D ERIVATIONS IN S ENTENTIAL L OGIC
Tomoya Sato
Department of Philosophy
University of California, San Diego
Phil120: Symbolic Logic
Summer 2014
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
1 / 41
R EVIEW
Logic is the study of formal validity.
Formal validity is a property of arguments.
def
An argument is valid ⇐⇒ if the premises of the argument
are/were true, then the conclusion must be true.
def
An argument is valid ⇐⇒ it is impossible that all the premises
are true and the conclusion is false.
def
An argument is formally valid ⇐⇒ it is valid due to its logical
form.
1.
2.
If the sidewalks are wet,
then either it rained recently or the sprinklers are on.
The sidewalks are wet.
∴
Either it rained recently or the sprinklers are on.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
2 / 41
R EVIEW
Logic is the study of formal validity.
Formal validity is a property of arguments.
def
An argument is valid ⇐⇒ if the premises of the argument
are/were true, then the conclusion must be true.
def
An argument is valid ⇐⇒ it is impossible that all the premises
are true and the conclusion is false.
An argument is formally valid if it is valid due to its logical form.
1.
2.
If the sidewalks are wet,
then either it rained recently or the sprinklers are on.
The sidewalks are wet.
∴
Either it rained recently or the sprinklers are on.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
3 / 41
R EVIEW
The proof-theoretic method
⇑
⇑
The semantic method
Symbolization
⇑
Formal Validity
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
4 / 41
R EVIEW
S YMBOLIZATION
The game will be called off if it rains.
R : It rains.
G : The game will be called off.
R→G
The game will be called off only if it rains.
G→R
S YMBOLIZATION
If Adam leaves, then Bob stays if Carol sings.
A : Adam leave.
B : Bob leave.
C : Carol leave.
A → (C → B)
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
5 / 41
R EVIEW
D EFINITION : D ERIVATION
A derivation of an argument is a finite sequence of sentences such
that:
1
The first line is "Show (the conclusion of the argument)";
2
Each sentence is either a premise of the argument or a sentence
that can be derived from earlier sentences by an inference rule;
3
Each sentence is followed by a justification.
T HEOREM
If there is a complete derivation for an argument, then the argument is
formally valid.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
6 / 41
R EVIEW
T HE F IRST F OUR I NFERENCE RULES
mp: Modus Ponens;
mt: Modus Tollens;
dn: Double Negation;
r: Repetition.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
7 / 41
R EVIEW
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
Show R
P
P→Q
Q→R
Q
R
T OMOYA S ATO
P
P→Q
Q→R
R
pr
pr
pr
2 3 mp
4 5 mp dd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
8 / 41
C HAPTER 1, S ECTION 6
T HREE T YPES OF D ERIVATIONS
Direct derivation;
Conditional derivation;
Indirect derivation.
C ONDITIONAL D ERIVATION
The goal is to show a conditional ϕ → ψ.
Assume the antecedent ϕ.
Derive the consequent ψ.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
9 / 41
E XAMPLE
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
7.
8.
R → ∼S
∼S → ∼T
W→T
R → ∼W
Show R → ∼W
R
R → ∼S
∼S → ∼T
W→T
∼S
∼T
∼W
T OMOYA S ATO
ass cd
pr
pr
pr
2 3 mp
4 6 mp
5 7 mt cd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
10 / 41
E XAMPLE
1.
2.
∴
1.
2.
3.
4.
5.
6.
7.
P → (Q → ∼R)
R
P → ∼Q
Show P → ∼Q
P
P → (Q → ∼R)
R
Q → ∼R
∼ ∼R
∼Q
T OMOYA S ATO
ass cd
pr
pr
2 3 mp
4 dn
5 6 mt cd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
11 / 41
E XERCISES
1.
2.
3.
∴
(W → Z) → (Z → W)
(Z → W) → ∼X
P→X
P → ∼(W → Z)
T OMOYA S ATO
1.
2.
3.
∴
U → (U → V)
∼R → ∼(U → V)
R → ∼S
U → ∼S
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
12 / 41
D IFFERENCE B ETWEEN DD AND CD
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
∼P → (W → T)
P→T
∼T
W→T
1.
2.
∴
1.
2.
Show W → T
3.
∼P → (W → T) pr
pr
4.
P→T
pr
∼T
5.
∼P
3 4 mt
6.
2 5 mp dd 7.
W→T
T OMOYA S ATO
P → (Q → ∼R)
R
P → ∼Q
Show P → ∼Q
P
P → (Q → ∼R)
R
Q → ∼R
∼ ∼R
∼Q
ass cd
pr
pr
2 3 mp
4 dn
5 6 mt cd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
13 / 41
D IFFERENCE B ETWEEN DD AND CD
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
∼P → (W → T)
P→T
∼T
W→T
1.
2.
∴
1.
2.
Show W → T
3.
∼P → (W → T) pr
pr
4.
P→T
pr
∼T
5.
∼P
3 4 mt
6.
2 5 mp dd 7.
W→T
T OMOYA S ATO
P → (Q → ∼R)
R
P → ∼Q
Show P → ∼Q
P
P → (Q → ∼R)
R
Q → ∼R
∼ ∼R
∼Q
ass cd
pr
pr
2 3 mp
4 dn
5 6 mt cd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
14 / 41
C HAPTER 1, S ECTION 7
T HREE T YPES OF D ERIVATIONS
Direct derivation.
Conditional derivation.
Indirect derivation.
I NDIRECT D ERIVATION
The goal is to show a sentence.
Assume the opposite of the sentence.
Derive a contradiction.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
15 / 41
E XAMPLE
1.
2.
∴
1.
2.
3.
4.
5.
6.
P
∼W
∼(P → W)
Show ∼(P → W)
P→W
P
∼W
W
T OMOYA S ATO
ass id
pr
pr
2 3 mp
4 5 id
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
16 / 41
E XAMPLE
1.
2.
∴
1.
2.
3.
4.
5.
6.
7.
U→S
∼U → S
S
Show S
∼S
U→S
∼U → S
∼U
∼ ∼U
T OMOYA S ATO
ass id
pr
pr
2 3 mt
2 4 mt
5 6 id
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
17 / 41
I DEA OF ID
E XAMPLE
1 Show "Tomoya can speak Japanese fluently"
2
Tomoya has been working as a TA for Japanese courses.
(Premise 1)
3
If Tomoya cannot speak Japanese fluently, he cannot be a TA for
Japanese courses. (Premise 2)
4
Assume that Tomoya cannot speak Japanese fluently. (ass id)
5
Tomoya cannot be a TA for Japanese courses. (3 4 mp)
6
(2 5 id)
7
Therefore, Tomoya can speak Japanese fluently.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
18 / 41
E XERCISES
1.
2.
3.
∴
R→P
Q→R
∼Q → R
P
1.
2.
3.
∴
T OMOYA S ATO
∼Q → R
S → ∼R
∼S → Q
Q
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
19 / 41
E XERCISES
1.
2.
3.
4.
∴
Q→P
∼R → Q
R→S
S → ∼R
∼ ∼P
1.
2.
3.
4.
∴
T OMOYA S ATO
∼P → (R → S)
(R → S) → T
∼T
Q → (R → S)
∼(P → Q)
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
20 / 41
I NFERENCE RULES AND D ERIVATIONS OF C HAPTER 1
F OUR I NFERENCE RULES
mp: Modus Ponens.
mt: Modus Tollens.
dn: Double Negation.
r: Repetition.
T HREE T YPES OF D ERIVATIONS
Direct derivation.
Conditional derivation.
Indirect derivation.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
21 / 41
C HAPTER 1, S ECTION 8
S UBDERIVATIONS
Derivations within a derivation.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
22 / 41
E XAMPLE
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
7.
8.
9.
∼P → Q
∼P → ∼Q
P→R
R
Show R
∼R
∼P → Q
∼P → ∼Q
P→R
∼P
Q
∼Q
T OMOYA S ATO
ass id
pr
pr
pr
2 5 mt
3 6 mp
4 6 mp
7 8 id
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
23 / 41
E XAMPLE
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
∼P → Q
∼P → ∼Q
P→R
R
Show R
∼P → Q
∼P → ∼Q
P→R
Show P
∼P
Q
∼Q
R
T OMOYA S ATO
pr
pr
pr
ass id
2 6 mp
3 6 mp
7 8 id
4 5 mp dd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
24 / 41
E XAMPLE
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
∼P → Q
∼P → ∼Q
R → (P → S)
R→S
Show R → S
R
∼P → Q
∼P → ∼Q
R → (P → S)
Show P
∼P
Q
∼Q
P→S
S
T OMOYA S ATO
ass cd
pr
pr
pr
ass id
3 7 mp
4 7 mp
8 9 id
2 5 mp
6 11 mp cd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
25 / 41
E XERCISES
1.
2.
3.
∴
P → ∼Q
∼Q → R
S → ∼(P → R)
∼S
T OMOYA S ATO
1.
2.
3.
∴
∼(R → Q) → P
P → (∼Q → Q)
∼Q
∼R
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
26 / 41
AVAILABLE L INES
1.
2.
∴
1.
2.
3.
4.
5.
6.
7.
P → (Q → ∼R)
R
P → ∼Q
Show P → ∼Q
P
P → (Q → ∼R)
R
Q → ∼R
∼ ∼R
∼Q
T OMOYA S ATO
ass cd
pr
pr
2 3 mp
4 dn
5 6 mt cd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
27 / 41
AVAILABLE L INES
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
∼P → Q
∼P → ∼Q
P→R
R
Show R
∼P → Q
∼P → ∼Q
P→R
Show P
∼P
Q
∼Q
R
T OMOYA S ATO
pr
pr
pr
ass id
2 6 mp
3 6 mp
7 8 id
4 5 mp dd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
28 / 41
AVAILABLE L INES
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
∼P → Q
∼P → ∼Q
R → (P → S)
R→S
Show R → S
R
∼P → Q
∼P → ∼Q
R → (P → S)
Show P
∼P
Q
∼Q
P→S
S
T OMOYA S ATO
ass cd
pr
pr
pr
ass id
3 7 mp
4 7 mp
8 9 id
2 5 mp
6 11 mp cd
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
29 / 41
AVAILABLE L INES
AVAILABLE L INES
def
In a derivation, a line is available at a point ⇐⇒ that line is an
earlier line that is not an uncancelled show line and is not already in a
box.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
30 / 41
C HAPTER 1, S ECTION 9
Citing Premises without Rewriting Them
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
7.
8.
P→Q
∼P → R
∼Q → ∼R
Q
Show Q
∼Q
P→Q
∼P → R
∼Q → ∼R
∼P
R
∼R
T OMOYA S ATO
ass id
pr
pr
pr
2 3 mt
4 6 mp
2 5 mp id
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
31 / 41
C ONVENTION ON C ITING P REMISES
1.
2.
3.
∴
1.
2.
3.
4.
5.
6.
P→Q
∼P → R
∼Q → ∼R
Q
Show Q
∼Q
∼P
R
∼R
ass id
2 pr1 mt
3 pr2 mp
2 pr3 mp
4 5 id
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
32 / 41
C HAPTER 1, S ECTION 9
N OTE
"Mixed derivations" is not allowed in this course.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
33 / 41
E XERCISES
1.
2.
3.
∴
1.
2.
3.
∴
P→R
Q → ∼R
∼Q → Q
P→Q
1.
2.
3.
∴
P→Q
∼P → R
∼Q → ∼R
Q
U → (V → W)
X→U
∼X → W
V→W
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
34 / 41
E XERCISES
1.
2.
3.
∴
You get lunch at Hi Thai only if you can come back on time.
You cannot come back on time if you go to Price Center,
given that Price Center is a long way from YORK.
It is not the case that
if Price Center is a long way from YORK
then you don’t go to Price Center.
You don’t get lunch at Hi Thai.
S CHEME OF A BBREVIATION
P : You get lunch at Hi Thai.
Q : You can come back on time;
R : You go to Price Center.
S : Price Center is a long way from YORK.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
35 / 41
C HAPTER 1, S ECTION 10
BASIC S TRATEGY

Atomic Sentences 
=⇒ Use id!
Negations

Conditionals
=⇒ Use cd!
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
36 / 41
C HAPTER 1, S ECTION 11
1.
2.
3.
∴
(P → Q) → S
S→T
∼T → Q
T
1.
2.
3.
∴
ϕ
D EFINITION : T HEOREM
A theorem is a sentence that can be the conclusion of a valid argument
with no premises.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
37 / 41
E XAMPLE
∴ (P → Q) → (∼Q → ∼P)
1.
2.
3.
4.
5.
6.
Show (P → Q) → (∼Q → ∼P)
P→Q
ass cd
Show ∼Q → ∼P
∼Q
ass cd
∼P
2 4 mt cd
3 cd
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
38 / 41
E XERCISES
1
∴ P → (Q → P)
2
∴ [P → (Q → R)] → [Q → (P → R)]
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
39 / 41
C HAPTER 1, S ECTION 12
N OTE
Using previously proved theorems is not allowed in this course.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
40 / 41
H OMEWORK 1
Due at the beginning of the Monday lecture.
T OMOYA S ATO
L ECTURE 2: D ERIVATIONS IN S ENTENTIAL L OGIC
41 / 41