# Math Help - Truth Tables

1. ## Truth Tables

I think I understand how
(P Λ Q) ≡ ¬ (P ↓ Q) and
(P Λ Q) ≡ ¬P ↓ ¬Q.

However I can’t seem to make sense of
¬ (P ↓ Q) ≡ (P ↓ Q) ↓ (P ↓ Q) and
¬P ↓ ¬Q ≡ (P ↓ P) ↓ (Q ↓ Q).

Thanks

2. ## Re: Truth Tables

Originally Posted by Gayelle
I think I understand how
(P Λ Q) ≡ ¬ (P ↓ Q) and
(P Λ Q) ≡ ¬P ↓ ¬Q.
However I can’t seem to make sense of
¬ (P ↓ Q) ≡ (P ↓ Q) ↓ (P ↓ Q) and
¬P ↓ ¬Q ≡ (P ↓ P) ↓ (Q ↓ Q).
The chief difficulty here is that almost no one agrees on notation.
Although it was invented in the 1880's by CS Pierce, still today there is a disagreement.
Here is the way I learned it from C.I. Copi. He called IT Stroke and dagger.
$\begin{array}{*{20}{c}}P&{}&Q&{}&{}&{P|Q}\\\hline T&{}&T&{}&{}&F\\T&{}&F&{}&{}&T\\F&{}&T&{}&{}&T\\F& {}&F&{}&{}&T\end{array}$ and $\begin{array}{*{20}{c}}P&{}&Q&{}&{}&{P \downarrow q} \\ \hline T&{}&T&{}&{}&F\\T&{}&F&{}&{}&F\\F&{}&T&{}&{}&F\\F& {}&F&{}&{}&T\end{array}$

Willard Quine (the greatest American logician of the 20th century) calls it alternative denial.

3. ## Re: Truth Tables

Originally Posted by Gayelle
I think I understand how
(P Λ Q) ≡ ¬ (P ↓ Q) and
(P Λ Q) ≡ ¬P ↓ ¬Q.

However I can’t seem to make sense of
¬ (P ↓ Q) ≡ (P ↓ Q) ↓ (P ↓ Q) and
¬P ↓ ¬Q ≡ (P ↓ P) ↓ (Q ↓ Q).

Thanks
just methodically make the truth tables.

$\begin{array}{ccccc} P &Q &(P \downarrow Q) &\neg (P \downarrow Q)& (P \downarrow Q) \downarrow(P \downarrow Q) \\ 0 &0 &1 &0 &0 \\ 0 &1 &0 &1 &1 \\ 1 &0 &0 &1 &1 \\ 1 &1 &0 &1 &1 \end{array}$

you can see the last two columns are identical.

$\begin{array}{ccddcccc} P &Q &(P \downarrow P) &(Q\downarrow Q)& (P \downarrow P) \downarrow(Q\downarrow Q) &\neg P & \neg Q &\neg P \downarrow \neg Q \\ 0 &0 &1 &1 &0 &1 &1 &0\\ 0 &1 &1 &0 &0 &1 &0 &0\\ 1 &0 &0 &1 &0 &0 &1 &0\\ 1 &1 &0 &0 &1 &0 &0 &1\\ \end{array}$

you can see the 5th and 8th columns are identical

4. ## Re: Truth Tables

I’m sorry, I should have stated that I understood that they are equivalent and I could derive this via truth formulas. However although I know it’s correct I can’t understand the reasoning. The actual problem in its entirety was: Find formulas using only the connective ↓ that are equivalent to ¬P, P V Q and P Λ Q. I got that (P V Q) ≡ ¬ (P ↓ Q) [Sorry about the error as I originally posted this as (P Λ Q) ≡ ¬ (P ↓ Q)] The reasoning being that saying “either P OR Q is true” (P V Q), is the same as saying “it is not the case that neither P nor Q is true” ¬ (P ↓ Q). I must admit that I’m still haven’t completely wrapped my mind around this however how one gets from ¬ (P ↓ Q) to (P ↓ Q) ↓ (P ↓ Q) is to me completely unfathomable (reasoning wise that is). As ¬ (P ↓ Q) ≡ (P ↓ Q) ↓ (P ↓ Q). Hope I better explained myself.

5. ## Re: Truth Tables

Originally Posted by Gayelle
I’m sorry, I should have stated that I understood that they are equivalent and I could derive this via truth formulas. However although I know it’s correct I can’t understand the reasoning. The actual problem in its entirety was: Find formulas using only the connective ↓ that are equivalent to ¬P, P V Q and P Λ Q. I got that (P V Q) ≡ ¬ (P ↓ Q) [Sorry about the error as I originally posted this as (P Λ Q) ≡ ¬ (P ↓ Q)] The reasoning being that saying “either P OR Q is true” (P V Q), is the same as saying “it is not the case that neither P nor Q is true” ¬ (P ↓ Q). I must admit that I’m still haven’t completely wrapped my mind around this however how one gets from ¬ (P ↓ Q) to (P ↓ Q) ↓ (P ↓ Q) is to me completely unfathomable (reasoning wise that is). As ¬ (P ↓ Q) ≡ (P ↓ Q) ↓ (P ↓ Q). Hope I better explained myself.
There's nothing I can write down that's going to help you internalize these relationships. You just have to work them out yourself to get that lightbulb to come on.

6. ## Re: Truth Tables

Originally Posted by romsek
There's nothing I can write down that's going to help you internalize these relationships. You just have to work them out yourself to get that lightbulb to come on.
So I take it that there is no way to reason ¬ (P ↓ Q) ≡ (P ↓ Q) ↓ (P ↓ Q) or ¬P ↓ ¬Q ≡ (P ↓ P) ↓ (Q ↓ Q). If so then thank you.

You see I recently started doing this and had the impression that it all could be somehow “reasoned” hence I thought I was missing something as I could not put into words how one got from ¬ (P ↓ Q) to (P ↓ Q) ↓ (P ↓ Q) or ¬P ↓ ¬Q to (P ↓ P) ↓ (Q ↓ Q). I have already derived it myself via tinkering with the truth tables it’s the rationalizing into words that I can’t get.

Thanks

7. ## Re: Truth Tables

Originally Posted by Gayelle
So I take it that there is no way to reason ¬ (P ↓ Q) ≡ (P ↓ Q) ↓ (P ↓ Q) or ¬P ↓ ¬Q ≡ (P ↓ P) ↓ (Q ↓ Q). If so then thank you. You see I recently started doing this and had the impression that it all could be somehow “reasoned” hence I thought I was missing something as I could not put into words how one got from ¬ (P ↓ Q) to (P ↓ Q) ↓ (P ↓ Q) or ¬P ↓ ¬Q to (P ↓ P) ↓ (Q ↓ Q). I have already derived it myself via tinkering with the truth tables it’s the rationalizing into words that I can’t get.
As I wrote in my reply, the function is of great interest to professional logicians.
It turns out that all other logical functions can be written with the stroke, $\downarrow$, alone.

Quine called it the alternate denial, neither P nor Q is true.
So $\neg P \equiv \left( {P \downarrow P} \right)$ we have replaced not.

As for your questions above, those were just worked out as part of the replacement program.
If you have access to a good library Mathematical Logic by Quine, 1947 has a discussion.

8. ## Re: Truth Tables

Originally Posted by Plato
As I wrote in my reply, the function is of great interest to professional logicians.
It turns out that all other logical functions can be written with the stroke, $\downarrow$, alone.

Quine called it the alternate denial, neither P nor Q is true.
So $\neg P \equiv \left( {P \downarrow P} \right)$ we have replaced not
.
Absent reason, so what?

You could take appropriate combinations of existing symbols, give them new symbols, then replace the old symbols in symbolic formulas with the new symbols.

So what?

9. ## Re: Truth Tables

But there is an advantage to expressing combinations of logic symbols in terms of others when taking into account the capabilities of digital logic circuits. For example, I believe all truth tables can be expressed using only “and” “or” and “not.”

If the dagger provides such an advantage, or potential advantage, that is certainly a reason.

10. ## Re: Truth Tables

Originally Posted by Hartlw
Absent reason, so what?

You could take appropriate combinations of existing symbols, give them new symbols, then replace the old symbols in symbolic formulas with the new symbols.

So what?
Are you asking Plato to justify the history of logic? That's a tall request for anyone.

11. ## Re: Truth Tables

Originally Posted by Plato
As I wrote in my reply, the function is of great interest to professional logicians. It turns out that all other logical functions can be written with the stroke, $\downarrow$, alone. Quine called it the alternate denial, neither P nor Q is true. So $\neg P \equiv \left( {P \downarrow P} \right)$ we have replaced not. As for your questions above, those were just worked out as part of the replacement program. If you have access to a good library Mathematical Logic by Quine, 1947 has a discussion.
Thanks for the info. I tough I was missing something, however you've all helped clear that up .

Thanks again!

12. ## Formal Logic

Formal Logic

1) If temp is below 32F then water freezes.
2) If A then B.
3) A→B

By convention motivated by 1)
A B A→B
T T T
F T T
T F F
F F T

Beyond that are more definitions, rules, and symbol manipulation.