 For our purposes, a  language \(\mathbf{L}\) is identified with the set of all grammatical sentences it contains. Sentences will be denoted by lowercase letters Greek \(\alpha, \beta, \ldots\). The language \(\mathbf{L}\) is assumed to contain a set of  atomic sentences \(\alpha,\beta, \ldots\) which are not built out of any other sentences, as well as all the sentences generated by combining the atomic sentences with truth-functional connectives from propositional logic. In other words: if \(\alpha,\beta\) are sentences in \(\mathbf{L}\) then \(\neg \alpha\), \(\alpha\vee \beta\), \(\alpha\wedge \beta\), \(\alpha\rightarrow \beta\), and \(\alpha \leftrightarrow \beta\) are also sentences in \(\mathbf{L}\). These are meant to be read respectively as “not \(\alpha\)”, “\(\alpha\) or \(\beta\)”, “\(\alpha\) and \(\beta\)”, “if \(\alpha\), then \(\beta\)” and “\(\alpha\) if and only if \(\beta\)”. The symbol \(\bot\) (pronounced “falsum”) denotes an arbitrarily chosen contradiction (e.g. \(\alpha\wedge \neg \alpha)\) and the symbol \(\top\) (pronounced “top”) denotes an arbitrary tautology.