There are two arguments in question number 1. The first argument can be represented as disjunctive syllogism and the second in the Modus Ponens form (If X, then Y).
The first argument can be represented as disjunctive syllogistic as follows:
P1: P or Q (Either Billy-Bob said he had never had sexual relations with that Lewinsky woman or he said something else that sounded like “I never had sexual relations with that Lewinsky woman”)
P2: Not Q (However, it is false that he said something else that sounded like “I never had sexual relations with that Lewinsky woman”)
C: P (We must conclude that Billy-Bob said that he never had sexual relations with that Lewinsky woman).
The second argument from the same paragraph may be represented as follows:
P.1: If P, then Q (If he never had sexual relations with that Lewinsky woman, then he was wrongly accused by the Congress).
P2: Q (He was wrongly accused by the Congress)
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C: P (Hence, he never had sexual relations with that Lewinsky woman).
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However, it needs to be stated that the above argument is invalid and the conclusion does not follow the premises. The argument is fallacious. For example let us see this argument: P1. If it rains hard, it rains; P2. It rains; C: It rains hard. It is easy to see that we cannot conclude that it rains hard from it rains while we can conclude it rains from it rains hard.
The argument can be represented as follows: P1: Everyone loves my baby P2: My baby loves none but me. C. Therefore, I love myself alone.
It is evident that the conclusion doesn’t follow from the premises. It can be represented as: P1: All S is P/ P2: All P is Q/ Therefore, Conclusion: Q. Further, we can substitute information in P and Q as follows: All men are mortal/All mortals are living beings/ Therefore, All men are living beings. It can be observed that the conclusion “I love my self alone” is invalid because we cannot arrive at the conclusion ‘I love myself’ from ‘everyone loves my baby’ and ‘my baby loves me’.
The third question can be represented in a syllogistic format as follows:
P1: If food is not exposed to bacteria, it will not rot (If not A, not B)
P2: If it does rot, it has been exposed to bacteria (If B, then A)
P3: A (I have exposed the food to bacteria)
C: Therefore, B
The above is a legitimate substitution instance of modus ponens:
If not A, then not B
Not A
C: Not B.
However, the validity of the conclusion is doubtful. Why? Because we can infer B from ‘If not A, then B’ and ‘not A’ but cannot infer ‘not A from B’. That is to say, the consequent follows from the antecedent, but the antecedent does not follow from the consequent. However, this could have been true in an argument involving conditionals.
In question 4, a contradictory conclusion is arrived. In other words, this is an example of proof by contradiction. The argument can be represented as follows:
P1: Suppose the earth is hollow.
P2: If it were hollow, then it would collapse owing to the gravitational forces.
P3: But it is not collapsing, so it is not hollow.
C: Therefore, it is both hollow and not hollow.
In the example above a statement is being proved by showing the contradiction involved. The arguments of this form are often used in mathematical sciences. If we can arrive at a contradictory conclusion, we have proven that our supposition stated in premise 1 is wrong.
7. The seventh question can be stated syllogistically in the following form:
P1: Some vitamins are anti-oxidants.
P2: No anti-oxidants are harmful to your health.
C: Therefore, no vitamins are harmful to your health.
It is evident that the conclusion is wrong as it does not follow the premises. We know that no anti-oxidants are harmful to your health and also that some vitamins are anti-oxidants. From this the only legitimate conclusion we can arrive at is that some vitamins are not harmful to your health because only some vitamins have anti-oxidant properties.
