TR1413: Discrete Mathematics For Computer Science

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TR1413 Discrete Mathematics For Computer Science

• Lecture 4 System L

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Introduction

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

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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 ?,?

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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.

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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.

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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.

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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.

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Examples of proof in L
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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.

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Examples of deduction in L
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Derived Rules of deduction

• The Deduction Theorem
• The Inverse Deduction Theorem
• Hypothetical syllogism

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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.

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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.

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Hypothetical Syllogism

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

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Examples
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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)