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I should write the following sentence as a predicate logic formula:

"It is not true that all soccer players are fearless and it is not true that some sailors aren't fearless."

I would write it like this, but I'm not sure if it is correct: $\neg(\forall x)F(x)\land\neg (\exists y)\neg F(y)$ .

Especially I'm not sure about those variables and quantifiers since once we are talking about ALL soccer players and second we are talking about SOME sailors - in other words, two different object are quantified.

TKN
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  • NO; you have to use suitable predicates for Soccer players and Sailors. – Mauro ALLEGRANZA Jan 26 '20 at 17:42
  • Could you be more specific please? How should than the formula look like? – TKN Jan 26 '20 at 17:50
  • Your formula reads: "Not everything is fearless and everything is fearless" – Mauro ALLEGRANZA Jan 26 '20 at 17:52
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    Start with the simple parts: "all soccer players are fearless" that is $\forall x (\text {SocPlay}(x) \to \text {Fearless}(x))$. – Mauro ALLEGRANZA Jan 26 '20 at 17:54
  • Oh I see. So for example like this? $\neg(\forall x) (SocPlay(x)\implies Fear(x))\land\neg (\exists x) (Sailor(x)\implies\neg Fear(x))$ – TKN Jan 26 '20 at 18:01
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    "some sailors aren't fearless" is $\exists x (\text {Sailor} \land \lnot \text {Fearless}(x))$. See Categorical propositions. – Mauro ALLEGRANZA Jan 26 '20 at 18:07
  • Thank you very much. – TKN Jan 26 '20 at 18:08
  • I now has to find out, if the following sentence is a logical consequence of the previous formula: "Therefore some brave people are sailors." I know that in order to find out a logical consequence I has to find out wheter the conjunction of the previous formula with the negation of this sentence is a contradiction. But how should I formalize this consequence sentence? Is the following formalization of that sentence correct?: $\exists x (people(x)\land sailors(x))$ – TKN Jan 26 '20 at 18:26
  • I'm not an English native speaker, but I think that "brave" and "fearless" mean the same. – Mauro ALLEGRANZA Jan 26 '20 at 18:41
  • Yes, sorry, I meant to write "fearless" instead of "brave". – TKN Jan 26 '20 at 18:44
  • Is the formalization I proposed for the consequence sentence correct than? I am not sure since People is a superset for Sailors and Soccers. – TKN Jan 26 '20 at 19:43
  • No, it is not; it must be $\exists x (\text {Fearless}(x) \land \text {Sailor}(x))$. From a formal point of view, Brave and Fearless are not the same. – Mauro ALLEGRANZA Jan 26 '20 at 19:50

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