Symbols and notation in propositional logic

Question

I'm currently taking a course on mathematical logic. I'm now studying propositional logic and I'm having some trouble getting used to the symbolism used, as well as the general idea behind mathematical logic.

Anyway can someone explain to me what this notation means? $$\left \langle \left \{ t_1,t_2,...,t_n \right \}, t \right \rangle$$ , where $t_1,t_2,...t_n,t$ are propositional formulas.

Answer 1

It is an ordered pair where the first term is a set of $n$ formulas and the second term is a formula.

It is the general form of an inference rule, where the first term is the set of premises and the second one is the conclusion.

It is only a "notational variant" of $(\Gamma, \varphi)$; see Christopher Leary & Lars Kristiansen, A Friendly Introduction to Mathematical Logic (2nd ed.2015), page 42.

Following this notation, we may write the usual Modus Ponens rule : $\dfrac {A \ \ A \to B}{B}$, as follows :

$⟨ \{ A, A → B \},B ⟩$.