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St. Jaśkowski andG. Gentzen are the first authors of the logical systems based on rules. The logical systems based on the rules constructed by these authors had been presented in the papers:St. Jaśkowski.On the Rules of Suppositions in Formal Logic. “Studia Logica” 1. Warszawa 1934.G. Gentzen.Untersuchungen über das logische Schliessen. Mathematische Zeitschrift. Bd. 39. Berlin 1935. At the beginning of his paper Jaśkowski writes that he presents the solution of the problem formulated by J. Ŀukasiewicz in 1926. Jaśkowski presented the first results concerning this problem in the lectureTeoria dedukcji oparta na dyrektywach założeniowych, delivered at the meeting of the I Polish Mathematical Congress in Lwów, 1927. (The title of this lecture is given on the page 36 ofKsięga Pamiątkowa I Polskiego Zjazdu Matematycznego, Kraków 1929).
This system had been presented at the meeting of the Polish Mathematical Society in Wrocław on June 9, 1953 (CompareJ. Słupecki Sur une méthode de noter les démonstrations en symboles logiques. Colloquium Mathematicum III. 2 (1955), p. 210).
CompareT. Czeżowski.Logika. Warszawa 1949, p. 65.
The essential idea of the rule of constructing the direct suppositional proof is contained in the manner of constructing such proofs given by Jaśkowski who uses however another notation manner of such proofs. The circumstance that the notation manners of suppositional proofs applied by Jaśkowski and Gentzen deviate too much from the usual notation manner of proofs may be one of the main obstacle in using their systems in the didactic. Therefore in later papers of the manual character in which the rules of constructing the suppositional proofs appear we may observe the tendency to approach the notation manner of such proofs to the usual one. The formulations of constructing the (direct or apagogical) suppositional proofs presented here take into account also the possibility of adding to the proof the theses already proved as new lines while in other systems of “natural deduction” the authors aim at proving a given thesis without referring to theses already proved. This aim may be also achieved by such a formulation of the rules of constructing the suppositional proofs which allows us to include into the proof of a given thesis the proof of another thesis to which we should refer in the proof. From the didactic and practical point of view it seems rather the formulation of the rule of constructing the suppositional proofs to be more suitable which allows us to add immediately to the proof the theses already proved.
Φ1 is the antecedent of the expression (I), Φ2 is the antecedent of the expression we obtain from (I) by cancelling the expression Φ1 and the immediately following sign of implication, etc.
The abbreviation of the expression: the suppositions.
The casen=0 is here also taken into consideration.
The abbreviation of the expression: the supposition of the apagogical proof.
This rule, known in the usual systems as the deduction theorem, is used by Jaśkowski (Ch, II, on the page 10) and by Gentzen (FE, on the page 186) as one of the primitive rules. This rule together with primitive rules enables us to prove each thesis without referring to the theses previously proved.
This theorem is valid for each propositional calculus which is based on the rule of detachment and in which the axioms are adequate to prove the theses:p → p, p → (q → p), [p → (q → r)] → [(p → q) → (p → r)]. (Compare e.g.Rosser.Logic for Mathematicians, p. 75). It can be easily verified that the intuitionistic propositional calculus satisfies these conditions.
RA may be limited to the case 1≤n≤2. It is worth noticing that this remark concerns also the rule of constructing the apagogical suppositional proof accepted for the classic propositional calculus.
CompareLewis-Langford:Symbolic Logic, pp. 501, 493, 497, 123–126.
We take here into consideration the system S4 without definitions of some terms, for example the signs of alternation, of material implication, etc. It may be easily shown that supplementing the system SL with the suitable rules for these terms, given in Ch. I, we obtain the system equivalent to the system S4 supplemented with the definitions of these terms.
CompareRuth Barcan Marcus.Strict implication, deducibility and the deduction theorem. The Journal of Symbolic Logic. V. 18. N. 3, pp. 234–236.
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Borkowski, L., Słupecki, J. A logical system based on rules and its application in teaching mathematical logic. Stud Logica 7, 71–113 (1958). https://doi.org/10.1007/BF02548936
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DOI: https://doi.org/10.1007/BF02548936