If classical logic is inconsistent then so is classical mathematics.(adsbygoogle = window.adsbygoogle || []).push({});

A. x=y -> (Fx <-> Fy). This is an axiom of first order logic and it is a theorem of second order logic. (Leibnitz's Law)

A is a theorem of Principia Mathematica, *13.15.

1. x=y -> ([](x=x) <-> [](x=y)).

therefore,

2. x=y -> [](x=y)?? (Because [](x=x) is an axiom or a theorem)

3. Herkyl=Herkyl -> [](Herkyl=Herkyl) ??

Surely it is true that Herkyl is self identical, but, it is false to say that Herkyl=Herkyl is logically true.

B. y=(the x:Gx) -> (Fy <-> F(the x:Gx)).

B is a theorem of Principia Mathematica *14.15.

1a. (the x:Gx)=y -> ([](y=y) <-> []((the x:Gx)=y).

2a. (the x:Gx)=y -> []((the x:Gx)=y)??

3a. (the number of planets)=9 -> []((the number of planets)=9)??

It is clearly false to say that (Herkyl=Herkyl) is necessarily true, and,

It is clearly false to say that (the number of planets)=9, is necessarily true.

That is to say the logical deductions, 3 and 3a, are false.

Therefore classical logic is inconsistent!

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# Is classical logic inconsistent?

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