An Introduction to Paraconsistent Integral Differential Calculus: With Application Examples

Author(s)
João Inácio Da Silva Filho

ABSTRACT

In this paper we show that it is possible to integrate functions with concepts and fundamentals of Paraconsistent Logic (PL). The PL is a non-classical Logic that tolerates the contradiction without trivializing its results. In several works the PL in his annotated form, called Paraconsistent logic annotated with annotation of two values (PAL2v), has presented good results in analysis of information signals. Geometric interpretations based on PAL2v-Lattice associate were obtained forms of Differential Calculus to a Paraconsistent Derivative of first and second-order functions. Now, in this paper we extend the calculations for a form of Paraconsistent Integral Calculus that can be viewed through the analysis in the PAL2v-Lattice. Despite improvements that can develop calculations in complex functions, it is verified that the use of Paraconsistent Mathematics in differential and Integral Calculus opens a promising path in researches developed for solving linear and nonlinear systems. Therefore the Paraconsistent Integral Differential Calculus can be an important tool in systems by modeling and solving problems related to Physical Sciences.

KEYWORDS

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

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

Cite this paper

Da Silva Filho, J. I. (2014) An Introduction to Paraconsistent Integral Differential Calculus: With Application Examples.*Applied Mathematics*, **5**, 949-962. doi: 10.4236/am.2014.56090.

Da Silva Filho, J. I. (2014) An Introduction to Paraconsistent Integral Differential Calculus: With Application Examples.

References

[1] Da Costa, N.C.A. (2000) Paraconsistent Mathematics. In: Batens, D., Mortensen, C., Priest, G. and Bendegen van, J.P., Eds., I World Congress on Paraconsistency1998 Ghent, Belgium, Frontiers in Paraconsistent Logic: Proceedings, King’s College Publications, London, 165-179.

[2] Jas’kowski, S. (1969) Propositional Calculus for Contradictory Deductive Systems. Studia Logica, 24, 143-157.

http://dx.doi.org/10.1007/BF02134311

[3] Da Costa, N.C.A. (1986) On Paraconsistent Set Theory. Logique et Analyse, 115, 361-371.

[4] 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, 328.

[5] 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

[6] Da Silva Filho, J.I. (2012) Analysis of the Emissions Spectral line of the Paraquantum with Hydrogen Atom. Journal of Modern Physics, 3, 233-254.

http://dx.doi.org/10.4236/jmp.2012.33033

[7] Da Silva Filho, J.I. (2012) An Introductory Study of the Hydrogen Atom with Paraquantum Logic. Journal of Modern Physics, 3, 312-333.

http://dx.doi.org/10.4236/jmp.2012.34044

[8] 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

[9] Stroyan, K.D. and Luxemburg, W.A.J. (1976) Introduction to the Theory of Infinitesimals. Academic Press, New York.

[10] Bell, J.L. (1998) A Primer of Infinitesimal Analysis. Cambridge University Press, Cambridge.

[11] Baron, M.E. (1969) The Origins of the Infinitesimal Calculus. Pergamon Press, Hungary.

[12] Keisler, H.J. (1976) Elementary Calculus: An Infinitesimal Approach. 1st Edition, Prindle, Weber & Schmidt, Boston.

[13] 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

[14] Pl Tipler, A. and Llewellyn, R.A. (2007) Modern Physics. 5th Edition, W. H. Freeman and Company, New York.

[15] Kleene, S.C. (1952) Introduction to Metamathematics. North Holland/Van Nostrand, Amsterdam/New York.

[1] Da Costa, N.C.A. (2000) Paraconsistent Mathematics. In: Batens, D., Mortensen, C., Priest, G. and Bendegen van, J.P., Eds., I World Congress on Paraconsistency1998 Ghent, Belgium, Frontiers in Paraconsistent Logic: Proceedings, King’s College Publications, London, 165-179.

[2] Jas’kowski, S. (1969) Propositional Calculus for Contradictory Deductive Systems. Studia Logica, 24, 143-157.

http://dx.doi.org/10.1007/BF02134311

[3] Da Costa, N.C.A. (1986) On Paraconsistent Set Theory. Logique et Analyse, 115, 361-371.

[4] 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, 328.

[5] 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

[6] Da Silva Filho, J.I. (2012) Analysis of the Emissions Spectral line of the Paraquantum with Hydrogen Atom. Journal of Modern Physics, 3, 233-254.

http://dx.doi.org/10.4236/jmp.2012.33033

[7] Da Silva Filho, J.I. (2012) An Introductory Study of the Hydrogen Atom with Paraquantum Logic. Journal of Modern Physics, 3, 312-333.

http://dx.doi.org/10.4236/jmp.2012.34044

[8] 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

[9] Stroyan, K.D. and Luxemburg, W.A.J. (1976) Introduction to the Theory of Infinitesimals. Academic Press, New York.

[10] Bell, J.L. (1998) A Primer of Infinitesimal Analysis. Cambridge University Press, Cambridge.

[11] Baron, M.E. (1969) The Origins of the Infinitesimal Calculus. Pergamon Press, Hungary.

[12] Keisler, H.J. (1976) Elementary Calculus: An Infinitesimal Approach. 1st Edition, Prindle, Weber & Schmidt, Boston.

[13] 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

[14] Pl Tipler, A. and Llewellyn, R.A. (2007) Modern Physics. 5th Edition, W. H. Freeman and Company, New York.

[15] Kleene, S.C. (1952) Introduction to Metamathematics. North Holland/Van Nostrand, Amsterdam/New York.