Tableau Calculi for the Logics of Finite k-Ary Trees
Summary (1 min read)
1 Introduction
- In recent years there has been a growing interest (see [1,3,4,6,7,8,10,11]) in prooftheoretical characterization of propositional intermediate logics, that is logics laying between Intuitionistic and Classical Logic.
- Also the approach based on hypersequent calculi or hypertableau calculi (the dualized version of hypersequents presented in [6]) seems to be inadequate to treat some Intermediate Logics and further variations are needed.
- The authors proof-theoretical characterization is based on a hybrid tableau calculus that uses the two usual signs T and F, some tableau rules for Intuitionistic Logic and two rules formulated in a hypertableau fashion (a structural rule and a purely logical rule).
- In Section 3 the authors introduce the calculi TDk and they prove that they characterize the logics Dk. Finally, in Section 4 they use these calculi to prove the main properties of the logics Dk.
2 Preliminaries
- Here the authors consider the propositional language based on a denumerable set of atomic symbols and the logical constants ⊥,∧,∨,→.
- The above quoted properties of the logics.
- On the other hand, Weak has a non simple configuration as premise and a simple configuration as consequence; on the contrary, as the authors will see, the rule Dk properly characterizing TDk has a simple configuration as premise and (in general) a non simple configuration as consequence.
- A formula A is provable in TDk if there exists a closed TDkproof table for the configuration {FA} (the configuration consisting of the set {FA} only).
- Let us suppose that the assertion holds for every H ′ such that dg(H ′) < dg(H).
4 Properties of Dk
- The resulting calculus TD′k is trivially valid for Dk.
- The authors have chosen to present the main calculus for TDk with the rules F∨1 and F∨2 since they allow us to get an immediate and syntactical constructivity proof for Dk. Theorem 4. Proof.
- Dk extracted from the completeness theorem for their tableau calculus.
- The authors remark that lines 1-12 implement the cases N1−N8 of Section 3 and case N9 above of the construction of the node set, while lines 13-15 implement the construction of the successor sets (case S2 of Section 3).
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Frequently Asked Questions (12)
Q2. What is the main property of the logics Dk?
An intermediate propositional logic (see, e.g., [5]) is any set L of formulas satisfying the following conditions: (i) L is consistent; (ii) Int ⊆ L; (iii) L is closed under modus ponens; (iv) L is closed under propositional substitution (where a propositional substitution is any function mapping every propositional variable to a formula).
Q3. What is the definition of the structure KD(S)?
Then ∆ is a member of P and ∆ is an immediate successor of Γ in KD(S); 3. ≤ is the transitive and reflexive closure of the immediate successor relation; 4. For every atom p and for every Γ ∈ P , Γ p iff Tp ∈ Γ .
Q4. How many hypersets are there in the consequence of the rule?
Dk is intrinsically inefficient, indeed the number of hypersets in the consequence of the rule is exponential in the number of F-signed implicative formulas occurring in the premise.
Q5. What is the main set of swff’s of the rule?
The rule properly characterizing the calculus TDk is Dk that applies to the premiseS,F(A1 → B1), . . . ,F(An → Bn),T((C1 → D1) → E1), . . . ,T((Cm → Dm) → Em)Let U = {F(A1 → B1), . . . ,F(An → Bn)} V = {T((C1 → D1) → E1), . . . ,T((Cm → Dm) → Em)}and let U ∪ V be the main set of swff’s of the rule.
Q6. What is the force relation for every atomic symbol?
A (propositional) Kripke model (see, e.g., [5]) is a structure K = 〈P,≤, 〉, where 〈P,≤〉 is a poset (partially ordered set), and (the forcing relation) is a binary relation between elements of P and atomic symbols such that, for every atomic symbol p, α p implies β p for every β ∈
Q7. What is the meaning of the signs T and F?
The meaning of the signs T and F is as follows: given a Kripke model K = 〈P,≤, 〉 and a swff H, α ∈ P realizes H (in symbols α✄H) if H ≡ TX and α X, or H ≡ FX and α X. α H means that α✄H does not hold.
Q8. What is the definition of a constructive intermediate logic?
c© Springer-Verlag Berlin Heidelberg 2002we call constructive any intermediate logic L satisfying the disjunction property : A ∨ B ∈ L implies A ∈ L or B ∈ L.
Q9. What is the axiom schema of the Kripke models?
let Tk be the the family of all the Kripke models K = 〈P,≤, 〉 where: – 〈P,≤〉 is a finite tree; – Given α ∈ P , α has at most k immediate successors in 〈P,≤〉.
Q10. What is the construction of the structure KD(S)?
Given a finite and TDk-consistent set S of swff’s, the authors use the construction above to define the structure KD(S) = 〈P,≤, 〉 as follows: 1. S ∈ P , where S is a node set of S; 2.
Q11. What is the decidability of a tableau calculus?
In the following sections the authors introduce a tableau calculus for every TDk and then the authors use the properties of such a calculus to deduce the decidability and the disjunction property for Dk.3 The Sequence of Tableau Calculi TDk (k ≥ 2) A signed formula (swff for short) is an expression of the form TX or FX where X is any formula.
Q12. what is the swff of the type F(AB)?
(S2) If S does not contain swff’s of the kind F(A∨B) and contains at least one swff of the kind F(A→ B), then let:U = {F(A → B) | F(A → B) ∈ S} V = {T((C → D) → E) | T((C → D) → E) ∈