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Demonstrating the validity of an argument using syllogisms.

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1 Demonstrating the validity of an argument using syllogisms

2 P Q P  Q P Q ~ P  Q P Q ~ Q  ~ P   <
A SYLLOGISM is a valid argument – usually a basic pattern of reasoning that is frequently used. The following are examples of syllogisms. MODUS PONENS P Q P  Q DISJUNCTIVE SYLLOGISM P Q ~ P  Q MODUS TOLLENS P Q ~ Q  ~ P < An argument can be analyzed two premises at a time.

3 p  r ~ r p s < s  q q

4 p  r P  Q ~ p ~ r ~ Q p s <  ~ P s  q q Modus Tollens
Each premise is a piece of information. The first two premises yield a new piece of information, ~p. This can now be used with the third premise.

5 p  r ~ p ~ r p s < s  q q

6 p  r Disjunctive Syllogism ~ p s P Q < ~ r ~ P p s < Q s  q q

7 p  r ~ p ~ r s p s < s  q q

8 Modus Ponens p  r P  Q ~ p ~ r q s P p s <  Q s  q q

9 p  r ~ r p s < s  q q The premises will not necessarily
be arranged in order, with premises that fit together placed together. s  q q

10 p  r s  q ~ r ~ r p s < p s < s  q p  r q

11 s  q s  q ~ r ~ r p s < p s < p  r p  r q

12 s  q ~ r Number the premises for reference. p s < p  r q

13 1. s  q 2. ~ r 3. p s < 4. p  r q

14 1. s  q 2. ~ r Move the conclusion 3. p s < 4. p  r q q

15 1. s  q 2. ~ r 3. p s < 4. p  r q q

16 1. s  q P  Q 2. ~ r ~ Q 3. p s <  ~ P 4. p  r 5. ~ p 2 , 4 , MT
Modus Tollens P  Q 2. ~ r ~ Q 3. p s <  ~ P 4. p  r Every step of the process must be justified. The reason for writing statement 5 is clear when you combine statements 2 and 4 using the syllogism Modus Tollens. 5. ~ p , 4 , MT q

17 1. s  q 2. ~ r 3. p s < 4. p  r 5. ~ p , 4 , MT q

18 1. s  q P Q 2. ~ r < ~ P 3. p s < Q 4. p  r 5. ~ p 2 , 4 , MT
Disjunctive Syllogism 2. ~ r P Q < ~ P 3. p s < Q 4. p  r 5. ~ p , 4 , MT 6. s , 5 , DS q

19 1. s  q 2. ~ r 3. p s < 4. p  r 5. ~ p , 4 , MT 6. s , 5 , DS q

20 1. s  q P  Q 2. ~ r P 3. p s <  Q 4. p  r 5. ~ p 2 , 4 , MT
Modus Ponens P  Q 2. ~ r P 3. p s <  Q 4. p  r 5. ~ p , 4 , MT 6. s , 5 , DS 7. q , 6 , MP q

21 1. s  q 2. ~ r 3. p s < 4. p  r 5. ~ p 2 , 4 , MT 6. s 3 , 5 , DS
You are finished when you reach the given conclusion ( in this case “q” ). There is a reason given for each statement that is deduced from the given premises. 3. p s < 4. p  r 5. ~ p , 4 , MT 6. s , 5 , DS 7. q , 6 , MP q

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