In our opinion, the notion of an approximative consistency proof may capture the core of the conception of finitary metamathematical consistency proofs which Hilbert developed in his papers on

proof theory in the 1920s.

A

proof theory for a concurrent imperative language in general relates the "how," that is, the flow of control described by a program, to the "what," a specification of the program in some (usually first-order) logic.

As to the

proof theory, six rules are introduced which allow the derivation of a new diagram from two given ones: the rule of erasure of a diagrammatic object, the rule of erasure of part of an x-sequence, the rule of spreading x' s, the rule of introducing a basic region, the rule of conflicting information, the rule of unification of diagrams.

As mentioned above, Uniacke contrasts the recent recognition of subjective putative self-defense (sometimes a failure of

proof theory) with actual self-defense and worries that the former may "undermine self-defense as a justificatory defense" [44].

The fifth chapter,

Proof Theory and Decidability, might better be labelled

Proof Theory and Undecidability, although there are a few decidability results mentioned.

Gallier introduces mathematical logic, emphasizing

proof theory and procedures for constructing formal proofs of formulae algorithmically.

the theory of algorithms and the theory of formal systems which has led to the development of computers and computer languages, and advances towards artificial intelligence (Hofstadter, 1999); for the evolution of mathematical proof and

proof theory; and for the development of logic as it is taught today.

The papers, which include abstracts and references, cover type theory (including a dependent set theory), computational

proof theory (including methods of problem solving in elementary geometry), security (including highly efficient proofs of correctness of computations that preserve secrecy), timed and stochastic systems, verification, constraints, proof complexity, finite model theory, concurrency and process calculi, semantics of programming languages (including the algebraic theory of effects), game semantics (including categorical combinatories for "innocent" strategies), linear logic, and topology and computable mathematics.

For instance, the reader will not find any material on modal predicate logic or on the

proof theory of modal logic.

Recall that the branch of mathematical logic called

Proof Theory is only concerned with validity and varieties of constructivity.