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I need to write the following declarative sentence in propositional logic.

No shoes, no shirt, no service.

My solution is: ~p,~q, ~r , is it correct or do i need to use implication -> instead

Iffat Fatima
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    "No shoes, no shirt, no service" is not a proposition and any of "shoes", "shirt", "service" are propositions, so you can't deny them. It just doesn't have a precise meaning, you can interpret "no shoes" as "there are no shoes" or "it is not allowed to use shoes" or in many other different ways – la flaca Feb 27 '16 at 20:07
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    This is a sign that is commonly seen at gas stations and convenience stores in the U.S. It means that if you're not wearing shoes or if you're not wearing a shirt then you can't get service. – Dan Brumleve Feb 27 '16 at 20:08
  • You need a connective. What logical connective is ", " ? I don't know that one. It's not even incorrect. – BrianO Feb 27 '16 at 22:10

2 Answers2

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"," is not a symbol of propositional logic. So that in itself says you have it wrong.

You need to think about what the phrase means: "If you have no shoes, or you have no shirt, then you will receive no service." Try translating that into propositional logic.

Paul Sinclair
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  • I'm guessing OP is using "," to mean $\land$. – Dan Brumleve Feb 27 '16 at 20:10
  • @DanBrumleve - and I am 99.9% certain that the OP simply translated the "no"s in the original phrase to "$~$" and replaced the three elements with letters, without knowing what to do with the ",". So I pointed out that this isn't adequate, and gave the correct interpretation of the phrase, – Paul Sinclair Feb 27 '16 at 20:20
  • "," did not mean ∧ in my case, – Iffat Fatima Feb 27 '16 at 20:46
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"No shoes, no shirt, no service" is a phrase that states that both shoes and shirt are required, in order to receive service. Therefore, the statement should read $\neg \operatorname{Shoes} \vee \neg \operatorname{Shirt} \to \neg \operatorname{Service}$ in propositional logic (where you may replace "Shoes", "Shirt", "Service" with variables). Note that this is equivalent to $\operatorname{Service} \to \operatorname{Shoes} \wedge \operatorname{Shirt}$.

Stefan Mesken
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