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38Generalization of Kalmar's method for quasi-matrix logic1
YURIY V. IVLEV
abstract. Quasi-matrix logic is based on the generalization of the principles of classical logic: bivalency (a proposition take values from the domain {t (truth), f (falsity)}); consistency (a proposition can not take on both values); excluded middle (a proposition necessarily takes some of these values); identity (in a complex proposition, a system of propositions, an argument the same proposition takes the same value from domain {t, f}); matrix principle — logical connectives are defined by matrices. As a result of our generalization, we obtain quasi-matrix logic principles: the principle of four-valency (a proposition takes values from domain {tn,tc,fc,fz}) or three-valency (a proposition takes values from domain {n,c,i}); consistency : a proposition can not take more than one value from {tn,tc,f c,fr} or from {n,c,i}; the principle of excluded fifth or fourth ; identity (in a complex proposition, a system of propositions, an argument the same proposition takes the same value from domain {tn,tc,fc,fr} or domain {n,c,i}); the quasi-matrix principle (logical terms are interpreted as quasi-functions). Quasi-matrix logic is a logic of factual modalities.
Keywords: quasi-matrix logic, semantic completeness, decision problem, Kalmar's method
1 Kalmar's method
Well-known proof method for methateorem of semantic completeness of classical propositional calculus, which may be also treated as an approach to the solution of the decision problem, implies the proof of the following lemma:
xThis work is supported by Russian Foundation of Fundamental Research grant № 11-06-00296-a.
Lemma 1. Assuming that D is a formula, a\,... ,an are all different variables, occurring in D, b\,... ,bn are truth-values of these variables; let Ai be ai, —ai, depending on whether bi takes value t or f; let D' be D or —D depending on whether D takes value t or f with truth-values b\,... ,bn variables a\,... ,an. Then A\,... ,An ^ D'.
is here a sign (symbol) for logical entailment, — — for negation, t h f — truth and falsity, respectively.)
2 Generalization of Kalmar's method for many-valued matrix logic
At the end of the sixties of the 20-th century I was able to generalize this method for functionally complete many-valued matrix logics. (Probably the generalization of this kind had been done earlier by somebody else, but I have not heard of it up to now.)
Let's illustrate the basic principles underlying the generalization with one of the system of modal logic Sb- constructed by me.
Logical terms of language: —, D, □, 0. ('D', '□', '0' — are respectively signs for implication, necessity and possibility)
2.1 Semantics Definitions of logical terms
D tn tc f1 f c A —A □A OA
tn tn tc f1 f c tn f1 tn tn
tc tn tc fc f c tc f c f c tc
f1 tn tn tn tn f1 tn f1 f1
f c tn tc tc tc f c tc f c tc
tn,tc,fc,fl — are respectively truth-values 'necessary truth', 'contingent truth', 'contingent falsity', 'necessary falsity'. Designated values are tn and tc.
2.2 Formalisation
The calculus includes schemes of axioms of classical propositional calculus, modus ponens rule of inference and also following schemes of axioms:
□A D A; —D—A D OA; OA D —D—A; -OA D n(A D B); OB D □ (A D B); OB D O(A D B); O—A D O(A D B); O(A D B) D
(uA d oB); u(A d B) D (OA d ub ); uA D uuA; OuA D OA; OA D OuA; uA D uOA; uOA D uA; OOA D OA.
For the proof of meta-theorem of semantic completeness of calculi Sb- the following lemma is needed.
Lemma 2. Assuming that D is a formula, a1, ...,an are all different variables occurring in D, bi,bn are truth-values of these variables. Let Ai be Uai, ai&o—ai, —ai&oai, —Oai depending on whether bi is tn,tc,fc or fi. Let D' be UD, D&o—D, —D&oD or —OD depending on whether D takes value tn, tc, fc or fi with truth-values b1,..., bn of the variables a1,..., an. Then A1,..., An ^ D'. ( & — is here a sign for conjunction)
Lemma is proved by the use of recurrent mathematical induction.
If formula D takes designated value with all possible truth-values of its variables, then D' is UD or D&o—D. In each case Ai,...,An ^ D.
Let us substitute assumption Uai+1 with number i + 1 from the set of assumptions A1,..., An for the set of formulas ai+1, —O—ai+1, assumption ai+1&o—ai+1 for the set of formulas o—ai+1,ai+1, assumption —ai+1&Oai+1 for the set of formulas —ai+1, oai+1, assumption —Oai+1 for the set of formulas —ai+1, —oai+1. Then all assumptions with number i + 1 may be eliminated.
2.3 Illustration
1. A1, ...,Ai,ai+1, —o—ai+1 ^ D,
2. A1, ...,Ai,ai+1, o—ai+1 ^ D,
3. A1,...,Ai, —ai+1, oai+1 ^ D,
4. A1,...,Ai, —ai+1, —oai+1 ^ D,
5. A1,..., Ai, ai+1 ^ D - from 1, 2,
6. A1,..., Ai, —ai+1 ^ D - from 3, 4,
7. A1,..., Ai ^ D - from 5, 6.
In my doctoral thesis I brought forward 30 problems calling for solution. Later these ideas were published in monograph [8, p. 208-
217]. Many of these problems have been solved by now. The solutions were published in 13 PhD theses and publications. Some of the problems have not been solved yet. One of these problems (problem number 9) may be formulated as follows: if logic is functionally complete, then for any propositional variable a and any truth value i there is a formula fi(a) containing only this variable and taking some designated value if and only if a takes value i; suppose a1, a2,..., an are all different variables occurring in D; suppose b1,b2,..,bn are the truth-values of these variables; suppose As is fk(as), if bs is k; suppose D' is fr(D) (fr(D) is a formula formed on the basis of D and taking designated value with truth-values b1, b2,..., bn of the variables a1, a2,..., an). Then Ai, A2,..., An ^ D.
For example, 1, 2, 0 are the truth-values of three-valued modal logic of Lukasiewicz; f1(a) is □a, fi(a) is 0a&0—a, fo(a) is —0a; if formula takes value 1 with some truth-values of its variables, then fr(D) is □D, etc.; assumptions may be eliminated like it was stated for Sb-.
The ninth problem is the problem of finding the proof for metatheorem of semantic completeness of all known finite-valued matrix logics and finding sets of axioms for all logics of this kind stated semantically.
The seventeenth problem is the problem of generalization of this method for the proof of semantic completeness (and solution of the decision problem) of propositional quasi-matrix logics. This problem has not been solved for a long time. The solution is brought off in this article.
3 Quasi-matrix logic
Quasi-matrix is a set (Q,G,qf1 ,...,qfs), where Q and G are nonempty sets such that Q C G; qf1,..., qfs are quasi-functions.
If a function is a correspondence in virtue of which an object from some (functional) domain is related with certain object (from the range of the function) then a quasi-function is a correspondence in virtue of which an object from a certain subset of some set is related with some object from a certain subset of some or another set (from the range of the quasi-function).
3.1 Examples
Function: {(a,d), (b,k), (c,k)}.
Quasi-function: {(a, d)Y2(a, k), (c, m)} = {{(a, d), (c, m)}Y2{(a, k), (c,m)}},
Quasi-function: {Y4((a,k), (a,n), (c,k), (c,n)), (d,r) = Y4[{(a,k), (d, r)}, {(a, n), (d, r)}, {(c, k), (d, r)}, {(c, n), (d, r)}]},
Y2 and are two- and four-place (respectively ) metalinguistic exclusive disjunctions. Let us assume that disjunction may be degenerative, i. e. in this particular case quasi-function is just a function. Then a matrix is a particular case of quasi-matrix.
In the general case an object of application of a quasi-function, as well as truth-value of a quasi-function, are indefinite. Only subrange of the range of quasi-function, which includes this object, and sub-range of the range of values of a quasi-function, which contains a value of a quasi-function, are defined.
Such vagueness may be of a cognitive nature. It takes place, when the above-mentioned correspondence or relation is objectively functional, but this is not known to the researcher. For example, there are three probable variants of translation of a certain word in a dictionary, but the translator doesn't know, which of these three readings is the most appropriate in the present case (context). Such situations also appear in systems of automatic translation.
Another cause of indetermination is that reality may be indeterminate itself. For example, for planning of a production we have to take into account the following reasons. Suppose that we know the limits of alteration of a quantity of raw stuff, which will be factored next year. But it s impossible to figure out any rigid link between definite quantity of a factored raw stuff and a quantity of output, even if we knew a quantity of man-power, equipment etc.
For the first time some particular examples of quasi-functions were represented by H. Reichenbach (1932, 1935, 1936), Z. Zavarski (1936), F. Gonseth (1938, 1941), N. Rescher (1962, 1964, 1965, 1969). Rescher considers a material implication and defines it as follows:
A э B
t t t
t f f
f (t,f ) t
f (t,f ) f
(t, f ) is not a determinate truth-values. This bracketed entry (t, f ) means that either one of these two truth-values may occur in the various particular cases. Hence, depending on specific sense of propositions, the whole implication may be either true or false. Other logical terms are formulated in a usual way.
It is obvious that not all tautologies of a classical propositional logic of the form A э B take the truth-value 't' under any given assignment of truth-values to elementary propositions.
Rescher formulates the conception of quasi-tautology. He adopts t and (t, f ) in his quasi-functional system Q as designated truth-values. Then quasi-tautology is a formula which invariably does or can take either of this designated truth-values for every assignment of truth-values to its propositional variables. But if we bring to a logical end Rescher's reasoning we also have to treat as a quasi-tautology propositional variable p.
Then Rescher 'corrects' definitions of Lukasiewicz' three-valued logic.
A & B
i ( 2, o) 2
Independently of the above-mentioned and some other authors I came to the same considerations at the end of the sixties / beginning of the seventies. My ideas were concerned with the way of modal logic development. Though by that time a lot of different 'logical systems' had been constructed, it wasn't clear, what kind of modal operators and notions (either factual or logical necessity, possibility etc.) were defined by these systems. It made the application of modal systems to the natural reasoning analysis very difficult. This condition of modal logic seemed to me unsatisfactory and inadequate. On purpose to overcome these difficulties I distinguished two different branches of modal logical investigations: proper logic
(or logic itself) and an imitation of logic. Proper logic deals with the forms of thoughts. H. Curry called this kind of logic a philosophical one. Imitation of logic is a certain (formal) system, e. g. algebraic system, which in some respect resembles philosophical logic (usually with respect to some technical symbols and signs) [15].
In the following explanations I am treating modern logic as a philosophical logic in the sense of Curry.
In logic, as well as in each other science, it's possible to distinguish empirical and theoretical levels of development. An essential feature of a theory is its ability to explain phenomena. As I think, my approach to the analysis of logical modalities, elaborated by N. Arkhiereev, possesses this ability. Theory of factual modalities, which is to be based on quasi-matrix logic, has not been yet completely developed. (Fundamental ideas of theory of logical modalities are represented in [1, 2, 6, 7, 13, 14].)
I began to work out quasi-matrix logic with constructing the system of minimal modal logic.
3.2 Minimal modal logic Smin (Symbols of formalised language: u, O, —, D).
Lukasiewicz's well-known statement about impossibility of proper definitions of modal operators 'necessary (a) and 'possibly' (O) in terms of 'truth' and 'falsity' is valid only if these operators are interpreted as functions.
But if we interpret modal operators as quasi-functions, it becomes possible to define them in above-mentioned terms.
Let's consider formula uA. Assume A takes value f (falsehood). Then formula uA also takes value f, since not-existing state of affairs can not be necessary (both logically and factually). Assume formula A takes value t (truth). What truth-value takes formula uA in this case? The value is indeterminate. Formula uA takes either value t, or value f. Let's notify this situation by t/f.
By the same reasoning, we can conclude that truth-value of the formula OA is indeterminate, when formula A takes value f. Definitions of signs of negation and implication are usual. Designated truth-value is t.
Principles of classical propositional logic and logic S,
Classical propositional logic principles Principles of quasi-matrix logic Smin
(1) the principle of bivalency (propositions take values from the domain {t (truth), f (falsity)}) the principle of bivalency
(2) the principle of consistency (a proposition can not have both the values) the principle of consistency
(3) the principle of excluded middle (a proposition necessarily has some of these values) the principle of excluded middle
(4) the principle of identity (in a complex proposition, a system of propositions, an argument one and the same proposition has one and the same value from the domain {t,f}) the principle of identity
(5) the principle of specifying the truth value of a complex proposition by truth values of elementary propositions constituting it (in classical logic this principle acts as a matrix principle — logical connectives are interpreted as functions) the principle of specifying the truth value of a complex proposition by truth values of elementary propositions constituting it (in Smin this principle acts as a quasi-matrix principle — logical terms are interpreted as quasi-functions)
Smin — formalism which is adequate to the system constructed se-mantically. Smin-calculus is an extension of a classical propositional calculus with added new axiom schemes: □A D A, A D 0A. Smin-calculus is weaker than basic modal logic of Lukasiewicz, since the formula □A = — 0—A is not provable there.
For the proof of semantic completeness meta-theorem of Smin-calculus, we define alternative interpretation as follows.
Alternative interpretation is a function || || such as to: If P is — propositional variable then ||P|| € {t, f}.
If IIAll and IIB|| are defined, then II—All = t & ||A|| = f; IIA D Bll = f & 11 A|| = f or IIBII = t; || A|| = f ^ |M|| = f; ||A|| = t ^ ||□All € {t,f}; 11A|| = t ^ ||0A|| = t; ||A|| = f ^ ||0A|| € {t, f}. (& and ^ are here abbreviations for expression 'if and only if' ('iff') and 'if..., then...' respectively.)
Formula is satisfiable iff it takes the value 'true' in some alternative interpretation. Formula is valid iff it is true under each alternative interpretation.
3.3 Four-valued quasi-matrix logical systems
Truth-values tn, tc, fc, fi are interpreted as follows: proposition taking values tn describes a state of affairs which takes place in reality and which is strictly determined by certain circumstances; proposition taking values tc describes a state of affairs which takes place in reality and which is not strictly determined by either circumstances; proposition taking values fc describes a state of affairs which doesn't exist in reality and the absence of which is not strictly determined by either circumstances; proposition taking values fi describes a state of affairs which doesn't exist in reality and which absence is strictly determined by certain circumstances.
Four-valued quasi-matrix logic based on the following generalization of classical logic principles.
Classical logic principles Quasi-matrix logic principles
(1) the principle of bivalency (propositions take values from the domain {t (truth), f (falsity)}) the principle of four-valency (propositions take values from the domain {tn,tc,f c,f}
(2) the principle of consistency (a proposition can not have both the values) consistency: can not have more than one value from {tn,tc,f c,f *}
(3) the principle of excluded middle (a proposition necessarily has some of these values) the principle of excluded fifth
(4) the principle of identity (in a complex proposition, a system of propositions, an argument one and the same proposition has one and the same value from the domain {t, f} ) identity from the domain {tn,tc,f c,f *}
(5) the principle of specifying the truth value of a complex proposition by truth values of elementary propositions constituting it (in propositional logic this principle acts as a matrix principle — logical connectives are defined by matrices, in predicate logic it shows up in the interpretation of logical terms and predicates as truth functions).
the quasi-matrix principle (logical terms are interpreted as quasi-functions)
Logical terms are the same as those in the £TOin-system.
Definitions of logical terms:
A -A a b c d e
□A OA □A OA □A OA □A OA □A OA
tn f' t t tn tn tn tn tc tc tc tc
tc fc f t fc tc f' tn fc tc f' tn
f* tn f f f1 f' f1 f' fc fc fc fc
fc tc f t fc tc f' tn fc tc f' tn
A -A f g h i
□A OA □A OA □A OA □A OA
tn f' t t t t tn tn tc tc
tc fc f' tn fc tc f t f t
f' tn f f f f f1 f' fc fc
fc tc f' tn fc tc f t f t
B B
(-) D tn tc fz f c () D tn tc fz f c
tn tn tc f1 f c tn tn tc f1 f c
A tc tn tc f c f c A tc tn tn\tc f c f c
fr tn tn tn tn fr tn tn tn tn
f c tn tc tc tc f c tn tc tc tn\tc
B
(+) A
D tn tc f1 f c
tn tn tc f1 f c
tc tn tn\tc fc f c
fr tn tn tn tn
f c tn tn\tc tc tn\tc
t and tn\tc mean «either tn, or tc». f and f%\fc mean «either for f c ».
Following logical systems have been constructed on the basis of above-stated definitions: SaSa, Sa+, Sb-, Sb, Sb+, ScSc, Sc+, Sd-, Sd, Sd+, SeSe, Se+, S/S/, S/+, Sg-, Sg, Sg+, ShSh, Sh+, Si-, Si, Si+. Lower case letters occurring in the name of systems corresponds to the definition of modal terms, signs +, — and their absence correspond to the definition of implication.
tn and tc are distinguished truth-values.
The following considerations underlie the above-stated definitions of logical terms. Let us consider formula ooA. If the subformula A takes value t, then the value of a formula OA, as it has already been settled, is not determined, i. e. situation which is described by A takes place in reality but is determined itself either strictly or not. In the first case we have to assign to the formula OA value t, in the second one — the value /.
I.e. in the first case a proposition A is interpreted as being true and (factually) necessary (in our terms it takes value tn). What value in this case takes formula ooA? If A describes a state of affairs which is strictly determined by any circumstances, then these circumstances may in its own turn be either determined or not by some others. That is formula OA also takes value tn (or tc) etc.
Such situations occur both in subjective and objective reality.
Different kinds of distinct and fuzzy determination in biology were considered by V.Yu. Ivlev in [5,6].
Semantic-constructed systems are formalized by a number of calculi including as their general part all schemes of axioms of a classical propositional calculus, modus ponens — rule of inference and following schemes of axioms: OA D A; —O—A D OA; OA D —O—A; -OA D O(A D B); OB D O(A D B); oB D O(A D B); O—A D O(A D B); O(A D B) D (OA D OB).
We sign with letter S the calculus, which is obtained from classic propositional calculus by means of above-stated eight model schemes of axioms. The calculi corresponding to the semantic-constructed systems may be worked out by addition to S of the following schemes of axioms:
Sa-: O(A D B) D (oA D OB).
Sa: O(A D B) D (OA D OB); O(A D B) D (oA D oB); O(A D B) D (OA D (o—B D (—A D —B))).
Sa+: □(A D B) D (□A D □ B); □(A D B) D (0A D 0B). Sb-: □(A D B) D (0A D □B); □A D □□A; 0^A D 0A; 0A D 0^A; □A D ^0A; ^0A D □A; 00A D 0A.
Sb: □(A D B) D (□A D □B); □(A D B) D (0A D 0B); □(A D B) D (0A D (0—B D (—A D —B))); □A D □□A; 0^A D 0A; 0A D 0^A; □A D ^0A; ^0A D □A; 00A D 0A.
Sb+: □(A D B) D (□A D □B); □(A D B) D (0A D 0B); □A D □□A; 0^A D 0A; 0A D 0^A; □A D ^0A; ^0A D □A; 00A D 0A.
Calculi Sc-, Sd-, Se- ,Sf-, Sg-, Sh-, Si- include schemes of axioms □(A D B) D (0A D □B).
Calculi Sc, Sd, Se, Sf, Sg, Sh, Si include schemes of axioms
□ (A D B) D (□A D □B); □(A D B) D (0A D 0B); □(A D B) D (0A D (0—B D (—A D —B))).
Calculi Sc+,Sd+,Se+ ,Sf +,Sg+,Sh+,Si+ include schemes of axioms □(A D B) D (□A D □B); □(A D B) D (0A D 0B);.
Calculi, which have the same lower case letter occurring in the names (e. g. calculi Sc-,Sc,Sc+), differ from calculi, which have other lower case letters occurring in the names (e. g. calculi Si-, Si, Si+), by sets of schemes of axioms {□(A d B) D (0A d
□ B)}, {□(A D B) D (□A D □B); □(A D B) D (0A D 0B);
□ (A D B) D (0A D (0—B D (—A D —B)))}, {□(A D B) D (□A D
□ B); □(A D B) D (0A D 0B)}.
The other additional schemes of axioms of these calculi are the same:
Calculi Sc-, Sc, Sc+: □A d □□A; 00A D 0A; 0^A d □A; 0A D □0A.
Calculi Sd-, Sd, Sd+: 0A*, A* is modalized formula. Calculi Se-,Se,Se+: 00A; 0—□A; —0A d 0^A; □A d 0—0A; 0^A D (A D □A); 0^A D (0A D A); A D (0—A D ^0A); —A D (0A D ^0A).
Calculi Sf-,Sf,Sf+: 0^A D (A D □A); 0^A D (0A D A); A D (0—A D ^0A); —A D (0A D ^0A).
Calculi Sg-,Sg, Sg+: A D (—□A D 0^A); —A D (0A D 0^A);
□ 0A D (A D □A); ^0A D (0A D A).
Calculi Sh-,Sh, Sh+: □A D □□A; 0^A D 0A; □A D ^0A; 00A D 0A.
Calculi Si-, Si, Si+: 00A; 0—□A; —0A D 0^A; □A D 0—0A.
We use the rule of substitution of ——A with A and visa versa.
For the proof of metatheorem of semantic completeness of calculi Sb-, Sc-, Sd-, Se- (semantics for these calculi are of matrix sort) the following lemma is proved.
Lemma 3. Assuming that D is a formula, a\,...,an are all different variables, occurring in D, bi,...,bn are truth-values of these variables. Let Ai be □ai, ai&0—ai, —0ai, —ai&0ai depending on whether bi is tn,tc,fi or fc. Let D' be □D, D&0—D, —0D or —D&0D depending on whether D takes value tn, tc, fi or fc with truth-values bi,..., bn variables ai,..., an. Then Ai,..., An ^ D'.(^ is here a sign for entailment.)
Lemma is proved by the use of recurrent mathematical induction.
Semantics for others calculi are quasi-matrix (proper). For the proof of metatheorem of semantic completeness of these calculi the notion of alternative interpretation is used. We have the following definition of alternative interpretation for Sa+-system.
Alternative interpretation is a function || || satisfying the following:
If P is — propositional variable then ||P|| € {tn, tc, f%, fc}.
If IIAII and ||B|| are defined, then II—All = tn & ||A|| = fi; ||—A|| = tc & 11A|| = fc; ||—A|| = fi & 11A|| = tn; ||—A|| = fc & II A|| = tc;
IIA D Bll = fc & (||A|| = tn and IIB|| = fc) or (||A|| = tc and IIBII = fi);
IIA D Bll = fi & ||A|| = tn and IIBII = fi;
if either (||A|| = tn and ||B|| = tc) or (||A|| = fc and ||B|| = fi), then IIA D B|| = tc;
if IIAll = fi or IIB|| = tn, then IIA D B|| = tn;
if either IIAll = ||B|| = tc or (||A|| = fc and ||B|| = tc), or IIAII = IIBII = fc), then IIA D B|| € {tn,tc};
||A|| = tn => ||□All € {tn,tc}; if either A = tc or A = fc, or IIAII = fi, then II^AIl€{fc,fi};
||A|| = fi ^ ||0A|| € {fc,fi}; if either IIAll = tn or IIAll = tc, or IIAII = fc, then ||0A|| € {tn,tc}.
Sr — three-valued quasi-matrix logic.
(Symbols of formalised language are the same.) n,c,i — values of Sr-system — which are interpreted respectively as 'necessary', 'contingently', 'impossibly'. State of affairs is necessary if and only if (iff) it is distinctly determined by certain circumstances; state of affairs is contingent, iff neither its existence nor its absence is not strictly determined by some circumstances; state of affairs is impossible iff its absence is strictly determined by some circumstances. Actually, here and above the evaluations of state of affairs concern (to) propositions. (To my regret, I couldn't find proper terms for evaluation of propositions.)
Sr-logic is based on the following generalizations of principles of classic logic.
Classical logic principles Principles of quasi-matrix logic
(1) the principle of bivalency the principle of three-valency (propositions take values from the domain {n, c, i})
(2) the principle of consistency consistency: can not have more than one value from {n,c,i}
(3) the principle of excluded middle the principle of excluded fourth
(4) the principle of identity Identity (in a complex proposition, a system of propositions, an argument one and the same proposition has one and the same value from the domain { n, c, i} )
(5) the matrix principle the quasi-matrix principle (logical terms are interpreted as quasi-functions)
Definitions of logical terms:
A -A □A oA D n c i
n i n n n n c i
c c i n c n n\c c
i n i i i n n n
n\c is interpreted as 'either n or c'. n is a designated value.
Corresponding calculus includes all schemes of axioms of classical propositional calculus (note: in these schemes of axioms metasymbols A, B, C denote modalized formulas; the modalized formula definition: if A is a formula of classical propositional calculus, then □A and 0A are modalized formulas; if B and C are modalized formulas, then □B, 0B, —B, (B&C), (B V C), (B D C) are modalized formulas; nothing else is a modalized formula.), modus ponens, Godel's rule, all schemes of axioms of Sc+-calculus, and besides the following schemes: □A D 0A; —A D —□A; —0A D —A; A D 0A.
Alternative interpretation is a function || || for which the following helds:
If P is propositional variable then ||P|| € {n,c,i}.
If IIAII and IIB|| are defined, then II—All = n & ||A|| = i; II—All = c & ||A|| = c; ||—A|| = i & ||A|| = n;
if either ||^^L|| = i or ||B|| = n, then ||A d Bll = n;
if IIAll = ||B|| = c, then IIA D B|| € {n,c};
if either {||A|| = c and ||B|| = i} or {||A|| = n and ||B|| = c}, then IIA D B|| = c;
IIAll = n and IIB|| = i, iff IIA D B|| = i;
IP All = n iff II All = n; lp All = i, iff {either II All = c or
IIAII = i};
||0A|| = i, iff IIAII = i; ||0A|| = n, iff {either IIAll = n or IIAII = c}.
The formalisation and the proof of the meta-theorem of semantic completeness are the same as they were stated above.
3.4 Some peculiar properties of this logical system
First of all, it allows the use of the rule A ^ □A.
Besides, all derivable rule of inference of a classical propositional calculus are applicable to modalized formulas only. Some (at least some) direct rules of inference of a classical propositional calculus are also applicable to non-modalized formulas, for example: A V B, — A ^ B; but such indirect rules as rule of deduction:
r, A ^ B r ^ A D B
and rule reductio ad absurdum
r, A ^ B; r, A ^—B r ^-A
are not applicable to non-modalized formulas in derivation. However, so-called weakened rule of reductio ad absurdum
r, A ^ B;T,A ^—B
r^o-A
is applicable to any formula in derivation.
4 Generalisation for quasimatrix logic 4.1 For logic Smin
Lemma 4. suppose that D is a formula, a1, ...,an are all different variables, occurring in D, b1,...,bn are truth-values of these variables; let Ai be ai or —ai, depending on whether bi is t or f; let D' be D or —D depending on whether D takes value t or f with truth-values b\, ...,bn of the variables a\, ...,an in every alternative interpretation, formed on the basis of some initial interpretation. Let D' be D V —D depending on whether D takes value t under the truth assignment bi,...,bn of the variables ai,...,an in some alternative interpretation formed on the basis of the initial interpretation, or it takes value f under the truth assignment b1, ...,bn of the variables a1, ...,an in some alternative interpretation formed on the basis of the initial interpretation. Then A1,..., An ^ D'.
If in some alternative interpretations formula D takes value t and in some alternative interpretations it takes value f, then statement 'A1,..., An ^ D V —D' may be substituted for the statement 'Ai,..., An ^ D or A1,..., An ^ —D\
Proof. Lemma is proved by the use of recurrent mathematical induction.
Basis of induction. D does not contain any logical terms. Proof is obvious.
Assumption of induction. Proof holds for the formulas, containing k (k < n) occurrences of logical terms.
Step of induction. Proof holds for the formulas containing n + 1 occurrences of logical terms.
Case 1. n + 1-th occurrence of the logical terms is the occurrence of the sign of negation. Formula D is —B.
Suppose formula D takes value t in all alternative interpretations, formed on the basis of some initial interpretation. Then B takes value f in all these alternative interpretations. By the assumption of induction Ai,..., An — —B.
Suppose formula D takes value f in all alternative interpretations, formed on the basis of some initial interpretation. Then B takes value t in all these alternative interpretations and by the assumption of induction Ai,..., An — B. Then Ai,..., An — ——B.
Under the third possibility Ai,..., A
n —^ 1 B v-.-.B.
Case 2. n + 1-th occurrence of the logical terms is the occurrence of the sign of necessity. Formula D is □B. Suppose B takes value f in all alternative interpretations, formed on the basis of some initial interpretation. Then by the assumption of induction Ai, ...,An — —B. Since —B D —□B is a theorem scheme (contraposition of axiom scheme □B D B), then Ai,...,An — —□B. If B takes value t in all or some alternative interpretations, then formula □B takes value t in some alternative interpretations and in some other alternative interpretations it takes value f. Then it is obvious that Ai,..., An — □B V —□B.
Case 3. n + 1-th occurrence of the logical terms is the occurrence of the sign of possibility. Formula D is 0B. Suppose B takes value t in all alternative interpretations, formed on the basis of some initial interpretation. By the assumption of induction Ai,...,An — B. Since B D 0B is a theorem , Ai,...,An — 0B. If B takes value f in all or some alternative interpretations, then formula 0B takes value t in some alternative interpretations and it takes value f in some other alternative interpretations. Then Ai,..., An — 0B V —0B.
Case 4. n + 1-th occurrence of the logical terms is the occurrence of the sign of implication. Formula D is B D C. If formula D under above-mentioned truth-assignments of its variables takes value t in some alternative interpretations and in some other alternative interpretations it takes value f, then D is (B D C) V —(B D C). The entailment is obvious. If D takes value f, then D is —(B D C).
It is possible if in every alternative interpretation formula B takes value t and formula C takes value f. By the assumption of induction for every alternative interpretation holds that A1, ...,An ^ B and Ai,..., An ^ —C. Consequently Ai,..., An ^ —(B d C). Let's take into consideration the last case, then D takes value t in every alternative interpretation. It means that in every alternative interpretation formula B takes value f or formula C takes value t. Hence by the assumption of induction,
Ai,...,An ^ —B
or
Ai,...,An ^ C.
Analyzing all possible cases we conclude: A1,...,An ^ (B d C).
□
4.2 For logic Sr
Lemma 5. Suppose that D is a formula, a1,...,an are all different variables, occurring in D, b1,..., bn are values of these variables; let Ai be □ai , oai&o—ai, —oai, depending on whether bi is n, c, or i. Let D' be □D, oD&o—D or —oD, depending on whether D takes value n, c, or i with values b1,...,bn variables a1,...,an in all alternative interpretations, formed on the basis of some initial interpretation; suppose D' is □D V (oD&o—D), □D V — OD, (oD&o—D) V —oD, (□D V (oD&o—D)) V —oD, depending on whether D takes, respectively, value n in some alternative interpretations and in some other alternative interpretations it takes value c; D takes value n in some alternative interpretations and in some others it takes value i; D takes value c in some alternative interpretations and in some others it takes value i; D takes value n in some alternative interpretations or it takes value c in some other alternative interpretations, or it takes value i in some other alternative interpretations. Then A1,...,An ^ D'.
If D' is □Di V (oDi&o—Di), statement 'A1,...,An ^ D'' may be substituted for 'A1,..., An ^ □Di or A1,..., An ^ ODi&O—Di'. The substitution of the same kind is possible in case of other values in different alternative interpretations. I.e, logical entailment is based on alternative interpretations formed on the basis of some
initial interpretation. For example, if formula takes value n in every alternative interpretation, then the following holds for these alternative interpretations 'Ai,...,An — □Di or Ai,...,An — □Di, or Ai,..., An — □Di'. Hence Ai,..., An — □Di. Note that if there is no any ambiguity the only alternative interpretation that is possible is the initial one. In this case Ai,...,An — □Di also holds. The same holds for the other values.
Proof. Lemma is proved by recurrent mathematical induction on the number of occurrences of logical terms in formula D.
Step of induction.
Case 1. Formula D is —B.
Suppose D takes value n in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value i in every alternative interpretation formed on the basis of this initial interpretation. By the assumption of induction Ai,...,An — —0B. —0B D □—B is a theorem scheme. (Using theorem scheme —□—A D 0A.) Then Ai,..., An — □—B.
Suppose D takes value i in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value n in every alternative interpretation. By the assumption of induction Ai,...,An — □B. Then Ai,...,An — —0—B. Here we use the axiom scheme 0A D —□—A and the rule of substitution of ——A for A and vice versa.
Suppose D takes value c in every alternative interpretation formed on the basis of some initial interpretation. Then B also takes value c in every alternative interpretation. By the assumption of induction Ai,...,An — 0B&0—B. Hence Ai,...,An — (0—B&0——B).
Suppose D takes value n in some alternative interpretations and it takes value c in some others. By the assumption of induction: Ai,...,An — —0B or Ai,...,An — 0B&0—B. Since in the first case Ai,...,An — □—B and in the second one Ai,...,An — (0—B&0——B), the following holds: Ai,...,An — □—B V (0—B&0——B).
For other possible cases proof is analogous.
Case 2. Formula D is □B.
Suppose D takes value n in every alternative interpretation formed on the basis of some initial interpretation. Then B also takes value n in every alternative interpretation. By the assumption of induction Ai,...,An ^ OB. Then Ai,...,An ^ OOB. (Using axiom scheme OA D OOA.)
Suppose D takes value i in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value i in every alternative interpretation, or it takes value c in every alternative interpretation, or it takes value i in some alternative interpretation and it takes value c in some another alternative interpretation. Under the last possibility by the assumption of induction
Ah...,An ^ -oB or
A\,..., An ^ (OB&O-B).
In both cases A1,...,An ^ —oOB. (In the first case we use axioms schemes OoA d OA and OA D OA, and in second one -OoA d OA and OA D —O—A.) Formula D can not take value c.
If formula D takes different truth values in different alternative interpretations the proof may be concluded from the above-analyzed cases.
Case 3. Formula D is OB.
Suppose D takes value n in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value n in every alternative interpretation, or it takes value c in every alternative interpretation, or it takes value n in some alternative interpretation and it takes value c in another alternative interpretation. Under the last possibility by the assumption of induction
Al7..., An ^ OB or
Ai,...,An ^ (oB&o—B).
In both cases A1,...,An ^ OoB. (In the first case we use axioms schemes OA D OA and OA D OOA, and in the second case we need only the last axiom)
Suppose D takes value i in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value i in
every alternative interpretation. By the assumption of induction A1,...,An ^ —oB. Then A1,...,An ^ —OOB. (Using the axiom scheme OOA D OA.) Formula D can not take value c. If formula D takes different truth values in different alternative interpretations the proof may be concluded from the above-analyzed cases.
Case 4. n + 1-th occurrence of the logical terms is the occurrence of the sign of implication. Formula D is B D C.
Suppose D takes value n in every alternative interpretation formed on the basis of some initial interpretation. It is possible if B takes value i in every alternative interpretation or C takes value n in every alternative interpretation. By the assumption of induction for every alternative interpretation holds: A1,..,An ^ —oB or A1,..., An ^ UC. Hence: A1,..., An ^ U(B D C). (Using axiom schemes —OA D U(A D B); UB D U(A D B).)
Suppose D takes value i in every alternative interpretation formed on the basis of some initial interpretation. It is possible if B takes value n in every alternative interpretation and C takes value i in every alternative interpretation. By the assumption of induction for every alternative interpretation holds: A1, ...,An ^ UB u A1,...,An ^ —oC. Then A1,...,An ^ —o(B D C). (Using axiom schemes O(A D B) D (UA D oB).)
Suppose D takes value c in every alternative interpretation formed on the basis of some initial interpretation. It is possible if B takes value n and C takes value c in every alternative interpretation or B takes value c and C takes value i in every alternative interpretation. In the first case A1}..., An ^ UB and A1}..., An ^ oC&o—C. Then we have to prove: Ah ..., An ^ o(B D C)&o—(B D C).
A1,..,An ^ o(B D C) (using theorem scheme OB d o(A D B)). A1, ...,An ^ o—(B D C) (using axiom schemes U(A D B) D (UA D UB) and —U—A D oA, and rule of substitution of ——A for A and vice versa). In second case A1, ...,An ^ oB&o—B, and A1,...,An ^ —oC. Then A1,...,An ^ o(B D C) ( using axiom scheme o—B D O(A D B)). A1, ...,An ^ o—(B D C) (using axiom schemes U(A D B) D (oA D oB) and —U—A D OA).
Suppose D takes value n in some alternative interpretation formed on the basis of some initial interpretation and it takes value c in another interpretation. Then we have to prove: A1, ...,An ^
0(B D C)&0—(B D C) or Ai,..., An — □(B D C), or the equivalent statement Ai,...,An — 0(B d C). This case is possible if both B and C takes value c in all alternative interpretations. By the assumption of induction Ai,..., An — 0B&0—B and Ai,..., An — 0C&0—C. Then Ab ..., An — 0(B D C) (using axiom scheme 0B D 0(A D B)).
The proof of other possibilities may be concluded from the above-analyzed cases. □
Metatheorem 1. If formula D is universally satisfiable then it is provable.
Since for every truth-assignment of the variables holds Ai,..., An — □D, then the following holds:
1. Ai,..., An-i, □an — □D,
2. Ai,...,An-i, —0an — □D,
3. Ai,...,An-i, 0an&0—an — □D. Hence:
4. Ai,..., An-i, 0an, —0—an — □D, from 1,
5. Ai,..., An-i, —0an — □D, from 2,
6. Ai,..., An-i, 0an, 0—an — □D, from 3.
7. Ai,..., An-i, 0an — □D, from 4, 6,
8. Ai,...,An-i — □D, from 5, 7. etc.
As □D entails D, D is provable.
Remark 1. Since formula can take one of the seven values (n, c, i, n/c, n/i, c/i, n/c/i), the problem arises to construct 7-valued logic with this values (lets sign them with 1, 2, 3, 4, 5, 6, 7) and compare it with Sr.
4.3 For logic Sa-
Lemma 6. Suppose D is a formula, a1, ...,an are all different variables, occurring in D, b1,..., bn are truth-values of these variables; let Ai be □ai , ai&o—ai, —oai, —ai&oai, depending on whether bi is tn, tc, fi or fc. Let D' be □D, D&o—D, —oD or —D&oD, depending on whether D takes value tn, tc, fi or fc with values b1,..., bn of the variables a1,...,an in all alternative interpretations formed on the basis of some initial interpretation. Suppose D' is □D V (D&o—D), □D V—oD, (D&o—D) V—oD, (□D V (D&o—D)) V—oD and so on, depending on whether D takes respectively value tn in some alternative interpretations and in some other alternative interpretations it takes value tc; D takes value tn in some alternative interpretations and in some others it takes value fi; D takes value tc in some alternative interpretations and in some others it takes value fi; D takes value tn in some alternative interpretations or it takes value tc in some other alternative interpretations, or it takes value fi in some other alternative interpretations. Then A1, ...,An ^ D'.
Proof. Lemma is proved by recurrent mathematical induction on the number of occurrence of logical terms in formula D.
Step of induction.
Case 1. Formula D is —B.
Suppose D takes value tn in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value fi in every alternative interpretation formed on the basis of this initial interpretation. By the assumption of induction A1,...,A n ^ —oB. —oB d □—B is a theorem scheme. (Using theorem scheme —□—A d OA.) Then A1, ...,An ^ □—B.
Suppose D takes value fi in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value tn in every alternative interpretation formed on the basis of this initial interpretation. By the assumption of induction A1,...,An ^ □B. Then A1 ,..,A n ^ —o—B. Here we use an axiom scheme OA d —□—A and rule of substitution of ——A for A and vice versa.
Suppose D takes value tc in every alternative interpretation formed on the basis of some initial interpretation. Then B takes
value fc in every alternative interpretation formed on the basis of this initial interpretation. By the assumption of induction Ai,..., An — — B&0B. Hence Ai,..., An — —B&0——B.
Suppose D takes value fc in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value tc too in every alternative interpretation formed on the basis of this initial interpretation. By the assumption of induction Ai,..., An — B&0—B. Hence Ai,..., An — ——B&0—B.
Suppose D takes value tn in some alternative interpretations formed on the basis of some initial interpretation and it takes value tc in some other interpretations. By the assumption of induction B takes value f1 in some alternative interpretations and it takes value fc in other alternative interpretations. Then Ai,..., An — —0B or Ai,...,An — — B&0B.
Since in the first case Ai,...,An — □—B and in the second Ai,..., An — —B&0——B, the following holds: Ah ..., An — □—B V (—B&0——B).
For other possible cases proof is analogous.
Case 2. Formula D is □B.
Suppose D takes value tn or tc in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value tn in every alternative interpretation formed on the basis of this initial interpretation. By the assumption of induction Ai,..., A n — □B. Then we have to prove: Ai,...,An —
□□b v (□b&0—□b).
□ □B V (□B^^B) ^ (□□B V ^^(□□B V 0—□B).
(□□BV□B^(□□BV0—□B) ^ (□□BV^^(□□BV—^B).
(□□B V ^^(□□B V —□□B) ^ □B.
Proof is completed. (^ is a sign for metalanguage equivalence).
Suppose D takes value f1 or fc in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value tc or fl, or fc in every alternative interpretation formed on the basis of this initial interpretation. We have to prove: Ai, ...,An — —0^B V (—□B&0^B). That is, we have to prove: Ah ...,An — —□B.
In the first case by the assumption of induction Ai,..,An — B&0—B. The proof is evident.
In the second case A1,...,An ^ —oB. —oB ^ —□B. (Using axiom schemes —□—A d OA and □A d A.) The statement is proved.
In the third case A1,...,An ^ —B&oB. —B ^ —□B. The statement is proved.
Case 3. Formula D is oB.
Suppose D takes value tn or tc in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value tn or tc, or fc in every alternative interpretation formed on the basis of this initial interpretation. We have to prove: A1, ...,An ^ □ oB V (oB A o—oB). That is we have to prove: Ab ..., An ^ oB. By the assumption of induction in every of three cases A1,...,An ^ oB.
Suppose D takes value fi or value fc in every alternative interpretation formed on the basis of some initial interpretation. Then B takes value fi in every alternative interpretation. We have to prove: A1, ...,An ^ —ooB V (—oB&ooB). By the assumption of induction A1,..., An ^ —oB.
—ooB V (—oB&ooB) & —oB
So A1,..., An ^ —ooB V (—oB&ooB) is proved.
Cases when formula D takes different values in different alternative interpretations may be reduced to the above-analyzed cases.
Case 4. n + 1-th occurrence of the logical terms is the occurrence of the sign of implication. Formula D is B D C.
Suppose formula D takes value tn in every alternative interpretation. It is possible if either B takes value fi or C takes value tn. We have to prove: A1,..., An ^ □(B D C). The statement may be easily proved by axiom schemes —OA D □(A d B), □A D □(A d B).
Suppose formula D takes value fi in every alternative interpretation. Then B takes value tn and C takes value fi. We have to prove: A1,...,An ^ —o(B D C). By the assumption of induction A1,..., An ^ □B and A1,..., An ^ —oC. Hence, A1,..., An ^ —o(B D C). (Using axiom scheme o(A D B) D (□A D oB).)
Suppose formula D takes value tc in every alternative interpretation. It is possible if both B and C takes value tc in every alternative interpretation, or if B takes value tn and C takes value tc, or if B takes value fc and C takes one of the three values: tc or f or fc.
We have to prove: Ai, ..,An ^ (B D C)&O—(B D C). Under the first condition A1}..., An ^ B&o—B and Ai,..., An ^ C&o—C. C ^ B D C. B ^ oB. O—C ^ —DC. oB&—UC ^ o—(B D C). (Using axiom schemes U(A D B) D (OA D UB), —U—A D OA.)
Under the second condition Ai,...,An ^ UB and Ai,...,An ^ C&0-C. The proof is the same as in the previous case.
Under the third condition Ai,...,An ^ —B&OB and Ai,..., An ^ C&o—C or Ai,..., An ^ —oC, or Ai,..., An ^
—C&OC. In any case Ai, ...,An ^ —DC. The proof is completed.
□
Cases when formula D takes different values in different alternative interpretations may be reduced to the above-analyzed cases.
Metatheorem 2. If formula D is universally satisfiable then it is provable.
(Since for every truth-assignment of the variables holds Ai,...,An ^ UD or Ai,...,An ^ (D&o—D) then the following holds: Ai, ...,An ^ D.)
1. Ai,...,An-i, Dan ^ D,
2. Ai,...,An-i, —oan ^ D,
3. Ai, ...,An-i,an&o—an ^ D,
4. Ai,..., An-i, —an&oan ^ D, Hence
5. Ai,..., A n i, — o—an ^ D, from 1,
6. Ai,..., An-i, —an, —oan ^ D, from 2,
7. Ai,..., An-i, an, o—an ^ D, from 3,
8. Ai,..., An-i, —an, oan ^ D, from 4, And then:
9. Ai,..., An-i, an ^ D, from 5, 7,
10. Ai,..., An-i, —an ^ D, from 6, 8,
11. Ai, ...,An-i ^ D, from 9, 10, and so forth.
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