If I wanted to express the statement in predicate logic: 'There is exactly one apple that is green.' Would it be correct to say that if $x$ is an apple that is an element of $A$: {all apples} and that $G(x)$: $x$ is a green apple, that: 'There exists an apple $x$ that is an element of the set $A$, $|A|=1$, $G(x)$'?
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4$$\exists x(G(x)\land\forall y(G(y)\to y=x))$$ – Brian M. Scott Jan 30 '21 at 23:11
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1Your statement isn't really well-formed, but it seems to say "There is exactly one apple and it is green". That's not what you want. – Brian Moehring Jan 30 '21 at 23:15
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Introducing sets and their sizes is not necessary. If, for $x$ in our universe of apples, $G(x)$ means $x$ is green, then $\exists x\forall y(G(y) \leftrightarrow x = y)$ is true iff there is exactly one green apple.
Rob Arthan
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Thanks for your reply! I don't understand why it's not necessary to include the sets. Why don't we need to include that 'x' belongs to the set A? Also, am I reading your predicate statement correctly by saying, 'There exists a variable 'x', such that for all variables 'y', 'y ' are apples that are green if and only if x = y'? – kooner3 Jan 31 '21 at 04:56
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1@kooner3 In logic it is common to separate sentences from structures. Evaluating the sentence $(\exists x)(\forall y) \big( G(y) \leftrightarrow x=y \big)$ in the structure $A$ of all apples gives the required statement "there is exactly one green apple". In classical math you would rather write $(\exists x \in A)(\forall y \in A) \big( G(y) \leftrightarrow x=y \big)$ and not think about any particular structure. I wonder though - do all apples form a set or maybe a proper class?... – Adayah Jan 31 '21 at 08:14