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Single Idea 18958

[filed under theme 4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set ]

Full Idea

In the theory of types, 'x ∈ y' is well defined only if x and y are of the appropriate type, where individuals count as the zero type, sets of individuals as type one, sets of sets of individuals as type two.

Gist of Idea

In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type

Source

Hilary Putnam (Philosophy of Logic [1971], Ch.6)

Book Ref

Putnam,Hilary: 'Philosophy of Logic' [Routledge 1972], p.48

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The 14 ideas from 'Philosophy of Logic'

18949

The universal syllogism is now expressed as the transitivity of subclasses [Putnam]

18951

For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam]

18950

Physics is full of non-physical entities, such as space-vectors [Putnam]

18955

Having a valid form doesn't ensure truth, as it may be meaningless [Putnam]

18952

'⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)' [Putnam]

18953

Modern notation frees us from Aristotle's restriction of only using two class-names in premises [Putnam]

18954

Before the late 19th century logic was trivialised by not dealing with relations [Putnam]

18956

Asserting first-order validity implicitly involves second-order reference to classes [Putnam]

18957

Nominalism only makes sense if it is materialist [Putnam]

18958

In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam]

18959

Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam]

18960

Most predictions are uninteresting, and are only sought in order to confirm a theory [Putnam]

18962

Unfashionably, I think logic has an empirical foundation [Putnam]

18961

We can identify functions with certain sets - or identify sets with certain functions [Putnam]