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I am reading the second chapter of mathematical logic by Schoenfield and I am slightly confused

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In the second section later on he mentions something else ( as l see it ) and it Is a bit vague.

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I thought the negation, OR and the existential quantifier symbols were logical and also a part of first order language as mentioned in the earlier section.

What does he mean by completely determined ?

I understand logical symbols have a fixed meaning and don't need interpretation as opposed to non logical symbols and this has something to do with determining first order language but I'm only guessing at what he wants to convey

Aristotle Stagiritis
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Brief answer...

We have a "fixed part" made of logical symbols: connectives, variables, quantifier, equality (and auxiliary symbols: parentheses) and a "variable part" made of non-logical symbols that is specific of the theory we want to formalize.

For the mathematical theory of sets, e.g. $\mathsf {ZF}$ we need only one non-logical symbols: the symbol $\in$ for the binary relation "... is a member of...".

We may have also the individual constant $\emptyset$ for the empty set, or we may add later to the theory, having proved that it exists (using the Null Set axiom) and it is unique (by Estensionality).

An example of formula in the "pure" first-order language will be: $\forall x (x=x)$.

An example of formula in the language of first-order set theory will be: $\forall x \forall y [(x \in y) \lor \lnot (x \in y)]$.

Mauro ALLEGRANZA
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