# What's the difference between $P \to Q$ and $P \implies Q$?

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

• 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
• @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

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.