1. Chapter 8 Natural Deduction
PHIL 201
Chapter 8
Natural Deduction

2. A New Proof Method
Truth tables provide a mechanical way of testing validity, but they are too unwieldy to be of use for many arguments.
Indirect truth tables are less cumbersome, but in some cases, many possible solutions requires multiple applications of the method.
Neither method makes explicit the chain of reasoning which justifies the conclusion.
Natural deduction has distinct advantages over truth tables in that it is a much more compact and efficient proof method and it makes explicit the chain of reasoning.

3. Natural Deduction
Natural deduction: A proof procedure by which the conclusion of an argument is validly derived from the premises through the use of rules of inference.
Proof: A sequence of steps in which each step is either a premise or follows from earlier steps in the sequence according to the rules of inference.
Rules of inference: Justify the steps of a proof.
Rules of Implication
Rules of Replacement

4. Procedure
A proof in the system of natural deduction consists of a sequence of statements each written on a separate line.
Each line contains a statement that is either a premise of the argument or a valid deduction from the lines above it.
These non-premise lines are justified by the rules that make up the system.
There are two types of rules: Rules of Implication and Rules of Replacement.
The proof is complete when the conclusion is shown to be validly deducible from the premises by a sequence of steps (the non-premise lines of the proof), each one of which is an intermediate, validly deducible conclusion.

5. The Rules of Implication
The rules of implication are specific, valid argument forms that can be used to manipulate the information contained on individual lines of the proof.
Some of the rules of implication allow you to decompose more complex statements; some allow you to create new statements out of given information.
All of the rules of implication are operator specific. That is, they are applicable only in the presence of specific operators.
Paying attention to what operators are present in a proof will help you identify which implication rules would be useful.
Please note that the implication rules have to be used on entire lines of a proof. You can’t use one of these rules on a part of a line/statement.

6. First Four Implication Rules
The First Four Rules of Inference:
1) Modus Ponens (MP): a conditional rule that authorizes you to infer the consequent of a conditional if the conditional statement is present on one line of the proof and the antecedent is on another.
2) Modus Tollens (MT): a conditional rule that authorizes you to infer the negation of the antecedent of a conditional if you have the conditional statement on one line and the negation of the consequent on another.
3) Hypothetical Syllogism (HS): a conditional rule that authorizes you to create a new conditional made up of the antecedent of one conditional and the consequent of another if you have two seperate conditional statements that share a common, middle term [a>b/b>c//a>c].
4) Disjunctive Syllogism (DS): a disjunction rule that allows you to infer the right hand disjunct of a disjunction present in the proof if you have the negation of the left hand disjunct present as another line of the proof.
Review the argument forms of these first four rules in the text.

7. Modus Ponens (MP)
Any uniform substitution instance using simple or compound statements of the argument form: Valid Applications of MP: Misapplications of MP:

8. Compare
VALID INVALID
If it’s raining, then the streets are wet If it’s raining, then the streets are wet.
It’s raining The streets are wet.
The streets are wet It’s raining.

9. Modus Tollens (MT)
Any uniform substitution instance using simple or compound statements of the argument form:
Valid Applications of MT:
Misapplication of MT:

10. Compare
VALID INVALID
If it’s raining, then the streets are wet. If it’s raining, then the streets are wet.
The streets are NOT wet. It’s NOT raining.
It’s NOT raining. The streets are NOT wet.

11. Hypothetical Syllogism (HS)
Any uniform substitution instance using simple or compound statements of the argument form:
Valid Applications of HS:
Misapplications of HS:

12. Disjunctive Syllogism (DS)
Any uniform substitution instance using simple or compound statements of the argument form:
Valid Applications of DS:
Misapplication of DS:

13. Justification: Applying the Rules of Inference
Justification: Refers to the rule of inference that is applied to every validly derived step in a proof.
Lines 1–4: Premises and Conclusion
Derived from Lines 1 and 3 by DS
Derived from Lines 2 and 5 by MT Conclusion, derived from Lines 4 and 6 by MP

14. Exercise 8B Example I Answer 1. (T v S)  (P · ∼ R) 2. ∼ (P · ∼ R)
3. __________________ , 2, MT
Answer
1. (T v S)  (P · ∼ R)
3. ∼ (T v S) , 2, MT
Exercise

15. Exercise 8B Example II Answer 1. S v ∼ (P  Q) 2. ∼ S
3. __________________ , 2, DS
Answer
1. . S v ∼ (P  Q)
3. ∼ (P  Q) , 2, DS
Exercise

16. Applying Implication Rules I
General strategies and specific tactics for proofs
Strategy: locate conclusion ‘inside’ premises
Tactic 1: If consequent, try (MP)
Tactic 2: If antecedent, try (MT)
Tactic 3: If conditional statement, try (HS)
Tactic 4: If disjunct, try (DS)

17. Exercise 8C Example (8C I #6) Answer 1. Q 2. L ⊃ ( S ⊃ P )
3. Q ⊃ ( R ⊃ S )
4. L / R ⊃ P
Answer
4. L / R ⊃ P Tactic 3: try HS
5. R ⊃ S , 3, MP
6. S ⊃ P , 4, MP
7. R ⊃ P , 6, HS
Exercise

18. Implication Rules II
Simplification (Simp): a rule that allows you to free a conjunct of a conjunction.
Conjunction (Conj): a rule that allows you to create a new conjunction out of any two statements that are lines in the proof.
Addition (Add): a rule that allows you to create a new disjunction by disjunctively adding any information to an existing statement. Be careful! Add only information that is helpful to solving the proof.
Constructive Dilemma (CD): a rule that allows you to create a new disjunction from the conjunction of two conditionals and the disjunction of the antecedents of the conditionals. The disjunction created is of the consequents of the conditionals.

19. Simplification (Simp)
Any uniform substitution instance using simple or compound statements of the argument form:
Valid Applications of (Simp):
Misapplication of (Simp):

20. Conjunction (Conj)
Any uniform substitution instance using simple or compound statements of the argument form:
Valid Applications of (Conj):
Misapplication of (Conj):

21. Addition (Add)
Any uniform substitution instance using simple or compound statements of the argument form:
Valid Applications of (Add):
Misapplication of (Add):

22. Constructive Dilemma (CD)
Any uniform substitution instance using simple or compound statements of the argument form:
Valid Application of CD:
Misapplication of CD:

23. Applying Implication Rules II
Strategy: locate conclusion ‘inside’ premises
Tactic 5: If conjunct, try (Simp)
Tactic 6: If conjunction, try (Conj)
Tactic 7: If new letter, try (Add)
Tactic 8: If disjunction, try (CD)

24. Exercise 8D Example Answer 1. ( A  ∼ C ) v B 2. ∼ ( D · ∼ E )
3. __________________ , 2, Conj
Answer
3. [ ( A  ∼ C ) v B ] · ∼ ( D · ∼ E ) , 2, Conj (Tactic 6)
Exercise

25. Exercises 8D Example (8D III #8) Answer 1. P · F
2. ( P ⊃ Q ) · ( R ⊃ S ) / Q v S
Answer
2. ( P ⊃ Q ) · ( R ⊃ S ) / Q v S
3. P , Simp
4. P v R , Add
Q v S , 4, CD (Tactic 8)
Exercise

26. Replacement Rules
While the implication rules are made up of simple, valid argument forms, the replacement rules express relationships of logical synonymy between two or more statement forms.
Statement forms are logically synonymous when they always (under any possible truth conditions) have the same truth value.
Because they are logically synonymous, these statement forms can be swapped in for each other at any time and at any position in a proof.
Thus, unlike the implication rules, which operate only on entire lines of a proof, the replacement rules can operate on entire lines and parts of lines.
The new symbol “::” is used in all the replacement rules; it means “logically equivalent.”
The replacement rules are very useful in proofs where changing the form of statements is the key.

27. First Five The First Five Rules of Replacement
9) DeMorgan’s Rule (DM): a rule that operates on conjunctions and disjunctions with negations: when moving a negation in or out of a parentheses, a dot switches to a wedge or vice versa.
10) Double Negation (DN): familiar from grammar; pairs of tilde’s can be removed or added to expressions.
11) Commutation (Com): familiar from mathematics; the truth value of a conjunction or disjunction is unaffected by order.
12) Association (Assoc): familiar from mathematics, the truth value of sets of conjunctions or disjunctions is not effected by grouping order (punctuation) if the operator is always the same.
13) Distribution (Dist): also familiar from math; applies to sets of conjunctions and disjunctions allowing one to move inside and outside of parentheses, changing the major operator of the statement.

28. De Morgan (DM)
Conjunctions and disjunctions in the following statement forms are logically equivalent:
Valid Application of (DM):
Misapplications of (DM):
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29. Double Negation (DN)
Justifies the introduction or elimination of pairs of negation signs.
Valid Application of (DN):
~ ~ C C
Misapplication of (DN):
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30. Commutation (Com)
Conjunctions and disjunctions in the following statement forms are logically equivalent:
Valid Application of (Com):
Misapplication of (Com):
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31. Association (Assoc)
Allows the use of parentheses to group the component parts of conjunctions and disjunctions in the following statement forms without changing truth value:
Valid Application of (Assoc):
Misapplications of (Assoc):
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32. Distribution (Dist)
Conjunctions and disjunctions in the following statement forms are logically equivalent:
Valid Application of (Dist):
Misapplications of (Dist):
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33. Applying Replacement Rules I
Strategy: locate conclusion ‘inside’ premises
Tactic 9: Use (Conj) to establish basis for (DM)
Tactic 10: Use (Add) to establish basis for (DM)
Tactic 11: Use (CD) to establish basis for (DM)
Tactic 12: Use (Dist) to establish basis for (Simp)
Tactic 13: Use (Dist) to establish basis for (DS)
Tactic 14: Use (Com) to establish basis for (MP)
Tactic 15: Use (Com) to establish for (DS)
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34. Exercise 8E Example Answer ∼ [ ( S  R ) · B ]
__________________ , DM
Answer
∼ ( S  R ) v ∼ B , DM
Exercise
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35. Exercise 8E Example (8E III #7 ) Answer
1. [ S ⊃ ( L ∙ M ) ] ∙ [ P ⊃ ( M ∙ Q ) ]
2. S v P / M
Answer
3. ( L ∙ M ) v ( M ∙ Q ) , 2, CD
4. ( M ∙ L ) v ( M ∙ Q ) , Com
5. M ∙ ( L v Q ) , Dist (Tactic 12)
6. M , Simp
Exercise
© Oxford University Press

36. Second Five The Remaining Five Rules of Replacement
14) Transposition (Trans): swap antecedent and consequent of a conditional if negations are added (or removed) from both.
15) Material Implication (Imp): a conditional is equivalent to a disjunction if the antecedent term is negated in the disjunction.
16) Material Equivalence (Equiv): Our only rule for manipulating biconditionals; two forms: equivalent to the conjunction of two conditionals or equivalent to the disjunction of two conjunctions.
17) Exportation (Exp): a series of conditionals is equivalent to another conditional with a conjunctive antecedent.
18) Tautology (Taut): eliminates redundancy in conjunctions and disjunctions.
Again, make sure you familiarize yourself with all of the rules. Internalizing them is the key to using the method successfully.

37. Transposition (Trans)
Conditional statements in the following statement forms are logically equivalent:
Valid Application of (Trans):
Misapplication of (Trans):
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38. Material Implication (Impl)
Conditional statements in the following statement forms are logically equivalent:
Valid Application of (Impl):
Misapplications of (Impl):
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39. Material Equivalence (Equiv)
Biconditional and conditional statements in the following statement forms are logically equivalent:
Valid Application of (Equiv):
Misapplications of (Equiv):
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40. Exportation (Exp)
Conditional statements and conjunctions in the following statement forms are logically equivalent:
Valid Application of (Exp):
Misapplication of (Exp):
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41. Tautology (Taut)
Disjunctions and conjunctions in the following statement forms are logically equivalent:
Valid Application of (Taut):
Misapplication of (Taut):
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42. Applying Replacement Rules II
Strategy: locate conclusion “inside” premises
Tactic 16: Use (Trans) to establish basis for (HS)
Tactic 17: Use (Impl) to establish basis for (Dist)
Tactic 18: Use (Impl) to establish basis for (HS)
Tactic 19: Use (Exp) to establish basis for (MP)
Tactic 20: Use (Exp) to establish basis for (MT)
Tactic 21: Use (Equiv) to establish basis for (Simp)
Tactic 22: Use (Equiv) to establish for (DS)
‘ to ‘’
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43. Exercise 8F Example Answer ( N  ∼ M ) · ∼ Q
__________________ , Impl
Answer
( N  ∼ M ) · ∼ Q
(∼ N v ∼ M ) · ∼ Q , Impl
Exercise
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44. Exercise 8F Example (8F III #14) Answer
1. T · ( S v R )
2. ~ ( T · S ) / T · R
Answer
3. ( T · S ) v ( T · R ) 1, Dist
4. T · R , 3, DS
Exercise
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