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Is There a Hierarchy of Indirect Senses and References?
<draft version 3/10/2004>
The goal is to formulate a semantics for (a small fragment of) natural language, and to see if this can
be done without encountering a hierarchy of indirect senses and references. The semantics will be
formulated in a symbolic metalanguage. (This paper is thus written in a meta-metalanguage.) The
metalanguage has the syntax and semantics of the first-order predicate calculus with identity, relation
symbols, and function symbols. There are no non-extensional contexts in the metalanguage. I also
assume here that there are no empty terms in the metalanguage, so it is fully classical.
Structure of that-clauses: I assume that in English a combination of a verb and the word 'that' (such
as 'believe that') is not a syntactic unit. I am confident that this can be fully argued, but in the present
work I will merely assume it. I will assume instead that a that-clause is a syntactic unit which has a
sentence as a syntactic sub-unit. This can also be argued, but I will not argue it here. I will also
assume that a good semantic theory may be developed in which one takes the semantic units to be the
syntactic units. This is not obvious, but it's natural. An indirect confirmation of these assumptions
will be the development of a good semantic theory based on them. That is what this paper is about.
I assume that the syntactic structure of a sentence like 'Agatha believes that Chris speaks' contains at
least this structure:
S
DP
|
Agatha
VP
V
|
believes
CP
C
|
that
S
DP
|
Chris
V
|
speaks
where the meaning and reference of the embedded sentence may be distinct from the meaning and
reference of the that-clause containing the sentence.
There is a long tradition of seeing a that-clause as consisting of only one unit so far as semantics is
concerned. Specifically, one may see a that-clause as consisting of a sentence along with a word,
'that', whose role it is to make expressions in the sentence change from their customary to their
indirect references. Or perhaps 'that' is a word used to signal that these shifts have already occurred.
Or perhaps it is merely a punctuation sign. Thinking in this way, one does not clearly distinguish the
that-clause from the contained sentence when doing semantics. I have often assumed something like
this myself.1
Consider what the situation would be like if there were only one semantically significant unit present
in a that-clause. First, this unit would have to refer to the customary sense of the embedded sentence.
This is almost pure data -- it is what Frege observed when he said "In reported [indirect] speech one
talks about the sense, e.g., of another person's remarks". Second, the sentence, before it is put into
the that-clause, does not refer to its own customary sense. So in being put into the that-clause (or in
becoming a that-clause) the sentence must have changed reference! Voila -- a shift of reference (and
meaning)!
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I don't know of any good reason apart from a conflation of a that-clause with its contained sentence
for any shift of reference to take place.2 I will assume that no shifting occurs. This may or may not
lead to difficulties; we'll have to see.
Since there is no shifting, no expression ever changes meaning or sense. So there are no indirect
senses. So there is no hierarchy of indirect senses.
Order of formulation: I will formulate a semantic theory guided by the following paraphrase of
some remarks by Frege:
The regular connection between a sign, its meaning, and its reference is of such a kind that to
the sign there corresponds a meaning, and to that in turn a reference.
In particular, expressions are to be assigned meanings, and their references are stipulated to be
whatever those meanings present. This is the reverse of the tradition according to which one
produces a theory of reference (perhaps including a theory of truth, a la Davidson) and then uses the
"truth-conditions" employed in that theory of reference as the foundations of a theory of meaning. If
you take this latter approach, difficulties often arise in applying the theory of reference to thatclauses, difficulties which are sometimes addressed by invoking reference-shifting and indirect
senses. These problems do not arise if you take the path of beginning with an account of meaning.
Certain techniques will be adopted simply because they make the theory much simpler. One of these
is that sentences refer to truth-values.
I will also assume that functional signs have functions as their meanings and references. Again, this
makes the theory simpler.3
A number of important issues are postponed for later work. They include:
Empty singular terms.
Meanings that do not present anything.
Garbage control.
Non-extensional contexts other than that-clauses ("PDL sought the fountain of youth", "Possibly she
won't get a ride", etc.)
The "other" hierarchy: e.g. using names and definite descriptions in the object language to refer to
meanings of transitive verbs as if those meanings were themselves individuals.
<list to be continued!!!!!!!>
The Syntax of the Object Language
Stage 1: Crawl before you walk!
Vocabulary:
Singular terms: Chris, Agatha, Maria
Intransitive verb: speaks
Transitive verb: believes
Complex expressions:
A singular term followed by an intransitive verb is a sentence.
Example: Agatha speaks
A sentence preceded by that is a singular term.
Example: that Agatha speaks
A transitive verb flanked by two singular terms is a sentence.
Examples: Maria believes that Chris speaks,
Agatha believes that Maria believes that Chris speaks
Nothing else is in the language so far.
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The Metalanguage
This consists of familiar predicate calculus notation and ordinary logic. There are no indirect
contexts in the metalanguage.
Down-Arrow (Presentation): Each categorical expression of the object language expresses a
meaning. Each meaning presents a thing, which will be the reference of the sign whose meaning it is.
Presentation is the nonlinguistic function alluded to in the underlined part of the slogan "to a sign
there corresponds a meaning, and to that in turn a reference". We use the sign ‘’ for presentation.
That is, '()' stands for whatever  presents. We assume that it obeys this functionality principle
when F is a functional meaning and a, and b are meanings:

 (F(a, b)) = F (a, b)
where the parentheses for functional application surrounding a simple sign are omitted to avoid
clutter. Without this abbreviatory device, the equation would be '(F(a, b)) = (F)((a), (b))'.
(There may be additional limitations on this principle. Very serious issue. To be explored . . . )
Up-Arrow ("that"): I assume that the meaning of a sentence determines the meaning of any thatclause containing it. This is the principle called 'rigidity' in Parsons 1980. Defense of this principle
requires extensive argumentation, some of which is given in that paper, and in Parsons [San Marino].
See also Tyler's defense of the principle in his recent work.
I use the sign ‘’ to stand for the function which maps each sentence meaning to the meaning of any
that-clause containing it. There are two assumptions here: one (rigidity) is that each sentence
meaning determines a unique that-clause meaning, and the other is that one can use a sign to stand for
the unique function that maps a sentence-meaning to the that-clause meaning that it determines. I
make both assumptions.
For Stage 1, I assume that ‘(x)’ is defined whenever x is a sentence meaning (a proposition). To
understand better the significance of '' as used here, read Tyler's discussion of his C function, which
is the same thing (that is, (x) = C(x)), and his discussion of expressions that can be understood by
understanding what they present. (E.g. you understand the meaning of 'that Chris speaks' by
understanding the meaning of 'Chris speaks'.) We seem to have a built-in cognitive ability to get from
the meaning of the sentence to that of the that-clause; that is, we understand the latter on the basis of
understanding the former. This process determines a unique function from the sentence meanings to
the that-clause meanings. The sign '' is used to stand for that function.
Since a that-clause refers to the meaning of its contained sentence, the meaning of the that-clause
must present the meaning of that sentence. It follows that the down-arrow cancels the effect of the
up-arrow:

((x)) = x
whenever x is a sentence-meaning
The reverse principle: ((x)) = x does not hold in general. ['' does not provide "a road back"]
[N.B. '' does not stand for either a meaning or a reference of an expression of the object language, so the
functionality principle for  does not apply to it. Exercise for the reader: Show that applying the functionality
principle to : ((x)) = ()( (x)) leads to absurdity.] 4
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Semantics of the Object Language for Stage 1
Meanings of lexical items: For each vocabulary item of the object language there is a sign in the
metalanguage that stands for its meaning. This must be stipulated for each individual vocabulary
item that has a meaning. [N.B. 'that' is not a vocabulary item which has a meaning.]





mng{Chris} = c
mng{Agatha} = a
mng{Maria} = m
mng{speaks} = S
mng{believes} = B
Meanings of simple singular terms are individual concepts; meanings of verbs are functions that map
sequences of meanings to meanings.
Meanings of Complex Expressions: We have rules for the meanings of complex symbols (where
round parentheses always indicate functional application). (Curly brackets also indicate functional
application; they are used specifically in assigning a meaning or reference to an expression.)

If  is a singular term and  an intransitive verb, mng{} = mng{}(mng{}).

If  is a transitive verb, and ,  singular terms, mng{} = mng{}(mng{}, mng{})

If  is a sentence, mng{that } = (mng{}).
References: If  is any meaningful expression of the object language then

ref{} = (mng{})
That's all, folks. This completes the statement of the semantic theory for the object language of
Stage 1.
Theorem:
Proof:
ref{that } = mng{}
ref{that } = (mng{that }) = ((mng{})) = mng{}
Abbreviations for convenience only: we abbreviate the operation of a down arrow on a simple
italicized sign in the metalanguage by the same sign in roman type. So for the atomic vocabulary we
have:





ref{Chris}
ref{Agatha}
ref{Maria}
ref{speaks}
ref{believes}
=
=
=
=
=
c
a
m
S
B
=
=
=
=
=
c
a
m
S
B
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Some examples:
(i) mng{Chris runs} = mng{runs}(mng{Chris}) = R(c)
(ii) mng{Agatha believes that Chris runs} =
mng{believes}(mng{Agatha}, mng{that Chris runs}) =
B(a, (mng{Chris runs})) =
B(a, (R(c)))
(iii) ref{Agatha believes that Chris runs} = (mng{Agatha believes that Chris runs})
= ( B(a, (R(c))))
= B ( a , ((R(c))) )
= B ( a , R(c) )
= B(a, R(c))
(iv) mng{Maria believes that Agatha believes that Chris runs}
= mng{believes}(mng{Maria}, mng{that Agatha believes that Chris runs})
= B(m, (mng{Agatha believes that Chris runs}))
= B(m, (B(a, (R(c)))))
<using (ii) above>
(v) ref{Maria believes that Agatha believes that Chris runs}
= (mng{Maria believes that Agatha believes that Chris runs})
= (B(m, (B(a,(R(c))))))
<using (iv) above>
= B(m,((B(a,(R(c))))))
= B(m, B(a,(R(c))))
= B(m, B(a,(R(c))))
A Theory of Truth and Meaning
The above semantics already provides a materially adequate theory of meaning and of truth. This is
because for each sentence ‘S’ of the object language the theory entails analogues of:
‘S’ is true iff S
‘S’ means that S
Meaning: The material adequacy condition for meaning is that the theory entails for each sentence
 of the object language something of the form:
mng{} = *
where * is an expression of the metalanguage that stands for the meaning of . The "meaningtranslation" of  into the metalanguage (that is, *) can be precisely delimited as the result of doing
the following:
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(i) Rearrange the expressions of the object language into functional notation. For example,
change ‘ believes ’ into ‘Believes (, )’.
(ii) Replace each ‘that ’ by ‘()’.
(iii) Replace each vocabulary item by the symbol in the metalanguage that is used to identify
its meaning.
For example, the meaning-translation of ‘Agatha believes that Chris speaks’ is produced by the
following transformations:
(i) believes (Agatha, that speaks (Chris))
(ii) believes (Agatha, (speaks (Chris)))
(iii) B(a, (S(c)))
Claim: The material adequacy condition for meaning is met by the theory. We proved some
examples of this above, such as:
mng{Agatha believes that Chris runs} = B(a, (R(c)))
Truth
One form of the material adequacy condition for truth could be that the theory entails for each
sentence  of the object language something of the form:
ref{} = *
where * is the meaning-translation of  into the metalanguage. This is easy; it follows immediately
from what we have proved above together with the principle that ref{} = (mng{}). So in a sense
the job is done. However, this might be seen as containing a question-begging use of the symbol ‘’.5
Can we produce for each sentence of the object language a sentence of the metalanguage that stands
for its reference, and that does not contain ‘’ applied to propositions? Yes, this is possible.
For example, we proved above that:
ref{Agatha believes that Chris speaks} = B(a, S(c))
Also proved:
ref{Agatha believes that Maria believes that Chris speaks} = B(a, B(m, (S(c))))
One might object that the metalanguage proposition that we have produced still uses ‘’ as applied to
primitive names and verbs, since the roman symbols used above have been defined as the application
of ‘’ to the italic symbols. But this can be got around. We could reformulate our semantics so that
we introduce both a roman and an italic symbol of the metalanguage for the reference and meaning
respectively of each simple vocabulary item, not defining one in terms of the other. We would
thereby avoid the hidden use of '' in the metalinguistic sentences used above. (The definitions of
roman letters in terms of italic ones would then be replaced by axioms, one for each pair of symbols.)
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Is There a Hierarchy?
Is the hierarchy still with us? No, it is not. Our present theory makes no use of ever-higher levels of
senses associated with each expression. Consider this meaning-translation of a sentence with a
doubly embedded context, and consider the levels of the things that the meaningful parts stand for:
Believes(Agatha, (Believes(Maria, (Speaks(Chris))))
<proposition>
Believes(Agatha, (Believes(Maria, (Speaks(Chris))))
<propositional concept>
Believes(Agatha, (Believes(Maria, (Speaks(Chris))))
<proposition>
Believes(Agatha, (Believes(Maria, (Speaks(Chris))))
<propositional concept>
Believes(Agatha, (Believes(Maria, (Speaks(Chris))))
<
p r o p o s i t i o n >
The levels don’t keep going up. Every time a that-clause takes us up a level to a propositional
concept, the meaning of ‘believes’ takes us right back down to a proposition. And no expression ever
changes level from one context to another.
Perhaps this illustration is not directly pertinent to the issue since it deals with the structure of the
metalanguage. A similar point can be made directly for the object language. In the following list, we
state what sort of entity is the meaning of each indicated phrase:
Agatha believes that Maria believes that Chris speaks
<proposition>
Agatha believes that Maria believes that Chris speaks
<propositional concept>
Agatha believes that Maria believes that Chris speaks
<proposition>
Agatha believes that Maria believes that Chris speaks
<propositional concept>
Agatha believes that Maria believes that Chris speaks
< p r o p o s i t i o n >
How is the Burge argument avoided? The Burge argument applies to a Fregean theory that does not
distinguish the semantics of sentences in that-clauses from the semantics of whole that-clauses. So it
is not applicable here. Also, the Burge argument takes the form of an RAA of the claim that
a sentence in a doubly embedded context refers to its customary sense. That claim is not true in the
semantics given here, so refuting it is not relevant.
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1
See my "What Do Quotation Marks Name? -- Frege's Theories of Quotations and That-Clauses,"
Philosophical Studies 42 (1982) 315-28.
2
As David Kaplan has pointed out to me, Frege was probably motivated to preserve a principle of
extensionality for reference. I think this is not a good reason to invoke reference shifting, but that takes some
arguing, which I won't attempt here.
Some people dispute this; they think that the meaning of speaks is a non-function S, and the meaning of Chris
speaks is the composition of the meanings of speaks and Chris:
3
mngDUMMETT{Chris speaks} = Comp(S, c)
I will take the meaning of speaks instead to be the function S that maps any singular term meaning c to
whatever Comp maps S and c to:
mng{speaks} = the function which maps any c to Comp(S, c).
This issue is not important to the semantics, which can be formulated either way. Letting semantic values of
functional signs be functions is considerably simpler. (I believe that the "garbage control" problem will arise in
some form whether we take functional meanings to be functions or not. But that's a big investigation.)
Solution of exercise: We have that (P) = P whenever P stands for a proposition.
Functionality of  applied to P would yield:
P = ((P)) = (P) = ((truth value of P))
Let Q be any proposition with the same truth-value as P. Then:
Q = ((Q)) = (Q) = ((truth value of Q)) = ((truth value of P)) = P
This identifies any two propositions that have the same truth-value, which is absurd. QED
4
It may be thought to be question-begging because the limitation of ‘’ to propositions stands for propositional
truth; that is, it stands for the reference of ‘is true’ as applied to propositions. This is because the reference of
‘is true’ as applied to propositions is a function that maps true propositions to T and false ones to F. But that is
what  does when applied to propositions. So the schema laid out above defines sentential truth in terms of
propositional truth.
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