T-norm Fuzzy Logics - Logical Language

Logical Language

The logical vocabulary of propositional t-norm fuzzy logics standardly comprises the following connectives:

• Implication (binary). In the context of other than t-norm-based fuzzy logics, the t-norm-based implication is sometimes called residual implication or R-implication, as its standard semantics is the residuum of the t-norm that realizes strong conjunction.
• Strong conjunction (binary). In the context of substructural logics, the sign and the names group, intensional, multiplicative, or parallel conjunction are often used for strong conjunction.
• Weak conjunction (binary), also called lattice conjunction (as it is always realized by the lattice operation of meet in algebraic semantics). In the context of substructural logics, the names additive, extensional, or comparative conjunction are sometimes used for lattice conjunction. In the logic BL and its extensions (though not in t-norm logics in general), weak conjunction is definable in terms of implication and strong conjunction, by

The presence of two conjunction connectives is a common feature of contraction-free substructural logics.

• Bottom (nullary); 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 t-norm fuzzy logics). The proposition represents the falsity or absurdum and corresponds to the classical truth value false.
• Negation (unary), sometimes called residual negation if other negation connectives are considered, as it is defined from the residual implication by the reductio ad absurdum:

• Equivalence (binary), defined as

In t-norm logics, the definition is equivalent to

• (Weak) disjunction (binary), also called lattice disjunction (as it is always realized by the lattice operation of join in algebraic semantics). In t-norm logics it is definable in terms of other connectives as

• Top (nullary), also called one and denoted by or (as the constants top and zero of substructural logics coincide in t-norm fuzzy logics). The proposition corresponds to the classical truth value true and can in t-norm logics be defined as

Some propositional t-norm logics add further propositional connectives to the above language, most often the following ones:

• The Delta connective is a unary connective that asserts classical truth of a proposition, as the formulae of the form behave as in classical logic. Also called the Baaz Delta, as it was first used by Matthias Baaz for Gödel–Dummett logic. The expansion of a t-norm logic by the Delta connective is usually denoted by
• Truth constants are nullary connectives representing particular truth values between 0 and 1 in the standard real-valued semantics. For the real number, the corresponding truth constant is usually denoted by Most often, the truth constants for all rational numbers are added. The system of all truth constants in the language is supposed to satisfy the bookkeeping axioms:

etc. for all propositional connectives and all truth constants definable in the language.

• Involutive negation (unary) can be added as an additional negation to t-norm logics whose residual negation is not itself involutive, that is, if it does not obey the law of double negation . A t-norm logic expanded with involutive negation is usually denoted by and called with involution.
• Strong disjunction (binary). In the context of substructural logics it is also called group, intensional, multiplicative, or parallel disjunction. Even though standard in contraction-free substructural logics, in t-norm fuzzy logics it is usually used only in the presence of involutive negation, which makes it definable (and so axiomatizable) by de Morgan's law from strong conjunction:

• Additional t-norm conjunctions and residual implications. Some expressively strong t-norm logics, for instance the logic ŁΠ, have more than one strong conjunction or residual implication in their language. In the standard real-valued semantics, all such strong conjunctions are realized by different t-norms and the residual implications by their residua.

Well-formed formulae of propositional t-norm logics are defined from propositional variables (usually countably many) by the above logical connectives, 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)

First-order variants of t-norm logics employ the usual logical language of first-order logic with the above propositional connectives and the following quantifiers:

• General quantifier
• Existential quantifier

The first-order variant of a propositional t-norm logic is usually denoted by