# Need help with a proof

• Nov 26th 2012, 07:14 AM
wattskickin
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

(Doh)
• Nov 26th 2012, 08:11 AM
Plato
Re: Need help with a proof
Quote:

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 $\begin{array}{*{20}c} P &\vline & Q &\vline & {P \downarrow Q} \\\hline {T} &\vline & {T} &\vline & {F} \\ {T} &\vline & {F} &\vline & {F} \\ {F} &\vline & {T} &\vline & {F} \\ {F} &\vline & {F} &\vline & {T} \\ \end{array}$
• Nov 26th 2012, 08:28 AM
emakarov
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.

Quote:

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.
• Nov 26th 2012, 09:11 AM
wattskickin
Re: Need help with a proof
Attachment 25942I constructed the following truth table and looking at the final line it appears the statement is true
• Nov 26th 2012, 09:48 AM
Plato
Re: Need help with a proof
Quote:

Originally Posted by wattskickin
Attachment 25942I 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.
• Nov 26th 2012, 12:44 PM
emakarov
Re: Need help with a proof
Quote:

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)?
• Nov 26th 2012, 06:22 PM
wattskickin
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.(Thinking)