Abstract
In this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard \(\mathsf {BL}\)-algebras, we focus on a uniform approach to the following problem: given a left-continuous t-norm \(*\), find an axiomatic system (possibly with infinitary rules) which is strongly complete with respect to the standard algebra 
This system will be an expansion of Monoidal t-norm-based logic. First, we introduce an infinitary axiomatic system \(\mathsf {L}_*^\infty \), expanding the language with \(\Delta \) and countably many truth constants, and with only one infinitary inference rule, that is inspired in Takeuti–Titani density rule. Then we show that \(\mathsf {L}_*^\infty \) is indeed strongly complete with respect to the standard algebra 
. Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0, 1] satisfy some regularity conditions.
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As it is usual in the literature, the symbol \(\Delta \) denotes both the connective in the language and the corresponding operation in the algebra, as it happens with \(\wedge \) and \(\vee \).
The \(\Delta \) connective is used to prove semilinearity of the axiomatic system.
This is simply a subset, closed under substitutions, of \(\{ \langle \Gamma , \varphi \rangle : \Gamma \cup \{\varphi \} \subseteq Fm \}\). Notice that we allow rules with infinite premises \(\Gamma \).
Let us stress that we do not consider the smallest consequence operator closed under substitutions.
Let us point out that indeed this condition implies that for every natural number \(m \leqslant n\), it holds that either
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\(\xi _m \in \Sigma _{n+1}\), or
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\(\lnot \Delta \xi _m \in \Sigma _{n+1}\) and there is some \(\psi \in \Phi _m\) such that \(\lnot \Delta \psi \in \Sigma _{n+1}\).
Notice that this last disjunction is capturing the intuition that \(\Sigma _{n+1}\) (and all its extensions) is a model of all the derivations in \(\langle (\Phi _{m}, \xi _{m}): m \leqslant n \rangle \).
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This finiteness is employed to show that for each rule in the axiomatic system, there is at most a countable number of substitutions.
If for all two constants like above \(d_1 *d_2 \leqslant c\), applying residuation \(d_1 \leqslant d_2 \rightarrow c\) and so the supremum can be taken in the left side. Similarly, we get that \(\sup \mathcal {C}^-_a *\sup \mathcal {C}^-_b \leqslant c\) which contradicts the assumptions.
This can be also seen as a direct consequence of the fact that all \(L_*^\infty \)-chains are relatively simple.
For the reader interested in checking the details we suggest to start considering the following three elements in \(\varvec{A}\):
$$\begin{aligned} t_1:=\left( \frac{1}{2}\right) _{q \in [0,1)_\mathbb {Q}} \quad t_2:=(\widetilde{q})_{q \in [0,1)_\mathbb {Q}} \quad t_3:=(q)_{q \in [0,1)_\mathbb {Q}} \end{aligned}$$and checking that all possible combinations of these three elements under \(\wedge , \vee , \rightarrow , \Delta \) are also elements in our universe A. Indeed, all difficulties to provide a general proof that A is closed under the operations are illustrated in the previous particular case.
With the notation \(x \leqslant r \uparrow \) we mean that \(x \leqslant \overline{c}^{\varvec{A}}\) for every \(c \in (r,1]_\mathbb {Q}\). In an analogous way, \(r \downarrow \leqslant x\) stands for \(\overline{c}^{\varvec{A}} \leqslant x\) for all \(c \in [0,r)_\mathbb {Q}\).
As expected, \(\Upsilon _i^f(\vec {\varphi })\) stands for \(\{\lambda (\vec {\varphi })\}_{\lambda \in \Upsilon _i}\), and \(\vec {e(\varphi )} = \langle e(\varphi _1), \ldots e(\varphi _n) \rangle \).
Observe that if component k of region \(R_i\) has only one point, these premises are never met. Thus, the rule associated to that pair of region and component is equivalent to the rule \(\overline{0} \vdash \vartheta \), and so, it will not be used in order to “compute” the value of the function in that point (this will be done with a different component or a book-keeping axiom, depending on the case).
As already mentioned before concerning operations with regions with smaller dimensions, at this point of the proof, if f was an operation such that \(\{d \in A: (\Upsilon _i^f)^{\varvec{A}}(\langle \vec {b}, d \rangle _k \in \mathrm{{Int}}_f(R_i) \text { for some } \vec {b} \in A^{n}\} = \{\overline{c_0}^{\varvec{A}}\}\) for some \(c_0 \in \mathbb {Q}_*^{\mathrm{{OP}}}\), the proof would finish here. Indeed, \(c_0\) in \(C_1\) by definition, and so trivially \( \overline{f}^{\varvec{A}}(\overline{c_0}^{\varvec{A}}, a_2) \leqslant \sup \{\overline{f}^{\varvec{A}}(\overline{c}^{\varvec{A}}, a_2): c \in [0,b]_\mathbb {Q}, \overline{c}^{\varvec{A}} \leqslant a_1\} \).
This set is non-empty: \(x_1 > 0\) because \((\Upsilon _1^g)^{\varvec{A}}(x_1, x_2) = \{\overline{1}^{\varvec{A}}\}\) and the constants are dense in A.
Nevertheless, the case of the left-continuous t-norm operation \(*\) has a more direct proof, not needing any of the
, \(\mathrm {\vee M^{f U}_{\langle i,k\rangle }}\) nor 
rules, relying on the MTL-axiomatization of a residuated operation, as we saw in Lemma 4.7.
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Acknowledgments
The authors are thankful to an anonymous reviewer for his/her comments that have helped to improve the final layout of this paper. Vidal has been supported by the joint project of Austrian Science Fund (FWF) I1897-N25 and Czech Science Foundation (GACR) 15-34650L and by the institutional support RVO:67985807. Esteva and Godo have been funded by the FEDER/MINECO Spanish Project TIN2015-71799-C2-1-P and by the Grant 2014SGR-118 from the Catalan Government. Bou thanks the Grant 2014SGR-788 from the Catalan Government.
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Communicated by A. Di Nola, D. Mundici, C. Toffalori, A. Ursini.
This paper is dedicated to the memory of Professor Franco Montagna, an excellent researcher and better person to whom these authors will always be indebted to. He has been a pioneer in many research areas (not only in MFL), and among such pioneering contributions we can find his work on infinitary axiomatic systems for standard $$\mathsf {BL}$$ BL -algebras. The present paper has been inspired by the novel ideas introduced by Franco on that topic.
This is an expanded and fully revised version of the conference paper (Vidal et al. 2015). In particular some proofs have become more cumbersome than were presumed there.
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Vidal, A., Bou, F., Esteva, F. et al. On strong standard completeness in some MTL\(_\Delta \) expansions. Soft Comput 21, 125–147 (2017). https://doi.org/10.1007/s00500-016-2338-0
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DOI: https://doi.org/10.1007/s00500-016-2338-0