CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
SYNTACTICAL 'I'REATMENTS OF MODALITY
A thesis submitted in partial satisfaction
of the requirements for· the degree of
Master of Arts in
Philosophy
by
.Charles David Koenig
'-:~
.: '
June, 1981
The 1hesis of Charles David Koenig is approved:
JcL~LC~
u.
S~cha
Cornmi -t:t.ee Chairman
California State University, Northridge
l.l
To Christina
My Love a.nd My Inspiration
iii
TABLE OF CONTENTS
iii
Dedication .
Abstract
vi
SECTION
I
II
III
INTRODUC'l'ION
•
•
~
0
•
•
•
•
•
•
•
..
•
•
1
Part 1.
Introduction of Quine's
Position . . . . .
7
Part 2.
Quine's Argument for Considering
the Second Degree Reducible to that. of
the First. Degree
. . . . . . . . . . .
9
Part 3.
Quine's Remarks on the
Extensionality Policy • .
16
Part 4.
18
A Discussion of Metalanguages. .
Part 1.
Introduction of Montague's
Position
. . . .
23
Part 2.
Some of the Preliminary Notions
Necessary for Montague's Theorems and
Lenm1as are Set Up Here
. . . . . . . •
24
Part 3.
28
0
•
•
•
•
•
•
•
The Diag LE::mma is Established.
.
Part 4.
The Fixed Point Lemma is
Established • . . .
31
Part 5.
Lob's Theorem and Montague's
Lemmas 3 and 4 are Proved . . . . .
33
Part 6.
Montague's Theorems 1 and 4
are Proved
. • . . . • . . • . • .
40
Part 7.
A Restatement: of Theorem 1 with
Lon's Theorem but Without Montague's
Lemma 3 • . . • • • • • . • • . . • • •
43
o
iv
•
•
•
•
•
TABLE OF CONTENTS (continued)
SECTION
IV
Part 1.
Introduction to Skyrm's
Position . . . . . . . . . . .
Part 2. Laying out of the Rules for and
a Description of the Object Language(s)
and the Modal Language(s) as well as
the Function c . . .
. . . . . . .
v
45
.
46
Part 3. The Soundness and Completeness
Theorems for Skyrms' Metalanguages
are Established
. . . . . .
49
Part 4.
Skyrms' Criticism of Montague
is Made Here • . . . .
. • .
53
Part 1.
Introduction to the Final
Section . . . . . . .
. . .
56
Part 2. \i"Ve Flesh Out Skyrms' Theory in
Accordance with His Footnotes and
within His Parameters
. . . . . .
.
57
Part 3.
Show that if Skyrms does not
Restrict the Godel Numbering Function
then Montague Is Remarks Still s·tand
61
Part 4.
Show that Even Without the Godel
Numbering F'unction His Naming Function
Gets Us All We Need for Montague's
Criticisms to Go. . . .
. . • .
67
FOOTNOTES
71
BIBLIOGRAPHY
74
v
ABSTRACT
SYN'l,ACTICAL TREATHENTS OF MODALITY
by
Charles David Koenig
Master of Arts in Philosophy
In this paper I examine the syntactical interpretation of modality as a viable position.
I
look at the
position as presented by Quine as well as his remarks on
the subject.
I
look at the position that Montague takes
on the subject and his remarks which were directed at
Quine.
I
look at the position hold by Skyrms and his
criticism of Montague.
I
finally look at Skryms with a
critical eye and make a few critical remarks of my own.
The issue is important in ·thaJc without a cogent interpretation of our modal operators then our modal logic is just
an academic endeavor.
It would then make the task of
understanding and structuralizing our natural languages
much more complicated.
vi
SECTION I
INTRODUCTION
The terms "necessary," "possible," "contingent," as
well as many other related terms are used throughout the
philosophical literature freely.
It has been demonstrated
in the laying out of the various modal systems that these
terms are interrelated.
other.
They are definable terms of each
Thus we can supply rules for defining all the
other terms as some form of one of these terms.
For
perspicuity sake I will (without giving the appropriate
rules :for reductions) take all these terms to be reducible
to the term "necessity."
This still leaves us with a
basic problem, what are we to understand to be meant by
"necessity? 11
What we take "necessity" (or "necessary") t.o mean is
the problem of interpretation.
If we are going to provide
an interpretation for a given set of symbols, then we must
to.llow a certain set of ground rules that are laid out.
When we are providing an interpretation of an axiomatic
system vvhat we are doing is assigning a meaning to the
undefined terms of the system in such a way that the
axioms are all true for all values of the variables.
So,
as we can see, any interpretation of our undefined term
1
"necessity" must be one that turns all the given modal
axioms true for all the variables.
It is very important
that we recognize that if we are given a set of axioms
with at least one undefined term
and we provide an inter-
pretation that turns only a proper subset of that set of
axioms true, then we have not met the above condition for
an interpretation.
It is also important to note that, if
we are given this same set of axioms and provide an interpretation that turns some but not all those axioms true
as well as some of the axioms of some other set of axioms,
then we still have not done the job of providing an interpretation for the given set of axioms.
Providing an interpretation for the undefined term
"necessity" for the axioms of Sl-SS is basically what is
at issue here.
The above-noted point is a point that
comes to bear if the providing of the interpretation for
"necessity" forces us to change our set of axioms from
those of S4-S5 to something else.
With this in mind we
note that many at.tempts have been made to in·terpret t.he
term "necessity" in terms of some semantic property.
Those who favor this approach have tried sema::;.+--i.c properties like analyticit.y, validity, provability, etc.
There
are those on the other hand who feel this approach to be
not only nonproductive but impossible.
come in.
This is where we
Quine takes up the position of those wishing to
3
provide a semantic property and we lay out this position
from his paper '"rhree Grades of Modal Involvement. "
1
In section II, I lay out the problem as Quine sees
it.
The issue as it was originally presented in Quine's
paper pointed out three grades or degrees of the idea of
necessity which we can allow our logic to embrace.
I have
restricted this paper to the first two; the third involves
quantification which has other problems to be dealt with
and I felt that this paper vmuld
bE~
much to lengthy for
any thorough look into all three.
In section II, as well as setting out the initial
problem, we are also presented with a proposed solution.
Quine proposes that the second degree of modal involvement
be considered to be reducible to that of the first degree.
If the reduction of t:he second degree to that of ·the first
degree fails, then since the t:hird degree is not.hing but
an extension of the second degree, it will fail to be
reducible.
The issue in this case is ·to look at the
interpretation of "necessity" and this is best served by
the restriction of this paper to that of the first and
second degrees of modal involvement with little attention
paid to the third degree.
Montague takes issue with Quine's treatment of the
issue.
In section III, I deal with Montague's objections.
In order to understand Montague's objections and to
sE~e
the force behind them I had to first lay out some of the
background that led to them.
Montague not only feels that
Quine's solution to the problem of interpretation is wrqng
but his feeling is that degree one is not and cannot even
be a consideration without some very basic changes in our
modal systems.
Montague proposes that the correct inter-
pretation is that of degree two.
The understanding of Montague's critique of Quine is
facilitated by looking upon it as an extension of results
obtained by M. H. L6b.
taken.
This is the approach that I have
In section III, I proved those results by first
laying out and proving the basic theorems which lead to
Lob's t.heorem.
I further demonstrated that Montague's
results can be generated directly from Lob's theorem without going through Montague's Lemma 3.
Any objection to
Montague's results, and/or his critique, is one which, if
it is to be cogent, must go beyond Montague to Lob.
Any
such objection in essence must either prove Lob's theorem
incorrect or else demonstrate that it is not applicable to
modal logic.
Brian Skyrms has contributed to the issue by taking
Quine's position and defending it against Montague's criticism.
In section IV, we take up Skyrms
and the resulting critique of Montague.
1
defense of Quine
Skyrms, being
fully aware of the fact that he ultimately needs to
address himself to the Lob theorem, bypasses the actual
5
Montague Lemmas and ·theorems and goes directly to the
variation derived directly from the LBb theorem.
Skyrms 1 ·thr·ough a very complex series of moves
r
has
set up a metalinguis·tic approach to the Quine problem
which purports to avoid the L6b-Montague results.
He has
placed restrictions upon substitution and he has made use
of quotation as a naming function.
It is via ·the rigorous
description of the metalanguages Sk.yrms has in mind in
conjunction with the proposed restrictions on substitution
and the naming function that Skyrms feels he avoids the
Lob-Montague results (criticism).
If Skyrms• metalinguistic approach stands up to close
scrutiny 1 then it is obvious that among other things Quine
is correct in his feelings about the possible reduction of
degree t.wo to that of degree one.
Skyrms will have pro-
vided an avenue of escape around the LBb-Montague critique.
He in fact provides us with the semantic predi-
cates which are to be used as the interpretation of necessity.
The S4
predicate is that of provability and the
S5 predicate is that of validity.
He tells us exactly
what is to be understood by "provability" and "validity."
In section V, I do see if Skyrms' approach stands up
to close scrutiny.
I do just as he asks and flesh out his
system the way he specifies.
It is apparent that if his
position does not stand up under a closer look then he
6
has not effected a cure for what might be felt to be an
ailing position.
That ailment is the Quine position.
In the concluding section I point out that Skyrms
does not succeed in his search for a way around the LobMontague critique.
I show just why he does not succeed
and then try to point out that what he needs to succeed
will make his system so unuseful that it would be unworthy of consideration.
In conclusion I feel that the
Montague critique still stands and the position he puts
forward as to the interpretation of necessity as not being
able to be a semantic predicate (property) is true.
I
further feel that based upon these results the only viable
alternative is that the interpretation of necessity be
that of what Quine calls a statement operator.
These
last few comments are based upon results of the paper.
SECTION II
PART 1.
INTRODUCTION OF QUINE'S POSITION
Quine has provided much material of a critical nature
on modal logic.
The area and in particular the article of
most interest ·to us is his "Three Grades of Modal Involve'"'
ment."
2
In this paper Quine distinguishes among three
grades of modality.
Quine feels the least objectionable degree of modal
involvement is that which is a semantical predicate.
(The
semantical predicate involved we will leave unstated for
the moment so that we can ma.ke the major distinctions with
expediency.)
A semantical predicate is attached to a name
of a statement to form a statement.
such a predicate.
Let "Nee" stand for
Quine makes use of quotation marks as
a device to form names; hence for him something like Nee
"9 > 5" is a statemen·t wi t.h the first grade of modal involvement.
'rhere are other methods of forming names for
which Quine seemingly has no objections.
His characteri-
zation of the first degree should be equally as applicable
to· them since for him all that is required is that the
predicate 'Nee' attach to the name of a c,tatement to form
the modal statement.
7
8
The second grade of modal involvement is one of being
a statement operator.
This degree, by con·trast, is more
objectionable and hence less acceptable than the first.
It involves attaching a statement operator to an already
formed statement to form a new statement.
The operator
grammatically takes the form of an adverb as contras·ted
with the subject-predicate or name-verb grammatical form
of the first degree.
Quine and for us also.
'nee' is the statement operator for
It then seems that any statement
of the form "nee (9 > 5)" is one with the second degree
of modal involvement:
(9 > 5) is not a name of a s·tate-
ment as is "9 > 5" but is in fact the statement and "nee
is used and not "Nee".
The third degree and hence the most objectionable is
that of the sentence operator.
'I'he sentence operator is
like the statement operator in that it is an extension of
the statement opera·tor to open sentences; we will use
'nee' also as the sentence operator.
The distinction is
in the fact. that open sentences are not. st.atements i open
sentences have (anf unbound variable(s).
The sentence
operator, as Quine points out, is the precursor to adding
quantifiers thus yielding for
"(x) nee (x
5)" or "3x nee (x
ex<:~.mple
5)".
statements like
9
PART 2. QUINE'S ARGUMENT FOR CONSIDERING
THE SECOND DEGREE REDUCABLE TO THAT OF
THE FIRST DEGREE
Quine's concern is to consider the reduction of the
second degree to that of the first degree.
This is to be
done by showing the second degree capable of being reconstrued in terms of the semantic predicate.
The third
degree, on the other hand, even though it is an extension
of the second degree, he does not feel can be reconstrued
in terms of the semantic predicate.
3
In fact there are
so many problems with the third degree that Quine would
be much happier if it was eliminated altogether.
The background to this end must first be laid out.
Quine initiates this task by distinguishing between an
occurrence of a term being used as a purely referential
occurrence or its use as a non-purely referential occurrenee,
This distinction is drawn along the lines of sub-
stitutivity.
That is to say that in those cases of an
occurrence of a term being used where substitution of some
other name or occurrence of some other term referring to
the same object can be performed calus verite then the
occurrence of the term is purely referential.
"F'ailure
of substi tuti vi ty reveals merely tha·t the occurrence to
be supplanted is not purely referential, that is, that t.he
statement depends not only on the objects but on the form
of the name.
For it is clear ·that whatever can be
10
affirmed about the object remains true when we refer to
the object by any other name."
4
So a purely referential
occurrence of a singular term in a statement is one which
refers to the object of references and to the truth of the
statement is not altered by a substitution of some other
singular term referring to the same object.
A not-purely
referential occurrence of a singular term is obviously one
in which the truth of the statement is changed by a substitution of some other occurrence of a singular term
which refers to the same object.
Quine makes it clear at this juncture that, "Occurrences within quotations are not in general referen. 1.
t 1a
It
e
•
,.5
Quine, " ( 9)
(10)
This fact is exemplified by noting, as does
'Cicero contains six letters
'9 > 5' contains just three characters
6
•
•
says not h 1ng
a b ou t th e statesman C1cero
or th e nu.mb er 9 • II
It is obvious that in (9) if we were to substitute some
other occurrence of a singular term into (9) for "Cicero 11
say "Tully", which refers to the statesman in question
then the truth of (9) would change.
This is also true of
(10) for if for "9" we substituted say "the number of
planets" then the truth of (10) is not preserved.
This
fact would lead us to conclude, as Quine has stated, that
occurrences within quotation are not referential.
We now need to lay out what is meant by a referen·tially opaque context.
Quine says, " ... we may speak of
11
a context as referentially opaque when, by putting a
statement into that context, we can cause a purely referential occurrence of > to be not-purely referential in the
whole context ••.• Briefly a context is referentially opaque
if i t can render a referential occurrence non-referen. ~1 • "7
t ~0.
The referentially opaque context then takes us
from the purely referential occurrences to not-purely
referential occurrences and (or)
from the referential
occurrences to the non-referential occurrences.
8
To exemplify these distinctions we have wha·t Quine
calls
(15) Cicero has a six-letter name.
This has "Cicero" as a purely referential occurrence of
a term for which if we were to substitute "Tully" in its
stead the statement (15) would still be true.
(15) is
about the statesman and hence "Cicero" in (15) refers to
the statesman.
In (9) on the other hand "Cicero" does
not refer to the statesman and hence is not purely referential as was stated before.
Quine states, "Quotation is the referen.tially opaque
.context par-exce 1 lence." 9
An example of a·referentially
opaque context is for us to take a look at Quine's (10).
The statement 9 > 5 in which '9' is purely referential is
put inside of quotes which in turn make '9' not purely
referential and upon close inspection turns "9 > 5" into
a non-referring occurrence altogether.
Hence in (10)
12
"9 > 5" does not refer and this context is referentially
opaque because it has turned a referential occurrence into
a non-referential occurrence.
The "nee" context like the quotation context demonstrated to be a referentially opaque context.
As is done
with quotation we show that in the "nee" context truth is
not always preserved when statements with the truth values
are substituted for each other.
e.g.
!
(4)
nee ( 9 > 5)
(18) nee (the number of planets > 5)
It is obvious that, "the number of planets is identical"
with "9" and hence a purely referential occurrence of "9"
would allow such an interchange.
In the above example
(18) we note truth is not preserved with the substitution
made.
Hence we have gone from a purely referential to a
not purely referential occurrence of the term ng" by
attaching nee to the statement.
The attaching of nee to
the statement "9 > 5" then creates a referentially opaque
context.
Hence the "nee" context is referentially opaque.
Quine makes it clear that one cannot expect occurrences of statements contained within referentially opaque
contexts, like quotation and nee, to be truth-functional.
"An occurrence of a statement as part of a longer statement is called truth-functional if, whenever we supplant
the contained statement by another statement having the
same truth value, the containing statement remains
13
unchanged in truth value."
10
This quote is to be in light
of the further explanation, "One might not expect occurrences of statements within statements to be truthfunctional, in general, even when the contexts are not
referentially opaque; certainly not when the contexts are
referentially opaque."
11
Quine tries to make it clear in
these two statements that we have on the one hand some
occurrences of statements contained in longer statements
which are truth-functional.
On the other hand this situa-
tion is not a general rule of thumb for statements contained in longer statements regardless of their contexts.
That is to say that it is not to be expected generally for
these contained statements to be truth-functional.
He then
goes on to say that it is not the case that in referentially opaque contexts will these occurrences be truthfunctional.
The next item to be dealt with here is the policy of
extensj_onali ty.
Given p for example r two predicates which
are true of the same object then in a statement the truth.
of this statement is pressed when one predicate is substituted for the other (this is provided we are dealing
with non-referentially opaque contexts) this gives us an
idea of what an extension language is.
The policy of
extensionality is any extension, in a sense, of the notion
of an extensional language.
That is, it is a policy which
admits statements to be truth-functional within statements.
14
As Quine notes, the policy of extensionality is a parameter
or restriction placed upon our substitution operations in
mathematical logic.
12
Hence in mathematical logic, one of
the ways we limit substitution is via the policy of extensionality, we only allow the substi tut.ion of a statement
within a statement on the condition that the subst.ituting
statement be of the same truth value as the substituted
statements.
All of this is only in those non-referentially
opaque contexts, like quotation.
Hence since quotation is
a referentially opaque context then it is excusable from
this limitation placed upon subs·ti tution by the policy of
extensionalities.
The semantic predicate "Nee" is reconcilable with
this policy of extensionality since whatever apparent
infractions that would and do occur are those shared by
examples like (10) .
These are attribu-table to quotation
which is referentially opaque and hence excusable from
the policy of extensionality.
The sentential operator "nee" is a departure from
extensionality.
Since the "nee" context is referentially
opaque it does not square with extensionality.
Quine sees
the departure from or non-squaring of "nee" with extensionality as a two-horned dilemma.
On the one side vJe can
deny the policy of extensionality on the grounds that it
is either not needed or that i·t is not reflective of what
goes on in natural languages.
On the other side of this
15
dilemma we can accept the policy of extensionality and
excuse the "nee" context as being an opaque context hence
putting i t on a par with the quotational context.
If the
latter option is exercised then from the policy of extensionality point of view the quotational context is no worse
than the "nee" context for they are both on a par.
e.g. :
(1) Nee "9 > 5"
( 4 ) nee
(9 > 5 )
The only difference between (1) and (4) from the point of
view of the policy of extensionality is that (1) is
written with quotation marks,
(4) is not.
This leaves us with the other horn of the
the rejection of the policy of extensionality.
dile~na,
Quine makes
13 h
. c 1 ear 1n
.
.
. 1s
. not a 11 t h at
1t
an argumen t h e g1ves
t at 1t
easy to reject the policy of extensionality.
Hence if it
is not at all easy to reject the policy of extensionality
then that leaves us with but only one viable alternative,
that being to accept extensionality and to view "nee" as
being on a par with quotation from the point of view of
extensionality, they are both referentially opaque.
What
attaches more force to equating the "nee" context to that
of the quotational context is the fact that Quine seems
to have some sort of ancillary policy of reduction of all
referentially opaque contexts to those of quotational
form.
14
Now obviously if such a policy is put forward
then the "nee" context would be reducible en toto to the
16
quo~ational
form and not only reconcilable with the policy
of extensionality but also as a result made to appear no
different from those more stubborn "Nee" contexts which
are reconcilable because of their being quotational.
In
view of this there would be nothing ·to recommend the statement operator over the semantic predicate operator.
PART 3.
QUINE'S REMARKS ON THE
EXTENSIONALITY POLICY
It is not enough to show the semantic predicate and
the statement operator to be one on a par from t.he point
of view of the extensionality policy.
Quine provides us
with an ancillary policy of reducing the statement operator to that of the semantic predicate.
making use of the quotation marks.
This is done by
This resolves the use-
mention problem for modal interpretation for Quine.
We no
longer need worry about what is being said when we see
e.g., "All men are mortal implies all fat men are mortal"
... "It's necessary that if I have a 1964 Ford then I have
a car."
'l'he confusion originally came from Russell and
Whitehead in that they misread the material conditionalf
which connects statements to statements, as 'implies'
which is a predicate taking a name of a statemen·t as its
subject.
An attempted repair v;as done by int.roducing the
strict. implication sign which was supposed to be an
improved subst.i t.ute for the ma·t.erial conditional.
'l'his
17
attempt failed because it also did not distinguish
~
from mention, hence the strict implication though read
'implies' was actually functioning as a statement connective.
A validity operator 'nee' was later added to
facil~
itate the definition of the strict implication connective
but this move seemed to fail also since this only relegated
the validity operator to the level of s·tatement operator
instead of bridging the gap between the material conditional and 'implies'.
The motivation behind the validity
operator is very simple, if your material conditional is
true in every case then you have a valid statement, hence
you have logical implication.
So you see it was felt that
the strict implication was none other than the material
conditional with the validity operator.
Quine removes the use-mention problem by reducing the
statement operator to the semantic predicate thus making
the problem seemingly innocuous.
With the removal of the
use-mention confusion done in this way we have two good
reasons for converting the statement operator into the
semantic predicate.
The first of course is the fact that
we display a constant reminder of the referential opacity
in the quotes instead of hiding it in the statement operator.
A second reason, which we will say very little
about, is that the statement operator leads directly into
a sentence operator use and into problems of essentialism
for which Quine and others have no love.
18
PART 4.
A DISCUSSION OF METALANGUAGES
The semantic predicate is ordinarily applied in a
metalanguage.
This fact is obvious in that the names of
the objects named are usually contained in the metalanguage
hence any predicate applied to those names must be applied
in the metalanguage.
The fact that the semantics and the
names of the objects of a language are usually cont.ained
in the metalanguage makes Quine feel uneasy about iterated
modal forms.
He makes it clear that there is no real
problem in moving from the iterated form of the statement
operator to an iterated form of the semantic predicate as
long as care is taken in the proper placement of the
quotes.
The problem comes in making sense of this form,
i.e., the semantic predicate under iteration.
He does
feel the iterated form of statement operators is also hard
to make sense of but he concentrates his critical remarks
on the semantic predicate.
.
Q u1ne
. c 1 ear ln
.
" Two Dogmas o f Emp1r1C1sm
. . .
"15
rna k es 1t
that analycity will not do as a candidate for the semantic
predicate.
He states specifically in the T.G.M.I.
Grades of Modal Involvement)
choice.
16
(Three
that validity is his
It is reasonably clear that one can cite truth
conditions for the semantic predicate but one cannot
equate the one to the other for it is also clear that a
true statement is not a valid statement.
A valid
19
statement in truth-function theory is one where it and all
statements like it in structure (t.ruth-functionally) are
true. Hence it is clearly Quine's content.ions that validity be the sole interpretation of the semantic predicate.
,..
Q .
17 I n t h e quote rrom u.1.ne
"cnere are some matters
contained therein which need further discussion.
first get clear on what metalanguages are.
Let us
A metalanguage
is "A language used to talk about an object language ....
Derivatively, a proposition is said to be in the meta·language if, and only if, it is about an expression in
the object language."
18
The contrast of the metalanguage
to that of the object language is only clear if we know
what an object language is, so a definition of object
language is forthcoming.
Object language is "A language
used to talk about things, rather than about other languages."
Derivitively, a proposition is said to be in the
object language if, and only if, it is not about any
19
.
.
.
.
1 J.ngu1.st1.c express1.on.
Simple statements like, "Grass
is green" or "That car is a Ford" are statements in the
object languages.
Contrasted with the object language
examples are "'Ford' has four lett.ers" or "'That car is
a Ford 1 is a declarative sentence, which are both examples
of metalinguistic statements.
This distinction even
though not always clearly stated is present whenever one
makes statements about languages regardless of which
language it is in.
If one says of some English expression
20
in the German language that it is predicative or so forth,
then they are speaking in the metalanguage even though
this fact remains unst.ated.
A lot of the confusion in the
subjects comes from statements like, "English contains its
own metalanguage. '1
All tha·t tb.is means is that when we
speak in a metalanguage about some part of English as the
object language, we are saying it in English.
In the above same quote on metalanguage Quine talked
about the semantics in the same sentence with that of the
metalanguage.
There.m.ight be some confusion here about
Quine's feelings about the relationship of metalanguages
and semantics or syntax.
Before pointing out and then
trying to clear up the possible confusion let us first try
to distinguish between semantics and syntax.
The differ-
ence between the semantic and the syntax is, semantics
studies the interpretations of the symbols of the language.
Semantics gives an account of how things may be said with
the language, whereas the syntax is restricted to the
study of how the symbols combine to form well-formed
formulas.
The syntax of the language is not concerned
with the ways of interpreting these symbols.
Let us now draw the distinction between an object
language and a metalanguage.
In essence the metalanguage
is the language that contains the names of the objects
that are in the object-language.
It is the language in
which and through which talk is done about the objects
21
of the object language.
That is not to say that because
we draw a distinction between the objec'c and rne'talanguages
that a language cannot be both.
We need to note the difference between a semantic
metalanguage and a syntactical metalanguage.
The Robinson
arithmetic is an example of being set up in a mixed sernantical syntactical metalanguage :for it does no·t contain
"true" anywhere therein.
"True" is a semantical proper-ty
and an essential one at that so it need be present if one
is to set out a semantical metalanguage.
Another example
of a semantical syntactical metalanguage would be one like
that set out by Thomason.
20
Essentially a purely semantic
metalanguage is one in which we are dealing '\.Yi th the
purely semantic terms or names of those terms.
make this a bit clearer.
Let us
In semantics we deal with lin-
guistic relations and properties like satisfiability, true,
validity¥ etc., our purely semantic metalanguage has the
names of only those semantic terms.
Since our semantics
is the study of how to interpret the syntactical symbols,
there will always be a syntactical element in our semantic
metalang.:wge.
There is stric·tly speaking no such thing
as a purely semantic metalanguage but we can come mighty
close to one.
A purely syntactical metalanguage is one in which the
names of the syntactical objec·ts are contained.
This is
not hard ·to imagine in that any statement about these
22
syntactical objects which is not a semantic statement is
made in a syntactical metalanguage.
Most metalanguages
are a combination of both in that usually you wish to make
some semantic property known about, say one or more of
your syntactical objects like formulas, this is done generally in some form of a semantic-syntactical metalanguage.
All natural languages contain semantical-syntactical metalanguages.
This synthesis can be carried out in varying
degrees depending upon just how many semantical properties
one wishes in his metalanguage, i.e., just how rich one
wishes his metalanguage to be.
SECTION III
PART 1.
INTRODUCTION OF MONTAGUE'S POSITION
Montague has written a paper "Syntactical Treatments
of Modality"
21
which is in part critical of the position
taken by Quine as portrayed in the previous section.
It
is quite obvious that Montague cannot be accused of something quite as simple as not understanding Quine.
This is
made evident by the first paragraph of the article where
he says, "On several occasions it has been proposed that
modal terms ( ••. ) be treated as predicates of expression
rather than as sentential operators ...• The proposal thus
amounts to the following:
to generate a meaningful con-
text a modal term should be prefixed not to a sentence or
formula but to a name of a sentence of formula (or perhaps
a variable whose values are understood as including sentence.) "
22
In ·the footnote to this quote Montague points
out that he has read Quine's paper and given the accuracy
of the previous section of this paper the above quote is
a good statement of the general position held by Quine
(with the possible exception of the parenthetical claim,
but eliminating that will not hurt
Montague's position
so we should ignore that part of the above statement if we
23
24
have any reservations about it) .
So it is obvious that
Montague and Quine both have an understanding of the issue:
it is just that Montague t.hinks Quine is Hrong.
PART 2.
SOME OF THE PRELIMINARY NOTIONS
NECESSARY FOR MONTAGUE'S THEOREMS AND
LEMMAS ARE SET UP HERE
In order to get a good grasp of Montague, we will need
t.o sta·te his
lemma~
s and theorems (at least the crucial
ones) in some detail.
In order to do this, some common
explanations are needed.
In the first paragraph under
23 h
. .
.
.
.
pre 1 1ID1nar1es,
e exp 1 a1ns
wh at t ypes o f th eor1es
are
to be discussed, i.e., what the general parameters of the
theories are.
Basically a theory is a set of sentences of
some language such that if S is a logical consequence of T
(the theory), then
s
(a sentence) is also in T.
This, in
conjunction with his claim that all the logical axioms of
T are sentences, and these axioms are recursive, formulates our basic notions of theories under consideration
here.
The theories in quest:ion must also be the smallest .
set of sentences which is closed under logical consequence.
If a sentence is a member of 'I' then it is a theorem of 'l'
and this is generally indicated by
tence of T.
frr
A where A is a sen-
We will later on make use of the phrase "is
in T" to be synonymous vJith the notation
hji -.
Any theory
which is not closed under detachment or lo.gical consequence is not a theory under consideration here.
The
25
logical consequence relation is that of the first order
theories with identity; this point needs to be made to
avoid any confusion with second o~der theories and their
relationships.
Montague has narrowed the scope of theories down to
those which can be written in the language of arithmetic.
The two he focuses in on are that of Robinson Q and Peano
P, both of which we will discuss in a moment.
The lan-
guage of arithmetic has as non-logical constants the
symbols 0,
s, +,
P and Q both also have these non-
and·.
logical constants since both systems can be written in the
L.A.
{language of arithmetic).
Q is the system or theory with the following seven
axioms and the set of sentences which are their logical
consequences:
Ql (x) (y) (Sx=Sy x-y
Q2 (x)
(~Sx)
Q3 (x)
(x~Q-+Ey
(x=:sy))
Q4 ( x) ( x) {x+Q.=x)
Os
(x) (y)
(x+Sy=~(x
y))
Q6 (x) (x) •0=0)
Q7 {X) (y) (x·_§_y=(x•y)+x)
24
P (Peano's arithmetic) is an extension of Q.
This
is obvious from the fact that the first seven axioms of P
are the seven axioms stated above.
This makes the logical
consequences of Q a sub-set of the set of logical
26
consequences of P.
The axiom which differentiates P from
Q is best understood as the main induction axiom.
This is
the axiom from which all the other induction axioms are
derived along with the axioms
o1 -o 7 .
((A (0) & (x) ((A) x) -+A (Sx))) ·+ (x) A (x))
Both P and Q can be
The stated axiom is:
25
contrasted with what I will call
"arithmetic" as a point of reference.
Arithmetic is the
set of sentences of L.A. which are true under the standard
interpretation of L.A. which is (N).
Even though Montague
makes i·t perfectly clear that his theories can remain uninterpreted, i.e., his results are not dependent upon any
interpretation, he does however tell us what is meant by
the standard interpretation in order to give us a reference point.
mean:
By the standard interpretation (n) of L.A. we
0 is zero, S is the successor function, + is the
function of adding and • is the function of multiplication.26
Arithmetic then under (N) is an extension of P
and hence of Q since P is an extension of Q,
The differ-
ence is that even though P and Q are axiomatizable,
metic is not.
arith~
What is meant by this is that the set of
axioms, if they are to be axiomatizable, are recursive and
this is not and cannot be true of arithmetic.
The set of
axioms of P and Q can be recursive hence P and Q are
axiomatizable.
Arithmetic is a theory, though, for the
other requirement for theoryhood is that it be closed
under logical consequence and it is.
27
In Q the numeral corresponding to a number n is 0
preceded by n occurrences of S.
This means that the nu-
meral 6 looks like this:
~ ~ ~
(S S S
0) and this says
that 6 is the successor of the successor of,
of zero.
Further we notice that any number n+l will have
a corresponding numeral that can be written "Sa"
is the numeral corresponding to the number n);
(where a
~is
an
expression or sequence of symbols, a term of the abovecited sort.
We wish to set up the notion of representable in
by making use of the above distinctions.
natural numbers k 1 ,k 2 .•. ,kn' J.
!
We will indicate
The numerals for these
natural numbers in Twill be, k , k 2 , •.. ,k ,
1
11
notion of representability is this:
~·
The
if we have an n-place
function f such that f is representable then there is a
formula A(X ,
1
x2 , x3 , •.. ,X,
Xn+l) such that for any na-
tural numbers (k 1 , k 2 , .•. ,kn' j) if f(kl, k 2 r···kn):::: j;
then (x(n+l)) (A(k 1 , k 2 , ... kn,x(n+l))+-+x(n+l)=J) is
in T.
It is obvious that if (x(n+l) (A(k , k , ...
2
1
,k
11 ,x~+D)
+-+x(n+l)=J) is i n ! then A(k , k , •.. ,kn' J) is also be1
2
cause X(n+l)=J and we substitute J for X(n+l).
J~k
is in Q_ whenever
J~k
Now if
then we have it that if
f(k , k , ..• ,k )~j and A is the formula that represents
2
n
1
the funct.ion ! then A(k 1 k , .•. ,k , J) is in T.
2
-,
n
We must remember that theorems or sentences are
formulas.
What the notion of representability boils down
28
to is that we have the syntactical apparatus of our system
and it is sufficient to take the function and its values
and generate the theorems which correspond to it.
This
in Q is done with the apparatus of first order proof procedures.
At is turns out the axioms of Q are sufficient
to theorems that correspond to every recursive function.
Because of this, all recursive functions are representable
in
g_,
but since P is an extension of
9.
and arithmetic is
an extension of P then it is true of both of these
theories that every recursive function is representable.
PART 3.
THE DIAG LEJI.1MA IS ESTABLISHED
In Q we have a function that takes us from expression
(e) to numbers, i.e., it arithmetizes the expressions
involved (and those are as we will see all the expressions
of Q).
This function is Godel numbering.
Godel
nurr~ering
is an assignment of natural numbers to expression in
accordnace with these three conditions:
1) different
expressions receive different Godel numbers; 2) the Godel.
number of any expression of the set of expressions involved
must be calculable; 3) tl:".tere must be a means to decide
whether a number is a Godel number of some expression in
the set provided, and one must be provided with a means
for calculating which expression the Godel number is a
number of.
27
This method of Godel numbering, as we stated
above, arithmetizes the expressions in the set of
29
expressions in this case of Q and of any of its extensions.
What this accomplishes is to allow us to go from sentences
in an interpreted language, with the natural numbers as
the intended domain of that interpretation, which are about
the natural numbers of sentence about to numbered expressions.
Hence the function G (Godel numbering) will be
assumed to be a one-to-one relationship that goes from the
set of expressions into the set of natural numbers.
Hence
any sentence about the expression is a sentence about some
natural number which is obtained via the Godel numbering
function.
So G(e) is the Godel number of some expression
(e) •
Godel numbering is a recursive function and since it
is recursive, it is also representable in Q.
Since it is
representable, then we have in Q every numeral of every
Godel number.
If we are given the set of expressions of
Q and we Godel-number each one of them 1 then we have
present in Q the numeral of e.yery Godel number.
Hence in
a very real sense we have both the expressions of Q and
their names in Q.
It is important to note that the pres-
ence of the numerals for the Godel numbers or otherwise is
not as imperative here as the fact that we have a function
which takes us from expressions to terms.
Though it is
obvious that numerals are terms it is also obvious other
things are as well.
So t.he real import here lies with the
fact that there is a one-to-one relationship between
30
expressions and terms.
In most accounts of quotation,
quotation as a function takes us from expressions to terms;
so it does not matter what function is used as long as
there is a distinct term for every distinct Godel number.
This essentially goes back to our notion of representability in that the representability of some functions is
nothing more than the fact that your language is rich
enough to be able to represent, characterize and talk
about the object being represented.
Hence if you are
given some function, or any object, and its values and
arguments, you want to know if your language can characterize or talk about it (to be more precise you want
to know if you can talk about this object within the
language) ; if the answer is yes then the function is
representable, if the answer is no then the function is
not representable.
So in essence, to the notion of
representability, it does not matter what function you
use to go from expressions to ·terms so long as there is
such a function present and it assigns different t.erms
to different expressions.
There is also a function "diag" which takes us from
Godel number to Godel number.
If we are given a formula
say A(x) and we Godel number it, the result if G(A(x)).
Now if we substitute the Godel number G(A(x)) for x in
A (x) then we get A (G (A (x))) •
If we t.ake the Godel number
of this new formula then the result
i~
G(A{G{A(x)))).
31
This is what the fuction diag does, it takes us from the
Godel number of a formula with at least one free variable
to the Godel number of the formula with at least one free
variable substituted for the variable in the original
formula.
Diag is a recursive function which means that
diag is representable in Q.
Hence there is a term
assigned ·to the function diag, by some function .in Q,
which is present in Q.
So there is a two-place predicate
diag which is a term of Q such that:
X
G(Sub(G(A{x) ,))
if diag (G(A(x))=
28
X
then (y)(diag (G(A(x))ry)+-+y=G(Sub(G{A(x)),A) is in Q.
It follows then that we have:
x
diaQ (G(A(x)) ,G(Sub(G(Ax)) ,A))) is in Q,
i.e., diag (G(A{x)) ,G(A(G(A(x)))) is in Q.
It is well to point. out that diag and other functions might
well be representable in theories in which not all recursive functions are representable, i.e., the function
assigning terms to expressions in other theories might
assign a term to diag but not to all or any other recursive functions.
PART 4.
THE FIXED POINT LEMMA IS ESTABLISHED
We come here to the notion of a fixed··point and then
to a version of Montague's first two lemmas.
£Oint in T for predicate B (y)
able y free)
is a sentence
s
A fixed-
--..
(containing only ·the varisuch that
32
S++B(G(S)) is in T
The above is just a definition of a fixed-point, the
theorem which is the crucial one in Montague's lemma. 1
(we will concern outselves with some variation on it which
I
find to be a clearer statement of the lemma).
The
theorem is this:
~ES++ {B)
representable.
Bx if T is a theory in which diag is
Essentially this says that if T is a
theory in which diag is representable then there is a
fixed-point for every formuly B(y).
Let us assume diag is representable in T for all B(x)
X
(1)
~
(y) diag (G(B(x){y)++y=G(Sub(G(B(x)) ,B)))
X
Let (y) diag (G(B(x) ,y)++y=G(Sub(G(B(x) ,B)))=C
Let A(x) be (y)
(2)
(diag{x,y)+B(y))
h_r
A(G(A(x)))++(y} (diag(G(A(x)) ,y)+B(y)))
Bx
Sub (Ax,C) we get
(3)a ~ (y) diag(G(A(x) ,y)++y=G(Sub(G(A_(x~A
(3)b ~ (y) diag(G(A(x)) ,y)++y=G(A(G(A(x))))
From (3b) and (2) by replacement we get
(4)
hr
A(G(A(x} )++(y) (y=G(A(G(A(x))) )+B(y)
From (4) by universal instantation we get
(5)
IT
A(G(A{x))++B(G(A(G(A(x)))))
So the sentence A(G(A(x)) is a fixed-point since it
is in T if and only if the predicate B with the Godel
number of the original sentence being substituted for the
free variable y is in T.
33
so consequently:
(y) (diag (G(A(x)) ,y)-+B(y) is a "fixed-point of B(x)
Q.E.D.
With the establishment of lenuna l we move on to
lerr~a
2.
This lemma 1 will not prove but we will try to
state it in such a way that is clear that it is true.
What lemma 2 says is that if we have a theory T which is
an extension of Q then all recursive functions which are
representable in Q are representable in
•r.
This is made
obvious from noting that if you have a set of representable functions of some theory you add to that theory and
then you still have at least the original set of representable functions.
The set of representable functions
has not gotten smaller in size nor has any of the original
members been removed or replaced.
This fact would apply
no matter whether the set of representable functions is
recursive or not.
This lemma would apply to non-recursive
as well as recursive
PART 5:
functions~
L6B'S THEOREM AND MONTAGUE'S
3 AND 4 ARE PROVED
LE!~S
There are two crucially important elements other than
diag itself,. to be noted about the proof of lenuna 1.
One
is the naming functionf i.e+, the function which assigns
terms to
expn~ssion,
and t:he o"cher is substi tut:ion.
Q and
34
many other theories have these as part of the theory; so
it is because of this that we can show that these theories
cannot contain certain things either, along with showing
that they contain certain things.
us to lemma 3.
It is this that brings
Lemma 3 essentially tells us that, if a
theory cont.ains certain things in conjunction with other
certain things, then it is a consistent theory.
Let us first examine an inconsistency lernna before
moving on to Montague's third lemma.
This will help us
get a hold of the notion of inconsistency.
The lemma
actually states that the set of Godel numbers is not
definable in Q or any extension thereof without
inconsistency.
First we need a definition:
a set, C, of natural
numbers is definable in T if there is a formula B(x) ofT
such that for any n,
if n is an element of C, then B(n) is in T.
and
if n is not an element of C, then -B(n)
is in T.
With the preliminary definition out of the way, and the
fact that we know T to be an extension of Q, we launch
into our lemma.
We will assume C(y) to define the set of Godel numbers of theorems of T.
Hence by the diagonal lemma there
is a sentence of the language such as:
( 1)
trrT
S+-+-C(G(S))
35
Now if we suppose S not to be a theorem of T then
G(S) is not a member of C since C(y) defines the set of
Godel numbers ofT.
T.
So then by hypothesis -C(G(S)) is in
This and number (1) tell us
1-T s
( 2)
If S is in T then we know that it is a theorem of the
theory.
If S .is a theorem then G(S) .is the Godel number
of the theorem and since C(y) defines the Godel number of
T 1 then G(S) is a number of C(y).
of C(y) then C(G(S)) is in T.
Since G(S) is a member
We now have an inconsis-
tency in that both the statements of C(G(S)) and -C(G(S))
are in T.
This essentially shows that the set of Godel
numbers cannot be defined in T, but it also serves to
point out the notion of inconsistency we will work with.
Tarski's theorem says that the set of Godel numbers
or sentences true in N {the standard interpretation of the
language of arithmetic) is not definable in arithmetic.
To see this we first must assume the preliminary remarks
for the above lemma to be true here.
Now we show that if
the set G (e) such that. e is a sentence of the language of
arithmetic and e is true in N 1 is definable in arithmetic,
then arithmetic is inconsistent.
This is a straight-
forward application of the above lemma.
It shows that,
since we know arithmetic, which is an extension of Q and
by definition consistent, is inconsistent under the assumption that the set of Godel numbers of N is definable in
36
arithmetic, the assumption is wrong and the set of Godel
numbers of N is not definable in arithmetic.
Now I want to take a look atM. H. L6b's theorem
which is a refinement on Tarski's theorem and the results
of the previous lemma.
Lob's theorem shows in essence
that if we are to define a provability predicate, B(y) for
some theory T, which is an extension of Q, in a particular
way then one to define the provability predicate cannot be
the condition that ~ B(G(A)) A.
If this condition is
added to the other three then any sentence whatsoever would
be a theorem ofT, i.e., T would be inconsistent.
Let us
see how that goes:
~ B(G(A))
(1)
If!T A then
(2)
hr
B ( G (A) +G ( C ) ) + B ( G (A) ) + B ( G (C) ) )
(3)
hr
B(G(A))+B(G(B(~(A))))
These first three assumptions are the conditions for
B(y) to be a provability predicate for T.
That is, if for
all sentences A and C of the language of arithmetic B(y)
satisfies conditions (1),
(2) and (3) above, then it is a
provability predicate for theory T.
The theorem states:
if B(y) is a provability predi-
cate forT, then for any sentence A, if
then
hr
hr
B(G(A))+A,
A.
To see if
IT
B(G(A))+A is the fourth condition for
the definition of provability predicate, we will assume
it is step (4) in our direct proof.
This is while keeping
37
in mind that if condiiton four is a part of the definition
of provability predicate then the result of this proof
should show that
( 4)
Assume
hr
IT
A.
Let us see if it does.
B (G (A)) -+A
Let D(y)=(B(y)-+A).
(.5)
hr
C+--+(Bi_G(C))-+A is the result of applying the diag
lemma to D(y).
(6)
hr
Hence:
Hence:
C+--+(B(G(C2L)-~A)
From (1) and (6) we get:
(7)
hr
B(G{C-+B(G(C))--rA)))
From (2) and (7) we get:
(8)
hr
B(G(C(-+(B(§j_~))-+A))-+(B(G_(C))-+B(G(B(~))
From (7) and (8) we get:
(9)
hr
B(G(C))-+B(G_(B(§Jfl)-+A))
From (2} and (9) we get:
(10)
hr
B
(G (B (Q_~) -+A))-+ (B (G (B (Q..(_9_))) ->-.B (G (A))))
From (9) and (10) we get:
(11)
hr
(B (G (C))-+ (B (G (B (G (C))))
From (3) we get:
(12)
~T B(G(C))-+B(G(B(~)))
From (11) and (12) we get:
(13)
hr
B(G(C))-+B{G(A)l._
From (1) and (13) we get:
(14)
hj_, E-( G (C) ) -+A
From (2) and (14) we get:
(15)
hr
c
-+B (§_JA))))
-+A))
38
From (1) and (15) we get:
(16)
hr
B(G(C))
Finally from (14) and (16) we get:
(17)
hrA
Since A is a variable ranging over any sentence in the
language whatsoever and is one of the conditions for B(y)
to be a provability predicate and since it has been shown
that A is provable in T if h_r, B(G(A)) +A, it follows t:hat
any sentence can be proven to be in T.
To avoid the ob-
vious inconsistency generated by accepting the above as
one of the conditions for B(y) being a provability predicate and also since T is consistent by definition, we must
reject
IT
B(§jA))+A; hence B(G)A))+A is not a member ofT,
i.e., is not provable in T.
Montague in his lemma 3 makes use of Lob's theorem
and in fact his lemma 3 is a straightforward application
of Lob's theorem with a few refinements.
Let us see how
that goes:
(1)
(2)
hr
f-±
B(G(A))+A
B(G(A))+A then
1-T
hr
B(G(G(A) )+A))
{ 3)
~
A then
( 4}
br
B(G(A+C}} and hj_, B(§(A)) then
(5)
B (G (A))
hr
B(§_(C))
T is an extension of Q.
The claim then is, that T is inconsistent given the
above stated five conditions.
We will let C be a valid
sentence of Q from which all valid sentences of Q are
39
logically derivable.
There is at least one sentence of T
which is not derivable from C since T is an extension of Q.
So this amounts to some sentence D such that_ C->- -D.
know that D+-->-B(G(D))
D+--+B (G (C-+-D))
is in T.
is in T.
We
Since D is C -+-D we have
Here we come to the first: step i.n
the proof:
( 6)
D+--+ B ( G ( C -+ -D) )
Now C is a sub-set of L and L is the set of logically
valid sentences of T.
So we have
hr
C-+ (D+--~B (G (C-+·-D))) .
By a theorem of sentential logic we get:
( 7)
~
( B ( G ( C -D) ) -+ ( C-+-D) ) ·+ { C-+-D)
Letting H==(B{G(C-+.-D))-+(C->--D))
and from (3) and (4) and (7)
we get:
(8)
~
B(G(H-+(C-+-D)))
( 4)
From ( 8) ,
(9)
.
hr, B (G (C-+-D))
and ( 9) we get:
From ( 6)
(10)
and (2) we get:
~D
From (1) and (9) we get:
(11}
~ C-+-D
From the fact that T is an extension of Q (S) and ~Q C
then
~ C hence we have C and conjoin it with
(12)
~ -D
(10)
and (2) conjoined we get~
hr
(11}
to get:
D and -D, an inconsis·-
tency.
Hence T is inconsistent under the assumptions
(1-5).
Therefore if a provability predicate or any
40
predicate meets conditions (1-5) then the theory which
defines it is inconsistent.
If we know that T is by
definition consistent then we know that we cannot have
a predicate which meets these five conditions as a predicate of the theory.
Now lemma 4, even though not in the mainstream of
our interests here, still has some ancillary results of
interest.
I will just state these results and now show
the proof for lemma 4.
Essentially 4 tells us that given
the appropriate changes mentioned about lemma 3 we can
generate a statement in T of the form -B(G(H}).
The
changes needed are in condition four of lemma 3 and we
need not assume condition 2 at all.
reads:
if
hr
B(G(A+C)) then
hr
The new condition 4
B(G(H) )+B(G(C)).
There
should be no qualms with this change for this is only a
law of distribution over the conditional.
Now H in the
conclusion must be of the form B(G(C))+C.
PART 6.
MONTAGUE'S THEOREMS 1 AND 4 ARE PROVED
We now come to the application of the above lemma
that Montague makes.
This application is a consideration
of t.he modal notions and whether the semantic predicate N
is going to do the job.
Now if N is a predicate, which
the semantic view holds, then there must be a formula
which expresses it; hence we get N equaling a formula,
say, B(y) with (y) as its only free variable.
If we take
41
N to be B of our previous lenunas we will find the problem
Montague points out to be clear.
N(G(A))
when
N
We get a sentence A
is the semantic predicate.
In theorem 1 we suppose that T is any theory such
that
(1)
T is an extension of Q
( 2)
~ N(G(A))-rA
( 3)
hr
( 4)
~ N (G (A-rC)+{N {G (A)) +N (G (C))
( 5)
h:rN(G{A})
N ( G ( N ( ~} +A) )
if hrA
Then T is inconsistent.
Thie theorem goes through
by applying lemma 3 hence I wi 11 no·t prove it.
It seems
that with a little inspection the denial of any of {2-5)
would be unjustifiable.
Surely n1milier (2) is acceptable,
for wha·t could be meant by the necessity of A if we don't
mean to say that necessary A implies A is true.
Nu~ber
(3) also is a law of modal logic so we cannot eliminate
that without changing our whole modal system.
We are
trying to find a way to interpret necessity given the
present structure, not to change the structure to fit
some interpretation.
Number (4) is also a modal law of
distribution of the modal operator.
Number (5) is the
law of necessitation and also cannot be dispensed with.
So this trings us to number (1).
Number (1) can be dis-
pensed with but at th£ risk of certain other things.
42
Anyway the results of theorem 1 would directly apply to
all those extensions of Q.
In theorem 2 we have no·thing o·ther than an extension
of lenu'Ua 4 to modal logic.
Hence from the results of
lemma 4 theorem 2 generates a statement of the form:
hr
-N(~{N(~+A))
Theorem 3 tells us that if we are given a theory T
such that it satisfies conditions (1) and (2) of theorem 1
in conjunction with condition ( 6)
hr
N (G (A))
whenever ~.\ is
a sentence such that ~ A, then T is inconsistent.
This
result is obtained from lemma 3.
Theorem 4 follows from theorem 1 and says that if
you have any theory with the same constants as P and it
is a true theory and if you have a formula of P defining
the set of valid sentences of the theory, say U, then
N (G (N
(~_)
-l--A)) is not true for some sentences of P.
Theorems 2-4 are just like lemma 4, not really in
the mainstream of our interests here, so a further discussion might facilitat.e confusion and in essence is super·flous.
Theorem 1 is our mainstream interest and its
results are of the most importance to us and the issues.
Montague concludes the section of this paper we are
dealing with my making· some observations about theorem 1.
He says of conditions (2-5) basically what I have said
about them and I will not dwell any further on that.
says of condition (1) in addition to what I have said,
He
43
that condition (1) is not an unnaturally imposed c6ndition,
for after all modal logic should be able to be applied to
.
. .
t
an ar b 1trary
sUbJect
mat-er
even
. 'h met1c.
.
29
ar1~
Now we come to the controversial quote, "Thus if
necessity is to be treated syntactically, that is, as a
predicate of sentences, as Carnap and Quine have urged,
then virtually all of modal logic, even the weak system
Sl, must be sacrificed."
30
Whether this statement is true
or not remains to be seen but it is
~t
least true of Q or
any of its extensions, for as we remarked before, in conjunction with theorem 1, if we were to accept N as the
semant:ic predicate then we must give up one or more of
conditions (2-5) which would entail a change in our modal
systems.
Since we are trying to find an interpretation
of necessity that fits our systems as they stand, the
move to reject (2-5) is quite undesirable, in fact out
of the question.
PARrr 7. A RES'rATEN'LEN'I' OF THEOREM 1
WITH LOB'S THEOREM BUT WITHOUT
MONTAGUE'S LEMMA 3
Up to this point we have shown the similarities
between Montague's lemma 3 and that of Lob's theorem.
It seems then that one should be able to go directly from
Lob's theorem to Montague's theorem 1 while bypassing
Montague's lemma 3.
This is in fact the case.
There
would need to be a change in condition {3) on Montague's
44
theorem 1.
(3')
In place of this condition we will put:
N(G(A))+N(G(N(~)))
Even though Montague's condition three is a direct result
of an application of Lob's conditions 1 and 3 by modus
Ponems.
It is much more readily apparent that theorem 1
follows from Lob's theorem directly as well as from
Montague's lemma 3.
The change in condition (3) is equally
as indispensable as Montague's original condition (3).
Three (3') prime is also a law of modal logic (that being
NA+NNA which is a law of S3 and higher) .
The remarks made
by Montague are equally as applicable or not applicable
even given the proof of theorem 1 of Montague from Lob's
theorem.
Hence it is obvious that the results of
Montague's theorems can be obtained directly from Lob's
theorem.
If we disregard Montague's lemma 3 and take Lob's
theorem in its stead we find that a variation of
Montague's theorem 1 about modal logic follows.
So it
is apparent that if we are to have any cogent objections
to theorem 1 they cannot be objections which deal only
with Montague's results.
These objections must go even
deeper to at least Lob's results.
SECTION IV
PART l.
IN'l'RODUCTION TO SKYRMS' POSITION
We have seen Quine's criticisms of the problem and
his proposed solution.
We have seen Montague's criticisms
of Quine and noted the fact that Montague understood the
problem as Quine did.
We also noted that there was to be
no simple way to ignore Montague's remarks by assessing
that Montague was not aware of the initial problem in the
way Quine did, for he did.
We now come by Brian Skyrms'
article "An Immaculate Conception of Modality"
31
wherein
he takes exception to Montague's concluding remark in his
article.
The remark in question is this, "Thus if neces-
sity is to be treated syntactically, that is, as a predicate of a sentence, as Carnap and Quine have urged, then
virutally all of modal logic, even the weak system
must be sacrificed."
32
s1 ,
Skyrms not only takes exception
to this remark but also paints a picture which is supposed to shov; how Quine's posi·tion can avoid Montague's
objections.
It is apparent from the fact that Skyrms
cites Quine and Montague as references that he, like
Montague, understands the problem as Quine.
Hence any
objection to Skyrms must meet Skyrms' points head-on and
45
46
not sidestep them by charging Skyrms with not understanding
the issue.
PART 2.
LAYING OUT OF THE RULES FOR AND A
DESCRIPTION OF THE OBJECT LANGUAGE{S) AND
'rHE HODl-'I.L LANGUAGE ( S) AS WELL AS
THE FUNCTION C
" ... We start with a base language L
0
1
which we use
as an object language upon which to erect a cumulative
hierarchy of metalanguages L , L 21 ••• 1 which we then
1
33
collect t.ogether in one language Lw."
This is the
strategy Skyrms lays out for us for constructing his
metalinguistic system.
Upon a basic language we construct
our hierarchy by taking as part of L , the next level up
1
in the hierarchy, not only L
0
one item in L
0
not only L
0
of L
0
.
but the name of at least
and usually all items.
Hence L
1 contains
as part of it but the names of the objects
The rules Skyrms sets out for the construction of
each metalanguage are:
"The L
ns
The sentences of Ln+l are the smallest
set satisfying the following conditions:
( 1)
If S is a sentence of L , then S and *Q(S)
n
are sentences of Ln+l*
(2)
If S and T are sentences of Ln+l then (SvT),
(SAT)
Ln+l"
1
(S~T)
.. 34
,
-s
and -T are sentences of
47
The import of this is that the metalanguage contains the
names (which for Skyrms is signified by 'Q' while the
asterisk signifies the place for the semantic predicate
which either "is valid" or ''is provable") of the sentence
of the objects of the linguistic level just below is the
hierarchy.
The metalanguage also contains objects, i.e.,
the sentences of the lower language and sentences which
ascribe a semantic predicate, either validity or provability, to these names of the sentences of the object language.
So with each metalinguistic level up in the hier-
archy we have at least those three items contained therein.
The above construction makes the language Lw, which is the
union of all the metalanguages, equivalen·t to the last
member of the hierarchy when we randomly choose one as
the end point, i.e., if we randomly choose the 17th level
of the hierarchy to be the stopping point then Lw will be
equal to the language L
that is to say they will have
17
the same set of theorems.
L0 is the basis· for Lm which, for Skyrms, is
th~~
modal language which contains the set of modal theorems
which is part of the modal theory being set up.
So as we
can see the language of modal logic (Lm) is constructed
from the chosen language L
0
"L :
in conjuction with the rule:
the sentences of L are the smallest set satisfying
m
m
the following conditions:
48
( 1)
If S is a sentence of L
0
then S is a sentence
of L .
m
(2)
(S"'T)
1
If S and T are sentences of Lm then (SvT),
(·-S), and
L m· u35
·
S are 1n
The whole point of Skyrms' paper is to show how we
can set up a metalinguistic heirarchy and with a particular
function map the modal sentences of Lm into Lw' i.e., with
the function map the sentences of r.m into some one metalanguage or another.
In essence the function picks out
the metalinguistic level of the hierarchy which the modal
sentence corresponds to.
Since each sentence of Lw is a
member of some lowest level metalanguage, the process of
matching the members of Lw with the member of Lm is also
the process of picking out meta level corresponding to
each modal sentence.
This ftmction \¥hich maps the modal
sentences into Lw is designated as C and C(S) is known as
the metalinguistic counterpart of S.
'rhe parameters
governing the behavior of C are:
"C is a mapping from the sentences of Lm to the
sentences of Lw such that:
(1)
If S is free of modalities then C(S)=S
(2)
If Sis N(R), then C(S)=*Q(C(R))
(3)
If Sis (RVT),
(C (R) vc (T))
(R T)y R T), or-R then C(S) is
(C (R)
respectively.
n
36
c (T))
,
{C {R)
c (Tl)
r
or
-c (R),
49
Hence from the characterization of C we can see why Skyrms
must make certain that at each meta level there are statements predicating either validity or provability to the
names of the sentences in the object level of that meta
level language.
It is these sentences which turn out to
be the metalinguistic counterparts to the modalized sentences of Lm.
PART 3. THE SOUNDNESS AND COMPLETENESS
THEOREMS FOR SKYRMS' METALANGUAGES
ARE ESTABLISHED
The asterisk is meant by Skyrms to be treated as
either "is valid" or "is provable."
The notion of valid-
ity is that of truth in models, to be more specific a
sentence is valid if it is true in all models.
So in
order to get a feel for this definition we must understand
what a model is.
Essentially the modeling structure is
set up similarly to the metalinguistic hierarchy in that
we begin with a model f for the base language L
f 0 we extend it to build a model for the L
0
From
.
This is
ns
done in accordance with the rules stated below:
"Models for the L
induced by a model for L :
ns
o
(A) The model for L
0
just f
0
induced by a model f
0
of L
0
is
•
(B)
The model fn+l of Ln+l induced by model f 0 of L0
is the smallest extension of the model fn of Ln induced
by f 0 such that:
50
(1)
f
(Q (S)) ==S
(2)
f
(*x)=l if x is Q(S) and S is true in all
models of Ln and f
(3)
(*:x:)=O otherwise.
The Sentential connectives v, A,
~,
-, are
interpreted as denoting truth functions in
the usual way.
Model for Lw induced by a model for L0
the model f of
:
L\v induced by a model f 0 of L 0 is the union of the models
f
n
induced by f
of L
0
0
(n
E
w) . .. 37
Hence we see that each model
will induce one and only one chair of models for the
Ln and a corresponding model for the Lw.
models need not be the same every time.
This chain of
That is to say
that with each different model for Lo we will get a different chain of models for the Lns and a different model
for Lw.
All of this is done in accordan6e with the rules
of model building given above.
Not only is the notion of a model important to use in
the definition of validity but it is also important in the
proof of the soundness theorem for SS tha·t Skyrms uses for
his system.
The soundness theorem states that no matter
what you choose for the case language L
0
if S is a thesis
of propositional S-5, then C(S) is true in all models of
Lw.
'rhis means to say that C (S) is true in every model of
Lw indusced by a choice of L
0
.
L 0 has a large number of
models and based upon each one of these models of L0 is
created a chain of models for the L 's and hence a
n
51
different model f
for the L .
w
w
Lw for each model of L
these models of L
0
There is a different model
and C(S) must be true in each of
in order for S to be a thesis of S-5
w
(propositional).
The proof of the soundness theorem is accomplished by
first noting S-5 axioms and the inference rules in conjunction with the tautologies of the propositional system.
Next we check to see whether a sentence of L
m
is an axiom
of S-5, its metalinguistic counterpart is true in all
models of Lw.
Then we check to see if the inference rules
preserve validity of these metalinguistic counterparts.
The inference rules must preserve the truth in every model
of Lw of the sentences of Lm which are axioms of S-5 in
order for this proof to go.
It is a relatively straight-
forward procedure and does not depend upon anything any
more complicated nor technical than checking truth in a
model of these axioms under detachment and necessitation.
The next theorem of importance to us is completeness.
In order for us to understand the notion of completeness .
here, we need to get clear on a few points.
at the models for L
and the different choices we have for
0
L
0
When we look
then we can wonder just what model principles hold for
all the choices of L
0
are not decidable.
•
There are some choices of L
0
which
The fact that there are such L s means
0
that there are principles in some that are n:ot shared.
There is a set of shared principles among all the L s and
0
52
a set of principles t.hat is not held in common among all
the choices of L
0
.
It is this intersection of the modal
theories induced by the L s which is the set we are inter0
ested in.
It is the technique of taking t.he intersection
of all these L s that Kripke makes use of in his complete0
ness proof.
it is this technique which is used here to
also get completeness.
The set of principles of the
intersection of the modal ·theories induced by the L0 s
turns out to be proved to be at least the principle of SS
as shown by the soundness theorem.
The completeness
theorem on the other hand will show that it is the SS
principles at most v1hich are in this set.
The technique of putting all the L s into what he
0
calls a uniform notation is what eventually gets him this
set of principles o£ the intersection.
He does this via
a function which then generates a new language called L e.
0
The models for the L es
0
are the same models as those for the original L0 s and
hence the models for the, L es are the same. Now we come
w
e
to the Ltn s and the Lms, the Lms generated on the Les that
are undecidable will have some principles that are not
shared in common by all the other Lms.
The completeness theorem for SS is this:
every sen-
tence of L e which is in every modal theory induced by
m
every choice of L e is a theorem of propositional S5.
0
The
remaining point to be made here is the connection between
53
the metalinguistic semantics for S5 and the Kripke
semantics for S5.
This essentially is done by showing
that the metalinguistic counterpart for the Kripke model
is the same as the metalinguistic model.
What happens is
that we le·t Kripke do the work and show that our models
and his are the same.
We make use of the technique of
taking ·the intersection of the base language which is
shared in common with Kripke.
Hence it follows from these
stated facts that if Kripke semantics show completeness
then we have completeness here.
It is in essence relative
to the Kripke semantics that we get completeness.
PART 4.
SKYRMS' CRITICISM OF MONTAGUE
IS MADE HERE
In the first section of Skyrms' paper he addresses
himself to the points of Montague's paper t.hat are held
in contention, in particular the infamous quote.
Skyrms
says that Montague's quote is part of the reason that some
have insisted that Montague has proven the metalinguistic
conception of modality to be inconsistent.
He points out,
as we have, that Montague:s -theorem is a result of Lob's
theorem.
Skyrms sets out an example which purports to show that
his system can accommodate modal logic while providing a
semantic predicate to interpret the model operator.
He
does this my making use of the provability interpretation.
54
Both the provability interpretation and the validity
interpretation of
'*'
make '*' a provability predicate.
Since both interpretations yield a predicate then so long
as any example Skyrms constructs is not restricted to only
one of the predicates, the example should fly no matter
which interpretation of the semantic predicate we use when
setting up our example.
Let us take as our base language L , the theory Q or
0
Robinson arithmetic.
ity predicate.
'I'he semantic predicate is the valid-
The Lob theorem, the Montague, and the
Tarski theorem all make use of Godel numbering in such a
way that we cannot avoid self-reference.
S is in Q and given that G(S)
So if a sentence
is the Godel number of S
tha~
by Lob's theorem there is no provability predicate B such
that, for all sentences
s,
BG(S) S is in Q.
This Skyrms
feels does not conflict with his system for he can hold
that it is true that there is no such predicate for the
system \vhen we use the Godel numbers but ·the catch is that
t..ve are not using Godel numbering here to name our object
in the metalanguage.
The naming procedure is that of
quotation and i.:here is no reason to believe that there
cannot be more than one naming procedure.
It is appar-
ently obVious that if self-reference is at the bottom of
our problem and if self·-reference is embedded in the naming procedure ·then a change in the naming func·tion or
55
procedure should eliminate the problem.
Skyrms does
certainly feel that this is true.
The case for Skyrms becomes even clearer when we note
that a comparison between BG(S) and *Q(S) does not reinstitute the needed paradox that is the result of Lob's theorem.
The point made here is that, if BG and *Q agree in all respects, then there is no difference between the two functions
and hence the paradox would not be avoided; but if there is
a noticeable difference, then apparently there would be no
way of transferring the results of one directly onto the
other functions.
Skyrms admits to an agreement beb-.reen the
two functions when the sentenceS is derivable, thatis when
S is a theorem of Q under these circumstances.
The problem
is, in trying to further such a connection when S is not a
theorem (derivable) .
Skyrms points out that if S is not a
theorem and hence not in Q then neither is BG(S) nor is
*B(S).
Also, if
-s
is in Q then we know that -*Q(S) is in
Q but we do not know if 1 or we have no argument for the fact
that BG(S) is in Q and hence no argument for the fact t.hat
BG(S) S is in Q.
So i t seems that we have a breakdown in
our comparison between the two functions in that they seem
to do different things to sentences that are not theorems
and i t is this difference which seems to prevent the paradox.
Skyrms points out that the crux of this issue is the
.fact that non-provable sentences need not be provably nonprovable.
SECTION V
PART 1.
INTRODUCTION TO THE FINAL SECTION
Has Skyrms made the semantic view a viable system
with his remarks and adjustments?
Has Skyrms avoided
Montague's remarks when we take a close look at the system
as he presents it?
I take it that the mere fact ·that these
questions are still lurking about only indicate that
Skyrms, on the surface of it all, has failed to convince
us of the success of his remarks.
This may be due to
either the subtlety of the points he has made or to the
failure of them to obtain their objective.
In this section I wish to examine the subtlety of
Skyrms' remarks and see if they do what he wishes them to
do.
Skyrms is aware of the fact that these metalanguages
he has set up are not the normal or standard kind which we
are used to.
He tries to facilitate the understanding of
these nonstanoard metalanguages with a further example and
some footnotes at. the end of his paper.
He further wished
t.o_ address himself to the issue of the richness of these
nonstandard metalanguages.
"Les·t the skeletal and some-
what nonstandard me·talangua9es that I have used leave the
reader with the suspicion that paradox is avoided only at
56
57
the cost of very weak metalanguages, I would like to
indicate how paradox is kept at bay even when the skeleton
is fleshed out and fattened, so long as self-reference is
38
avoided."
This quotation notes, as I have pointed out,
that even Skyrms feels there to be a few questions linger-.
ing about.
It is in light of this that I feel his remarks
to be subtle and in need of a closer look to see these
subtleties.
PART 2. WE FLESH OU'I' SKYRMS 1 THEORY IN
ACCORDANCE WITH HIS FOOTNOTES AND
WITHIN HIS PARAMETERS
Let us do as Skyrms says, flesh out and fatten his
system in just the way he indicates.
base language (or L) is chosen.
First a very rich
Upon this base language
will be built not only the metalanguages but also the
modal languages.
After these have been built up, with
the help of our correspondence function C, we will be able
to find the counterpart in the metalanguages which correspond to the modal statement(s).
Hence for C to do its job
in mapping the modal statements onto these metalinguistic
counterparts there must be at least one metalinguistic
statement for each modal statement.
The base language L
language of arithmetic.
0
which we have chosen here is the
L
0
contains a name for every
natural number, operation and relation symbols for every
operation and relation on the natural numbers.
L
0
contains
58
quantifiers and variables interpreted as ranging over the
natural numbers.
The first meta level, L
1
will be constructed to be a
very rich syntactical metalanguage on L
for every sentence of L
0
.
L
1
0
L
.
has a name
1
has a name for every indi-
vidual variable and individual constant of L
0
•
L
1
has a
name for every relation symbol, operation symbol, connective, quantifier, formula, and sequence of formulas of L .
0
The domain of L
on L
0
1
is the natural numbers which is the domain
plus the syntactical objects of L
above {what this amounts to is this:
0
which are noted
the domain of Ln+l
is the domain of Ln with the syntactical objects of Ln
added to it).
So far with the above-noted construction in mind it
is hard to note the difference that makes the difference.
It is hard to put your finger on what it is about these
metalanguages that makes them avoid the Montague remarks,
but this fact is the point of a footnote of Skyrms.
"Notice that, al·though these metalanguages are not closed·
under substitution and quotation (which would lead to selfreference) , they contain the restriction of the correspending relations to the object languages."
39
This is
supposed to shed light upon the issue and it does.
Jl.iost
standard metalanguages name everything in their object
languages; i.e., they have a name for all the objects of
their object language.
So here is one subtle difference.
59
Another difference lies in the fact that we do not have a
closure under substitution.
This means that we do get
theorems from theorems every time upon substitution.
question then is:
The
If this is true, then what constitutes
an appropriate substitution in which theorems are gotten
from theorems?
This question is answered by looking fur-
ther at the footnote.
If we restrict our relations to
their object language then and only then will we get
proper substitution.
To see this and how the subtle changes and restrictions work we will look at an example.
Before we do this,
though, let us make a few more observations about the
system.
First among all the relations on arithmetic at
least some are represented in arithmetic since some are
recursive.
We know that any function is recursive if it
is effectively decidable, i.e.
1
has an algorithm.
So if
a function is not recursive, then there is not even in
principle a way to give an effective procedure for calculation or determining all the values of the function or
for determining if something is not a value of the function.
All recursive functions are mechanically calculable.
Skyrms' naming function Q is a calculable function in that
he gives us what appears to be the standard function of
naming by applying quotes and we can determine with little
trouble if the value of the function is the quoted name of
the value of the variables.
Hence this function is
60
representable in arithmetic.
Nothing really seems to hang
on this fact but it is an interesting aside.
Let us look at the example.
f(x,y)=x+y.
Let us take a function
This is a recursive function; hence it is
representable.
Since it is representable in arithmetic
then there is a formula A(x 1 •.. ,xn+l) such that
(x(n+l))) · (A(Kl, · • · ,Kn,x(n+l)) )+-+x(n+l)=j) is in arithmetic (note that K , ... ,K and j are all named.
n
1
Hence
this function requires that its representing formula must
have the names of the objects of the function in ·the domain of the formula.
Since in the domain of the formulas
of L0 are the numerals, then in L0 the function can only
generate true sentences when the numerals are the values
of the variables of these formulas.
above restriction comes in:
This is where the
the relations, functions, and
.properties, even though they are representable in arithmetic by some formula, if the names of these formulas are
not present in their own domains, cannot become values of
their own variables.
This is not to say that these func-.
tions, i.e., formulas, do not apply to other formulas; it
is certainly true that ever so many of these relations
apply to formulas as well as to numbers and it would be a
serious shortcoming of Skyrms' system if it were unable to
express these facts.
The point is that these further
applica·tions take place in some one of the meta levels;
that is, they take place on the first level in which the
61
names of these formulas become a part of the domain of
these formulas and not before.
Hence the restriction
becomes clear and the notion of theorems from theorems
seems to have been defined.
You get a theorem from a
theorem when the relation involved is restricted to those
things in its domain.
PART 3. SHOW THAT IF SKYID~S DOES NOT RESTRICT
THE GODEL NUMBERING FUNCTION THEN
MONTAGUE 2 S REMARKS STILL STAND
Among the functions representable in arithmetic is
G, the Godel numbering function.
from expressions to nmnbers.
This function takes us
What G does is to take the
formulas, which represent the functions, of arithmetic and
map them into the natural numbers as names.
in L
0
Since we have
a name corresponding to every natural number (it is
unclear whether the names of the natural numbers are the
results of applying the function Q or are ·the numeral names
of numbers) , what the formula representing G needs is the
names of all the formulas of L
0
and the names of all the
nmnbers to be present in the same meta level.
This hap-
pens in L , even though the statement of G's may be in L ,
1
2
the relationship of these statements is in L .
1
Hence the
statements of L 2 are about what goes in L etc.
1
These formulas correspond to some na·tural numbers and
since we have the names of these natural numbers in the
domain of L0
,
·then they are in the domain of L .
1
The
62
formulas in L
are able to express the fact that they can
1
be values of their own variables since some of the names
of the natural nUTI'.bers pick out these formulas.
Specific-
ally there is one natural number which picks out one formulai hence the name(s)
of the natural number picks out
(names) that formula.
Since all recursive f1.mctions are
---
representable in arithmetic, t:hen all recursive functions
are present in L
0
(represented in L
0
) .
Since all recur-
sive functions are represented in L , then all recursive
0
functions are in L 1 .
Since all the recursive functions
are names in L , then all the recursive functions fit the
1
above-noted observation.
So far we still seem to be well
within the prescribed pararneters for substitution and quotation in that these names are members of the domain of
these formulas.
Diag is a representable f1mction mapping the set of
Godel numbers (a subset of the natural numbers) into the
Godel numbers; i.e., Diag maps the set of Godel numbers
into itself.
Diag ·takes the Godel number of a formula and
maps it to another Godel number of a formula.
The rela-
tionship between these two formulas of the diagonalization
function is that the value of the function is the result
of substi t.uting the number which is the Godel number of
the formula back in as the value of the variable of the
formula.
The value of this formula happens to be the Godel
63
number of this function.
All this seems to be able to be
accomplished in L .
1
If we are to prevent the Lob-Montague results, the
failure of some one or more of the steps to this end must
be effected.
If the above observations are correct then
this failure must come at some point beyond the generation
of the diag function.
The next step in the process, as
outlined in the Montague section, is the fixed-point
theorem.
What about the system is unique enough to create
a failure of the fixed-point theorem?
This is obviously
answered by looking to see if there is something required
by this theorem that is not present in the system.
The
theorem states that if T is a theory in which diag is
representable~
ula B(y).
then there is a fixed point for every form-
Essentially all the requirements for the fixed-
point ·theorem are met if diag is represented in the theory.
So the proof goes through basically on the force of the
representability of diag.
Since diag is representable
(given the truth of the above) , then the fixed-point
theorem is established and from this we can conclude that
the failure of the Lob-Montague resul·t must come about at
some place further along.
Tarski's theorem is the next in the progression of
steps to the Lob-Montague theorem.
If we look at the
'l'arski theorem with the same basic question (s)
\\Te
had with
regard to the fixed-point theorem, we note that the
64
theorem might fail only if the fixed-point t'heorem fails.
It surely seems that the rest of this theorem is straightforward.
So in. essence the 'l'arski theorem hangs upon the
nail of the fixed-point theorem.
This nail though is only
as strong as the diag function representability and the
diag function is representable.
Hence the Tarski theorem
seems to not present us with the weakness Skyrms needs.
There is only ohe basic step in the line of reasoning
to the Lob-Montague theorem left.
This step is the Lob
theorem.
If this step does not fail, then where is the
failure?
The Lob and of course Montague theorems are none
other than extensions of Tarski's theorem.
for these theorems seems to stand.
So the basis
What then about these
theorems is required for them to go, that is not required
by any of the steps in the reasoning process, up to this
point, that might make them fail?
Without going through
the proof again, the weakness in this theorem is essent:iall:y the fixed-:point theorem and hence the diag functions
representability.
Since the final results of the Montague
paper can be obt.ained directly from the Lob theorem, if
the·Lob theorem goes then so will the Montague results.
All ·this seems to hang upon the hook of the diag function
and its representability or usefulness.
It seems that it
has been demonstrated satisfactorily tha·t Skyrms has not
prevented the diag function from being represent:able and
hence useful.
Hence under this reading he has not
65
generated a failure in any of the steps of reasoning
leading to the Lob-Montague results with regard to the
diag function.
In the Montague section on the proof for the fixedpoint theorem it showed that the main ingredients for this
proof other than diag were the naming function and substitution; we have a naming function in Q and G.
is still some confusion
There
about the naming function here.
In the normal metalinguistic system used by Montague the
naming function is essentially that of Godelization.
Skyrms has not as far as I can see totally prevented the
use of this function.
In the above section the Godel
numbering i.s working in concert. with his naming function
Q to produce the results, undesirable to Skyrms, of the
Lob-Montague theorem(s).
Skyrms may have intended some-
thing other than the concerted efforts of both these naming functions.
We will look at this a little later.
The final point to be made in this part is that we
still have substitution, that is to say we still are ableto get theorems from theorems within the restrictions
placed upon the somewhat unlimited substi tut.ion of the
standard metalinguistic systems we are used to.
re~trictions
These
placed on substitution by Skyrms do not pre-
vent the question of theorems from theorems so long as
these entities taken as the values of the variables are
in the domain of these formulas which represent the
66
theorems.
There are some vary dramatic results obtained
if one does not have the names of the formulas within the
domain of the formulas.
One such result is best exempli-
fied by taking a basic theorem P+P.
theorem of reflexivity.
P+P is the logical
It stands to reason that, given
a formula that is deducible in Skyrms' system (any formula), if it is deducible in his system then it is deducible
in his system.
In order for us to get a theorem from this
·theorem, we at least need the names of t.he formulas to
stand in for the sentential variables.
In the case of the
logical·formulas if we do not have the names of the formulas in the domain of these formulas, then even simple
laws like the law of reflexibity will not generate other
theorems~
To complicate the issue we notice that many of
the arithmetic formulas stand in relationship to each
other which relationships are expressed by other arithmetic formulas.
To eliminate t.he names of the formulas
from the domain of these formulas
(which as I see it
Skyrms has not done, but I bring the issue up to point out
that it does not seem to be a viable alternative) would
severely restrict the expressibility of the system.
There
seems to be little doubt that, since the formulas of the
system takes names as arguments or values of the variables, then as long as these names are present in the domain of these formulas the restriction on substitution
will not be violated.
We get theorems from theorems
67
within the restric·tions placed upon substitution and
hence are able to represent the Lob-Montague theorem(s)
even with the naming function as Skyrms has set it out.
PART 4.
SHOW THAT EVEN WITHOUT THE GODEL
NUMBERING FUNCTION HIS N~.MING FUNCTION
GETS US ALL WE NEED FOR MONTAGUE'S
CRITICISMS TO GO
Can Skyrms opt to restrict the Godel numbering func·tion such that it is not representable in his system?
Will it help the situation any to place restrictions
either on the Godel numbering function or on the system
such that either Godel numbering never gets started or the
results of Godel numbering are of no import to the system?
Let us start by assuming the other possibility of naming
which was expressed in the prior part:
naming of all the
nu~bers
quotation function.
that is, that the
and formulas is done via the
Let us further assume that the Godel
function is not representable in the systems as Skyrms
has set it. out.
(It seems that Skyrms has not prevented
the representability of the Godel function.
functiu~
The Godel
is still able to help generate the diag function
in concert with Skyrms quotation function.)
The condition
set up by elimination of the representability of the Godel
function (however this is to be effected) is the extreme
condition for the theorems and steps of reasoning towards
the Lob-Montague theorem(s).
This is so because the
68
burden of naming lies with the quotation function.
Hence
these funct.ions must go with the quotations of names.
We effect a beginning by again examining the steps
along the part of reasoning towards the LBb-Montague
theorem(s) and this will be done with the above parameters
in effect.
As has been pointed out before the diagonal
function is a two place between the names of formulas.
It maps the names of the formula into the names of the
other formulas.
The names in this case take the form of
quotes but are nonetheless names.
For the reasons given
in the previous partf there is no reason to think that the
names of the formulas are not in the domain of these formulas at some point.
The diagonalization function takes
us from the names of a formula with free variable X. to
~
the name of a sentence which is the original formula's
name substituted in for that very same formula's variable.
Skyrms has provided us with the names of the numbers in
L
0
and has guaranteed us that the names of the formulas
are present in L
sentences of L
0
•
1
as well as all the names of all the
In L
1
the names of t.he formulas in L
are in the domain of the formula of L .
1
0
The recursive
functions will all be represented by formulas in L0 as
guaranteed by the fact that all recursive functions are
representable in arithmetic.
In L
1
the formulas can now
·take as arguments the names of these formulas since they
are in their domain.
These formulas even though they are
69
still formulas are now also sentences since there is no
longer a variable.
The system guarantees that each Ln+l
the names of the sentences of L
n
will be present.
So now
we have the names of the sentences of L in L along with
1
2
the formulas of L .
1
It now seems that we have the requi-
site stuff to represent the diagonal function in the
system.
With the generation of the formula representing the
diagonal function, if everything falls into place as it
did in the previous part, the fixed-point theorem should
be able to be generated.
Remember that the main items
needed for the fixed-point theorem is first the representability of the diagonal function.
been accomplished.}
and substitution.
(This seems to have
Secondly we need a naming function
The naming function Skyrms has pro-
vided us with in his function Q, and substitution even
with the required parameters, still does not seem to have
been violated.
So with the diagonal function in hand we
have quickly moved on to establish the fixed-point theorem
and we have so far found no failure of any of the steps in
·the reasoning process towards ·the Lob-Nontague theorem(s).
It should seem perfectly clear by now that once we
have represented the diagonal func·tion in our system then
even the
'J~arski
and Lob theorems present no trouble at all
for us to generate.
To cut short the discussion and to
avoid possible redundancy, I will assume that it is in
70
fact obvious that we can now generate the Lob-Montague
theorem(s) even with the full use of quotation as the
naming function instead of the Godel functions alone or
in concert with Q.
·rhe Lob--Montague theorem(s) if they are to be generated in a theory, would seem to be provable regardless
of the naming function in use.
It surely seems that what
has been demonstrated here does not constitute a counterexample to the claim even if it does not prove it either.
The restriction upon substitution, on the other hand, may
be able to make the system such that the diagonal function
is not representable but this will be done at a great
expense to the system.
It will diminish the ability of
the system to express.
Without this expressibility the
usefulness of the system becomes suspect.
All this as.ide,
though, as Skyrms has stated his system, I do not see that
he has blocked
th~:.~
Lob-1o.1ontague theorem and hence he has
not accomplished his original goal.
In this even·t it
would seem that Mont.ague' s remarks still stand.
FOOTNOTES
1
w. v. Quine, "Three Grades of Modal Involvement,"
Ways of Paradox (Harvard University Press, 1977),
pp. 158-176.
~he
2
3
Ibid.
Ibid., p. 176.
Quine, "Reference and Modality," From a~ogical
Point of View (Harper Torch Books, Harper and Row, 1961),
140.
4
p.
SQ u~ne,
.
"Three Grades," p. 160.
6
Ibid.
7
Quine, "Three Grades," pp. 160-161. Quine in this
quoted paragraph makes it clear that a referentially
opaque context takes us from a purely referential occurrence of ~ to a not purely referential occurrence of ~.
He also says that it can render a referential occurrence
non-referential. ·This is not to say that non-referen·tiality is co-extensive with not purely referential. That is
to say that t.he set of not-referential occurrences is a
subject of the set of not purely referential occurrences.
It is important to note that there are some mertibers of the
set of not purely referential occurrences which do in fact
refer and hence do not qualify as non-referential but all·
these are not purely referential.
8
It is not clear what would be an example of a purely
referential occurrence which was taken to the not purely
referen·tial via the opaque context, and still is referential. Quine gives us examples of non-referential occurrences but not of referential occurrences. Heney my noting the and (or) situation, for even though it is not central to my topic here it is not clear just. what would
qualify as a purely referential occurrence taken to not
purely referential while being referential by the opaque
context.
9
Quine, "Three Grades," p. 161.
71
72
10
rbid.
11
rbid., p. 162.
12
13
14
Ibid.
rbid., p. 163-164.
Ibid.
1
P• 163.
15
Quine, "Two Dogmas of Empiricism," Phil Review 60
( 19 51 ) , pp . 2 0- 4 3 •
.
16Q_ulne,
.
"Three Grades," p. 169.
17 Ibid.
(last whole paragraph)
l8B. A. Brody, "Glossary of Logical Terms,"
Macmillan, 1967, vol. 5),
~ncyclo:e~dia of Phil (New York:
p. 68.
19
Ibid., p. 70.
20
Richmond Thomason, Symbolic Logic!
{New York: Macmillan Co., 1970).
An Introduction
21
R.ichard Montague, "Syntactical Treatments of
Modality with Corollaries on Reflexion and Finite
Axiomatizability, in Formal Philosophy, Selected Papers
of Richard Montague, Richmond H. Thomason, ed. (Yale
University Press, 1974), pp. 286-302.
22
Ibid., p. 286.
23.rb.
d· •
-~ l
,
p. 28 7.
24
George S. Boolos and R. C. Jeffrey, Compatibility
and ~ogic (Cambrige University Press, 1974), p. 161. A
statement of ·the seven axioms of R. Robinson's system are
found here but I have had to make some adjustments in the
notational form used to make them conform with that of
Montague's.
25
26
Boolos and Jeffrey, Compatibility, p. 184.
Montague,. "Syntactical Treatments," p. 287.
27 Boo los and ,Jeffrey,
C~!!}patibili ty,
p. 17 3.
73
28
The term G(A(X)) is underlined here to avoid confusion between the expression and the term.
Sub (
·A) is
to be understood as the proper substitution of the term
for the variable x in the formula A and since G(A(x)) is
the term we will have G(A{x)) for x in A.
29
30
Montague,
11
Syntactical Treatments," p. 294.
Ibid.
31
Brian Skyrms, "An Immaculate Conception of
Modality," The Journal of _Philosophy 75 (July 1978):
368-387.
.
32
Montague, "Syntactical Treatments," p. 294.
33Sk yrms,
34
35
36
37
38
39
"Immaculate Conception." p.
Ibid.
Ibid.
rbid.
p.
370.
Ibid., p.
385.
Ibid., p.
386, footnote 20.
I
Ibid.
369.
BIBLIOGIU\.PHY
Boolos, George S. and Jeffrey, R. C. Co~Eatibility and
~ogic.
Cambridge University Press, 1974.
Brody, B. A.
"Glossary of Logical Terms." Encyclopedia
of PhilosoEhy. New York: Macmillan Co., 1967.
Montague, Richard.
"Syntactical 'l'reatments of Hodality
with Corollaries on Reflexion and Finite Aniomatizabi. li ty." Formal PhilosQp.J.TI_. Selected Papers of
Richard Montague, Richmond H. Thomason, ed. Yale
University Press, 1974.
Quine, w. v. D. "Reference and Modality." From A Logical
Point ot_yiew. Harper 'l'orch Books, Harper and Row,
'1961.
"Three Grades of Modal Involvement." The
Ways of Paradox. Harvard University Press, 197-,:"Two Dogmas of Empiricism."
(1951).
~view 60
Philosophical:._
Skyrms, Brian.
"An Immaculate Conception of Modality."
The Journal of Philo~..2E.h.Y 75 (July 1978).
Thomason, Richard H. Symbolic Logic:
New York: Macmillan Co., 1970.
74
An Introduc·tion