Table of Contents - 21.38.6. Propositions from _Begriffsschrift_
In 1879, Frege introduced notation for documenting formal reasoning about
propositions (and classes) which covered elements of propositional logic,
predicate calculus and reasoning about relations. However, due to the
pitfalls of naive set theory, adapting this work for inclusion in set.mm
required dividing statements about propositions from those about classes and
identifying when a restriction to sets is required. For an overview
comparing the details of Frege's two-dimensional notation and that used in
set.mm, see mmfrege.html. See ru for discussion of an example of a
class that is not a set.
Frege introduced implication, negation and the universal quantifier as
primitives and did not in the numbered propositions use other logical
connectives other than equivalence introduced in ax-frege52a,
frege52b, and ax-frege52c. In dffrege69, Frege introduced
to say that relation , when restricted to operate
on elements of class , will only have elements of class in its
domain; see df-he for a definition in terms of image and subset.
In dffrege76, Frege introduced notation for the concept of two sets
related by the transitive closure of a relation, for which we write
, which requires to also be a set.
In dffrege99, Frege introduced notation for the concept of two sets
either identical or related by the transitive closure of a relation, for
which we write , which is a superclass of
sets related by the reflexive-transitive relation .
Finally, in dffrege115, Frege introduced notation for the concept of a
relation having the property elements in its domain pair up with only one
element each in its range, for which we write (to ignore
any non-relational content of the class ). Frege did this without
the expressing concept of a relation (or its transitive closure) as a
class, and needed to invent conventions for discussing indeterminate
propositions with two slots free and how to recognize which of the slots
was domain and which was range. See mmfrege.html for details.
English translations for specific propositions lifted in part
from a translation by Stefan Bauer-Mengelberg as reprinted in
From Frege to Goedel: A Source Book in Mathematical Logic,
1879-1931. An attempt to align these propositions in the larger
set.mm database has also been made. See frege77d for an example.