It is high time to take the 'must' seriously when considering what entities must be taken to be values of the bound variables in order for the theory to be true.
Not only the place of the bound variable is different, but ⌇ builds in contraction for n−1 variables (which are separated by ∣− and other symbols in the left-hand expression).
When showing all ingredients of an object are its elements, Leśniewski instantiates the bound variable by saying "Using the expression 'x' with the meaning of the expression 'ingredient of the object A' …".
We can then say: one is ontologically committed to those entities that are needed as values of the bound variables for this chosen epistemically best theory to be true.
According to Quinean criteria, this sentence is committed to entities that swim, but not to a property of swimming; for no property of swimming need be among the values of the bound variables in order for the sentence to be true.
(The modern notation shows that in the proposed definition of fαˆ in PM notation, we shouldn't expect α in the definiens, since it is really a bound variable in fαˆ; similarly, we shouldn't expect φ in the definiendum because it is a bound variable in the definiens).
In the process of the elimination of the bound variables, UXY is obtained from an expression that contains 'Xx' and 'Yx', where x does not occur in X or Y, and if X and Y happen to be n-ary predicates with n ≥ 2, then x occurs (only) in their last argument place.
where is the density of competitive species; is the control variable; ; bounded sequences,,,, and ; and are positive integer; denote the sets of all integers and all positive real numbers, respectively; is the first-order forward difference operator ;.
It provides a rigorous and clear framework to compare competing models, avoiding the calculation of significance levels and without depending upon asymptotic properties of frequentist estimators [ 40], Bayes factor behaves well even when the bounded variable to be tested is either close to the boundary of the parametric space [ 18].
The key simplification is that the nonlinear terms δ i d k =: z ik in (3) (the strong duality theorem equality) are exactly linearizable as follows: (4) 0 ≤ z ik ≤ δ i max d k (5) δ i − δ i max (1 − d k ) ≤ z ik ≤ δ i where δ i max is the upper bound for the dual variable δ i (chosen arbitrarily big in the implementation).
