Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic

This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. The generalization of Kalmar’s proof of deducibility for two valued tautologies inside classical propositional logic gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of loading into two valued logic. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and may be for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.


Introduction
Many-valued logic (MVL) as a separate subject was created and developed first by Łukasiewicz [4]. His intention was to use a third, additional truth value for "possible" (or "unknown"). The outcomes of these investigations are, however, the Łukasiewicz's systems, and a series of theoretical results concerning these systems. Essentially parallel to the Łukasiewicz approach, Post [5] introduced the basic idea of additional truth degrees, and applied it to problems of there presentation of functions. Later on, Gödel [6], Jaskowski [7], Kleene [8], Bochvar [9], Belnap [10] and many others continued investigation in this area. In the earlier years of development, this caused some doubts about the use fullness of MVL. In the mean time, however, many interesting applications were found in such fields as logic, mathematics, hardware design, artificial intelligence and some other area sofinformation technologies, therefore the investigations in area of MVL are very actual. Main theoretical results concern to formal systems, which can present different versions of MVL and questions on logical completeness of such systems.
In this paper we focus on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. Generalization of Kalmar's proof of deducibility for two valued tautologies inside classical propositional logic [1] gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of usually loading into two valued logic [11]. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and maybe for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics. First of constructed system based on the logic with one designated value and conjunction, disjunction, implication, defined by Gödel, and negation, defined by permuting the truth values cyclically. Axioms of this system_are generalizations of formulas, using in Kalmar's proof of deducibility for two valued tautologies. Second system is obtained from first one by some restrictions, based on the notions of determinative conjunct and determinative disjunctive normal form (dDNF), introduced by first coauthor for two-valued Boolean functions in [2] and generalized in [3] for 3-valued logic.

k-Valued Logic
Let E k be the set 0, , … , , 1 . We use the well-known notions of propositional formula, which defined as usual from k-valued propositional variables p, q, p i (i≥1) with values from E k, parentheses (,), and logical connectives & , ∨ , ⊃ , ¬ , defined as follow: In considered logic we fix 1 as designated value, so a formula φ with variables p 1 , p 2 ,…p n is called k-tautology if for every 8 9 = 8 , 8 , … , 8 : ∈ < = : assigning 8 j (1≤j≤n) to each p j gives the value 1 of φ.
Our investigations will be focus on the 3-valued logic, but can be generalize for k-valued (k≥3) logics in the future.

Determinative Disjunctive Normal Form for 2-Valued Logic
Here we will use the current concepts of the unit Boolean cube ( < : ) for E= >0,1? , a propositional formula and a classical tautology. The particular choice of a language for presented propositional formulas is immaterial in this consideration. However, because of some technical reasons we assume that the language contains the propositional 2valued variables ( ≥ 1 and (or) ( A ≥ 1; C ≥ 1 , logical connectives ¬, &, ˅, ⊃ and parentheses (,). Following the usual terminology we call the variables and negated variables literals for 2-valued logic. The conjunct K(term) can be represented simply as a set of literals (no conjunct contains a variable and its negation simultaneously).
In [2] the following notions were introduced. We call a replacement-rule each of the following trivial identities for a propositional formulaψ : Here must be deleted repetition of previous sentence Application of a replacement-rule to some word consists in replacing some its subwords, having the form of the left-hand side of one of the above identities, by the corresponding right-hand side.
Let φ be a propositional formula, 1   conjunct are called also φ -determinative or determinative for φ .

Determinative Disjunctive Normal Form for3-Valued Logic
Here we recall the notion of determinative disjunctive normal form for 3-valued logic, given in [3] as follow.
For the other cases we have introduced the following auxiliary relations for replacement Where by ¬HIE is denote a function, which is equal to 0, if value of E is more than 0 and 1, in the opposite case, and by HIE is denote the function, which is equal to 1, if value of E is more than 0, and 0, in the opposite case.
Let φ be a propositional formula of 3-valued logic, be the set of all variables of φ , and (1 m n ≤ ≤ ) be some subset of P .  For example the conjuncts р 1, ¬ р 1 и ¬¬ р 1 , are 1determinative for the formula р 1 ⊃ р 2 ⊃ р 3 ⊃р 1 )). For two first we use only replacement-rules, but for the last one we 1 As usual, given a propositional variable p and 1 E σ ∈ , by p σ we denote the function , must use the auxiliary relations for replacement also. Definition of dDNF for3-valued logic gives by analogy. It is easy proved that 1) if for some tautology φ the minimal number of literals, containing in φ -determinative conjunct, is m ,then φ -determinative DNF has at least 3 m conjuncts; 2) if for some tautology φ there is such m that every conjunct with m literals is φ -determinative, then there is φ -determinative DNF with no more than 3 m conjuncts. By analogy can be define determinative conjuncts and dDNF for k-valued logic with the mentioned properties also. For k-valued logic we must introduce the corresponding replacement-rules and auxiliary relations for replacement.

Definitions and Properties of Main Systems
Here the two main proof systems for above version of 3valued logic are described. The completeness every of them will be proved also.

The System CN3 (CyclicallyNegation3-Valued)
For every formula A, B, C of 3-valued logic and each J , J from the set {0,1/2,1} the following formulas are axioms schemes of CN3 The notions of derivation and derivation from premises are defined as usually.
It is easy to prove the following statement for CN3.
Deduction theorem. Let Γ be a set of some formulas and NandObe some formulas. If the formula B is derived in the system CN3 from the premises Γ and N(Γ, N ⊢ ghM O ) then the formula N ⊃ O is derived in the system CN3 from the premises Γ (Γ ⊢ ghM N ⊃ O).
Really, for proving this theorem it must be use the axioms 1. and 2. , and also the formula N ⊃ N, which in his turn can be derived using the formulas 1. and 2.
Further we will omit the abbreviation CN3 from the notion ⊢ ghM sometimes.
Main Lemma Let This statement is generalization of corresponding Lemma for 2-valued logic (see for example [1]).
Proof is given by induction on number n of variables in the formula A. Theorem. Any formula is derived in CN3 iff it is 3-valued tautology.
Proof. It is obvious that every formula, which is derived in CN3, is 3-valued tautology. Let So, the number of premises is now n-1. Repeating above steps, we obtain finally the derivation of tautology A in CN3.⧠ Note, that this proof is the full analogy to proof of corresponding theorem for the2-valued logic. It is obvious, that after proving the corresponding Main Lemma for any kvalued logic for k≥4, the proof of the theorem, corresponding above, will be the same. Some hardness for generalization of above results is contained in definition of the functions a * N, O, J , J for * ∈ >&,∨, ⊃? to describing the corresponding axioms schemas 3.-5. for the systems CNk.

The System CN3-Cut-Free
This system is defined by analogy to the cut-free Frege system F − , which is described and investigated in [2]. The schematic axioms of the system CN3-cut-free are the following

Rule of inference is modus ponens
Note that in spite of rule modus ponens, the restrictions1.-5."insist on repeats" steps of Main Lemma for derivation of any 3-tautolgy in the system CN3-cut-free. It is obvious that this system is complet.
Note also that on the base of the systems CNk for k≥4 can be constructed the corresponding systems CNk-cut-free.

Conclusion
Two types of complete propositional proofs systems for some version of many valued propositional logic are introduced here. In [2] were proved that for 2-tautologies the systems F − , resolution and cut-free sequent are polynomially equivalent by proof sizes and by proof steps (polynomial equivalence means, that transformation of any proof in one system into a proof in the other system can be done with no more than polynomial increase of proof complexity). We hope that analogous results can be proved for k≥3 between the systems CNk-cut-free and E k, which are constructed in [3]. It is posible to describe for above version of k-valued logic the corresponding cut-free sequent with the same properties. We hope that the systems, correspondings to all systems, which are described here and in [3], can be introduced for other versions of k-valued logic.