The Measures of Verisimilitude of the Similarity Approach

Abstract

I have already introduced the basic ideas of the similarity approach to truthlikeness^l in Sections 2.4. and 2.5., and explained the way cognitive problems and hypotheses concerning them are represented in this monograph in Section 1.2. I shall begin the detailed discussion of applying the similarity approach to multidimensional, quantitative cognitive problems by introducing a few additional notational conventions. In what follows, the set ℝ ⁿ is thought of as being equipped with its familiar structure of a linear space so that the operations of scalar multiplication and addition are defined in the usual way, i.e. by

$$\begin{gathered} r({x_1},...,{x_n})\; = (r{x_1},...,r{x_n}), \hfill \\ whenever\;({x_1},...,{x_n}) \in {\mathbb{R}^n}\;and\;r \in \mathbb{R},\;and\;\;by \hfill \\ ({x_1},...,{x_n}) + ({y_1},...,{y_n})\; = \;({x_1} + {y_1},...,{x_n} + {y_n}), \hfill \\ \end{gathered}$$

whenever (x ₁,...,x _n ), (y ₁,...y _n ) ∈ ℝⁿ, respectively. I shall also use the familiar notations x+A = {x+y | y∈A} and rA = {ry | y∈A} when x ∈ ℝⁿ, r ∈ ℝ, A ⊑ ℝⁿ. I shall denote the ith component of each x ∈ ℝⁿ by x _i so that x = (x _l,...,x _n ). The symbol | x | will refer to the Euclidian norm of x ∈ ℝⁿ so that | x | = (x ₁ ² +...+ x _n ²)^1/2. Finally, in what follows the symbol 0 will refer to the n-dimensional zero vector so that 0=(0,...,0) ∈ℝⁿ.