I am currently reading Rautenberg's book on mathematical logic, in it he defines a propositional language ##\mathcal{F}##, set theoretically, as the smallest (i.e. the intersection) of all sets of strings ##S## built from propositional variables (##\ p_1,p_2,\ldots##) as well as any binary connectives on those variables, with the properties:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] (1)\ p_1,p_2,\ldots\in{}S;\quad{}(2)\ \alpha,\beta\in{}S\Rightarrow(\alpha\circ\beta)\in{}S [/tex]

How does this definition work? Is ##S## supposed to be representing strings or a set? Is ##S## supposed to be the set of all strings with those properties? Or is ##\mathcal{F}##? Also, what am I supposed to be intersecting there?

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# Difference between propositional language and set of all formulas

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