The Differential Semantics of Łukasiewicz Syntactic Consequence

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

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

$$ W=\{U_\fancyscript{H}\mid \fancyscript{H}\subseteq \fancyscript{X}, \,\,\fancyscript{H} \, \mathrm{finite}\} $$

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

$$\begin{aligned} \mathfrak p_W=\bigcup \left\{ \mathfrak p_{U_\fancyscript{I}} \mid \fancyscript{I}\subseteq \fancyscript{X}, \,\,\fancyscript{I} \, \text {finite}\right\} . \end{aligned}$$

(7.5)

Conversely, suppose $\mathfrak p$ is a prime ideal of ${{\mathrm{\fancyscript{M}}}}([0, 1]^{\kappa }) ={{\mathrm{\fancyscript{M}}}}([0, 1]^{\fancyscript{X}})$. Then

$$ \mathfrak p = \bigcup \left\{ \mathfrak p\cap {{\mathrm{\fancyscript{M}}}}([0, 1]^{\fancyscript{H}}) \mid \fancyscript{H} \, \text {a finite subset of} \, \fancyscript{X}\right\} . $$

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

$$ \mathfrak p_{U_{\fancyscript{I}\cup \fancyscript{J}}} \cap {{\mathrm{\fancyscript{M}}}}([0, 1]^\fancyscript{I}) = \mathfrak p_{U_\fancyscript{I}} \,\,\text { and }\,\, \mathfrak p_{U_{\fancyscript{I}\cup \fancyscript{J}}} \cap {{\mathrm{\fancyscript{M}}}}([0, 1]^\fancyscript{J}) = \mathfrak p_{U_\fancyscript{J}}. $$

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. $\square$