2. Introduction
• classical logic that are necessary for the
understanding of formal development methods
• begin by presenting the propositional logic
– which deals with simple truthvalued
– statements that can be combined according to a set of
rules
• Further, present the predicate logic, which is an
essential tool needed in the formal specification
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3. Propositions
In classical logic, propositions are
statements that are either
TRUE or FALSE.
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4. Propositions Examples (True)
• The following are examples of propositions
that evaluate to TRUE:
There are seven days in a week
Paris is the capital of France
2 + 4 = 6
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5. Propositions Examples (False)
• The following propositions evaluate to FALSE:
The angles of a triangle add up to 360
London is the capital of France
2 - 4 = 7
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6. Examples with Symbol
• In mathematics we often represent a
proposition symbolically by a variable name
such as P or Q.
• For example:
P: I go shopping on Wednesdays
Q: 102.001 > 101.31
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7. • Occasions arise when it is not possible to
evaluate expressions precisely – maybe a
program terminated incorrectly, or perhaps
somebody tried to evaluate the square root of
a negative integer.
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8. • It is possible to account for such situations by
defining a three-valued logic, which allows a
proposition to take the value UNDEFINED as
well as TRUE or FALSE.
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9. Logical Connectives
• Simple propositions can be combined into
compound statements by operators called
logical connectives.
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10. Truth Tables
• The purpose of defining these connectives is
to provide a rigorous framework that gives
precise meaning to such words as ‘AND’ and
‘OR’ that occur in the natural language. The
way we give semantic meaning to these
connectives is to provide tables known as
truth tables,
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11. The AND operator
• The operator known as AND is represented by
the symbol “^”.
• The statement P AND Q is therefore
represented by:
P ^ Q
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12. Cont.
• The precise meaning of this operator is given
in the following truth table, where TRUE and
FALSE are represented by T and F, respectively.
• you can see that the first two columns of the
truth table provide all the possible
combinations of the values of P and Q.
• The final column shows the corresponding
value of the combined statement P Q.
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13. Truth table “AND” Operator
P Q P ^ Q
T T T
T F F
F T F
F F F
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14. Conjunction
• Combining two propositions with the AND
operator is known as conjunction.
• Individual proposition in the compound
statement is known as a conjunct.
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15. The OR operator
• The operator known as or is represented by
the symbol .
• The statement P or Q is therefore represented
by:
P V Q
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16. OR Example
• Thus if P represented the statement It is
raining and Q represented the statement
Today is Tuesday then:
• P v Q would represent the statement It is
raining or today is Tuesday.
• The precise meaning of this operator is given
in the following truth table.
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17. Truth table “OR” Operator
P Q P Q
T T T
T F T
F T T
F F F
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18. Disjunction
• Combining two propositions with the or
operator is known as disjunction.
• individual proposition in the compound
statement is known as a disjunct.
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19. The implication operator
• In defining an implication operator we
attempt to give meaning to the expression P
implies Q.
• The implication operator is represented by the
symbol ⇒. The statement P implies Q is
therefore represented by:
P ⇒ Q
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20. Cont.
• An alternative way of expressing implication is
if P then Q.
• Thus if P represented the statement “It is
Wednesday and Q represented the statement
I do the ironing” then:
• P ⇒ Q would represent the statement “If it is
Wednesday I do the ironing”.
• The truth table for implication appears next,
and requires some explanation.
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21. Truth table “implication” Operator
P Q P⇒Q
T T T
T F F
F T T
F F T
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22. Cont.
• The first two rows of the table capture the
central idea of implication: if the first and
second statements are both TRUE, then the
statement that the first implies the second is
also TRUE.
• whereas if the first is TRUE but the second is
not, then the statement that the first implies
the second is FALSE.
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23. Further Readings
• Palgrave Macmillan Formal Software
Development From VDM to Java by Quentin
Charatan and Aaron Kans (Chapter 2)
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24. Questions
Any Question Please?
You can contact me at: umber@uosahiwal.edu.pk
Your Query will be answered within one working day.
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