Homework 3
True or False?
True or false? For each of the following WFFs, determine whether it is
true or false on the given evaluation.
Evaluation: P = T, Q = T, R = T
1. ~(P ↔ Q)
2. ~((P v (Q → R)) & ~P)
True or False?
True or false? For each of the following WFFs, determine whether it is
true or false on the given evaluation.
Evaluation: P = T, Q = T, R = T
1. ~(P ↔ Q)
2. ~((P v (Q → R)) & ~P)
Write Down the Evaluation
P
T
Q
T
~
(P
↔
Q)
Copy the Evaluations beneath the Sentence
Letters
P
T
Q
T
~
(P
T
↔
Q)
T
Consult Biconditional Truth-Table
φ
T
T
F
F
ψ
T
F
T
F
(φ ↔ ψ)
T
F
F
T
Consult Biconditional Truth-Table
φ
T
T
F
F
ψ
T
F
T
F
(φ ↔ ψ)
T
F
F
T
Copy Down the Answer
P
T
Q
T
~
(P
T
↔
T
Q)
T
Consult the Negation Truth-Table
φ
T
F
~φ
F
T
Consult the Negation Truth-Table
φ
T
F
~φ
F
T
Copy Down the Answer
P
T
Q
T
~
F
(P
T
↔
T
Q)
T
The Answer is “False”
P
T
Q
T
~
F
(P
T
↔
T
Q)
T
True or False?
True or false? For each of the following WFFs, determine whether it is
true or false on the given evaluation.
Evaluation: P = T, Q = T, R = T
1. ~(P ↔ Q)
2. ~((P v (Q → R)) & ~P)
From My Inbox
Dear Michael,
Are there too many or too little parentheses in this formula:
~((P v (Q → R)) & ~P)
Thanks,
--Concerned Student
Nope!
~((P v (Q → R)) & ~P)
Set #1
~((P v (Q → R)) & ~P)
Set #2
~((P v (Q → R)) & ~P)
Set #3
~((P v (Q → R)) & ~P)
Definition of WFF
i. All sentence letters are WFFs.
ii. If φ is a WFF, then ~φ is a WFF.
iii. If φ and ψ are WFFs, then (φ & ψ), (φ v ψ), (φ → ψ), (φ ↔ ψ) are
also WFFs.
iv. Nothing else is a WFF.
Demonstration
Using the definition we can show that certain sequences of symbols are
WFFs. For example ~((P v (Q → R)) & ~P) is a WFF.
All Sentence Letters Are WFFs
By (i), all sentence letters are WFFs. So:
P is a WFF
Q is a WFF
R is a WFF
All Sentence Letters Are WFFs
By (i), all sentence letters are WFFs. So:
P is a WFF
Q is a WFF
R is a WFF
~((P v (Q → R)) & ~P)
By (iii) if φ and ψ are WFFs, then (φ → ψ) is a WFF.
~((P v (Q → R)) & ~P)
~((P v (Q → R)) & ~P)
All Sentence Letters Are WFFs
By (i), all sentence letters are WFFs. So:
P is a WFF
Q is a WFF
R is a WFF
~((P v (Q → R)) & ~P)
~((P v (Q → R)) & ~P)
By (iii) if φ and ψ are WFFs, then (φ v ψ) is a WFF.
~((P v (Q → R)) & ~P)
~((P v (Q → R)) & ~P)
All Sentence Letters Are WFFs
By (i), all sentence letters are WFFs. So:
P is a WFF
Q is a WFF
R is a WFF
~((P v (Q → R)) & ~P)
~((P v (Q → R)) & ~P)
By (ii) if φ is a WFF, then ~φ is a WFF.
~((P v (Q → R)) & ~P)
~((P v (Q → R)) & ~P)
By (iii) if φ and ψ are WFFs, then (φ & ψ) is a WFF.
~((P v (Q → R)) & ~P)
~((P v (Q → R)) & ~P)
By (ii) if φ is a WFF, then ~φ is a WFF.
~((P v (Q → R)) & ~P)
~((P v (Q → R)) & ~P)
Scope
Every occurrence of a connective in a WFF has a scope. The scope of
that occurrence is the smallest WFF that contains it. For example The
scope of “&” in “(~(~P&Q)→P)” is (~P&Q)
• (~(~P & Q) → P) is not a WFF.
• (~(~P & Q) → P) is not a WFF.
• (~(~P & Q) → P) is a WFF, but is bigger than (~P&Q)
Occurrences
Notice that the same symbol can occur different times in the same
formula, and that its different occurrences can have different scopes.
• ~((~P & Q) & (R ↔ Q))
• ~((~P & Q) & (R ↔ Q))
• ~((~P & Q) & (R ↔ Q))
• ~((~P & Q) & (R ↔ Q))
Scope of →
~((P v (Q → R)) & ~P)
Scope of v
~((P v (Q → R)) & ~P)
Scope of &
~((P v (Q → R)) & ~P)
Scope of Main Connective
~((P v (Q → R)) & ~P)
Main Connective
The main connective in a formula is the occurrence of the connective
whose scope is widest (the scope of the occurrence of the connective =
the formula).
A formula is a conditional if its main connective is →, a conjunction if
it’s &, a negation if it’s ~, etc.
The Answer
The answer of an evaluation or truth-table is always located under the
main connective.
True or False?
True or false? For each of the following WFFs, determine whether it is
true or false on the given evaluation.
Evaluation: P = T, Q = T, R = T
1. ~(P ↔ Q)
2. ~((P v (Q → R)) & ~P)
Copy Down Evaluation
P
T
Q
T
R
T
((P
v
(Q
→
R))
&
~
P)
Copy beneath Sentence Letters
P
T
Q
T
R
T
((P
T
v
(Q
T
→
R))
T
&
~
P)
T
Consult the Negation Truth-Table
φ
T
F
~φ
F
T
Consult the Negation Truth-Table
φ
T
F
~φ
F
T
Copy Value from Truth-Table
P
T
Q
T
R
T
((P
T
v
(Q
T
→
R))
T
&
~
F
P)
T
Consult Truth-Table for Arrow
φ
T
T
F
F
ψ
T
F
T
F
(φ → ψ)
T
F
T
T
Consult Truth-Table for Arrow
φ
T
T
F
F
ψ
T
F
T
F
(φ → ψ)
T
F
T
T
Copy Down Value
P
T
Q
T
R
T
((P
T
v
(Q
T
→
T
R))
T
&
~
F
P)
T
Consult Truth-Table for Wedge
φ
T
T
F
F
ψ
T
F
T
F
(φ v ψ)
T
T
T
F
Copy Down Value
P
T
Q
T
R
T
((P
T
v
T
(Q
T
→
T
R))
T
&
~
F
P)
T
Consult Truth-Table for &
φ
T
T
F
F
ψ
T
F
T
F
(φ & ψ)
T
F
F
F
Consult Truth-Table for &
φ
T
T
F
F
ψ
T
F
T
F
(φ & ψ)
T
F
F
F
Copy Down Value
P
T
Q
T
R
T
((P
T
v
T
(Q
T
→
T
R))
T
&
F
~
F
P)
T
The Answer is “True”
P
T
Q
T
R
T
((P
T
v
T
(Q
T
→
T
R))
T
&
F
~
F
P)
T
The Answer is “True”
P
T
Q
T
R
T
((P
T
v
T
(Q
T
→
T
*Note: This is because I forgot the ~.
R))
T
&
F
~
F
P)
T
Part 2: Full Truth-Tables
For each of the following WFFs, write out their full truth-table.
3. ~(P v Q)
4. (((P → Q) → P) → P)
Part 2: Full Truth-Tables
For each of the following WFFs, write out their full truth-table.
3. ~(P v Q)
4. (((P → Q) → P) → P)
Write Down ALL POSSIBLE Evaluations
P
T
T
F
F
Q
T
F
T
F
~
(P
v
Q)
Copy Values beneath Sentence Letters
P
T
T
F
F
Q
T
F
T
F
~
(P
T
T
F
F
v
Q)
T
F
T
F
Consult Truth Table for Wedge
φ
T
T
F
F
ψ
T
F
T
F
(φ v ψ)
T
T
T
F
Insert Values Appropriately
P
T
T
F
F
Q
T
F
T
F
~
(P
T
T
F
F
v
T
T
T
F
Q)
T
F
T
F
Consult Truth-Table for Negation
φ
T
F
~φ
F
T
Insert Values Appropriately
P
T
T
F
F
Q
T
F
T
F
~
F
F
F
T
(P
T
T
F
F
v
T
T
T
F
Q)
T
F
T
F
Michael’s Favorite Logic Formula
For each of the following WFFs, write out their full truth-table.
3. ~(P v Q)
4. (((P → Q) → P) → P)
Write Down All Possible Evaluations
P
T
T
F
F
Q
T
F
T
F
(((P
→
Q)
→
P)
→
P)
Copy beneath Sentence Letters
P
T
T
F
F
Q
T
F
T
F
(((P
T
T
F
F
→
Q)
T
F
T
F
→
P)
T
T
F
F
→
P)
T
T
F
F
Consult Truth-Table for Arrow
φ
T
T
F
F
ψ
T
F
T
F
(φ → ψ)
T
F
T
T
Insert Appropriate Values
P
T
T
F
F
Q
T
F
T
F
(((P
T
T
F
F
→
T
F
T
T
Q)
T
F
T
F
→
P)
T
T
F
F
→
P)
T
T
F
F
Insert Appropriate Values
P
T
T
F
F
Q
T
F
T
F
(((P
T
T
F
F
→
T
F
T
T
Q)
T
F
T
F
→
T
T
F
F
P)
T
T
F
F
→
P)
T
T
F
F
Insert Appropriate Values
P
T
T
F
F
Q
T
F
T
F
(((P
T
T
F
F
→
T
F
T
T
Q)
T
F
T
F
→
T
T
F
F
P)
T
T
F
F
→
T
T
T
T
P)
T
T
F
F
Valid or Fails the Test?
Valid or invalid? For each of the following arguments, use the truthtable test for validity to determine whether it’s valid or invalid.
5. P Ⱶ (Q → P)
6. Q Ⱶ (Q → P)
7. ~P Ⱶ (P → Q)
8. ~Q Ⱶ (P → Q)
Valid or Fails the Test?
Valid or invalid? For each of the following arguments, use the truthtable test for validity to determine whether it’s valid or invalid.
5. P Ⱶ (Q → P)
6. Q Ⱶ (Q → P)
7. ~P Ⱶ (P → Q)
8. ~Q Ⱶ (P → Q)
Write Down All Evaluations
P
T
T
F
F
Q
T
F
T
F
P
(Q → P)
Write Down Truth Table For Premises
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
(Q → P)
Write Down Truth Table For Conclusion
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
(Q → P)
T
T
F
T
Ignore Lines Where Premises Are False
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
(Q → P)
T
T
F
T
Make Sure Conclusion is True on the Other
Lines
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
(Q → P)
T
T
F
T
Always True with True Conclusion
φ
T
T
F
F
ψ
T
F
T
F
(φ → ψ)
T
F
T
T
Valid or Fails the Test?
Valid or invalid? For each of the following arguments, use the truthtable test for validity to determine whether it’s valid or invalid.
5. P Ⱶ (Q → P)
6. Q Ⱶ (Q → P)
7. ~P Ⱶ (P → Q)
8. ~Q Ⱶ (P → Q)
Write Down All Evaluations
P
T
T
F
F
Q
T
F
T
F
Q
(Q → P)
Write Down Truth Table For Premises
P
T
T
F
F
Q
T
F
T
F
Q
T
F
T
F
(Q → P)
Write Down Truth Table For Conclusion
P
T
T
F
F
Q
T
F
T
F
Q
T
F
T
F
(Q → P)
T
T
F
T
Ignore Lines with False Premises
P
T
T
F
F
Q
T
F
T
F
Q
T
F
T
F
(Q → P)
T
T
F
T
Make Sure Conclusion is All T on Other Lines
P
T
T
F
F
Q
T
F
T
F
Q
T
F
T
F
(Q → P)
T
T
F
T
Sometimes False with True Premise
φ
T
T
F
F
ψ
T
F
T
F
(φ → ψ)
T
F
T
T
Valid or Fails the Test?
Valid or invalid? For each of the following arguments, use the truthtable test for validity to determine whether it’s valid or invalid.
5. P Ⱶ (Q → P)
6. Q Ⱶ (Q → P)
7. ~P Ⱶ (P → Q)
8. ~Q Ⱶ (P → Q)
Write Down All Evaluations
P
T
T
F
F
Q
T
F
T
F
~P
(P → Q)
Write Down Truth-Tables
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
(P → Q)
T
F
T
T
Ignore Lines with False Premises
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
(P → Q)
T
F
T
T
Make Sure Conclusion is T if Premises are T
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
(P → Q)
T
F
T
T
Always True with False Antecedent
φ
T
T
F
F
ψ
T
F
T
F
(φ → ψ)
T
F
T
T
Valid or Fails the Test?
Valid or invalid? For each of the following arguments, use the truthtable test for validity to determine whether it’s valid or invalid.
5. P Ⱶ (Q → P)
6. Q Ⱶ (Q → P)
7. ~P Ⱶ (P → Q)
8. ~Q Ⱶ (P → Q)
Write Down Evaluations, Truth-Tables
P
T
T
F
F
Q
T
F
T
F
~Q
F
T
F
T
(P → Q)
T
F
T
T
Ignore Lines with False Premises
P
T
T
F
F
Q
T
F
T
F
~Q
F
T
F
T
(P → Q)
T
F
T
T
Ignore Lines with False Premises
P
T
T
F
F
Q
T
F
T
F
~Q
F
T
F
T
(P → Q)
T
F
T
T
Sometimes False with False Consequent
φ
T
T
F
F
ψ
T
F
T
F
(φ → ψ)
T
F
T
T