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Uniqueness is usually asserted as: ∃x(Px & ∀y(Py → y=x)). But this only asserts the uniqueness of x. Say I had the phrase "I have only once seen the sun rise". Here I would have to assert the uniqueness of the expression SEEN(me,sunrise) to say that SEEN(me,sunrise) happened only once. How would I go about doing this? I can't really write something like ∀z ∈ SetOfPredicates, z = SEEN(x,y)

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What you probably want to do is model this a different way. For example, you might use a function or predicate symbol to model the number of times a person has seen something. Alternatively, you might use a predicate symbol that states that a person saw something at a particular time, and then you can assert the uniqueness of that time.

You might be able to produce something essentially equivalent to the last statement in some temporal logics. More dramatically, you could switch to a substructural logic like linear logic where there is a difference between something being "true" once and being "true" more than once, i.e. the linear logic equivalent of $A\equiv A\land A$ does not hold. It's unlikely that you want to move to these non-standard logics though.

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