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I was given a question by my professor during lecture today, to translate into predicate logic statements,

"Every human loves a reindeer, but every human loves at most only 1 reindeer",

without using the $\exists !$ statement.

So far, I've come up with

$$∀ \,\text{person} \,(\text{Loves}(\text{person}, \text{reindeer}))$$

Where $\text{Loves}(x,y)$ means $x$ loves $y$. However, I am having trouble coming up with an addition to the statement that shows that "every human only loves ONE reindeer".

Any one has any ideas?

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    $\begingroup$ I'm not sure this is even a valid statement but what about something like $L(x,y)\ \land\ L(x,z)\Rightarrow y=z?$ $\endgroup$ – AnotherJohnDoe Sep 18 '18 at 14:33
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First, notice that being a reindeer$-$as well as being a human$-$is a property of the individuals in the universe: you are not talking about an individual called "Reindeer". So, it is more natural to use the predicates:

  • $H(x) : x$ is a human;
  • $R(x) : x$ is a reindeer;
  • $L(x,y) : x$ loves $y$.

The first half of the sentence ("Every human loves a reindeer") can be translated as follows:

$$ \forall x \big(H(x) \to \exists y (R(y) \land L(x,y)) \big) $$

indeed, its meaning can be pedantically reformulated as

"For every individual $x$, if $x$ is a human then there is an individual $y$ such that $y$ is a reindeer and $x$ love $y$".

The second half of the sentence ("Every human loves at most only on reindeer") can be translated as follows:

$$ \forall x \big(H(x) \to \forall y \forall z ((R(y) \land R(z) \land L(x,y) \land L(x,z)) \to y = z \big) $$ indeed, its meaning can be pedantically reformulated as

"For every individual $x$, if $x$ is a human then, for every individuals $y$ and $z$, if $y$ and $z$ are both reindeers that $x$ loves then $y$ and $z$ are the same".

(To express that there is at most one individual satisfying a given property, the idea is that if two individuals satisfy such a property, then they are the same.)

Finally, the two sentences are connected by "but", whose logical meaning is equivalent to the connective "and". So, the whole translation in predicate logic of "Every human loves a reindeer, but every human loves at most only one reindeer" is:

$$ \forall x \big(H(x) \to \exists y (R(y) \land L(x,y)) \big) \land \forall x \big(H(x) \to \forall y \forall z ((R(y) \land R(z) \land L(x,y) \land L(x,z)) \to y = z \big) $$

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Uniqueness can be expressed by some variation on

If condition(a) and condition(b) then a = b.

There is also the fact that

For all a, p

is equivalent to

Not (there exists a s.t. not p)

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