# Translating an argument into symbolic logic

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...

Let:

• $A$ = Gets office position
• $B$ = Works hard
• $C$ = Gets a bonus
• $D$ = Go on a trip

Then:

$$((A \land B) \to C) \land (C \to D) \land (\neg D),\therefore ((\neg A) \lor (\neg B))$$

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Homework? The question is worded nicely, but there -is- a homework tag. – Mitch Apr 23 '11 at 0:15
Now you need to make a list of sentences, like in your other question, where each one is derived from the earlier ones by rules of logic. You should start from ((A and B) -> C) and (C -> D) and (~D) (or the same broken into three) and end with ((~A) or (~B)) if the argument is valid. – Ross Millikan Apr 23 '11 at 0:16

• '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 $\neg W \land (W \lor Z)$ with $Z$. 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).