1

Propositional Logic

2

PropositionsA proposition is a declarative sentence that is either

true or false.Examples of propositions:

The Moon is made of green cheese. Trenton is the capital of New Jersey. Toronto is the capital of Canada. 1 + 0 = 1 0 + 0 = 2

Examples that are not propositions. Sit down! What time is it? x + 1 = 2 x + y = z

3

Propositional Logic (or Calculus)Constructing Propositions

Propositional Variables: p, q, r, s, …The proposition that is always true is denoted

by T and the proposition that is always false is denoted by F.

Compound Propositions: constructed from other propositions using logical connectives Negation ¬ Conjunction ∧ Disjunction ∨ Implication → Biconditional ↔

4

Compound Propositions: NegationThe negation of a proposition p is denoted by ¬p and has this truth table:

Example: If p denotes “The earth is round” then ¬p denotes “It is not the case that the earth is round,” or more simply “The earth is not round.”

p ¬p

T F

F T

5

ConjunctionThe conjunction of propositions p and q is

denoted by p ∧ q and has this truth table:

Example: If p denotes “I am at home” and q denotes “It is raining” then p ∧q denotes “I am at home and it is raining.”

p q p ∧ q T T T

T F F

F T F

F F F

6

DisjunctionThe disjunction of propositions p and q is

denoted by p ∨q and has this truth table:

Example: If p denotes “I am at home” and q denotes “It is raining” then p ∨q denotes “I am at home or it is raining.”

p q p ∨qT T T

T F T

F T T

F F F

7

The Connective Or in EnglishIn English “or” has two distinct meanings.

Inclusive Or: For p ∨q to be T, either p or q or both must be T

Example: “CS202 or Math120 may be taken as a prerequisite.” Meaning: take either one or both

Exclusive Or (Xor). In p⊕q , either p or q but not both must be TExample: “Soup or salad comes with this entrée.” Meaning: do not expect to get both soup and salad

p q p ⊕qT T F

T F T

F T T

F F F

8

ImplicationIf p and q are propositions, then p →q is a conditional

statement or implication, which is read as “if p, then q ”p is the hypothesis (antecedent or premise) and q is the

conclusion (or consequence).

Example: If p denotes “It is raining” and q denotes “The streets are wet” then p →q denotes “If it is raining then streets are wet.”

p q p →qT T T

T F F

F T T

F F T

9

Understanding ImplicationIn p →q there does not need to be any

connection between p and q. The meaning of p →q depends only on the truth

values of p and q. Examples of valid but counterintuitive

implications:“If the moon is made of green cheese, then you

have more money than Bill Gates” -- True“If Juan has a smartphone, then 2 + 3 = 6” --

False if Juan does have a smartphone, True if he does NOT

10

Understanding ImplicationView logical conditional as an obligation or

contract:“If I am elected, then I will lower taxes”“If you get 100% on the final, then you will earn an

A”If the politician is elected and does not lower

taxes, then he or she has broken the campaign pledge. Similarly for the professor. This corresponds to the case where p is T and q

is F.

11

Different Ways of Expressing p →q if p, then qif p, qp implies q q if p q when pq whenever pNote: (p only if q) ≡ (q if p) ≢ (q only if p)

It is raining → streets are wet: TIt is raining only if streets are wet: TStreets are wet if it is raining: TStreets are wet only if it’s raining: F

q follows from pq unless ¬p p only if q p is sufficient for q q is necessary for p

12

Sufficient versus NecessaryExample 1:

p →q : get 100% on final → earn A in classp is sufficient for q:

getting 100% on final is sufficient for earning A

q is necessary for p: earning A is necessary for getting 100% on final

Counterintuitive in English: “Necessary” suggests a precondition Example is sequential

Example 2:Nobel laureate → is intelligentBeing intelligent is necessary for being Nobel

laureateBeing a Nobel laureate is sufficient for being

intelligent (better if expressed as: implies)

13

Converse, Inverse, and ContrapositiveFrom p →q we can form new conditional

statements .q →p is the converse of p →q ¬ p → ¬ q is the inverse of p →q¬q → ¬ p is the contrapositive of p →q

How are these statements related to the original?

How are they related to each other?Are any of them equivalent?

14

Converse, Inverse, and ContrapositiveExample:

it’s raining → streets are wet1. converse: streets are wet → it’s raining2. inverse: it’s not raining → streets are not wet3. contrapositive: streets are not wet → it’s not

raining

only 3 is equivalent to original statement1 and 2 are not equivalent to original statement:

streets could be wet for other reasons1 and 2 are equivalent to each other

15

BiconditionalIf p and q are propositions, then we can form the

biconditional proposition p ↔q, read as “p if and only if q ”

Example: If p denotes “You can take a flight” and q denotes “You buy a ticket” then p ↔q denotes “You can take a flight if and only if you buy a ticket”True only if you do both or neitherDoing only one or the other makes the proposition

false

p q p ↔q

T T T

T F F

F T F

F F T

16

Expressing the BiconditionalAlternative ways to say “p if and only if q”:

p is necessary and sufficient for qif p then q, and converselyp iff q

17

Compound Propositionsconjunction, disjunction, negation, conditionals,

and biconditionals can be combined into arbitrarily complex compound proposition

Any proposition can become a term inside another propositionPropositions can be nested arbitrarily

Example: p, q, r, t are simple propositionsp ∨q , r ∧t , r →t are compound propositions

using logical connectives(p ∨q) ∧ t and (p ∨q) →t are compound

propositions formed by nesting

18

Precedence of Logical Operators

Operator Precedence

1

23

45

With multiple operators we need to know the orderTo reduce number of parentheses use precedence rules

Example: • p ∨q → ¬ r is equivalent to (p ∨q) → ¬ r• If the intended meaning is p ∨(q → ¬ r ) then

parentheses must be used

19

Truth Tables of Compound PropositionsConstruction of a truth table:Rows

One for every possible combination of values of all propositional variables

ColumnsOne for the compound proposition (usually at

far right)One for each expression in the compound

proposition as it is built up by nesting

20

Example Truth TableConstruct a truth table for

p q r p q r p q → rT T T T F F

T T F T T T

T F T T F F

T F F T T T

F T T T F F

F T F T T T

F F T F F T

F F F F T T