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What exactly does "Types as ranges of significance of propositional functions. In modern 
terminology, types are domains of predicates" mean?

Update: I found in this paper (Pag 14 or 234) by Russell, where he defines what is ranges of significance, not exactly including types else propositions.

A function is said to be significant for the argument $x$ if it has a value for this argument. Thus we may say shortly $\phi x$ is significant, meaning the function $\phi$ has a value for the argument $x$. The range of significance of a function consists of all the arguments for which the function is true, together with all the arguments for which it is false.

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    $\begingroup$ Can you give more context? Where did you read about "ranges of significance"? I don't believe this is a standard term. $\endgroup$ – jmite Apr 30 '16 at 5:47
  • $\begingroup$ I saw this phrase in the slides of a course of my teacher. But after digging I found the reference. I'll put an answer with the exact explanation of Russell of that . $\endgroup$ – jonaprieto May 2 '16 at 5:46
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I believe this is simply referring to old terminology, as the quoted text implies.

When using a formula such as $\sqrt{x-5}$, it is common to specify what $x$ is intended to be so that the formula makes sense. That is, we could let $x$ to be a real number $\geq 5$, so that the square root is defined. Only when $x$ "ranges" over the values satisfying such condition the formula is meaningful, or "has significance". Hence, the "range of significance" of a formula essentially is the class of the values the variables must be in so that the formula makes sense.

This is true in propositions, too. If a propositional formula involves variables, as in $f(x) < 4 \implies f(x) < 0$, then we can regard the formula as a function mapping the value of its (free) variables $f,x$ to a proposition. This is why it can be called a "propositional function". Its "range of significance" are the values $(f,x)$ which make the proposition meaningful. We can't let e.g. $f = 5$ and $x = \mathbb{N}$, so that pair is outside the range.

In modern terms, indeed, this is what we usually call "domain of a predicate".

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  • $\begingroup$ Can I say that a propositional function is indeed a predicate? $\endgroup$ – jonaprieto May 3 '16 at 4:46
  • $\begingroup$ @jonaprieto Yes, they are the same thing. $\endgroup$ – chi May 3 '16 at 7:19
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Comes from Russell, and it's actually about the domain, not the codomain, if I understand ol' Bertie correctly. A propositional function like "x>3" only has significance if we limit the possible values of x to numbers. Today we would call that the domain, but his choice of words is defensible: the range of significance of a propositional fn is the set of values that make it meaningful - i.e. true or false - as opposed to meaningless nonsense, like "Pegasus>3". The range of significance of a numeric fn is the set of values for which it yields a value, rather than crashing, so to speak.

The relation to types is fairly obvious: the range of significance of a fn $f: A\rightarrow B$ is just the "values" of type $A$ to which $f$ can be meaningfully applied. The concept and term "type" is just a more commodious way of expressing the same basic concept. Google "propositional function range of significance" and you'll get links to his original paper and lots more.

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