On expansions of WNM t-norm based logics with truth-constants.

*(English)*Zbl 1193.03049The paper deals with expansions of Weak Nilpotent Minimum propositional logics (WNM, for short) by a set of truth-constants. A WNM t-norm \(*\) has the finite partition property if its associated negation \(n_* : [0,1] \to [0,1]\), \(n_*(x) = x \to_* 0\) (where \(\to_*\) is the residuum of \(*\)) is constant over a finite number of subintervals of \([0,1]\). Let \(L_*\) denote the logic whose Blok-Pigozzi equivalent algebraic semantics is the variety generated by the standard algebra \([0,1]_* = ([0,1],*,\to_*,0)\), and let \(C\) be any denumerable subalgebra of \([0,1]_*\).

The authors study the logics \(L_*(C)\) obtained by extending the language of \(L_*\) by a countable set of truth-constants \(\{\bar{r} \mid r \in C\}\), and by adding suitable book-keeping axioms reflecting the operations in the subalgebra \(C\). A standard \(L_*(C)\)-algebra is then defined as a standard \(L_*\)-algebra equipped with a countable set of truth-constants \(\{\bar{r} \mid r \in C\}\) such that each \(\bar{r}\) is interpreted in \(r\) itself.

The authors prove a set of completeness results for logics \(L_*(C)\) where 1) \(*\) has the finite partition property and 2) \(C\) is such that each interval in the partition of \(n_*\) contains at least one element of \(C\) in its interior. It is first shown that for any countable subalgebra \(C\) of \([0,1]_*\) and any proper filter \(F\) of \(C\), there exists a standard \(L_*(C)\)-algebra \(A\) having the property that \(F\) coincides with the set of elements \(r \in C\) such that the interpretation in \(A\) of the constant \(\bar{r}\) is the top of \(A\). Such an algebra \(A\) is called a standard \(L_*(C)\)-chain of type \(F\).

The authors prove that for every WNM t-norm \(*\) and every truth-constant set \(C\) enjoying properties 1) and 2) above, the logic \(L_*(C)\) enjoys strong standard completeness with respect to the family of standard \(L_*(C)\)-chains of type \(F\), for \(F\) ranging over the proper filters of \(C\). They then prove that the WNM logics with finite partition property enjoying canonical standard completeness (i.e., completeness w.r.t. a single standard algebra) are exactly those ones whose negation on the set of positive elements (i.e., strictly greater that their negation) is either involutive and continuous, or is constantly zero.

The paper considers expansions of \(L_*(C)\) by the \(\Delta\) projection operator, proving that, for all WNM t-norms \(*\) with finite partition property and all countable \(C\), the obtained systems enjoy strong standard completeness but they are not conservative expansions of \(L_*(C)\) (while \(L_*(C)\) is a conservative expansion of \(L_*\)). The paper ends with some complexity results on the satifiability, tautologousness, and logical consequence problems for these logics.

The authors study the logics \(L_*(C)\) obtained by extending the language of \(L_*\) by a countable set of truth-constants \(\{\bar{r} \mid r \in C\}\), and by adding suitable book-keeping axioms reflecting the operations in the subalgebra \(C\). A standard \(L_*(C)\)-algebra is then defined as a standard \(L_*\)-algebra equipped with a countable set of truth-constants \(\{\bar{r} \mid r \in C\}\) such that each \(\bar{r}\) is interpreted in \(r\) itself.

The authors prove a set of completeness results for logics \(L_*(C)\) where 1) \(*\) has the finite partition property and 2) \(C\) is such that each interval in the partition of \(n_*\) contains at least one element of \(C\) in its interior. It is first shown that for any countable subalgebra \(C\) of \([0,1]_*\) and any proper filter \(F\) of \(C\), there exists a standard \(L_*(C)\)-algebra \(A\) having the property that \(F\) coincides with the set of elements \(r \in C\) such that the interpretation in \(A\) of the constant \(\bar{r}\) is the top of \(A\). Such an algebra \(A\) is called a standard \(L_*(C)\)-chain of type \(F\).

The authors prove that for every WNM t-norm \(*\) and every truth-constant set \(C\) enjoying properties 1) and 2) above, the logic \(L_*(C)\) enjoys strong standard completeness with respect to the family of standard \(L_*(C)\)-chains of type \(F\), for \(F\) ranging over the proper filters of \(C\). They then prove that the WNM logics with finite partition property enjoying canonical standard completeness (i.e., completeness w.r.t. a single standard algebra) are exactly those ones whose negation on the set of positive elements (i.e., strictly greater that their negation) is either involutive and continuous, or is constantly zero.

The paper considers expansions of \(L_*(C)\) by the \(\Delta\) projection operator, proving that, for all WNM t-norms \(*\) with finite partition property and all countable \(C\), the obtained systems enjoy strong standard completeness but they are not conservative expansions of \(L_*(C)\) (while \(L_*(C)\) is a conservative expansion of \(L_*\)). The paper ends with some complexity results on the satifiability, tautologousness, and logical consequence problems for these logics.

Reviewer: Stefano Aguzzoli (Milano)

##### MSC:

03B52 | Fuzzy logic; logic of vagueness |

03B50 | Many-valued logic |

03G25 | Other algebras related to logic |

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

##### Keywords:

monoidal t-norm based logic (MTL); nilpotent minimum logic (NM); weak nilpotent minimum logics (WNM); completeness results
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\textit{F. Esteva} et al., Fuzzy Sets Syst. 161, No. 3, 347--368 (2010; Zbl 1193.03049)

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