1.6.3—
The Scope of Formal Symbol-Manipulation Techniques

It turns out that formal inference techniques have a surprisingly wide scope. In the nineteenth and early twentieth century it was shown that large portions of logic and mathematics are subject to formalization. And this is true not only in logic and number theory, which some theorists hold to be devoid of semantic content, but also in such domains as geometry, where the terms clearly have considerable semantic content. Hilbert (1899), for example, demonstrated that it is possible to formulate a collection of syntactic types, axioms, and derivation-licensing rules that is rich enough to license as valid all of the geometric derivations one would wish for on semantic grounds while excluding as invalid any derivations that would be excluded on semantic grounds.

Similarly, many problems lying outside of mathematics that involve highly context-specific semantic information can be given a formal characterization. A game such as chess, for example, may be represented by (1) a set of symbols representing the pieces, (2) expressions representing possible states of the board, (3) an expression picking out the initial state of the board, and (4) a set of rules governing the legality of moves by mapping expressions representing legal states of the board after a move m to the set of expressions representing legal successor states after move m + 1. Some games, such as tic-tac-toe, also admit of algorithmic strategies that assure a winning or nonlosing game. In addition to games, it is

also possible to represent the essential features of many real-world processes in formal models of the sorts employed by physicists, engineers, and economists. In general, a process can be modeled if one can find an adequate way of representing the objects, relationships, and events that make up the process, and of devising a set of derivation rules that map a representation R of a state S of the process onto a successor representation R^* of a state S^* just in case the process is such that S^* would be the successor state to S . As a consequence, it is possible to devise representational systems in which large amounts of semantic information are encoded syntactically, with the effect that the application of purely syntactic derivation techniques can result in the production of sequences of representations that bear important semantic relationships: notably, sequences that could count as rational, cogent lines of reasoning.