In Elements of Modern Mathematics by Kenneth O. May, the author writes:
A symbol that in a particular context is the name of just one specific thing is called a constant. Grammarians call constants proper nouns. The thing that a constant names is called its
value. A constant stands for, or names, its value.
Then goes on to explain variables:
A symbol that, in a particular context, is not a constant but for which any one of certain constants may be substituted is called a variable. For example, 'x' is a variable in the context x/1 = x, and 'a chair' is a variable in the context a chair has four legs. The constants that may be substituted for a variable in a particular context are called significant substitutes for it. The value of a significant substitute is called a value of the variable in a particular context. In our examples, '2' is a significant substitute for 'x' and 'the first chair in the first row' is a significant substitute for 'a chair.' The number 2 and the first chair in the first row are the corresponding values.
If I understand correctly, the author is trying to say that if we have a sentence in variable 'x', we can substitute this variable by constants, in which case we would get a sentence that makes a proposition about the value of that constant (proposition about the thing that that constant denotes).
However, the author says that when we replace a variable by a constant, the value of that constant also becomes the value of the variable.
Now given the definition of a variable as a symbol that denotes an unspecified thing (unlike a constant that denotes a particular thing), can we meaningfully ever say that a variable has taken a particular value?
I mean the very meaning of 'x' in any sentence is that it denotes an unspecified arbitrary thing, so even if we ever want to talk about a variable having a "value" it will be an unspecified thing. Saying that upon replacing the variable by a constant that variable gets the value of that constant, won't contradict the definition of a variable?
Please help me with adding or removing any tags. Thank you.
Best Answer
Mathematical discourse, while introduces its own distinctions, abstracts away those insignificant for itself. Some of them surface in foundational and philosophical studies. Even then, several are of practical consequence, others are not.
Our usage of variables are mainly in three ways:
All three senses share the core idea that the symbol as the variable has no intrinsic semantic value; we shall focus on the second and the third ones for the present question.
The distinction between (2) and (3) can be illustrated as follows:
Suppose we calculate $y$ such that $y = x + 5$ as $x$ takes integral values in the range of $\{1, 2,\ldots, 10\}$. We may view $x$ as assigned to the number $1$ so that $x$ refers to $1$ and $y$ refers to $6$, and so on. Thus, what $x$ refers to varies with the range of the values.
Alternatively, we may view $x$ as a place-holder that marks a place in the expression. Thus, $x$ refers to the dots (the gap) in $y =\ldots + 5$ to be filled in by appropriate symbols. We do not actually assign values to $x$ to refer to, but substitute numbers (in fact, numerals, but we skip over such details) from the set $\{1, 2,\ldots, 10\}$ for $x$ and those substitutes are the values of the variable $x$ as the author states. We write the numbers in the place to which $x$ points at (i.e., the dots). It might be helpful to compare this sense of variable to place-holder as the linguistic term. Consider the sentence:
“It is a conjecture that $n+1, n+2,\ldots, n+k$ being composite numbers, there are $k$ distinct primes $p_{i}$ such that $p_{i}$ divides $n+i$ for $1\leq i\leq k$.”
The occurrence of ‘It’ in the preceding sentence is not the genuine subject of the sentence; it is a placeholder.
In the usual mathematical parlance, these two perspectives is conflated. It is up to one's cognitive choice to shift from one viewpoint to the other. If the set $x$ ranges over is large, one might well conceive of the place-holder $x$ “as if” $x$ were substituted by the specified values in abstracto. Since we fix a set $\{1, 2,\ldots, 10\}$ in both cases and obtain the set $\{6, 7,\ldots, 15\}$, it does not matter which one we choose. Thus, the terms ‘variable’ and ‘place-holder’ are held as synonymous.
Likewise, in the sentence $\forall x(x\in\mathbb{R}\rightarrow P(x))$, $x$ is taken with varying referents. However, when we instantiate it to a constant in the language of logic, say $\alpha$ and so $P(a)$, $x$ is taken with varying occurrence (i.e., its occurrence is replaced with the occurrence of $\alpha$).
Indeed, the distinction perspectives can be translated into substitutional interpretation of quantifiers as contrasted to the familiar objectual interpretation and thus the truth-conditions of quantified sentences can be differentiated in logic. Let us briefly touch on this.
The usual interpretation of the standard quantifiers $\forall$ and $\exists$ is objectual; values are regarded as ‘objects’ to which the variables refer. This requires us to fix a domain of values, otherwise, the variables cannot refer or refer to $\emptyset$.
If a quantifier is interpreted substitutionally, appropriate expressions are substituted for the variable, and the resultant sentence is evaluated to true or false. Notice that substitutional quantifiers do not require us to fix a domain to provide the referents of the variables, for the variables actually do not refer, but leave their places.
The usual symbols for substitutional universal and existential quantifiers are $\Pi$ and $\Sigma$, but I think it is better to denote them by $\overline{\forall}$ and $\overline{\exists}$ in order both to reserve the former symbols for their other uses and to facilitate the association with the familiar quantifiers, given the contemporary ease we have for typographical variations. So, let us consider the sentences:
and their instantiation to ‘Pegasus is a horse’.
From the received point of view, the sentence ‘Pegasus is a horse’ is false, because the sentence (1) seeks its referent in the domain of horses of which Pegasus is not a member. But from the substitutional point of view, it is true, because the resultant sentence expresses a cultural fact independently of a domain of discourse to seek a referent for Pegasus. But ‘the table is a horse’ would be false substitutionally as it would be objectually. Thus, we make sense of the author's use of the phrase “significant substitute”: ‘Pegasus’ is a significant substitute, but ‘the table’ is not.
For those interested, I recommend Gabriel Uzquiano’s SEoP article Quantifiers and Quantification to gain more insight about the topic.
To conclude, we sometimes imply the referents of a variable, and sometimes, its occurrences. We call both the referents and the substitutes (of occurrences) the values of a variable. We have to take caution: Taking the former sense as the sole one may cause confusion when the latter one is also used in the context.