canonical model (original) (raw)

The canonical model of Λ based on ℱΛ is the pair MΛ:=(ℱΛ,VΛ), where

The main result regarding the canonical model of Λ is:

Theorem 1.

MΛ⊧A iff Λ⊢A, where A is any wff.

Since the logic Λ is the intersectionMathworldPlanetmath of all maximally consistent sets (see here (http://planetmath.org/LindenbaumsLemma)), the theorem is the result of the following:

Proposition 1.

MΛ⊧wA iff A∈w.

which is the result of the following:

Lemma 1.

For any world w in MΛ, □⁢A∈w iff A∈u for all worlds u such that w⁢RΛ⁢u.

Proof.

Suppose □⁢A∈w and w⁢RΛ⁢u. Then A∈u by the definition of RΛ. Conversely, suppose A∈u for all u such that w⁢RΛ⁢u. In other words, A∈u for all u such that Δw⊆u, or A∈⋂{u∣Δw⊆u}. But ⋂{u∣Δw⊆u}=Ded⁢(Δw), the deductive closure of Δw, so Δw⊢A, and therefore □⁢Δw⊢□⁢A (see here (http://planetmath.org/SyntacticPropertiesOfANormalModalLogic)), or w⊢□⁢A (since Δw⊆w), or □⁢A∈w (since w is maximally consistent). ∎

Proof of PropositionPlanetmathPlanetmath 1. We do inductionMathworldPlanetmath on the number n of logical connectives in A. If n=0, then A is either a propositional variable or ⟂. The former is just the definition of VΛ. The later case is just the definition of Λ-consistency. Next, if A is B→C, then MΛ⊧wA iff either MΛ⊧̸wB or MΛ⊧wC iff B∉w or C∈w iff ¬⁢B∈w or C∈w iff ¬⁢B∨C∈w iff A∈w. Finally, if A is □⁢B, then MΛ⊧w□⁢B iff □⁢B∈w iff B∈u for all u such that w⁢RΛ⁢u iff MΛ⊧uB for all u such that w⁢RΛ⁢u.

Recall that a logic is completePlanetmathPlanetmathPlanetmathPlanetmath in a frame if it is complete in every model based on the frame. As a corollary to Theorem 1, we have

Corollary 1.

Λ is complete in its canonical frame FΛ.

Proof.

Any wff valid in every model based on ℱΛ is valid in MΛ in particular, and therefore a theorem of Λ by Theorem 1. ∎

The converse is not true. There are in fact normal modal logics that are sound in no frames at all.

Canonical models are useful in proving the completeness theorems for many common normal modal logics. To prove that a logic is complete in a class of frames, by the corollary above, it is enough to show that the canonical frame is in the class. Here are two examples:

    1. Let Λ be the smallest normal logic containing the schema □⁢A. Then Λ is complete in the class of null frames.
    1. Let Λ be the smallest normal logic containing the schema A→□⁢A. Then Λ is complete in the class of weak identityPlanetmathPlanetmathPlanetmathPlanetmath frames (a binary relationMathworldPlanetmath R is weak identity it is satisfies the condition ∀x∀y(xRy→x=y)).
Proof.

Again, we show that RΛ is weak identity. Suppose w⁢RΛ⁢u. Then for any A, □⁢A∈w implies that A∈u. Now, if A∈w, then applying modus ponens to A→□⁢A, we get that □⁢A∈w since w is closed under modus ponens. But this means that A∈u. So w⊆u. But since both w and u are maximal, they are the same. ∎