Ch.

10

Formal Semantics
John I Saeed

Sane M Yagi

1
 Formal Semantics
• Formal Semantics is a label usually used for a
family of denotational theories which use logic in
semantic analysis. It has also been called
Conditional Semantics, Model–Theoretic
Semantics, Montague Grammar, and Logical
Semantics.
• In discussing formal semantics, we touch on an
important philosophical divide between
representational and denotational approaches to
meaning.
• For a representational semanticist like Jackendoff,
for example, meaning is the search for mental
representations; hence, semantic analysis
involves discovering the conceptual structure
which underlines language.
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Deonotational vs Representational
Semantic Approaches
• In the representational approach: semantic analysis
involves discovering the conceptual structure (the
representation) which underlies language. For
representationalists (eg, Jackendoff), the search for
meaning is the search for mental representations.
• For denotationalists, a primary function of language
is that it allows us to talk about the world around us,
to describe, or model, facts and situations. Hence,
understanding the meaning of an utterance is being
able to match it with the situation it describes.
3
 Formal Semanticists’ Views
• Formal semanticists employ the
correspondence theory of truth. Speakers
are held to be aware of the situation that
an utterance describes and to be able to
tell whether or not the utterance and the
situation match up. A successful match is
called true but an unsuccessful match is
false. The listener who understands the
sentence is able to determine the truth
conditions of the uttered sentence.
• In the basic version of this approach,
there is no modification using ‘almost’ or
‘nearly’, etc.
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 Objection to this Approach
• On a practical level, this characterization
applies only to statements, since intuitively it
is hard to see how questions and orders can
be viewed as descriptions of situations.
• On a more general level, the idea of correct or
incorrect matches seems to remove the
subjectivity of the speaker. The certainty
shown by a statement might be just one of a
range of speaker attitudes to , or confidence in
, a proposition. This is discussed in terms of
‘modality’ and ‘evidentiality’.

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 Advantages of Formal Semantics
1. It uses logic as a metalanguage;
2. Metalanguage brings into linguistics the
economy and formality of logic.
3. Metalanguage has the advantage of precision.
4. Formal Semantics escapes the problem of
circularity. The metalanguage is semantically
grounded by the link to real word situations.
5. It allows us to see more clearly the
connection between human language and the
simpler sign systems of other primates.

6
Disadvantage of Formal Semantics
• One temporary, practical disadvantage
for students new to formal semantics is
itsapproach being very technical and
highly formalized.

7
 Model-Theoretical Semantics
• The predicate logic described here owes
much to the logician and mathematician
Gottlob Frege. The notion of truth owes
much to Alfred Tarski and the notion of
model-theoretical semantics owes much
of its popularity to the logician Richard
Montague.

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 Model-Theoretical Semantics.
• A model is a formal structure
representing linguistically relevant
aspects of a situation. Linguists’ and
computer scientists’ application of this
approach to linguistic description has led
both to further development of the
model–theoretical approach and the
emergence of a number of related
approaches: eg, discourse representation
theory and situation semantics.

9
 Model-Theoretical Semantics..
• In model-theoretic semantics, analysis
consists of three stages: (1) a translation
from a natural language into a logical
language whose syntax and semantics
are explicitly defined; (2) the
establishment of a mathematical model
of the situation; and (3) a set of
procedures for checking the mapping
between the expressions in the logical
language and the modeled situations.
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 Model
Gamut, L.T.F. (1991)

• Model is the device which makes it possible to

interpret formal systems in model-theoretic
semantics. The expressions of a formal language
are then interpreted with respect to a model. In
propositional logic, this model is an assignment of
truth values to the basic propositional letters of
the language. EXAMPLE: the following example
shows how complex expressions are interpreted in
terms of the truth values that the model assigns to
the propositional letters p and q.
• (i) VM(p & q) = 1 if and only if VM(p) = 1 and VM(q) = 1
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 Model.
Gamut, L.T.F. (1991)

• In predicate logic, the model M consists of a

universe of discourse (D) and a mapping I from
the individual constants and predicate letters
to the universe of discourse. As the example
shows, the interpretation of the formula P(c) is
determined by the denotations that P and c
get from the model.
• (ii) VM( P(c) ) = 1 iff IM(c) in IM(P)

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 Translating English into a Logical
Metalanguag
• We can translate from a sentence in an
individual language like English into an
expression in a universal metalanguage.
• One such metalanguage is propositional
logic, which is focused on sentence
connectives.
• Another is predicate logic which builds on
propositional logic’s sentence
connectives and investigates the internal
structure of sentences.
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 Propositional Logic
Gamut, L.T.F. (1991)

• Propositional logic is the logical system which takes

sentences and their combinations as primitives. The
logical constants of the language are negation and
the connectives &, v, ->, and <->. Propositional
letters (also atomic propositions) are combined with
these connectives into more complex propositional
formulas. The semantics interprets the meaning of
the logical constants in terms of truth-values.
• Propositional logic characterizes a particular class of
valid arguments, like the one in (i) as (iii):

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 Propositional Logic..
Gamut, L.T.F. (1991)
– (i) If the sun is shining, then John is happy
– The sun is shining
– Therefore, John is happy

• When we translate the natural language statements in (i)

into propositional logic (as in (ii)) we get the schema in
(iii).
– (ii) p: the sun is shining
– q: John is happy
– (iii) p -> q
– p
– ------
– q

• Translation into propositional logic makes it clear that the

argument in (i) is valid because of certain logical constants.
The validity of the schema in (iii) can be demonstrated with
a formal syntactic deduction or by means of a truth-table.
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 Predicate Logic
Gamut, L.T.F. (1991)

• Predicate logic is the logical system in which the

atomic propositional letters of propositional
logic are analyzed in terms of combinations of
predicates and individual terms. The basic
expressions are predicates and individual
constants and variables instead of propositions.
EXAMPLE: sentence (i) would be translated in
propositional logic with a mere p (for
proposition).
• (i) John walks

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 Predicate Logic.
Gamut, L.T.F. (1991)

• In predicate logic, we analyze John as an individual

constant j, representing an entity or individual, and
walks as a predicate constant W, representing a
property that is attributed to the individual. Translation of
(i) into predicate logic would result in the proposition
W(j), an atomic formula in predicate logic. The
individual variables allow formulas to be
quantified (into) by means of the existential
quantifier and the universal quantifier.

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 Steps in Translating English into a
Logical Metalanguage
1. Represent the predicate by a capital letter,
e.g.: ‘is asleep’= A; ‘smokes’=S
2. subject argument by a
Represent the
lower case letter from a to t (this is called an
individual constant), e.g: Mulligan=m; Bill=b
3. Create a meaning postulate beginning with
predicate and then the subject
the
constant. Eg, Mulligan is asleep=A(m)

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 Rules in Translating English into a
Logical Metalanguage
• If we want to leave the identity of the subject
unspecified we can use variables,
lower case letters from the
end of the alphabet (eg, w,x,y,z)
– Bill resembles Edie: R(b,e)
– Libby adores Morgan: A(l,m)
– Pete crazier than Ryan: C(p,r)

order of constant terms

• The

mirrors that in English sentences

– Fatima prefers Bill to Henry : P(f,b,h)

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 Rules in Translating English into a
Logical Metalanguage.
•
We can reflect negative and compound sentences by making use of
connectives:
– Fred smokes an Kate drinks : S(f) ^ D(k)
– If bill drinks , jenny gets anger : P(b) ^A(j)
•
complex sentences containing
We can represent

relative clauses by viewing them as a form of conjunction ,

i.e. by using ^ ‘and’’, as in:

– Carrick, who is a millionaire, is a socialist: M(c) ^ S(c)

– Emile is a cat that doesn’t purr: C(
e) ^ ~ P(e)

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 Quantifier
Gamut, L.T.F. (1991)

• Quantifier, in predicate logic, is the logical

constant indicating whether a statement is
universal or particular. The universal quantifier
‘All’ indicates that all entities in the universe
have a given property while the existential
quantifier ‘ThereIs’ indicates that at least one
entity has the property:
– (i) a All(x) [ P(x) ]
– "Every x has property P"
– b ThereIs(y) [ Q(y) ]
– "At least one y has property Q"

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 Quantifier.
Gamut, L.T.F. (1991)

• The term quantifier can either be used for the

symbols All and ThereIs themselves or for the
combination with the variable they bind:
All(x) and ThereIs(y). A more complex use of
quantifiers is shown in (ii):
– (ii) All(x) [ P(x) -> ThereIs(y) [ Q(y) & R(x,y) ]
• which might be the translation of a sentence
like “Every teenage girl adores a rock star”, for
all x, if x is a teenage girl, then there is y, such that y is a rock star and x
adores y.

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 Quantifiers in Predicate Logic
• All language have strategies for allowing a
proposition to be generalized over ranges or
sets of individuals. In English, for example,
quantifiers include words like one, some, a
few, many, a lot, most, and all.
• Some/Many/Most/All students will pass the
exam.
• The simple logical representation we have
developed isn’t able to reflect this ability to
generalize statements over a set of
individuals.
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 Quantifiers in Predicate Logic.
• One way to generalize statements over
a set of individuals is to follow a
proposal of Frege’s that statements
containing quantifiers be divided into
two sections: the quantifying
expression which gives the range of the
generalization; and the rest of the
sentence, which will have a place-holder
element for the quantified nominal.

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 Quantifiers in Predicate Logic..
• Every, All, and Each are represented in
predicate logic by the universal
quantifier (symbolized as ∀).
• We can translate as follows:
• Every student wrote a paper.
• ∀x (S(x) → W(x, p))
• For every thing x, if x is a student then x wrote
a paper.
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 Quantifiers in Predicate Logic…
• Using the same variable letter in the
quantified noun phrase and for the place
holder shows that the noun phrase is
associated with the right position in the
predicate expression . see for example:
• Every student read: (Ax: Sx) Rx
• Every examination is difficult: (Ay: Ey)
Dy

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 5
• We should note that in English some quantifiers can stand
alone , e.g. every thing , everybody , everywhere, These will
have to be translated into complex expression in predicate
logic, as in
• Everything every thing (Ax : Tx)
• Some is represented in predicate logic by the existential
quantifier , symbolized as E. We can thus translate the phrase
some student as in 10.35:
• Ex: Sx)
• Key: S: is a student
• This expression corresponds to the English noun phrases a
student , some student , and at least one student.
• Something some thing (Ex: Tx)

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 6
• Some advantages of predicate logic translation
• Form a linguist’s perspective there are a number of
advantages to the representations have introduced,
We can take as an example the way that the
representations of quantifiers, as introduced above
clarifies some ambiguities found in natural
languages. One of these is scope ambiguity, which
can occur when there is more than one has
quantifier in a sentences. for example
• 10.41: a. everyone loves someone.
• c. There is some Pearson who is loved by
everyone.

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 7
• Version 10.41 is described as having a wide scope
interpretation and it involves a many-to-many
relationship of loving.
• Narrow scope interpretation, and involves a many-
to-one relationship. While the English sentence is
structurally ambiguous between these tow
interpretations, the difference is explicitly shown
in predicate logic by the ordering of quantifiers.
• a. (Ax: Ox) (Ey: py) Lxy wide scope
• b. (Ey: py) (Ax: Ox) Lxy narrow scope

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 8
• Similarly, a second advantage to this from of
representation is that it allows us to
disambiguate some sentences which contain
combinations of quantifiers and negation.
• Everybody didn’t like the concert.
• (Ax:Px)-Lxc
• A third advantage is that this from of
representation allows us to distinguish between
tow uses of definite noun phrases: one of which
is a hidden general or universal statement.

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 The Semantics of the Logical
Metalanguage
• After translating natural language into a logical
formula, we have to relate the set of symbols to
something outside, to the situation described. To do
this we need to add three further elements:
1. Semantic interpretation for the symbols of predicate logic
2. A domain: this is a model of a situation which identifies
the linguistically relevant entities, properties and relations
3. A denotation assignment function: a procedure which
matches the logical symbols for nouns, verbs, etc. with
the items in the model that they denote. This function is
also sometimes called a naming function.
• The domain & naming function are called a model.

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1. Semantic Interpretation of Predicate
Logic Symbols
• Sentences: Following the correspondence theory
of truth, we will take the denotatum of a whole
sentence to be the match or lack of it with the
situation it describes. A match will be called
true(T); a mismatch will be called false (F),
• Constant terms: We will assume the denotation of
constant terms to be individuals or sets of
individuals in the situation.
• Predicates: We will assume that predicates
identify sets of individuals for which the predicate
holds. Thus, a one-place predicate like be
standing will pick out the set of individuals who
are standing in the situation described.
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 2. Domain
• The domain is a representation of the
individuals and relationships in a
situation, which we will call v.
– E.g., Domain can be: A situation in the Cavern
Club, Liverpool in 1962 where the Beatles are
rehearsing for that evening’s performance. We can
identify several individuals in this situation: the Beatles themselves,
John, Paul, George and Ringo, their manager Brian Epstein and one stray fan we’ll call Bob.

• The situation v contains a set of individuals,

U, such that in this case U = {John, Paul,
George, Ringo, Brian Epstein, Bob}.
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 3. Denotation Assignment Function
• This function matches symbols from the
logical representation with elements of the
domain. In our simple example, we can divide
denotation assignment into two parts: (a) the
matching of constant terms with individuals
in the situation v; and (b) the matching of
predicate letters with sets of individuals in v.
a. Matching constant terms: The assignment is a function, which
we can symbolize as F(x). Our function F(x) will return this
extension of arguments in the Beatles’ situation.
– F(j) = John
– F(p) = Paul
– F(g) = George
• i.e., The individual constant j denotes the entity John in
situation v, the constant p denotes Paul, and so on.

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 3. Denotation Assignment Function
• (b) Matching predicate letters: Our function
F(x) will return the extension of predicates
in the Beatles’ situation as follows:
– F(B) = was a Beatle = {John, Paul ,George, Ringo}
– F(M) = was a manager = {Brian Epstein}
– F(S) = sang = {John, Paul}
– F(D) = played the drums = {Ringo}
– F(J) = joked with = {<John, George>}
– F(G) = played guitar = {John, Paul, George}
• Thus, the extension of F ‘joked with’ in the
situation is the set of the ordered pair, John
and George (i.e., John joked with George).
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 Denotation Assignment Function..
• That is precisely how we define the denotational
behavior of logical constituents and establish a
model.
• A model is a combination of a domain and the
assignment function.
• A model is schematically described as:
– Mn=<Un, Fn>, where M = the model; U = the set of
individuals in the situation; and F is our denotation
assignment function. The subscript n (for 1, 2, 3, …n)
on each element relatives the model to one particular
situation.

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 Checking the Truth Value of
Sentences
• Our procedures for checking the truth
value of a sentence must reflect the
compositionality of meaning.
• If this is done correctly, then we will
have shown something of how the
constituents of a sentence contribute
to the truth value of the whole
sentence.

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 Evaluating a Simple Statement
• If we take our model M, we might construct some
relevant sentences in predicate logic as below:
– Dr
– Gb
• To evaluate these sentences, the reader may
translate them back into English. For example, Dr
could be read as “Ringo played the drums”.
• The procedure for checking if Gb is true is based
on the denotational definitions given earlier and
can be schematized as follows:
– [Sj]M1 = 1 iff [j] M1 e [S] M1 = “The sentence ‘John sang’ is
true if and only if the extension of ‘John’ is part of the
set defined by ‘sang’ in the model M1.

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 Evaluating a compound sentence
with ^ ‘and’
• Evaluating a compound sentence follows the same
procedure.
• We can create sentences like these for our model M1:
– Sj ^ Sp
– Fjg ^ Jrb
– Me ^ Fb
– Sj ^ Ibe
• To evaluate any compound sentence P ^ Q, we first
establish the independent truth value of P and then of q.
Afterwards, we evaluate the effect of joining them with ^.
• A compound with ^ is only true when p is true and q is
true.
– [p^q]=1 iff [p] =1 and [q] = 1

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 Evaluating sentences with the
quantifiers A or E
• The same procedure with some modification can be used to evaluate
sentences with the universal and existential quantifiers A and E.
– All the cats hunt Jerry. (Ax: Cx)Hxj
• We reflect the meaning of A, all, by establishing the rule that a sentence
with this quantifier is true if the generalization is true for each
denotation of x. Thus, we need to test the truth of the expression ‘x
hunts Jerry’ for each individual in the situation that x can denote.
• We need now a function that matches variable terms with their
denotation in the situation. Let’s call it gn. This function will tell us that
the following matches are possible since x is a cat.
– x = Tom
– x = Felix
– x = Korky
• All we need to do then is to test the generalization with each value for x,
ie, to use the procedure we used for simple statements earlier to
evaluate each of the following versions:
– x = Tom : is Hxj true/false?
– x = Felix : is Hxj true/false?
– x = Korky : is Hxj true/false?

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 Evaluating sentences with the
quantifiers A or E..
• Sentences containing the existential
quantifier E can be evaluated in the
same way, except the rule for this
quantifier is that:
• If the generalization is true of at least
one individual in the range, the
quantified sentence is true.
– Some cat hunts Jerry. (Ex: Cx)Hxj

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 Intentionality
• Languages contain a whole range of verbs which describe
different mental states. Instead of a flat statement S, we can say
in English for example:
– Frank knows that S.
– Frank believes that S
– Frank doubts that S.
– Frank regrets that S
• One disadvantage of the simple version of the denotational approach
is that it downplays the speaker-hearer's subjectivity. The
procedures we have been outlining allow a mechanical matching
between statements and situations. However, natural languages
communicate interpretations between speakers and hearers.
• If a speaker chooses between different sentences which share the
same proposition, the choice reflects a difference in propositional
attitude between certainty and degrees of lack of certainty:
– John misrepresented his income.
– John probably misrepresented his income.
– John may have misrepresented his income.
• In other words, sentences which reveal this interpretive or cognitive
behavior are said to be intensional and the property is called
intensionality.
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 Challenge for Formal Semantics
• The challenge for formal semantics is to develop
a semantic model that reflects the interpretation
and calculation that is so central to language.
• One strategy has been to enrich the formal
devices in certain areas where intensionality
seems most clearly exhibited in natural
languages. Such areas include modality, tense,
aspect, and verbs of propositional attitude.

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 Modality
• Modality is often described in terms of two related aspects of
meaning:
• 1- Epistemic modality which concerns the resources available to the
speaker to express judgment of facts versus possibility.
• 2- Deontic modality which allows the expression of obligation and
permission in terms of morality and law.
• A two-fold division of epistemic modality was developed: fact
versus possibility, or 'situation as is' versus 'situation as may be'.
One way of discussing this distinction between the actual and the
non-actual is to talk of possible worlds. One way of dealing with
this is to see truth as being relativized to possible situations, or
possible worlds.
• To reflect this, logicians introduce two logical operators  'it is
possible that' and  'it is necessary that'.

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 Deontic Modality
• Deontic modality has been treated in similar
ways: as a projection from the world as it is to the
world as it should be under some moral or legal
code, i.e. as the speaker entertaining an idealized
world. Deontic modal operators have been
suggested for logic, including OØ 'obligatorily that
Ø' and PØ 'permitted that Ø'. The former can be
interpreted denotaionally as 'true in all morally or
legally ideal worlds' and the latter as 'true in
some morally or legally ideal worlds.'

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 Modality
• The first, epistemic modality, concerns the resources available to the speaker to
express judgment of fact versus possibility. The second, deontic modality, allows the
expression of obligation and permission, often in terms of morality and law.
• In response to these facts about modality, modal logics were developed. The simplest
approach employs a tow – fold division of epistemic modality into faced versus
possibility or ‘situation as is’ versus ‘situation as may be’. One way of discussing this
distinction between the actual and the non-actual is to talk of possible worlds a
phrase derived from Leibniz and formally developed by Kripke can recognize the idea
that a speaker, in moving away from certainty, can envisage tow or more possible
scenarios. So if we say Fritz may be on the last train , we entertain two situation ; one
where Fritz is on the train and another where he is not, Thus we imagine one situation
where the statement fritz is on the last train is true and another, where it is not.
• To reflect this, logicians introduce two logical operators ít is possible that’ and ‘it is
necessary that’. These can be put in front of any formula of the predicate logic, e.g.
• It is possible that
• It is necessary that
• Semantic definition of these relies on this new ontology of possible worlds: means
‘true in all possible world’(i.e. no alternatives are envisaged by the speakers) and
mean ‘true in some possible world’(i.e. the speaker does envisage alternative
scenarios)

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 Tense and Aspect
• These two further important intentional categories are, related to the
speaker’s view of time.
– Charles admires Diana.
– Charles admired Diana.
• These sentences might differ in truth value when you read them, and
the only difference between them is their tense. We saw that an
utterance can only be given a truth value relative to a situation: it
seems that part of the character of situations may be their location in
time.
• It seem that part of the character of situations may be their location in time. One
response to this has been to incorporate time into model-theoretic semantics. One
way to do this is to include tense operators. We might include three operators:
Past (Ø), Present (Ø), and Future (Ø). This would allow formulae like:
– Past(Ct j) Tom chased Jerry.
– Present(Ct j) Tom is chasing Jerry.
– Future(Ct j) Tom will chase Jerry.
– Key: C: chase
– t: Tom
– j: Jerry

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 Tense and Aspect..
• Formal approaches have to cope with the various aspectual and
situation-type distinctions.
• Cann proposes, for example, a perfective aspect operator Perf and
an imperfective operator Impf for predicate logic, which will further
relativize the truth of logical formulae. These operators rely on the
idea of intervals of time.
• A perfective formula will be true if both the start and end instances
are included before the reference time point, thus reflecting the
complete interpretation of the perfective aspect, as in sentence 1.
While an imperfective formula will be true if the activity overflows
the time interval that is being interpreted, as in sentences 2.
–  Tom chased Jerry.
– Past (Perf(Ctj))
–  Tom was chasing Jerry (when I opened the door).
– Past (Impf(Ctj))

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 Meaning Postulate
Gamut, L.t.F. (1991)

• Meaning postulate is a device used in logical semantics

to stipulate semantic relations between lexical items.
Meaning postulates were introduced in Carnap (1947)
in order to account for the fact that a sentence like (i) is
an analytic truth, true in every model. The meaning
postulate in (ii) captures this analyticity:
– (i) Bachelors are unmarried
– (ii) For all x, if x is a bachelor, then x is unmarried
• Meaning postulates can be seen as an alternative for
decomposition of word meaning (see Componential
analysis). They are extensively used in Montague
Grammar.

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 Word Meaning: Meaning Postulates
• This is consistent with this approach’s general assumption that the focus of semantic enquiry is
sentence meaning ,the meaning of the word is something best not pursued in isolation but in
terms of their contribution to sentence meaning . Thus most formal approaches define a word’s
meaning as the contribution it makes to the truth value of a sentence containing it.
• However, the original structuralist position that words gain their significance from a combination
of their denotation (reference) and their sense still seems to have force. We can return to our
example from chapter 3: that if an English speaker hears 10.74 below ,he knows 10.75
• I saw my mother just now.
• Speaker saw a woman.
• The meaning postulates approach would recognize that 10.75 follow automatically from
knowledge of 10.74 but rather than state this in terms of components of meaning of either word.
This approach simply identifies relationship as a form of knowledge.
• Let’s look at some lexical relations in this approach, hyponymy. The hyponymy relation ship
between for example, dog and can be represented using, the ‘if…then’ connective,by writing a rule
like 10.77: Az{ DOG (x)…..ANIMAL(x)}
• Binary antonyms Here we can use the “not” symbol as in 10.78 below.
• Ax {DEAD(x)…. ALIVE(x)}
• This approach thus allow the formal semanticist to reflect the network of sentence relation that we
detect in the vocabulary of language, in a format consistent with translation into predict logic and
interpretation via model theory.
• These meaning postulates can be seen as a way of restricting or constraining denotation, e.g. if
something is a dog, then it is an animal ‘tells us something about the denotation behavior of the
word dog.

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