 The weakest logic with connectives interpreted by truth-functions of the type described above is MTL (Monoidal T-norm based Logic, Esteva &amp; Godo 2001). It is a logic with the primitive binary connectives \(\mathbin{\&amp;}, \to, \wedge\) and a truth-constant \(\overline{0}\), and derivable connectives defined as:   \[ \begin{align} \varphi \lor \psi &amp;= ((\varphi \to \psi) \to \psi) \land ((\psi \to \varphi) \to \varphi)\\ \neg \varphi &amp;= \varphi \to \overline{0} \\ \varphi \leftrightarrow \psi &amp;= (\varphi \to \psi) \land (\psi \to \varphi)\\ \overline{1} &amp;= \neg \overline{0}. \end{align} \]  MTL is defined as a consequence relation by means of the semantics given by all left-continuous t-norms. Namely, given a particular left-continuous t-norm \(\ast\) and a mapping from propositional variables to \([0,1]\), we define the \(\ast\)-evaluation of all formulas by interpreting \(\&amp;\) as \(\ast\), the implication \(\to\) as its residuum \(\Rightarrow\), and \(\land\) and \(\overline{0}\) as \(\min\) and \(0\), respectively.