Semantics and Inference
Part I
Johan Bos
Overview of this lecture
• Inferences on the sentence level
– Entailment
– Paraphrase
– Contradiction
• Using logic to understand semantics
– Introduction to propositional logic
– Syntax and Semantics
– Different kinds of logics
Making inferences
• Meaning relations between expressions
in a language
– Entailments
– Paraphrases
– Contradictions
Entailment, definition
• A sentence expressing a proposition X
entails a sentence expressing a
proposition Y if
– it is not possible to think of a situation
where X is true and Y is false
– Or, put alternatively: the truth of Y follows
necessarily from the truth of X
Entailments, examples
Sentences 2, 3, and 4 are entailed by
sentence 1:
1) Dylan stroked the cat and hugged the dog.
2) Dylan hugged the dog.
3) Someone stroked the cat.
4) Dylan hugged an animal.
Paraphrases, definition
• Two sentences are paraphrases of
each other if they entail each other
• Put differently: whenever one is true,
the other must also be true
Paraphrases, examples
The following sentences are
paraphrases of each other:
1) Dylan stroked the cat and hugged the dog.
2) Dylan hugged the dog and stroked the cat.
3) The cat was stroked by Dylan and the dog
was hugged by Dylan.
Contradiction, definition
• Two sentences are contradictory if it is
impossible to think of a situation where
both sentences can be true
Contradiction, examples
Sentence 1 and 2 are contradictions,
and so are sentence 1 and 3:
1) Dylan stroked the cat and hugged the dog.
2) The dog wasn’t hugged.
3) Nobody stroked anything.
Side remark
• Usually references and contexts are kept
constant in natural language inferences!
• Example:
– Dylan likes Groucho.
– Dylan hates Groucho.
• Contradiction or not?
Side remark
• Usually references and contexts are kept
constant in natural language inferences!
• Example:
– Dylan likes Groucho.
– Dylan hates Groucho.
• Contradiction or not?
– “Of course the sentences are contradictory. You
can’t hate and like someone at the same time.”
– “The sentences are not contradictory. I meant
Dylan Dog in the first sentence, and Bob Dylan in
the second…”
Formal Semantics
• Study of meaning with the help of
(mathematical) logic
• Has been controversial for some time,
but now widely accepted
 "Aren`t human languages imperfect
and illogical anyway?"
 "Human languages have their own
internal logic!"
Human vs. logical languages
• Languages such as Italian, English and
Dutch are human languages (natural
or ordinary languages)
• Logical systems are also referred to as
languages by logicians; these are of
course artificial languages; to avoid
confusion they are sometimes called
calculi
Basic idea of formal semantics
• Provide a mapping from ordinary
language to logic
Human
Language
Logical
Language
(ambiguous)
(unambiguous)
• But what are logical languages or
calculi ?
Logical languages
propositional logic
modal logic
description logic
first-order logic (predicate logic)
second-order logic
higher-order logic
expressive power
•
•
•
•
•
•
This lecture
• In this lecture we will try to map
English to Propositional Logic
• Propositional logic is suitable to model
the basics of sentence semantics
• As we will see it is not very useful for
modelling sub-sentential semantics, for
which usually more expressive logics
are used
Why is this a useful exercise?
• Description of some aspects of
meaning in language
• Detect ambiguities or imprecisions
• Most of the literature in formal
semantics presuppose familiarity with
propositional and first-order logic
Propositions
• What is a proposition?
– Something that is expressed by a
declarative sentence making a statement
– Something that has a truth-value
• Propositions can be true or false
– There are only two possible truth-values
– True, T or 1
– False, F or 0
Propositional logic
• Propositional logic is a language
• So we will look at its ingredients
• We will define the syntax, or in other
words, the grammar
• We will define the semantics
Ingredients of propositional logic
• Propositional variables
– Usually: p, q, r, …
• Connectives
– The symbols: , ,, , 
– Often called logical constants
• Punctuation symbols
– The round brackets ( )
Propositional variables
• Variables are used to stand for
propositions
• Usually, the letters p, q, r are used for
propositional variables
• Example
p  “It is raining outside."
q  “Eva Kant is reading a newspaper.“
• Note:
– the internal structure of propositions is not
of our concern in this lecture
Syntax of propositional logic
• All propositional variables are
propositional formulas
• If  is a propositional formula,
then so is 
• If  and  are propositional formulas,
then so are (), (), () and
()
• Nothing else is a propositional formula
Which of these are propositional formulas?
1)
2)
3)
4)
5)
6)
7)
8)
9)
(pp)
p
q
((pq)(q))
((pq)qr)
(p(p(pp)))
(rq)
(((pq)q))
(pp))
Which of these are propositional formulas?
1)
2)
3)
4)
5)
6)
7)
8)
9)
(pp)
p
q
((pq)(q))
((pq)qr)
(p(p(pp)))
(rq)
(((pq)q))
(pp))
•
•
•
•
•
•
•
•
•
Yes
Yes
Yes
No
No
Yes
No
No
No
Logicians are only human
• Even though logicians and mathematicians
are usually very precise in their formulations,
they sometimes drop punctuation symbols if
this does not give rise to confusion
• Often outermost brackets are dropped; also
other brackets if no confusion arises
Logicians are only human
• Even though logicians and mathematicians
are usually very precise in their formulations,
they sometimes drop punctuation symbols if
this does not give rise to confusion
• Often outermost brackets are dropped; also
other brackets if no confusion arises
• Examples:
pq
instead of (p  q)
p  (q  r) instead of (p  (q  r))
(p  q  r) instead of (p  (q  r))
Negation
• Symbol: 
• Pronounced as: “not”
•  is called the negation of 
• Truth-table:


True
False
False
True
Negation: examples
•
p  “It is raining."
p  “It is not raining.”
•
p  “Eva said something."
p  “Eva didn’t say anything.”
•
p  “Diabolik sometimes lies."
p  “Diabolik never lies.”
Conjunction
• Symbol: 
• Pronounced as: “and”
• () is called the conjunction of the
conjuncts  and 
• Truth table:

True
True
False
False

True
False
True
False
()
True
False
False
False
Conjunction: examples
• p  “The plan is simple.”
q  “The plan is effective."
(pq)  “The plan is simple and effective."
• p  “Diabolik was in trouble.”
q  “Diabolik managed to escape."
(pq)  “Although he was in trouble,
Diabolik managed to escape."
Disjunction
• Symbol: 
• Pronounced as: “or”
• () is called the disjunction of the
disjuncts  and 
• Truth table:

True
True
False
False

True
False
True
False
()
True
True
True
False
Disjunction: examples
• p  “Eva has a gun."
q  “Eva has a knife."
(pq)  “Eva has a gun or a knife (or both)."
• p  “He is a fool."
q  “He is a liar."
(pq)  “He is a fool or a liar (or both)."
(Material) Implication
• Symbol: 
• Pronounced as: “implies” or “arrow”
• Truth table:

True
True
False

True
False
True
()
True
False
True
False
False
True
Implication: examples
• p  “Ginko shot Diabolik."
q  “Diabolik is wounded."
(pq)  “If Ginko shot Diabolik
then Diabolik is wounded."
• p  “The paper will turn red."
q  “The solution is acid."
(pq)  “If the paper turns red,
then the solution is acid."
Equivalence (biconditional)
• Symbol: 
• Pronounced as: “if and only if”
• Truth table:


()
True
True
False
True
False
True
True
False
False
False
False
True
Equivalence: examples
• p  “The number is even."
q  “The number is divisible by two."
(pq)  “The number is even precisely if
it is divisible by two."
• p  “The company has to be registered."
q  “The annual turnover of the company
is above Euro 5,000."
(pq)  “The company has to be registered
just if its annual turnover is
above Euro 5,000."
Problematic cases
“Eva wants a black and white cat.”
“Eva wants a black cat”  “Eva wants a white cat”
“Geller can read your mind or he can bend spoons”
“Geller can read your mind” v “Geller can bend spoons”
“If you want to wash your hands, the bathroom is first on the left”
“You want to wash your hands”  “the bathroom is first on the left”
Problematic cases
“Eva wants a black and white cat.”
“Eva wants a black cat”  “Eva wants a white cat”
“Geller can read your mind or he can bend spoons”
“Geller can read your mind” v “Geller can bend spoons”
“If you want to wash your hands, the bathroom is first on the left”
“You want to wash your hands”  “the bathroom is first on the left”
Problematic cases
“Eva wants a black and white cat.”
“Eva wants a black cat”  “Eva wants a white cat”
“Geller can read your mind or he can bend spoons”
“Geller can read your mind” v “Geller can bend spoons”

“If you want to wash your hands, the bathroom is first on the left”
“You want to wash your hands”  “the bathroom is first on the left”
Problematic cases
“Eva wants a black and white cat.”
“Eva wants a black cat”  “Eva wants a white cat”
“Geller can read your mind or he can bend spoons”
“Geller can read your mind” v “Geller can bend spoons”

“If you want to wash your hands, the bathroom is first on the left”
“You want to wash your hands”  “the bathroom is first on the left”
Translate these in logic
a) Neither I nor my wife speak Russian.
b) If I am not Italian then I am not
allowed to play for the Italian football
team.
c) You will get a room provided you have
no pets.
d) Diabolik will not fail to find the
diamonds.
Semantic relations
Relation
between
sentences
Relation
between pair of
words
entailment
…
paraphrase
…
contradiction
…
Fill in the dots:
hyponymy, synonymy, antonymy
Semantic relations
Relation
between
sentences
Relation
between pair of
words
entailment
hyponymy
paraphrase
synonymy
contradiction
antonymy
Different Logics
propositional logic
modal logic
description logic
first-order logic (predicate logic)
second-order logic
higher-order logic
expressive power
•
•
•
•
•
•
Logics and how they relate
  v
propositional
Logics and how they relate
  v
propositional
[]
<>
modal
Logics and how they relate
  v
propositional
[]
<>
x
x
modal
first-order
Logics and how they relate
  v
propositional
[]
<>
x
x
x
modal
first-order
higher-order
Why different logics?
• Why don’t we take the most expressive logic
and use that to analyse semantics?
• Answer: different logics have different
computational properties
– There is an algorithm to decide whether a formula
is a validity (a theorem) for propositional and
modal logic
– But there is no such algorithm for first-order logic
(or higher-order logic)
A note on notation…
•
•
•
•
•
Negation:
Conjunction:
Implication:
Equivalence:
Brackets:
 or 
 or &
 or 
 or 
(…) or […]
Further reading
• Cann (1993):
Formal Semantics; An introduction, Chapter 7
• Hodges (1977):
Logic. An introduction to elementary logic.
• Hurford & Heasley (1983):
Semantics. A coursebook, Unit 10
• Lyons (1977):
Semantics, Volume 1, Chapter 6