Semantics: Handout 5
Analysis of Propositions into Predicates and Constituents
Yılmaz Kılıçaslan
October 15, 2009
1 Need for a more expressive language
Now that we have a system of propositional logic at our disposal, we can translate (indicative)
sentences of English into that system. In this way, we can formally capture the validity of
argument (1) and the validity argument (2) (cf. Handout 3) (along with those of (7) and (8)).
Below are the representations of these arguments in L0, respectively:
(1)
p q
¬p
------q
(2)
p q
¬r  ¬ p
¬r
------q
(1) is an instance of the valid argument schema (11) in Handout 3, which is repeated here as
(3):
(3)
A or B
Not A
-------B
and (2) comprises an instance of the well-known modus ponens:
(4)
If A then B
A
----B
where the premises are the second and third premises of (2) and the conclusion is ¬ p, and an
instance of (3), where the premises are the first premise of (2) and the sub-conclusion
obtained by (4) and the conclusion is q.
However, translating third, fourth and fifth arguments in Handout 3 into L0 would result in the
loss of some parts of their meanings upon which a valid argumentation would be based. Such
valid arguments will be left to be treated later on (cf. Handout 6). Let us now look at another
case of a valid argument whose translation into L0 will bring about the same sort of intolerable
loss in meaning:
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(5)
Muhammad Ali is taller than Richard Nixon.
Richard Nixon is taller than Noam Chomsky.
------------------------------------------------------Muhammad Ali is taller than Noam Chomsky.
This is a valid argument. However, that cannot be shown if we translate the premises and
conclusion into propositional letters:
(6)
p
q
--r
Given that propositional letters are logical variables which could, in principle, stand for any
statement, (6) cannot be considered a valid argument schema (or, more precisely, an instance
of a valid argument schema). The problem with L0 when dealing with arguments like (5) is
that propositions are treated as the most detailed level of analysis. What we need is an
analysis of the propositions constituting (5) into a structure where the individuals and the
relation between them are represented. In what follows, we will see how statements can be
given such a deeper analysis.
2 Predicate-Constituent Analysis
Let us start with simple statements with a clear subject-predicate structure:
(7)
Jacques Chirac is bald.
(8)
Noam Chomsky snores.
Each of the sentences has one part which refers to a property (being bald and snoring) and
another part which refers to some entity (Jacques Chirac and Noam Chomsky). Accordingly,
we need to have (individual) constants / names which are always interpreted in such a way
that they refer to an entity (that is, an individual or an object) and predicate constants which
are always interpreted such that they refer to all kinds of properties which entities (of some
particular sort) may or may not have. Note that individual constants and predicate constants
are logical variables.1 Let us use lowercase letters for individual constants and capital letters
for predicate constants. A well-formed formula corresponding to a sentence can be made by
prefixing a predicate letter to a constant occurring in brackets. If we have some particular
interpretation of the sentences in mind, then we may choose suggestive letters. So, (7) and (8)
might, for example, be represented as (9) and (10), respectively:
(9)
B(j)
(10)
S(n)
1
Recall that the expressions which the validity of argument schemata of a logical system depends on are called
the logical constants of that logical system, as their meaning is completely fixed in the system. Linguistic items
whose meaning does not take part in determining the validity of arguments are treated as logical variables. The
denotation of items of the latter type varies from one model to another.
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Until the end of the nineteenth century, statements with subject-predicate structure were the
only individual statements which were taken seriously. This was the only structure which
Aristotle ascribed to propositions. Besides these, however, there are other kinds of individual
statements which from a logical point of view cannot profitably be analyzed in terms of
subjects and predicates. Sentences which say that two entities bear some particular
relationship to each other are a case in point. The premises and conclusion of argument (5) are
examples of such sentences. These sentences too could be translated in the subject-predicate
schema. The first premise would be translated as R(m), with m translating Muhammad Ali,
and R standing for is taller than Richard Nixon, whereas the second would come out as N(r),
with r for Richard Nixon and N for is taller than Noam Chomsky, and the conclusion would
read N(m). But the argument schema:
(11)
R(m)
N(r)
------N(m)
cannot be shown to be valid.
What we need is an approach to the structure of propositions where not only subject-predicate
forms like:
(12)
a is P
but also relational forms like:
(13)
a R b (a bears the relation R to b)
can find an equally important place. It was Gottlob Frege who first appreciated the importance
of relational propositions. He removed the grammatical notion of subject from the central
place it had previously occupied in logic. This gives way to the concept of a constituent, a
term which refers to an entity. Any number of different constituents may appear in relational
propositions, and none enjoys a privileged position above the others. There is no need to
single out a unique subject. Frege’s own example motivating this departure from traditional
practice is still quite instructive. He notes that the sentence
(14)
The Greeks defeated the Persians at Plataea.
which would appear to be about the Greeks (as subject), is in fact synonymous with the
passive construction:
(15)
The Persians were defeated by the Greeks at Plataea.
If subjects are to be singled out, the Persians would seem to be the subject of (15). The lesson
to be drawn is that neither of the constituents the Greeks and the Persians is logically more
important than the other. There may be differences between (14) and (15), but they do not
belong to logic.
Having relaxed the structure of propositions in the way indicated above, (5) can now be
turned into the schema:
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(16)
T(m, r)
T(r, n)
--------T(m, n)
where T stands for is bigger than and m, r, and n for Muhammad Ali, Richard Nixon and
Noam Chomsky, respectively. Once we add an extra premise stating the transitivity of the
relation R for all constituent entities, we will be able to show that the argument is valid. The
construction of this latter premise requires us to employ quantifiers and variables which will
be introduced later on (cf. Handout 6). We will now introduce the formal specifications of two
simple languages, L1 and L1E, which are essentially the propositional calculus with
propositions analyzed into predicates and constituents.
3 The Language L1
In this new language, the propositional letters will disappear and individual and predicate
symbols will be our new basic expressions. The rest of the syntax will be formulated
accordingly.
Syntax of L1
A. Basic Expressions:
Category
Basic Expressions
Names
d, n, j, and m
One-place predicates M, B
Two-place predicates K, L
B. Formation Rules:
1. If δ is a one-place predicate and α is a name, then δ(α) is a sentence.
2. If γ is a two-place predicate and α and β are names, then γ(α, β) is a sentence.
3. If φ is a sentence, then ¬φ is a sentence.
4. If φ and ψ are sentences, then [φ  ψ] is a sentence.
5. If φ and ψ are sentences, then [φ  ψ] is a sentence.
6. If φ and ψ are sentences, then [φ  ψ] is a sentence.
7. If φ and ψ are sentences, then [φ  ψ] is a sentence.
L1 has two kinds of formation rules. There are rules used to produce atomic sentences by
combining predicates with an appropriate number of names. These are rules 1. and 2. above.
There are also rules which form a sentence out of one or more other sentences (3.-7. above).
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Semantics of L1
We assume that names take individuals as their semantic values. This valuation is certainly
congruent with our common sense. The way we determine the valuations of predicates is
probably a less intuitive one. One-place and two-place predicates are assumed to take as their
semantic values sets of individuals and sets of pairs of individuals, respectively. For the sake
of simplicity, our universe will consist of just four individuals, namely, Richard Nixon, Noam
Chomsky, John Mitchell and Muhammad Ali. Each semantic valuation will be a function. A
particular model set up in this way is given below:
A. Basic Expressions:
[d] = Richard Nixon
[n] = Noam Chomsky
[j] = Jacque Chirac
[m] = Muhammad Ali
[M] = the set of all living people with moustaches
[B] = the set of all living people who are bald
[K] = the set of all pairs of living people such that the first knows the second
[L] = the set of all pairs of living people such that the first loves the second
Our strategy for determining the semantic values of the sentences will strictly follow the
Principle of Compositionality. Having assigned a semantic value to each basic expression
(e.g., as above), the semantic rules will determine the semantic value of each larger
constituent in terms of the semantic values of its components by taking into consideration
how these constituents are syntactically combined:
B. Semantic Rules:
1. If δ is a one-place predicate and α is a name, then δ(α) is true iff [α]  [δ].
2. If γ is a two-place predicate and α and β are names, then γ(α, β) is true iff <[α], [β]>  [δ].
3. If φ is a sentence, then ¬φ is true iff φ is not true.
4. If φ and ψ are sentences, then [φ  ψ] is true iff φ and ψ are true.
5. If φ and ψ are sentences, then [φ  ψ] is true iff either φ or ψ is true.
6. If φ and ψ are sentences, then [φ  ψ] is true iff either φ is false or ψ is true.
7. If φ and ψ are sentences, then [φ  ψ] is true iff either φ and ψ are both true or else φ and
ψ are both false.
Note that we have introduced the connectives ¬,, , , and  syncategorematically this
time. We could, however, have treated them as basic expressions in, for example, a category
of sentence conjunctions in a way similar to what we did when specifying L0 (cf. Handout 5).
Similarly, several other aspects of our syntax or semantics could have received alternative
formulations. A good way of illustrating which features of L1 are crucial and which are
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matters of conventions is to design a different language. This will be a language that
resembles English to a much greater degree than L1. The syntactic design of the new
language, which we will call L1E, will be somehow different but it will be a semantically
similar language:
4 The Language L1E
Syntax of L1E
Instead of recursive definitions, let us use a (context-free) phrase-structure grammar of the
sort linguists are accustomed to in order to specify the syntax of L1E.
(17)
N  Sadie
N  Liz
N  Hank
Vi  snores
Vi  sleeps
Vi  is-boring
Vt  loves
Vt  hates
Vt  is-boring
Conj  and
Conj  or
Neg  it-is-not-the-case-that
S  S Conj S
S  Neg S
S  N VP
VP  Vi
VP  Vt N
Semantics of L1E
Even though we have used a phrase-structure grammar to formulate the syntax of L1E we will
still adopt the Principle of Compositionality in our interpretation of L1E in much the same way
as with L1. We will assign a semantic value to each lexical item as we did to each basic
expression in L1, and for each syntactic constituent there will be a rule for determining its
semantic value from the semantic value(s) of its sub-constituents. Though we may think of the
phrase-structure rules as defining a tree “from top to bottom,” our semantic rules will be
formulated as proceeding from the bottom of the tree (its terminal nodes) to the top.
Nevertheless, there will be a semantic rule corresponding to each phrase structure rule in the
semantics of L1E, just as there was a one-to-one correspondence between syntactic and
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semantic rules in L1. More formally, for each phrase structure rule α  β1 β2 … βn, there will
be a semantic rule for determining the semantic value of the constituent labeled α in a tree in
terms of the semantic values of the constituents β1 β2 … βn which the node α dominates.
We will first assign semantic values to the names. As with L1, we will want names to denote
individuals, so we may assign the names Sadie, Liz and Hank values as follows:
(18)
[Sadie] = Anwar Sadat
[Liz] = Queen Elizabeth II
[Hank] = Henry Kissinger
We wish to give the semantic values of intransitive verbs, Vi , the effect of singling out a set of
individuals, just as we did with the one place-predicates of L1. We will achieve this effect in a
different way, however.
Sets of individuals are in a one-to-one correspondence with functions that map individuals to
0 or 1. If A is the set of individuals and S is any subset of A, we define a function on the set A
by letting
(19)
fs(a) =
1 if a  S
0 if a  S
The semantic values of Vi’s in L1E will all be characteristic functions of sets of individuals.
Assuming for the sake of simplicity that the three individuals mentioned in (18) are the only
individuals in the world, we might for example stipulate that the semantic values of our Vi’s
are the following functions:
(20)
Anwar Sadat
[snores] =
[sleeps] =
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
[is boring] =
1
Queen Elizabeth II
Henry Kissinger
0
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Let us now ask what sort of values the Vt’s should have. Semantically they seem to express
relations between individuals. According to the grammar, a Vt followed by an N forms a VP.
Therefore, a VP can have as its semantic value a function from individuals to truth values. We
need the semantic value of a Vt to be something that maps the semantic value of an N (i.e. an
individual) into the semantic value of a VP (i.e. a function from individuals to truth values).
Thus, we take the value of a Vt to be a function which yields other functions as its “outputs”.
Its domain will be the set of individuals, and its co-domain (i.e. set within which its “outputs”
must lie) will be the set of all functions from individuals to truth values. The following are the
semantic values of the Vt’s:
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(21) [loves] =
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
Anwar Sadat
Anwar Sadat
0
1
Queen Elizabeth II
Henry Kissinger
Anwar Sadat
0
1
Queen Elizabeth II
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
9
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
Anwar Sadat
Anwar Sadat
0
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
[hates] =
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
10
0
[is-taller-than] =
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
Anwar Sadat
Anwar Sadat
0
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
0
Anwar Sadat
1
Queen Elizabeth II
Henry Kissinger
11
0
The remaining lexical items are the negation operator it-is-not-the-case-that and the twocoordinating conjunctions and and or. We will assume that these have semantic values as
defined in the semantic specification of Lo. Below are the functions assigned as semantic
values to these items:
(22)
[it-is-not-the-case-that] =
1
0
0
1
<1, 1>
[and] =
1
<0, 1>
<1, 0>
[or] =
<0, 0>
0
<1, 1>
1
<0, 1>
<1, 0>
<0, 0>
0
As for the semantic rules, they should be something like the following:
(23)
If α is γ, where γ is any lexical category and β is any lexical item, then [α] = [β].
β
(24)
If α is VP and β is Vi, then [α] = [β].
β
(25)
If α is N and β is VP, and if γ is S, then [γ] = [β]([α]).
α
(26)
β
If α is Vt and β is N, and if γ is VP, then [γ] = [α]([β]).
α
β
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(27)
If α is Neg and φ is S , and if ψ is S, then [ψ] = [α]([φ]).
α
(28)
β
If α is Conj and φ is S , and if ψ is S, and  is S , then [] = [α](<[φ], [ψ]>).
φ
α ψ
5 A Comparison of L1 and L1E
The syntax and semantics of the two languages we have just described are similar in the
overall effect of their semantic interpretations but differ in a number of details. The
differences between the two languages may be summarized as follows:
1. While the basic expressions of L1 look like those of the formal languages found in
logic textbooks, as does the ordering of these in formulas, the lexical items of L1E are
deliberately designed to resemble English, and the word order in L1E is much like that
of English sentences.
2. We used recursive definitions to specify the syntax of L1, but we used a context-free
phrase structure grammar to specify that of L1E.
3. Our semantic rule for L1 classified sentences as “true” or “false” in the meta-language,
while those of L1E simply assigned 1 or 0 as the semantic value of a sentence.
4. The semantic values of one-place predicates were sets in L1, but the semantic values of
Vi’s were characteristic functions of sets in L1E.
5. The semantic values of two-place predicates were sets of pairs in L1E, but the semantic
values of Vt’s in L1E were functions from individuals to the kind of semantic values
assigned to Vi’s.
6. The logical symbols ¬,, , , and  were treated syncategorematically in L1E but the
corresponding lexical items it-is-not-the-case-that, and, etc., in L1E were introduced as
members of the category Neg or Conj, and their semantic values were therefore
defined as functions on truth values (or pairs of them), independently of the semantic
rules corresponding to S  Neg S and S  S Conj S.
However, these six characteristics are not in any essential way tied to the differences between
natural languages and the formal languages of logicians, but rather are independent and
somewhat arbitrary choices which we made for convenience. We could in fact alter the syntax
or semantics of L1 or L1E with respect to almost any one of these characteristics while keeping
the other five the same.
6 Some Features Essential to Truth-Conditional Semantics
Let us now summarize the essential points that are common to the applications of truthconditional semantics we have presented so far:
1. Any truth-conditional semantics is always tightly interconnected with the syntax of the
language in question.
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2. The syntax is required to have at least a set of syntactic categories, one of which is the
category “sentence” or something of the sort – the category associated with truth or
falsity.
3. There must be some initial assignment of expressions of the language to the syntactic
categories and, since in every interesting case we will be dealing with an infinite
language, there must e rules which effect the assignment of the remaining well-formed
expressions to their respective categories.
4. There must be a set of things which can be assigned as semantic values. In the system
assumed thus far these are (i) a set of individuals (ii) a set of truth values, and (iii)
various functions constructed out of these by means of set theory.
5. A specification is needed for each syntactic category of the type of semantic value that
is to be assigned to expressions of that category (e.g., names are to have individuals
assigned, etc.) Sentences are to be assigned truth values.
6. There must be a set of semantic rules specifying how the semantic value of any
complex expression is determined in terms of the semantic values of its components.
7. Each of the basic expressions must be assigned a specific semantic value of the
appropriate type.
7 The Notion of Truth Relative to a Model
If we look at the list of essential components of a truth conditional semantics given above, we
see that the items in the list fall into two broad classes corresponding to the kinds of factors
which go into determining the semantic values of a sentence. We could say, roughly, that a
particular sentence gets the semantic value that it does – is either true or false – because of
certain formal structural properties that it has on the one hand, and on the other hand because
of certain facts about the world. To take an example from L1E, the sentence Liz snores gets the
semantic value 1 (true) because all the following considerations interact to yield this result:
1. Liz has as its semantic value Queen Elizabeth II.
2. snores has as its semantic value a function from individuals to truth values which
maps Queen Elizabeth II into 1.
3. Liz is a name and snores is an intransitive verb, and the truth value of any sentence
composed of a name plus an intransitive verb is determined by applying the function
which is the semantic value of the intransitive verb to the argument which is the
semantic value of the name.
Of these, the last is a theory-internal condition which is specified as a part of our semantic
theory for L1E in particular. The first two, however, are conditions which have to do with
assumed facts about the connections between the language and the world and facts about the
way the world is.
All this discussion is by way of introducing the notion of a model. Formally, a model is an
ordered pair < A, F >, where A is a set, the set of individuals, and F is a function which
assigns semantic values of the appropriate sort to basic expressions. All the rest is taken as the
fixed part of the semantics for a particular language, and we may then examine the effect on
the semantic values of expressions in the language as we allow the model to vary.
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PS: The content and examples of this handout are adopted from Dowty et al (1981) and
Gamut (1991).
References
Dowty D.R., Wall R. E. and Peters S. (1981) Introduction to Montague Semantics. Dordrecht,
The Netherlands: Kluwer Academic Publishers.
Gamut, L.T.F. (1991) Introduction to Logic: Logic, Language, and Meaning (Volume 1).
Chicago: The university of Chicago Press.
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