# 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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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
Could you provide an example of using the equivalences as per my answer in 'a' so I can get an idea of what modus tollens might look like? I am having trouble figuring out how to manipulate the equivalences. – user109594 Nov 18 '13 at 1:36

• 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).

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Could you provide an example of using the equivalences as per my answer in 'a' so I can get an idea of what modus tollens might look like? I am having trouble figuring out how to manipulate the equivalences. – Pete Apr 23 '11 at 2:18
@Pete: Example added. – Mitch Apr 23 '11 at 15:01
@Mitch: I think you meant Y is false in the last paragraph. I fixed it-please check. – Ross Millikan Apr 24 '11 at 1:46
@Ross: Argh...yes...thanks for fixing that. My first response was going to be:"C'mon...true...false...really what's the big deal?". – Mitch Apr 24 '11 at 1:48
@Pete: I saw your edit and fixed the problem. Thanks for pointing it out. I'll reject your edit but only to avoid the comment on how to make edits. Generally, once you have the reputation, you would leave that as a comment for the answerer to fix. – Ross Millikan Apr 24 '11 at 1:53