Propositional Logic
6) If (Again)
Copyright 2008, Scott Gray
1
Difficult Conditionals
□ My Honda is a motorcycle only if it has
two wheels
□ How would you symbolize this?
M → T or
T→M
□ These symbolizations are not
equivalent
Copyright 2008, Scott Gray
2
The “only if” Rule
□ “only if” introduces consequents
□ P only if Q
is symbolized
P→Q
□ Do not confuse this with “if and only if”
□ The word “only” should act as a flag
indicating thinking must occur!
Copyright 2008, Scott Gray
3
Necessary & Sufficient
□ If A is a sufficient condition for B, then
A’s occurrence ensures B occurrence
□ If C is a necessary condition for D, then
D cannot occur with C’s absence
□ Sufficient conditions become
antecedents
□ Necessary conditions become
consequents
Copyright 2008, Scott Gray
4
Arrow Reminder
□ The arrow conveys no temporal
significance
□ Example:
If the grill cover is chewed, then Coda
used his chompers
Copyright 2008, Scott Gray
5
Arrow In
□ This is essentially the “chain argument”
□ A → B, B → C, C → D ∴ A → D
□ Arrow In Rule:
From the derivation of Z from
assumption X (and perhaps other
assumptions) derive X → Z
Copyright 2008, Scott Gray
6
Assumption Rule
□ Any statement may be introduced as
an assumption at any point in a proof
□ We will call these “provisional
assumptions”
□ To support provisional assumptions we
must introduce another column in our
proofs, the “assumption dependence”
column
Copyright 2008, Scott Gray
7
Assumption Dependence Column
□ This column goes on the left of the
proof
□ This column indicates the assumptions
(both given and provisional) which the
line depends upon
Copyright 2008, Scott Gray
8
Arrow In Example
□ “If the Twins win this year’s series, they
will have acquired better pitching.”
□ If the Twins win this year’s series, they
will have won the division and made
the playoffs
□ If the Twins have won the division, they
will have acquired better pitching
Copyright 2008, Scott Gray
9
Arrow In Example, cont
S → (D & P), D → B ∴ S → B
Copyright 2008, Scott Gray
10
Arrow In Example, cont
1
2
3
1,3
1,3
1,2,3
1,2
(1)
(2)
(3)
(4)
(5)
(6)
(7)
S → (D & P)
D→B
S
D&P
D
B
S→B
Copyright 2008, Scott Gray
A
A
PA
1,3 →O
4 &O
2,5 →O
3-6 →I
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Arrow In Comments
□ Assumptions depend upon themselves
□ Arrow Out, Ampersand In, and
Ampersand Out depend upon the
assumption dependencies of the
“used” lines
□ Arrow In depends on all of the
assumptions of the corresponding
dependence statements, minus the
assumption of the antecedent
Copyright 2008, Scott Gray
12
Arrow In Comments, cont.
□ A proof isn’t complete if the last line
depends on a provision assumption
□ So, Arrow In reduces dependency
□ But this reduction in dependency
allows us to make “any” provisional
assumption
Copyright 2008, Scott Gray
13
Arrow In Mistakes
1
2
3
4
3
(1)
(2)
(3)
(4)
(5)
(6)
S → (D & P)
D→B
A&C
S
C
S→C
A
A
A
PA
3 &O
4-5 → I
wrong!
Copyright 2008, Scott Gray
14
Arrow In Mistakes, cont.
□ The consequent must depend on the
antecedent
□ The antecedent must be an
assumption or provisional assumption
Copyright 2008, Scott Gray
15
Arrow In Sample #2
P → R, S, (R & S) → Q ∴ P → Q
1
(1) P → R
A
2
(2) S
3
(3) (R & S) → Q
4
(4) P
1,4
(5) R
1,2,4
(6) R & S
1,2,3,4
(7) Q
1,2,3
(8) P → Q
Copyright 2008, Scott Gray
A
A
PA
1,4 →O
2,5 &I
3,6 →O
4-7 →I
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Arrow In Tactics
□ If the goal line is a condition use the
Arrow In strategy
□ Provisionally assume the antecedent
of the conditional
□ try to derive the consequent of the
conditional
□ Use Arrow In to get the conditional
Copyright 2008, Scott Gray
17
Arrow In Example #3
1
2
3
2,3
1,2,3
1,2
1
(1) (P & Q) → R
A
[∴ P → (Q → R)]
(2) P
PA
(3) Q
PA
(4) P & Q
2,3 &I
(5) R
1,4 →O
(6) Q → R
3-5 →I
(7) P → (Q → R)
2-6 →I
Copyright 2008, Scott Gray
18
Arrow In Example #4
1
2
(1)
(2)
3
1,3
5
1,3,5
1,2,3,5
1,2,3
1,2
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(P → Q) → T
A
(T & R) → S
A
[∴ (P → Q) → (R → S)]
P→Q
PA
T
1,3 →O
R
PA
T&R
4,5 &I
S
2,6 →O
R→S
5-7 →I
(P → Q) → (R → S)
3-8 →I
Copyright 2008, Scott Gray
19
Assignments
□ Read Chapter 4
□ Do all of the exercises
□ Be sure to ask me questions if you don’t
understand something or can’t solve a
problem
Copyright 2008, Scott Gray
20