Logic

[MattF]

∧ ∨ Are logical opertions, how many different logical operations can there be?

There are obviously at least 4, and Im guessing I would combine these to show other logical operations, but Im unclear as to which ones I can or cannot combine

 

[SlipEternal]

Assuming the following:

1. You are dealing with a binary system of logic (true and false are the only logical values)
2. The Law of Excluded Middles (This is usually assumed with a binary system of logic)
3. You are only looking at binary operators

Any binary operator is used in the form:
\(\displaystyle A \text{ operator } B\)
where both \(\displaystyle A\text{ and }B\) can be one of two possible values (true or false).
So, a truth table for this operator would look like this:

A	B	A operator B
true	true	?
true	false	?
false	true	?
false	false	?

Two logical operators can be considered equivalent if they take \(\displaystyle A\text{ and }B\) to the same truth value for all possible truth values of \(\displaystyle A\text{ and }B\). This implies that there are \(\displaystyle 2^4=16\) possible operators (since there are two possible choices for each question mark, either true or false).

 

[pka]

∧ ∨ Are logical opertions, how many different logical operations can there be?

I understand reply #2. It is probably what the question means.
However, here is a different reading.
Willard Quine writes that these are operators in a functionally complete logic:
\(\displaystyle \begin{array}{*{20}c} {{\rm{not}}} &\vline & \neg \\
{{\rm{or}}} &\vline & \vee \\{{\rm{and}}} &\vline & \wedge \\{{\rm{imply}}} &\vline & \Rightarrow \\\end{array}\)

Then he points that Charles Sanders Peirce found in the 1880's that one can write each those in terms of a single operator: the stroke operator.
\(\displaystyle \begin{array}{*{20}c} A & B &\vline & {A|B} \\\hline T & T &\vline & F \\ T & F &\vline & T \\ F & T &\vline & T \\ F & F &\vline & T \\\end{array}\)

 

[MattF]

Thank You

 

[SlipEternal]

I understand reply #2. It is probably what the question means.
However, here is a different reading.
Willard Quine writes that these are operators in a functionally complete logic:
\(\displaystyle \begin{array}{*{20}c} {{\rm{not}}} &\vline & \neg \\
{{\rm{or}}} &\vline & \vee \\{{\rm{and}}} &\vline & \wedge \\{{\rm{imply}}} &\vline & \Rightarrow \\\end{array}\)

Then he points that Charles Sanders Peirce found in the 1880's that one can write each those in terms of a single operator: the stroke operator.
\(\displaystyle \begin{array}{*{20}c} A & B &\vline & {A|B} \\\hline T & T &\vline & F \\ T & F &\vline & T \\ F & T &\vline & T \\ F & F &\vline & F \\\end{array}\)

That is not quite correct. Charles Sanders Pierce found that one can write each in terms of the NOR operator with the following truth table:

\(\displaystyle \begin{array}{*{20}c} A & B &\vline & {A\downarrow B} \\\hline T & T &\vline & F \\ T & F &\vline & F \\ F & T &\vline & F \\ F & F &\vline & T \\\end{array}\)

The truth table you gave cannot yield all possible operators.
The common symbol of the NOR operator is \(\displaystyle \downarrow\)
So \(\displaystyle A\downarrow A = \neg A\)
\(\displaystyle (A\downarrow B)\downarrow (A\downarrow B) = A \lor B\)
\(\displaystyle (A\downarrow A)\downarrow (B \downarrow B) = A \land B\)
\(\displaystyle ((A\downarrow A)\downarrow B)\downarrow ((A\downarrow A)\downarrow B) = A \Rightarrow B\)

 

[pka]

That is not quite correct. Charles Sanders Pierce found that one can write each in terms of the NOR operator with the following truth table:

See page 266 of Irving M. Copi's Symbolic Log​ 3ed.

 

[SlipEternal]

See page 266 of Irving M. Copi's Symbolic Log​ 3ed.

Then, I believe the book is wrong. Everything I have read stated that Pierce used the NOR operator. I just showed you several examples. I can get negation from the stroke operator you listed, but I cannot get AND nor OR. If you can, by all means, please prove me wrong.

Edit: I just tried a Google search, and everything I am seeing supports that Pierce used the NOR operator. Again, I would be glad to be proven wrong, but I think that book is rather outdated.

 

[pka]

Then, I believe the book is wrong. Everything I have read stated that Pierce used the NOR operator.

Are you really saying that you think that Copi and Quine are both mistaken about a point about the history of logic?

Have you read Peirce's 1888 essay on this topic?

Of course there are those who still say "Those who quote Peirce quotes trash".

 

[SlipEternal]

What I am saying is that I don't have access to a library at the moment. I was able to generate any of the 16 operators from the NOR operator. Using the stroke operator, I was not able to get all 16 operators. I was able to get only 8 of them using the stroke operator, and I had to use tricks like \(\displaystyle A|\text{True} = \neg A\). This is because \(\displaystyle A|A = \text{ False}\). So, I ask again, can you show me how you can use the stroke operator to achieve the AND symbol? I believe you are mistaken still. And yes, perhaps that means I think that both Copi and Quine are wrong. No, I have not read Pierce's original 1888 essay. I have simply figured out the logic myself, and I don't see how it is possible. That is why I am asking you to prove me wrong.

 

[SlipEternal]

Ok, I see where the problem is. You corrected the truth table you listed. Now it makes sense. That truth table also suffices (it is the NAND operator).