Question about the possible non equivalent logical statements

[iceblits]

Homework Statement

I'm trying to figure out how many unique (non-equivalent) logical statements are possible with n statements...I think I could figure that out as soon as I see a pattern so I would like to know how many are possible for 2 statements and for 3 statements

Homework Equations

The Attempt at a Solution

I know that for 1 statement there are two possibilities, namely, P and ~p. I think for two there are twelve possibilities: P,Q,~P,~Q, PVQ, P^Q, ~(PVQ),~(P^Q),P=>Q,Q=>P,Q<=>P,~(Q<=>P)..are those all the possibilities for 2 statements?

 

[Dick]

I think you mean the number of inequivalent statements containing n logical variables, right? Think about the number of possible ways to fill out a truth table.

 

[iceblits]

Hey its you! :)
uhh yes I think...like for n=3 it would be the statements P,Q, R and all the possible ways that they are connected..
I know that a truth table can be filled out 2^n ways..like that's how many rows there will be in the truth table

 

[Dick]

Hey its you! :)
uhh yes I think...like for n=3 it would be the statements P,Q, R and all the possible ways that they are connected..
I know that a truth table can be filled out 2^n ways..like that's how many rows there will be in the truth table

Right. Now how many ways can you fill out the truth of the statement entry in the truth table?? For each line in the truth table you can write T or F, right?

 

[iceblits]

yep so...its...2^n*(2^n)?..is it like if u have a true false test of say..ten questions then you have 2^10 different possibilities?..but this time the number of rows are 2^n

 

[iceblits]

no wait...its 2^(2^n)

 

[Dick]

no wait...its 2^(2^n)

Yes. You don't have to write out explicit statements for each case to know how many there are.

 

[iceblits]

wait ...does that mean that for n=1 there's 2^2=4 different possibilities..I thought there was only 1 (P and ~P)

 

[Dick]

wait ...does that mean that for n=1 there's 2^2=4 different possibilities..I thought there was only 1 (P and ~P)

There are four. P, ~P, T and F. Or P, ~P, (P and ~P) and (P or ~P) if you prefer. See?

 

[iceblits]

AHH I see I didnt think about that!..but are (P and ~P) and (P or ~P) logically...valid? like what do they mean..do such things exist ..such that (P and ~P) for example

 

[Dick]

AHH I see I didnt think about that!..but are (P and ~P) and (P or ~P) logically...valid? like what do they mean..do such things exist ..such that (P and ~P) for example

Of course they exist. (P or ~P) or (P V ~P) is always true. Isn't it? What about (P and ~P)?

 

[iceblits]

hmm the truth table would be:
P Q P^Q
T F F
F T F

So P^Q is false all the time...I guess that makes sense because u can't have something and "not something" at the same time

 

[Dick]

hmm the truth table would be:
P Q P^Q
T F F
F T F

So P^Q is false all the time...I guess that makes sense because u can't have something and "not something" at the same time

It's just like any other logical statement. It just happens to be false all the time. It's the opposite of a tautology.

 

[iceblits]

ooo..a contradiction? Thank-you so much again! Haha I wish you were my teacher!