# Math Help - Simple logic truth table question

1. ## Simple logic truth table question

Here is a truth table for the new symbol *

 P * Q T F T T T F F T T F T F

Given this truth-table, what ordinary English expression can this new truth-functional connective (*) be used to symbolize?

Suppose there are two sentences a and b. On every interpretation that a is true, b is false. What can you conclude (if anything) about the following argument? Briefly explain.

~(a -> b)
.:. ~b v a

-> :material conditional
v :disjuction/or

Many thanks for help

2. ## Re: Simple logic truth table question

Originally Posted by apterosss
Here is a truth table for the new symbol *
P Q P * Q
T T F
T F T
F T T
F F T

Given this truth-table, what ordinary English expression can this new truth-functional connective (*) be used to symbolize?

I have most often have seen this as $\begin{array}{*{20}{c}} P&Q&\;&{P|Q} \\ \hline T&T&\;&F \\ T&F&\;&T \\ F&T&\;&T \\ F&F&\;&T \end{array}$.

WVO Quine called this the alternative denial operator, first discovered by CS Peirce.
These are equivalent:
$\begin{array}{*{20}{c}} {\neg P}&{\mathop = \limits^{df} }&{P|P} \\ {P \wedge Q}&{\mathop = \limits^{df} }&{(P|Q)|(P|Q)} \\ {P \vee Q}&{\mathop = \limits^{df} }&{(P|P)|(Q|Q)} \\ {P \to Q}&{\mathop = \limits^{df} }&{P|(Q|Q)} \end{array}$

3. ## Re: Simple logic truth table question

Plato - apologies, something happened with my spacing.
Yes your table is equivalent to what I tried to put; please see edit.

What would the English phrase be for this? Is there a sentence this could be symbolised with?

Thanks

4. ## Re: Simple logic truth table question

Originally Posted by apterosss
Given this truth-table, what ordinary English expression can this new truth-functional connective (*) be used to symbolize?
A * B can be read, "A and B cannot both be true" or "A contradicts B" (note that this is the same as "B contradicts A", i.e., * is symmetric).

Originally Posted by apterosss
Suppose there are two sentences a and b. On every interpretation that a is true, b is false. What can you conclude (if anything) about the following argument? Briefly explain.

~(a -> b)
.:. ~b v a
This argument is valid without assuming anything about truth of a and b in interpretations.

5. ## Re: Simple logic truth table question

Originally Posted by apterosss
What would the English phrase be for this? Is there a sentence this could be symbolised with?

Actually there are probably many names for this.

As I said, WVO Quine called this the alternative denial operatorr, first discovered by CS Peirce.
I don't remember is Perice had a name for it.
It also goes by the name stroke function,

The chart I posted shows that the other functions can be defined using this operator.

It in effect: "It is false if both statements are true and true otherwise".

6. ## Re: Simple logic truth table question

Hello, apterosss!

Here is a truth table for the new symbol $\star$

. . $\begin{array}{ccc} P & \star & Q \\ \hline T & F & T \\ T & T & F \\ F & T & T \\ F & T & F \end{array}$

Given this truth-table, what ordinary English expression
can this new operation $(\star)$ be used to symbolize?

$\text{We have: }\:\sim(P \wedge Q) \:\Longleftrightarrow\: \sim\!P\:\vee \sim\!Q$

. . $"P\text{ or }Q\text{ is false."}$

7. ## Re: Simple logic truth table question

Thank you all for your help

8. ## Re: Simple logic truth table question

Sorry I mistyped this one - still don't understand the answer.

Suppose there are two sentences a and b. On every interpretation that a is true, b is false. What can you conclude (if anything) about the following argument? Briefly explain.

~(a -> c)
.:. ~b v c

-> :material conditional
v :disjuction/or

9. ## Re: Simple logic truth table question

Originally Posted by apterosss
Suppose there are two sentences a and b. On every interpretation that a is true, b is false. What can you conclude (if anything) about the following argument?

~(a -> c)
.:. ~b v c
This argument is valid in this sense. The formula ~(a -> c) is equivalent to a /\ ~c. Then use the assumption.