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What's the difference between a predicate and a propositional function, both in mathematical aspect and logical aspect?

I also see some texts in wikipedia, which made me confused. "Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed two theories to try to get at this question: the zig-zag theory and the ramified theory of types."

Eric
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In modern terms, i.e. using in first order logic, we can say that a predicate is symbolized with a predicate letter $P(x)$ while a propositional function is expressed by an open formula $\varphi(x)$.

For a brief overview, see Classical Logic: Language.

Regarding the evolution of Russell's theories, from paradoxes to type theory, you can see: Nicholas Griffin (editor), The Cambridge Companion to Bertrand Russell (2003) :

  • Ch.5 Bertrand Russell’s Logicism by Martin Godwyn and Andrew Irvine,

  • Ch.6 The Theory of Descriptions by Peter Hylton,

  • Ch.7 Russell’s Substitutional Theory by Gregory Landini,

  • Ch.8 The Theory of Types by Alasdair Urquhart;

as well as :

  • Alasdair Urquhart, "Russell's zig-zag path to the ramified theory of types" (1988), Russell,8: 82–91.
Mauro ALLEGRANZA
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  • Thanks for answering! It's my first time to hear the noun "open formula". I searched wikipeida just now, seeing that "In mathematical logic, a non-closed formula is a formula which contains free variables. (Note that in logic, a "sentence" is a formula without free variables, and a formula is "open" if it contains no quantifiers, which disagrees with the terminology of this article.) Unlike closed formulas, which contain constants, non-closed formulas do not express propositions; they are neither true nor false. " – Eric Jan 29 '16 at 11:35
  • It makes me confused again. What is the critical difference between a propositional function P(x) and a open formula phi(x) ? wikipedia's description is a little bit messing. – Eric Jan 29 '16 at 11:35
  • @Eric - in First order arithmetic with $<$ as a binary predicate (i.e. $<(x,y)$) the following is an open formula : $\exists z (x < z)$; as you can see, the variable $x$ is not quantified in the formula, i.e. it is free. – Mauro ALLEGRANZA Jan 29 '16 at 19:35