Paraconsistent Differential Calculus (Part II): Second-Order Paraconsistent Derivative

João Inácio Da Silva Filho

Group of Applied Paraconsistent Logic, Santa Cecília University-UNISANTA, Santos, Brazil;Institute for Advanced Studies of the University of S?o Paulo-IEAUSP, S?o Paulo, Brazil.

DOI: 10.4236/am.2014.58107                

Abstract

The Paraconsistent Logic (PL) is a non-classical logic and its main property is to present tolerance for contradiction in its fundamentals without the invalidation of the conclusions. In this paper, we use the PL in its annotated form, denominated Paraconsistent Annotated Logic with annotation of two values-PAL2v. This type of paraconsistent logic has an associated lattice that allows the development of a Paraconsistent Differential Calculus based on fundamentals and equations obtained by geometric interpretations. In this paper (Part II), it is presented a continuation of the first article (Part I) where the Paraconsistent Differential Calculus is given emphasis on the second-order Paraconsistent Derivative. We present some examples applying Paraconsistent Derivatives at functions of first and second-order with the concepts of Paraconsistent Mathematics.

Keywords

Paraconsistent Logic, Paraconsistent Annotated Logic, Paraconsistent Mathematics, Paraconsistent Differential Calculus

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Da Costa, N.C.A. (1974) On the Theory of Inconsistent Formal Systems. Notre Dame Journal of Formal Logic, 15, 497-510. http://dx.doi.org/10.1305/ndjfl/1093891487
[2]	Arruda, A.I. (1989) Aspects of the Historical Development of Paraconsistent Logic. In: Priest, G., Routley, R. and Norman, J., Eds., Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, 99-130.
[3]	Da Silva Filho, J.I., Lambert-Torres, G. and Abe, J.M. (2010) Uncertainty Treatment Using Paraconsistent Logic: Introducing Paraconsistent Artificial Neural Networks. IOS Press, Amsterdam.
[4]	Jas’kowski, S. (1969) Propositional Calculus for Contradictory Deductive Systems. Studia Logica, 24, 143-157. http://dx.doi.org/10.1007/BF02134311
[5]	Da Silva Filho, J.I. (2011) Paraconsistent Annotated Logic in Analysis of Physical Systems: Introducing the Paraquantum hψ Factor of Quantization. Journal of Modern Physics, 2, 1397-1409. http://dx.doi.org/10.4236/jmp.2011.211172
[6]	Da Costa, N.C.A. (2000) Paraconsistent Mathematics. In: Batens, D., Mortensen, C., Priest, G. and van Bendegen, J.P., Eds., I. World Congress on Paraconsistency, 1998 Ghent, Belgium. Frontiers in Paraconsistent Logic: Proceedings, King’s College Publications, London, 165-179.
[7]	Da Silva Filho, J.I. (2011) Paraconsistent Annotated Logic in Analysis of Physical Systems: Introducing the Paraquantum γψ Gamma Factor. Journal of Modern Physics, 2, 1455-1469. http://dx.doi.org/10.4236/jmp.2011.212180
[8]	Stroyan, K.D. and Luxemburg, W.A.J. (1976) Introduction to the Theory of Infinitesimals. Academic Press, New York.
[9]	Pl Tipler, A. and Llewellyn, R.A. (2007) Modern Physics. 5th Edition, W. H. Freeman and Company, New York.
[10]	Diethelm, K. and Ford, N. (2004) Multi-Order Fractional Differential Equations and Their Numerical Solution. Applied Mathematics and Computation, 154, 621-640. http://dx.doi.org/10.1016/S0096-3003(03)00739-2
[11]	Bell, J.L. (1998) A Primer of Infinitesimal Analysis. Cambridge University Press, Cambridge.
[12]	Baron, M.E. (1969) The Origins of the Infinitesimal Calculus. Pergamon Press, Hungary.
[13]	Keisler, H.J. (1976) Elementary Calculus: An Infinitesimal Approach. Prindle, Weber & Schmidt, Boston.