Logic - clarification needed about implication

In summary, the conversation discusses the concept of material conditional and how it can be true even when P is false and Q is true. This is because the truth table for P→Q is the same as for (not Q)→(not P), meaning that P→Q must be defined as true when both P and Q are false. This is necessary for logic to work properly.
  • #1
autodidude
333
0
If P→Q, and P is false but Q is true, then why is P→Q true? To me, it seems as though we shouldn't be able to do proceed because there isn't enough information. Same goes when P and Q are both false, how does that suggest P→Q is true?
 
Mathematics news on Phys.org
  • #2
"If it rains, the street gets wet"
This statement is true, even if I spill water on the street (without rain).
More general: It cannot be false, if it does not rain. It just does not give any information about the street in that case.
 
  • #3
Another reason for those definitions is so that logic "works" the way it should, for every combination of "true" and "false".

For example, "P implies Q" means the same (in ordinary English) as "if P is true, then Q is true", which means the same as "if Q is false, then P is false".

So the truth table for P→Q must be the same as for (not Q)→(not P),

That means P→Q must be defined as true, when P and Q are both false.

You can create a similar argument to show how P→Q must be defined with P is false and Q is true.
 
  • #5


In logic, the statement "P→Q" means "if P, then Q." This is known as an implication, where the truth of one statement (P) leads to the truth of another statement (Q). In the case where P is false and Q is true, the statement "if P, then Q" is still true because there is no contradiction. While P may be false, the statement is still logically valid because Q is true, thus fulfilling the condition of the implication. In other words, the implication is not dependent on the truth values of P and Q individually, but rather on the logical relationship between them.

Similarly, when both P and Q are false, the statement "if P, then Q" is still true because there is no contradiction. In this case, since both P and Q are false, the statement cannot be disproved and therefore is considered true. This may seem counterintuitive, but in logic, a statement is only considered false if it can be proven false. Since there is no evidence to disprove the statement "if P, then Q," it is considered true.

In summary, the truth value of an implication is not determined by the individual truth values of P and Q, but rather by the logical relationship between them. In both cases, there is no contradiction, and therefore the implication is considered true.
 

FAQ: Logic - clarification needed about implication

1. What is the definition of implication in logic?

Implication is a logical relationship between two statements, where the truth of one statement (known as the antecedent) guarantees the truth of another statement (known as the consequent).

2. How is implication represented in logical notation?

In logical notation, implication is represented by the symbol "→" or "⊃". The antecedent is placed to the left of the symbol and the consequent is placed to the right.

3. What is the difference between implication and equivalence?

Implication is a one-way relationship, where the truth of the antecedent guarantees the truth of the consequent. Equivalence, on the other hand, is a two-way relationship, where the truth of both statements are dependent on each other.

4. Can a statement be true if its consequent is false?

No, in an implication, if the consequent is false, then the statement as a whole is considered false. The only way for an implication to be true is if the antecedent is true and the consequent is also true.

5. How is implication used in everyday life?

Implication is commonly used in everyday life to make logical deductions and inferences. For example, if it is raining outside (antecedent), then the ground will be wet (consequent). This allows us to make predictions and decisions based on logical reasoning.

Back
Top