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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?

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 $q$.

X Leo
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    [...]as "if p then q" is used to show the relationship between p and q[.]: No, because in logic the only thing left is the structure of reasoning, not our daily logic. The statement is true. – linear_combinatori_probabi Sep 19 '18 at 17:12
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    In logic, the conditional "if..., then..." does not express a causal link. In logic there is no "cause": logic is not about nature. – Mauro ALLEGRANZA Sep 19 '18 at 19:22

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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.

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

Taroccoesbrocco
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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".

Christoph
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  • From truth table, this is definitely right. The question is if it is true, the basic logic, i.e. the causal link, included in $P\Rightarrow Q$ is lost. And also, this makes the statement "$P$ is the sufficient condition of $Q$" weird, as obviously now $P$ is not the condition of $Q$. – X Leo Sep 19 '18 at 17:36
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    "$P\implies Q$" is also true whenever $P$ is false, regardless of whether $Q$ is true or false. – Dan Christensen Sep 19 '18 at 21:47
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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.

Doug Spoonwood
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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.

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.)

Dan Christensen
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  • So this means that the truth table, or the classical logic can not say everything about "If $p$ then $q$", especially the causal link is not necessary. However, this is wired as we call $p$ the sufficient condition of $q$. – X Leo Sep 19 '18 at 17:30
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    @Xliu The following statements are all logically equivalent: (1) If p then q, (2) p implies q, (3) p is sufficient for q, and (4) q is necessary for p. In classical logic, neither of them suggest any causal link. That is just a common misconception about the rules of logic. These rules have been successfully applied to every field of human endeavour for millennia. – Dan Christensen Sep 19 '18 at 17:45
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    @Xliu There is no notion of causality in mathematics. Causality is in the realm of science. Scientists now know that smoking causes cancer, but someone can smoke without ever having cancer. – Dan Christensen Sep 19 '18 at 18:04
  • @Xliu See edit. – Dan Christensen Sep 19 '18 at 20:55