# First Order Logic : Tableau

#### Plato

MHF Helper
This formulae gives me some troubles.
View attachment 25545

Lets say we are dealing with $$\displaystyle \mathbb{N}$$ and $$\displaystyle p(m,n)$$ means $$\displaystyle m\le n$$.
Then $$\displaystyle (\exists x)(\forall y)[p(x,y)]$$ says "some natural number precedes every natural number". Does that imply that "every natural is preceded by some natural number"?

Can you use IE, UI, UG & EU in a valid way to get what you need?

#### Razoor

Lets say we are dealing with $$\displaystyle \mathbb{N}$$ and $$\displaystyle p(m,n)$$ means $$\displaystyle m\le n$$.
Then $$\displaystyle (\exists x)(\forall y)[p(x,y)]$$ says "some natural number precedes every natural number". Does that imply that "every natural is preceded by some natural number"?

Can you use IE, UI, UG & EU in a valid way to get what you need?
Im not quite sure what you mean with your last question. But this is how i understand the sentence.

$$\displaystyle \exists x \forall y p(x,y)$$

"There exist an X where all Y is in the same function p(X,Y) which implies that for all Y is there an X in the same function p(X,Y)"

First of all i need to remove the imply arrow by negating the right side and split up the left and right part like this:

$$\displaystyle \exists x \forall y p(x,y) , \neg ( \forall y \exists x p(x,y) )$$

You can then remove $$\displaystyle (\exists x)$$ by introducing a new constant for X.

$$\displaystyle \forall y p(A,y) , \neg ( \forall y \exists x p(x,y) )$$

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