Hybrid Logic and its Proof-Theory by Torben Braüner

By Torben Braüner

This is the 1st book-length therapy of hybrid common sense and its proof-theory. Hybrid common sense is an extension of normal modal common sense which permits specific connection with person issues in a version (where the issues symbolize instances, attainable worlds, states in a working laptop or computer, or whatever else). this is often valuable for lots of purposes, for instance while reasoning approximately time one usually desires to formulate a chain of statements approximately what occurs at particular occasions. there's little consensus approximately proof-theory for traditional modal good judgment. Many modal-logical evidence structures lack very important houses and the relationships among facts structures for various modal logics are frequently uncertain. within the current booklet we display that hybrid-logical proof-theory treatments those deficiencies by way of giving a spectrum of well-behaved evidence structures (natural deduction, Gentzen, tableau, and axiom platforms) for a spectrum of alternative hybrid logics (propositional, first-order, intensional first-order, and intuitionistic).

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245) The method of reasoning in natural deduction systems is called “forwards” reasoning: When you want to find a derivation of a certain formula you start with the rules and try to build a derivation of the formula you have in mind. This is contrary to T. V. 2011 21 22 2 Proof-Theory of Propositional Hybrid Logic tableau systems which are backward reasoning systems since you explicitly start with a particular formula and try to build a proof of it using tableau rules, cf. 1. A derivation in a natural deduction system has the form of a finite tree where the nodes are labelled with formulas such that for any formula occurrence φ in the derivation, either φ is a leaf of the derivation or the immediate successors of φ in the derivation are the premises of a rule-instance which has φ as the conclusion.

2. Below we give some examples of derivations in the system NH . The end-formula of the first example derivation is the standard modal axiom K prefixed by a satisfaction operator (it is assumed that the nominal a in the derivation is new). @b (φ → ψ )3 @b ♦a1 @a (φ → ψ ) @b φ 2 ( E) @a ψ @b ψ @b ♦a1 @a φ ( (→ E) I)1 @b ( φ → ψ ) (→ I)2 @b ( (φ → ψ ) → ( φ → ψ )) (→ I)3 ( E) 26 2 Proof-Theory of Propositional Hybrid Logic @a φ @a ψ @a (φ ∧ ψ ) [@a φ ] · · · @a ψ @a (φ ∧ ψ ) (∧I) @a φ @a φ @a ⊥ @c @a φ @a φ @a φ @a φ [c/b] @a ∀bφ [@a c] · · · @c φ [c/b] @a ♦e @a ∀bφ (∀I)† @a φ [e/b] @a ↓ bφ ( E) (∀E) @a e @e φ [e/b] (↓ I)‡ (→ E) (@E) @e φ ( I) (∧E2) (⊥2) @c ⊥ (@I) @c @a φ @a φ @a ψ (→ I) [@a ¬φ ] · · · @a ⊥ (⊥1)∗ @a φ @a φ @a ψ @a (φ → ψ ) @a (φ → ψ ) [@a ♦c] · · · @c φ @a (φ ∧ ψ ) (∧E1) (↓ E) @a ↓ bφ ∗ φ is a propositional symbol (ordinary or a nominal).

Note in the proposition above that φ can be any formula; not just a propositional symbol. Thus, the rule (⊥) generalizes the rule (⊥1) whereas (Nom) generalizes (Nom1) (and the rule (Nom2) as well). 4). In the case with (⊥1), it is well-known from the literature that the subformula property does not hold without the side-condition, cf. Prawitz (1965, 1971). We shall return to the subformula property later. 3 Soundness and Completeness The aim of this section is to prove soundness and completeness of the natural deduction system for propositional hybrid logic.

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