Logic

Reading: Chapter 4 (44-59) from the text book

1

Propositions

• In arithmetic we work with numbers

• Similarly, the fundamental objects in logic are propositions

• Definition: A proposition is a statement that is either true or false. Whichever of these is the case, it is called truth value of the proposition.

2

Examples

1. “1 Omani Rial = 300 Bisas”2. “The pass mark of IDM course is 50”3. “7 is greater than 5”4. “There are 8 days in a week”(2) & (3) are true – (1) & (4) are false

1. “Stop talking”2. “What day of the week is it today”(1) is a command & (2) is a question so neither is proposition.

3

Predicates

• ‘x > 5’ is an example of a predicate.

• A predicate is a statement which contain one or more variables; it cannot be assigned a truth value until the value of the variables are specified.

• e.g. if x=7, the sentence is true, but if x=2, it is false.

4

Logical Structure of Propositions

• In logic, we will study propositions such as:‘If the door and the window are not both closed, then the door is

not closed or the window is not closed’

• This proposition is true because of its logical structure.

• As well, any sentence with the same logical structure must also be true.

5

Connectives

• The statement: ‘If the door and the window are not both closed, then the

door is not closed or the window is not closed’

is a compound proposition.

• It is built up from atomic proposition: ‘the door is closed’ & ‘the window is closed’

using the words and, or, not & if-then

• These words are called connectives.6

Notation for Basic Connectives

• To identify the connectives and, or, not, etc.., we will underline them (and, or, not, etc…)

• Note that if-then is also known as implies and if-and-only-if is also known as is-equivalent-to

• The 5 connectives we will use, and their symbols are given in the following table:

7

Table of Connectives

8

Connective Symbol

and

or

not

If-then (or implies)

if-and-only-if (or is-equivalent-to)

Example of Notation• Let p be the proposition

‘Cows are animals’ • And let q denote:

‘Cows produce milk’. • Then:

p q denotes the proposition ‘Cows are animals and cows produce milk’

p means ‘Cows are not animals’

Exercise : Let p and q denote respectively the propositions 'It is snowing' 'I will go skiing'.

Write down English sentence corresponding to the proposition Express the proposition: ‘If I will not go skiing then it is snowing’

in symbolic form.9

qp

The Connective ‘and’

• The everyday understanding of the term ‘and’ is that for p q to be true, p & q must both be true.

• If p is false, or q is false, or both are false, then p q is false

• This everyday understanding leads to the following truth table

10

Truth Table for ‘and’

11

p q p q

T

T

F

F

T

F

T

F

T

F

F

F

The Connective ‘or’

In everyday language, ‘or’ has two meanings

• In ‘Ali will pass or fail’, it means the result will be either a pass or a fail, but not both.

• Here, ‘or’ is used exclusively –the possibility of both outcomes is excluded

• In logic, or means the inclusive ‘or’ –i.e. p q means ‘p or q or both’

12

Truth Table for ‘or’

13

p q p q

T

T

F

F

T

F

T

F

T

T

T

F

The Connectives ‘not’ & ‘if-then’

• The truth table for not is straightforward. i.e. ifp is T then p is F & if p is F then p is T

• Now look at the connective if-then• Suppose p is the propn ‘Fatima attends tutorials’,

& q is the propn ‘Fatima will pass Discrete Maths’

• Then p q is the propn ‘If Fatima attends tutorials, then she will pass Discrete Maths’

14

The Connective ‘if-then’

• The claim is certainly true if Fatima attends tutorials & passes, so p q is T if p is T & q is T

• However, if Fatima attends tutorials and doesn’t pass, the claim is false, so p q is F if p is T & q is F

• What if p is false –i.e. Fatima doesn’t attend tutorials?

• The claim is true, because it says only what would happen if Fatima attended tutorials.

• Thus p q is true whenever p is false. 15

Truth Table for ‘if-then’

16

p q p q

T

T

F

F

T

F

T

F

T

F

T

T

The Connective ‘if-then’

• The truth table can be interpreted as:‘The only time the implication is false is if the premise is true & the conclusion is false’

• Are the following propositions true or false?:‘If the sky is yellow then cows have 6 legs’‘If the sky is blue then cows have 6 legs’

17

The Connective ‘if-and-only-if’

• The connective is defined to be true precisely when the 2 propositions it connects have the same truth value.

• i.e. p q is true whenever p & q are both true or p & q are both false.

• Thus the proposition ‘Al-Aqsa is in India if and only if apples are blue’ is true.

18

Truth Table for ‘if-and-only-if’

19

p q p q

T

T

F

F

T

F

T

F

T

F

F

T

Compound Propositions

• A compound proposition is a proposition built up from atomic (i.e. basic) propositions

• Example: Write in symbols: ‘Either Thomas Edison or Alexander Bell invented the telephone’

• Solution: Firstly define the atomic propositions:p is ‘Thomas Edison invented the telephone’q is ‘Alexander Bell invented the telephone’

Then the proposition is (p q) (p q)20

Logical Expressions

• With (p q) (p q) we could find its truth value if we knew the truth values of:

p = ‘Thomas Edison invented the telephone’q = ‘Alexander Bell invented the telephone’

• In logic, we will study expressions such as (p q) (p q), where p & q are regarded as variables, rather than specific propositions

21

Logical Expressions and Truth Tables

• If this is done, (p q) (p q) is a logical expression –it becomes a proposition only when p & q are replaced by propositions.

• This is exactly the same as algebra, x2 – 4 becomes a number only when x is replaced by a number

• The logical expression(p ∨ q) ¬(∧ p ∧ q)

can be analyzed using a truth table 22

Truth Table – Basic Connectives

23

p q p q p q p p q p q

T

T

F

F

T

F

T

F

T

F

F

F

T

T

T

F

F

F

T

T

T

F

T

T

T

F

F

T

Truth Table for (p q) (p q)

24

p q p q p q (p q) (p ∨ q) ¬(∧ p ∧ q)

T

T

F

F

T

F

T

F

T

T

T

F

T

F

F

F

F

T

T

T

F

T

T

F

Truth Tables

• If a logical expression has 2 variables, its truth table will have 4 rows (as in the example)

• If an expression has 3 variables (p, q, r), its truth table will have 8 rows – i.e. 1 row for each of the 8 ways of allocating truth values to p, q & r

25

Tautologies

• Recall the proposition:‘If the door and the window are not both closed, then the door is not closed or the window is not closed’

• If p denotes ‘the door is closed’ & q is ‘the window is

closed’, this proposition is ¬(p ∧ q) → (¬p ¬∨ q)

• The truth table for this logical expression shows that it is always true, irrespective of the truth values of p and q

• Such an expression is called a tautology 26

Contradictions• Consider the proposition

p ¬(∧ p ∨ q)

• The truth table for this logical expression shows that it is always false, irrespective of the truth values of p and q.

• Such an expression is called a contradiction.

• Exercise: Is the proposition p (¬∧ p ¬∨ q)a contradiction?

27

Logical Equivalence

• The table shows that for every combination of the truth values of p & q, the truth values of:

p ∧ q and q (∧ p ¬∨ q) are the same.

• The expressions: p ∧ q and q (∧ p ¬∨ q)

are said to be logically equivalent.28

Definition of Logical Equivalence

• Definition: 2 expressions are logically equivalent if they have the same truth values for every combination of the truth values of the variables.

• Exercise: Use a truth table to show that ¬(p ∨ q) and ¬p ¬∧ q are logically equivalent (so it doesn’t matter which expression is used for theproposition) 29

Logical Equivalence & Tautology

• If the two expressions A and B are logically equivalent, they always have the same truth values, this means that A ↔ B is always true.

• Therefore A ↔ B is a tautology

• So another way of saying that A and B are logically equivalent is to say that A ↔ B is a tautology

30

Implications

• Recall that if-then is also known as implies• i.e. p → q can be read as ‘if p then q’ or ‘p

implies q’• Expressions of the type p → q are called

implications

• The converse of p → q is q → p• The contrapositive of p → q is ¬q → ¬p

31

The Converse and Contrapositive

• Example: Consider the implication ‘If I am in Salalah, then I am in Oman’

• Its converse is: ‘If I am in Oman, then I am in Salalah’

• Its contrapositive is: ‘If I am not in Oman, then I am not in Salalah’

• In this example, the original proposition is a true sentence, its converse is false, and its contrapositive is true

32

Truth Table for Converse & Contrapositive

• Now use a truth table to investigate the expressions p → q, q → p and ¬q → ¬p

33

p q p q q p q p ¬q p

T

T

F

F

T

F

T

F

T

F

T

T

T

T

F

T

F

T

F

T

F

F

T

T

T

F

T

T

Converse and Contrapositive

• The truth table shows (by columns 3 & 7):an implication and its contrapositive are logically equivalent.

• However (by columns 3 & 4): an implication and its converse are not logically equivalent.

• Thus the results of the Salalah/Oman example aren’t surprising. 34

Arguments in Logic

• The argument consists of some premises & a conclusion which is supposed to be a consequence of the premises

• An argument has the logical expression(P1 ∧ P2) → Q

• In (P1 ∧ P2) → Q, P1 and P2are the premises and Q is the conclusion 35

Validity of an Argument

• If the argument is valid, the conclusion should be true whenever all the premises are true

• This means that if P1 ∧ P2 is true, then Q must also be true

• Thus the argument is valid provided that(P1 ∧ P2) → Q is a tautology

36

Testing the Validity of an Argument

• Let’s test the validity of the argument:[(p → q) ¬∧ p] → ¬q

• Is [(p → q) ¬∧ p] → ¬q a tautology?

• To answer this, we’ll use a truth table, though the laws of logic could also be used

37

Truth Table for the Argument

38

p q

P1p q

P2p

P1 P2(pq)p

Q

q(P1 P2) Q

[(pq) p] q

T

T

F

F

T

F

T

F

T

F

T

T

F

F

T

T

F

F

T

T

F

T

F

T

T

T

F

T

Conclusion from the Truth Table

• By the truth table, (P1 ∧ P2) → Q is not always true (i.e. it’s not a tautology), so the original argument is not valid

• In fact, row 3 of the truth table tells us why the argument is not valid:

• The premises of the argument are satisfied, but the conclusion is not satisfied

39

Laws of Logic• In a previous slide we showed:

q (p ¬q) ∧ ∨ and p q ∧ are logically equivalent

• Thus the more complicated expression can be replaced by the simpler expression without affecting the truth value

40

Laws of Logic

• Aim to simplify logical expressions effectively• Example: q (p ¬q) ≡ p q∧ ∨ ∧• The symbol ≡ means ‘is logically equivalent to’

• To do this, we establish a list of key pairs of expressions that are logically equivalent.

• The most important laws of logic follow

41

Laws of Logic

42

Law(s) of Logic Name

p ↔ q ≡ (p → q) (q → p)∧

p → q ≡ ¬p q∨

¬¬p ≡ p

equivalence

implication

double negation

p p ≡ p

p q ≡ q p

(p q) r ≡ p (q r)

p (q r) ≡ (p q) (p r)

¬(p q) ≡ ¬p ¬q

p T ≡ p

p F ≡ F

p ¬p ≡ F

p (p q) ≡ p

p p ≡ p

p q ≡ q p

(p q) r ≡ p (q r)

p (q r) ≡ (p q) (p ∧ r)

¬(p q) ≡ ¬p ¬q

p F ≡ p

p T ≡ T

p ¬p ≡ T

p (p q) ≡ p

idempotent

commutative

associative

distributive

de Morgan’s

identity

annihilation

inverse

absorption

The Laws of Logic

• The first two laws allow for the connectives ↔ and → to be removed from any expression

• All remaining laws involve just and, or & not

• Apart from the double negation law, all these remaining laws occur in pairs

43

Using the Laws of Logic• We could use the laws of logic to simplify logical

expressions

• Example: Use the laws of logic to simplify the expression: (p ¬∧ q) ∨ q

• Solution: (p ∧ ¬q) ∨ q≡ q ∨ (p ∧ ¬q) (2nd commutative

law)≡ (q ∨ p) ∧ (q ∨ ¬q) (2nd distributive law)≡ (q ∨ p) ∧ T (2nd inverse law)≡ q ∨ p (1st identity law)≡ p ∨ q (2nd commutative

law)

Therefore (p ∧ ¬q)∨ q ≡ p∨ q 44

Why not Use Truth Tables?

• Note that we could not have used truth tables in the previous example.

• Truth tables can be used to verify logical equivalences, but the laws of logic are needed to determine the equivalences in the first place.

• Thus truth tables could be used to answer the question “Verify (p ¬∧ q) ∨ q ≡ p ∨ q”

45

How to decide which law(s) to use

• There are no fixed rules to determine which law(s) to use when simplifying expressions

• However, begin by eliminating ↔ and → (if they appear) using the first 2 laws

• After this, try a law to see if it helps to simplify theexpression – if it doesn’t, then try another law

• The process gets easier with practice!46

Another Example

• Example: Simplify the expression ¬(p → ¬q) ∧ p

• Solution: ¬(p → ¬q) ∧ p≡ ¬(¬p ¬∨ q) ∧ p (implication law)≡ (¬¬p ¬¬∧ q) ∧ p (2nd de Morgan’s law)≡ (p ∧ q) ∧ p (double negation law)≡ p (∧ p ∧ q) (1st commutative law)≡ (p ∧ p) ∧ q (1st associative law)≡ p ∧ q (1st idempotent law)

47

Yet More Examples

• Exercise: Simplify the logical expression¬(p → ¬q) ¬∧ p

• Example: Use truth tables to verify thatp → q ≡ ¬p ∨ q

• Using truth tables may be a lengthy method, but it is a mechanical process that will always work.

• Using the laws of logic is usually shorter, but often it’s not easy to know which law to apply.

48