Basic Proof Theory

Abstract

This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.

Using Proofs by Coinduction to Find “Traditional” Proofs

“…7 However, it turns out that these additional axioms rules are admissible in cREG 0 in the following sense: if their use is limited to situations in which subderivations do not contain open assumptions, then no more theorems (than those of cREG 0 ) become derivable. 8 This is an easy consequence of the fact, which is stated formally below, that the theorems of cREG 0 are precisely the identities of regular expression equivalence.…”

“…the description of "N-systems" in [8]), namely the use of assumptions that may be "closed" (discharged) at a later stage in a deduction, derivations in cREG 0 (Σ) are prooftrees such that: the leaves at the Possible assumptions in cREG0(Σ) and the inference rules of cREG0(Σ) : Figure 2: A coinductively motivated, natural-deduction style proof system cREG 0 (Σ) for regular expression equivalence, given that Σ = {a 1 , . .…”

Zenon Modulo: When Achilles Outruns the Tortoise Using Deduction Modulo

“…Zenon's logic is classical and expressed in a formalism very close to sequent calculus [14]. As a consequence, using Dedukti as a backend requires two steps: the first one is to translate classical proofs of Zenon into proofs of constructive sequent calculus with equality, which we discuss here, and the second one is a standard translation from sequent calculus to natural deduction.…”

Sequent Systems for Lewis’ Conditional Logics

“…(A 1 , A 2 ⇒ B)σ = A 1 σ, A 2 σ ⇒ Bσ. We use the system G3cp of [18] with axioms Γ, A ⇒ ∆, A (where A ranges over the set of formulae) as basis for all systems that extend classical propositional logic and denote its proof rules by G. We adopt the standard structural rules…”