11
ARGUMENT FORM IN PREDICATE LOGIC
Paal Antonsen
[email protected]
https://sites.google.com/site/paalantonsen/teaching/logic
a
Formal Logic
The story so far. . .
I In the last lecture we introduced some new concepts:
Names
Variables
Predicates
Existential Quantifier
Universal Quantifier
a, b, c, . . .
x, y , z, . . .
P, Q, R, . . .
∃
∀
Singular terms
To be bound by quantifiers
With a fixed number of places
There is (at least one) . . .
For all (every) . . .
The story so far. . .
I In the last lecture we introduced some new concepts:
Names
Variables
Predicates
Existential Quantifier
Universal Quantifier
a, b, c, . . .
x, y , z, . . .
P, Q, R, . . .
∃
∀
Singular terms
To be bound by quantifiers
With a fixed number of places
There is (at least one) . . .
For all (every) . . .
Anakin is an evil jedi
Pa & Qa
There is an evil jedi
(∃x)(Px & Qx)
All jedi are evil
(∀x)(Px ⊃ Qa)
a
Px
Qx
=
=
=
Anakin
x is evil
x is a jedi
The story so far. . .
I In the last lecture we introduced some new concepts:
Names
Variables
Predicates
Existential Quantifier
Universal Quantifier
a, b, c, . . .
x, y , z, . . .
P, Q, R, . . .
∃
∀
Singular terms
To be bound by quantifiers
With a fixed number of places
There is (at least one) . . .
For all (every) . . .
Anakin is an evil jedi
Pa & Qa
There is an evil jedi
(∃x)(Px & Qx)
All jedi are evil
(∀x)(Px ⊃ Qa)
a
Px
Qx
=
=
=
Anakin
x is evil
x is a jedi
The story so far. . .
I In the last lecture we introduced some new concepts:
Names
Variables
Predicates
Existential Quantifier
Universal Quantifier
a, b, c, . . .
x, y , z, . . .
P, Q, R, . . .
∃
∀
Singular terms
To be bound by quantifiers
With a fixed number of places
There is (at least one) . . .
For all (every) . . .
Anakin is an evil jedi
Pa & Qa
There is an evil jedi
(∃x)(Px & Qx)
All jedi are evil
(∀x)(Px ⊃ Qa)
a
Px
Qx
=
=
=
Anakin
x is evil
x is a jedi
The story so far. . .
I In the last lecture we introduced some new concepts:
Names
Variables
Predicates
Existential Quantifier
Universal Quantifier
a, b, c, . . .
x, y , z, . . .
P, Q, R, . . .
∃
∀
Singular terms
To be bound by quantifiers
With a fixed number of places
There is (at least one) . . .
For all (every) . . .
Anakin is an evil jedi
Pa & Qa
There is an evil jedi
(∃x)(Px & Qx)
All jedi are evil
(∀x)(Qx ⊃ Px)
a
Px
Qx
=
=
=
Anakin
x is evil
x is a jedi
The story so far. . .
Vocabulary
NAMES ( INDIVIDUAL
CONSTANTS )
INDIVIDUAL VARIABLES
PREDICATES
CONNECTIVES
QUANTIFIERS
PARENTHESES
a, b, c, . . .
x, y , z, . . .
P, Q, R, . . .
∼, &, ∨, ⊃, ≡
∃, ∀
(, )
Grammar
ATOMIC
QUANTIFIERS
CONNECTIVES
CLOSURE
If P is a predicate with n-places and a1 , . . . an are names
then Pa1 . . .an is a formula.
If A is a formula then (∃x)A(a := x) and (∀x)A(a := x)
formulas, where A(a := x) is the result of replacing all
occurences of a with x.
If A is a formula then ∼A is a formula.
If A and B are formulas then (A & B), (A ∨ B), (A ⊃ B)
and (A ≡ B) are formulas.
Nothing else counts as a formula.
Argument form in predicate logic
I Today we’re going to use this language to identify the agument form of
particular arguments. This is needed if we want to analyze their validity.
Argument form in predicate logic
I Today we’re going to use this language to identify the agument form of
particular arguments. This is needed if we want to analyze their validity.
I Notice how the following arguments share a common form:
All jedi are evil
Bastila is a jedi
————————Therefore, Bastila is evil
All vampires are charming
Sebastian is a vampire
————————
Sebastian is charming
Argument form in predicate logic
I Today we’re going to use this language to identify the agument form of
particular arguments. This is needed if we want to analyze their validity.
I Notice how the following arguments share a common form:
All vampires are charming
Sebastian is a vampire
————————
Sebastian is charming
All jedi are evil
Bastila is a jedi
————————Therefore, Bastila is evil
(∀x)(Px ⊃ Qx)
Pa
——–
Qa
Holberg Erasmus Montanus
Montanus Little mother, I will turn you into a stone.
Mother Nille Oh, nonsense! That is more than even learning can do.
Montanus You shall hear whether it is or not. A stone cannot fly.
Mother Nille No, indeed it can’t unless it is thrown.
Montanus You cannot fly.
Mother Nille That is true, too.
Montanus Ergo: little mother is a stone. [Nille cries.] Why are you
crying, little mother?
Mother Nille Oh! I am so much afraid I shall turn into a stone. My legs
already begin to feel cold.
Holberg Erasmus Montanus
A stone cannot fly
Mother Nille cannot fly
——————————–
Mother Nille is a stone
Holberg Erasmus Montanus
A stone cannot fly
Mother Nille cannot fly
——————————–
Mother Nille is a stone
a
Px
Qx
=
=
=
Mother Nille
x is a stone
x is can fly
(∀x)(Px ⊃ Qx)
∼Qa
———————Pa
Holberg Erasmus Montanus
A stone cannot fly
Mother Nille cannot fly
——————————–
Mother Nille is a stone
a
Px
Qx
=
=
=
Mother Nille
x is a stone
x is can fly
(∀x)(Px ⊃ Qx)
∼Qa
———————Pa
Holberg Erasmus Montanus
A stone cannot fly
Mother Nille cannot fly
——————————–
Mother Nille is a stone
a
Px
Qx
=
=
=
Mother Nille
x is a stone
x is can fly
(∀x)(Px ⊃ ∼Qx)
∼Qa
———————Pa
Holberg Erasmus Montanus
A stone cannot fly
Mother Nille cannot fly
——————————–
Mother Nille is a stone
a
Px
Qx
=
=
=
Mother Nille
x is a stone
x is can fly
(∀x)(Px ⊃ ∼Qx)
∼Qa
———————Pa
Holberg Erasmus Montanus
A stone cannot fly
Mother Nille cannot fly
——————————–
Mother Nille is a stone
a
Px
Qx
=
=
=
Mother Nille
x is a stone
x is can fly
(∀x)(Px ⊃ ∼Qx)
∼Qa
———————Pa
Holberg Erasmus Montanus
A stone cannot fly
Mother Nille cannot fly
——————————–
Mother Nille is a stone
a
Px
Qx
=
=
=
Mother Nille
x is a stone
x is can fly
(∀x)(Px ⊃ ∼Qx)
∼Qa
———————Pa
Holberg Erasmus Montanus
A stone cannot fly
Mother Nille cannot fly
——————————–
Mother Nille is a stone
a
Px
Qx
=
=
=
Mother Nille
x is a stone
x is can fly
(∀x)(Px ⊃ ∼Qx)
∼Qa
———————Pa
Holberg Erasmus Montanus
Montanus Will you admit that the man who drinks well is blessed?
Jeppe I think rather he is accursed, for a man can drink himself out of
both reason and money.
Montanus I will prove that he is blessed... Whoever drinks well, sleeps
well. Isn’t that so?
Jeppe That’s true enough, for when I am half-drunk I sleep like a horse.
Montanus He who sleeps well does not sin. Isn’t that true, too?
Jeppe True too; so long as a man’s asleep he doesn’t sin.
Montanus He who does not sin is blessed.
Jeppe That is also true.
Montanus Ergo: he who drinks well is blessed.
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Holberg Erasmus Montanus
Everyone who drinks well sleeps well
None who sleep well sin
Everyone who doesn’t sin is blessed
—————————————
Everyone who drinks well is blessed
Px
Qx
Rx
Sx
=
=
=
=
x
x
x
x
drinks well
sleeps well
sins
is blessed
(∀x)(Px ⊃ Qx)
∼(∃x)(Qx & Rx)
(∀x)(∼Rx ⊃ Sx)
———————(∀x)(Px ⊃ Sx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Rx
Rx
Rx
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Philosophical argument: seeing is existing
Material objects can be perceived
Only sensations can be perceived
There are no sensation that can’t be perceived.
——————————————————————
There are no material objects that can’t be perceived
Px
Qx
Rx
=
=
=
x is a material object
x can be perceived
x is a sensation
(∀x)(Px ⊃ Qx)
(∀x)(Qx ⊃ Rx)
∼(∃x)(Rx & ∼Qx)
—————————
∼(∃x)(Px & ∼Qx)
Doyle The Dog that Didn’t Bark
Holmes I had grasped the significance of the silence of the dog, for one
true inference invariably suggests others. The Simpson incident had
shown me that a dog was kept in the stables, and yet, though someone
had been in and had fetched out a horse, he had not barked enough to
arouse the two lads in the loft. Obviously the midnight visitor was
someone whom the dog knew well.
Doyle The Dog that Didn’t Bark
Someone was in the stables but the dog didn’t bark at them
The dog barked at everyone it didn’t know
——————————————————————
The dog knew everyone who was in the stables
a
b
Pxy
Qxy
Rxy
=
=
=
=
=
The stables
The dog
x was in y
x barks at y
x knows y
(∃x)(Pxa & ∼Qbx)
(∀x)(∼Rbx ⊃ Qbx)
—————————
(∀x)(Pxa ⊃ Rbx)
Doyle The Dog that Didn’t Bark
Someone was in the stables but the dog didn’t bark at them
The dog barked at everyone it didn’t know
——————————————————————
The dog knew everyone who was in the stables
a
b
Pxy
Qxy
Rxy
=
=
=
=
=
The stables
The dog
x was in y
x barks at y
x knows y
(∃x)(Pxa & ∼Qbx)
(∀x)(∼Rbx ⊃ Qbx)
—————————
(∀x)(Pxa ⊃ Rbx)
Doyle The Dog that Didn’t Bark
Someone was in the stables but the dog didn’t bark at them
The dog barked at everyone it didn’t know
——————————————————————
The dog knew everyone who was in the stables
a
b
Pxy
Qxy
Rxy
=
=
=
=
=
The stables
The dog
x was in y
x barks at y
x knows y
(∃x)(Pxa & ∼Qbx)
(∀x)(∼Rbx ⊃ Qbx)
—————————
(∀x)(Pxa ⊃ Rbx)
Doyle The Dog that Didn’t Bark
Someone was in the stables but the dog didn’t bark at them
The dog barked at everyone it didn’t know
——————————————————————
The dog knew everyone who was in the stables
a
b
Pxy
Qxy
Rxy
=
=
=
=
=
The stables
The dog
x was in y
x barks at y
x knows y
(∃x)(Pxa & ∼Qbx)
(∀x)(∼Rbx ⊃ Qbx)
—————————
(∀x)(Pxa ⊃ Rbx)
Doyle The Dog that Didn’t Bark
Someone was in the stables but the dog didn’t bark at them
The dog barked at everyone it didn’t know
——————————————————————
The dog knew everyone who was in the stables
a
b
Pxy
Qxy
Rxy
=
=
=
=
=
The stables
The dog
x was in y
x barks at y
x knows y
(∃x)(Pxa & ∼Qbx)
(∀x)(∼Rbx ⊃ Qbx)
—————————
(∀x)(Pxa ⊃ Rbx)
Doyle The Dog that Didn’t Bark
Someone was in the stables but the dog didn’t bark at them
The dog barked at everyone it didn’t know
——————————————————————
The dog knew everyone who was in the stables
a
b
Pxy
Qxy
Rxy
=
=
=
=
=
The stables
The dog
x was in y
x barks at y
x knows y
(∃x)(Pxa & ∼Qbx)
(∀x)(∼Rbx ⊃ Qbx)
—————————
(∀x)(Pxa ⊃ Rbx)
Doyle The Dog that Didn’t Bark
Someone was in the stables but the dog didn’t bark at them
The dog barked at everyone it didn’t know
——————————————————————
The dog knew everyone who was in the stables
a
b
Pxy
Qxy
Rxy
=
=
=
=
=
The stables
The dog
x was in y
x barks at y
x knows y
(∃x)(Pxa & ∼Qbx)
(∀x)(∼Rbx ⊃ Qbx)
—————————
(∀x)(Pxa ⊃ Rbx)
Doyle The Adventure of the Blue Carbuncle
Watson I can see nothing.
Holmes On the contrary, Watson, you can see everything. You fail,
however, to reason from what you see. You are too timid in drawing you
inferences.
Watson Then pray tell me what it is that you can infer from this hat?
Holmes That the man was highly intellectual is of course obvious upon
the face of it.
Watson I have no doubt that I am very stupid, but I must confess that I
am unable to follow you. How did you deduce this man intellectual?
Holmes It is a question of cubic capacity ... so large a brain must have
something in it.
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Doyle The Adventure of the Blue Carbuncle
The murderer owns a large hat
Anyone who owns a large hat owns large head
Anyone who owns large head is intellectual
———————————————————
The murderer is intellectual
a
Px
Qx
Rx
Sxy
=
=
=
=
=
The murderer
x is an intellectual
x is a large hat
x is a large head
x owns y
(∃x)(Qx & Sax)
(∀x)((∃y )(Qy & Sxy ) ⊃ (∃z)(Rz & Sxz))
(∀x)((∃y )(Ry & Sxy ) ⊃ Px)
—————————
Pa
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Rxy
Rxy
Rxy
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))
Philosophical argument: something caused this mess
Everything is a physical event or an abstract object, but nothing is both
All physical events have a cause
There is an ultimate cause for physical events that is uncaused
——————————————————————
There is an abstract object that is the ultimate cause for physical events
Px
Qx
Rxy
=
=
=
x is a physical event y
x is an abstract object
x causes y
(∀x)(Px ∨ Qx) & ∼(∃y )(Py & Qy )
(∀x)(Px ⊃ (∃y )Ryx)
(∃x)((∀y )(Py ⊃ Rxy ) & ∼(∃z)Rzx)
—————————
(∃x)(Qx & (∀y )(Py ⊃ Rxy ))