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Concept
Monoidal t-norm logic
Summary
In mathematical logic, monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of the 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 (known as Höhle's monoidal logic, Ono's FLew, or intuitionistic logic without contraction) by the axiom of prelinearity.
In fuzzy logic, rather than regarding statements as being either true or false, we associate each statement with a numerical confidence in that statement. By convention the confidences range over the unit interval , where the maximal confidence corresponds to the classical concept of true and the minimal confidence corresponds to the classical concept of false.
T-norms are binary functions on the real unit interval [0, 1], which in fuzzy logic are often used to represent a conjunction connective; if are the confidences we ascribe to the statements and respectively, then one uses a t-norm to calculate the confidence ascribed to the compound statement ‘ and ’. A t-norm has to satisfy the properties of
commutativity ,
associativity ,
monotonicity — if and then ,
and having 1 as identity element .
Notably absent from this list is the property of idempotence ; the closest one gets is that . It may seem strange to be less confident in ‘ and ’ than in just , but we generally do want to allow for letting the confidence in a combined ‘ and ’ be less than both the confidence in and the confidence in , and then the ordering by monotonicity requires . Another way of putting it is that the t-norm can only take into account the confidences as numbers, not the reasons that may be behind ascribing those confidences; thus it cannot treat ‘ and ’ differently from ‘ and , where we are equally confident in both’.
Because the symbol via its use in lattice theory is very closely associated with the idempotence property, it can be useful to switch to a different symbol for conjunction that is not necessarily idempotent.
Official source
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