Abstract
This chapter offers a review of Petr Hájek’s contributions to first-order axiomatic theories in fuzzy logic (in particular, ZF-style fuzzy set theories, arithmetic with a fuzzy truth predicate, and fuzzy set theory with unrestricted comprehension schema). Generalizations of Hájek’s results in these areas to MTL as the background logic are presented and discussed.
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Notes
- 1.
At the same time, we add the axiom \(y\notin \emptyset \) to the theory; see Theorem 4.3. Henceforth, whenever we add new constants and function symbols, we also add the corresponding axioms implicitly.
- 2.
In fact, in any logic that proves the schema \((\varphi \rightarrow \varphi ^2)\rightarrow (\varphi \vee \lnot \varphi )\); cf. Remark 4.1.
- 3.
Note the semantics of the existential quantifier: mere validity of a formula \((\exists {x})\varphi (x)\) in a model M does not guarantee that there is an object \(m\) for which \(\Vert \varphi (m)\Vert _\mathbf{M}=1\).
- 4.
One can also take \(\le \) to be a defined symbol, relying on axiom (Q8).
- 5.
An analogous statement can be formed for weaker theories, including \(\mathrm {Q}\).
- 6.
In fact, Restall does not prove the crispness axiom in \(\mathrm {PA}\L \) but rather verifies it as a semantic consequence of the theory \(\mathrm {PA}\L \) in the standard MV-algebra; note that this is a weaker statement since \({\mathrm {\L }\forall }\) is not complete w.r.t. the standard MV-algebra. Still, each of the steps can be reconstructed syntactically in \(\mathrm {PA}\L \).
- 7.
In fact, Hájek et al. (2000) proved a stronger statement, for a variant of \(\mathrm {PA}\L \mathrm {Tr}\) allowing the predicate symbol \(\mathrm {Tr}\) to occur in formulae the induction rule is applied to.
- 8.
The consistency status of naïve comprehension over these logics is not known, either. Still, being weaker, they have better odds of consistency even if naïve comprehension turns out to be inconsistent over Łukasiewicz logic.
- 9.
Hájek (2005 and subsequent papers) denoted the theory by \(\mathrm {C{\L }}_0\), whereas by \(\mathrm {C{\L }}\) he denoted an inconsistent extension of \(\mathrm {C{\L }}_0\). In this paper we shall use a systematic symbol \(\mathrm {C}_L\) for naïve set theory over the logic \(L\). The corresponding theory over standard \([0,1]_{{\L }}\)-valued Łukasiewicz logic is called \(\mathrm {H}\) by White (1979) and Yatabe (2007).
- 10.
See Theorems 4.2–4.3 for the conservativeness of these (and subsequent similar) definitions in \(\mathrm {C_{{\L }}}\). The symbol \(\oplus \) denotes the ‘strong’ disjunction of Łukasiewicz logic, defined in \({\L }\) as \( \varphi \oplus \psi \equiv _{\mathrm {df}}{\lnot }({\lnot }\varphi \mathbin { \& }{\lnot }\psi )\).
- 11.
The schematic translation of propositional tautologies into theorems of elementary fuzzy set theory Běhounek and Cintula (2005) only relies on certain distributions laws for quantifiers, and so works for \(\mathrm {C_{{\L }}}\) (as well as \(\mathrm {C_{\mathrm {MTL}}}\) introduced in Sect. 4.5.3). The converse direction (disproving theorems not supported by propositional tautologies), however, cannot be demonstrated as in elementary fuzzy set theory (namely, by constructing a model from the counterexample propositional evaluation), since no method of constructing models of \(\mathrm {C_{{\L }}}\) or \(\mathrm {C_{\mathrm {MTL}}}\) is known. In fact, it is well possible (esp. for \(\mathrm {C_{\mathrm {MTL}}}\)) that the comprehension schema does strengthen the logic of the theory (as it does exclude some algebras of semantic truth values, see comments following Theorem 4.21 and preceding Corollary 4.4 in Sect. 4.5.3).
- 12.
In order to become a full-fledged theory of fuzzy sets, some kind of (preferably, conservative) extension of naïve fuzzy set theories would be needed (cf. Běhounek 2010; (Hájek 2013b, Sect. 3)). Such extensions, however, make the comprehension axioms restricted to the formulae in the original language, and so lose the intuitive appeal of the unrestricted comprehension schema. Cf. Remark 4.4 below.
- 13.
- 14.
In fact, as proved by Hájek (2013a), if \(\mathrm {C_{{\L }}}\vdash t\notin t\) for a term \(t\), then there is a term \(t'\) such that \( \mathrm {C_{{\L }}}\vdash t\approx t'\mathrel { \& }t\ne t'\). Moreover, he also proved that if \(\mathrm {C_{{\L }}}\vdash (\forall {u})(u\approx t\mathrel {\rightarrow }u\notin t)\) for a term \(t\), then there are infinitely many terms \(t_i\) such that \(\mathrm {C_{{\L }}}\) proves \(t\approx t_i\) and \(t_i\ne t_j\), for each \(i,j\in \mathbb N\). (Thus, for instance, there are infinitely many Leibniz-different empty sets.) The above terms \(t',t_i\) are defined by the fixed-point theorem (i.e., Theorem 4.13).
- 15.
This is a corollary of the theorem given in footnote 14, as \(\omega \) satisfies its conditions.
- 16.
Recall that an element \(x\) of an \(\mathrm {MTL}\)-algebra is called n-contractive if \(x^{n-1}=x^n\). Equivalently, \(x\) is \(n\)-contractive if \(x^{n-1}\) is idempotent. An \(\mathrm {MTL}\)-algebra is called \(n\)-contractive if all its elements are \(n\)-contractive.
- 17.
Owing to the existence of a fixed point \(\rho _1\) of negation, naïve comprehension is furthermore inconsistent in logics with strict negation, i.e., in \(\mathrm {SMTL}\) and any of its extensions, which include \(\mathrm {\Pi MTL}\), \(\mathrm {SBL}\), \(\mathrm {\Pi }\), and \(\mathrm {G}\).
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Acknowledgments
The work was supported by grants No. P103/10/P234 (L. Běhounek) and P202/10/1826 (Z. Haniková) of the Czech Science Foundation.
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Běhounek, L., Haniková, Z. (2015). Set Theory and Arithmetic in Fuzzy 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_4
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