## Peirce’s 1870 “Logic Of Relatives” • Comment 11.18

An order-preserving map is a special case of a structure-preserving map and the idea of preserving structure, as used in mathematics, means preserving some but not necessarily all of the structure of the source domain in the transition to the target domain. In that vein, we may speak of structure preservation in measure, the suggestion being that a property able to be qualified in manner is potentially able to be quantified in degree, admitting answers to questions like, “How structure-preserving is it?”

Let’s see how this applies to the “number of” function $v : S \to \mathbb{R}.$ Let $-\!\!\!<\!"$ denote the implication relation on logical terms, let $\!\!\le\!\!"$ denote the less than or equal to relation on real numbers, and let $x, y$ be any pair of absolute terms in the syntactic domain $S.$ Then we observe the following relationships: $\begin{array}{lll} x ~-\!\!\!< y & \Rightarrow & v(x) \le v(y) \end{array}$

Equivalently: $\begin{array}{lll} x ~-\!\!\!< y & \Rightarrow & [x] \le [y] \end{array}$

Nowhere near the number of logical distinctions that exist on the left hand side of the implication arrows can be preserved as one passes to the linear ordering of real numbers on the right hand side of the implication arrows, but that is not required in order to call the map $v : S \to \mathbb{R}$ order-preserving, or what is known as an order morphism.

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