Download 10) Odds and Ends

Propositional Logic
10) Odds & Ends
Copyright 2008, Scott Gray
1
Proof Strategy
□ Proofs are creative, not mechanical,
so these strategies are incomplete.
□ Some proofs can be done from the
“top down” Others require working
backward from the goal.
□ Some proofs require both forward and
backward approaches.
Copyright 2008, Scott Gray
2
Arrow In Strategy
□ If the goal line is a conditional, make a
PA of the antecedent and add the
consequent as a goal line
Copyright 2008, Scott Gray
3
Arrow Out Strategy
□ If the premise line is a conditional, look
for the antecedent in the other
premise lines. If found, use arrow out.
If not found, add antecedent as a
goal.
Copyright 2008, Scott Gray
4
Ampersand In Strategy
□ If the goal is a conjunction, search for
the conjuncts. If found, use
ampersand in. For unfound conjuncts,
add them as goals.
Copyright 2008, Scott Gray
5
Ampersand Out Strategy
□ If the premise is a conjunction, use
ampersand out.
Copyright 2008, Scott Gray
6
Wedge In Strategy
□ If the goal is a disjunction, search for
one of the disjuncts. If found, do
wedge in.
Copyright 2008, Scott Gray
7
Wedge Out Strategy
□ If a premise is a disjunction, look for
two conditionals whose antecedents
match the disjuncts and whose
consequents match the goal line. If
found, apply wedge out. If not found,
make the necessary conditionals as
goals.
Copyright 2008, Scott Gray
8
Double Arrow In
□ If the goal is a biconditional, look for
the associated conditionals. If found,
do the double arrow in. If on or both
not found, add them as goals.
Copyright 2008, Scott Gray
9
Double Arrow Out
□ If a premise is a biconditional, apply
double arrow out.
Copyright 2008, Scott Gray
10
Tilde In Strategy
□ If the goal is a negation, make a PA of
the statement minus the tilde and try
to derive a standard contradiction.
Copyright 2008, Scott Gray
11
Tilde Out
□ If the goal line is affirmative (tilde-free),
make a PA of its negation and try to
derive a standard contradiction.
Copyright 2008, Scott Gray
12
Derived Rules
□ These are rules which can be proven
from the primitive rules we have
learned so far. But the logic they
capture are somewhat common and
will save us time in building proofs.
Copyright 2008, Scott Gray
13
Modus Tollens (MT)
□ From A → B and ~B, derive ~A
□ If it is snowing then it is 32 degrees or
colder. It is not 32 degrees or colder,
therefore it is not snowing.
Copyright 2008, Scott Gray
14
Disjunctive Argument (DA)
□ From A v B and ~A, derive B
From A v B and ~B, derive A
□ Either Coda is black or Bear has four
legs. Coda is not black, therefore Bear
has four legs.
Copyright 2008, Scott Gray
15
Conjunctive Argument (CA)
□ From ~(A & B) and A, derive ~B
From ~(A & B) and B, derive ~A
□ It is not true that both the Dodge and
Buick are a truck. The Dodge is a
truck, therefore the Buick is not a truck.
Copyright 2008, Scott Gray
16
Chain Argument
□ From A → B and B → C, derive A → C
□ If we plant pickling cucumbers, then
we will have something to pickle. If we
have something to pickle, then we will
be eating homemade pickles. We
plant cucumbers, therefore we will eat
homemade pickles.
Copyright 2008, Scott Gray
17
Double Negation
□ From A, derive ~~A and vice versa
□ It is not true that Bear does not have a
tail. Therefore, Bear has a tail.
Copyright 2008, Scott Gray
18
DeMorgan’s Law (DM)
□ From A & B, derive ~(~A v ~B) and vice
versa
From ~(A & B), derive ~A v ~B, etc.
From ~A & ~B, derive ~(A v B), etc.
From ~(~A & ~B), derive A v B, etc.
□ Bear is black and Coda is yellow.
Therefore it is not true that Bear is not
black or Coda is not yellow.
Copyright 2008, Scott Gray
19
Arrow (AR)
□ From A → B, derive ~A v B and vice
versa
From ~A → B, derive A v B, etc.
(your book has more of these)
□ If it is snowing, then it is colder than 33
degrees. Therefore it is either not
snowing or it is colder than 33 degrees.
Copyright 2008, Scott Gray
20
Contraposition (CN)
□ From A → B, derive ~B → ~A, and vice
versa
(your book has more of these)
□ If it is snowing then it is colder than 33
degrees. Therefore, If it isn’t colder
than 33 degrees, then it isn’t snowing.
Copyright 2008, Scott Gray
21
Assignments
□
□
□
□
□
Read 8.2: Proof Strategy
Understand the Russell argument proof
Do exercises 89-98
Read 9.1 & 9.2 on Derived Rules
Do exercises 99-100
Copyright 2008, Scott Gray
22