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Single Idea 18952

[filed under theme 4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC ]

Full Idea

The symbol '⊃' (read 'if...then') is used with the definition 'Px ⊃ Qx' ('if Px then Qx') is short for '¬(Px & ¬Qx)'.

Gist of Idea

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

Source

Hilary Putnam (Philosophy of Logic [1971], Ch.3)

Book Ref

Putnam,Hilary: 'Philosophy of Logic' [Routledge 1972], p.26

A Reaction

So ⊃ and → are just abbreviations, and not really a proper part of the language. Notoriously, though, this is quite a long way from what 'if...then' means in ordinary English, and it leads to paradoxical oddities.

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]