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I'm currently studying logical propositions through distance education for a college course and I'd like some assistance and critique on translating simple sentences into propositional logic statements.

These statements are based around trying to reach the trophy by obtaining different fruit.

  1. Neither apples nor bananas will give you the trophy
  2. Strawberries will give you the trophy but grapes will make you sick

The first one had me quite confused because of the nor word. I'm confused as to whether I should join them with an and or an or because they don't seem dependent on each other and it doesn't say anything about what happens when they are both true. My current thinking is this $\sim apples \land \sim bananas$

For the second one, I understand the strawberries part but I'm struggling with trying to represent the grapes part. I know that it will always be false. I started off with $strawberries \lor (grapes \land 0) $ but I'm not sure if this is correct. I decided to and the grapes with a 0 to make it always false but should I be using $\sim grapes$ instead?

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  • $\begingroup$ For 1) you are right : "A nor B" is true only when both A and B are false; thus, it is equivalent to $\lnot A \land \lnot B$ i.e. "not-A and not-B". $\endgroup$
    – Mauro ALLEGRANZA
    Oct 14 '14 at 11:10
  • $\begingroup$ For 2) a "reasonable" translation of "A but B" is : "A and B". $\endgroup$
    – Mauro ALLEGRANZA
    Oct 14 '14 at 11:13
  • $\begingroup$ Thank you for your response. That helps me understand it a lot more. $\endgroup$
    – Jade L
    Oct 14 '14 at 11:27
  • $\begingroup$ Regarding the second one, what if the but was taken out of the statement so it read "Strawberries will give you the trophy, grapes will make you sick". Does that still imply that there is an and between the two statements? $\endgroup$
    – Jade L
    Oct 14 '14 at 11:28
  • $\begingroup$ I assume that but and and "works" in the same way; if we omit but, we are deleting the "structure" of the complex proposition. Without a connective, we cannot "model" it as "P or Q" or "P and Q", but only as "P". $\endgroup$
    – Mauro ALLEGRANZA
    Oct 14 '14 at 11:38
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Neither x nor y simply means "Not x and not y" when thinking about real life situations. So when writing this down logically, this will simply become $\neg x \wedge \neg y$, as it would be in any other situation (I am neither 20 nor 21 years old is the same as "I am not 20 years old and I am not 21 years old".

For the second, $grapes\wedge 0$ is just the same as $0$, so that's not what you want. Instead, take $\neg grapes$

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