Propositional Logic

1
Propositional Logic

• From Chapter 4
• Formal Specification using Z
• David Lightfoot

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Propositional calculus

• Propositional calculus is also known as Boolean
  algebra. Propositions in Z are either true or
  false. Negation can be written using bar notation
  . In Z negation is written as
  (pronounced not)
• p p
• false true
• true false

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Propositional calculus

• Conjunction is pronounced and and is written
  as L.
• p q p L q
• false false false
• false true false
• true false false
• true true true

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Propositional calculus

• Disjunction is pronounced or and is written as
  v.
• p q p v q
• false false false
• false true true
• true false true
• true true true

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Implication Definition

• If p and q are propositions, the compound
  proposition
• if p then q
• is called a conditional proposition (implies)
  and is denoted as
• The proposition p is called the hypothesis (or
  antecedent )
• and the proposition q is called the conclusion
  (or consequence). Can also be written as

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Truth Table For Implication (Conditional
Proposition)
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Implication P Q

• If it is raining I will wear a raincoat.
• Statement does not say what I will do if it is
  not raining. Rule only covers first two cases,
  must apply logic of first two cases to second two
  cases, i.e. when its not raining
• raining raincoat (true)
• raining raincoat (false)
• raining raincoat (true)
• raining raincoat (true)
• To get truth value for last two
• cases apply (P v Q)

(P v Q) P Q P Q F F
T F T T T F
F T T T
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Implication

• P Q
• is a predicate that is true if (P v Q)
• Example
• (11 lt 3) (225) is true
• (11lt3) (224) is true
• (11 gt 3) (225) is false
• (11gt3) (224) is true

P Q P Q F F T F T
T T F F T T
T
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Equivalence

• If p and q are propositions, the compound
  proposition
• p if and only if q
• (sometimes written p iff q)
• it is also is called a bi-conditional
  proposition and is denoted by

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Equivalence

• An alternative ways to state the equivalence (or
  a bi-conditional proposition) are
• p is equivalent to q
• p is a necessary and sufficient condition for q
• p if and only if q

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Truth Table For Equivalence
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Equivalence

• (P Q) (P v Q)
• ((P Q) L (Q P)) (P Q)

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DeMorgans Laws

• (P L Q) P v Q
• (P v Q) Q L P

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Demonstrating Laws of Logic

• A law is a relationship which holds good
  irrespective of the propositions involved. The
  truth tables can be used to demonstrate the
  validity of a law. For example, to show the
  validity of the first of DeMorgans laws
• (P L Q) P v Q
• We complete the truth table, building towards the
  expressions to be compared.
• Write the truth table for DeMorgans laws in
  Word, using Z fonts.

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Using Laws of Logic

• Laws are used to prove that two statements in the
  propositional calculus, that may not necessarily
  be identical, are equivalent. In formal
  specifications laws that are used in chains of
  transformations are called proofs which can
  verify that a specification is consistent and
  makes deductions about behaviour of a system from
  its specification.

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Order of evaluation

• 1. Logical connectives within brackets.
• 2. Negation
• 3.
• 4
• 5
• Where you have a choice use brackets.
• Associativity is left except for the conditional
  which is right.

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Logic Terminology

• and is often called a Conjunction
• or is often called a Disjunction.
• A tautology is a proposition that is always true
  e.g.
• (B v B ) (Shakespear).
• A contradiction is always false e.g.
• (B LB )

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Logic Terminology(Not Core)

• A Well Formed Formula (WFF).
• Let p,q,r.., be propositions. If we have some
  compound proposition (or formula) called W
  involving p,q,r.., whenever these variables are
  replace by their truth values and W becomes a
  proposition. Then W is a well formed formula.

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Logic Terminology(Not Core)

• In some cases, two different propositions may
  have the same truth values no matter what truth
  values their constituent propositions have. Such
  propositions are said to be logically equivalent .

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Logic Terminology(Not Core)

• Suppose that the compound proposition P and Q
  are made up of the propositions p1pn. We say
  the P and Q are logically equivalent and write
• given any truth values of p1pn, either P and Q
  are both true or both false.

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Logic Terminology(Not Core)

• If P and Q are wffs, we say that P logically
  implies Q if any assignment of truth values to
  the propositions which make P true also make Q
  true. We write
• P Q
• Contratrast this definition with implies, which
  can be defined in terms of a truth table.
  Difffers form Stimulus/Response and
  Condition/Response.

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Value Variable

• Value a constant,no location in time or space
• Variable holder for value, has location in time
  space

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Mathematical proof

• A mathematical system consists of
• Axioms which are assumed true.
• Definitions which are used to create new concepts
  in terms of existing ones
• Undefined terms are not explicitly defined but
  are implicitly defined by axioms.
• A theorem is a proposition that has been proved
  to be true.
• An argument that establishes the truth of a
  theorem is called a proof.
• Logic it the tool for the analysis of proof.

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Logical Argument

• A logical argument is a sequence of propositions
• p1 Punch is a cat
• p2 All cats are clever
• pn
• ------ -------
• q Punch is clever

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Exercise 0

• Draw the truth table for
• p XOR q
• Show that the following two definitions are
  equivalent.
• p XOR q (p v q) L (p L q)
• p XOR q (p L q) v (p L q)

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Exercise 1

• Show by truth table that
• (P Q) (P v Q)

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Exercise 2

• Show by truth table that
• (P Q) L (Q P) (P Q)

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Exercise 3

• By using the laws from chapter 4 simplify
• (p onboard L onboard lt capacity )

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Exercise 3 Answer

• By using the laws from chapter 4 simplify
• (p onboard L onboard lt capacity )
• (p onboard) v (onboard lt capacity )
• p onboard v onboard capacity )

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Exercise 4

• By using the laws from chapter 4 simplify
• (a L b) v (a L c) v(a L c)

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Exercise 4 Answer

• By using the laws from chapter 4 simplify
• (a L b) v (a L c) v(a L c)
• a L (b v c v c)
• a L (b v true) a L true
  a

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Exercise 5

• Recall from chapter 2
• 1. Certain people are registered as users of a
  computer system. At any given time, some of these
  users are logged in to the computer. There is a
  limit (unspecified) to the number of users logged
  in at any one time. All users are either staff
  users or customers.

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Exercise 5

• Given
• p loggedIn p user
• Check that
• p loggedIn L p user
• can be simplified to
• p loggedIn
• user truth table

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Exercise 5 Answer

• Given
• p loggedIn p user
• ----
• p loggedIn L p user
• Can only be true is if both sub-expressions are
  true
• Because of the given implication if
• p loggedIn
• then so is
• p user

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Exercise 6

• Use DeMorgans Laws to simplify

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Exercise 6Answer

• Use DeMorgans Laws to simplify
• (x2 and x6)
• Tricky
• (x2) V (x6)
• any number is either different from 2 or
  different from 6
• Moving towards variables

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Exercise 7

• Simplify
• st L s EOF L t EOF

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Exercise 7 Solution

• Simplify
• st L s EOF L t EOF
• st L s EOF

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Exercise 8

• Simplify
• xx L (x y v xy)

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Exercise 8 Solution

• Simplify
• xx L (x y v xy)
• x y

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Exercise 9

• Simplify
• x0 L x 0
• x0

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Exercise 10

• Simplify
• (age 16 v student)
• age lt 16 L student)