## Abstract

I argue that the definition of a logic by preservation of all degrees of truth is a better rendering of Bolzano’s idea of consequence as truth-preserving when “truth comes in degrees”, as is often said in many-valued contexts, than the usual scheme that preserves only one truth value. I review some results recently obtained in the investigation of this proposal by applying techniques of abstract algebraic logic in the framework of Łukasiewicz logics and in the broader framework of substructural logics, that is, logics defined by varieties of (commutative and integral) residuated lattices. I also review some scattered, early results, which have appeared since the 1970’s, and make some proposals for further research. 2010 *Mathematics Subject Classification*: 03B50, 03G27, 03B47, 03B22

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## Notes

- 1.
In this chapter I will represent logics as consequences by the symbol \(\vdash \), independently of the way they are defined, be it of semantical or syntactical origin, and will add sub- or superscripts when needed. The symbol \(\vDash \) will only be used for satisfaction of equations in (classes of) algebras.

- 2.
In whatever mechanism; one need not assume truth-functionality for this discussion to make sense.

- 3.
- 4.
That the truth degrees can be compared (i.e., ordered) seems to be another essential ingredient motivating fuzzy logic: “

*We shall understand [fuzzy logic in the narrow sense] as a logic with a comparative notion of truth*” ((Hájek 1998, p. 2)). - 5.
As we now know, coincidence of this way of expressing semantical consequence with the truth-preserving one also holds in other, non-classical cases, see Theorems 6.1, 6.2, 6.3 and 6.5.

- 6.
Pavelka develops his proposal for \(\mathbf {L}\)-valued fuzzy sets, where \(\mathbf {L}\) is an arbitrary complete residuated lattice, but proves his completeness result for the cases where \(\mathbf {L}\) is \(\mathbf{[0, 1]}\) and all its finite subalgebras.

- 7.
- 8.
Among the main results, he proved that the two consequences coincide for the finite subalgebras but not for the denumerable one or for the whole interval, in which cases the truth-preserving consequences are not finitary. However, they do coincide on finite sets of assumptions; thus, if one considers only the associated finitary consequences, then the two fully coincide.

- 9.
For \(\xi \leqslant \aleph _0\), \(\xi \) is the cardinality of the subalgebra of \(\mathbf{[0, 1]}\) taken as the model; in the \(\aleph _0\) case, it is the rational subalgebra. \(\aleph _1\) is used to refer to the whole algebra \(\mathbf{[0, 1]}\).

- 10.
Later results, see Theorem 6.4 and the comments before Theorem 6.5, will make it clear that the logics presented in Gil et al. (1993) also coincide with \(\vdash ^{\!{\leqslant }}_n\), but this was not explicit at the time of its publication.

- 11.
In Wójcicki (1973) only the “if” part is proved, and only for \(\vdash _{\!\mathbf {S}}\).

- 12.
As a matter of fact, finitarity is part of the definition of a logic in most studies outside abstract algebraic logic, and also in some inside it.

- 13.
- 14.
In accordance with most of the literature starting with Ward and Dilworth (1939), here

*residuated lattices*are algebras of similarity type \((\wedge ,\vee ,\star ,\rightarrow ,1,0)\) such that \(\wedge \,,\vee \) are lattice operations, \(\star \) is a commutative monoidal operation (usually called “fusion”, “intensional conjunction” or “multiplicative conjunction”) with unit \(1\) also being the maximum of the lattice, and \(\rightarrow \) is its residuum. A constant \(0\) is included in the type but in the general case there is no need to postulate anything about it; so these residuated lattices coincide with the FL\(_{ei}\)-algebras of Galatos et al. (2007), where the term “residuated lattice” denotes in turn a much larger class. The smaller class of FL\(_{ew}\)-algebras is found when postulating that \(0\) is the minimum of the order, and includes the algebras of most well-known substructural logics such as MTL, BL, Ł\(_{\infty }\), \(\text{ G }\), \(\Pi \), etc. - 15.
Observe that, in a lattice, an order relation \(\alpha \mathbin {\preccurlyeq }\beta \) holds if and only if the equation \(\alpha \wedge \beta \mathbin {\approx }\alpha \) holds; thus, using \(\mathbin {\preccurlyeq }\) is just a more intuitive way of writing identities of that particular form.

- 16.
- 17.
Generalized Heyting algebras can be described informally as “Heyting algebras without minimum”; a residuated lattice is a generalized Heyting algebra if and only if the fusion operation \(\star \) coincides with the lattice conjunction \(\wedge \).

- 18.
For a finitary logic this set can be taken finite; if moreover the logic has a conjunction, then the set can be reduced to a single formula.

- 19.
A residuated lattice is \(n\)

*-contractive*, also called “\(n\)-potent” in the literature, when it satisfies the equation \(x^n\mathbin {\approx }x^{n+1}\). The associated logics are also called “\(n\)-contractive”. - 20.
In principle the general theorem concerns an arbitrary set of formulas acting collectively as an implication, but since in the present case all logics are finitary and have a conjunction, one can directly speak about a single formula.

- 21.
The symbol \(\mathrel {\,\vartriangleright \,}\) is here just a neutral replacement for other symbols like \(\vdash \) or \(\Rightarrow \), which might lead to misunderstanding if used to describe sequents or rules in the present context.

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## Acknowledgments

I would like to thank Félix Bou, Norbert Preining and two anonymous referees for some bibliographical references, and the latter also for a number of useful observations on the first version of the chapter. While writing it I was partially funded by the research project MTM2011-25747 from the government of Spain, which includes feder funds from the European Union, and the research grant 2009SGR-1433 from the government of Catalonia.

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Font, J.M. (2015). Consequence and Degrees of Truth in Many-Valued Logic. In: Montagna, F. (eds) Petr Hájek on Mathematical Fuzzy Logic. Outstanding Contributions to Logic, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-06233-4_6

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