Empirical vs Analytical Expressions in Natural Language Processing
Lecture 3
Natural Language Processing
Restricted quantifiers
& literal analysis
Empirical vs. analytical expressions
Partiality
; requisites vs. pre-requisites
Presuppositions vs. mere entailment
Two kinds of negation
https://elogika.vsb.cz/evidence/
1
Restricted quantifiers
All students are clever.
w
w
t
t
x
x
[[
[[
0
0
Student
Student
wt
wt
x
x
]
]
[
[
0
0
C
C
lever
lever
wt
wt
x
x
]]
]]
Some
students are lazy
.
w
w
t
t
x
x
[[
[[
0
0
Student
Student
wt
wt
x
x
]
]
[
[
0
0
L
L
azy
azy
wt
wt
x
x
]]
]]
No student is retired.
w
w
t
t
x
x
[[
[[
0
0
Student
Student
wt
wt
x
x
]
]
[
[
0
0
Retired
Retired
wt
wt
x
x
]]
]]
These analyses are
not the literal
ones
,
because they are not in harmony with the
so-called
‘
Parmenid
es
princip
le
’
and with our method of literal analysis;
the sentences do not mention
,
.
Hence,
we introduce restricted quantifiers:
All
All
,
,
Some
Some
,
,
No
No
,
,
Most
Most
, …
, …
of type
of type
((
((
(
(
))(
))(
))
))
[
0
All
M
]
,
M
v
(
),
v
-
constructs the set of all supersets of
M
[
0
Some
M
]
,
M
v
(
),
v
-
constructs the set of those sets
that have a non-empty
intersection with
M
[
0
No
M
]
,
M
v
(
),
v
-
constructs the set of all sets that have an empty intersection
with
M
w
w
t
t
[[
[[
0
0
All
All
0
0
Student
Student
wt
wt
]
]
0
0
C
C
lever
lever
wt
wt
]
]
w
w
t
t
[[
[[
0
0
Some
Some
0
0
Student
Student
wt
wt
]
]
0
0
L
L
azy
azy
wt
wt
]
]
w
w
t
t
[[
[[
0
0
No
No
0
0
Student
Student
wt
wt
]
]
0
0
Retired
Retired
wt
wt
]
]
2
Empiric
al expressions
Empiric
Empiric
al expressions denote
al expressions denote
non-trivial
non-trivial
(
(
i.e. non-constant
i.e. non-constant
)
)
intens
intens
ions
ions
‘King of France’
,
‘the
prezident
of
ČR
’
,
‘the highest mountain’
denote
denote
offices (roles)
offices (roles)
of type
of type
,
◦
a
nd
actually refer
actually refer
(
in this
w
a
nd
t
of evaluation
)
to nothing
,
Petr Pavel
a
nd
Mount Everest
, respectively
.
◦
Yet the referred object must be found out empirically; it is beyond the logical
analysis
Predicates like
‘to be a student’
,
‘to be red’
,
‘to be happy’
,
‘to be of the
age 60’
denote
denote
properties of type
(
)
◦
a
nd
refer
refer
to their population
,
i.e. to a set of individuals who happen to be
students, red
,
happy
a
nd of the age 60, respectively
.
Sentences like
“Prague is greater than Brno”
denote
denote
propositions of
type
◦
and
refer
refer
to a truth-value
(
this sentence to
T
).
Hence
these expressions are empirical
,
because the object to which
they happen to
refer
in a given state of the world is
beyond logical
analysis
;
it can be determined only by empirical investigation of the
state of the world at a given time moment
.
3
Analytic
expressions
Analytic
Analytic
expressions denote
expressions denote
extens
extens
ions
ions
or
or
trivial (i.e.
trivial (i.e.
constant)
constant)
intensions
intensions
The objects they
denote
and
refer to
can be determined
by
mere understanding a given expression
,
without
empirically investigating the state of the world
.
Mathematical
Mathematical
and
logical
logical
expressions are analytical
,
they
denote extensions
Expressions that contain empirical subexpressions are
analytical
,
if they denote
constant intensions
constant intensions
.
◦
Hence the referred object is
necessarily
,
in all
worlds
w
and times
t
one and the same
.
“No bachelor is married”
“No bachelor is married”
“Whales are mammals”
“Whales are mammals”
◦
Denote the constant proposition
True/
, that takes the
value
T
in all
w
and
t
4
Analytic
expressions
“No bachelor is married”
“No bachelor is married”
T
rue
Bachelor
,
Married
/(
)
;
No
/((
(
))(
))
w
t
[[
0
No
0
Bachelor
wt
]
0
Married
wt
]
How to prove its analyticity? Let us define and refine
:
m
,
n
/
1
v
(
),
x
v
:
0
No =
m
n
x
[[
m x
]
[
n x
]], [[
0
No m
]
n
] =
x
[[
m x
]
[
n x
]].
0
Bachelor
=
w
t
x
[
[
0
Married
wt
x
]
[
0
Man
wt
x
]].
◦
(
to be unmarried and man are
re
re
quisites
quisites
of the property of being a
bachelor
)
[[
0
No
0
Bachelor
wt
]
0
Married
wt
]
=
x
[[
0
Bachelor
wt
x
]
[
0
Married
wt
x
]] =
x
[
[
0
Married
wt
x
]
[
0
Man
wt
x
]
[
0
Married
wt
x
]
].
◦
For
every
every
valuation of
w, t v-
constructs
T
,
thus we can generalize
:
w
w
t
t
x
x
[
[
[
[
0
0
Married
Married
wt
wt
x
x
]
]
[
[
0
0
Man
Man
wt
wt
x
x
]
]
[
[
0
0
Married
Married
wt
wt
x
x
]]
]]
.
5
Quine’s paradox
Necessarily
, 8
> 5
The number of planets = 8
???
Necessarily
,
the number of planets
> 5
w
w
t
t
[
[
0
0
>
>
0
0
8
8
0
0
5]
5]
(analytic
al necessity
)
w
w
t
t
[
[
[
[
0
0
Number_of
Number_of
0
0
P
P
lanet
lanet
wt
wt
] =
] =
0
0
8]
8]
(empiric
al fact
)
w
w
t
t
[
[
0
0
> [
> [
0
0
Number_of
Number_of
0
0
P
P
lanet
lanet
wt
wt
]
]
0
0
5]
5]
(empiric
al fact,
not necessary
)
Types. Number_of
/(
(
)):
the number of elements of a finite set;
Planet
/(
)
;
>
, =/(
)
Proof
:
1)
[
0
>
0
8
0
5]
1. assumption,
E
2)
[
[
0
Number_of
0
P
lanet
wt
] =
0
8]
2. assumption,
E
3)
[
0
>
[
0
Number_of
0
P
lanet
wt
]
0
5
]
1, 2 Leibniz
, subst. of identicals
w
t
[
0
>
[
0
Number_of
0
P
lanet
wt
]
0
5
]
3,
I
Comment.
In the last step we must not introduce
, because the variables
w
,
t
were bound
by
rather than
6
Partiality, v
-
improper constructions
Application of a function
f
to
an argument
a
such that
f
is not
defined at
a
◦
[
0
:
x
0
0]
is
v-
improper for any valuation
v
◦
x
[
0
:
x
0
0]
is not v-improper
,
as it constructs a
degenerate function
degenerate function
◦
0
[
0
:
x
0
0]
is not v-improper
,
as it constructs the Composition
[
0
:
x
0
0]
◦
[
0
Improper
0
[
0
:
x
0
0]
]
constructs
T
Improper
/(
1
):
the class of constructions
v-
improper for any
v.
Incoherent typing
◦
If
X
is not a construction of order
n
(
n
1),
then
1
X
is improper
;
(a
non-procedural object cannot be executed)
◦
If
X
is not a construction of order
n
(
n
2),
then
2
X
is
improper
;
(a
non-procedural object cannot be executed)
◦
If
X
,
X
1
, …,
X
n
are not constructions typed to produce objects of
types according to
Def
.
,
then
[
X
X
1
…
X
n
]
is improper by failing to
v-
construct anything for any valuation
v
.
◦
Example
.
Tom
/
; 5/
;
Student
/(
)
w
t
[
0
Student
wt
0
5],
2
[
w
t
[
0
Student
wt
0
Tom
]]
7
Par
t
iality and Compositionality
„
If five divided by zero equals five
,
then
Tom
is the
Pope
“
◦
denotes the degenerate proposition undefined at all
w,t
!
w
t
[[[
0
:
0
5
0
0] =
0
5]
[
0
Tom =
0
Pope
wt
]]
.
◦
0, 5/
;
:
/(
);
Tom
/
;
P
ope
/
; [
0
:
0
5
0
0]
; [[
0
:
0
5
0
0] =
0
5]
;
0
Tom
;
0
P
ope
wt
v
; [
0
Tom =
0
P
ope
wt
]
v
; [[[
0
:
0
5
0
0] =
0
5]
[
0
Tom =
0
P
ope
wt
]]
v
.
◦
The relation
=
does not get the first argument
,
hence the
implication truth-function
also does not get the first
argument;
therefore, the whole Composition is
v-
improper
by failing to construct anything
.
◦
Partiality is propagated up
Partiality is propagated up
.
◦
The Closure
degenerate proposition
8
Par
t
iality and Compositionality
◦
If you intuitively feel that the sentence could be true,
because the speaker wanted to say that it is not true that
five divided by zero is five, we must analyse this sentence like
this
:
„
If
it is true that
it is true that
five divided by zero equals five
,
then
Tom
is the Pope
“.
True*
True*
/(
/(
n
n
)
)
:
the class of those constructions that
v-
construct
T
for every valuation
v
.
[
0
True*
0
C
]
v-
constructs
T
,
iff
C v-
constructs
T
for any
valuation
v
,
otherwise
otherwise
F
F
.
w
w
t
t
[[
[[
0
0
True*
True*
0
0
[[
[[
0
0
:
:
0
0
5
5
0
0
0] =
0] =
0
0
5]]
5]]
[
[
0
0
Tom =
Tom =
0
0
Papež
Papež
wt
wt
]]
]]
Now the Closure constructs the proposition
T
rue
.
The sentence is
analytically true
analytically true
9
Par
t
iality and Compositionality
False*
/(
n
) a
nd
Improper*
/(
n
)
are classes of
constructions
that
v-
construct
F
or are
v-
improper for all valuations
v
,
respectively
:
[
0
True*
0
C
] =
[
0
False*
0
C
]
[
0
Improper*
0
C
]
[
0
False*
0
C
] =
[
0
True*
0
C
]
[
0
Improper*
0
C
]
[
0
Improper*
0
C
] =
[
0
True*
0
C
]
[
0
False*
0
C
]
Similarly,
and frequently,
we often need these
properties of
propositions
:
True
,
False
,
Undef
/(
)
10
properties of
properties of
propositions
propositions
True
True
,
,
False
False
,
,
Undef
Undef
/(
/(
)
)
P,Q
[
0
True
wt
P
]
v-constructs
T
if
[
P
wt
=
0
T
]
,
otherwise
F
[
0
False
wt
P
]
v-constructs
T
if
[
P
wt
=
0
F
]
,
otherwise
F
[
0
Undef
wt
P
]
=
[
0
True
wt
P
]
[
0
False
wt
P
]
11
Requisites
Requisite
is an analytically necessary relation-in-extension
between two
-
intensions such that, necessarily, any
-
object
that instantiates one intension also instantiates the other
(though not necessarily the other way around).
For instance, if the office of Head of State of the Vatican is a
requisite of the office of Bishop of Rome (i.e. the office of
Pope) then, necessarily, whoever occupies the papal office
must also occupy the requisite office.
Req
/(
);
Between properties (
)
;
True
/(
)
[
0
Req
[
0
Unmarried
0
Man
]
0
Bachelor
] =
[
w
t
x
[[
0
True
wt
w
t
[
0
Bachelor
wt
x
]
]
[
0
True
wt
w
t
[
[
0
Unmarried
0
Man
]
wt
x
]
]]]
12
requisities
[
0
Req
F G
] =
w
t
x
[[
0
True
wt
w
t
[
G
wt
x
]]
[
0
True
wt
w
t
[
F
wt
x
]]]
F, G
(
)
Gloss.
The property F is a requisite of the property G iff necessarily, for all x holds:
if x happens to be a G then x is an F
Examples
.
“No bachelor is married”
“All whales are mammals”
[
0
Req
0
Mammal
0
Whale
] =
w
t
[
x
[[
0
True
wt
w
t
[
0
Whale
wt
x
]]
[
0
True
wt
w
t
[
0
Mammal
wt
x
]]]]
“The president of CR is a Czech citizen”
w
t
[[
0
Exist
wt
w
t
[
0
Pres-of
wt
0
CR
]]
[
0
True
wt
w
t
[[
0
Czech
0
C
itizen
]
wt
[
0
Pres-of
wt
0
CR
]]]]
Being a Czech citizen is a requisite of the Czech presidential office
Types.
Czech
/((
)
(
)
): property modifier;
Citizen
/(
)
;
[
0
Czech
0
C
itizen
]
(
)
13
Pre-requisites; presuppositions
1.
Tom never smoked
2.
Tom stopped smoking
◦
Can the second sentence be true or false under the
assumption of the first sentence?
If Tom never smoked, he couldn’t stop smoking
the
sentence (2) is false (?)
But then it is true that Tom did not stop smoking
he
still smokes (?)
The sentence (2), or rather the proposition denoted by
(2) has
no truth-value
Being an ex-smoker is a
prerequisite
prerequisite
of the property Stop-
smoking
The
presupposition
presupposition
of (2) is the proposition that Tom
previously smoked
14
Pre-requisites; presuppositions
Prereq
/(
(
)
(
)
)
Definition
(
prerequisite relation
)
Let
X
,
Y
be constructions
such that
X
,
Y
/
n
(
)
;
x
. Then
[
0
Prereq Y X
] =
w
t
[
x
[[[
0
True
wt
w
t
[
X
wt
x
]]
[
0
False
wt
w
t
[
X
wt
x
]]]
[
0
True
wt
w
t
[
Y
wt
x
]]]].
Gloss
definiendum
as, “
Y
is a prerequisite of
X
”, and
definiens
as,
“Necessarily, any
x
for which it is true or false that
x
instantiates
X
at
w
,
t
then
x
also instantiates
Y
at
w
,
t
.”
Corollary.
If it is not true that
x
instantiates the prerequisite
Y
of the property
X
then the proposition that
x
instantiates
X
is
neither true not false, it has not truth-value. Hence, the
proposition that
x
instantiates
Y
is a
presupposition
presupposition
of the
proposition that
x
instantiates
X.
The property of being a previous smoker is not only a requisite
The property of being a previous smoker is not only a requisite
of the property of having stopped smoking, it is its
of the property of having stopped smoking, it is its
prerequisite
prerequisite
.
.
15
Presupposition vs. mere entailment
Analytical entailment
is defined as follows (
P
,
S
/
n
,
╞
/(
)).
(
S
╞
P
) iff
w
t
[[
0
True
wt
S
]
[
0
True
wt
P
]]
The logical difference between a presupposition and mere entailment
is this:
P
is a
presupposition
of
S
iff (
S
╞
P
) and (
non-S
╞
P
)
P
is
merely
entailed
by
S
iff
(
S
╞
P
) and neither
(
non-S
╞
P
)
nor
(
non-S
╞
non-P
)
Comments.
If
P
is a presupposition of
S
and
P
is not true at a given
w
,
t
-pair, then
neither
S nor
non-S
is true. Hence,
S
has no truth-value at
such a
w
,
t
-pair at which its presupposition is not true.
On the other hand, if
P
is merely entailed by
S
, then if
S
is not true we
cannot deduce anything about the truth-value, or lack thereof, of
P.
Definition
(
presupposition vs. mere entailment
) Let
P
and
S
be
propositional constructions (
P
,
S
/
n
). Then
P is
entailed
entailed
by S
iff
w
t
[[
0
True
wt
S
]
[
0
True
wt
P
]]
P
is a
presupposition
presupposition
of
S
iff
w
t
[[[
0
True
wt
S
]
[
0
False
wt
S
]]
[
0
True
wt
P
]]
(
Question.
Do you know why we need here the property
True
?)
16
Presupposition vs. mere entailment
Examples.
“
The King of Germany is wise
”
presupposes
presupposes
that
“
the King of Germany exists
”
◦
If the King of Germany does not exist, then
“the King of Germany is (isn’t) wise” has a
truth-value gap, neither true nor false
“
Police found the murderer of JFK
”
merely
merely
entails
entails
that
“
the murderer of JFK exists
”
◦
If the unique murderer of JFK does not exist,
then it
can be true that police did not find
the
murderer of JFK
17
finding
Recall the tragedy in Dallas on Friday, November 22,
1963
John Fitzgerald Kennedy, the 35th President of the
United States, was assassinated at 12:30 p.m. Central
Standard Time in Dallas, Texas, while riding in a
presidential motorcar through Dealey Plaza.
Lee Oswald was arrested by the Dallas Police
Department 70 minutes after the initial shooting.
Oswald was charged under Texas state law with the
murder of Kennedy, as well as that of Dallas
policeman J. D. Tippit, who had been fatally shot a short
time after the assassination.
At 11:21 a.m. November 24, 1963, as live television
cameras were covering his transfer from the city jail to
the county jail, Oswald was fatally shot in the basement
of Dallas Police Headquarters by Dallas nightclub
operator Jack Ruby.
18
finding
After a 10-month investigation, the Warren
Commission concluded that Oswald
assassinated Kennedy, that
Oswald had acted
entirely alone
, and that Ruby had acted alone in
killing Oswald.
Question (during the Warren commision
investigation).
Did the police find the only
murderder of JFK
?
Answer
.
No,
not yet.
True
If the existence of the only murder of JFK
were a presupposition of Police’s finding,
could the answer be truly NO?
Hence, the existence is here
merely
entailed
but
not presupposed
.
19
Restricted quantifiers and literal analysis in natural language processing reveal the distinctions between empirical and analytical expressions. While empirical expressions refer to non-trivial intensions that require empirical investigation, analytical expressions denote constant intensions that can be understood without empirical evidence. Examples and definitions illustrate these concepts, shedding light on the different types of expressions used in linguistic analysis.
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Lecture 3 Natural Language Processing Restricted quantifiers & literal analysis Empirical vs. analytical expressions Partiality; requisites vs. pre-requisites Presuppositions vs. mere entailment Two kinds of negation https://elogika.vsb.cz/evidence/ 1
Restricted quantifiers w t x [[0Studentwtx] [0Cleverwtx]] w t x [[0Studentwtx] [0Lazywtx]] w t x[[0Studentwtx] [0Retiredwt x]] All students are clever. Some students are lazy. No student is retired. These analyses are not the literal ones, because they are not in harmony with the so-called Parmenides principle and with our method of literal analysis; the sentences do not mention , . Hence,we introduce restricted quantifiers: All, Some, No, Most, of type (( ( ))( )) [0All M], M v ( ),v-constructs the set of all supersets of M [0Some M], M v ( ),v-constructs the set of those sets that have a non-empty intersection with M [0No M], M v ( ),v-constructs the set of all sets that have an empty intersection with M w t [[0All0Studentwt] 0Cleverwt] w t [[0Some0Studentwt] 0Lazywt] w t [[0No0Studentwt] 0Retiredwt] 2
Empirical expressions Empirical expressions denote non-trivial (i.e. non-constant) intensions King of France , the prezident of R , the highest mountain denote offices (roles) of type , and actually refer (in this w and t of evaluation) to nothing, Petr Pavel and Mount Everest, respectively. Yet the referred object must be found out empirically; it is beyond the logical analysis Predicates like to be a student , to be red , to be happy , to be of the age 60 denote properties of type ( ) and refer to their population, i.e. to a set of individuals who happen to be students, red, happy and of the age 60, respectively. Sentences like Prague is greater than Brno denote propositions of type and refer to a truth-value (this sentence to T). Hence these expressions are empirical, because the object to which they happen to refer in a given state of the world is beyond logical analysis; it can be determined only by empirical investigation of the state of the world at a given time moment. 3
Analytic expressions Analytic expressions denote extensions or trivial (i.e. constant) intensions The objects they denote and refer tocan be determined by mere understanding a given expression, without empirically investigating the state of the world. Mathematical and logical expressions are analytical, they denote extensions Expressions that contain empirical subexpressions are analytical, if they denote constant intensions. Hence the referred object is necessarily, in all worlds w and times t one and the same. No bachelor is married Whales are mammals Denote the constant proposition TRUE/ , that takes the value T in all w and t 4
Analytic expressions No bachelor is married TRUE Bachelor, Married/( ) ; No/(( ( ))( )) w t [[0No 0Bachelorwt]0Marriedwt] How to prove its analyticity? Let us define and refine: m, n/ 1 v( ), x v : 0No = m n x [[m x] [n x]], [[0No m] n] = x [[m x] [n x]]. 0Bachelor = w t x [ [0Marriedwtx] [0Manwtx]]. (to be unmarried and man are requisites of the property of being a bachelor) [[0No 0Bachelorwt] 0Marriedwt] = x [[0Bachelorwtx] [0Marriedwtx]] = x [ [0Marriedwtx] [0Manwtx] [0Marriedwtx]]. For every valuation of w, t v-constructs T, thus we can generalize: w t x [ [0Marriedwtx] [0Manwtx] [0Marriedwtx]]. 5
Quines paradox Necessarily, 8 > 5 The number of planets = 8 Necessarily, the number of planets > 5 ??? w t [0> 08 05] w t [[0Number_of0Planetwt] = 08] w t [0> [0Number_of0Planetwt] 05] (analytical necessity) (empirical fact) (empirical fact, not necessary) Types. Number_of/( ( )): the number of elements of a finite set; Planet/( ) ; >, =/( ) Proof: 1) [0> 08 05] 2) [[0Number_of0Planetwt] = 08] 3) [0> [0Number_of0Planetwt] 05] 4) w t [0> [0Number_of0Planetwt] 05] Comment. In the last step we must not introduce , because the variables w, t were bound by rather than 1. assumption, E 2. assumption, E 1, 2 Leibniz, subst. of identicals 3, I 6
Partiality, v-improper constructions Application of a function f to an argument a such that f is not defined at a [0:x00] is v-improper for any valuation v x [0:x00] is not v-improper, as it constructs a degenerate function [0Improper0[0:x00]] constructs T Improper/( 1): the class of constructions v-improper for any v. Incoherent typing If X is not a construction of order n (n 1), then 1X is improper; (a non-procedural object cannot be executed) If X is not a construction of order n (n 2), then 2X is improper; (a non-procedural object cannot be executed) If X, X1, , Xnare not constructions typed to produce objects of types according to Def., then [XX1 Xn] is improper by failing to v- construct anything for any valuation v. Example. Tom/ ; 5/ ; Student/( ) w t [0Studentwt05], 2[ w t [0Studentwt0Tom]] 0[0:x00] is not v-improper, as it constructs the Composition [0:x00] 7
Partiality and Compositionality If five divided by zero equals five, then Tom is the Pope denotes the degenerate proposition undefined at all w,t ! w t [[[0:0500] = 05] [0Tom = 0Popewt]] . 0, 5/ ; :/( ); Tom/ ; Pope/ ; [0:0500] ; [[0:0500] = 05] ; 0Tom ; 0Popewt v ; [0Tom =0Popewt] v ; [[[0:05 00] = 05] [0Tom =0Popewt]] v . The relation = does not get the first argument, hence the implication truth-function also does not get the first argument; therefore, the whole Composition is v-improper by failing to construct anything. Partiality is propagated up. The Closure degenerate proposition 8
Partiality and Compositionality If you intuitively feel that the sentence could be true, because the speaker wanted to say that it is not true that five divided by zero is five, we must analyse this sentence like this: If it is true that five divided by zero equals five, then Tom is the Pope . True*/( n): the class of those constructions that v- construct T for every valuation v. [0True* 0C] v-constructs T, iff C v-constructs T for any valuation v, otherwise F. w t [[0True*0[[0: 0500] = 05]] [0Tom =0Pape wt]] Now the Closure constructs the proposition TRUE. The sentence is analytically true 9
Partiality and Compositionality False*/( n) and Improper*/( n) are classes of constructions that v-construct F or are v- improper for all valuations v, respectively: [0True* 0C] = [0False* 0C] [0Improper* 0C] [0False* 0C] = [0True* 0C] [0Improper* 0C] [0Improper* 0C] = [0True* 0C] [0False* 0C] Similarly, and frequently, we often need these properties of propositions:True, False, Undef/( ) 10
properties of propositions True, False, Undef/( ) P,Q [0Truewt P] v-constructs T if [Pwt= 0T], otherwise F [0Falsewt P] v-constructs T if [Pwt= 0F], otherwise F [0Undefwt P] = [0Truewt P] [0Falsewt P] 11
Requisites Requisite is an analytically necessary relation-in-extension between two -intensions such that, necessarily, any -object that instantiates one intension also instantiates the other (though not necessarily the other way around). For instance, if the office of Head of State of the Vatican is a requisite of the office of Bishop of Rome (i.e. the office of Pope) then, necessarily, whoever occupies the papal office must also occupy the requisite office. Req/( ); Between properties ( ) ; True/( ) [0Req [0Unmarried 0Man] 0Bachelor] = [ w t x [[0Truewt w t [0Bachelorwt x]] [0Truewt w t [[0Unmarried 0Man]wtx]]]] 12
requisities [0ReqF G] = w t x [[0Truewt w t [Gwtx]] [0Truewt w t [Fwtx]]] F, G ( ) Gloss. The property F is a requisite of the property G iff necessarily, for all x holds: if x happens to be a G then x is an F Examples. No bachelor is married All whales are mammals [0Req 0Mammal 0Whale] = w t [ x [[0Truewt w t [0Whalewt x]] [0Truewt w t [0Mammalwt x]]]] The president of CR is a Czech citizen w t [[0Existwt w t [0Pres-ofwt 0CR]] [0Truewt w t [[0Czech 0Citizen]wt[0Pres-ofwt 0CR]]]] Being a Czech citizen is a requisite of the Czech presidential office Types. Czech/(( ) ( ) ): property modifier; Citizen/( ) ; [0Czech 0Citizen] ( ) 13
Pre-requisites; presuppositions Tom never smoked Tom stopped smoking Can the second sentence be true or false under the assumption of the first sentence? If Tom never smoked, he couldn t stop smoking the sentence (2) is false (?) But then it is true that Tom did not stop smoking he still smokes (?) The sentence (2), or rather the proposition denoted by (2) has no truth-value Being an ex-smoker is a prerequisite of the property Stop- smoking The presupposition of (2) is the proposition that Tom previously smoked 1. 2. 14
Pre-requisites; presuppositions Prereq/( ( ) ( ) ) Definition (prerequisite relation) Let X,Y be constructions such that X,Y/ n ( ) ; x . Then [0Prereq Y X] = w t [ x [[[0Truewt w t [Xwtx]] [0Falsewt w t [Xwtx]]] [0Truewt w t [Ywtx]]]]. Gloss definiendum as, Y is a prerequisite of X , and definiens as, Necessarily, any x for which it is true or false that x instantiates X at w, t then x also instantiates Y at w, t . Corollary. If it is not true that x instantiates the prerequisite Y of the property X then the proposition that x instantiates X is neither true not false, it has not truth-value. Hence, the proposition that x instantiates Y is a presupposition of the proposition that x instantiates X. The property of being a previous smoker is not only a requisite of the property of having stopped smoking, it is its prerequisite. 15
Presupposition vs. mere entailment Analytical entailment is defined as follows (P, S/ n , /( )). (S P) iff w t [[0TruewtS] [0TruewtP]] The logical difference between a presupposition and mere entailment is this: P is a presupposition of S iff (S P) and (non-S P) P is merely entailed by S iff (S P) and neither (non-S P) nor (non-S non-P) Comments. If P is a presupposition of S and P is not true at a given w, t -pair, then neither S nor non-S is true. Hence, S has no truth-value at such a w, t -pair at which its presupposition is not true. On the other hand, if P is merely entailed by S, then if S is not true we cannot deduce anything about the truth-value, or lack thereof, of P. Definition (presupposition vs. mere entailment) Let P and S be propositional constructions (P, S/ n ). Then P is entailed by S iff w t [[0TruewtS] [0TruewtP]] P is a presupposition of S iff w t [[[0TruewtS] [0FalsewtS]] [0TruewtP]] (Question. Do you know why we need here the property True?) 16
Presupposition vs. mere entailment Examples. The King of Germany is wise presupposes that the King of Germany exists If the King of Germany does not exist, then the King of Germany is (isn t) wise has a truth-value gap, neither true nor false Police found the murderer of JFK merely entails that the murderer of JFK exists If the unique murderer of JFK does not exist, then it can be true that police did not find the murderer of JFK 17
finding Recall the tragedy in Dallas on Friday, November 22, 1963 John Fitzgerald Kennedy, the 35th President of the United States, was assassinated at 12:30 p.m.Central Standard Time in Dallas, Texas, while riding in a presidential motorcar through Dealey Plaza. Lee Oswald was arrested by the Dallas Police Department 70 minutes after the initial shooting. Oswald was charged under Texas state law with the murder of Kennedy, as well as that of Dallas policeman J. D. Tippit, who had been fatally shot a short time after the assassination. At 11:21 a.m. November 24, 1963, as live television cameras were covering his transfer from the city jail to the county jail, Oswald was fatally shot in the basement of Dallas Police Headquarters by Dallas nightclub operator Jack Ruby. 18
finding After a 10-month investigation, theWarren Commission concluded that Oswald assassinated Kennedy, that Oswald had acted entirely alone, and that Ruby had acted alone in killing Oswald. Question (during the Warren commision investigation). Did the police find the only murderder of JFK? Answer. No, not yet. True If the existence of the only murder of JFK were a presupposition of Police s finding, could the answer be truly NO? Hence, the existence is here merely entailed but not presupposed. 19
