1. The document discusses key concepts in propositional logic including propositions, truth functionality, and sentential connectives. 2. A proposition is defined as the invariant meaning of a sentence, devoid of modality. Propositions involve the meanings of declarative, interrogative, and imperative sentences. 3. Sentential connectives like conjunction, disjunction, implication, equivalence, and negation determine the truth value of complex propositions based on the truth values of their component simple propositions.

1. 1
Group 7
Sentence meaning and propositional content
• What is a proposition
• Proposition content
• Notational representation of propositions
• Truth functionality
• Sentence types and their meaning

2. 2
Proposition (Revisited)
• A Proposition is defined as the invariant meaning
expressed by a sentence, devoid of any modality.
e.g. She is probably right
• Proposition: She is right
• Modality: probable - impossible
• In semantics, the letters ‘p, q, r’ are often used as
symbols of propositions.
• Propositions involve in the meanings of not only
declarative, but also interrogative and imperative
sentences.
e.g. Is she right? (You) be careful

3. 3
Truth - value vs. truth - conditions
• The truth value of a proposition should be
distinguished from the truth conditions of a sentence
e.g. Mary married a rich man.
• Truth value of a proposition: The proposition can be
either true or false.
• Truth conditions of the sentence:
Mary must be a woman
Mary is of a marriageable age

4. 4
Propositional content
1. Propositional content
2. Semantic roles
3. Predicates & Arguments

5. 5
Propositional content
• A proposition usually consists of (a) something which is
named or talked about known as ARGUMENT or entity,
and (b) an assertion or predication made about the
arguments expressed by the PREDICATE.
e.g. The man bit the dog.
The dog bit the man.
• Predicate: BITE
• Two arguments: MAN & DOG
• The meaning of a sentence consists of the predicate,
argument(s), and the role of each argument. When we
specify all these elements, we are talking about the
propositional content of the sentence.

6. 6
Arguments
• Not all entities are arguments
e.g. It rained heavily
• The arguments may fall into two sub-groups: participant
and non-participant.
• Participants are those necessitated by the predication,
and answer the question: Who does what to whom?
• Non-participants are optional and answer the questions:
why, when, where, how?
e.g. The woman hit the man (with a ruler)
• There are three arguments: the woman, the man, ruler
• In standard grammatical treatement, participant
arguments surface as subject, direct or indirect object
whereas non-participant arguments occur as adverbials

7. 7
Predicator - Predicate
• A PREDICATE is any word, or sequence of words, which ,
in a given single sense, can function as the predicator of a
sentence.
e.g. Hungry, in, asleep, hit, show, bottle: are predicates
And, or, but, not: are not predicates
• Predicate and predicator are terms of quite different sorts.
The term ‘predicate’ identifies elements in the language
system, independently of particular example sentences.
• The term ‘predicator’ indentifies the semantic role played
by a particular word (or groups of words) in a particular
sentence.
• A simple sentence has one predicator, although it may
well contain more than one instance of a prediate.

8. 8
Degrees of predicates
• The combination of predicate and arguments can be
defined in terms of degree. The DEGREE of a predicate is
a number indicating the number of arguments it is normally
understood to have in a simple sentence.
• A predicate of degree one (often called a one-place
predicate) is used with one argument
e.g. Asleep, beautiful
• A prediate of degree two (often called a two-place
predicate) is used with two arguments
e.g. Kill, see
• A predicate of degree three (often called a three-place
predicate) is used with three arguments
e.g. Give, make

9. 9
Arguments vs. predicates
• Arguments refer to entities while predicates deal
with events, properties, attributes and states.
• Those individuals that are independent and can
stand alone are arguments.
• Things like qualities, relations, actions and
processes that are dependent and cannot stand
alone are termed predicates.
e.g. My computer
break down, fast, new

10. 10
Predication
• The relationship between entities as arguments and events,
qualities, states as predicates is predication.
• Frawley (1992) defines predication as the way that
individuals instantiate – embody, carry out, take on or are
linked to – properties, actions, events, attributes or states.
e.g. My wife is writing a report
• The event is “writing” because “writing” must be done by
someone, an entity, and of something, another entity.
• Each of the entities “my wife, and a report” is the arguments
of the predicate because they instantiate the “writing”
• However, not all events are predicates.
• In English, not only verbs, but also nouns, adjectives and
propositions can function as predicates

11. 11
Semantic roles
1. Definition
2. Levels of generality
3. Types of semantic roles

12. 12
Definition
• Semantic roles are a means to represent
sentence meaning in logical terms. They are
usually assigned to nouns and noun phrases
according to the relation they hold with the
predicate.
e.g. John is writing a letter
Mary kicked the dog
My mother bought me a car

13. 13
Levels of Generality
• Semantic roles can exist at three levels of generality
1. “Verb-specific” roles
Runner, killer, hater, smeller, receiver, located, sent to…
2. Relation or thematic roles: are generalizations across
the verb-specific roles like:
Agent (giver, speaker, dancer, runner), instrument,
experiencer (liker, thinker, feeler, presumer, lover),
patient…
3. Generalized or macro-roles: are generalizations
across thematic relations.
Actor (agent, instrument, recipient…), Undergoer
(experiencer, patient, stimulus…)

14. 14
Levels of Generality
• Van Valin, Jr (1999) claims that there are only two macro-
roles: ACTOR and UNDERGOER.
• ACTOR is generlization across agent, experiencer,
instrument and other roles, which surface as the
grammatical subjects.
e.g. Mary opened the door
They liked the play very much
The key opened this door
• UNDERGOER is a generalization subsuming patient,
theme, recipient, stimulus, and other roles, which surface
as the grammatical object.
e.g. She made me a cake
You love her because of her money

15. 15
SVO
Examples Roles
surfacing as
S
Roles
surfacing
as O
The wind opened the door Instrument Patient
The girl opened the door Agent Patient
Bob received a mail Recipient Theme
She robbed me of my love Agent Source

16. 16
Notational representation of
propositions
1.
Arguments: single lower-case letters
Predicate: capital letters
• Tom loves Mary  tLOVEm

17. 17
Notational representation of
simple propositions
2.
Argument: bracketed full words
Predicate: capital letters
• Joe was in Hanoi  (Joe) IN (Hanoi)

18. 18
Notes
1. Argument has to be a particular
someone/something
a. Someone get me a drink, please
b. Someone called me last night
2. Predicate can be Adj, V, Prep, N
• She is nice.
• She smiles.
• She is in New York.
• She is a bartender.

19. 19
Notes
3. The less arguments, the less informative the
proposition
My mother wrote a letter.
• My mother wrote me.
• My mother wrote me a letter.
4. Arguments of a proposition may be expressed
by another proposition.
• She said that she was well taken care of.
• Whoever did it must be brought to course.

20. 20
Embedded propositions
• The arguments of a proposition may be expressed by
another proposition, not a referring expression.
e.g. I know that she is a smart person
Do you want me to go there?
I doubt if the film will start on time
She did not tell me what to do
• According to Kreidler (1998), predicates that have
embedded propositions as theme arguments may
include the following
a. Knowledge or ignorance of a possible fact.
e.g. I know that she’s right
I doubt if she will come.

21. 21
b. An attitude or orientation towards a fact or possible fact.
e.g. I am happy that she was able to pass the test
I like to become a doctor
c. Causing, allowing, or preventing the occurence of a fact
e.g. She stopped me from going out
I’ll have the students rewrite their essays
d. Perception of a fact
e.g. I watched the boys playing football
I heard her cry
e. Saying something about a fact or possible facts
e.g. She said that she had lost her belongings
She told us what to do
f. The beginning, continuing, or termination of an event
e.g. I started to smoke
We will continue to have perfect leaders.

22. 22
• Propositions as arguments can occupy subject or object
positions, and they can surface as nominal clauses in the
following ways.
a. Wh- clauses
e.g. What happened last night was appalling
I didn’t like what I saw
b. That- clauses
e.g. That she cried in public was unthinkable
She said that she would come
c. To infinitive clauses
e.g. I want her to succeed.
It is rather hard to get a job these days
d. Ing- clauses
e.g. I saw the teacher coming
Treating her like that is unacceptable.

23. 23
Notational representation of
embedded propositions
• At the first level of representation, the proposition as
argument is symbolized by x.
• At the next level, this x-proposition will be presented in
the usual way for a simple proposition.
e.g. She said that she would wait for me
 At the first level we can have:
sSAYx
 At the next level down we have:
x = sWAITm
 Putting the two together we have: sSAYx
x = sWAITm

24. 24
Truth Functionality
• Definition
• Terminologies
• Five operations

25. 25
Simple and composite sentences
• Simple sentences: one clause
• Composite sentences: more than one clause
 Compound sentences
Either he did not pass his driving test or I am a
Dutchman.
 Complex sentences
If he passes his driving test, I am a Dutchman.

26. 26
Definition
• The truth value of a composite proposition is a
function (in a mathematical sense) of the truth
values of its component propositions (i.e. simple
propositions).
E.g., She is married and she is pregnant

27. 27
Compositional variables
f(p,q)
She dances and he sings.
p q

28. 28
Logical variables
Negation
~
Conjunction &
Disjunction V
Implication 
Equivalence

29. 29
Conjunction
• The English words and and or correspond roughly to
logical connectives. Connectives provide a way of joining
simple propositions to form complex propositions. A
logical analysis must state exactly how joining
propositions by means of a connective affects the truth of
the complex propositions so formed.
• Any number of individual well-formed formulae can be
placed in a sequence with the symbol “&” between each
adjacent pair in the sequence. The result is a complex
wellformed formula.
• E.g. The three simple formulae: jGREETm (John greeted
Mary), jHUGm (John hug Marry) and jKISSm (John
kissed Mary) can be joined together to form:
(jGREETm) & (jHUGm) & (jKISSm)

30. 30
Conjunction
• Truth table for &
p q p & q
T T T
T F F
F T F
F F F
She left and I stayed

31. 31
Conjunction
• This operation generates a composite proposition,
symbolized as p & q, which is true if and only if both p and
q are true. For example:
• In a situation in which Henry died and Terry resigned is
both true, then Henry died and Terry resigned is true
• In a situation where Henry died is true, but Terry
resigned is false, then Henry died and Terry resigned is
false
• Where Henry died is false, but Terry resigned is true,
then Henry died and Terry resigned is also false.
• Where Henry died and Terry resigned are both false,
then Henry died and Terry resigned is also false

32. 32
Disjunction
• Any number of well-formed formulae can be placed in a
sequence with the symbol V between each adjacent pair
in the sequence. The result is a complex well-formed
formula.
• For example, from the simple propositions:
hHERE: Harry is here
cDUTCHMAN Charlie is a Dutchman
A single complex formula can be formed:
(hHERE) V (cDUTCHMAN)
• More examples:
Dorothey saw Bill or Alan
Either John or Peter has used my computer

33. 33
Disjunction
p Q p V q
T T T
T F T
F T T
F F F
You can get there (either) by train or
by bus.
• Truth table for V-inclusive

34. 34
Disjunction
p q p V q
T T F
T F T
F T T
F F F
Either you are her mother or that lady
is
• Truth table for V-exclusive

35. 35
Disjunction
• Disjunctions creates a composite proposition: p V q, which
is true
a. If and only if either p or q is true and
b. If and only if both p and q are true
• In a situation in which Henry died and Terry resigned is
both true, then (Either) Henry died or Terry resigned is
true
• Where Henry died is true, but Terry resigned is false,
then (Either) Henry died or Terry resigned is true
• Where Henry died is false, but Terry resigned is true,
then (Either) Henry died or Terry resigned is also true.
• Where Henry died and Terry resigned are both false,
then (Either) Henry died or Terry resigned is also false

36. 36
Implication
• The logical connective symbolized by →
corresponds roughly to the relation between
an ‘if’ clause and its sequel in English. The
linking of two propositions by → forms what
is called a conditional
• If Alan is here, Clive is a liar

37. 37
Material Implication
• Truth table
p q p  q
T T T
T F F
F T T
F F T
If Mary invited John, he will go.

38. 38
Material Implication
• This operation creates a composite proposition whereby p →
q (p implies q). p → q is true if and only if: (a) Both p and q
are true. (b) Both p and q are false. (c) p is false and q is true
• It is false if p is true and q is false. For example
If she has married him, they are honeymooning in HL now
• This composite proposition can be true if (1) it is true that
she has married him and they are honeymooning in HL now,
or (2) it is false that she has married him and it is also false
that they are honeymooning in HL now. If it is true that she
has married him but it is false that they are honeymooning in
HL now, then the proposition is false because p does not
imply q. The last case is when she has not married him but
they are honeymooning in HL now, which is found by most
people to be paradoxical.

39. 39
• Entailment
p is true and q is necessarily true (i.e. true in all
possible worlds)
If dogs are mammals, they are animals.
• Implicature
The truth of q can be inferred from p in certain
contexts in which p is made.
If Trang’s cellphone is on, she must be writing a
message or making a phone.

40. 40
Equivalence
• Truth table for ≡
p q p ≡ q
T T T
T F F
F T F
F F T
If the horse is a mammal, the shark is a fish.
The conjunction of two implications

41. 41
Equivalence
• The logical connective symbolized by ≡ expresses the
meaning ‘if and only if’ in English. The linking of two
propositions by ≡ produces what is called a
‘biconditional’
• E.g. The meaning of “John is married to Mary if and only if
Mary is married to John” could be represented as:
(jMARRYm) ≡ (mMARRYj)
• The biconditional connective is aptly named because it is
equivalent to the conjunction of two conditionals, one
‘going in each direction’. Inother words, there is a general
rule: p ≡ q is equivalent to (p → q) & (q →p)

42. 42
Negation
• The connective ~ used in propositional logic is
paraphrasable as English ‘not’. Strictly speaking, ~
does not CONNECT propositions, as do (&) and
(V). ~ is prefixed to the formula for a single
proposition, producing its negation. ~ is sometimes
called the ‘negation operator’, rather than ‘negation
connective’.
• E.g.
Alice didn’t sleep can be represented as ~ aSLEEP
Clare is not married to Bill = ~cMARRYb

43. 43
Negation
• Truth table for ~
P ~ P
T F
F T
I love you.

44. 44
Exercise
Find out the truth-value of the following composite
propositions in the following cases (a, b, c, d)
1. I passed the driving test and got the license a week later.
(a). It was true that I passed the driving test, but I received the
license four weeks later.
(b). It was true that I passed my driving test, and I got the
license a week later.
(c). It was true that I did not pass my driving test, but it was
true that I got the license a week later.
(d). Neither of the situations was true.

45. 45
Exercise
2. If you marry her, I will kill my self
(a). If it is true: ”you marry her”, and it is also true: “I will kill
myself”.
(b). If it is true: ”you marry her”, but it is not true: “I will kill
myself”.
(c). If it is not true: ”you marry her”, but it is true: “I will kill
myself”.
(d). Neither is true.