Question about the possible non equivalent logical statements

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In summary, the conversation discusses the number of unique logical statements possible with n statements. The speaker mentions using a truth table to determine the number of possibilities and clarifies that for n=1, there are four possibilities. The conversation also touches on the validity of statements such as (P and ~P) and (P or ~P).
  • #1
iceblits
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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?
 
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  • #2
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.
 
  • #3
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
 
  • #4
iceblits said:
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?
 
  • #5
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
 
  • #6
no wait...its 2^(2^n)
 
  • #7
iceblits said:
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.
 
  • #8
wait ...does that mean that for n=1 there's 2^2=4 different possibilities..I thought there was only 1 (P and ~P)
 
  • #9
iceblits said:
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?
 
  • #10
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
 
  • #11
iceblits said:
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)?
 
  • #12
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
 
  • #13
iceblits said:
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.
 
  • #14
ooo..a contradiction? Thank-you so much again! Haha I wish you were my teacher!
 

Question 1: What are non-equivalent logical statements?

Non-equivalent logical statements are two or more statements that have different truth values. This means that one statement can be true while the other is false, or vice versa.

Question 2: How can I determine if two logical statements are equivalent?

To determine if two logical statements are equivalent, you can use truth tables or logical equivalences. Truth tables show all possible combinations of truth values for the statements, while logical equivalences are rules that show when two statements are equivalent.

Question 3: Can non-equivalent logical statements be used interchangeably?

No, non-equivalent logical statements cannot be used interchangeably. They have different truth values and, therefore, cannot be substituted for each other in a logical argument.

Question 4: What is the difference between logically equivalent and logically consistent statements?

Logically equivalent statements have the same truth values, while logically consistent statements can be true at the same time. In other words, logically equivalent statements are always consistent, but logically consistent statements may or may not be equivalent.

Question 5: Are all logical statements either equivalent or non-equivalent?

No, there is also the concept of logical independence, where two statements have no relation to each other and cannot be categorized as equivalent or non-equivalent.

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