4.1211 Thus one proposition 'fa' shows that the object a occurs in its sense, two propositions 'fa' and 'ga' show that the same object is mentioned in both of them. If two propositions contradict one another, then their structure shows it; the same is true if one of them follows from the other. And so on.

Well... not really I am afraid: The first statement is totally trivial, apart from the notation, which is the Principia Mathematica's and unexplained. The second statement is simply false: Yes, "this is green" and this is not green" presumably "show" the "structure" of a contradiction, viz. the particle "not", but "that number is even" and "that number is odd" don't, so "their structure" does not "show" their contradictoriness (and that they would if e.g. "odd" were replaced by "not even" is hardly relevant in view of what W. says). Finally, "And so on." is hardly helpful after a triviality and a falsehood.