Abstract
The classical condition “c is a semantic consequence of a set P of premises” in infinite-valued propositional Łukasiewicz logic Ł\(_{\infty }\) is refined using enriched valuations that take into account the effect on the formula c of the stability of the truth-value of each formula p in P under small perturbations (or, measurement errors) of the models of P. The differential properties of the functions represented by c and by all p in P naturally lead to a new notion of semantic consequence that turns out to coincide with syntactic consequence.
Dedicated to Petr Hájek
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Acknowledgments
The author is grateful to Manuela Busaniche and two anonymous referees for their valuable criticism and suggestions. The author’s major debt of gratitude is due to Petr Hájek for his outstanding scientific work, his inspiring example, and his friendship.
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7.7 Appendix: Stable Consequence for Arbitrary Sets of Sentences
7.7 Appendix: Stable Consequence for Arbitrary Sets of Sentences
Up to isomorphism, MV-algebras are the Lindenbaum algebras of sets of formulas on unlimited supplies of variables. So let \(\fancyscript{X}=\{X_1,X_2,\dots ,X_\alpha ,\ldots \mid \alpha <\kappa \}\) be a set of variables of infinite, possibly uncountable cardinality \(\kappa \), indexed by all ordinals \(0\le \alpha <\kappa .\) Letting \(\mathsf {FORM}_\fancyscript{X}\) be the set of formulas \(\psi (X_{\alpha _1},\dots ,X_{\alpha _t})\) whose variables are contained in \(\fancyscript{X}\), in 7.7.5 below we will routinely extend Definition 7.3.7 to arbitrary subsets \(\Theta \) of \( \mathsf {FORM}_\fancyscript{X}\) and formulas \(\psi \in \mathsf {FORM}_\fancyscript{X}\).
Throughout, we will tacitly identify the valuation space \([0, 1]^\fancyscript{X}\) with the Tychonov cube \([0, 1]^\kappa \) equipped with the product topology.
7.7.1 The free MV-algebra over \(\kappa \) free generators is the MV-algebra \({{\mathrm{\fancyscript{M}}}}([0, 1]^\kappa )\) of all functions on \([0, 1]^\kappa \) obtainable from the coordinate functions \(\pi _\beta (x)=x_\beta ,\,\,\,(x\in [0, 1]^\kappa ,\,\,\, 0\le \beta <\kappa )\) by pointwise application of the operations \(\lnot \) and \(\oplus \), (Cignoli et al. (2000), Theorem 9.1.5).
7.7.2 For any finite set \(\fancyscript{K}=\{X_{\alpha _1},\dots , X_{\alpha _d}\} \subseteq \fancyscript{X}\) we further identify \([0, 1]^{\fancyscript{K}}\) with the set of all \(x\in [0, 1]^\fancyscript{X}\) such that every coordinate \(x_\beta \) of \(x\) vanishes, with the possible exception of \(\beta \in \{\alpha _1,\dots ,\alpha _d\}.\) If \(\,\,\emptyset \not = \fancyscript{H} \subseteq \fancyscript{K} \subseteq \fancyscript{X},\,\,\) the MV-algebra \({{\mathrm{\fancyscript{M}}}}([0, 1]^\fancyscript{H})\) is canonically identified with an MV-subalgebra of \({{\mathrm{\fancyscript{M}}}}([0, 1]^\fancyscript{K})\) via cylindrification.
7.7.3 Suppose we are given two finite subsets \(\fancyscript{H} \subseteq \fancyscript{K}\) of \( \fancyscript{X}\) and two differential valuations \(U=(u_0,u_1,\dots ,u_d)\) in \(\mathbb R^{\fancyscript{H}}\) and \(V=(v_0,v_1,\dots ,v_e)\in \mathbb R^{\fancyscript{K}}\). We then have two prime ideals \(\mathfrak p_U\) of \({{\mathrm{\fancyscript{M}}}}([0, 1]^\fancyscript{H})\) and \(\mathfrak p_V\) of \({{\mathrm{\fancyscript{M}}}}([0, 1]^\fancyscript{K})\supseteq {{\mathrm{\fancyscript{M}}}}([0, 1]^\fancyscript{H}).\) We say that \(V\) dominates \(U\), in symbols, \(V\succeq U,\) if \(\mathfrak p_U=\mathfrak p_V\cap {{\mathrm{\fancyscript{M}}}}([0, 1]^\fancyscript{H})\). When this is the case, the coordinates of \(v_0\) corresponding to the variables of \(\fancyscript{H}\) agree with those of \(u_0.\) Further information on the relationship between \(U\) and \(V\) can be found in (Busaniche and Mundici (2007), Sect. 4).
The following definitions extend 7.3.3, 7.3.6 and 7.3.7 to any set \(\fancyscript{X}\) of variables of infinite cardinality \(\kappa \):
7.7.4 By a differential valuation in \(\mathbb R^\kappa \) we understand a \(\succeq \) -direct system
of differential valuations \(U_\fancyscript{H}\) in \(\mathbb R^\fancyscript{H}\), in the sense that for any finite \(\fancyscript{I}, \fancyscript{J}\subseteq \fancyscript{X},\,\,\,\) \(U_{\fancyscript{I}\cup \fancyscript{J}}\) dominates both \(U_{\fancyscript{I}}\) and \(U_{\fancyscript{J}}.\) We say that \(W\) satisfies a formula \(\varphi \in \mathsf {FORM}_\fancyscript{X}\) if \(U_{{{\mathrm{\mathrm var}}}(\varphi )}\) satisfies \(\varphi \) in the sense of 7.3.6, i.e., \(1-\hat{\varphi }\) belongs to the prime ideal \(\mathfrak p_{U_{{{\mathrm{\mathrm var}}}(\varphi )}}\). (As usual, \({{\mathrm{\mathrm var}}}(\varphi )\) denotes the set of variables occurring in \(\varphi \).)
7.7.5 For \(\Theta \subseteq \mathsf {FORM}_\fancyscript{X}\) and \(\psi \in \mathsf {FORM}_\fancyscript{X}\) we say that \(\psi \) is a stable consequence of \(\Theta \) and we write \(\Theta \models _\partial \psi ,\) if \(\psi \) is satisfied by every differential valuation \(W\) in \(\mathbb R^{\kappa }\) that satisfies each \(\theta \in \Theta \).
The “strong completeness” result 7.3.9 is now extended to arbitrary sets of variables:
7.7.6 Let \(\fancyscript{X}\not =\emptyset \) be an arbitrary (finite or infinite) set of variables, \(\Theta \subseteq \mathsf {FORM}_\fancyscript{X}\), and \(\psi \in \mathsf {FORM}_\fancyscript{X}.\) Then \(\Theta \models _\partial \psi \,\) iff \(\,\Theta \vdash \psi \).
Proof
In the light of 7.3.9 we have only to consider the case when \(\fancyscript{X}\) has infinite cardinality \(\kappa \).
By construction, every differential valuation \(W=\{U_\fancyscript{I} \mid \fancyscript{I}\subseteq \fancyscript{X}, \,\,\fancyscript{I} \, \text {finite}\}\) determines the prime ideal
Conversely, suppose \(\mathfrak p\) is a prime ideal of \({{\mathrm{\fancyscript{M}}}}([0, 1]^{\kappa }) ={{\mathrm{\fancyscript{M}}}}([0, 1]^{\fancyscript{X}})\). Then
By 7.3.4 (iv), for any finite subset \(\fancyscript{H}\) of \(\fancyscript{X}\) there is a differential valuation \(U_\fancyscript{H}\) in \(\mathbb R^\fancyscript{H}\) such that \(\mathfrak p\cap {{\mathrm{\fancyscript{M}}}}([0, 1]^{\fancyscript{H}})= \mathfrak p_{U_\fancyscript{H}}\). For any two finite subsets \(\fancyscript{I}, \fancyscript{J}\) of \(\fancyscript{X}\) we have
We then obtain a differential valuation \(W_\mathfrak p=\{U_\fancyscript{H}\mid \fancyscript{H}\subseteq \fancyscript{X}, \,\,\fancyscript{H} \, \text {finite}\}\) such that \( \mathfrak p = \mathfrak p_{W_\mathfrak p} \) (notation of 7.5).
Having thus shown that the map \(W\mapsto \mathfrak p_W\) sends the set of differential valuations in \(\mathbb R^\kappa \) onto the set of prime ideals of \({{\mathrm{\fancyscript{M}}}}([0, 1]^{\kappa })\), we get the desired result arguing as in the proof of 7.3.9. \(\square \)
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Mundici, D. (2015). The Differential Semantics of Łukasiewicz Syntactic Consequence. 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_7
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