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In Enderton's Mathematical logic, at the end of page 69, when he defines First-Order language syntax, he categorized its symbols into two groups 1) logical symbols 2) parameters.

I understand that function and predicate symbols are parametric for each First order language (they differ in each application in terms of quantity and arity of each symbol), so they go under parameters. But why he puts the quantifier in this group?

Can anybody comment what is the precise criterion for this distinction (logical symbols and parameters)?

Update: The categorization of variables as either logical or non-logical symbols are also discussed here (noted by @ Mauro ALLEGRANZA below). It seems there is no unified agreement in classifying FOL symbols. I leave the question as it is, with the hope to get more feedback from the community.

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    $\begingroup$ You can see here a related discussion : there is some "variability" in the practice, regarding the definition (if any) of logical symbols. $\endgroup$ – Mauro ALLEGRANZA Mar 23 '15 at 16:35
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My (very) personal understanding of this issue is related to the (not obviuos) concept of Logical Constants.

According to the "traditional" view :

The most venerable approach to demarcating the logical constants identifies them with the language's syncategorematic signs: signs that signify nothing by themselves, but serve to indicate how independently meaningful terms are combined.

In my reading, Enderton uses "logical symbols" with the same meaning of "logical constants", i.e. symbols like $\lnot$ or $=$ that do not change meaning according the context.

Obviously, the meaning of a predicate or constant symbol is specified only through an interpretation.

If we follow this approach, we can say that also the quantifiers are specified through an interpretation; the meaning of $\forall$ is obviously "all", but this "all" changes if we are speaking of natural numbers : in this case $\forall$ means "for all $n \in \mathbb N$", or if we are speaking of Greek philosophers : in this case $\forall$ means "Plato and Aristotle and ..."


But you can see other equally authoritative textbooks (like van Dalen's one) where a similar distinction is not present, and there is only one list of symbols.

You can see also :

The logical symbols will have a fixed interpretation. In particular, “$=$” will always be expected to mean equals.



Regarding variables, we can compare :

proper names [i.e. constants] and variables [ ...] we call proper symbols, and we regard them as having meaning in isolation, the primitive names as denoting (or at least purporting to denote) something, the variables as having (or at least purporting to have) a non-empty range. But in addition to proper symbols there must also occur symbols which are improper - or in traditional (Scholastic and pre-Scholastic) terminology, syncategorematic - i.e., which have no meaning in isolation but which combine with proper symbols (one or more) to form longer expressions that do have meaning in isolation. [...] Connectives are combinations of improper symbols [...].

with :

we divide all our signs into logical and descriptive (or non-logical). Descriptive signs are those constants which serve to refer to objects, properties, relations, etc., in the world; they include the individual constants, the predicates, and the sentential constants. Logical signs include all the variables and the logical constants. Logical signs do not themselves refer to something in the world (the world of things has nothing like negation, disjunction, etc.); rather, they bind together the descriptive constants of a sentence and thereby contribute indirectly to the sense of a sentence.

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  • $\begingroup$ But, using your given criterion --logical symbols' meaning (i.e., their semantic interpretation) does not change according to the context-- variable should then go under the parameters section too: if we take that the quantifier symbol interpretation depends on the context, it seems variable interpretation definitely varies according to the choice of universe of discourse too. Enderton listed variables as logical symbols. $\endgroup$ – qartal Mar 23 '15 at 17:37

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