# Thread: Need help with a proof

1. ## Need help with a proof

-(p q)⇐⇒(-p ↑-q)

I need to prove this but are not sure what the down and up arrows mean.

Do I have to construct a truth table?

Thanks,
Chris

2. ## Re: Need help with a proof

Originally Posted by wattskickin
[SIZE=2]-(p q)⇐⇒(-p ↑-q)
I need to prove this but are not sure what the down and up arrows mean.
Do I have to construct a truth table?

So far as I know the definitions vary here. I can give you the one used by WVO Quine:
$\begin{array}{*{20}c} P &\vline & Q &\vline & {P \uparrow Q} \\\hline {T} &\vline & {T} &\vline & {F} \\ {T} &\vline & {F} &\vline & {T} \\ {F} &\vline & {T} &\vline & {T} \\ {F} &\vline & {F} &\vline & {T} \\ \end{array}$ and $\begin{array}{*{20}c} P &\vline & Q &\vline & {P \downarrow Q} \\\hline {T} &\vline & {T} &\vline & {F} \\ {T} &\vline & {F} &\vline & {F} \\ {F} &\vline & {T} &\vline & {F} \\ {F} &\vline & {F} &\vline & {T} \\ \end{array}$

3. ## Re: Need help with a proof

I agree. The notations $\downarrow$ and $\uparrow$ probably refer to Peirce's arrow (NOR) and Sheffer stroke (NAND), respectively.

Originally Posted by wattskickin
Do I have to construct a truth table?
This depends on the assignment. There are many ways of proving a propositional formula, from constructing a truth table to giving a derivation in one of many formal systems.

4. ## Re: Need help with a proof

I constructed the following truth table and looking at the final line it appears the statement is true

5. ## Re: Need help with a proof

Originally Posted by wattskickin
I constructed the following truth table and looking at the final line it appears the statement is true
If we guessed correctly, then I agree with you.

6. ## Re: Need help with a proof

Originally Posted by wattskickin
I constructed the following truth table and looking at the final line it appears the statement is true
Why does the table have the last two columns if those formulas don't occur in the formula from post #1? Also, how can you judge the validity of a formula by looking only at the last line (presumably, last row)?

7. ## Re: Need help with a proof

Sorry, I also had to prove the formula ¬(p ↑ q)⇐⇒(¬p ↓¬q) so I added that information to the columns. since the columns for ¬(p ↓ q) equal the columns for (¬p ↑¬q) that formula is true. Since the columns for ¬(p ↑ q) equal the columns for (¬p ↓¬q) that formula is true.

Thank you for challanging me to finish thinking this problem through. I have suffered a recent brain injury and my thought process suffers from this.