FORMAL SYSTEMS
A formal system consists of
a Vocabulary,
a set of Formation
Rules and a set of Proof Rules
as described below.
VOCABULARY.
A specification of ALL the symbols to be used in the system.
FORMATION RULES.
A specification of which combinations of symbols in the
vocabulary are properly formed expressions of the system. This is
usually done by specifying the various types of expressions, and then
by specifying for each type of expression how to use the vocabulary to
generate expressions of that type. Using the Formation Rules one should
be able to tell, for any combination of symbols in the vocabulary,
whether that combination is a proper expression of the system and, if
so, of which type.
PROOF
RULES. A specification of which combinations of properly
formed expressions of the system constitute a "proof" or a "derivation"
and, if so, what is proved or derived. Using the Proof Rules one should
be able to tell, for any combination of properly formed expressions,
whether that combination is a "proof" or a "derivation" and, if so, of
what. The Proof Rules, however, may not enable us to tell, given a
formula of the system, whether that formula is provable (or derivable
from certain others).
Conditional
Logic as a Formal System. (This links to a pdf file.)