 → (A2→ · · · (An →
B ) · · · ) is a tautological instance, then Γ ⊢Σ B .
The proof is an easy exercise. Part (5) of Proposition 3.36
gives us that, for instance, if Γ ⊢Σ A ∨ B and Γ ⊢Σ ¬A, then
Γ ⊢Σ B . Also, in what follows, we write Γ, A ⊢Σ B instead of Γ ∪ {A} ⊢Σ B .
Definition 3.37. A set Γ is deductively closed relatively to a sys- tem Σ if and only if Γ ⊢Σ A implies A ∈ Γ .
3.13
Consistency
Consistency is an important property of sets of formulas. A set of formulas is inconsistent if a contradiction, such as ⊥, is deriv- able from it; and otherwise consistent. If a set is inconsistent, its formulas cannot all be true in a model at a world. For the com- pleteness theorem we prove the converse: every consistent set is true at a world in a model, namely in the “canonical model.”
Definition 3.38. A set Γ is consistent relatively to a system Σ or, as we will say, Σ -consistent, if and only if Γ ⊬Σ ⊥.
So for instance, the set {□(p → q ), □p, ¬□q } is consistent rel- atively to propositional logic, but not K-consistent. Similarly, the set {♢p, □♢p → q, ¬q } is not K5-consistent.
Proposition 3.39. Let Γ be a set of formulas. Then:
1. A set Γ is Σ -consistent if and only if there is some formula A such that Γ ⊬Σ A.
60 CHAPTER 3. AXIOMATIC DERIVATIONS
3. If Γ is Σ -consistent, then for any formula A, either Γ ∪ {A} is Σ -consistent or Γ ∪ {¬A} is Σ -consistent.
Proof. These facts follow easily using classical propositional logic. We give the argument for(3). Proceed contrapositively and sup- pose neither Γ ∪ {A} nor Γ ∪ {¬A} is Σ -consistent. Then by
(2), both Γ, A ⊢Σ ⊥ and Γ, ¬A ⊢Σ ⊥. By the deduction theorem
Γ ⊢Σ A → ⊥ and Γ ⊢Σ ¬A → ⊥. But (A → ⊥) → ((¬A → ⊥) → ⊥) is
a tautological instance, hence byProposition 3.36(5), Γ ⊢Σ ⊥. □
Problems
Problem 3.1. ProveProposition 3.7.
Problem 3.2. Find derivations in K for the following formulas:
1. □¬p → □(p → q ) 2. (□p ∨ □q ) → □(p ∨ q ) 3. ♢p → ♢(p ∨ q )
Problem 3.3. Prove Proposition 3.19 by proving, by induction on the complexity of C , that if K ⊢ A ↔ B then K ⊢ C [A/q ] ↔ C [B /q ].
Problem 3.4. Show that the following derivability claims hold:
1. K ⊢ ♢¬⊥ → (□A → ♢A); 2. K ⊢ □(A ∨ B) → (♢A ∨ □B); 3. K ⊢ (♢A → □B) → □(A → B).
Problem 3.5. Show that for each formula A in Definition 3.26: K ⊢ A ↔ A♢.
61 3.13. CONSISTENCY
Problem 3.7. Give an alternative proof ofTheorem 3.34using a model with 3 worlds.
Problem 3.8. Provide a single reflexive transitive model showing that both KT4 ⊬ B and KT4 ⊬ 5.
CHAPTER 4
Completeness
and Canonical
Models
4.1
Introduction
If Σ is a modal system, then the soundness theorem establishes that if Σ ⊢ A, then A is valid in any class C of models in which all instances of all formulas in Σ are valid. In particular that means that if K ⊢ A then A is true in all models; if KT ⊢ A then A is true in all reflexive models; if KD ⊢ A then A is true in all serial models, etc.
Completeness is the converse of soundness: that K is com- plete means that if a formula A is valid, ⊢ A, for instance. Prov- ing completeness is a lot harder to do than proving soundness. It is useful, first, to consider the contrapositive: K is complete iff whenever ⊬ A, there is a countermodel, i.e., a model M such that
M ⊮ A. Equivalently (negating A), we could prove that whenever
⊬ ¬A, there is a model of A. In the construction of such a model, we can use information contained in A. When we find models for specific formulas we often do the same: E.g., if we want to
63 4.1. INTRODUCTION
find a countermodel to p → □q , we know that it has to contain a world where p is true and □q is false. And a world where □q is false means there has to be a world accessible from it where q is false. And that’s all we need to know: which worlds make the propositional variables true, and which worlds are accessible from which worlds.
In the case of proving completeness, however, we don’t have a specific formula A for which we are constructing a model. We want to establish that a model exists for every A such that ⊬Σ ¬A.
This is a minimal requirement, since if ⊢Σ ¬A, by soundness, there is no model for A (in which Σ is true). Now note that ⊬Σ ¬A iff A is Σ -consistent. (Recall that Σ ⊬Σ ¬A and A ⊬Σ ⊥
are equivalent.) So our task is to construct a model for every Σ -consistent formula.
The trick we’ll use is to find a Σ -consistent set of formulas that contains A, but also other formulas which tell us what the world that makes A true has to look like. Such sets are complete Σ - consistent sets. It’s not enough to construct a model with a single world to make A true, it will have to contain multiple worlds and an accessibility relation. The complete Σ -consistent set contain- ing A will also contain other formulas of the form □B and ♢C . In all accessible worlds, B has to be true; in at least one, C has to be true. In order to accomplish this, we’ll simply take all possible complete Σ -consistent sets as the basis for the set of worlds. A tricky part will be to figure out when a complete Σ -consistent set should count as being accessible from another in our model.
We’ll show that in the model so defined, A is true at a world— which is also a complete Σ -consistent set—iff A is an element of that set. If A is Σ -consistent, it will be an element of at least one complete Σ -consistent set (a fact we’ll prove), and so there will be a world where A is true. So we will have a single model where every Σ -consistent formula A is true at some world. This single model is the canonical model for Σ .
64 CHAPTER 4. COMPLETENESS AND CANONICAL MODELS