patternModerate
Can I use ellipses in first order logic
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canorderellipseslogicfirstuse
Problem
I ask, because I have to come up with a first-order logic sentence that shows that there are exactly N objects in the universe. What I've been able to come up with is:
$$ \forall x \; \exists y_1, y_2, \dots , y_{n-1} \; (x \neq y_1) \land (x \neq y_2) \land \dots \land (x \neq y_{n-1})$$
I'm not sure if this is a valid first-order logic sentence or not.
$$ \forall x \; \exists y_1, y_2, \dots , y_{n-1} \; (x \neq y_1) \land (x \neq y_2) \land \dots \land (x \neq y_{n-1})$$
I'm not sure if this is a valid first-order logic sentence or not.
Solution
Strictly speaking, your statement is invalid because $\ldots$ is not part of the syntax of first-order logic. However, your statement is an abbreviation of a statement in first-order logic. For example, when $n = 3$, your statement is an abbreviation of the bona fide statement
$$ \forall x, \exists y_1, y_2 (x \neq y_1) \land (x \neq y_2). $$
As long as you understand this difference, you can use the statement. Your statement is stated in the metalanguage, and it corresponds to a bona fide statement of the language.
Here is a similar example. The following defines a number, for every integer $n \geq 1$:
$$
\overbrace{11\cdots 1}^{\text{$n$ times}}
$$
For example, when $n = 5$, this number is $11111$. While the notation above, which belongs to the metalanguage, is not in itself a valid number, for every specific $n$ it satisfies a valid number. You can think of it as a pattern, a number scheme (in your case, a statement scheme), or a function that takes $n$ and outputs a number (in your case, a statement).
That said, your statement (for every $n > 1$) states that there are at least two different objects in the universe. So while it is a valid statement, it doesn't state what you intended it to.
$$ \forall x, \exists y_1, y_2 (x \neq y_1) \land (x \neq y_2). $$
As long as you understand this difference, you can use the statement. Your statement is stated in the metalanguage, and it corresponds to a bona fide statement of the language.
Here is a similar example. The following defines a number, for every integer $n \geq 1$:
$$
\overbrace{11\cdots 1}^{\text{$n$ times}}
$$
For example, when $n = 5$, this number is $11111$. While the notation above, which belongs to the metalanguage, is not in itself a valid number, for every specific $n$ it satisfies a valid number. You can think of it as a pattern, a number scheme (in your case, a statement scheme), or a function that takes $n$ and outputs a number (in your case, a statement).
That said, your statement (for every $n > 1$) states that there are at least two different objects in the universe. So while it is a valid statement, it doesn't state what you intended it to.
Context
StackExchange Computer Science Q#41935, answer score: 10
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