 The first-order logic MTL\(\forall\) is then defined as the consequence relation given by the preservation of total truth (value \(1\)) in \(\ast\)-structures (allowing \(\ast\) to run over all left-continuous t-norms). More precisely, we say that a first-order formula \(\varphi\) is a consequence of a set of formulas \(\Gamma\) (in symbols: \(\Gamma \models_{\mathrm{MTL}\forall} \varphi\)) if, for each left-continuous t-norm \(\ast\) and each \(\ast\)-structure \({\mathbf M}\), we have \(\|\varphi\|^{\mathbf M}_{\mathrm v} = 1\) for each valuation v, whenever \(\|\psi\|^{\mathbf M}_{\mathrm v} = 1\) for each valuation v and each \(\psi \in \Gamma\).