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If I wanted to create a sentence like:

African elephants can carry coconuts; Asian elephants cannot.

How would I do that with propositional logic?

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If you're willing to consider predicate logic (with predicates and quantifiers), there's a plausible way to express this. Let $P(x)$ stand for the proposition "$x$ can carry coconuts". Let $S$ denote the set of African elephants and $T$ denote the set of Asian elephants. Then one interpretation of your sentence is

$$\forall x \in S . P(x) \quad \land \quad \forall y \in T . \neg P(y).$$

Propositional logic doesn't allow quantifiers or predicates; I don't see any reasonable way to express that sentence in pure propositional logic.

Important caveat: logic is not a way of encoding all English sentences. It is not intended as a model of conversational language. Rather, it is a way of making precise certain mathematical subjects, and supporting a particular type of mathematical reasoning. So don't expect it to be more than it is.

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    $\begingroup$ This is knows as predicate logic, since you have a predicate $P$ and quantifiers. In propositional logic predicates and quantifiers are not allowed. $\endgroup$ – Yuval Filmus Sep 27 '17 at 6:38
  • $\begingroup$ @YuvalFilmus, oh, right. Doh, I overlooked that. Thank you! $\endgroup$ – D.W. Sep 27 '17 at 7:19
  • $\begingroup$ @YuvalFilmus, how the logic used in TQBF is called then? $\endgroup$ – rus9384 Sep 27 '17 at 21:51
  • $\begingroup$ @rus9384 First-order logic. $\endgroup$ – Yuval Filmus Sep 27 '17 at 22:02
  • $\begingroup$ @YuvalFilmus, but it has no predicates, only boolean values, so, it's first-order logic over $\mathbb{F}_2$? $\endgroup$ – rus9384 Sep 27 '17 at 22:03
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lacking a better context and restraining this to propositional logic you can only do this:

p: African elephants carry coconuts q: Asian elephants carry coconuts

Then what is given can be restated symbolically as $p; \neg q$
or
$p \land \neg q$ depending on how you interpret the original prompt.

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You can divide a sentence into three core statements:

$a$: "$X$ is African elephant";

$b$: "$X$ is Asian elephant";

$c$: "$X$ can carry coconuts".

So, now you can make a formula from this: $(a\rightarrow c)\land(b\rightarrow\overline c)$.

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