I am trying to validate the following argument using a truth table, but I get stuck:
If you invest in this business, then you get the profit. You didn't invest in this business. Therefore, you didn't get the profit.
Asked
Active
Viewed 135 times
-1
-
Not valid: $P \to Q, \lnot P \nvDash \lnot Q$. See Denying the antecedent. – Mauro ALLEGRANZA Nov 06 '17 at 07:35
-
The argument is valid when, in all cases when the premises are TRUE, also the conclusion is. Consider the row of the tuth table enetred for $P$ FALSE and $Q$ TRUE. – Mauro ALLEGRANZA Nov 06 '17 at 07:37
-
2What does your truth table look like? – Graham Kemp Nov 06 '17 at 09:10
-
I've aready seen this. – Nov 06 '17 at 19:39
2 Answers
1
You say $A \implies B$, so $\neg A \implies \neg B$. We conclude the first statement is false if only if we observe '$A \wedge \neg B$', while we conclude the second one is false if and only if we observe '$\neg A \wedge B$', therefore the two are not equivalent and do not yield the same truth table. Consider a statement like 'all cows are animals, so if we observe something that is not a cow, then it is not an animal', which would be true according to your logic. But the observed object might still be a chicken, for example. You're statement if true if we would have $A \iff B$, so using your example, 'you get rich if and only if you invest in business'.
Václav Mordvinov
- 3,268
