 We now introduce a deductive system, \(D\), for our languages. As above, we define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. If there are any other sentences in the argument, they are its  premises.[1]  By convention, we use “\(\Gamma\)”, “\(\Gamma'\)”, “\(\Gamma_1\)”, etc, to range over sets of sentences, and we use the letters “\(\phi\)”, “\(\psi\)”, “\(\theta\)”, uppercase or lowercase, with or without subscripts, to range over single sentences. We write “\(\Gamma, \Gamma'\)” for the union of \(\Gamma\) and \(\Gamma'\), and “\(\Gamma, \phi\)” for the union of \(\Gamma\) with \(\{\phi\}\).