a. Write the following argument in symbolic logic.
If Ryan gets the office position and works hard, then he will get a bonus. If he gets a bonus, then he will go on a trip. He did not go on a trip. Therefore, either he did not get the office position or he did not work hard.
b. Use logical equivalences to determine if the argument is valid or invalid.
So.... I have an answer for a, but I am having troubles understanding what they are looking for in b, any ideas? The following is my answer for a...
Answer for a:
Let:
- $A$ = Gets office position
- $B$ = Works hard
- $C$ = Gets a bonus
- $D$ = Go on a trip
Then:
$$((A \land B) \to C) \land (C \to D) \land (\neg D),\therefore ((\neg A) \lor (\neg B))$$
first, do you believe the the argument is true? And how did you come to that belief? (that might help later when you do symbolic manipulation)
'using logical equivalences' means replace parts of the sentence with equal parts. e.g. $X \rightarrow Y$ can be replaced by $\neg X \lor Y$
the [kinds of equivalences you might use here...modus tollens: replace $X\rightarrow Y$ with $\neg Y \rightarrow \lnot X$ (that's a true equivalence, right?) and $W \land (W \lor Z)$ with $W$. Repeat until you get what you want.
For example of modus tollens, if as part of a larger statement, you can replace $(X\rightarrow Y) \land \neg Y$ with $\neg X$ because they are equivalent (because given that $X$ implies $Y$, if you also know that $Y$ is false then you can infer that $X$ cannot be true, so $\neg X$ is true).