1 8 Ju l 2 00 5 Chapter 1 IN THE BEGINNING WAS GAME SEMANTICS

This chapter presents an overview of computability logic — the game-semantically constructed logic of interactive computational tasks and resources. 1. Introduction In the beginning was Semantics, and Semantics was Game Semantics , and Game Semantics was Logic. 1 Through it all concepts were conceived, for it all axioms are written, and to it all deductive systems should serve... This is not an evangelical story, but the story and philosophy of com-putability logic (CL), the recently introduced [12] mini-religion within logic. According to its philosophy, syntax — the study of axiomatiza-tions or any other, deductive or nondeductive string-manipulation sys-Through him all things were made; without him nothing was made that has been made. " — John's Gospel. 2 In the beginning was game semantics... tems — exclusively owes its right on existence to semantics, and is thus secondary to it. CL believes that logic is meant to be the most basic, general-purpose formal tool potentially usable by intelligent agents in successfully navigating real life. And it is semantics that establishes that ultimate real-life meaning of logic. Syntax is important, yet it is so not in its own right but only as much as it serves a meaningful semantics, allowing us to realize the potential of that semantics in some systematic and perhaps convenient or efficient way. Not passing the test for sound-ness with respect to the underlying semantics would fully disqualify any syntax, no matter how otherwise appealing it is. Note — disqualify the syntax and not the semantics. Why this is so hardly requires any explanation: relying on an unsound syntax might result in wrong beliefs, misdiagnosed patients or crashed spaceships. Unlike soundness, completeness is a desirable but not necessary condition. Sometimes — as, say, in the case of pure second-order logic, or first-order applied number theory with + and × — completeness is impossible to achieve in principle. In such cases we may still benefit from continuing working with various reasonably strong syntactic constructions. A good example of such a " reasonable " yet incomplete syntax is Peano arithmetic. Another example, as we are going to see later, is affine logic, which turns out to be sound but incomplete with respect to the semantics of CL. And even when complete axiomatizations are known, it is not fully unusual for them to be sometimes artificially downsized and made incomplete for efficiency, simplicity, convenience or even esthetic considerations. Ample examples of …