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### Single Idea 21555

#### [catalogued under 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory]

Full Idea

In his 1903 theory of types he distinguished between individuals, ranges of individuals, ranges of ranges of individuals, and so on. Each level was a type, and it was stipulated that for 'x is a u' to be meaningful, u must be one type higher than x.

Gist of Idea

For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x

Source

Bertrand Russell (The Principles of Mathematics [1903], App)

A Reaction

Russell was dissatisfied because this theory could not deal with Cantor's Paradox. Is this the first time in modern philosophy that someone has offered a criterion for whether a proposition is 'meaningful'?

Book Reference

Russell,Bertrand: 'Essays in Analysis', ed/tr. Lackey,Douglas [George Braziller 1973], p.129

Related Idea

Idea 21554
Sets always exceed terms, so all the sets must exceed all the sets **[Lackey]**