# quantifier: A quantifier is a function that computes a fuzzy truth value... In beerda/lfl: Linguistic Fuzzy Logic

## Description

A quantifier is a function that computes a fuzzy truth value of a claim about the quantity. This function creates the <1>-type quantifier. (See the examples below on how to use it as a quantifier of the <1,1> type.)

## Usage

 1 2 3 4 5 quantifier( quantity = c("all", "almost.all", "most", "many", "some", "at.least"), n = NULL, alg = c("lukasiewicz", "goedel", "goguen") ) 

## Arguments

 quantity the quantity to be evaluated. 'all' computes the degree of truth to which all elements of the universe have the given property, 'almost.all', #' 'most', and 'many' evaluate whether the property is present in extremely big, very big, or not small number of elements from the universe, where these linguistic expressions are internally modelled using the lingexpr() function. 'at.least' quantity requires the 'n' argument to be specified, as it computes the truth value that at least n elements from the universe have the given property. n the number of elements in the 'at.least n' quantifier alg the underlying algebra in which to compute the quantifier. Note that the algebra must have properly defined the order function, as in the case of 'goedel', 'goguen', or 'lukasiewicz' algebra, (see the algebra() function) or as in the dragonfly() or lowerEst() algebra.

## Value

A two-argument function, which expects two numeric vectors of equal length (the vector elements are recycled to ensure equal lengths). The first argument, x, is a vector of membership degrees to be measured, the second argument, w, is the vector of weights to which the element belongs to the universe.

Let U be the set of input vector indices (1 to length(x)). Then the quantifier computes the truth values accordingly to the following formula: \vee_{z \subseteq U} \wedge_{u \in z} (x[u]) \wedge measure(m_z), where m_z = sum(w) for "some" and "at.least and m_z = sum(w[z]) / sum(w) otherwise. See sugeno() for more details on how the quantifier is evaluated.

Setting w to 1 yields to operation of the <1> quantifier as developed by Dvořák et al. To compute the <1,1> quantifier as developed by Dvořák et al., e.g. "almost all A are B", w must be set again to 1 and x to the result of the implication A \Rightarrow B. To compute the <1,1> quantifier as proposed by Murinová et al., e.g. "almost all A are B", x must be set to the result of the implication A \Rightarrow B and w to the membership degrees of A. See the examples below.

Michal Burda

## References

Dvořák, A., Holčapek, M. L-fuzzy quantifiers of type <1> determined by fuzzy measures. Fuzzy Sets and Systems vol.160, issue 23, 3425-3452, 2009.

Dvořák, A., Holčapek, M. Type <1,1> fuzzy quantifiers determined by fuzzy measures. IEEE International Conference on Fuzzy Systems (FuzzIEEE), 2010.

Murinová, P., Novák, V. The theory of intermediate quantifiers in fuzzy natural logic revisited and the model of "Many". Fuzzy Sets and Systems, vol 388, 2020.

sugeno(), lingexpr()
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  # Dvorak <1> "almost all" quantifier q <- quantifier('almost.all') a <- c(0.9, 1, 1, 0.2, 1) q(x=a, w=1) # Dvorak <1,1> "almost all" quantifier (w set to 1) a <- c(0.9, 1, 1, 0.2, 1) b <- c(0.2, 1, 0, 0.5, 0.8) q <- quantifier('almost.all') q(x=lukas.residuum(a, b), w=1) # Murinová <1,1> "almost all" quantifier (note w set to a) a <- c(0.9, 1, 1, 0.2, 1) b <- c(0.2, 1, 0, 0.5, 0.8) q <- quantifier('almost.all') q(x=lukas.residuum(a, b), w=a) # Murinová <1,1> "some" quantifier a <- c(0.9, 1, 1, 0.2, 1) b <- c(0.2, 1, 0, 0.5, 0.8) q <- quantifier('some') q(x=plukas.tnorm(a, b), w=a)