Natural Deduction 1

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Natural Deduction (1)
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• Besides using truth-table to prove the validity
  of an argument, we can use another method, called
  natural deduction, to do the same.
• Using this method, we can deduce step-by-step the
  conclusion from the premises with the help of
  some rules.

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Rules of Inference

• There are altogether 18 rules 8 rules of
  implication and 10 rules of replacement.
• Dont worry some of them you have already learnt
  and some are very obvious.

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Rules of Implication

• The 8 rules of implication are themselves valid
  argument forms.

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1. Modus Ponens (MP)

• This we have learnt, apart from the name.
• It simply says that given a conditional, and the
  premise, we can deduce the consequent or the
  conclusion.
• p ? q
• p____
• q

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2. Modus Tollens (MT)

• It simply says that given a conditional and the
  denial of the consequent, we can deduce the
  denial of the antecedent/premise.
• p ? q
• q__
• p

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3. Hypothetical Syllogism (HS)

• Given two conditionals with a common element, we
  can make up a new conditional without the common
  proposition.
• p ? q
• q ? r
• p ? r

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4. Disjunctive Syllogism (DS)

• It says given a disjunction, and the denial of
  either one disjunct, we can arrive at the other
  disjunct.
• p v q
• p__
• q

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• Up till now, we can see that the rules are okay
  because they observe the results of the 5 truth
  tables that we have learnt. But more importantly,
  the rules feel natural to us.
• But note that the method of natural deduction can
  only prove validity, not invalidity. This is
  different from the truth table method, which can
  work both ways.

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Method

• 1. Symbolize the argument.
• 2. List the individual premises one by one on the
  left, number them consecutively, then use a slash
  to separate the conclusion to the right of the
  last premise.

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Example

• If Arsenal wins the FA cup, then Bolton will lose
  the chance to play UEFA. If Arsenal does not win
  the FA cup, then either Chelsea or Derby will be
  eligible to buy Henry. Bolton will not lose the
  chance to play UEFA. Furthermore, Chelsea will
  not be eligible to buy Henry. Therefore, Derby
  will be eligible to buy Henry.

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• Translating the above argument, we have
• 1. A ? B
• 2. A ? (C v D)
• 3. B
• 4. C / D

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• Now we want to prove that the conclusion does
  follow the premises.
• We derive more lines by applying the rules we
  have learnt.
• We write the rules we have used and the lines
  involved on the right of the derived lines.

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• 1 A ? B
• 2. A ? (C v D)
• 3. B
• 4. C / D
• 5. A 1, 3, MT
• 6. C v D 2, 5, MP
• 7. D 4, 6, DS

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• There are no rules constraining us to apply which
  rule at which point.
• But we should be aware that the best way to
  tackle this sort of problem is to look for the
  conclusion in the premises.
• If we cannot find it, then try to look for the
  components of the conclusion in the premises.

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Strategies

• 1. If the conclusion contains a letter that
  appears in the consequent of a conditional
  proposition in the premises, consider obtaining
  that letter by MP.

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Example

• 1. A ? B
• 2. C v A
• 3. A / B
• 4. B 1, 3, MP

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Strategies

• 2. If the conclusion contains a negated letter
  which appears in the antecedent of a conditional
  proposition in the premises, consider obtaining
  the negated letter by MT.

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Example

• 1. C ? B
• 2. A ? B
• 3. B / A
• 4. A 2, 3, MT

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Strategies

• 3. If the conclusion is a conditional
  proposition, consider obtaining it by HS.

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Example

• 1. B ? C
• 2. C ? A
• 3. A ? B / A ? C
• 4. A ? C 1, 3, HS

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Strategies

• 4. If the conclusion contains a letter that
  appears in a disjunctive proposition in the
  premises, consider obtaining that letter by DS.

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Example

• 1 A ? B
• 2. A v C
• 3. A / C
• 4. C 2, 3, DS

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• Once it is proved valid, then it is valid, we
  dont need to use all the lines.

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Trial

• 1. A ? (B ? C)
• 2. D ? (C ? A)
• 3. D v A
• 4. D / B
• If I give you all the derived lines, can you tell
  me which rules I have used?

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• 1. A ? (B ? C)
• 2. D ? (C ? A)
• 3. D v A
• 4. D / B
• 5. A ______
• 6. B ? C ______
• 7. C ? A ______
• 8. B ? A ______
• 9. B ______

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Example

• 1. A ? (E ? F)
• 2. H v (F ? M)
• 3. A
• 4. H / E ? M

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• No conclusion in the premises.
• Since the conclusion is a conditional
  proposition, we can get it by HS.
• (E ? F)
• (F ? M)
• Use HS then get (E ? M)
• How can we get (E ? F) and (F ? M)?

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Example

• 1. N ? ((B ? D) ? (N v E))
• 2. (B ? E) ? N
• 3. B ? D
• 4. D ? E / D
• Again no conclusion in premises.
• But 3 and 4 are obviously in HS.
• Therefore, we have B ? E

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• B ? E is the antecedent of 2.
• Therefore, we apply MP, then we have
• N
• With N, we can use MP with 1 to get
• (B ? D) ? (N v E)
• But (B ? D) is in line 3 already. So we can use
  it with MP to get (N v E)
• Again we have obtained N earlier, so we can use
  DS to get E
• Finally, we use E and 4 with MT, therefore, we
  get D

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5. Constructive Dilemma (CD)

• (p ? q) (r ? s)
• p v r
• q v s

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5. Constructive Dilemma (CD)

• A special form of CD
• p v p
• (p ? q) (p ? q)
• q
• Either you are going to recover from your illness
  or you arent. If you are going to recover, there
  is no use to see a doctor. If you arent going to
  recover, there is no use to see a doctor.
  Therefore, there is no use to see a doctor when
  you are ill.
• Whats wrong with the argument?

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6. Simplification (Simp)

• p q
• p
• This is also obvious since if both conjuncts are
  true, then either one of them must be true.

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7. Conjunction (Conj)

• p
• q___
• p q
• Similar to simplification, conjunction means if
  we have two letters on separate lines, then their
  conjunction must be true.

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8. Addition (Add)

• p____
• p v q
• If a letter is true, then adding another
  component with a disjunction must be true.

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Example

• 1. K ? L
• 2. (M ? N) S
• 3. N ? T
• 4. K v M / L v T

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Example

• 1. M N
• 2. P ? M
• 3. Q R
• 4. (P Q) ? S / S v T

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Strategies

• 5. If a conclusion contains a letter that appears
  in a conjunctive proposition in the premises,
  consider obtaining that letter by simplification.

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• 6. If the conclusion is a conjunctive
  proposition, consider obtaining it by conjunction
  by obtaining the individual conjuncts.

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• 7. If the conclusion is a disjunctive
  proposition, consider obtaining it by
  constructive dilemma or addition.

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• 8. If the conclusion contains a letter not found
  in the premises, addition must be used to obtain
  that letter.