Predicate logic question: "Everything is either fun or not fun"

[LewisH]

Hi guys. New to this forum and could do with your help on something.

I have been trying to summarise "Everything is either fun or not fun" in predicate logic.

I thought it would be fairly easy, but now I'm thinking there are some ambiguities in what I have that I may have to take into account.

So far I have:

Universe of discourse = Everything
F(x) = is fun
∀ x [ F(x) v ¬F(x) ]

I am not confident about this, because the or surely means both are possible, which shouldn't be the case judging by the sentence. I am not sure how it would be possible to express that both are not possible in this form.

Please advise if I'm on the right lines and what I need to do further

 

[LewisH]

Hi guys. New to this forum and could do with your advice.

I am trying to express "Everything is either fun or not fun" in predicate logic.

I have this so far:

Universe of discourse = everything
F(x) = is fun
∀ x[ F(x) v ¬F(x) ]

I am just not sure if it is correct, simply because the or could mean both are possible, and judging by the sentence that is not supposed to be the case. Do I need to express this differently?

 

[stapel]

I have been trying to summarise "Everything is either fun or not fun" in predicate logic.

I thought it would be fairly easy, but now I'm thinking there are some ambiguities in what I have that I may have to take into account.

So far I have:

Universe of discourse: Everything (all "x")
F(x): x is fun
∀ x [ F(x) v ¬F(x) ]

I am not confident about this, because the or surely means both are possible...

Hm... Yes, if you don't have an "exclusive 'or'" operator, then your "F(x) v ¬F(x)" statement would be true for x being both fun and not fun, which isn't what you want. Maybe you could use "if-then" statements...?

. . . . .\(\displaystyle \forall\, x\, \Bigg\{\, \bigg[\, F(x)\, \Rightarrow\, \neg\bigg(\neg F(x)\bigg)\, \bigg]\, \land\, \bigg[\, \neg F(x)\, \Rightarrow\, \neg \bigg(F(x)\bigg)\, \bigg]\, \Bigg\}\)

My logic is a bit rusty, so don't take the above as gospel. If somebody else replies with other suggestions, take that post more seriously.

 

[LewisH]

Isn't ¬(¬F(x)) simply equivalent to F(x)?

Why would this form be required?

 

[stapel]

Isn't ¬(¬F(x)) simply equivalent to F(x)?

Why would this form be required?

It might not be required. Like I said, my logic is a bit rusty.

Let's rename "not fun" as "N(x)". Then try it this way:

. . . . .\(\displaystyle \forall\, x\, \bigg\{\, \bigg[\, \neg F(x)\, \land\, N(x)\, \bigg]\, \lor\, \bigg[\, F(x)\, \land\, \neg N(x)\, \bigg]\, \bigg\}\)

The above takes its lead from this page. Scroll down a little past halfway, to the "Exclusive Or: Expression from Truth Table" section.