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Lets say we have a proposition:

There is exactly one car parked out side that is black.

How can I express this in the universal discourse?

GivenPie
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2 Answers2

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There are many ways to say it. For one thing, it very much depends on the predicate symbols available. Let $C(t)$ stand for $t$ is a car that is parked outside, and $B(t)$ be the predicate that says $t$ is black. So we want to say that there is an $x$ which is a car parked outside, and which is black, and such that for any $y$, if $y$ is a car that is parked outside, and $y$ is black, then $y=x$. Here goes: $$\exists x((C(x)\land B(x))\land \forall y((C(y)\land B(y))\longrightarrow (y=x))).$$

André Nicolas
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  • Hmm so wording is key, then how would you say "There is no car parked outside that is black?"

    The logic of that would be for all that exists at least one, by using $y$=$x$, then exactly one exists?

     – GivenPie Sep 20 '13 at 00:32
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    $\lnot\exists x(C(x)\land B(x))$ or $\forall x(\lnot C(x)\land \lnot B(x))$ or $\forall x(C(x)\longrightarrow\lnot B(x))$ and many other equivalent ways. There may still be typos, comments are not fun to edit. – André Nicolas Sep 20 '13 at 01:35
  • Ok thanks Andres, is it possible to do a exactly one statement with exclusive or? Say the domain is all cars $c$ and the $C(c)$ and $B(c)$ are car parked outside and is black. – GivenPie Sep 20 '13 at 21:30
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    Maybe this will work: $\exists x\forall y((C(x)\land B(x))\text{XOR} \lnot(y=x))$. – André Nicolas Sep 20 '13 at 22:11
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Often you will see $\exists !$ for "there exists a unique." Can you use this on your question?

ncmathsadist
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