LOGIC

Question 1

What is the purpose of logic?

Answer 1

Purpose of logic is to deduce the validity of a statement from a set of premises.

Question 2

What is propositional logic?

Answer 2

Statements can only be either true or false
e.g. “I have a pen” you can either have a pen or not

Question 3

What is compound propositional logic?

Answer 3

Use of logical connectives to join different atomic propositions.
e.g. “we’re in paris” or “we’re in Berlin”
“we’re not in paris” (negation)

Question 4

How can we denote different propositions?

Answer 4

Using single letters
e.g. F is “Today is Friday” and H is “I’m happy” then
not F is “Today is not friday”

Question 5

What is negation?

Answer 5

¬ (not)
- 2 outcomes

Question 6

Using negation, how to calculate when “I am not cold” equate to true?

Answer 6

1. P = “I am cold”
2. Check whether P is true
3. Use truth table for negation

Question 7

What is conjunction?

Answer 7

∧
(and)

Question 8

Using conjunction, how to show “It is snowing and I am cold”?

Answer 8

1. Set atomic propositions as letters P = It is snowing Q = I am cold
2. Individually check P and Q
3. Truth table for conjunction

Question 9

What is disjunction?

Answer 9

∨
(or)
.. same question solve as add

Question 10

What is implication?

Answer 10

⇒
(implies)
(if …… then ……)

Question 11

How does implication truth table work?

Answer 11

For propositions P and L, P implies L if;
- If P = true, assumption is not held so it does not matter what conclusion is. So proves P ⇒ L
- if L = true , conclusion is already met, no matter what the assumption. So proves P ⇒ L
- if P = true and L = false. the assumption is held but the conclusion is not so P ⇒ L is false.

Question 12

What is double implication?

Answer 12

⇔
(iff - if and only if)

Question 13

How does double implication work?

Answer 13

P ⇔ Q can be written as:
P ⇒ Q ∧ Q ⇒ P or
If P then Q AND if Q then P
case 1: both True =
if P = T and Q = T then P ⇒ Q = T
if Q = T and P = T then Q ⇒ P = T
T ^ T = T
case 2/3: T and F (both ways) =
if P = T and Q = F then P ⇒ Q = F (based on implication truth table
if Q = F and P = T then Q ⇒ P = T
(based on implication truth table
F ^ T = F
case 4: both F =
if P = F and Q = F then P ⇒ Q = T
if Q = F and P = F then Q ⇒ P = T
T ^ T = T

Question 14

What is the purpose of the precedence hierarchy?

Answer 14

Brackets are used to resolve any ambiguity in arguments. Hierarchy helps reduce number of brackets.
e.g. P ^ Q v R = (P ^ Q) v R
P ⇔ Q ⇒ R is ambiguous (meaning can change depending on brackets)

Question 15

What is the precedence hierarchy?

Answer 15

Tight bind: NOT
AND
OR
Loose bind: IMPLIES, IFF

Question 16

What is logical equivalence?

Answer 16

P1 is logically equivalent to P2 iff they have identical truth values under all possible assignments of truth values.
This allows substitution.

Question 17

What is commutativity?

Answer 17

P v Q ⇔ Q v P
P ^ Q ⇔ Q ^ P

Question 18

What is associability?

Answer 18

(P v Q) v R ⇔ P v (Q v R)
(P ^ Q) ^ R ⇔ P ^ (Q ^ R)

Question 19

What is distributivity?

Answer 19

P ^ (Q v R) ⇔ (P ^ Q) v (P ^ R)
P v (Q ^ R) ⇔ (P v Q) ^ (P v R)

Question 20

What is double negation?

Answer 20

¬(¬P) ⇔ P

Question 21

What is idempotence?

Answer 21

P ^ P ⇔ P
P v P ⇔ P

Question 22

What is de morgan law?

Answer 22

¬(P ^ Q) ⇔ ¬P v ¬Q
¬(P v Q) ⇔ ¬P ^ ¬Q

Question 23

What are the 2 propositional constants?

Answer 23

true and false
true is always T
false is always F

Question 24

What is a tautology?

Answer 24

Proposition that is true for ALL possible combinations
e.g. P v ¬P and (P v ¬Q) v (¬P ^ Q)

Question 25

what is a contradiction?

Answer 25

Proposition that is false for ALL possible combinations
e.g. false and P ^ ¬P

Question 26

What is a Valid argument?

Answer 26

A valid argument is an argument which has a valid conclusion where both premises are valid.

Question 27

What is an argument, premise and conclusion?

Answer 27

ARGUMENT = collection of propositions
PREMISES = all other propositions to meet conclusion. Last column of truth table
CONCLUSION= proposition justified by the premises

Question 28

How to check for a valid argument?

Answer 28

For every instance of each atomic proposition, a premise is found to be T or F.
If both premises are T, consider the conclusion.
Valid argument if final conclusion is also T
Invalid argument if final conclusion is F