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14
The Foundation of a Generic Theorem Prover
 Journal of Automated Reasoning
, 1989
"... Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabell ..."
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Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabelle is now based on higherorder logic  a precise and wellunderstood foundation. Examples illustrate use of this metalogic to formalize logics and proofs. Axioms for firstorder logic are shown sound and complete. Backwards proof is formalized by metareasoning about objectlevel entailment. Higherorder logic has several practical advantages over other metalogics. Many proof techniques are known, such as Huet's higherorder unification procedure. Key words: higherorder logic, higherorder unification, Isabelle, LCF, logical frameworks, metareasoning, natural deduction Contents 1 History and overview 2 2 The metalogic M 4 2.1 Syntax of the metalogic ......................... 4 2.2 ...
Proof theory of reflection
 Annals of Pure and Applied Logic
, 1994
"... The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. Th ..."
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The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. This leads to consistency proofs for the theories KP + Πn–reflection using a small amount of arithmetic (PRA) and the well–foundedness of a certain ordinal notation system with respect to primitive recursive descending sequences. Regarding future work, we intend to avail ourselves of these new cut elimination techniques to attain an ordinal analysis of Π 1 2 comprehension by approaching Π1 2 comprehension through transfinite levels of reflection. 1
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
Conservatively extending classical logic with transparent truth
 Review of Symbolic Logic
, 2012
"... Abstract. This paper shows how to conservatively extend a classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truthinvolving) vocabulary. However, not all classical met ..."
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Cited by 5 (2 self)
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Abstract. This paper shows how to conservatively extend a classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truthinvolving) vocabulary. However, not all classical metainferences are preserved; in particular, the resulting logical system is nontransitive. Some limits on this nontransitivity are adumbrated, and two proof systems are presented and shown to be sound and complete. (One proof system features admissible Cut, but the other does not.) §1. Introduction. Adding a truth predicate to a language governed by classical logic is not easy. It is particularly tricky when the truth predicate is intended to be transparent— such that T 〈A 〉 and A are everywhere intersubstitutable. The trouble, as is wellknown, comes from such paradoxes as the liar; because of them, theories of truth typically either use a nontransparent truth predicate (Halbach, 2011; Maudlin, 2004) or a logic weaker than
Intuitionistic fixed point theories over Heyting arithmetic
 Proofs, Categories and Computations. Essays in honor of Grigori Mints
, 2010
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On Mathematical Instrumentalism
, 2005
"... In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragm ..."
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In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peano’s Arithmetic known as IΣ1 is a conservative extension of the equational theory of Primitive Recursive Arithmetic (P RA), IΣ1 has a superexponential speedup over P RA. On the other hand, theories studied in the Program of Reverse Mathematics that formalize powerful mathematical principles have only polynomial speedup over IΣ1. 1
A Decidable Case of the SemiUnification Problem (Draft Version)
, 1991
"... Semiunification is a common generalization of unification and matching. The semiunification problem is to decide solvability of finite sets of equations s = t and inequations ˜s ≤i ˜t between firstorder terms, with different inequality relations ≤i, i ∈ I. A solution consists of a substitution T0 ..."
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Semiunification is a common generalization of unification and matching. The semiunification problem is to decide solvability of finite sets of equations s = t and inequations ˜s ≤i ˜t between firstorder terms, with different inequality relations ≤i, i ∈ I. A solution consists of a substitution T0 and residual substitutions Ti, i ∈ I, such that, respectively, T0(s) = T0(t) and Ti(T0(˜s)) = T0(˜t). The semiunification problem has recently been shown to be undecidable [9]. We present a new subclass of decidable semiunification problems, properly containing those over monadic languages. In our ‘quasimonadic ’ problems, function symbols may be of arity> 1, but only terms with at most one free variable are admitted. 1
APPLICABILITY OF MOTOHASHI’S METHOD TO MODAL LOGICS
"... Motohashi [8] showed that the intuitionistic predicate logics can be faithfully embedded in the classical predicate logic by a prooftheoretic method. In the present paper, we show that Motohashi’s method can be applied to a wide range of modal logics. We prove correspondence theorems based on the s ..."
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Motohashi [8] showed that the intuitionistic predicate logics can be faithfully embedded in the classical predicate logic by a prooftheoretic method. In the present paper, we show that Motohashi’s method can be applied to a wide range of modal logics. We prove correspondence theorems based on the standard translation between classical predicate logic and the quantified versions of S4 and S5 and some subsystems of them, in a uniform way. 1.