truth value of "if...then..."

Question

From the truth table, when both $p$ and $q$ are true, then "if $p$ then $q$" is true. However, this is a little weird as "if $p$ then $q$" is used to show the relationship between $p$ and $q$. If $q$ is independent of $p$, even both $q$ and $p$ are true, how is about the truth value of "if $p$ then $q$"?

For example, let $p$ and $q$ are unrelated and both are always true. We have a statement:

If $1+1=2$, then Paris is the capital of France.

Is this statement true or false?

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EDIT:

From truth table, this statement is definitely true. However, if it is true, it seems that the basic property of $p\Rightarrow q$, the causal relationship, is lost. It also makes the definition "$p$ is the sufficient condition of $q$" weird, as obviously in my example, $p$ is definitely not a condition of $q$.

Answer 1

If q is independent of p, even both q and p are true, how is about the truth value of "if p then q"?

Since both q and p are true, "if p then q" is true in the language of two-valued logic.

if 1+1=2, then Paris is the capital of France.

Both statements are true, therefore the conditional is true in two-valued logic.

Answer 2

The statement

If $1+1=2$, then Paris is the capital of France.

is true and stays true as long as Paris is the capital of France.

A statement "$P \Rightarrow Q$" is only false when $P$ is true but $Q$ is false, in other words, for the statement to be true you need "When $P$ is true, $Q$ has to be true as well".

Answer 3

In classical logic, "If p then q" does not suggest any causal link or other relationship between logical propositions p (the antecedent) and q (the consequent). It says only that it is not the case that both p is true and q is false. In your statement, both the antecedent and consequent are true, so the statement is true.

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EDIT: The truth table for implication may seem counter-intuitive to the beginner. (It certainly did to me when first introduced to propositional logic.) It can, however, be derived from other more self-evident rules of logic. For details, see my blog posting on material implication. (Requires some knowledge of the basic methods of proof, e.g. proof by contradiction.)

Answer 4

There are many kinds of implication in the ordinary language. We may use the conditional "if..., then..." construction to indicate definitional, causal, or logical relations; not all of them are equivalent. We name material implication the specific kind of implication we use in logic. Precisely because it is just one kind of implication, material implication cannot capture every use of the English "if...then" or "implies". As correctly pointed out by @MauroAllegranza in his comment, the conditional "if..., then..." in logic does not express a causal link:

In logic there is no cause: logic is not about nature.

The (definition and) meaning of the material implication is in its truth table. So, in logic, implication is treated as a truth functional connective: we can figure out the truth-value of the conditional statement solely on the basis of the truth-values of its components. Very roughly, we can think of a material implication $p \to q$ as expressing a promise that whenever a certain condition is met (viz., that the antecedent $p$ is true), then the consequent $q$ is true; if the condition $p$ turns out not to be met, then the promise stands unbroken, regardless of $q$.

There are two paradoxes of material implication, which are evident from its truth table:

1. whenever the antecedent is false, the whole conditional is true (e.g. "if the moon is made of green cheese, then life exists on other planets");
2. whenever the consequent is true, the conditional is true (e.g. "if life exists on other planets, then life exists on earth").

Strange as it may seem, in both cases material implication is true, even though the antecedent and the consequent are completely unrelated. These are paradoxes in the sense of violations of our intuition about implication (they are not contradictions), but just because we want to interpret material implication as another$-$and not equivalent$-$kind of implication in use in ordinary language, notably the causal one. The fact that the material implication cannot capture other uses of implication in the ordinary language is the price to pay for adopting a truth functional approach.

Said differently, since the truth-value of a material implication is a function of the truth-values of its antecedent and consequent alone, then we will look only to the truth-values, not to the content, of the antecedent and consequent. This is consistent with the aim of disregarding content and representing only the logical form of statements and arguments. But if the content of the antecedent and consequent is irrelevant, then they may be utterly unrelated to one another. We have abandoned the requirement of ordinary implication that antecedent and consequent be mutually relevant or somehow connected. Truth-functionality requires the loss of relevancy.

But why would we adopt a type of implication with such counter-intuitive results? The answer is that the kind of implication used in a mathematical context is essentially the logic one (i.e. the material implication), which is truth functional. So, to formalize and mimic the kind of implication used by mathematicians in their theorems, we have to adopt the material implication. Notice that the truth functional approach has several benefits, such as the truth-table method to determine the validity of arguments that contain implication statements, and to determine whether certain implication statements are tautologies.

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For more details about material implication, see here, here, here, here and here.