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A partial answer might come through an introduction. Well, we know that Russell's efforts to understand the contradictory appearance of the class of all classes not members of themselves (a notion also apparent in Frege's analysis) produced the theory of types. But how is this useful? There are difficulties with Russell's theory of types (if indeed he is the original proponent of types). For instance, Aristotelian categories are said to falsely set limits for meaning; therefore, these are not proper subject matter for a theory. Following Russell, Wittgenstein first proposed this. In his early work, Tractatus..., Wittgenstein probably was the first to suggest the limits of reason or what can be demonstrated, but not necessarily rationally spoken. Wittgenstein enjoyed silence. Everyone knows the stories. Rational clarity is to be achieved through an ideal language. Some call this language, mathematics. But, mathematics, too, does not transcend the certain necessary limits in cognitive meaning. In other words, there are no answers. Wittgenstein knew this failure in both logic (symbols) and reason (intellectual movement) and it led him to obvious psychological despair. In philosophy, after about the mid-1930s, there is no universal scheme of categories (Aristotelian epistemology). This condition has remained. Why? Are absolutes unnecessary? Wittgenstein's influence may be seen in the present-day analytical philosophers claim that categories should have absolute universality (Aristotle) onward to the theory of types (Russell). But, this is like chasing specificity to the last millimeter. What is the difference between Aristotle's categories and Russell's types?

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The main difference is that Aristotle's Categories is an ontological theory, i.e. a system of classification of beings, while Whithead and Russell's Type Theory is a logico-mathematical theory.

The interpretation of Aristotle's theory is not an easy task (see also Aristotle' logic) :

The word “category” (katêgoria) means “predication”. Aristotle holds that predications and predicates can be grouped into several largest “kinds of predication”. [...] the categories may be kinds of predicate [or] may be seen as kinds of entity. [...] Which of these interpretations fits best with the [original texts] ? The answer appears to be different [...] : the Categories lists substance (ousia) in first place, while the Topics list what-it-is (ti esti). A substance, for Aristotle, is a type of entity, suggesting that the Categories list is a list of types of entity.

On the other hand, the expression “what-it-is” suggests most strongly a type of predication.

The original version of Type Theory contained into Principia Mathematica has also "ontological implications", but they are quite obscure, due to the intoduction of the ramified hierarchy and the reducibility axiom.

The subsequent development of the theory, produced a "streamlined" theory (see : Church's Type Theory) :

type theory is a formal logical language which is particularly well suited to the formalization of mathematics and other disciplines and to specifying and verifying hardware and software. It also plays an important role in the study of the formal semantics of natural language.

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    Nice summary with Web citations. Adjunct questions: Is 'being' always ontologic and therefore contingent? Thus, is 'becoming' always rationalistic-empiric and therefore necessary (must)? – Dallas-Rey Davis Jan 14 '15 at 20:20
  • @Dallas-ReyDavis - see Potentiality and actuality in Aristotle. – Mauro ALLEGRANZA Jan 14 '15 at 20:23
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    @Mauro: Yea, but yet, "Is being always ontologic and therefore contingent; and, is becoming always empiric and therefore a necessary must?" Remember the existentialist dichotomy detailed in the book, Being and Nothingness. They actually meant being/becoming, but made an appeal to Nietzsche at the last, thus becoming metamorphosed into nothingness, Nietzsche's code for the ontologic... Yea, in dichotomy, the moibus eventually folds into itself. – Darcy Davis Jan 14 '15 at 21:02
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    @DarcyDavis: In Hegel, it is Being and Non-Being that sublates into becoming. – Mozibur Ullah Jan 17 '15 at 11:00
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    @Mauro: you wrote: "The original version of Type Theory contained in Russell's Principia has also 'ontological implications', but they are quite obscure, due to the introduction of the ramified hierarchy and the reducibility axiom." Where in Russell are these passages? The Encyclopaedia of Philosophy does account for Russell's 'ontology' but does not relate it to the Principia. Would it be possible to develop these 'ontological implicatons' into a Masters thesis or Doctoral dissertation, tentatively titled, say for instance--"Russell's Ontology"? Now, that would be irony! – Dallas-Rey Davis Jan 21 '15 at 21:08
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I'd have to second Mauros answer on the main difference between Russells theory and Aristotles as being that of ontology and logic.

Russells was attempting to understand Freges theory, which was a first attempt to extend mathematical techniques into logic. The Russell Set, showed the limitation of his theory (using a different paraconsistent logic, the Universal Set and the Russell Set can be grounded). So its not a fatal flaw.

Russells theory of types has a new avatar in Category Theory, which is based in a different logic than the classical one; its based on the intuitionistic logic of Brower; and integrates mathematical techniques such as forcing as a geometrical technique; and this is in line with a major paradigm of mathematics - 'geometrisation' is the large sense of the Pythagorean school.

Wittgenstein wasn't the first to suggest the limits of reason; this was done by Hume; and before him by Al-Ghazali, in his The incoherence of Philosophy.

Much further back, if one steps outside 'pure philosophy', and re-enters the world of antiquity, with what was then the favoured discourse of sophia, that is poetry, and that in its form as the verse-drama of tragedy which considered ethical/moral dilemmas; then Oedipus Rex can be considered too in that light - Oedipus was warned away by Tiresias in his reasoned search and demand 'for truth'.

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