If $X$ then $Y$ unless $Z$
is represented by which of formula in propositional logic?
I am getting confused between $X\rightarrow \left ( Y\wedge \sim Z \right )$ and $\left ( X \rightarrow Y\right )\wedge \sim Z$.
Is it represented by any other formula than the two?
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Romy
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I would write $X\to(Y\lor Z)$ or $(X\to Y)\lor Z$ -- which are equivalent, so I'd not spend a lot of time choosing between them. – hmakholm left over Monica Dec 14 '15 at 01:00
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Note that these are also equivalent to my answer. (The first part, that is; ignoring the second.) – Noah Schweber Dec 14 '15 at 01:01
1 Answers
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Neither.
"If X, then Y unless Z" means that if X is true and Z is false, then Y is true; but if X is true and Z is true, then all bets are off.
That is, $$(X\wedge \neg Z)\implies Y.$$
Actually, there is some ambiguity here, as there often is with natural language: maybe "If X, then Y unless Z" means that if X is true and Z is false, then Y is true, but if X is true and Z is true, then Y is false. Then this would be represented as $$[(X\wedge \neg Z)\implies Y]\wedge [(X\wedge Z)\implies \neg Y],$$ or equivalently $$X\implies (Y\iff \neg Z).$$
Noah Schweber
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2For example, consider: "If $p$ is even, then $p$ is composite unless $p=2$." Here, $X=$"$p$ is even," $Y=$"$p$ is composite," and $Z=$"$p$=2. This is a true sentence, and diagrammed in propositional logic it would read $(X\wedge\neg Z)\implies Y$. – Noah Schweber Dec 14 '15 at 01:01
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Thank you for your explanation, I am convinced with you, but I am still in a state of confusion as I am not able to think what will be the representations for $X\rightarrow \left ( Y \wedge \sim Z \right )$ and $\left ( X \rightarrow Y\right )\wedge \sim Z$ using If, then & unless.Can you please help me in figuring this out? – Romy Dec 14 '15 at 01:11