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I know that there are many publications that talk about this topic, but it is still not entirely clear to me.

I had previously reflected on this topic, and trying to find some logic I came to the following conclusion:

If P, then Q (P and Q, are true)

From every true proposition only can be derivate another true proposition.

If P, then Q (P are False, and Q true)

From a false proposition can be derivate a true proposition.

If P, then Q (P are True, and Q False)

From a true proposition don't can implies a false proposition.

If P, then Q (P and Q, are false)

From a false proposition can implies another false proposition, but the implication can be true.

This makes sense to me, even if a statement is false it can derive a valid logical connection, either another false statement or a true statement, but this is complicated when they tell you that the value of an implication is independent of the relationship that the statements have, depending only on the truth value that they have.

Summarizing in a few words, why does the implication truth table have that shape?, and, why is its value independent of the relationship that the statement have?

Mauro ALLEGRANZA
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JuanJesús
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  • See also the post What is the relation between the material conditional in logic and conditionals that we use every day? as well as the post Why do people use the material implicaton over alternatives?. – Mauro ALLEGRANZA Dec 16 '22 at 07:20
  • For implications we expect its contrapositive to be also true. In order to satisfy this condition "false then false" must be true in order for "true then true"'s contrapositive to be true. For the rest of the cases one of them has to be false - "true then false" - in order for "false then true"'s contrapositive to be false. – Baris Aytekin Dec 16 '22 at 08:07
  • The pair of truth values assigned to two statements is a relationship between the statements. But I think you have a different relationship in mind, something like whether one statement can be derived from the other via rules of inference. That's an entirely different level of logic. The implication symbol $\implies$ is for when you just want a simple statement about truth values, which is useful to have in the context of many proofs. – David K Dec 16 '22 at 16:07
  • Mmm, but if the logical implication is not an abstraction of what is commonly known as "implication", then what is? – JuanJesús Dec 17 '22 at 01:43
  • This causes me a problem, for example, with the definitions, I have been studying some set theory and there a subset is defined as: A ⊆ B ⇐⇒ (∀x)(x A ⇒ x B).

    but being formulated in the language of mathematical logic allows me to deduce things as follows:

    Let A be any set. Let x be any object. Because the antecedent is false, the sentence is true. Therefore, Ø⊆ A.

    that although in the language of logic it is valid, it seems very unnatural to me, since it does not follow a chain of thoughts as such, based solely on a property of the implication that I do not quite understand

     – JuanJesús Dec 17 '22 at 01:53

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