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It is my understanding that relations are best described with predicate logic. I have a homework question that asks me to convert English sentences into propositional logic. The following list of sentences are similar to the homework, but do not reflect the actual assignment.

a) Humans have two legs.

b) If Humans had four legs, they would be related to Mutants.

Writing something like $P \land L$, where $P = \text{Human}$ and $L = \text{has two legs}$ does not make much sense to me. Is it even possible to show a relation like this with propositional logic?

Typically, I would assume that one would represent the first sentence by stating $P = \text{Humans have two legs}$

As for the second sentence, it makes more sense to let $P = \text{Humans had eight legs}$ and $Q = \text{Humans are related to Mutants}$. Then you could state $P \implies Q$.

So my question is this: Is there a way to break these sentences down even further using propositional logic?

Alex
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  • In order to deal with the predicate "has two legs" separately from the complete sentence, one needs predicate logic. Propositional logic can't do it. – Trevor Wilson Oct 09 '13 at 00:34

2 Answers2

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Propositional logic can only handle complete sentences which are either true or false. So it is invalid to assign a propositional letter to an element of a sentence like $P=\mathrm{Human}$. In your case, your typical translation is correct.

Kaa1el
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There are many ways to translate these into propositional logic. P="Humans have two legs" is one possibility. P(n)="Humans have $n$ legs" is another one; so P(2) is for 2 legs and P(8) is for 8 legs.

vadim123
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  • Just to clarify, you're using $P(2)$ and $P(8)$ as propositional variables, just like you might use $Q$ and $R$ as propositional variables, right? And there is no meaning to the individual "$P$", "$($", "$2$", and "$)$" as far as propositional logic can see? – Trevor Wilson Oct 09 '13 at 00:39
  • Correct; these are indexed propositions; infinitely many propositions $P(1), P(2), \ldots$, indexed by $n$ for convenience. – vadim123 Oct 09 '13 at 02:47