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background: I am trying to fully understand the meaning of implication which i understand intuitively . I learned that $P \to Q$ is a connective , which means that $P$ and $Q$ don't have a logical connection or any reason why $P$ being true should MAKE $Q$ be true and it's just a representation of $\neg P \vee Q$ .

question: $P \implies Q$ means that $P \to Q$ is a tautology , what does that mean ? any mathematical examples ?

in other words: What's the difference between $P \to Q$ and $P \implies Q$ ?

thanks

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In many cases the difference between single and double arrows is just a stylistic choice. A common use in metamathematics is to use $\to$ in formulas in the object language and $\Rightarrow$ for implications at the metalevel. – Henning Makholm Oct 22 '11 at 13:30
    
@Henning: Exercising your mjolnir? – Asaf Karagila Jul 16 '15 at 15:56
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@AsafKaragila: Apparently so; I didn't know it worked in that direction too. Basically I don't think a question about the difference between symbols should be a duplicate of one about the difference between the words "material" and "logical", when there's no strong convention that these symbols correspond bijectively to those words. – Henning Makholm Jul 16 '15 at 16:22
up vote 7 down vote accepted

Let $P$ and $Q$ be two propositions. In some logic texts, they say that $P \to Q$ is a new proposition, also written $\neg P \vee Q$. But $P \implies Q$ is a relation between the two propositions, not a proposition itself.

Maybe an analogy will help. Let $x$ and $y$ be two real numbers. Then $x+y$ is a new real number. But $x \le y$ is a relation between the two real numbers, and is not itself a real number.

The confusion is that in logic, we talk about some objects called "propositions", but in the language we are using we may also think that we are writing propositions. So you have to keep these two levels separate somehow in your mind.

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It's not a big deal, pal. $P \implies Q$ is a statement, i.e., a formula which has no free variables. But you may say it has free variables, namely P and Q, but in predicate calculus(forming new formulas from more simple (atomic) formulas), P and Q are just formulas, and by using "$\implies$" we mean that the relation $\to$(P,Q) (which is again by predicate calculus, a formula) is already known as truth, that is, it's a mathematical statement.

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