## well formed formulas

Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.

### well formed formulas

Are the following formulas wff??

1) $\exists 1[1.a=a.1=a]$

2) $\forall x[(\forall y(y.x=y))\Longrightarrow x=1]$
chris12

### Re: well formed formulas

I doubt.
Yours
HL

Hendrick Laursen
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### Re: well formed formulas

Where do you support your doubts
chris12

### Re: well formed formulas

chris12,

We can definitely read those formulas and understand their intent, so I guess the question's whether or not they conform to a specific set of rules in one of your textbooks, right?

For the first one I'd be suspicious about the x=y=z syntax, which has four possible meanings that I can think of:
1. $x=y$ and $y=z$;
2. $x=\left{{\begin{tabular}{ccc}\text{true}&\text{if}&y=z\\\text{false}&\text{if}&y{\neq}z\end{tabular}}\right.$;
3. $z=\left{{\begin{tabular}{ccc}\text{true}&\text{if}&x=y\\\text{false}&\text{if}&x{\neq}y\end{tabular}}\right.$;
4. assign the value of z to variables x and y.
I'm not sure how it would read in Formal Logic, if at all.
Natural ChemE
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### Re: well formed formulas

Natural ChemE » April 5th, 2014, 3:28 pm wrote:chris12,

We can definitely read those formulas and understand their intent, so I guess the question's whether or not they conform to a specific set of rules in one of your textbooks, right?

For the first one I'd be suspicious about the x=y=z syntax, which has four possible meanings that I can think of:
1. $x=y$ and $y=z$;
2. $x=\left{{\begin{tabular}{ccc}\text{true}&\text{if}&y=z\\\text{false}&\text{if}&y{\neq}z\end{tabular}}\right.$;
3. $z=\left{{\begin{tabular}{ccc}\text{true}&\text{if}&x=y\\\text{false}&\text{if}&x{\neq}y\end{tabular}}\right.$;
4. assign the value of z to variables x and y.
I'm not sure how it would read in Formal Logic, if at all.

The rules of wff are universal and do not depend on a particular book.

Now in the 2nd formula if we put x=2 and y=0 we get : (2.0=0)=> 2=1,which is a false result.

Can therefor the 2nd formula be a wff??
chris12

### Re: well formed formulas

chris12,

As I read it, $\forall x[(\forall y(y.x=y))\Longrightarrow x=1]$ is something like, "If multiplying any number $y$ by $x$ produces $y$, then $x$ must be one. This is true for all $x$."

"This is true for all $x$." $\left({\forall}x\right)$ is weird. Since the inner logic constrains $x$ to one, why go on to say "for all $x$"?

Perhaps the grammar that you're supposed to use doesn't allow ${\forall}x$ when $x$ is already constrained to a single value?

PS - Does a WFF need to be factually correct? This is, if the statement is correct grammatically but not logically, then is it still a WFF?
Natural ChemE
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