D ERIVATIONS IN S YMBOLIC L OGIC I
Tomoya Sato
Department of Philosophy
University of California, San Diego
Phil120: Symbolic Logic
Summer 2014
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W HAT IS LOGIC ?
L OGIC
Logic is the study of formal validity.
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C HAPTER 3, S ECTION 6
The proof-theoretic method
⇑
⇑
The semantic method
Symbolization
⇑
Formal Validity
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S COPE OF Q UANTIFIER
T WO T YPES OF O CCURRENCES OF VARIABLES
Bound Variables.
Free Variables.
∀x(Fx → ∃y(Gy → Hz)) ∨ (∼Hy ↔ (Fx ∧ ∃x(∼Fx → Gx)))
S CORE OF Q UANTIFIER
∀x (
)
∃y (
)
D EFINITION : S CORE OF Q UANTIFIER
The scope of an occurrence of a quantifier includes itself and
its variable along with the formula to which it was prefixed when
constructing the whole formula.
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S COPE OF Q UANTIFIER
E XAMPLE
∀xFx
∀x(Fx → Gx)
∃xFx ∧ ∃y(Gy ∧ Hy)
∃x(Fx ∧ ∀yGy)
∃x(Fx ∧ ∃y(∃zGz ∧ Hy))
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B OUND VARIABLES AND F REE VARIABLES
D EFINITION : B OUND VARIABLES
A variable α in a formula ϕ is bound if
1
the variable is within the scope of a quantifier;
2
the variable is the same as the one that accompanies the
quantifier;
3
the variable is not already bound by another quantifier
occurrence within the scope of the first quantifier.
∀x(Fx → Gx)
∃xFx ∧ ∃y(Gy ∧ Hy)
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B OUND VARIABLES AND F REE VARIABLES
∃x(Fx ∧ ∀yGy)
∃x[(Fx ∧ ∃y(∃zGz ∧ Hy)) → Iw]
∀x(Fx → ∃y(Gy → Hz)) ∨ (∼Hy ↔ (Fx ∧ ∃x(∼Fx → Gx)))
D EFINITION : F REE VARIABLES
def
A variable α in a formula ϕ is free ⇐⇒ it is not bound by any
quantifier.
D EFINITION : S ENTENCES
def
A formula is a sentence ⇐⇒ it contains only bound variables.
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E XERCISES
E XERCISES
1 ∃x(Fx → ∼Gx)
2 ∼∀x(Fx ∨ ∼Gy)
3
∀x∃y(Fx → ∼Fy)
4
∀x∃y[∀z(Fz ∧ Gy) → (Gx ↔ ((Fx ∧ ∃wHw) → Hx))]
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I NFERENCE RULES
Universal Instantiation (ui)
∀xϕ(x)
ϕ(a)
I DEA
Everything is interesting.
=⇒ Philosophy is interesting.
=⇒ Logic is interesting.
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E XAMPLE
1.
2.
∴
1.
2.
3.
Fa
∀x(Fx → Gx)
Ga
Show Ga
Fa → Ga
Ga
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pr2 ui
2 pr1 mp dd
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P ROBLEM
1.
2.
Socrates is a human being.
All human beings are mortal.
∴
Socrates is mortal.
1.
2.
∴
P
Q
R
Invalid!!
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E XAMPLE
1.
2.
Socrates is a human being.
All human beings are mortal.
∴
Socrates is mortal.
S CHEME OF A BBREVIATION
a : Socrates.
Fx : x is a human being.
Gx : x is mortal.
1.
2.
∴
Fa
∀x(Fx → Gx)
Ga
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I NFERENCE RULES
U NIVERSAL I NSTANTIATION ( UI )
1
Start with a universally quantified formula;
2
Remove the quantifier phrase;
3
Replace all and only the variable occurrences bound to the
quantifier phrase with occurrences of the same letter or variable.
4
Restriction: The occurrence of the variable of instantiation must
be free in the symbolic formula generated by UI.
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I NFERENCE RULES
G OOD UI
∀x(Fx → Gx)
Fa → Ga
G OOD UI
∀x(Fx → Gx)
Fy → Gy
G OOD UI
∀x(Fx → Gx)
Fx → Gx
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I NFERENCE RULES
BAD UI
∀x(Fx → Gx)
Fa → Gb
BAD UI
∀x(Fx → Gx)
Fx → Gy
BAD UI
∀xFx → Ga
Fb → Ga
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I NFERENCE RULES
Restriction: The occurrence of the variable of instantiation must
be free in the symbolic formula generated by UI.
∀x∃y(Fx ∧ Gy)
∃y(Fy ∧ Gy)
You cannot do this!
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E XERCISES
1.
2.
∴
∀x(Fx → Gx)
∀y(Hy ∨ ∼Gy)
Fa → Ha
T OMOYA S ATO
1.
2.
∴
∀x ∼Fx
∀y(∼Fy → Gy) ∨ Fa
Ga
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E XERCISES
1.
2.
∴
∀x∀y(Fx ∨ Gy)
∀y(Gy → Hy)
∼Fa → Ha
T OMOYA S ATO
1.
∴
∀x(Fx ↔ Gx)
∼Ga ∨ Fa
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I NFERENCE RULES
Existential Generalization (eg)
ϕ(a)
∃ϕ(x)
I DEA
Hanzo Hattori is a Ninja.
There is a Ninja.
For some x, x is a Ninja.
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E XAMPLE
1.
2.
∴
1.
2.
3.
4.
∀x(Fx → Gx)
Fa
∃yGy
Show ∃yGy
Fa → Ga
Ga
∃yGy
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pr1 ui
2 pr2 mp
3 eg dd
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I NFERENCE RULES
E XISTENTIAL G ENERALIZATION ( EG )
1
Start with any symbolic formula;
2
(Choice) Pick free occurrences of a term of instantiation;
3
(Choice) Pick the variable of generalization α (any variable is
fine);
4
Add ∃α to the left of the symbolic formula, and replace all
picked occurrences of the term of instantiation with occurrences
of α;
5
Restriction: Don’t bind a free occurrence of a term other than the
term of instantiation.
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I NFERENCE RULES
G OOD EG
Fa → Ga
∃x(Fx → Gx)
G OOD EG
Fx → Gx
∃y(Fy → Gy)
G OOD EG
Fa → Ga
∃y(Fy → Ga)
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I NFERENCE RULES
G OOD EG
Fx → Ga
∃y(Fy → Ga)
G OOD EG
Fx → Ga
∃y(Fx → Gy)
G OOD EG
∀x(Fx → Ga)
∃z∀x(Fx → Gz)
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I NFERENCE RULES
BAD EG
Fa → Gb
∃x(Fx → Gx)
BAD EG
Fa → Gy
∃x(Fx → Gx)
BAD EG
∀x(Fx → Ga)
∀x∃z(Fx → Gz)
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I NFERENCE RULES
Restriction: Don’t bind a free occurrence of a term other than the
term of instantiation.
Fx → Ga
∃x(Fx → Gx)
You cannot do this!
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E XERCISES
1.
∴
1.
2.
∴
∀xFx
∃xFx
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∀x[(∃y ∼Fy) ↔ Gx]
∼Fa
Gb
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I NFERENCE RULES
Existential Instantiation (ei)
∃ϕ(x)
ϕ(y)
I DEA
There is a Ninja.
Nick is a Ninja.
(“Nick" is a name that is temporarily assigned to that Ninja)
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E XAMPLE
1.
2.
∴
1.
2.
3.
4.
5.
∀x(Fx → Gx)
∃yFy
∃zGz
Show ∃zGz
Fw
Fw → Gw
Gw
∃zGz
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pr2 ei
pr1 ui
2 3 mp
4 eg dd
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I NFERENCE RULES
E XISTENTIAL I NSTANTIATION ( EI )
1
Start with an existentially quantified formula;
2
Remove the quantifier phrase;
3
Replace all and only the variable occurrences bound to the
quantifier phrase with occurrences of a new variable.
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I NFERENCE RULES
G OOD EI
∃xFx
Fw
G OOD EI
∃x(Fx → Ga)
Fz → Ga
G OOD EI
∃x∀y(Fx → Gy)
∀x(Fz → Gy)
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I NFERENCE RULES
BAD EI
∃x(Fx → Qb)
Fa → Qb
BAD EI
∃x(Fx → Qy)
Fy → Qy
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I NFERENCE RULES
BAD EI
∀x∃y(Fx → Gy)
∀x(Fx → Gw)
BAD EI
∃x(Fx → Qx)
Fy → Qz
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E XERCISES
1.
∴
∃x ∼Fx
∼∀yFy
1.
∴
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∀x ∼Fx
∼∃yFy
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E XERCISES
1.
∴
∃x∀y(Fx → Gy)
∃z∃w(Fw → Gz)
T OMOYA S ATO
1.
2.
3.
∴
∀x[(Bx ∧ Dx) → Ex]
∃x(Dx ∧ Fx)
∀x(Fx → Bx)
∃y(Dy ∧ Ey)
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C HAPTER 3, S ECTION 7
N OTE
Universal derivation (ud) is not allowed in this course.
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C HAPTER 3, S ECTION 9
Quantifier Negation (qn)
I DEA
It is not case that, for all x, x is not meaningful.
There exists x such that x is meaningful.
∼∀x ∼Mx.
∃xMx.
I DEA
It is not case that there exists x such that x is not meaningful.
For all x, x is meanigful.
∼∃x ∼Mx.
∀xMx.
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I NFERENCE RULES
I DEA
“ ∼∀x ∼” ⇐⇒ “∃x”.
“ ∼∃x ∼” ⇐⇒ “∀x”.
∼∀αϕ
∃α ∼ϕ
∼∃αϕ
∀α ∼ϕ
∼∀α ∼ϕ
∃αϕ
∼∃α ∼ϕ
∀αϕ
∃α ∼ϕ
∼∀αϕ
∀α ∼ϕ
∼∃αϕ
∃αϕ
∼∀α ∼ϕ
∀αϕ
∼∃α ∼ϕ
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E XERCISES
1.
2.
∴
∀x(Fx → Gx)
∀y(∼Fy → Gy)
∀zGz
T OMOYA S ATO
1.
∴
∀x(∼Fx → Ga)
∀xFx ∨ Ga
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A LPHABETIC VARIANCE ( AV )
N OTE
"Alphabetic variance" is not allowed in this course.
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N OTE
N OTE
In this course, it is not allowed to apply inference rules to a part of
quantified formulas.
BAD DM
∼∀x(Fx ∨ Qx)
∃x ∼(Fx ∨ Gx)
∃x(∼Fx ∧ ∼Gx)
BAD CDJ
∼∀x(Fx → Qx)
∃x ∼(Fx → Gx)
∃x ∼(∼Fx ∨ Gx)
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