Are variables logical or non-logical symbols in a logic system?

Question

Are variables logical or non-logical symbols in a logic system? I understand constants are 0-ary logical operation symbols. I think variables are non-logical symbols.

But here are two contrary examples:

It seems that variables are logical symbols in a propositional logic system, according to http://en.wikipedia.org/wiki/First-order_logic#Logical_symbols

Logical symbols

An infinite set of variables, often denoted by lowercase letters at the end of the alphabet x, y, z, … . Subscripts are often used to distinguish variables: x0, x1, x2, … .

It seems that variables are non-logical symbols in a propositional logic system, according to http://en.wikipedia.org/wiki/Propositional_logic#Generic_description_of_a_propositional_calculus

• The alpha set is a finite set of elements called proposition symbols or propositional variables. Syntactically speaking, these are the most basic elements of the formal language $\mathcal{L}$, otherwise referred to as atomic formulæ or terminal elements. In the examples to follow, the elements of are typically the letters $p, q, r$, and so on.

• The omega set $\Omega$ is a finite set of elements called operator symbols or logical connectives.

Thanks.

Answer 1

We can agree that there is some "variability" in the practice, regarding the definition (if any) of logical symbols in first-order logic.

According to the definition in Herbert Enderton, A Mathematical Introduction to Logic (2nd - 2001), of First-Order Languages [page 69], we have :

A. Logical symbols

$0$. Parentheses: $(,)$.

$1$. Sentential connective symbols: $\rightarrow, \lnot$.

$2$. Variables (one for each positive integer $n$):

$v_\, v_2, \ldots$.

$3$. Equality symbol (optional): $=$.

B. Parameters

$0$. Quantifier symbol: $\forall$.

$1$. Predicate symbols: For each positive integer $n$, some set (possibly empty) of symbols, called $n$-place predicate symbols.

[...]

Compare with Dirk van Dalen, Logic and Structure (5th ed - 2013), page 56 :

The alphabet consists of the following symbols:

1. Predicate symbols: $P_1, \ldots, P_n, =$

2. Function symbols: $f_1, \ldots, f_m$

3. Constant symbols: $c_i$ for $i \in I$

4. Variables: $x_0, x_1, x_2, \ldots$ (countably many)

5. Connectives: $∨,∧,→,¬,↔,⊥,∀,∃$

6. Auxiliary symbols: $(, )$.

Note that the expression "logical symbols" has been avoided.

The issue is related to that of Logical Form and Logical Constants. Traditionally :

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.

Roughly speaking, syncategorematic signs are the logical constants or logical symbols, i.e. those symbols (like conncetives) which are not interpreted or, according to modern semantics of first-order logic, do not "change meaning" when we vary the interpretation of the language.

According to this criteria, variables are ambiguos, because they have no "fixed meaning"; but also, their meaning is not fixed by the interpretation.

They are placeholders; in principle (see Frege) we can dispense with them. We can index argument places writing, instead of $Q(x, y)$ :

$Q[( \quad )_i , ( \quad )_j]$.

Thus, we can say that variables are another category of auxiliary symbols.

Answer 2

As Thomas Andrews pointed out in a comment, this terminology is not something where one can expect consistency from book to book.

However, as a practical matter, when one specifies that the non-logical language of a particular theory is such-and-such, it is highly unusual to have to say, "oh, and there are variables too". Variables are just supposed to be there, implicitly, when we say that what we're speaking about is a first-order theory.

As such, variables are treated as logical symbols, no matter whether the official definitions in any given book call them out as being so.

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Note that your second quote is about propositional logic where there are no object variables at all and there's no meaningful distinction between logical and non-logical vocabulary.