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).

Please help.

Thanks

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- March 16th 2014, 05:52 PMGayelleTruth 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).

Please help.

Thanks - March 16th 2014, 07:37 PMPlatoRe: Truth Tables
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. - March 16th 2014, 07:41 PMromsekRe: Truth Tables
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 - March 16th 2014, 08:15 PMGayelleRe: 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.

- March 16th 2014, 08:22 PMromsekRe: Truth Tables
- March 17th 2014, 04:35 AMGayelleRe: Truth Tables
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 - March 17th 2014, 05:43 AMPlatoRe: Truth Tables
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, , alone.

Quine called it the alternate denial, neither P nor Q is true.

So 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. - March 17th 2014, 06:56 AMHartlwRe: Truth Tables
- March 17th 2014, 08:12 AMHartlwRe: 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. - March 17th 2014, 09:36 AMromsekRe: Truth Tables
- March 17th 2014, 11:31 AMGayelleRe: Truth Tables
- March 18th 2014, 05:19 AMHartlwFormal 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.