Manual Interpolation and Definability: Modal and Intuitionistic Logics

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Email Address. Library Card. Gabbay L. Section 7 contains the construction of the pair of models proving non-interpolation, and a detailed proof of the failure of interpolation for CD. Model theory.

Interpolation and Definability in Modal Logics: Modal and Intuitionistic Logic

The atomic formulas are of the form P x1 ,. We use the notation A[x1 ,. We employ the notation L P, Q for the sublanguage of L in which the only predicate symbols are the one-place predicates P and Q. We shall use the second-order semantics only in the one-quantifier form. The counterexample. In this section, we produce the counterexample used in refuting the interpolation theorem for CD.

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Lemma 4. We give two proofs, the first model-theoretic, the second a deductive proof in a sequent calculus. First, we give a model-theoretic argument for the Lemma. However, we can argue directly for this in a multiple succedent sequent calculus formulation of CD see, e. Bm for sequents. It is possible to shed some further light on Lemma 4. In the following Lemma, we extract the content of these two sentences in the form of semantical conditions on G-models.

Let M be a G-model with base point v. Conversely, assume that M satisfies I P, Q.

By assumption, E is non-empty. Assume that M satisfies J P, Q. For the converse, let us assume that M does not satisfy J P, Q. Let u be an arbitrary state accessible from w. Existence of a modal interpolant. Part 1. Theorem 5. It is for this reason that we are forced to introduce a new concept, that of asimulation, specifically adapted to the context of intuitionistic predicate logic.

In fact, this new idea is the main conceptual innovation of this paper.

Larisa Maksimova on Implication, Interpolation, and Definability | Sergei Odintsov | Springer

In this section, we introduce the basic concept of CD- asimulation, an asymmetric counterpart of the concept of bisimulation familiar from the literature of modal logic [3, Chapter 2]. The definition given below can be considered as a more general version of the notion of bisimulation for modal predicate logic defined by Johan van Benthem [20].

Definition 6. The concept of asimulation is due to the second author of this paper. The version we are using here is a simplified version of the general notion; the sim- plifications are possible because we are operating in the context of the constant domain semantics. In the present case, however, we do not need the full characterization theorems; the following Lemma is sufficient for our purpose of refuting the interpolation theorem for the logic CD.

Lemma 6. For atomic formulas, the lemma holds by the definition of CD-asimulation. By the fourth condition in Definition 6. Let g be an arbitrary individ- ual in Dj.

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By the fifth condition in Definition 6. One might wonder, why we employ asimulation instead of the much more standard bisimulation relation from [20] which, in our present context, could be defined in the following way: Definition 6. By Lemma 4. Then we will construct an asimulation between M1 and M2 connecting their base states and use Lemma 6. This strategy can be carried through with asimulation but it fails with bisim- ulation. Although one can prove the analogue of Lemma 6. More precisely, we can prove the following fact: Lemma 6. Assume that M1 satisfies I P, Q. Then, by Lemmas 4. By condition 3 of Definition 6.

By condition 4 of Definition 6. Since u was arbitrary, this means that M2 satisfies J P, Q as well. Another way to see this same fact comes from combination of Theorem 5. Refuting interpolation. In this section, we define two G-models for L P, Q , M1 and M2 , together with a CD-asimulation between them these definitions and the ideas of the proofs given below are due to the second author of this paper.

This will enable us to carry out the strategy outlined in the comments following Lemma 4.

The states in both models are quasi-partitions, given by the following definition. Definition 7.

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Two quasi-partitions that will serve as the base points of our two models are defined as follows. The base points for M1 and M2 are v and w respectively; 2. The set of states for the models are as follows. The ordering on both M1 and M2 is E; 4. The relation E is clearly reflexive and transitive.

Hence, both M1 and M2 are G-models.

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Lemma 7. Hence, the condition J P, Q fails in the model M2. The following notations are useful in stating the definition of Z and in demonstrating its properties. Relative to the models M1 and M2 given in Definition 7. The relation Z in Definition 7. The first condition in Definition 6. The proof for atomic formulas Q xl is similar.

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