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1) "If interest rates are low, then housing starts are up. If housing starts are up, then marriage rates are high. If interest rates are low, then the economy is good. The economy is not good. Therefore marriage rates are not high."

2) "If Sajee is not on team A, then Marlon is on team B. If Marlon is not on team B, then Sajee is on team B. Therefore Sajee is not on team A or Marlon is not on team B."

Shaun
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Marlon Abeykoon
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2 Answers2

1

We use the knowledge that the contrapositive is logically equivalent to the positive.

Question 1:
$I \Longrightarrow H$. Contrapositive: $\lnot H \Longrightarrow \lnot I$.
$H \Longrightarrow M$. Contrapositive: $\lnot M \Longrightarrow \lnot H$.
$I \Longrightarrow E$. Contrapositive: $\lnot E \Longrightarrow \lnot I$.

Then, we have $\lnot E$. This is a contrapositive statement, $\lnot E \Longrightarrow \lnot I$. However, $\lnot I$ does not let us conclude anything further.

Question 2: this looks like homework, and is reasonably similar to the case above. You should do this one yourself: use a truth table or the method I outlined above.

Newb
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1

An argument is valid if and only if when all the premises are true, the conclusion is true also. An argument is invalid if and only if when all the premises are true, there exists some value assignment which can consistently render the conclusion false.

For the first problem let I stand for interest rates being low, H for housing starts are up, M for marriage rates being high, and E for the economy being good. Then the argument has form

(I$\rightarrow$H)

(H$\rightarrow$M)

(I$\rightarrow$E)

$\lnot$E

$\lnot$M

Suppose $\lnot$M false. Then M is true. So, (H$\rightarrow$M) is true also. Suppose E false. Then $\lnot$E is true. Suppose I false also. Then (I$\rightarrow$E) is true, and (I$\rightarrow$H) is true also. Consequently, all of the premises can hold true and the conclusion can qualify as false. Therefore, the argument is invalid.

For the second problem let S-A stand for Sajee being on team A. Let S-B stand for Sajee being on team B. Let M-A stand for Marlon being on team A. Let M-B stand for Marlon being on team B. Then the argument has form

$\lnot$[S-A]$\rightarrow$[M-B]

$\lnot$[M-B]$\rightarrow$[S-B]

$\lnot$[S-A] $\lor$ $\lnot$[M-B]

Suppose $\lnot$[S-A] $\lor$ $\lnot$[M-B] false. Then, $\lnot$[S-A] is false, and $\lnot$[M-B] is false also. Thus, [S-A] is true, and so is [M-B]. Since [M-B] is true, $\lnot$[S-A]$\rightarrow$[M-B] is true also. Since [M-B] is true, $\lnot$[M-B] is false. Consequently, $\lnot$[M-B]$\rightarrow$[S-B] is true. Therefore, this argument is also invalid.

Doug Spoonwood
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