Need help with truth table for P->Q and it's inverse

In summary: If both rows 2 and 3 were true, then Q would be false,and P would be true. However, since Q is false, Q\rightarrow P is always a truestatement. So if you don't live in France, then the statement "If I live in France, then I live in Paris" is true.
  • #1
Mofasa
1
0
Hi!

I'm struggling with the bellow truth tables:

P = I live in Paris
Q = I live in France

A B C
P → Q Q → P ¬ Q → ¬ P
S S S S S S F S F
S F F F S S S F F
F S S S F F F S S
F S F F S F S S S

Table A and C I'm fully clear with. However I don't understand the truth table for B (the converse of P → Q).

Table B:

Q → P
S S S
F S S
S F F
F S F

If I live in France there is a possibility I live in Paris S (Understand)
If I don't live in France there is a possibility I live in Paris S (Don't understand)
If I live in France there is not a possibility I live in Paris F (Don't understand)
I I don't live in France there is not a possibility I live in Paris S (Understand)

I want line 2 to be false and line 3 to be true.

Can someone explain why this should not be the case?
 
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  • #2
Q → P ("If I live in France, I live in Paris") is wrong in reality, why do you expect correct results for this?
 
  • #3
mfb said:
Q → P ("If I live in France, I live in Paris") is wrong in reality, why do you expect correct results for this?

If Q is false, then [itex]Q\rightarrow P[/itex] is always a true statement. So if you don't live in France, then the statement "If I live in France, then I live in Paris" is true.

The only way [itex]Q\rightarrow P[/itex] can ever be false is if [itex]Q[/itex] is true and [itex]P[/itex] is false.
 
  • #4
micromass said:
If Q is false, then [itex]Q\rightarrow P[/itex] is always a true statement. So if you don't live in France, then the statement "If I live in France, then I live in Paris" is true.

The only way [itex]Q\rightarrow P[/itex] can ever be false is if [itex]Q[/itex] is true and [itex]P[/itex] is false.
Okay, to be more precise: Q → P is not true in general (=for all) in reality.
 
  • #5
Mofasa said:
Hi!

I'm struggling with the bellow truth tables:

P = I live in Paris
Q = I live in France

A B C
P → Q Q → P ¬ Q → ¬ P
S S S S S S F S F
S F F F S S S F F
F S S S F F F S S
F S F F S F S S S

Table A and C I'm fully clear with. However I don't understand the truth table for B (the converse of P → Q).

Table B:

Q → P
S S S
F S S
S F F
F S F

If I live in France there is a possibility I live in Paris S (Understand)
If I don't live in France there is a possibility I live in Paris S (Don't understand)
If I live in France there is not a possibility I live in Paris F (Don't understand)
I I don't live in France there is not a possibility I live in Paris S (Understand)

I want line 2 to be false and line 3 to be true.

Can someone explain why this should not be the case?

The truth table for Q→P is exactly the same as that for P→Q , since they are both

plain/standard conditionals. Notice that the tables are exactly the same except

rows 2 and 3 have been exchanged.
 

Related to Need help with truth table for P->Q and it's inverse

1. What is a truth table?

A truth table is a mathematical table used to represent the possible combinations of truth values for logical expressions. It shows how the truth value of a compound statement is affected by the truth values of its individual components.

2. How do you create a truth table for P->Q?

To create a truth table for P->Q, start by listing all possible combinations of truth values for P and Q. Then, use the logical operator "->" to determine the resulting truth value for the compound statement. Finally, fill in the truth values for the compound statement in the last column of the table.

3. What is the inverse of P->Q?

The inverse of P->Q is ~P->~Q, which means "if not P, then not Q". It is formed by negating both the antecedent (P) and consequent (Q) of the original conditional statement.

4. Can a truth table help determine the validity of an argument?

Yes, a truth table can help determine the validity of an argument by showing all possible combinations of truth values for the premises and conclusion. If there is a row in the truth table where all the premises are true and the conclusion is false, then the argument is invalid.

5. How can a truth table be helpful in solving logical problems?

A truth table can be helpful in solving logical problems by providing a systematic way to analyze complex logical expressions and determine their truth values. It can also help identify patterns and relationships between different propositions, making it easier to find a solution or prove a statement.

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