Answer to Question #290627 in Discrete Mathematics for Yaku

Question

Question 2:

Consider the following narration:

If equality of opportunity is to be achieved then those people previously disadvantaged should be

given special opportunities. If those people previously disadvantaged should be given special

opportunities then some people should receive preferential treatment. If some people receive

preferential treatment then equality of opportunity is not to be achieved. Some people are not going

to receive preferential treatment. Therefore, equality of opportunity is not to be achieved.

a) Encode the argument in Propositional Calculus.

b) As a result of a) above, use logical rules of inference to show that the given argument is sound/valid.

HINT: For any statement forms, P and Q P \rightarrow Q = ¬Q \rightarrrow ¬P.

Answer 1

Let $p$ represents "Equality of opportunity", $q$ represents "people previously disadvantaged" and $r$ represents "people should receive preferential treatment"

a.)

$1.~~~~p \rightarrow q\\ 2.~~~~q \to r\\ 3.~~~~r \to \neg p\\ 4.~~\neg r\\ ~~~~~~~~\therefore \neg p$

b.)

$5. ~~~\neg q ~~~~~~(4,2)~~\text{Modus Tollens}\\ 6.~~~~ \neg p~~~~~~(5,1)~~\text{Modus Tollens}\\ 7,~~~~r~~~~~~(6,3)~~~\text{Modus Ponens}$

Hence the argument is not valid. Since we can't arrive at the conclusion