What is the difference between asserting "$\phi(a)$" and asserting "$\phi(a)$ is true" in Whitehead and Russell's PM?

Question

The first edition of Principia Mathematica clearly distinguishes "Socrates is a man" and "'Socrates is a man' is true." Judging from the context, the distinction is neither a primitive idea nor a definition; it is something the reader is supposed to "see." For some reason I can't "see" it, although I can deliberately memorize the distinction the same way I memorize multiplication table. I have an inkling that this distinction should directly appeal to the sense, like the distinction between red and blue. If you can "see" this distinction, please kindly help me out.

The following passage is from page 41, Introduction, Chapter II, section II, The Nature of Propositional Functions. https://i.stack.imgur.com/ejGD1.png

These symbols are for your convenience:

propositional function: $\phi(\hat{z})$

Ambiguous value: $ \phi(z) $

A function with itself as an argument: $\phi(\phi(\hat{z}))$

Answer 1

A possible interpretation of the passage must take into account the context, that is : Chapter II : THE THEORY OF LOGICAL TYPES.

In particular, the passage is taken from section II. The Nature of Propositional Functions.

We must remember the lack of clear and systematic distinction in W&R's Principia between object-language and meta-language (and meta-theory); in this chapter, they are explaining the "syntactical" restrictions involved in type theory.

See page 40 :

Now given a function $ \phi(\hat{x})$, the values for the function are all propositions of the form $\phi x$. It follows that there must be no propositions, of the form $\phi x$, in which $x$ has a value which involves $\phi \hat x$.

That is to say, the symbol " $\phi (\phi \hat x$) " must not express a proposition, as " $\phi a$ " does if $\phi a$ is a value for $\phi \hat x$. In fact " $\phi (\phi \hat x$) " must be a symbol which does not express anything: we may therefore say that it is not significant.

Roughly, the formula $\phi(\phi)$ is not true nor false; it is simply meaningless, because it violates what we today will call the "rules of formation"; see page 41 :

Since "$(x) . \phi x$ " involves the function $\phi \hat x$, it must, according to our principle, be impossible as an argument to $\phi$. That is to say, the symbol " $ \phi \{(x) . \phi x \} $ " must be meaningless.

Gregory Landini, in Russell's Hidden Substitutional Theory (1998), page 279, proposes a reading that starts from Frege's hierarchy of "levels" of functions. According to Landini :

The idea is that a "function" (i.e., a predicate variable) can occur in a subject position (argument position) of another predicate variable only if this position represents a predicate position in the semantics [Landini points to pages 47 and 48 of PM].

Answer 2

The only thing I can think of is that "Socrates is a man" is a sentence in the sense that it has a truth value, but we do not make any claim as to what that truth value may be. The claim that "Socrates is a man" is true, is asserting not only that the same sentence has a truth value, but also that that truth value is True.

Just to round things off: "Socrates slept quickly" is not even a sentence since it has no truth value at all, it is simply nonsense written in a grammatically plausible fashion. However, "Socrates had exactly 100,000 hairs on his head by his 18th birthday" is a sentence. It may be true or false, but we'll never know.

I hope this is at all in the right direction to answer your question.

Answer 3

I have never read or otherwise studied the Principia; however, I think the general distinction to which Russell is alluding is still very much a recognized principle in modern (formalized) mathematics. Its basically the difference between a sentence $\varphi,$ versus the metasentence $\vdash \varphi$.

Conceptually, the distinction is best explained with reference to partially ordered sets (hereafter poset). In a poset, we can assert $x \leq y$ (intuitively, $x$ entails $y$). We may also have a meet-semilattice structure, in which case our assertions can be more sophisticated: we may write $x \wedge x' \leq y,$ intuitively asserting that $x$ and $x',$ taken together, entail $y$.

Note that $\wedge$ is a function, $\leq$ a relation.

Now furthermore, any given meet-semilattice may or may not admit the existence of a function $\rightarrow$ with the following property.

• $x \wedge x' \leq y$ iff $x \leq x' \rightarrow y$.

If such a function exists, it is unique, by this result. (If it is not clear what the above definition has to do with Galois connections, please comment and I will clarify.)

Anyway, if there is such a function (which I will call "implication"), then it can be added to the language (alongside $\wedge$ and $\leq$) to get a more expressive language. And we can prove the basic facts we expect from implication, such as modus ponens:

$$(x \rightarrow y) \wedge x \leq y$$

By the way, I recommend saying $\leq$ as "entails", and $\rightarrow$ as "implies", although this is not standardized.

Anyway, the point is that $\rightarrow$ can be conceived as an internalization of $\leq.$ Note that $\rightarrow$ is a function, while $\leq$ is a relation. Thus, spiritually, we can think of $x \rightarrow y$ as a statement internal to the language, while a formula like $x \leq y$ can (kind of) be viewed as part of the metalanguage. I am speaking very informally, here, of course.

Now I haven't really explained how $\rightarrow$ is an internalization of $\leq$, so lets do that. It turns out that if a function $\rightarrow$ with the property of interest exists, then so too does a top element, so long as we're working in a non-empty poset. (Hint: consider the expression $x \rightarrow x$). Denote the top element $\top;$ we can think of this as denoting unadulterated truthood. Furthermore, it can be shown that $x \leq y$ is equivalent to $\top \leq (x \rightarrow y)$. This is the sense in which $\rightarrow$ is an internalization of $\leq$.

Finally, lets switch to more logical notation. Instead of $\leq$, write $\vdash$ (this can also be articulated: "entails"). And lets move to greek letters, which can be thought of as denoting logical formulae. Furthermore, as shorthand for $\top \vdash \varphi$, let us write $\vdash \varphi.$

Then there is a clear difference between $\varphi$, and $\vdash \varphi$.

However, oftentimes $\varphi$ can be used as shorthand for $\vdash \varphi$, if the meaning is clear in context. Similarly, sometimes $\varphi \rightarrow \psi$ can be used as shorthand for $\vdash \varphi \rightarrow \psi$, or in other words $\varphi \vdash \psi$.

I think this is (at least conceptually) the distinction to which Russell is alluding.