The Gentzen systems for a sequent Γ⇒Δ have been proposed in the propositional logic, the predicate calculus and other logics. In this paper, based on the Gentzen system for the predicate calculus, we propose the Gentzen system for a sequent Γ⇒Δ in the description logic, where Γ and Δ are two sets of assertions in ALC: Assertions with universal qualification and inclusion assertions are decomposed as those of universal quantification formulas in the Gentzen system for the predicate calculus. It is proved that the Gentzen system for description logic is sound and complete; but it is not decidable.

Proof Theory

A Gentzen system for the description logic

We introduce a notion of syntactical truth predicate (s.t.p.) for the second order arithmetic PA2. An s.t.p. is a set T of closed formulas such that:(i) T(t = u) if and only if the closed first order terms t and u are convertible, i.e., have the same value in the standard interpretation(ii) T(A → B) if and only if (T(A) ⇒ T(B))(iii) T(∀xA) if and only if (T(A[x ← t]) for any closed first order term t)(iv) T(∀X A) if and only if (T(A[X ← ∆]) for any closed set definition ∆ = {x ∣ D(x)}).S.t.p.'s can be seen as a counterpart to Tarski's notion of (model-theoretical) validity and have main model properties. In particular, their existence is equivalent to the existence of an ω-model of PA2, this fact being provable in PA2 with arithmetical comprehension only.

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Syntactical Truth Predicates for Second Order Arithmetic

We present a simplified and streamlined characterisation of provably total computable functions of the system ID1 of non-iterated inductive definitions. The idea of the simplification is to employ the method of operator-controlled derivations that was originally introduced by Wilfried Buchholz and afterwards applied by the second author to a streamlined characterisation of provably total computable functions of Peano arithmetic PA.

A Simplified Characterisation of Provably Computable Functions of the System ID_1 of Inductive Definitions

Abs t r ac t . Quasitautologies are formulas valid solely by the properties of identity and of propositional connectives. In connection with Herbrand's theorem the quasitautologies are central to the automated theorem proving with identity. It is a matter of folklore in logic that the predicate of being a quasitautology is decidable. We present a fmitary proof of this fact by a powerful method of transformation of tableaux with identity miles. This should shed some fight on the subtleties of tableaux with identity 9 We have extended this method in a separate paper to a much harder finitary proof of the conservativity of Skolem axioms. The question of why we should prefer fmitary over model-theoretic proofs occurs frequently in logic. The answer is always simple: we obtain more information from a finitary proof than from a model-theoretic one. In our case, we get that quasitautologies can be proved by tableaux with a subterm property 9

On Quasitautologies

This paper discusses the distinctions indicated in its title. It is argued that the distinction between syntax and semantics is much more important for the present situation in logic than other distinctions. In particular, doing formal syntax and formal semantics requires the use of an informal melanguage based on ordinary mathematics.

Proof Theory

Citations

A Gentzen system for the description logic

Syntactical Truth Predicates for Second Order Arithmetic

A Simplified Characterisation of Provably Computable Functions of the System ID_1 of Inductive Definitions

On Quasitautologies

Logic as Calculus Versus Logic as Language, Language as Calculus Versus Language as Universal Medium, and Syntax Versus Semantics

Related Papers (5)

Untersuchungen über das logische Schließen. II

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The collected papers of Gerhard Gentzen

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