1
Introduction
Propositional Logic
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Propositional Logic
22
Use of Logic
In mathematics and rhetoric:
square4
Give precise meaning to statements.
square4
Distinguish between valid and invalid arguments.
square4
Provide rules of `correct’ reasoning.
Natural language can be very ambiguous
`If you do your homework, then you’ll get to watch the game.’
`You do your homework, or you’ll fail the exam.’
=
`If you don’t do your homework, then you’ll fail the exam.’
≈
`If you don’t do your homework, then you will not get to watch ...’
Discrete Mathematics – Propositional Logic
23
Use of Logic (cntd)
In computing:
square4
Derive new data / knowledge from existing facts
square4
Design of computer circuits.
square4
Construction of computer programs.
square4
Verification of correctness of programs and circuit design.
square4
Specification
What the customer
really needed
How the Programmer
understood it
What the customer
got
Discrete Mathematics – Propositional Logic
24
Statements (propositions)
Propositional logic deals with
statements
and their
truth
values
A
statement
is a declarative sentence that can be
true
or
false
Truth values
are
TRUTH
(
T
or
1
)
and
FALSE
(
F
or
0
).
Examples:

1 + 1 = 2
(statement,
T)

The moon is made of cheese
(statement,
F)

Go home!
(not statement, imperative)

What a beautiful garden!
(not statement,
exclamation)

Alice said, `What a beautiful garden!’
(statement,
depends on Alice)

y
+ 1 = 2
(not statement,
uncertain)

The God exists
(statement,
?)
Discrete Mathematics – Propositional Logic
25
Compound Statements
We cannot decide the truth value of a primitive statement.
This is
not what logic does.
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 Spring '08
 PEARCE
 Logic, Gone with the Wind, Logical connective, connectives

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