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

2. 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.

3. Examples “1 Omani Rial = 300 Bisas”
“The pass mark of IDM course is 50”
“7 is greater than 5”
“There are 8 days in a week”
(2) & (3) are true – (1) & (4) are false
“Stop talking”
“What day of the week is it today”
(1) is a command & (2) is a question
so neither is proposition.

4. 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.

5. 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.

6. Connectives The statement: is a compound proposition.
‘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.

7. 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:

8. if-and-only-if (or is-equivalent-to)
Table of Connectives
Connective
Symbol
and or
not
If-then (or implies)
if-and-only-if (or is-equivalent-to)     

9. 'It is snowing' 'I will go skiing'.
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.

10. 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

11. Truth Table for ‘and’ p q
p  q T F

12. 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’

13. Truth Table for ‘or’ p q
p  q T F

14. The Connectives ‘not’ & ‘if-then’
The truth table for not is straightforward. i.e. if
p 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’

15. 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.

16. Truth Table for ‘if-then’p q
p  q T F

17. 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’

18. 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.

19. Truth Table for ‘if-and-only-if’p q
p  q T F

20. 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)

21. 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

22. 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

23. Truth Table – Basic Connectivesp q
p  q
p  q p
p  q
p  q T F

24. Truth Table for (p  q)  (p  q)

25. 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

26. 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

27. 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?

28. 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.

29. 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 the
proposition)

30. 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

31. 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

32. 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

33. Truth Table for Converse & Contrapositive
Now use a truth table to investigate the
expressions p → q, q → p and ¬q → ¬p p q
p  q
q  p q p
¬q  p T F

34. 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.

35. 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

36. 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

37. 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

38. Truth Table for the Argumentp q P1
p  q P2 p
P1  P2
(pq)p Q q
(P1  P2)  Q
[(pq) p] q T F

39. 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

40. 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

41. 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

42. Laws of Logic Law(s) of Logic Name p ↔ q ≡ (p → q) ∧ (q → p)
¬¬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

43. 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

44. 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

45. 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”

46. 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 the
expression – if it doesn’t, then try another law
The process gets easier with practice!

47. 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)

48. Yet More Examples Exercise: Simplify the logical expression
¬(p → ¬q) ∧ ¬p
Example: Use truth tables to verify that
p → 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.