What is a Formal System?

Is a Human Language an Example?

        Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48-64.

formal system is like a game in which tokens are manipulated according to rules in order to see what configurations can be obtained.  (Examples: chess, checkers, go, tic-tac-toe. Nonexamples: marbles, billiards, baseball).   All formal games have three essential features:
They are token manipulation games;
they are digital; and
they are finitely playable.

1.  The tokens ... are just the ``pieces'' (eg, markers or counters) with which the game is played. (Eg, chessmen, go stones, bits, 0s and Xs of tic-tac-toe).

2.  Manipulating tokens means one or more of the following:

3.  A digital system is a set of positive and reliable techniques (methods, devices) for producing and reidentifying tokens, or configurations of tokens, from some prespecified set of types.
A positive technique is one that can succeed absolutely, totally and without qualification ... has the possibility of succeeding perfectly.

We can substitute ``writing'' and ``reading'' for ``producing'' and ``reidentifying'', but only with two warnings: (1) writing isn't just making pen or pencil marks, but rather any kind of token manipulation that changes the formal position ... and (2) ``reading'' implies nothing about understanding (or even recognition) but only differentiation by type and position.

4. Finitely playable ... no infinite or magical powers required.  A finite player is assumed to have some finite repertoire of specific primitive operations (e.g., move a chess piece, write a bit, read a bit) any finite number of times.
Is a language a real formal system?

Chomsky and Halle and many other linguists assume that language IS such a system. Thus they typically assume that:
  1. Phonetic segments are discrete formal tokens. And since they are, it follows that phonological segments (or phonemes) as well as words, phrases, sentences, etc. are formal objects as well since they are discrete combination of the apriori phonetic tokens.
  2. The tokens are manipulated by formal rules or formally specified constraints, etc.
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  4. These tokens can be stored, read and manipulated essentially without error (thus, how many rules or constraints are involved is not an issue).
These are powerful assumptions that not only are not fully defended, they appear to have no evidence at all except the intuitions of linguists.