Reviewing Frege's techniques for formalising functions as rules

Frege’s Grundgesetze
Reviewing Frege’s techniques for
formalising functions as rules
Ruy J.G.B. de Queiroz
Centro de Inform´atica
Universidade Federal de Pernambuco (UFPE)
Recife, Brazil
III Latin American Analytic Philosophy Conference
Fortaleza, CE
May 2014
Ruy J.G.B. de Queiroz Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Reviewing Frege’s techniques for formalising functions as rules

Functions as Rules vs. Functions as Extensions
Value-ranges
In the wake of the publication of a complete English translation of
Volumes I and II of Frege’s Grundgesetze der Arithmetik (by Philip
Ebert & Marcus Rossberg, Oxford Univ Press, Dec 2013), it seems
appropriate to reﬂect on the pioneering techniques brought about by
Frege in his magnum opus for the formalisation of functions as
rules.

Functions as Rules vs Functions as Extensions
Propositional Equality: A Weak Version of Function Extensionality
With an appropriate formulation of propositional equality that
we have deﬁned somewhere else, we can actually prove a
weakened version of function extensionality, namely
A → ∀fA→B
∀gA→B
(∀xA
.IdB(f(x), g(x)) → IdA→B(f, g))
which asserts that, if the domain A is nonempty, then in case
f and g agree on all points they must be considered to be
propositionally equal elements of type A → B.

Value-ranges
The device of variable-binding, and the idea of having terms
representing incomplete ‘objects’ whenever they contain free
variables, were both introduced in a systematic way by Frege in his
Grundgesetze.
As early as 1893 Frege developed in his Grundgesetze I what can be
seen as the early origins of the notions of abstraction and application,
when showing techniques for transforming functions (expressions
with free variables) into value-range terms (expressions with no free
variables) by means of an ‘introductory’ operator of abstraction
producing the Werthverlauf expression, e.g., ‘´εf(ε)’, and the effect of
its corresponding ‘eliminatory’ operator ‘∩’ on a value-range
expression.

Value-ranges
Expressing how important he considered the introduction of a
variable-binding device for the functional calculus (recall that the
variable-binding device for the logical calculus had been introduced
earlier in Begriffsschrift), Frege says:
“The introduction of a notation for value-ranges seems to
me to be one of the most consequential additions to my
concept-script that I made since my ﬁrst publication on this
matter.” (Grundgesetze I, §9, p. 15f.)

Value-ranges
Both Church’s λ-Calculus and Sch¨onﬁnkel–Curry’s Combinatory
Logic are well established theories of functions as rules, but as it
turns out, not only did Frege introduce such notions of:
(later recast by Alonzo Church) abstraction (Vol. I, §9),
application (Vol. I, §34) and substitution (Vol. I, §9, p. 15),
but he also came up with
a rule of β-contraction (Vol. I, §34),
a rule of α-conversion (Vol. I, §9),
a rule of η-contraction (Vol. I, §39),
a rule of µ-equality (Vol. I, §52): (cont.)

Value-ranges
a procedure to transform a function of many arguments into
a chain of one-argument functions (usually, and unfairly,
called currying in reference to Haskell Curry) (Vol. I, §35),
as well as representing numbers as closed value-range
terms (anticipating the so-called “Church numerals”).

Church and the λ-Calculus
Notice that in the “ofﬁcial” history of λ-Calculus, it is often said
that Church’s “λ” notation came out of a minor typographic
incident, and not to a possible inspiration on Frege’s notation:
´εf(ε)
which starts with a backslash notation to indicate that it had the
force of a deﬁnite article.

On §11, p. 19 of Grundgesetze I Frege says:
(...) “´Φ( )” denotes the object falling under the
concept Φ(ξ) if Φ(ξ) is a concept under which falls one
and only object; in all other cases “´Φ( )” denotes the
same as “´Φ( )”.

In his paper “Lambda Calculus: some models, some
philosophy” (The Kleene Symposium, J. Barwise, H. J. Keisler
& K. Kunen (eds.), North-Holland, 1980, pp.223-265, Dana
Scott says (p.224):
According to CURRY (1968), Alonzo Church prepared
a manuscript in 1928 on a system with λ-abstraction,
and the publication, CHURCH (1932), indicates that it
was work done as a National Research Fellow
1928-1929.

Later in the next paragraph Scott says:
Unfortunately Church, to my knowledge, has never
explained as fully as Curry has how he was led to his
theory. He must have been strongly inﬂuenced by
Frege (via Russell), and he hoped to solve the
paradoxes-not through the theory of types, but by the
rejection of the law of the excluded middle. In
CHURCH (1932, p. 347), it is stated that such
combinations as occur in the Russell Paradox (namely,
(λxnot (x(x)))(λx.not (x(x))), which converts to its
own negation) simply fail to have a truth value.

Thus, we do not have here an intuitionistic theory, but
a failure of excluded middle because functions are
only partially deﬁned. Alas, in KLEENE and ROSSER
(1935) it was shown that Church’s system (which was
employed by Kleene in KLEENE (1934) (written in
1933) and his thesis, KLEENE (1935) (accepted in
September, 1933)) is inconsistent.

The proof was later very much simpliﬁed by Curry and
can be found in CURRY and FEYS (1958,
pp.258-260). It applies to various systems proposed
by Curry, also, but not to his thesis, which is just the
“equational theory of combinators”. This is essentially
the system of CHURCH (1941), and the “system of
symbolic logic” in that monograph is condensed to a
very few pages (§21, pp.68-71). The consistency of
these systems is very forcibly demonstrated by the
well-known theorem of CHURCH and ROSSER
(1936).

Notes added in Proof (February 1990):
I am much obliged to Professors Church, Curry and Seldin who
wrote me comments and corrections to the original manuscripts.
In particular, Professor Church wrote brieﬂy to the editors on 2
June 1979 as follows:
“To the best of my recollection I did not become acquainted
with Frege in any detail until somewhat than the period
about which Scott is writing, say 1935 or 1936. No
guarantee for this, it is just a recollection of something
never accurately recorded. But I was attracted to Frege
because he does not give priority to functions over sets,
and his system can be made consistent (presumptively) by
imposing a simple type theory. To this I would now add that
no doubt such a system can be given as much set-theoretic
strength as desired by adjoining strong axioms of inﬁnity.”

See, for example, Church’s former student J. Barkley Rosser in
his paper “Highlights of the History of the Lambda-Calculus
(Annals of the History of Computing, Volume 6, Number 4,
October 1984, pp. 337–349):
Church was struck with certain similarities between his
new concept and that used in Whitehead and Russell
(1925) for the class of all x’s such that f(x); to wit,
ˆxf(x). Because the new concept differed quite
appreciably from class membership, Church moved
the caret from over the x down to the line just to the
left of the x; speciﬁcally, ∧xf(x). Later, for reasons of
typography, an appendage was added to the caret to
produce a lambda; the result was λxf(x). (p.338)

Church’s Numerals and the Representation of
Arithmetic
Numbers as Closed Terms. Successor as a Closed Term
0 is associated with the closed term λx.λy.y
The successor function is associated with the closed term
λu.λx.λy.x((ux)y)

Frege’s Numerals and the Representation of
Arithmetic
Numbers as Werthverlauf Terms. Successor as a Werthverlauf Term
On paragraph 40 of Grundgesetze Vol. I, Frege deﬁnes an
association of numbers with Werthverlauf terms:
On §41 0 is associated with the Werthverlauf term (see Furth
(ed.))
On §42 1 is associated with the Werthverlauf term (see Furth
(ed.))
On §43 The successor function is associated with the
Werthverlauf term

Formalising Equality
Equality in Type Theory
Martin-Löf’s Intuitionistic Type Theory:
Intensional (1975)
Extensional (1982(?), 1984)
Remark (Definitional vs. Propositional Equality)
definitional, i.e. those equalities that are given as rewrite
rules, orelse originate from general functional principles
(e.g. β, η, ξ, µ, ν, etc.);
propositional, i.e. the equalities that are supported (or
otherwise) by an evidence (a sequence of substitutions
and/or rewrites)

Deﬁnitional Equality
Deﬁnition (Hindley & Seldin 2008)
(α) λx.M = λy.[y/x]M (y /∈ FV(M))
(β) (λx.M)N = [N/x]M
(η) (λx.Mx) = M (x /∈ FV(M))
(ξ)
M = M
λx.M = λx.M
(µ)
M = M
NM = NM
(ν)
M = M
MN = M N
(ρ) M = M
(σ)
M = N
N = M
(τ)
M = N N = P
M = P

Frege’s equalities between value-ranges
On §34 Frege deﬁnes a (sort of) analogue of β-equality:
In the ﬁrst instance it is a matter only of designating
the value of the function Φ(ξ) for the argument ∆, i.e.
Φ(∆), by means of “∆” and “´Φ( )”. I do so in this way:
“∆ ∩ ´Φ( )”,
which is to mean the same as “Φ(∆)”.
On §39, Frege gives a (sort of) analogue of η-equality:

Reviewing Frege's techniques for formalising functions as rules

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