[FOM] Foundational Challenge

[Lukasz T. Stepien]

Neil Tennant says: 

"Do not blame the _strictly classical_ part of logic for any of the
paradoxes.

They all arise for the constructivist. 
They can all be derived using only Core Logic (the relevant fragment of
Intuitionistic Logic). 
And Core Logic suffices for constructive mathematics." 

As you know, there in classical logic are paradoxes of material
implication related to problem of entailment. 

 So, this is interesting issue, whether one can base classical
Arithmetic System on Core Logic mentioned by you. My remark is caused by
this, what Bernd Buldt has written in his paper: The Scope of Gödel's
First Incompleteness Theorem, _Logica Universalis_, vol. 8, 499 - 552,
(2014), on page 531, namely, that Relevant Arithmetic will be more weak
than Classical Arithmetic. 

In the paper: T. J. Stepien and L. T. Stepien, "Atomic Entailment and
Atomic Inconsistency and Classical Entailment", _ J. Math. Syst. Sci. _
5, No.2, 60-71 (2015)) ; arXiv:1603.06621, a new solution of the problem
of entailment has been given - there has been defined atomic entailment
and atomic logic has been  formulated. In the next paper: T. J. Stepien
and L. T. Stepien, "The Formalization of The Arithmetic System on The
Ground of The Atomic Logic", _J. Math. Syst. Sci._ 5, No.9, 364-368
(2015)) ; arXiv:1603.09334, it has been showed than one can base
Classical Arithmetic on atomic logic. 

                                                                        
                                  Lukasz T. Stepien
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