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In mathematical logic, basic fuzzy logic (or shortly BL), the logic of the
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersectio ...
s, is one of the
t-norm fuzzy logics T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval , 1for the system of truth values and functions called t-norms for permissible interpretations of conjunctio ...
. It belongs to the broader class of substructural logics, or logics of residuated lattices;Ono (2003). it extends the logic MTL of all left-continuous t-norms.


Syntax


Language

The language of the propositional logic BL consists of countably many propositional variables and the following primitive logical connectives: * Implication \rightarrow ( binary) * Strong conjunction \otimes (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation \otimes follows the tradition of substructural logics. * Bottom \bot ( nullary — a propositional constant); 0 or \overline are common alternative signs and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL). The following are the most common defined logical connectives: * Weak conjunction \wedge (binary), also called lattice conjunction (as it is always realized by the lattice operation of
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in algebraic semantics). Unlike MTL and weaker substructural logics, weak conjunction is definable in BL as ::A \wedge B \equiv A \otimes (A \rightarrow B) * Negation \neg ( unary), defined as ::\neg A \equiv A \rightarrow \bot * Equivalence \leftrightarrow (binary), defined as ::A \leftrightarrow B \equiv (A \rightarrow B) \wedge (B \rightarrow A) : As in MTL, the definition is equivalent to (A \rightarrow B) \otimes (B \rightarrow A). * (Weak) disjunction \vee (binary), also called lattice disjunction (as it is always realized by the lattice operation of join in algebraic semantics), defined as ::A \vee B \equiv ((A \rightarrow B) \rightarrow B) \wedge ((B \rightarrow A) \rightarrow A) * Top \top (nullary), also called one and denoted by 1 or \overline (as the constants top and zero of substructural logics coincide in MTL), defined as ::\top \equiv \bot \rightarrow \bot Well-formed formulae of BL are defined as usual in propositional logics. In order to save parentheses, it is common to use the following order of precedence: * Unary connectives (bind most closely) * Binary connectives other than implication and equivalence * Implication and equivalence (bind most loosely)


Axioms

A Hilbert-style deduction system for BL has been introduced by Petr Hájek (1998). Its single derivation rule is
modus ponens In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
: :from A and A \rightarrow B derive B. The following are its axiom schemata: :\begin \colon & (A \rightarrow B) \rightarrow ((B \rightarrow C) \rightarrow (A \rightarrow C)) \\ \colon & A \otimes B \rightarrow A\\ \colon & A \otimes B \rightarrow B \otimes A\\ \colon & A \otimes (A \rightarrow B) \rightarrow B \otimes (B \rightarrow A)\\ \colon & (A \rightarrow (B \rightarrow C)) \rightarrow (A \otimes B \rightarrow C)\\ \colon & (A \otimes B \rightarrow C) \rightarrow (A \rightarrow (B \rightarrow C))\\ \colon & ((A \rightarrow B) \rightarrow C) \rightarrow (((B \rightarrow A) \rightarrow C) \rightarrow C)\\ \colon & \bot \rightarrow A \end The axioms (BL2) and (BL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2012) and (Cintula, 2005). All the other axioms were shown to be independent (Chvalovský, 2012).


Semantics

Like in other propositional
t-norm fuzzy logics T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval , 1for the system of truth values and functions called t-norms for permissible interpretations of conjunctio ...
, algebraic semantics is predominantly used for BL, with three main classes of algebras with respect to which the logic is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
: * General semantics, formed of all ''BL-algebras'' — that is, all algebras for which the logic is sound * Linear semantics, formed of all ''linear'' BL-algebras — that is, all BL-algebras whose lattice order is linear * Standard semantics, formed of all ''standard'' BL-algebras — that is, all BL-algebras whose lattice reduct is the real unit interval , 1with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any continuous
t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersectio ...
.


Bibliography

* Hájek P., 1998, ''Metamathematics of Fuzzy Logic''. Dordrecht: Kluwer. * Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, ''Trends in Logic'' 20: 177–212. * Cintula P., 2005, "Short note: On the redundancy of axiom (A3) in BL and MTL". ''Soft Computing'' 9: 942. * Chvalovský K., 2012,
On the Independence of Axioms in BL and MTL
. ''
Fuzzy Sets and Systems ''Fuzzy Sets and Systems'' is a peer-reviewed international scientific journal published by Elsevier on behalf of the International Fuzzy Systems Association (IFSA) and was founded in 1978. The editors-in-chief (as of 2010) are Bernard De Baets ...
'' 197: 123–129, {{doi, 10.1016/j.fss.2011.10.018.


References

Fuzzy logic