Introduction to Formal Logic

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Introduction to Formal Logic
Richard.Pettigrew_at_bris.ac.uk
Lecture 12
More on logical form Still digging
deeperquantifiers
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Recap
Yesterday, we learned how to express the form of
a sentence that asserts of an individual that it
has a particular property, or of a more than one
individual that they stand in a particular
relation to one another.
Examples
Michiel is Dutch or Johannes is Dutch
Dictionary a Michiel b Johannes Px x is
Dutch
(Pa ? Pb)
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Recap
Yesterday, we learned how to express the form of
a sentence that asserts of an individual that it
has a particular property, or of a more than one
individual that they stand in a particular
relation to one another.
Examples
Michiel is married to Carrie, and she is English
Dictionary a Michiel b Carrie Px x is
English Rxy x is married to y
(Rab Pb)
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Recap
And we learned the new grammatical rules by which
these translations may be added to the language
of the propositional calculus
4) If P is a predicate and a is a constant, Pa is
a wff.
5) If R is a relation of n places and a1, , an
are constants, then Ra1an is a wff.
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Today, we learn how to translate the following
sorts of sentence
There is an omnipotent being.
Everything is material.
There is a possum in Australia.
Every possum is a marsupial.
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There is an omnipotent being
Yesterday, we treated sentences that said, of a
specific individual, that it has a particular
property.
This sentence says that there is at least one
individual with a particular property, but
doesnt specify which individual.
The sentence gives no name for an individual with
this property. It states only that at least one
such individual exists.
So we introduce the existential quantifier
?
which is read there exists,
we give the following dictionary,
and we translate the sentence above as
(?x) Px
Dictionary Px x is omnipotent
Read There exists x such that x is omnipotent.
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Examples
There is a tennis player
Dictionary Px x plays tennis
(?x)Px
There is a red car
Dictionary Px x is a red car
(?x)Px
Dictionary Px x is a car Qx x is red
(?x)(Px Qx)
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Examples
There is a bald man wearing glasses
Dictionary Px x is a man Qx x is bald Rx x
is wearing glasses
(?x)((Px Qx) Rx)
There is a blonde woman and a brunette woman
Dictionary Px x is a woman Qx x is blonde Rx
x is brunette
(?x)(Px Qx) (?x)(Px Rx)
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Examples
There is a tall man outside Buckingham Palace
Dictionary a Buckingham Palace Px x is a
man Qx x is tall Rxy x is outside y
(?x)((Px Qx) Rxa)
There is a woman who loves John
Dictionary a John Px x is a woman Rxy x
loves y
(?x)(Px Rxa)
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Everything is material
Yesterday, we treated sentences that said, of a
single individual that it has a particular
property.
This sentence says of all individuals that they
each have a particular property namely, the
property of being material.
So we introduce the universal quantifier
?
which is read for all,
we give the following dictionary,
and we translate the sentence above as
(?x) Mx
Dictionary Mx x is material
Read For all x, x is material
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Examples
Everything is an illusion
Dictionary Px x is an illusion
(?x)Px
All possums are marsupials
Dictionary Px x is a possum Qx x is a
marsupial
(?x)(Px ? Qx)
Buses are red
Dictionary Px x is a bus Qx x is red
(?x)(Px ? Qx)
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Examples
Every train in Britain runs late
Dictionary Px x is a train Qx x is in
Britain Rx x runs late
(?x)((Px Qx) ? Rx)
All mammals have warm blood and hair
Dictionary Px x is a mammal Qx x has warm
blood Rx x has hair
(?x)(Px ? (Qx Rx))
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Examples
All possums are in Australia
Dictionary a Australia Px x is a possum Rxy
x is in y
(?x)(Px ? Rxa)
Every mammal indigenous to Australia is marsupial
Dictionary a Australia Px x is a mammal Qx
x is a marsupial Rxy x is indigenous to y
(?x)((Px Rxa) ? Qx)
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Cases with more than one quantifier
There is someone who loves Bob
Dictionary b Bob Lxy x loves y
(?x)Lxb
Bob loves someone
Dictionary b Bob Lxy x loves y
(?x)Lbx
There is someone who loves someone
Dictionary Lxy x loves y
(?x)(?y)Lxy
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Cases with more than one quantifier
Everyone loves Bob
Dictionary b Bob Lxy x loves y
(?x)Lxb
Bob loves everyone
Dictionary b Bob Lxy x loves y
(?x)Lbx
Everyone loves everyone
Dictionary Lxy x loves y
(?x)(?y)Lxy
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Cases with more than one quantifier
Everyone loves Bob
Dictionary b Bob Lxy x loves y
(?x)Lxb
Bob loves someone
Dictionary b Bob Lxy x loves y
(?x)Lbx
Everyone loves someone
Dictionary Lxy x loves y
(?x)(?y)Lxy
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Cases with more than one quantifier
Someone loves Bob
Dictionary b Bob Lxy x loves y
(?x)Lxb
Bob loves everyone
Dictionary b Bob Lxy x loves y
(?x)Lbx
Someone loves everyone
Dictionary Lxy x loves y
(?x)(?y)Lxy
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Cases with more than one quantifier
Every country has a capital city
Px x is a country Cxy x is the capital city
of y
(?x)(Px ? (?y)Cyx)
Every calf has a mother
Px x is a calf Mxy x is the mother of y
(?x)(Px ? (?y)Myx)
There is an ultimate cause.
Ex x is an event Cxy x causes y
(?x)(?y)(Ey ? Cxy)
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The language of the predicate calculus
components
1) Propositional letters p, q, r,
2) Propositional connectives , ?, , ?,
Propositional calculus
3) Punctuation ( and )
4) Constants a, b, c,
Predicate calculus
5) Predicates P, Q, F, G, H,
6) Relations R, S,
7) Quantifiers ? and ?
7) Variables x, y, z,
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The language of the predicate calculus grammar
1) Any proposition letter is a wff. 2) If A is a
wff, then A is a wff. 3) If A and B are wff,
then (A B), (A ? B), (A ? B), (A B) are
wffs.
Propositional calculus
4) If P is a predicate and a is a constant, Pa is
a wff.
Predicate calculus
5) If R is a relation of n places and a1, , an
are constants, then Ra1an is a wff.
6) If A is a wff and a is a constant,
then (?x)Ax / a and (?x)Ax / a are
wffs, where Ax / a is obtained from A by
replacing all occurrences of a by x.
e.g. If A is (Pa Qa Fb), then Ax / a is
(Px Qx Fb)