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TR1413: Discrete Mathematics For Computer Science

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Title: TR1413: Discrete Mathematics For Computer Science


1
TR1413 Discrete Mathematics For Computer Science
  • Lecture 4 System L

2
Introduction
  • One of the famous propositional formal system was
    developed by Lukasiewicz, known as System L.

3
Alphabets in System L
  • Consists of
  • The infinite set of propositional variables
    p1,p2,,pn
  • The set of punctuation symbols ,,(,)
  • The set of logical operators ?,?

4
wffs in System L
  • A propositional statement P is a wff in System L
    if it conforms to one of the following
    conditions
  • P is a propositional variables or
  • P is of the form (?Q) where Q is a wff or
  • P is of the form (Q ? R) where Q and R are wffs.

5
Axioms in System L
  • Three axioms
  • L1 (U ? (V ? U))
  • L2 ((U ? (V ? W)) ? ((U ? V) ? (U ? W)))
  • L3 (((?U) ?(?V)) ? (V ? U))
  • where U, V and W are wffs.

6
The Inference Rule of System L
  • System L has only one rule of inference, known as
    modus ponens (MP).
  • The rule
  • If V and (V ? W) are wffs of L which are members
    of a proof sequence the the wff W can be added to
    the proof sequence.

7
Proof in System L
  • A proof is defined to be a sequence of wffs of L
    such that each wff is either
  • An instance of an axiom of L or
  • Derivable from two earlier wffs in the sequence
    using the rule MP. The two earlier wffs must be
    of the form P and (P ? Q).
  • Every steps in a proof must be justified clearly.

8
Examples of proof in L
9
Deduction in System L
  • A deduction from the set of wffs T in System L is
    defined to be a sequence of wffs such that each
    wff is either
  • An instance of one of the axioms of L, or
  • One of the hypotheses, or
  • Is derivable from two earlier wffs in the
    sequence using the rule MP.
  • If such a sequence exists, the the sequence of
    wffs leading to a given wff P is said to be a
    deduction of P from T in L.

10
Examples of deduction in L
11
Derived Rules of deduction
  • The Deduction Theorem
  • The Inverse Deduction Theorem
  • Hypothetical syllogism

12
The Deduction Theorem
  • Can be stated as
  • If T ? U ? V where U and V are wffs, and T is a
    set of wffs, then T ? (U?V)
  • Meaning
  • If V is deducible from a set of hypotheses which
    include U, then U?V is deducible from the set
    with U removed.

13
The Inverse Deduction Theorem
  • Can be stated as
  • If the deduction T ? (U?V) where U and V are
    wffs, and T is a set of wffs, can be established,
    then the result T ? U ? V holds.

14
Hypothetical Syllogism
  • Can be stated as
  • (U?V), (V?W)? (U?W)

15
Examples
16
Notation
  • In the textbook, rules are written by using a
    horizontal bar.
  • For example, Hypothetical Syllogism is written as
  • (U?V), (V?W)
  • (U?W)
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