Relational predicates
h = Holmes; w = Watson; m = Moriarty;
Gx º x is a gangster; Fxy º x fears y;
Hxyz º x hires y to find z
- example with one-place predicate:
Moriarty is a gangster.
Gm
- examples with two-place predicate
Holmes fears Moriarty.
Fhm
Moriarty fears Holmes.
Fmh
- example with three-place predicate
Holmes hires Watson to find Moriarty.
Hhwm
Singular sentences
h = Holmes; w = Watson; m = Moriarty;
Dx º is a detective; Gx º x is a gangster;
Fxy º x fears y
(a) Holmes is not feared by Moriarty.
(b) Not both Holmes and Moriarty are gangsters.
h = Holmes; w = Watson; m = Moriarty;
Dx º is a detective; Gx º x is a gangster;
Fxy º x fears y
(c) Moriarty fears Holmes only if Holmes is a detective and Moriarty is a gangster.
(d) Holmes fears but is not feared by Moriarty.
h = Holmes; w = Watson; m = Moriarty;
Cx º is a cop; Gx º x is a gangster;
Fxy º x fears y
(a) Every cop fears Moriarty
(b) Only gangsters are feared by Holmes.
(c) Some gangster is feared by Holmes.
(d) Some gangsters do not fear Holmes.
e = the Empire State Building;
w = the Woolworth Building
c = the Chrysler Building
Txy º x is taller than y
(e) Nothing is taller than the Empire State Building
(f) Not everything is taller than the Woolworth Building.
(g) Something is taller than the Chrysler Building
h = Holmes; w = Watson; m = Moriarty;
Cx º is a cop; Gx º x is a gangster;
Fxy º x fears y
(a) Not all cops are feared by Moriarty.
(b) Watson fears all gangsters.
(c) No cop fears Moriarty.
h = Holmes; w = Watson; m = Moriarty;
Cx º is a cop; Gx º x is a gangster;
Fxy º x fears y; Rx º x is crooked
(d) Every crooked cop who fears Moriarty fears Holmes.
(e) All crooked cops are gangsters.
(f) Some cops fear Moriarty, but every gangster fears Holmes
h = Holmes; w = Watson; m = Moriarty;
Dx º x is a detective; Gx º x is a gangster;
Fxy º x fears y
(g) Neither Moriarty nor Holmes fears the other.
(h) Holmes is fearless.
j = John Wayne; m = Marlene Dietrich;
Mx º x is a movie; Lxy º x loves y
(a) John Wayne does not love every movie.
(b) Marlene Dietrich loves only movies.
(c) Marlene Dietrich loves all movies.
Symbolizations of quantifier sentences with multiple quantifiers
- we've had symbolizations with multiple quantifiers
-----> for example:
Some cats and dogs are good house pets.
Cx º x is a cat; Dx º x is a dog
Gx º x is a good house pet
($x)(Cx × Gx) × ($x)(Dx × Gx)
- none of these have been quantifier sentences
- however, quantifier sentences can have multiple
quantifiers
- consider: so far, when we have symbolized sentences with a relational predicate such as the two-place predicate 'x fears y', we've either:
- replaced both of the free variables with
individual constants
or
- quantified over one variable and replaced the
other with an individual constant
- so, for example, we symbolized:
Holmes is not feared by Moriarty.
as: ~Fmh
- and we symbolized:
Every cop fears Moriarty
as: (x)(Cx É Fxm)
But we can also quantify over both variables:
----> for example, symbolize:
Px º x is a person; Fxy º x fears y
(a) Everyone fears someone.
- in this case we end up with a quantifier sentence with two quantifiers
- this will be the case when we want to say of a group or individual specified in terms of ONE variable that IT relates in a certain way to a group or individual specified in terms of a SECOND variable
- so the second variable will be in the scope of the first variable
Px º x is a person
Cx º x is a cop; Gx º x is a gangster;
Fxy º x fears y
(b) Everyone fears everyone.
(c) Some cops fear no gangsters.
(d) Not every cop is feared by a gangster.
- general advice:
- if the English sentence has more than one quantifier word in it ('every', 'all', 'some', 'nothing', etc.), its symbolization will have multiple quantifiers
- be careful: sometimes quantifier words
won't be explicit
- then figure out if the sentence is truth functional sentence or a quantifier sentence (that is, whether a sentential operator or a quantifier is the major operator in the symbolization)
- if it is a quantifier sentence, figure out which quantifier word gets symbolized as the major operator (this is usually the first quantifier word in the sentence)--this will be symbolized first and it will determine whether the sentence is universal or existential
- when you introduce a new quantifier, use a new variable
Symbolizations of general sentences with multiple quantifiers
Restricting the domain to people; Axy º x admires y
compare
(x)(y)Axy For all x and for all y, x admires y.
Every x admires every y.
Everyone admires everyone.
(y)(x)Axy For all y and for all x, x admires y.
Every y is admired by every x.
Everyone admires everyone.
- if two or more universal quantifiers appear next to each other at the beginning of a sentence, the order of these quantifiers doesn't matter
- the same is true of existential quantifiers
Restricting the domain to people; Axy º x admires y
compare
($x)($y)Axy There is some x such that there is some y such that x admires y.
Some x admires some y.
Someone admires someone.
($y)($x)Axy There is some y such that there is
some x such that x admires y.
Some y is admired by some x.
Someone admires someone.
- if two or more existential quantifiers appear next to each other at the beginning of a sentence, the order of these quantifiers doesn't matter
- but with mixed universal and existential quantifiers, the order of the quantifiers matters:
(x)($y)Axy For all x there is some y such that x
admires y.
Everyone admires someone.
(Each of us admires someone or other-
-not necessarily the same person.)
($y)(x)Axy There is some y such that for all x
x admires y.
Someone is admired by everyone.
(At least one particular person inspires
universal admiration.)
($x)(y)Axy There is some x such that for all y
x admires y.
Someone admires everyone.
(At least one particular person admires
everyone.)
(y)($x)Axy For all y there is some x such that
x admires y.
Everyone is admired by someone.
(Each of us is admired by someone or
other-not necessarily the same person.)
- generally, dictionaries alphabetically order the variables in many-place propositional functions:
Axy º x admires y
- but the 'x' and 'y' just stand for places defined by the propositional function
-----> example:
Someone admires everyone.
($x)(y)Axy
- we could also symbolize this as:
($y)(x)Ayx
- what's important is the PLACE that variables occupy after the predicate letter
Lxy º x loves y
Compare:
(1) Everyone loves someone (not necessarily the same person).
(2) Someone is loved by everyone. (At least one particular person is loved by everyone.)
- QN equivalences for multiple quantifier general sentences
-----> example:
Restrict domain to people; Axy º x admires y
No one admires everyone.
~($x)(y)Axy
- is equivalent to:
(x)~(y)Axy
For every person, it's not the case that he/she
admires everyone.
- is equivalent to:
(x)($y)~Axy
For every person, there's someone that he/she
doesn't admire.
Ex. 7
Sx º x is a student; Bx º x is a book; Cx º x is a comic; Gx º x gets goods grades; Rxy º x reads y
(a) Every student reads some books.
(b) Some students read books and comics.
(c) No student reads all books.
(d) No student reads only comics.
(e) Some students who read no books will still get good grades.
Ex. 4.
Ixyz º x introduced y to z
p. Someone introduced himself to everyone.
Ex. 6.
p. There was no one who didn't introduce someone to John.
r. There are some people who introduced no one to John.
t. No one introduced everybody to everybody.
Ex. 7
Sx º x is a student; Bx º x is a book;
Rxy º x reads y; Lxy º x listens to y;
Fx º x is a professor; Hxy º x has y;
Px º x is poetry; Wxy º x writes y;
Axyz º x assigns y to z
f. Some students listen to some of their professors.
h. There is no student who listens to none of his professors.
l. Some students read some books assigned by some of their professors.
n. Some students write poetry.
Symbolize the following sentences using the dictionary provided:
w = William Faulkner; a = Amy
Cx º x is a comic strip; Nx º x is a novel;
Sx º x is Southern; Wx º x is a writer;
Ex º x is newsworthy; Px º x is person;
Wxy º x writes y; Lxy º x loves y
(a) No comic strips are written by William Faulkner.
(b) There is no novel that Amy doesn't love.
(c) Some Southern writers are newsworthy and some writers who aren't Southern aren't newsworthy.
w = William Faulkner; a = Amy
Cx º x is a comic strip; Nx º x is a novel;
Sx º x is Southern; Wx º x is a writer;
Ex º x is newsworthy; Px º x is person;
Wxy º x writes y; Lxy º x loves y
(d) There is someone who is loved by everyone. (That is, at least one particular person is loved by everyone.)
(e) Amy loves herself.
Symbolize the following sentences using the dictionary provided:
j = John Dos Passos
Nx º x is a novel; Sx º x is Southern; Wx º x is a writer; Ex º x is newsworthy; Px º x is person; Wxy º x writes y; Lxy º x loves y
(a) Not all novels are written by John Dos Passos.
(b) There is no writer that Dos Passos doesn't love.
(c) Some Southerners write novels.
(d) No one loves everyone.
For the following, use the dictionary entry 'Lxy º x loves y' and restrict the domain to people:
(e) Everyone loves someone.
(f) Everyone is loved by someone.
Symbolizing identity
- with identity statements, both subject and
predicate are individuals
-----> examples:
Superman is Clark Kent.
Mark Twain is Samuel Clemens.
- distinguish identity statements from singular statements that categorize individuals:
Superman is a fictional character.
Mark Twain is a writer.
- also distinguish identity statements from categorical statements that relate categories
Comic book characters are fictional characters.
Writers are artists.
- we would symbolize the singular and categorical sentences as follows:
Superman is a fictional character.
Fs
Mark Twain is a writer.
Wm
Comic book characters are fictional characters.
(x)(Cx É Fx)
Writers are artists.
(x)(Wx É Ax)
- typically identity statements are symbolized with the '='
Superman is Clark Kent.
s = c
Mark Twain is Samuel Clemens.
m = a
- Some logicians symbolize identity statements with the propositional function 'Ixy º x is identical with y'
Superman is Clark Kent.
Isc
Mark Twain is Samuel Clemens.
Ima
- uses of identity: numerical statements and definite descriptions
- symbolizing identity allows us to symbolize
numerical statements
-----> example:
Mx º x is a mountain; Ex º x is in England
There are at least two mountains in England
($x)($y)(Mx × My × Ex × Ey × x ¹ y)
Note: ($x)($y)(Mx × My × Ex × Ey)
isn't sufficient to symbolize this sentence; even though flagging rules require us to NOT ASSUME that the value of 'x' is the same as the value of 'y', these values COULD be the same; so 'x ¹ y' is needed to rule this out
-----> another example indicating that 'x ¹ y' is
necessary:
Domain restricted to people
Both 'The Godfather' and 'Raging Bull' had directors.
Gx º x directed 'The Godfather';
Rx º x directed 'Raging Bull'
($x)($y)(Gx × Ry)
- this says: some x directed 'The Godfather' and some y directed 'Raging Bull'
- we can't assume that the value of 'x' is the same as the value of 'y' (due to flagging rules)
- but as far as the sentence says, the person who directed 'The Godfather' could be the same as the person who directed 'Raging Bull'
- to rule out the possibility that the value of 'x' and the value of 'y' are the same, we add 'x ¹ y':
($x)($y)(Gx × Ry × x ¹ y)
-----> example:
Mx º x is a mountain; Ex º x is in England
There are at least three mountains in England.
($x)($y)($z)(Mx × My × Mz × Ex × Ey × Ez ×
x ¹ y × x ¹ z × y ¹ z)
-----> example:
Mx º x is a mountain; Ex º x is in England
There are at most two mountains in England.
(x)(y)(z)((Mx × My × Mz × Ex × Ey × Ez) É
(x = y v y = z v x = z))
-----> example:
Mx º x is a mountain; Ex º x is in England
There are exactly two mountains in England.
($x)($y)(Mx × My × Ex × Ey × x ¹ y × (z)((Mz × Ez) É (z = x v z = y)))