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Double Knot
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This is from the same reference as yours:

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation.

So just like in common math, logical symbols in FOL is like universal operational math symbols like >, =, and variables which always have the same meaning in a certain realm of math, say, arithmetic. While non-logical symbols are like abstract functional math symbols waiting to be interpreted semantically.

The non-logical symbols represent predicates (relations), functions and constants on the domain of discourse. It used to be standard practice to use a fixed, infinite set of non-logical symbols for all purposes. A more recent practice is to use different non-logical symbols according to the application one has in mind. Therefore, it has become necessary to name the set of all non-logical symbols used in a particular application. This choice is made via a signature.

In this approach, every non-logical symbol is of one of the following types.

A predicate symbol (or relation symbol) with some valence (or arity, number of arguments) greater than or equal to 0. These are often denoted by uppercase letters such as P, Q and R...

A function symbol, with some valence greater than or equal to 0. These are often denoted by lowercase roman letters such as f, g and h.

In summary, a non-logical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation. Consequently, a sentence containing a non-logical symbol lacks meaning except under an interpretation, so a sentence is said to be true or false under an interpretation. Using a more familiar math analogy, for a sentence y=f(x) in real analysis, the variables (logical symbol) x and y are always meant to be some (maybe unknown) specific real variables in R, while the abstract function f (non-logical symbol) has no specific meaning until we interpret it as certain relation like y=sin(x)...

Double Knot
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