Logic Review
FORMAT
Format
Part I
• 30 questions
• 2.5 marks each
• Total 30 x 2.5 = 75 marks
Part II
• 10 questions
• Answer only 5 of them!
• Total 5 x 5 marks each = 25 marks
Part I
The questions in Part I are multiple choice.
About 8-10 out of the questions in Part I (30
total) are about argument analysis and logic.
~20-25% of the total marks.
Part II
Part II contains mostly logic questions, about 9
out of 10.
You are only required to answer 5 of the 10
questions. If you answer more than 5, I will only
mark the first 5.
Part II is much harder than Part I.
LOGICAL CONCEPTS
Logical Concepts on the Final
About 6/30 multiple choice questions deal with
argument analysis (identification of premises
and conclusions) and logical concepts (validity,
soundness, logical equivalence, etc.).
Therefore mastering these concepts can go a
long way toward the grade you get on the Final,
up to about 15%.
What to Study
• Read Joe Lau’s modules on Argument Analysis
and Meaning Analysis (recommended reading,
Week 14).
• Read the powerpoint slides from 25/4
• Read the logic slides in Week 5.
Arguments
Arguments (in philosophy) are a collection of
premises (possibly zero) given in support of a
conclusion.
Argument Analysis
You will be asked to identify premises and
conclusions in questions, and to identify hidden
assumptions.
These questions are not hard if you think about
them and know the ideas.
Validity
To say that an argument is valid is to say:
If all of the premises of the argument are true,
then the conclusion must be true.
Validity
For the final you should memorize the definition
of validity. Furthermore, you should be able to
answer some basic questions about the
definition.
Examples
Can a valid argument have false premises?
Can it have a true conclusion?
Can it have false premises and a false conclusion?
Can it have true premises and a false conclusion?
Inductive Validity
You need to know the definition of inductively
valid/ inductively strong arguments. First:
inductively valid = inductively strong.
An argument is inductively strong if when all the
premises are true, the conclusion is very likely to
be true (has a high probability of being true).
Note
If the final asks you about a “valid argument”
assume that it means “deductively valid
argument.” We will always say “inductively valid
(strong)” when we mean that.
Logical Equivalence
Two sentences A and B are logically equivalent
when A├ B and B├ A.
In other words, A and B are logically equivalent
when “A therefore B” and “B therefore A” are
both valid arguments.
In other words A is true if and only if B is true.
Soundness
To say that an argument is sound is to say:
The argument is valid AND it has true premises.
You need to memorize this definition and be
able to answer some basic questions about it.
Can a sound argument have a false conclusion?
Etc.
SENTENTIAL LOGIC
SL on the Final
There aren’t a lot of multiple choice questions
about SL (there are only 30 questions total), but
there are several short answer questions (Part II)
that involve (or can involve) SL.
Knowing SL perfectly can count for up to about
20% of the marks on the final.
What to Study
• Read Joe Lau’s Sentential Logic Module (on
the readings list for Week 5).
• Read all the powerpoint slides from Weeks 5
and 6.
If you know all that material, you are as well
prepared for the logic questions as you can be.
Difficulty
Some of the questions are hard, and even made
me stop and think about them.
They are supposed to be like that.
Difficult questions help differentiate students.
The exam will be curved and the same number
of students will get A’s in this class as in any
other class.
Proofs
There are two proofs on the optional section.
They are not easy, but if you are good at proofs,
you might consider answering them.
If you don’t like proofs don’t bother studying
them. You can always choose 5 questions on
Part II that do not involve proofs.
Translations
You need to memorize the standard translations
for the logical connectives into English, both
because you will be asked questions about
which English sentence translates some
predicate logic sentence and because you will be
asked which predicate logic sentence translates
some English sentence.
Standard Translations
~A: not A, A is false, A is not true, it’s not the
case that A
(A & B): A and B, A but B, although A, B
(~A & ~B): neither A nor B
(A v B): A or B, A unless B
(A → B): if A then B, A only if B, B if A
(A ↔ B): A if and only if B
“If” and “Only if”
The most difficult part of translations is “if” and
“only if” and “if and only if”
“If A then B” “A only if B” “B if A”: (A → B)
“If B then A” “B only if A” “A if B”: (B → A)
“A if and only if B” “B if and only if A”: (A ↔ B)
Example Problem
(((P v Q) v R) → (S v (T & U)))…
a.
b.
c.
d.
…is true if S is true
…is true if T and U are false
…is true if R is false
…is false only if S is false
Truth-Table Won’t Work
This seems like a hard problem at first sight.
A truth-table would take too much time to
construct. It would have to have 2 x 2 x 2 x 2 x 2
x 2 = 64 rows!
Let’s consider it intelligently.
True Conditionals
When is (((P v Q) v R) → (S v (T & U))) true?
It’s a conditional, so it’s true whenever either
• ((P v Q) v R) is false
• Or (S v (T & U)) is true
The Material Conditional
φ
T
T
F
F
ψ
T
F
T
F
(φ → ψ)
T
F
T
T
False Conditionals
Additionally (((P v Q) v R) → (S v (T & U))) is false
when:
• ((P v Q) v R) is true
• And (S v (T & U)) is false
Example Problem
(((P v Q) v R) → (S v (T & U)))…
a.
b.
c.
d.
…is true if S is true
…is true if T and U are false
…is true if R is false
…is false only if S is false
Disjunction
If S is true, then (S v (T & U)) is true, because
disjunctions are true when either disjunct is
true.
Disjunction
φ
T
T
F
F
ψ
T
F
T
F
(φ v ψ)
T
T
T
F
Not (a)
So if S is true, (S v (T & U)) is true.
But then (((P v Q) v R) → (S v (T & U))) could be
false, if ((P v Q) v R) is false.
So (a) is not the right answer.
Example Problem
(((P v Q) v R) → (S v (T & U)))…
a.
b.
c.
d.
…is true if S is true
…is true if T and U are false
…is true if R is false
…is false only if S is false
Not (b)
If T and U are false, then (T & U) is false.
But this tells us nothing about whether (S v (T &
U) is true or false, because if S is false, it is false
and if S is true it is true.
Therefore we don’t know whether (((P v Q) v R)
→ (S v (T & U))) is true or false. So (b) is wrong.
Example Problem
(((P v Q) v R) → (S v (T & U)))…
a.
b.
c.
d.
…is true if S is true
…is true if T and U are false
…is true if R is false
…is false only if S is false
Not (c)
If R is false, then ((P v Q) v R) can be either true
or false. For example, it will be true if P is true or
if Q is true, and only false when P and Q are
both false.
Assuming R is false, ((P v Q) v R) might be true,
and therefore the whole sentence could be false
(antecedent true, consequent false).
Example Problem
(((P v Q) v R) → (S v (T & U)))…
a.
b.
c.
d.
…is true if S is true
…is true if T and U are false
…is true if R is false
…is false only if S is false
Why (d)?
By process of elimination, you know the answer is
(d). But why is it the right one?
“A only if B” means if A, then B.
(((P v Q) v R) → (S v (T & U))) is false only if S is false
If (((P v Q) v R) → (S v (T & U))) is false, then S is
false.
Why (d)?
If (((P v Q) v R) → (S v (T & U))) is false
Then ((P v Q) v R) is true
And (S v (T & U)) is false.
If (S v (T & U)) is false then
(T & U) is false
And S is false.
Example Problem 2
Use a truth-table to determine whether a
logically entails b.
a. (A & B)
b. ~(~A v ~B)
Make Truth-Table for (A & B)
A
T
T
F
F
B
T
F
T
F
(A
&
B)
Copy A, B
A
T
T
F
F
B
T
F
T
F
(A
&
B)
Copy A, B
A
T
T
F
F
B
T
F
T
F
(A
T
T
F
F
&
B)
T
F
T
F
Use TT for &
A
T
T
F
F
B
T
F
T
F
(A
T
T
F
F
&
B)
T
F
T
F
Use TT for &
A
T
T
F
F
B
T
F
T
F
(A
T
T
F
F
&
T
F
F
F
B)
T
F
T
F
Now Make TT for ~(~A v ~B)
A
T
T
F
F
B
T
F
T
F
~
(~
A
v
~
B)
Copy A, B
A
T
T
F
F
B
T
F
T
F
~
(~
A
v
~
B)
Copy A, B
A
T
T
F
F
B
T
F
T
F
~
(~
A
T
T
F
F
v
~
B)
T
F
T
F
Use TT for ~
A
T
T
F
F
B
T
F
T
F
~
(~
A
T
T
F
F
v
~
B)
T
F
T
F
Use TT for ~
A
T
T
F
F
B
T
F
T
F
~
(~
F
F
T
T
A
T
T
F
F
v
~
F
T
F
T
B)
T
F
T
F
Use TT for v
A
T
T
F
F
B
T
F
T
F
~
(~
F
F
T
T
A
T
T
F
F
v
~
F
T
F
T
B)
T
F
T
F
Use TT for v
A
T
T
F
F
B
T
F
T
F
~
(~
F
F
T
T
A
T
T
F
F
v
F
T
T
T
~
F
T
F
T
B)
T
F
T
F
Use TT for ~
A
T
T
F
F
B
T
F
T
F
~
(~
F
F
T
T
A
T
T
F
F
v
F
T
T
T
~
F
T
F
T
B)
T
F
T
F
Use TT for ~
A
T
T
F
F
B
T
F
T
F
~
T
F
F
F
(~
F
F
T
T
A
T
T
F
F
v
F
T
T
T
~
F
T
F
T
B)
T
F
T
F
Truth-Table Test
Now we can do the truth-table test for validity.
Remember that an argument is valid when IF
the premises are true, THEN the conclusion is
true.
Write Down Possibilities
A
T
T
F
F
B
T
F
T
F
Write Down TT for Premises
A
T
T
F
F
B
T
F
T
F
(A & B)
T
F
F
F
Write Down TT for Conclusion
A
T
T
F
F
B
T
F
T
F
(A & B)
T
F
F
F
~(~A v ~B)
T
F
F
F
Make Sure Conclusion T when
Premises T
A
T
T
F
F
B
T
F
T
F
(A & B)
T
F*
F*
F*
~(~A v ~B)
T
F
F
F
Consistency
A set of sentences is consistent if all of the
sentences can be true at the same time. (Some
or all of them might in fact be false, but they all
CAN be true together).
A set of sentences is inconsistent if they
CANNOT all be true at the same time.
Example Problem #3
Consider the following three sentences.
1. Sam will be sad unless we come to his party.
2. We will come to Sam’s party if and only if
there is food.
3. Sam won’t have food at the party.
Are they logically consistent all together?
Using SL
Translate:
S = Sam is sad
C = We come to Sam’s party
F = There is food at the party
Using SL
Translate:
1. (S v C)
2. (C ↔ F)
3. ~F
Solution
S
C
F
(S v C)
(C ↔ F)
~F
T
T
T
T
T
T
F
F
T
F
T
F
T
T
T
T
T
F
F
T
F
T
F
T
F
F
F
T
T
F
T
F
T
T
T
F
T
F
F
F
T
F
F
F
F
F
T
T
Solution
S
C
F
(S v C)
(C ↔ F)
~F
T
T
T
T
T
T
F
F
T
F
T
F
T
T
T
T
T
F*
F*
T
F*
T
F*
T
F
F
F
T
T
F
T
F
T
T
T
F*
T
F*
F*
F*
T
F*
F
F
F
F*
T
T
Proofs
I advise you to avoid the proofs unless you are
already pretty confident about your ability to do
them.
Advance warning: the two optional proofs both
involve using ~I or ~E. These rules are tricky, so if
you plan on considering the proofs, commit
them to memory.
~I (Negation Introduction)
If you have assumed ψ, and you have derived
(φ&~φ),
then you can write down ~ψ,
depending on everything (φ&~φ) depends on
except ψ.
(P v P) ├ P
1
2
1,2
1,2
2
1. (P v P)
2. ~P
3. P
4. (P & ~P)
5. P
A
A (for ~E)
1 vE
2,3 &I
2,4 ~E
PREDICATE LOGIC
Predicate Logic on the Final
There are fewer predicate logic questions on the
final than SL questions. About 2 multiple choice
and 2 short answer. So between 5% and 15% of
your grade, depending on which questions you
choose on the short answer.
What to Study
• The lecture slides on predicate logic.
• Joe Lau’s module on predicate logic, especially
PL02 Quantifiers.
The predicate logic questions are surprisingly
not as difficult as you would think, so studying
PL is a good investment.
Equivalences
I’d highly recommend committing to memory
these very easy-to-remember logical
equivalences:
• ∀yDy ≡ ~∃y~Dy
• ~∀yDy ≡ ∃y~Dy
• ∀y~Dy ≡ ~∃yDy
• ~∀y~Dy ≡ ∃yDy