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The extended propositional calculus is a system containing quantifiers and variables representing truth-functors apart from propositional variables and truth-functors.
In my paperOn Proper Quantifiers I show that: (a) The universal quantifier with the restricted range belongs to a different semantic category than the usual universal quantifier does; expressions of this semantic category I call quantifiers of two arguments. (b) There exist 215 proper (as distinguished from the numerical) quantifiers of two arguments and among them 212 suffice to define all remaining terms of the propositional and functional calculi only by means of individual variables as bound variables. (The universal disjunctional quantifier “ПD” considered here is one of them). Among the rest there exist 211 proper quantifiers of two arguments such that each of them is sufficient for that purpose provided that we also use propositional or functorial variables as bound variables (one of them is the universal quantifier with the restricted range). If we take into account all proper quantifiers ofn arguments (wheren is an arbitrary natural number) then the set of terms which enable us to define all remaining terms of the propositional and functional calculus is denumerable. The reader might doubt whether in the systems considered here we are really dealing with one term or with a class of typically ambiguous terms, i. e., whether in the expressions: „\(\mathop \Pi \limits_x fx\)“ and „\(\mathop \Pi \limits_p p\)“ the same quantifier has been used. In my paperOn Proper Quantifiers I present arguments for the view according to which the semantic category of a quantifier is independent of the semantic category of the bound variable (and is determined only by the number of arguments belonging to the scope of the quantifier), and this standpoint does not seem to lead to any difficulties or contradictions. From this point of view we may assert that in the examples above there appears the same universal quantifier which in one expression binds the individual variable and in the other—the propositional variable. It is so in the case of the universal quantifier with the restricted range and in the case of the universal disjunctional quantifier. At the same time I should like to emphasize that reference to my paperOn Proper Quantifiers in case of difficulties mentioned above does not presuppose its results to be necessary for understanding the present paper. This one constitutes the separate whole, and to make it intelligible it is sufficient to give the interpretation of the primitive term and to formulate primitive rules and (eventually) axioms of the system. Usually while constructing a system we do not inquire into the semantic category of its primitive terms and, for instance, quantifiers with the restricted range are commonly introduced without considerations concerning their semantic category. (c) In the above mentioned paper I also give some examples of proper quantifiers of two arguments; these are different from the universal quantifier with the restricted range and the universal disjunctional quantifier and make it possible for us to define all the remaining terms of the propositional and functional calculus. The results of the paper consisting in defining all remaining terms of the propositional and functional calculus by means of the universal disjunctional quantifier (and also by means of some other proper quantifier of two arguments) were presented for the first time in my lectureLogical Systems Based on One Primitive Term delivered at the meeting of the Polish Philosophical Society in Wrocław on December 5, 1950. I presented my paperLogic Based on One Primitive Term containing these results at the meeting of Société des Sciences et des Lettres de Wrocław on March 1, 1951. This paper had not been published. Its summary appeared inComptes rendus de la Société des Sciences et des Lettres de Wrocław, 6, 1951 p. 52. Meanwhile I have obtained some new results. These are included in the present paper and in my paperOn Proper Quantifiers, which will appear in “Studia Logica”. I express here my sincere gratitude to Prof. Dr J. Słupecki for his numerous remarks and suggestions.
The expressions of the systems will be written down inŁukasiewicz's symbolic notation without brackets, supplemented with the signs of the universal disjunctional quantifier, of the universal quantifier with the restricted range (see p. 8) and of the universal—negative quantifier (see p. 10 and DI. 7 on p. 20).
Compare, for example,A. Church:Introduction to Mathematical Logic, pp. 107–108.
I use quasi-quotation corners and quotation marks in the same way asQuine does. Compare, for example,W. V. Quine:Logic based on inclusion and abstraction. J. S. L. vol. 2, p. 147, andMathematical Logic I ed. 1947, pp. 33–37.
By the expression we mean here either the class of inscriptions equiform to an inscription occurring inside the quotation so that the quotation name is a singular name of expression thus conceived or by the expression we mean an individual inscription and then the quotation name is a general name denoting all inscriptions equiform to one appearing inside the quotation.
In axiomatic systems we (a) give axioms or axiomatic rules determining the schemata of axioms, and (b) we formulate primitive deductive rules allowing us to deduce new theses on the basis of axioms or theses already proved. The systems based on the rules differ from those based on the axioms in that no axioms or axiomatic rules are given in the former. In the systems based on the rules (in the meaning of the term used in this paper) there appear (a) rules stating that an expression is a thesis provided a suitably constructed direct or indirect suppositional proof exists for it, and (b) rules allowing us to add new lines of proof and possibly to derive new theses on the basis of theses already proved. The rules of the first kind differ from the axiomatic rules in that expressions are reckoned among theses not by simply stating that they have an appropriate form but only after showing that there exists a suitably constructed suppositional proof (direct or indirect). To go into further details it is necessary to give a precise definition of the concept of quasi-axiomatic rules and distinguish them from proper rules. As regards the systems based on the rules compare, for instance, the papers:S. Jaśkowski:On the rules of Suppositions in Formal Logic, 1934.G. Gentzen:Untersuchungen über das logische Schliessen, 1934.R. Suszko:O analitycznych aksjomatach i logicznych regułach wnioskowania, 1948.
CompareK. Ajdukiewicz:Die Definition p. 1, and my paperÜber analytische und synthetische Definitionen. Studia Logica, vol. IV, p. 9.
CompareJ. Łukasiewicz:On Variable Functors of Propositional Arguments. Proceedings of the Royal Irish Academy. Volume 54 (1951). Section A, No. 2, pp. 28–30.
The definition D1.1 is a rule of replacement (compare p. 13, point e) which allows us to replace the expression ⌈DΦψ⌉ by any expression ⌈∏DΦψ⌉, i. e. any expression belonging to the class of expressions ⌈\(\mathop {\Pi D}\limits_\alpha \Phi \psi \)⌉ such that α is a free variable neither in Φ nor in ψ. Definitions of this kind (containing an operator with a variable which does not occur as a free variable in its scope) are sometimes formulated as rules of translating each expression containing the defined term to a single expression which does not contain this term (compare, for instance,Quine:Logic based on inclusion and abstraction p. 146), for example, by suppelementing a provision that a given variable is the first (in a suitably established alpha-betic order) variable which does not occur in the scope of the operator. On the ground of appropriate rules of a system (e. g. the rule of relettering the bound variables) the rule of translating a given expression to any one from among all the expressions (mutually equivalent) of an appropriate form is a derived rule. The definition D1. 1 (or D3. 1), however, is also a correct definition since it satisfies the condition of translatability essential for definitions, enabling us to eliminate the defined term and nothing forbids us to accept as a primitive rule the rule of replacement of a given expression by any one from among all the expressions belonging to an appropriate class of mutually equivalent expressions instead of the rule of replacement of a given expression by a single expression (the first rule being a derived rule in another system). The formulation given here has some advantage over the usual one; namely, in some cases proofs are simplified and in other cases we avoid the necessity of accepting the rule of relettering the bound variables as a primitive rule (compare, for example, the proof of the thesis T 4.6 on the page 43, etc.).
Compare, for instance,L. Chwistek:Granice nauki, p. 96.
It is easy to notice that AI. 3, AI. 4, RI. 2, written down by means of the sign of implication, and DI.8 are equivalent to the axioms and rules of functional calculus given in the book:Hilbert-Ackermann:Grundzüge der theoretischen Logik.
Compare, for example,Hilbert-Bernays:Grundlagen der Mathematik, Bd. I. 1934, pp. 151–155 orA. Church:Introduction to Mathematical Logic, 1944, pp. 9–11.
The above method of constructing definitions of functors dependent upon parameters is known to me from Prof.J. Słupecki's lectures. The rule of substitution for functional expressions is known to have its history, since from the first formulation given in the I edition ofHilbert-Ackermann's bookGrundzüge der theoretischen Logik it has been formulated incorrectly several times. As to correct formulation compare, for example,A. Church:Introduction to Mathematical Logic,Hilbert-Ackermann, op. cit.,Grundzüge der theoretischen Logik. III. ed.,W. V. Quine:Methods of Logic. It is worth noticing that the correct formulation of the rule of substitution for functional expressions—in form of a derived rule with respect to the rule of substitution for variable functors and the rule of constructing correct definitions of functors dependent upon parameters—would have presented no difficulties had the rules for definitions given by Leśniewski been used. In this way we may also get the strongest formulation of the rule just considered which embraces the cases of substitution for functional expressions containing variable functors of an arbitrary number of variable or constant arguments. The formulation of necessary and sufficient conditions of the correctness of the rule just considered becomes then easy and the role of the respective restrictions is quite clear.
A II. 2 and R II. 3 written down by means of the sign of implication constitute the axiom and the rule for the universal quantifier given inHilbert-Ackermann's paperGrundzüge der theoretischen Logik. (compare note 12).
The system SIII (without the definition DIII. 1) is a part of an unpublished system based on the rules constructed by Prof.J. Słupecki (apart from the rules given here, in Prof. J. Słupecki's system there appear also the rules of adding and of omitting the existential quantifier, the latter having been formulated in a very simple and natural manner).
System SIV, is based onŁukasiewicz's implicational axiom (compareJ. Łukasiewicz:W obronie logistyki. Studia Gnesnensia. XV. 1937, p. 22) which together with the rules of substitution, detachment, omitting and adding the universal quantifier to the consequent of the implication and with the definitions of the signs of negation and of particular quantifier gives us usual propositional and functional calculi.
C. A. Meredith:On an Extended System of the Propositional Calculus (Proceedings of the Royal Irish Academy, V. 54, S. A. No. 3).
The rule R V. 1b is the rule of substitution for functional expressions applied to the extended propositional calculus (compare pp. 28–29). CompareJ. Łukasiewicz:On Variable Functors of Propositional Arguments, pp. 27–28 and my reporting articleZ nowszych badań nad rachunkiem zdań, “Studia Logica”, vol. V. The formulation of the rule of substitution for functional expressions presented in this paper enables us, for example, to substitute “r” for “fp” (compare above p. 29).
CompareJ. Łukasiewicz:On Variable Functors of Propositional Arguments, p. 30 and my articleZ nowszych badań nad rachunkiem zdań, “Studia Logica”, vol. V.
This rule was discussed more fully in Ch. IV. § 1. pp. 38–39.
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Borkowski, L. Systems of the propositional and of the functional calculus based on one primitive term. Stud Logica 6, 7–55 (1957). https://doi.org/10.1007/BF02547930
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DOI: https://doi.org/10.1007/BF02547930