Table Of ContentAN INTERPRETATION
FOR
free m & sm m
by
David E* van TIJn
**►
Submitted to the Faculty of the Graduate School
in partial fulfillment of the requirements
for the degree* Doctor of Philosophy*
In the Department of Mathematics*
Indiana University
*fune* X9SX
ProQuest Number: 10296442
All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
uest
ProQuest 10296442
Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author.
All rights reserved.
This work is protected against unauthorized copying under Title 17, United States Code
Microform Edition © ProQuest LLC.
ProQuest LLC.
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, Ml 48106 - 1346
X wish to express my gratitude to Dr, Max Zorn without
whose valuable criticise and more valuable encouragement this
thesis would not have been completed.
Introduction
Given a formal system F , and a model W for F , it has been
customary to assign as "independent meaning" to the constant terms and
closed formulas of F the elements into which they are mapped in the
model ( [ij , definition llj OsJ , p. 5 ffi Oil , P. 73 ff). Since
this method fail® for free variables of F , it has been equally cus
tomary not to assign any "independent meaning" - or in the terminology
of Carnap [$} "designation" - to these variables. We quote from 6
§6t "Roughly speaking the letters "x" and ”y" may be described as
[$J
ambiguous names of numbers •••"; and from , p. *>2i ..because
formulas containing free variables cannot denote any fixed thing." ^
In Practical mathematics on the other hand, it is usual to act as if free
variables do "designate" something, eg. in [pi , p. 1* "une lettre peut
designer un element generique," and in (j?l , p. 33* "A variable is
simply a collection of numbers."
X define in this paper a relation p called "naming relation"
between a system F and a set W , which is to give the "independent
meaning" 0 to an element of F if " " holds. But we do
not require that 1 be a model for F , merely that it be univocal * and?
(1) that W contains the truth-values, and that for any theorem
M of F p truth" holds, and
(2) that the equlvalence-relation generated in F between elements
with the same upvalue satisfies certain substitutivity-properties.
1 See also &J t P* 16, \1 .
* ' i.e. that the associated function be single valued*
A naming relation will satisfy Carnap's conditions on a "desig-
natlon-relation" ( [3] , lSfL2) with condition (2) slightly stronger than
the Interchangeability of closed sentences Carnap demands ( [3] , ^10),
demanding as it does substitutivity of free occurrence of elements with
the same deaignattm* If one restricts Carnap's ([$) condition 2l*-2
to free occurrences of terms or formulas, and asserts that a formula
"is about" the truth-values of the formulas of which it contains free
occurrences p satisfies 2i*.2. If In addition one restricts principle
2iu3 ( ) to free occurrences, p satisfies 2U»3 (a.)J It is this
that leads us to call the designationrrelatlon \l a "name-relation."
The first theorem (2.2) shows that the classical relation between
closed formulas and terms of a formal system F , and a model W satisfies
this definition.
Theorem (2.3) shows that dropping the requirement that W be a
model represents a real gain. Here we have a system F , a set W and
an intermediate system 1 • There is a naraing-relation u between L
and W , and a mapping i 1 F I» which satisfies the following condi
tions!
(1) if the elements of F are included in L as names for their
images under i , the extended system becomes a "true inter
pretation" of F . (Carnap QjJ , p. 20U),
(2) 1 and satisfy certain substitutivity relations.
(3) i satisfies the "principle of subject-matter" ( Carnap [2],
p. 98 .)
* "and of course p satisfies Carnap's condition 2I4-I of univocality.
- 3 -
We prove that the composition-mapping |xi i F —* W , where it is
defined, is a n&ming-relation between F and W . We call such map
pings i "Interpretations” of F In L ,
Together these two theorems show, that in order to find a naming
relation for free variables of F it is sufficient to seek a system L
which has a model, and an interpretation of F in L which maps free
variables of F into constant terms of L * We indicate how such a
system might be constructed, from the assumption that F has a models
Given a model for F , by £l] , definition 3-6 , we are given a set of
ranges for the elements of F , and (by definitions It and 6 in [l] )
the set, S , of all "proper” substitutions of elements of these ranges
for variables of F . If we construct L as a set theory with these
ranges and the set S as elements, the assumption that L is consis
tent Is no stronger than the two assumptions that F was consistent
(i.e. by [l] definition 8 had a model) and that the set-theory used
for the construction of a model is consistent. If we make these assump
tions, then we have a system 1$ with a model, and the only remaining
problem is to define an Interpretation i t F —* L * We indicate only
what the value i (3) will be for free variables cP of F , This
value is the "natural" one of a function on the set S of substitutions,
to the "proper” range of • This identification of variables with
functions is by no means new. (Monger [lo] , [ll] 3 Heyting [l§ , $7).
k -
I have constructed ouch a system L and a companion Inter
pretation i , which In the paper is called M , for a special system
F which is like the applied predicate ealculue F , of Church ( [3] )•
M maps only individual variables of F into constant terms of L , and
makes no provisions for the functional variables, and possible dependent
variables of F , The images under U of distinct Individual variables
of F are independent functions in L (theorem In this fashion
we show that if one is willing to accept the existence of a name-relatlon
as a definition of "independent meaning”, the free individual variables
of F have "independent meaning" in the same sense in which closed
fomiulas and constant terms have meaning* Thus we hope to settle an
old controversy as to the "independent meaning” of free variables in
their favour*
An additional feature of this method is, that while the name-
relation on F attaches the same designation (a truth-value) to a
formula of F and its closure, their images under M are different*
For example the formulas*
(x^ y v y { x ) and (x) (y) (x $ y ^ y^x)
are mapped by M respectively into*
(t) ( x(t)4 y(t) v y(t)\< x(t)} and (a) (b) ( a^b 'Z b 4 a) .
-5 -
Let L be a formal system. We use A, Ap ,,., B, as
syntactic variables for formulas; a, o^, as syntactic variables
for terns.
"Ap An*"A0B *® to mean that Aq is provable in L from
the hypotheses A^, A)1 .
The predicates (A, B| Aj, B^J , [A, Bj a, {3 hold if 3 is ob
tained from A by substituting (reap a) for some free occurrences
of (resp g) in A , and no occurrence of a variable is free in
and bound in B •
Let W be a set of elements, which includes the truth-values T
and F . We use tth f* )" ' as syntactic variables for elements of W ,
Definition 1.1 Let \l be a relation between certain formulas
and terms of L and elements of W . Then (4 generates an equivalence-
relation on L , defined by*
N
(1.111) a eq g if and only if, if and (J^y* , then 0 » 't
(1.112) A eq B if and only if, if Ap$ and B|aY , then <t ■
p is called a "naming-relation" between L and W , for those
elements on which it is defined, *f. the following axioms are satisfied*
(1.121) if A is a formula and A|i^ , then either 0 ■ T or 0 = F
if
(1.122) if for all n 3^ , An»"Ao) and A^p T $ A^ T #
then A0p, T *
(1.123) there are two formulas A and B such that Au T and Bp, F ,
A fcterb:
"if (i is univocal, and if, in addition;"
- L -
(1.12k) a eq p 0 * ^ Bj a, p] 3 A «q B ,
(1*125) A^ eq 3 * [A, Bj A^, B^I 0 A eq B .
Where "©q" 1® the relation defined by (1,11). A naming relation
03 / V
is a generalization of Romany's definition 11 for A expresses p >
In that W i® not required to be a model* We shall prove (thm. 2*1) that
the values of functions which "express" A> Bj *1* Bi* a and p in a model
satisfy the conditions (1*12). In future we use p> also as a function*
U A - 0 if and only if A ^ 0 holds*
Definition 1*2 Let L and W be as in definition (1*1), p a
"naming relation" between L and W and suppose that F is a formal
system like L * We use as syntactic variables for
formulas of F , Cl, CLt)- ; J?,bA,• ■ • as syntactic variables for terns* A
function i on formulas of F to formulas of L and terms of F to terms
of L is called an "interpretation" of F in L if it satisfies the
following condition®s
(1.21) i is 1 -1
- (U jjM- l nj
(1.22) 3sf for all nj if n r *** in F , then
i(^^), .... i ( \ ) 1(3/Q) In I .
(1.23) if G-,!?] in F and i 4 eqi'b in L ,
then i v eq i 13 in L .
(1.2U) if in F and i Cl eq ib in L , then
iW eq ilS in L .
(1.25) there are two formulas ^ and '25 in F , suoh that
(i® eq i© ) is false in L •
(1.26) if & occurs free in “W , i CL occurs free in i ^
The first five of these conditions ensure that the relation defined
between F and W byt
(1.31) Cl if and only if id. p 0
(1.32) ^ if and only if i*
is a naming-relation. This will be proved in Tbm (2.3), The additional
requirement will ensure that i is a satisfactory "naae-relation" as
defined in Carnap p, 98, As was pointed out, in the introduction, if
CL?l are adjoined to L as names for iA , 1*11 , the extended system
OS
will furnish a "true interpretation" for F in the sense of Carnap ,
p. 209.
$2, Some Theorems
mg' issasiii'.'. M.,r lawaBas
1, We start the section with a lemma about semi-models and models.
Cl]
The proof is based on Kenedy's definitions 2 through 6, The terminology
employed will be his,
(2.1) Lemma* If [&0$ Bqj A^, B^] and for all assignments of appropriate
value® in a model to the free variables of A and the values of the
o a
functions f * and f -a associated with A„ and B in this model
fla a a
are equal, then for all assignments of appropriate values to the free
variables of Aft and B_ the values of the functions associated with A
w w 0
and BQ are equal.
Proof;
(2.11) Lemma! The value of the function assigned to a composition depends
only on the values of the formulas assigned to its special formula parts
