SUSZKO S CONTRIBUTION TO THE THEORY OF NONAXIOMATIC PROOF SYSTEMS

Transcription

1 Bulletin of the Section of Logic Volume 38:3/4 (2009), pp Andrzej Indrzejczak SUSZKO S CONTRIBUTION TO THE THEORY OF NONAXIOMATIC PROOF SYSTEMS Abstract The paper is an exposition of the early works of Suszko devoted to the construction of nonaxiomatic proof systems. He provided a formalization of firstorder logic with identity and definite descriptions of quite an original character, based on the application of proper inference rules. A rule is proper, in Suszko s sense, iff schemata of premisses and conclusion are schemata of contingent formulae. According to Suszko, neither of nonaxiomatic systems proposed so far (e.g. Jaśkowski s and Gentzen s work on natural deduction and sequent calculi) satisfies this requirement. The proposed system formalizes, in a proper way the first-order logic but without identity; for the latter one axiomatic rule is needed. Suszko s system may be seen as an early attempt to provide a kind of sequent calculus which is in a sense opposite to Gentzen s approach. In the latter only one primitive sequent is postulated and all logical constants are characterised by means of rules of transformation of sequents. In Suszko s system the only rules transforming sequents are structural whereas all constants are characterised by primitive sequents representing (proper) inference rules. 1. Introduction It is well known that in 1930 s Jaśkowski [14] and Gentzen [5] independently started their succesfull investigations on natural deduction and sequent calculi. In the earlier works of Hertz [9] and Herbrand [8] one may also find some evidence for the need of more natural and inference oriented kind of formalizations of logic. After the Second World War a plethora of several types of nonaxiomatic proof systems were invented with different properties

2 152 Andrzej Indrzejczak and for different applications. One of the early proposal, of quite an original character, was due to young Roman Suszko, and presented in [19] and [20] based on his PhD thesis. His attempts resulted from the dissatisfaction with the earlier systems of Herbrand, Gentzen and Jaśkowski. The problem he posed is called by him the problem of logic without axioms and may be expressed as follows: Is it possible to build a calculus for classical logic in which all the logical axioms are dispensed with? Instead of logical axioms only proper inference rules must be assumed as primitive. Theses of logic (including axioms) may be deduced by means of these rules from arbitrary formulae. It is a way of expressing the idea that analytical sentences (= logical theses) are implied by any sentences. In Suszko s opinion all nonaxiomatic systems proposed so far comprise some rules which are not proper. But what is a proper inference rule? According to Suszko s view, a rule is proper if it is represented by a schema, where all premises and conclusion are schemata of contingent formulae only. He claims that, e.g. proof construction rules comprising Deduction Theorem are not proper in this sense because a conclusion of its application is an implicational thesis. The same applies to other rules of this sort like Indirect Proof or Proof by Cases. Suszko is referring here to Ajdukiewicz s view from [1] and treats such rules as the examples of axiomatic postulates. It seems that the objection against such proof construction rules like Conditional Proof is exagerated since their schemata do not exhibit schemata of theses in conclusion, moreover not all their applications lead to theses (e.g. if not all assumptions are discharged). In fact, it is difficult to compare Suszko s proper rules with proof construction rules since the latter are rather transformations of proofs not of formulae. It is a difference between G-rules and H-rules in the terminology of [22], and although for every H-rule we may provide corresponding G-rule, the opposite does not hold. Anyway, Suszko rejected such a kind of rules and his system is not a system of natural deduction in the strict sense (c.f. a characterisation of different forms of ND in [12]) but may be rather seen as a kind of sequent calculus; we will return to this issue after presenting a system. 2. The System The system presented in [20] is called by Suszko LS. The objects considered by Suszko as formal representant of rules are called by him logistic sequences and displayed by means of the following schemata:

3 Suszko s Contribution to the Theory of Nonaxiomatic Proof Systems 153 ϕ 1.. ϕ n ψ where, ϕ 1,..., ϕ n, ψ are schemata of formulae of suitable language. Informally, corresponds to deducibility relation generated by axiom calculus. In what follows we identify Suszko s sequences with Gentzen s sequents of the form: ϕ 1,..., ϕ n ψ where, ϕ 1,..., ϕ n represents an antecedent (of a sequent) being a list of formulae, and ψ is a succedent always containing only one formula. In [19] Suszko presented only an adequate formalization of CPL (Classical Propositional Calculus) in the language with implication and negation, comprising the following 7 sequents: ϕ ψ ϕ (ϕ ψ) χ ψ χ ϕ (ψ χ) (ϕ ψ) (ϕ χ) ϕ ψ (χ ϕ) ((ψ γ) (χ ψ)) ϕ, ϕ ψ ψ (ϕ ψ) χ ϕ χ ϕ ψ, ϕ ψ ψ In [20] the rest of the rules are supplied. For other propositional constants we have the following sequents: ϕ ψ ϕ χ ψ ϕ ψ χ ϕ ψ ϕ (ψ χ), γ (ψ δ) ϕ (γ (ψ χ δ)) (ϕ ψ) χ (ϕ ψ) χ (ϕ ψ) χ (ψ ϕ) χ ϕ (ψ χ), γ (χ ψ) ϕ (γ (ψ χ)) ϕ ψ ϕ ψ χ ϕ ψ ϕ χ ψ ϕ (ψ χ), γ (δ χ) ϕ (γ (ψ δ χ))

4 154 Andrzej Indrzejczak Alternatively, instead of sequents , Suszko considers a set of 8 sequents which are also sufficient to capture the (classical) meaning of,, : ϕ (ψ χ) ϕ ψ χ ϕ ψ χ ϕ (ψ χ) ϕ (ψ χ), ϕ (χ ψ) ϕ (ψ χ) ϕ (ψ χ) ϕ (ψ χ) ϕ (ψ χ) ϕ (χ ψ) ϕ ψ, χ ψ ϕ χ ψ ϕ ψ χ ϕ χ ϕ ψ χ ψ χ Suszko preferred this shorter set of rules in the later publications, e.g. in [21], where he used a revised LS. Also, instead of he used there a more readable: ϕ ψ (ψ χ) (ϕ χ) For quantifiers he provided: 2.1. ϕ(α) xϕ(x) x(ϕ ψ(x)) ϕ xψ(x) x(ψ(x) ϕ) xψ(x) ϕ ϕ xψ(x) ϕ ψ(α) xϕ(x) ψ ϕ(α) ψ where α represents apparent variable substituted properly by a real variable x bounded by a quantifier, and in schemata and ϕ has no occurence of x. Finally, for identity and definite descriptions we have 3 sequents: 3.1. ϕ x, x = α 3.2. ϕ(τ 1 ) τ 1 = τ 2 ϕ(τ 2 ) where τ i represents any term and ϕ in 3.1. is any formula 4. x(ϕ(x) y(ψ(y) y = x)) (ϕ(a) z v(ψ(v) v = z)) ϕ(ιxψ(x)) where a is a special name phrase representing a descript for which a

5 Suszko s Contribution to the Theory of Nonaxiomatic Proof Systems 155 substitution of other terms is not allowed. One should note that in Suszko s system the theory of definite descriptions due to Hilbert and Bernays is comprised. In order to derive other sequents from the primitive ones Suszko uses several forms (actually 6) of the rule of substitution from axiomatic calculus but applied on the whole sequent, and 4 structural rules taken from Gentzen s sequent calculus. These are the well known rules of weakening, permutation and contraction (in antecedents) and cut represented by the following schemata: ϕ 1,..., ϕ n ψ ϕ 1,..., ϕ n, ϕ n+1 ψ ϕ 1,..., ϕ i,..., ϕ j,..., ϕ n ψ ϕ 1,..., ϕ j,..., ϕ i,..., ϕ n ψ ϕ 1,..., ϕ i,..., ϕ i,..., ϕ n ψ ϕ 1,...,..., ϕ i,..., ϕ n ψ ϕ 1,..., ϕ n ψ ψ, χ 1,..., χ k γ ϕ 1,..., ϕ n, χ 1,..., χ k γ These rules are explicitly stated in [20], whereas in [19] we have only an implicit application of them. 3. Proofs Suszko did not pay an attention to the form of representation of proofs in his system. In [19] only informal (although detailed) description of derivations of three Lukasiewicz axioms for CPL is provided. In [20] more than one hundred derivable sequents is stated with hints concerning their justification, but no formal proof is provided. It seems quite natural to treat his system as a kind of sequent calculus and define proofs as binary trees with leaves being (instances of) primitive sequents and a root being a derivable sequent. Clearly, the instances of all previously derived sequents may be used as leaves of proof trees in further development of the theory. Let us provide some simple example: p (q r) (p q) (p r) (p q) (p r) q (p r) p (q r) q (p r) cut This simple proof tree provides a justification for a schema of commutation rule: ϕ (ψ χ) ψ (ϕ χ)

6 156 Andrzej Indrzejczak In what follows for the sake of compactness we display proofs in linear format, however, since in contrast to Gentzen s system trees in Suszko s system tend to use quite long sequents already as leaves and it is hard to put them in the page. It is an interesting feature of this system that only sequents with nonempty antecedent may be deduced from the primitive sequents of the system by means of these structural rules, again, in contrast to Gentzen s system, where theses are sequents of the form ϕ. Suszko s sequents naturally represent inference rules but in what way can we derive theses? After all, his system is devised for classical logic not for purely inferential logics like e.g. Kleene s K3. Theses of classical logic are derivable as the so called absolute sequents of the form ϕ ψ, where ϕ is an atomic formula not occuring as a subformula of ψ. Let us illustrate a proof of a thesis (i.e. an absolute sequent), this time in a linear format: 1. p p (p p) ((p q) (p q)) p p, (p p) ((p q) (p q)) (p q) (p q) p p, p p (p q) (p q) 1, 2, cut 4. p p (p q) (p q) 3, contraction 5. p p p p (p q) (p q) 4, 5, cut 7. (p q) (p q) p ((p q) q) commutation 8. p p ((p q) q) 6, 7, cut 9. p, p ((p q) q) (p q) q p, p (p q) q 8, 9, cut 11. p (p q) q 10, contraction 12. (p q) q q q p q q 11, 12, cut Note, by the way, that a proof of q q is straightforward in Suszko s system (one cut on and ), but we cannot apply deduction theorem to obtain an absolute sequent from line 13. above. In practice, LS treated as a kind of sequent calculus has a serious drawback concerning the proof display. It may be partly overcome if we apply a linear format but still proofs tend to be not only quite lengthy but also operate on often long sequents. It may be simplified a bit if we define a proof

7 Suszko s Contribution to the Theory of Nonaxiomatic Proof Systems 157 in LS not as a linear sequence of sequents, but a linear sequence of formulae, where the first lines contain elements of the antecedent (of a proved sequent) and the last line is a succedent. Hence, the primitive sequents are not items in proof but rather metalevel descriptions of primitive inference rules working on formulae. Thus the proof of p q q displayed above may be rewritten in a slightly more compact and readable way: 1. p assumed antecedent 2. p p 1, (p p) ((p q) (p q)) 2, (p q) (p q) 2, 3, p ((p q) q) 4, commutation 6. (p q) q 1, 5, q q 6, In justification column we only refer to the number of primitive (or a derivable) sequent, and the lines where its antecedents are displayed. The application of cut, contraction e.t.c. are implicit. This form of displaying proofs makes LS more manageable (shorter proofs made of shorter lines) and the obtained system is more in natural deduction spirit. But it is not a natural deduction system; there are no subordinate proofs and no rules for discharging assumptions, which is essential in this type of systems. 4. Adequacy of LS Suszko provided a syntactical proof of adequacy of his system. As for [19], in one direction it consists in noting that inference rules expressed by sequents are derivable in Lukasiewicz s axiomatization of CPL in implicational-negational language. In the other direction, Suszko provides a detailed account of how to prove in his system the following three absolute sequents: 1. s (p q) ((q r) (p r)) 2. s p ( p q) 3. s ( p p) p which are the counterparts of Lukasiewicz s axioms; the rule of detachement (Modus Ponens) is already present as a sequent In [20] Adequacy of the full system LS is proved by demonstration of equivalence with axiomatic

8 158 Andrzej Indrzejczak system LF which is Hilbert-Ackermann formalization of first-order logic with definite descriptions. It is exposed in 3 theorems: 1. If LF ϕ, then LS ψ ϕ, for any absolute sequent ψ ϕ 2. LS ϕ 1,..., ϕ n ψ iff LF ϕ 1 (...(ϕ n ψ)...), provided no real variables are present in ϕ 1,..., ϕ n 3. LF ϕ iff LS ψ ϕ, for any absolute sequent ψ ϕ We omit a proof. By the end of [20] Suszko made some remarks concerning the formalization of some nonclassical propositional logics. 1. A system comprising only sequents provides a formalization of PPL (Positive Propositional Logic). 2. Addition of: (ϕ ψ) ϕ ϕ to LS-PPL yields implicational CPL. 3. Replacement of in propositional part ( ) by any of: ϕ ψ (ϕ ψ) ψ ϕ ψ (ϕ ψ) ϕ provides a formalization of IPL (Intuitionistic Propositional Logic). 4. Finally, if in LS-IPL we replace also by: (ϕ ψ) χ ϕ χ we obtain a formalization of MPL (Minimal Propositional Logic). 5. Final Remarks Suszko himself was not wholly satisfied with LS because the project of total elimination of axioms in favor of proper inference rules was not fully realised. Namely, sequent 3.1. is an absolute sequent expressing cryptoaxiomatic rule. Hence, although, for LS, it holds that if a sentence is derivable from any sentence, then it is analytic (i.e. a thesis), the converse does not

9 Suszko s Contribution to the Theory of Nonaxiomatic Proof Systems 159 hold, at least when identity is concerned. But Suszko was mistaken in this respect. One may use e.g. sequents corresponding to Kalish/Montague [15] rules for identity: x(x = τ ϕ(x)) ϕ(τ) ϕ(τ) x(x = τ ϕ(x)) These sequents express proper rules in the sense of Suszko and are sufficient to obtain a complete system for first-order logic with identity. So, in fact, for LS with these rules replacing 3.1. and 3.2. it holds, as well, that if a sentence is analytic, then it is derivable from any sentence whatsoever. Our interpretation of LS as a kind of sequent calculus is quite a natural solution. Seen in this way LS and the original Gentzen s LK provide the extreme solutions with respect to the proportion between primitive sequents and primitive rules of inference (of sequents from sequents). Let us recall that Gentzen [5] uses only one kind of (structural) sequents ϕ ϕ as a starting point (leaves of proof-trees); all logical constants are defined by rules. Suszko, on the contrary, defines all logical constants by means of primitive sequents and uses only a small number of structural rules. Although Suszko does not mention Hertz [9], his solution is similar to Hertz idea of defining purely structural calculus (in the sense of the set of rules), where logical content is expressed by suitable sequents. In fact, also Gentzen in the earlier work [4] has made an interesting contribution to the theory of such sequent calculi. The approach of Suszko was mentioned and developed in some works of Polish logicians, namely [3], [13], [18] (cf. Pogonowski [16]). One may note, however, that later on a lot of sequent calculi were proposed that stay in betweeen Gentzen s and Suszko s approaches, e.g. systems of Bernays [2], Hasenjaeger [7], Rieger [17]. In all of them the logical constants are characterised partly by some primitive sequents and partly by sequential rules.

10 160 Andrzej Indrzejczak References [1] K. Ajdukiewicz, Sprache und Sinn, Erkenntniss, IV (1934), pp [2] P. Bernays, Betrachtungen zum Sequenzen-Kalkül, in: Contributions to Logic and Methodology ed. A. T. Tymieniecka, North-Holland (1965). [3] D. Bobrowski, Bezaksjomatyczne systemy rachunku zdań (1951), MSC thesis, Poznań. [4] G. Gentzen, Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen, Mathematische Annalen, 107 (1932), pp [5] G. Gentzen, Untersuchungen über das Logische Schliessen, Mathematische Zeitschrift 39 (1934), pp and 39: [7] G. Hasenjaeger, Introduction to the Basic Concepts and Problems of Modern Logic, Springer (1971). [8] J. Herbrand, Recherches sur la theorie de la demonstration, in: Travaux de la Societe des Sciences et des Lettres de Varsovie, Classe III, Sciences Mathematiques et Physiques, Warsovie, [9] P. Hertz, Über Axiomensysteme für beliebige Satzsysteme, Mathematische Annalen, 101 (1929), pp [10] W. Hetper, Rachunek zdań bez aksjomatów, Archiwum Towarzystwa Naukowego we Lwowie, Dzia l III, Tom X, Lwów (1938). [11] D. Hilbert, W. Ackermann, Grundzüge der theoretischen Logik, Berlin (1938). [12] A. Indrzejczak, Natural Deduction, Hybrid Systems and Modal Logics, Trends in Logic, vol. 30 (2010), Springer. [13] J. Iwanicki, Dedukcja naturalna i logistyczna, Warszawa: Nak ladem Polskiego Towarzystwa Teologicznego (1949). [14] S. Jaśkowski, On the Rules of Suppositions in Formal Logic, Studia Logica 1(1934) pp [15] D. Kalish, R. Montague, Logic, Techniques of Formal Reasoning, Harcourt, Brace and World, New York (1964). [16] J. Pogonowski, Okres Poznański w twórczości Romana Suszki, [in:] J. Pelc (ed.) Sens, Prawda, Wartość, Warszawa [17] L. Rieger, Algebraic Methods of Mathematical Logic, Academia, Prague (1967).

11 Suszko s Contribution to the Theory of Nonaxiomatic Proof Systems 161 [18] J. S lupecki, O w laściwych regu lach inferencyjnych, Kwartalnik Filozoficzny, 18(1949), [19] R. Suszko, W sprawie logiki bez aksjomatów, Kwartalnik Filozoficzny, 17(1948), [20] R. Suszko, O analitycznych aksjomatach i logicznych regu lach wnioskowania, Poznańskie Towarzystwo Przyjació l Nauk, Prace Komisji Filozoficznej, 7/5 (1949), [21] R. Suszko, Formalna teoria wartości logicznych I, Studia Logica, VI (1957), [22] R. Wójcicki, Theory of Logical Calculi, Kluwer (1988). Department of Logic University of Lódź Kopcińskiego 16/ Lódź indrzej@filozof.uni.lodz.pl