This document provides an overview of propositional logic, including: - Propositions are statements that can be true or false. Compound propositions combine simpler statements with logical connectives like "and" and "or". - Truth tables show the truth values of compound propositions based on the truth values of their variables. - Common logical connectives include conjunction, disjunction, negation, implication, and equivalence. - Tautologies and contradictions are types of statements that are always true or false regardless of variable values. - Quantifiers like "for all" and "there exists" can be used to define propositional functions on a domain. - Valid arguments are those where the conclusion is necessarily true

1. Propositional Logic
By
Mamata Pandey

2. Proposition
 A proposition (or statement) is a
declarative statement which is true or
false, but not both.
 Following statements are propositions
◦ Ice floats in water.
◦ 2 + 2 = 4
◦ China is in Europe
 Following statements are not
propositions
◦ Where are you going?

3. Compound and Primitive
Propositions
 Compound propositions are
composed of two or more sub-
propositions and various connectives
or logical operators (eg. And, or,
not…)
◦ Ex: Roses are red and violets are blue
 If a proposition cannot be broken into
simpler propositions, it is called
primitive preposition.
◦ Ex: it will rain

4. Truth Table
 A compound proposition can be denoted as P(p, q, . . .)
denote an expression constructed from logical variables
p, q, . . ., which take on the value TRUE (T) or FALSE (F),
and the logical connectives ∧, ∨, and ￢ (and others
discussed subsequently).
 The main property of a proposition P(p, q, . . .) is that its
truth value depends exclusively upon the truth values of
its variables, that is, the truth value of a proposition is
known once the truth value of each of its variables is
known.
 A simple concise way to show this relationship is
through a truth table.
 With n variables there are 2n rows, each with unique
combination of T and F values of variables and
corresponding truth value for proposition P.

5. Basic Logical Operations
(Logical Connectives)
 Conjunction, p ∧ q
 Disjunction, p ∨ q
 Negation, ￢p
 Implication, p → q
 Equivalence, p ↔ q

6. Conjunction, p ∧ q
 Any two propositions p, q can be
combined by the word “and” to form a
compound proposition called the
conjunction of the original
propositions.
 Symbolically, p ∧ q
 p ∧ q is true if both
p and q are true,
otherwise it is false.

7. Disjunction, p ∨ q
 Any two propositions can be combined
by the word “or” to form a compound
proposition called the disjunction of
the original propositions.
 Symbolically, p ∨ q
 p ∨ q is false if
both p and q are false,
otherwise it is true.

8. Negation, ￢p
 Given any proposition p, another
proposition, called the negation of p,
can be formed by inserting the word
“not” before p .
 Symbolically, the negation of p, read
“not p,” is denoted by ￢p or ~p
 ￢p is true if p is false and
￢p is false if p is true

9. Implication, p → q
 p → q is false only when the first part
p is true and the second part q is
false.
 Accordingly, when p is false, the
conditional p → q is true regardless of
the truth value of q
 i.e. If p then q.
 p → q is also called
conditional statement

10. Equivalence, p ↔ q
 p ↔ q is true whenever p and q have
the same truth values and false
otherwise.
 i.e. p if and only if q
 p ↔ q is also called biconditional
statement

11. Tautologies And
Contradictions
 A proposition P(p, q, . . .) contain only T in the last
column of its truth tables i.e. it is true for any truth
values of their variables. Such propositions are
called tautologies.
 A proposition P(p, q, . . .) is called a contradiction if
it contains only F in the last column of its truth table
i.e. if it is false for any truth values of its variables.
Tautology Contradiction

12. Algebraic Properties of
Propositions

13. Properties of implications

14. Propositional Function P(x)
 Let A be a given set. A propositional function
is an open sentence or condition defined on
expressed as P(x) which has the property
that p(a) is true or false for each a ∈ A.
 P(x) becomes a statement with a truth value
whenever any element a ∈ A is substituted for
the variable x.
 The set A is called the domain of P(x), and
 the set Tp of all elements of A for which P(a)
is true is called the truth set of P(x).
Tp = {x | x ∈ A, P(x) is true} , or
Tp = {x | P(x)}

15. Propositional Function
P(x):Example
 propositional function P(x) defined on
the set N of positive integers such
that
 P(x) be “x + 2 > 7” , then
 Tp={x| x ꞓ N and x + 2 > 7}
 i.e. truth set Tp= {6, 7, 8, . . .}
consisting of all integers greater than
5.

16. Universal Quantifiers
 The symbol ∀ which reads “for all” or “for
every” is called the universal quantifier.
 We can define a proposition function as
(∀x ∈ A)p(x) or ∀x p(x)
 If {x| x ∈ A, p(x)} = Athen ∀x p(x) is true;
otherwise, ∀x p(x) is false
 Example:
◦ The proposition (∀n∈ N)(n + 4 > 3) is true
since {n | n + 4 > 3} = {1, 2, 3, . . .} = N.
◦ The proposition (∀n∈ N)(n + 2 > 8) is false
since {n | n + 2 > 8} = {7, 8, . . .} = N

17. Existential Quantifier
 The symbol ∃ which reads “there exists”
or “for some” or “for at least one” is
called the existential quantifier
 We can define a propositional function
as (∃x ∈ A)p(x) or ∃x, p(x)
 If {x | p(x)} ≠ ф then ∃x p(x) is true;
otherwise, ∃x p(x) is false.
 Example:
◦ The proposition (∃n ∈ N)(n + 4 < 7) is true
since {n | n + 4 < 7} = {1, 2} = .
◦ The proposition (∃n ∈ N)(n + 6 < 4) is false
since {n | n + 6 < 4} = . x) or ∃x, p(x) (4.3)

18. Properties of Quantifiers

19. Well Formed Formula(WFF)
 WFF has following properties
◦ Every primitive proposition or atomic
statement is a WFF
◦ If proposition p is WFF then ￢p is also
WFF
◦ if p and q are WFF then p ∧ q, p ∨ q and
p → q are WFF

20. Example 1:
 Prove that : p → q ≡ ￢p ∨ q
 From above truth tables LHS= RHS
proved.
LHS: p → q RHS: ￢p ∨ q

21. Example 2:
 Prove that: (p → q)↔(￢q → ￢ p)
 From above truth table
(p → q)↔(￢q → ￢ p) is tautology ,
hence proved.

22. Arguments
 An argument is an assertion that a given set
of propositions P1, P2, . . . , Pn, called
premises or hypotheses, yields (has a
consequence) another proposition Q, called
the conclusion.
 Argument is denoted by P1, P2, . . . , Pn ͱ Q
 An argument P1, P2, . . . , Pn ͱ Q is said to
be valid if Q is true whenever all the premises
P1, P2, . . . , Pn are true.
 An argument which is not valid is called
fallacy.
 The argument P1, P2, . . . , Pn ͱ Q is valid if
and only if the proposition (P1∧P2 . . .∧Pn) →
Q is a tautology.

23.  Argument p1, p2, . . . , pn ͱ q can also be
written as
 Validity depends on only form of
statement not on the truth values of
statements
 Validity sates “if all pi’s are true then q is
true” not that “q is true”

24. Rules of Inference
 Each of following is Tautology

25. Arguments based on
tautologies
 Rules of inference are used to validate
arguments
 For example following tautology,
“if p implies q and q implies r then p implies r”
i.e.
((p → q) ∧(q →r)) →(p →r), or
Can be used to prove that an argument is
valid

26. Example 3
 Prove following argument is valid
 Let p = “He is a bachelor,” q = “He is
unhappy” and r = “He dies young;
 S1: p →q
 S2:q →r
 S: p →r
 Hence from rule of inference this is valid
argument

27. Example 4
 Prove if following argument is valid

28. Example 5
 Let n be an integer. Prove that n2 is
odd if n is odd.
 It solved by proof by contradiction
using tautology
(p → q)↔(￢q → ￢ p)