Basic fuzzy Logic the logic of continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 t-norm

T-norm

In mathematics, a t-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 intersection in a lattice and conjunction in logic...

s, is one of t-norm fuzzy logics

T-norm fuzzy logics

T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics which takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction...

. It belongs to the broader class of substructural logic

Substructural logic

In logic, a substructural logic is a logic lacking one of the usual structural rules , such as weakening, contraction or associativity...

s, or logics of residuated lattice

Residuated lattice

In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\z and z/y loosely analogous to division or implication when x•y is viewed as multiplication or conjunction respectively...

s; it extends the logic of all left-continuous t-norms MTL

Monoidal t-norm logic

Monoidal t-norm based logic , the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of commutative bounded integral residuated lattices by the axiom of...

.

Language

The language of the propositional logic BL consists of countably many propositional variable

Propositional variable

In mathematical logic, a propositional variable is a variable which can either be true or false...

s and the following primitive logical connective

Logical connective

In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

s:

• Implication (binary
  Arity
  In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
  
  )
• Strong conjunction (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation follows the tradition of substructural logics.
• Bottom (nullary — a propositional constant); or 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 (binary), also called lattice conjunction (as it is always realized by the lattice
  Lattice (order)
  In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
  
   operation of meet
  Meet (mathematics)
  In mathematics, join and meet are dual binary operations on the elements of a partially ordered set. A join on a set is defined as the supremum with respect to a partial order on the set, provided a supremum exists...
  
   in algebraic semantics). Unlike MTL
  Monoidal t-norm logic
  Monoidal t-norm based logic , the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of commutative bounded integral residuated lattices by the axiom of...
  
   and weaker substructural logics, weak conjunction is definable in BL as

• Negation (unary
  Unary operation
  In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....
  
  ), defined as

• Equivalence (binary), defined as

As in MTL, the definition is equivalent to

• (Weak) disjunction (binary), also called lattice disjunction (as it is always realized by the lattice
  Lattice (order)
  In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
  
   operation of join in algebraic semantics), defined as

• Top (nullary), also called one and denoted by or (as the constants top and zero of substructural logics coincide in MTL), defined as

Well-formed formula

Well-formed formula

In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...

e 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

Hilbert-style deduction system

In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert...

 for BL has been introduced by Petr Hájek

Petr Hájek

Petr Hájek is a Czech scientist in the area of mathematical logic and a professor of mathematics. He works at the Institute of Computer Science at the Academy of Sciences of the Czech Republic and worked as a lecturer at the Faculty of Mathematics and Physics at the Charles University in Prague...

 (1998). Its single derivation rule is modus ponens

Modus ponens

In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...

:

from and derive

The following are its axiom schemata:

The axioms (BL2) and (BL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2008) and (Cintula, 2005). All the other axioms were shown to be independent (Chvalovský, 2008).

Semantics

Like in other propositional t-norm fuzzy logics

T-norm fuzzy logics

T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics which takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction...

, algebraic semantics

Algebraic semantics

An programming language theory, the algebraic semantics of a programming language is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner....

 is predominantly used for BL, with three main classes of algebras

Algebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

 with respect to which the logic is complete

Completeness

In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...

:

• 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
  Lattice (order)
  In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
  
   order is linear
  Total order
  In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
  
  
• Standard semantics, formed of all standard BL-algebras — that is, all BL-algebras whose lattice reduct is the real unit interval [0, 1] with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any continuous t-norm
  T-norm
  In mathematics, a t-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 intersection in a lattice and conjunction in logic...