 The propositional logic MTL can be given a first-order counterpart MTL\(\forall\) in a predicate language \(\mathcal{P\!L}\) (defined as in the classical case) in the following way. The semantics is given by structures in which predicate symbols are interpreted as functions mapping tuples of domain elements into truth-values. More precisely, given a left-continuous t-norm \(\ast\), a \(\ast\)-structure \({\mathbf M}\) consists of a non-empty domain of elements \(M\), a function \(f_{\mathbf M}\colon M^n\to M\) for each \(n\)-ary function symbol \(f\in \mathcal{P\!L}\), and a function \(P_{\mathbf M}\colon M^n\to [0,1]\) for each \(n\)-ary predicate symbol \(P\in \mathcal{P\!L}\). Fixing a valuation \({\mathrm v}\) of object variables in \(M\), one defines values of terms (\(\|f(t_1,\dots,t_n)\|^{\mathbf M}_{\mathrm v} = f_{\mathbf M}(\|t_1\|^{\mathbf M}_{\mathrm v},\dots,\|t_n\|^{\mathbf M}_{\mathrm v})\)) and truth-values of atomic formulas (\(\|P(t_1,\dots,t_n)\|^{\mathbf M}_{\mathrm v} = P_{\mathbf M}(\|t_1\|^{\mathbf M}_{\mathrm v},\dots,\|t_n\|^{\mathbf M}_{\mathrm v})\)). Truth-values of a universally/existentially quantified formula are computed as infimum/supremum of truth-values of instances of the formula where the quantified variable runs over all elements of the domain \(M\). Formally:   \[ \begin{align} \|\forall x\, \varphi\|^{\mathbf M}_{\mathrm v} &amp; = \inf\{\|\varphi\|^{\mathbf M}_{{\mathrm v}[x{:}a]} \mid a\in M\}\\   \|\exists x\,\varphi\|^{\mathbf M}_{\mathrm v} &amp; = \sup\{\|\varphi\|^{\mathbf M}_{{\mathrm v}[x{:}a]}\mid a\in M\},\\   \end{align} \]   where \({\mathrm v}[x{:}a]\) is the valuation sending \(x\) to \(a\) and keeping values of other variables unchanged. The values of other formulas are computed by interpreting, as in propositional semantics, \(\&amp;\) as \(\ast\), the implication \(\to\) as its residuum \(\Rightarrow_\ast\), and \(\land\) and \(\overline{0}\) as \(\min\) and \(0\), respectively.