• Korean Journal of Logic
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• v.12 no.2
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• pp.115-139
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• 2009

This paper investigates an extension of the uninorm (based) logic UL, which is obtained by adding (t-weakening, $W_t$) (($\phi$ & $\psi$) ${\wedge}$ t) $\rightarrow$ $\phi$ to UL introduced by Metcalfe and Montagna in [8]. First, the t-weakening uninorm logic $UL_{Wt}$ (the UL with $W_t$) is introduced. The algebraic structures corresponding to $UL_{Wt}$ is then defined, and its algebraic completeness is established. Next standard completeness (i.e. completeness on the real unit interval [0, 1]) is established for this logic by using Jenei and Montagna-style approach for proving standard completeness in [3, 6].

• Korean Journal of Logic
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• v.12 no.1
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• pp.89-110
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• 2009

This paper investigates generalizations of weakening-free uninorm logics not assuming associativity of intensional conjunction (so called fusion) &, as non-associative fuzzy-relevance logics. First, the strong t-associative monoidal uninorm logic StAMUL and its schematic extensions are introduced as non-associative propositional fuzzy-relevance logics. (Non-associativity here means that, differently from classical logic, & is no longer associative.) Then the algebraic structures corresponding to the systems are defined, and algebraic completeness results for them are provided. Next, predicate calculi corresponding to the propositional systems introduced here are considered.

• Korean Journal of Logic
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• v.19 no.3
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• pp.437-461
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• 2016

This paper investigates weakening-free fuzzy logics with three weak forms of associativity (of multiplicative conjunction &). First, the wta-uninorm (based) logic $WA_tMUL$ and its two axiomatic extensions are introduced as weakening-free weakly associative fuzzy logics. The algebraic structures corresponding to the systems are then defined, and algebraic completeness results for them are provided. Next, standard completeness is established for $WA_tMUL$ and the two axiomatic extensions with an additional axiom using construction in the style of Jenei-Montagna.

• 한국논리학회:학술대회논문집
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• 2009.05a
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• pp.38-66
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• 2009

Weakening-free non-associative fuzzy logics, which are based on mica-norms, are introduced as non-associative substructural logics extending $GL_{e\bot}$ (Non-associative Full Lambek calculus with exchange and constants T, F) introduced by Galatos and Ono (cf. see [10, 11]). First, the mica-norm logic MICAL, which is intended to cope with the tautologies of left-continuous conjunctive mica-norms and their residua, and several axiomatic extensions of it are introduced as weakening-free non-associative fuzzy logics. The algebraic structures corresponding to the systems are then defined, and algebraic completeness results for them are provided. Next, standard completeness (i,e. completeness with respect to algebras whose lattice reduct is the real unit interval [0, 1]) is established for these logics by using Jenei and Montagna-style approach for proving standard completeness in [7, 18].

• Korean Journal of Logic
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• v.16 no.3
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• pp.437-452
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• 2013

This paper verifies the following two: First, I verify the standard completeness proof for the system $ULw_t$ is not correct in the sense that t-weakening uninorms are t-norms, but not weakening-free uninorms. Second, I verify that the proof for $ULw_t$ can be used for the system MTL. That is, I provide a new standard completeness proof for it.