# Errata in the Handbook of Mathematical Fuzzy Logic

## Volume 1

### Chapter I

• Page 22, Definition 1.3.1, item 2: the condition of monotony of $\displaystyle \&$ with respect to the lattice order is missing
• Page 52, the axiom (S$\displaystyle _n$ ) should read: $\displaystyle \neg(\varphi^{n-1})\vee\varphi$
• Page 55, Theorem 2.4.2: the condition should be $\displaystyle m - 1$ divides $\displaystyle n - 1$ .
• Page 92, lines 15–16: the claim of the consistency of naive comprehension over IMTL was in fact a conjecture (which is still unresolved by 2014)

### Chapter II

• Page 115, Proposition 2.2.11: "Let L be a weakly implicative logic L" should be "Let L be a weakly implicative logic in language $\displaystyle \cal{L}$ ".
• Page 116, first line after Definition 2.3.1: "$\displaystyle \langle \bf{A},F\rangle\in \mathbf{MOD}(L)$ " should be "an $\displaystyle \cal{L}$ -algebra $\displaystyle {\bf \it A}$ ".
• Page 117, Definition 2.3.5: In the first item there should be 'coarsest' instead of 'finest'.
• Page 122, Proof of Proposition 2.4.5, last line in the page: the sentence should read "... surjective and it can be easily shown to preserve meets".
• Page 124, Proof of Proposition 2.4.7: The last sentence of the proof "The fact that the set..." should be placed at the beginning of the proof.
• Page 129, proof of (PSL12): in the first line of the proof there is a wrong bracket, the formula shoulb be $\displaystyle \chi \& ((\chi \to \varphi) \land (\chi \to \psi)) \to \varphi$ . Analogously in the next line.
• Page 130, Theorem 2.5.7: For SLe, the second and the third formula should have $\displaystyle \psi$ instead of $\displaystyle \chi$ .
• Page 144, Definition 2.7.6: It should be "lattice-disjunctive" with hyphen.
• Page 145, Example 2.7.11: In the sixth line, L in the subindex should be IPC.
• Page 162, Corollary 3.2.5: L_1 should be assumed to be finitary.
• Page 166, Footnote 15: It should be "Then we construct $\displaystyle K_{i+1}$ ".
• Page 168, Theorem 3.3.8: It should be "Let L be a finitary semilinear finitely disjunctional logic".
• Page 169, Theorem 3.3.13: It should be "Let L be a finitary semilinear finitely disjunctional logic".
• Page 173, line 20: It should be "$\displaystyle a \rightarrow^\mathbf{A} b\in B'$ ".
• Page 173, line 29: It should be "By Theorem 2.7.18".
• Page 175, proof of Proposition 3.4.16: In 2 implies 3 it should be "If all the chains".
• Page 177, Convention 4.0.1: In the second item Nabla must be assumed to be a finite protodisjunction, without parameters.
• Page 181, Example 4.1.18: The example is wrong (the lattice is not distributive). A correct one can be found in page 404 of
A. Horn. Logic with truth values in a linearly ordered Heyting algebras. Journal of Symbolic Logic, 34(3):395–408, 1969.
• Page 197, line 5: $\displaystyle c$ should be $\displaystyle c_\mu$ .
• Page 198, Theorem 4.5.7: For the implication from 2 to 1 one should add the extra hypothesis that the class of L-chains admits regular completions, i.e. every L-chain can be embedded into a completely ordered L-chain preserving all existing suprema and infima. This ensures that the model obtained in the proof can be taken safe (if it was not, one would embed it into another over a completely ordered chain).
• Page 198, proof of Theorem 4.5.7: All occurrences of CM$\displaystyle '_T$ should be CM$\displaystyle _{T'}$ .

## Volume 2

### Chapter XI

• Page 899: Fact 5.1.1 points to a wrong reference. The correct reference is Lemma 3 of Petr Hájek, Making fuzzy description logic more general, Fuzzy Sets and Systems 154(1), 1-15, 2005.