In Whitehead & Russell's PM, what makes anything true or false?

Question

It seems that truth and falsehood are fundamental to meanings and types. A proposition is defined as anything that is true or that is false.

PM defines truth as "consisting in the fact that there is a complex corresponding to the discursive thought which is the judgement."* If I'm not mistaken, truth is not one of PM's primitive ideas. In Chapter IV of 1st ed introduction, the reason for pronouncing "$\{(x).\phi(x)\}$ is a man" as meaningless appears rather arbitrary.

I wonder if W&R assumed that the notion of truth is well-known. If so, whose definition were they referring to?

*Chapter II.Tuth and Falsehood.

Answer 1

From the point of view of "logical machinery", we can say that PM assumes "true" and "false" as two undefined concepts that we are able to manage in the context of logic and mathematics.

See the definition of truth-values, page 8 :

The "truth-value" of a proposition is truth if it is true and falsehood if it is false.

In pages 45-on, there is an attempt to define the concepts : truth and falsehood.

The context, as you already noted, is the "act of judgement", involving a mind and several interconnected objects. True is a "property" of the judgement :

When the judgement is true, [...] there is a corresponding complex of the objects of the judgement alone. Falsehood, [...] consists in the absence of a corresponding complex composed of the objects alone.

This seems an enunciation of The Correspondence Theory of Truth :

Narrowly speaking, the correspondence theory of truth is the view that truth is correspondence to a fact — a view that was advocated by Russell and Moore early in the 20th century.

The correspondence theory is often traced back to Aristotle's well-known definition of truth (Metaphysics 1011b25):

“To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true”

but virtually identical formulations can be found in Plato (Cratylus 385b2, Sophist 263b). It is noteworthy that this definition does not highlight the basic correspondence intuition. Although it does allude to a relation (saying something of something) to reality (what is), the relation is not made very explicit, and there is no specification of what on the part of reality is responsible for the truth of a saying.

Aristotle sounds much more like a genuine correspondence theorist in the Categories (12b11, 14b14), where he talks of “underlying things” that make statements true and implies that these “things” (pragmata) are logically structured situations or facts (viz., his sitting, his not sitting). Most influential is his claim in De Interpretatione (16a3) that thoughts are “likenessess” (homoiosis) of things. Although he nowhere defines truth in terms of a thought's likeness to a thing or fact, it is clear that such a definition would fit well into his overall philosophy of mind.

Apart from all metaphysical implications regarding objects (what they are ?) and their "interconnections" into complex (what are relations ?), this basic account - as noted in Kevin's comment above - can be compared to well-known Tarski's Truth Definition.