LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
One more reason why truth-conditions
are a central part of meaning:
 They allow for a simple account for “logical”
words like not, and, or
LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
The meaning of “NOT” and worlds
The sentences (1) The circle is NOT (fully) inside the
square means/refers to the set of worlds in which the
circle does not contain the square. In all the
remaining worlds, the sentence without negation (2)
The circle is (fully) inside the square would be true.
(The green set W contains all the infinite possible worlds.)
w1
W
(2)
w2
w3
The meaning of negation “NOT”
w4
w5
SEMANTIC INTUITIONS:
Truth table for “NOT”
w6
S1 = The circle is inside the square.
not S1 = The circle is not inside the square.
S1
True
False
not S1
False
True
(1)
w7
w8
w10
w11
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LIGN 130
w9
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Lecture 6 – Logical Words
CAPONIGRO
LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
The meaning of conjunction “AND”
“NOT”
X
W
not-X
not-X is the negation (or, technically, the
“complement set”) of X
that is,
X  not-X =  and X  not-X = W
(the set of worlds X refers to and the set of
worlds not-X refers to do not share any member
(= their intersection is the empty set) and they
together (=their union) form the set of all
possible worlds W)
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BASIC INTUITIONS: Truth table for “AND”
S1 = The circle is (fully) inside the square.
S2 = The triangle is (fully) inside the square.
S1 and S2= The circle is (fully) inside the square
and the triangle is (fully) inside the
square (too).
S1
True
False
True
False
S2
True
True
False
False
S1 and S2
True
False
False
False
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LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
The meaning of “AND” and sets of worlds
 The complex sentence (1) The circle is (fully) inside
the square AND the triangle is (fully) inside the square
(too) means/refers to the (infinite) set of worlds in
which the circle and the triangle are both inside the
square, and therefore the sentence is true. (w4, w5
below)
 Notice that such set of worlds is equivalent to the
intersection of the set of worlds in which the sentence
(2) The circle is (fully) inside the square is true with
the set of worlds in which the sentence (3) The triangle
is (fully) inside the square is true.
(2)
w1
w2
“AND”
X-and-Y = X  Y
the set of worlds X-and-Y refers to is identical
to the set of the worlds both X and Y refer to,
that is the set of the worlds in which both X and
Y are true (=intersection)
w3
(3)
w6
w4
w5
(1)
w7
w9
w10
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LIGN 130
Lecture 6 – Logical Words
6
CAPONIGRO
LIGN 130
Conjunction “OR”: one or two?
 The complex sentence (1) The circle is (fully)
inside the square OR the triangle is (fully)
inside the square means/refers to the
(infinite) set of worlds in which the circle or
the triangle is inside the square.
 And if they are both inside the square?
CAPONIGRO
(i) Exclusive “OR”
1. Truth table for exclusive “OR”
S1 = The circle is (fully) inside the square.
S2 = The triangle is (fully) inside the square.
S1 ORexc S2 = The circle is (fully)inside the
square or the triangle is (fully)
inside the square (but not both).
S1
True
False
True
False
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Lecture 6 – Logical Words
S2
True
True
False
False
S1 ORexc S2
False
True
True
False
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LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
2. Exclusive “OR” and sets of worlds
(2)
w2
w3
w6
w4
Lecture 6 – Logical Words
CAPONIGRO
“OR” exclusive
 (1) The circle is (fully) inside the square OR the
triangle is (fully) inside the square means/refers to the
set of worlds in which either the triangle or the circle
is inside the square, but not both.
 This is equivalent to the set of worlds to which either
the sentence (2) The circle is (fully) inside the square
or the sentence (3) the triangle is (fully) inside the
square refers to, but not both.
(1)
w1
LIGN 130
X-orEXCLUSIVE-Y = (X  Y) – (X  Y)
the set of worlds X-orEXCL-Y refers to is
identical to the set containing the worlds X
refers to or the worlds Y refers to, but not the
worlds both X and Y refer to, that is the union
of X and Y minus the intersection of X and Y
w5
w7
w9
w10
(3)
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LIGN 130
Lecture 6 – Logical Words
10
CAPONIGRO
(ii) Inclusive “OR”
SOME FACTS:
(a) If the circle is (fully) inside the square OR
the triangle is (fully) inside the square on
the blackboard behind the screen, the lecture
is over.
[Do you except the lecture to be over if the screen is
lifted and you see both figures are inside the square?]
(b) Every time the circle is (fully) inside the
square OR the triangle is (fully) inside the
square on the blackboard behind the screen,
the lecture is over.
LIGN 130
Lecture 6 – Logical Words
1. Truth table for inclusive “OR”
S1 ORinc S2= The circle is (fully) inside the
square or the triangle is (fully) inside the square
(or both).
S1
True
False
True
False
S2
True
True
False
False
S1 ORinc S2
True
True
True
False
[Do you except the lecture to be over every time the
screen is lifted and you see both figures are inside the
square?]
(c) It is not true that the circle is (fully) inside
the square OR the triangle is (fully) inside
the square on the blackboard behind the
screen.
[Do you think the sentence would be true if they are
both in the square?]
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CAPONIGRO
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LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
2. Inclusive “OR” and sets of worlds
 (1) The circle is (fully) inside the square OR the
triangle is (fully) inside the square means/refers to the
set of worlds in which either the triangle or the circle
are inside the square, or both.
 This is equivalent to the set of worlds to which either
the sentence (2) The circle is (fully) inside the square
or the sentence (3) the triangle is (fully) inside the
square refers to, or both.
(1)
w1
(2)
w2
w3
“OR” inclusive
X-orINCLUSIVE-Y = X  Y
the set of worlds X-orINCL-Y refers to is identical
to the set containing the worlds X refers to
and/or the worlds Y refers to, that is the union
of X and Y
w6
w4
w7
w5
w9
w10
(3)
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LIGN 130
Lecture 6 – Logical Words
14
CAPONIGRO
Two “OR” or just one?
 Are there two homophonous words “or”
with two different meanings?
LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
SUMMARY:
Basic semantic intuitions and basic logical
words can be represented as relations
between sets of worlds
 Or is there just one “OR” and which one?
 Inclusive “OR” is enough if pragmatics is
taken into account.
 But see the posts from the Language Log
(a popular blog for linguists) on the class
webpage
(1) Truth conditions
Sentence X refers/means
sets of worlds/situations
Sentence Y refers/means
(2) Synonymy
X and Y are synonymous
that is, X = Y
(the set of worlds X refers to is identical to the set of worlds Y refers to)
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LIGN 130
Lecture 6 – Logical Words
CAPONIGRO
LIGN 130
Lecture 6 – Logical Words
(3) Contradiction
(6) “AND”
X and Y are contradictory
X-and-Y = X  Y
that is, X  Y = 
CAPONIGRO
(the set of worlds X-and-Y refers to is identical to the set of the worlds
both X and Y refer to, that is the set of the worlds in which both X and Y
are true (=intersection))
(the set of worlds X refers to and the set of worlds Y refers to do not
share any member)
(4) Entailment
(7) “OR” inclusive
X-orINCLUSIVE-Y = X  Y
Y entails X
(the set of worlds X-orINCL-Y refers to is identical to the set containing
the worlds X refers to and/or the worlds Y refers to, that is the union of
X and Y)
that is, Y  X
(the set of the worlds Y refers is contained [= is a subset of] of the set of
worlds X refers to)
W
X
(5) “NOT”
not-X is the negation of X
not-X
that is, X  not-X =  and X  not-X = W
(the set of worlds X refers to and the set of worlds not-X refers to do not
share any member (= their intersection is the empty set) and they
together (=their union) form the set of all possible worlds W)
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(8) “OR” exclusive
X-orEXCLUSIVE-Y = (X  Y) – (X  Y)
(the set of worlds X- orEXCL -Y refers to is identical to the set containing
the worlds X refers to or the worlds Y refers to, but not the worlds both
X and Y refer to that is the union of X and Y minus the intersection of X
and Y)
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