Classical predicative logic-enriched type theories

Classical Predicative Logic-Enriched Type Theories

A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTT0 and LTT ∗ 0, which we claim correspond closely to the classical predicative systems of second order arithmetic ACA0 and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these...

Comments on Predicative Logic

The syntax of intuitionistic propositional second-order logic PSOLi consists of a set of propositional constants P , Q, R, etc, a denumerable set of propositional variables F , G, H, etc, together with the primitive logical signs of the conditional and the universal propositional quantifier, and ponctuation signs (parenthesis). The formulas of PSOLi are the smallest class of expressions of the ...

Predicative Logic and Formal Arithmetic

After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell’s Principia Mathematica with the axiom of infinity but without the axiom of reducibility. 1 Historical introduction After discovering the inconsistency in Frege’s Grundgesetze der Arithmetik, Russell proposed two changes: first, dropping the assumption that to every h...

Temporal Logic, Automata, and Classical Theories -- An Introduction

A Type-Theoretic Framework for Formal Reasoning with Different Logical Foundations

A type-theoretic framework for formal reasoning with different logical foundations is introduced and studied. With logic-enriched type theories formulated in a logical framework, it allows various logical systems such as classical logic as well as intuitionistic logic to be used effectively alongside inductive data types and type universes. This provides an adequate basis for wider applications...