A Historical and Comparative Study of Prominent Theories in Natural Deduction

Natural deduction is a formal method for proving and inferring the propositions in logic. In 1934, Jaskowski and Gentzen, the pioneers of modern logic, proposed this method independently of each other as an alternative to the axiomatic theory. Jaskowski by presenting graphical and annotation writing methods, and Gentzen by presenting a tree tableaus model, proved their deductive methods based on several presumptions and a number of rules that attracted the attention of logicians. However, Quine, with his special sharpness and acuteness, found out some shortcomings and put some restrictions on the rules laid down for quantifiers. Later, inspired by Quine, thinkers such as Copi, Fitch, and Lemmon offered other readings of natural deduction. Explaining the theory of natural deduction through a descriptive, historical and analytical method, this paper has examined the most important views of logicians in this field so that by comparing these views, their strengths and weaknesses are revealed.