Is a propositional variable a proposition?

Question

I understand a proposition is an assertion that is true or false (not both) and propositional variable is used to stand for an arbitrary and unspecified proposition. A propositional variable is analogous to using "$x$" in place of a number in algebra. To be sure, is it then formally correct to call a propositional variable a proposition?

Answer 1

Firstly, calling a propositional variable a proposition is prone to induce complications, at least, conceptually, if not operationally, for there is a clear difference between stating that (a variable) p and stating that $2$ is the smallest prime number; hence, better to avoid it.

Secondly, however, it is formally correct to assign a truth-value to a propositional variable just like a proposition, but dispensable: It depends on what structure one chooses to set out for propositional calculus (we shall restrict ourselves to propositional calculus, but the view discussed can be generalised).

We can make sense of these two seemingly contrary points by viewing propositional calculus in model-theoretic setting. It should be remarked that there are some variations among writers on model theory in the definitions of such preliminary concepts as language, model, assignment and interpretation, but this should not be troublesome, once we see that they are kindred manifestations of the same core ideas. The following presentation is fairly mainstream one over those ideas, stripping away inessential details.

Ordinarily, structures of model theory are defined to treat non-logical vocabulary of a language and a deductive apparatus is assumed (usually, first-order predicate logic). I shall follow Wolfgang Rautenberg's A Concise Introduction to Mathematical Logic (3rd edition, p. 42), subjecting propositional calculus itself to the definition of a structure, since this approach helps making the points better. Later, I shall mention also an ordinary structure assuming propositional calculus.

Let us consider a general structure form in the $n$-tuple notation:

$$\mathfrak{A}=\langle\mathrm{A}, F^{\mathfrak{A}}, R^{\mathfrak{A}}, C^{\mathfrak{A}}\rangle$$

where $F^{\mathfrak{A}}$, $R^{\mathfrak{A}}$ and $C^{\mathfrak{A}}$ represent function, relation and constant term symbols, respectively. Associated with the structure is a set of variables and a variable assignment function $\alpha$ that assigns to each variable an element of $\mathrm{A}$.

A familiar example in this form is

$$\mathfrak{N}=\langle\mathbb{N}, <, +, ·, 0, 1\rangle$$

in which we work with a set of variable symbols $\{w, x, y, x\}$ (to be indexed and extended if needed).

Similarly, $$\mathfrak{P}=\langle\mathrm{P}, \underbrace{\neg, \wedge, \vee}_{F^{\mathfrak{P}}}, \underbrace{\top, \bot}_{C^{\mathfrak{P}}}\rangle$$

with a set of variable symbols $\{p, q, s, t\}$ (to be indexed and extended if needed). $\mathrm{P} = \{0, 1\}$ is the universe of the structure (domain of discourse).

An interpretation $\mathfrak{P}, \alpha$ is a model of $\phi$:

$$\mathfrak{P}, \alpha\vDash\phi$$

Suppose $\phi$ is $(p\wedge q)\vee\neg r$. Then, each row of its truth table represents a separate assignment function. The rows that have T in the resultant column indicate the models of $\phi$:

If, for every $\alpha$, $\mathfrak{P}$ is a model of a formula $\phi$, then $\phi$ is said to be a tautology. For example,

$$\vDash p\vee\neg p$$" />

We see that the predicate 'is true' is actually just an alias of 'is assigned $1$'. As regards the present discussion, 'p is true' is not of a different character than '$x_{1}$ is assigned $3$'. The indifference originates from the fact that we study truth conditions of propositions in artificial (as opposed to natural) language with formal semantics. What is the position of formal semantics in the present context? I shall refer to Bas van Fraassen's Formal Semantics and Logic (p. 12f.) for this:

Metalogic can in turn be roughly divided into two parts: proof theory and formal semantics. In proof theory, the logical systems are treated as abstract mathematical systems, and the questions dealt with relate directly to the specific set of axioms and rules used to formulate the system. In formal semantics, the logical systems are studied from the point of view of their possible interpretations—with special reference to their intended interpretation, if such there be.

Ian Chiswell and Wilfrid Hodges' Mathematical Logic gives an alternative structuring. The universe of the structure is a set of propositions, instead of Boolean values. Thus, propositional variables range over propositions. Hence, ascribing a truth-value to a propositional variable p remains out of question in this setting. The propositions in the universe are assigned a truth-value by an additional assignment function.

Each choice of propositional structure has its own advantages and disadvantages. Ascription of truth does not present an epistemological problem, since in any case, we deal with not truth simpliciter itself, but a delegate of it.