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According to my book, the essential difference between a logical implication $\{p\} \Rightarrow q$ and the statement $p \to q$ is that $p \to q$ is part of the propositional language, and $\{p\} \Rightarrow q$ is not. My question is, to which language does the logical implication $\{p\} \Rightarrow q$ belong?

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Which book are you using? Other people may be able to look it up. – Carl Mummert Jul 11 '13 at 1:13
@CarlMummert I'm using this book: , page 32. But it is written in dutch. – Kasper Jul 11 '13 at 1:16
up vote 3 down vote accepted

The way that this book appears to be using the notation, if $A$ is a set of formulas and $\phi$ is a formula, the relation $A \Rightarrow \phi$ means that in the appropriate truth table, every row that makes all formulas in $A$ true also makes $\phi$ true.

So, for example, $p \land q \Rightarrow q$ holds because every row that makes $p \land q$ true also makes $q$ true. This is different than $p \land q \to q$, which is not true or false on its own, but only true or false for each row in the truth table. The logical implication arrow $\Rightarrow$ has the effect of quantifying over all the rows. So $p \land q \to q$ has "4 separate truth values", one for each row of a 2 variable truth table, while $p \land q \Rightarrow q$ is simply true.

Similarly, $\{p, p \to q\} \Rightarrow q$ holds, but this is not even of the right syntactic form to be true or false in each row.

The language of formulas is called the object language; the language of $\Rightarrow$ is called the metalanguage, which is another language used to study the object language. The need to keep these distinct is not obvious at first, but the distinction turns out to be vital for many aspects of logic.

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Great answer, really appreciate it :) – Kasper Jul 11 '13 at 1:31

The authors clarify the matter in the next sentence:

Schrijven we $\varphi\Rightarrow\psi$, dan stelt dit dat $\psi$ inderdaad volgt uit $\varphi$, terwijl als we $\varphi\to\psi$ opschrijven, deze formule best onwaar kan zijn.

In other words, $\varphi\Rightarrow\psi$ is the assertion that $\psi$ really is a logical consequence of $\varphi$, while $\varphi\to\psi$ is a proposition that may be true or false.

The connective $\to$ is part of the language in which we build propositions; $\Rightarrow$ is part of the language in which we talk about these propositions. Thus, if $A$ is a set of propositions, $A\Rightarrow\varphi$ is the assertion that if every proposition in the set $A$ has the truth value TRUE, then so must $\varphi$: $\varphi$ is a logical consequence of the propositions in $A$. In particular, $\{\varphi\}\Rightarrow\psi$ is synonymous with $\varphi\Rightarrow\psi$: it’s the assertion that if $\varphi$ has the truth value TRUE, then so must $\psi$.

In contrast, $\varphi\to\psi$ is just a proposition that may be true or false; the formula does not in itself make any assertion about the truth or falsity of any proposition.

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