
Projectively Modal Ontology: between worlds of St.Lesniewski and W.Soloviov Vyacheslav Moiseev, Russia, vimo@vmail.ru AbstractThe paper asserts that every ontology presupposes a basic structure, Ontological Tetrade, which consists of source of predications (“modus”), different predications of the source (“modas”), restricted conditions, under which the predications are formed (“models”), and operation of forming of the predications (“projector”). V.Soloviov used projective intuition of Ontological Tetrade comparing predications with projections of the body. It seems, St.Lesniewski also used a similar intuition in a non explicit form. A new axiomatic system, Projectively Modal Ontology, is offered, where logical means, similar to Lesniewski’ Ontology, are used for the expression of idea of Ontological Tetrade. Key words. Ontology, Ontological Tetrade, modus, moda, model, projector
The paper is devoted to the description of one axiomatical system, which can be called as ^ (PMO). This system has two main foundations: 1) one important philosophical concept from the philosophy of Vladimir Soloviov, and 2) logical form similar to logical form of St.Lesniewski’ Ontology. Breafly speaking PMO = Soloviov ‘ Content + Lesniewski’ Form Therefore, I shall say some words about Soloviov approach first of all. Further I shall explain some logical ideas of PMO.
Soloviov philosophy is a sort of Platonism. There exists a Highest Being (“Unity”) and there exist infinite set of principles, which are different aspects of Unity. Together Unity and its aspects form AllUnity (therefore the title of Soloviov philosophy is also “Russian Philosophy of AllUnity”). This is the case of an ierarchial Ontology with maximal and minimal (nonbeing) elements. Let us see a typical part of the ierarchy: one more ontologically strong principle (S) and, for example, two its aspects (A_{1} and A_{2}) – see fig.1. Soloviov used a projective intuition here, he interpreted aspects A_{1} and A_{2} as “projections” of the principal S (see also my book^{1}). To clear this idea let us see an example of geometrical projections. For example, we have a 3dimensional body B and two 2dimensional projections P_{1} and P_{2} (see fig.2). Every projection P_{i} is made in the framework of a plane: P_{1} in plane _{1}, P_{2} in plane _{2}. We can speak that every projection is the body B under the condition of the plane of projectivity, i.e., P_{i} is Bunderthecondition_{i} “Under the condition” is a functor, which can be called as projector. Finally we obtain P_{i} = B_{i} , where is projector This structure can be generalised and we might to write in general case A_{i} = SC_{i} , where S is a synthesis A_{i} is an aspect of S C_{i} is a restricted condition under which A_{i} is formed is projector, operation of forming of aspects from synthesis and restricted conditions I shall call these four principles, syntesis, aspect, condition and projector, as ^ . One of my basic assumptions is a such that any Ontology presupposes an Ontological Tetrade in a definite form. I shall use special terms for all elements of Ontological Tetrade: “modus” for synthesis, “moda” for aspect, “model” for restricted condition and “projector” for projector (see fig.3). Modus is a principle of variety, space of possibilities Model is a principle of restriction of variety Moda is an element of variety, one of the possibilities Projector is an act of restriction (transformation) of variety to an element One need to notice that the term “model” is used not in trivial sense here. I wanted to use one Latin root “mod”: modus, model, moda. Therefore I shall use the term “modal” in the ancient sense of this word expressing an idea of any variation, modification. To differ this sense from the contemporary using of the term “modal” in different modal logics I add word “projectively” to the word “modal”. I think Ontological Tetrade is a very old philosophical structure. We can find it in Plato, in East Philosophy, etc. For example the following realisations of Ontological Tetrade in some philosophical systems can be demonstrated here
Soloviov connects the idea of Ontological Tetrade with the idea of an Ontological Order () SC_{i} S , i.e. moda of modus less or equals modus in some ontological sense. In particular, modus can be represented as moda of itself, i.e., there exists a such model 1_{S} that S1_{S} = S. Model 1_{S} can be called as model unity. It is the case of absence of restricted conditions as a limited case of its zero presence.
It is well known that Stanislaw Lesniewski was nominalist. Therefore the structure of his Ontology as follows (see fig.4). There are many strong principles (“things”), every thing is a maximum of own ierarchy. Non maximal elements of ierarchy are more weak kinds of being (general properties of things). I shall use symbol _{L} for Lesniewski’ functor (“jest”). Then we have If (a _{L} b) is true, then a is a thing (strong being), b is either 1) thing also (then (b _{L} a) and (a =_{L} b) here, where (a =_{L} b) is (a _{L} b b _{L} a)), or 2) property of thing (and ( b _{L} a) in this case) We find an example of a relation of modus and moda here: if (a _{L} b) is true, then a is modus, b is moda (see fig.5). However Lesniewski does not deal explicitely with models and projectors in his Ontology. Nevertheless, it seems, his Ontology (LOntology) consists of intuition of Ontological Tetrade also. Therefore we can try to modify LOntology to express this idea.
PMO is an effort to construct a such more universal Ontology. Further I shall describe basic logical aspects of PMO using the term “Ontology” for PMO.
^ I use a primary 4placed predicate Mod, where Mod(a,b,c,f) can be read as “a is mode of modus b in model c with projector f” Variables a,b,c have categorial type N, variable f has type Language of OntologyI accept here almost all the logical means of the language of S.Lesniewski’s “Ontology”. Namely I accept Prothotetics without any changes, syntax of expressions of different categorial types, rules of inference with the exeption of Rule of Extensionality (see below). Prothotetical definitions will be used without any changes. Forms of Ontological definitions will be discussed below. Instead of Lesniewski’s functor I shall use the predicate Mod. ^ 1. Definitions of CoOrdination (In all 14 definitions) Dk_{1}…k_{m}. Mod^{k1…km}(x_{k1},…,x_{km}) х_{p1}...х_{pn}Mod(x_{k},x_{p}), where Mod^{k1…km}(x_{k1},…,x_{km}) is a definable expression of type S. Expression Mod^{k1…km}(x_{k1},…,x_{km}) contains only free variables x_{k1},…,x_{km} of the type , where 1≤k_{j}≤4, j=1,..,m, and type is either type N, when k_{j} < 4, or type (N,N)/N, when k_{j} = 4. I shall mean under the symbol х_{p1}...х_{pn}Mod(x_{k},х_{p}) an expression formed by adding of existential quantifiers х_{p1}...х_{pn} to predicate Mod, where variables х_{p1},...,х_{pn} have type , and is either type N, when p_{s} < 4, or type (N,N)/N, when p_{s} = 4, while 1≤p_{s}≤4 and s=1,..,n. Predicate Mod contains only variables x_{k1},…,x_{km} and х_{p1},...,х_{pn} such that variable x_{kj} stands at the place number k_{j} in predicate Mod, variable x_{ps} stands at the place number p_{s}, and m+n = 4. For example: D123. Mod^{123}(a,b,c) : Mod(a,b,c) fMod(a,b,c,f), where “Mod(a,b,c)” is read as “a is mode of modus b under model c” D12. Mod^{12}(a,b) : Moda(a,b) cfMod(a,b,c,f), where “Moda(a,b)” is read as “a is mode of modus b” D23. Mod^{23}(b,c) : Model(c,b) af Mod(a,b,c,f), where “Model(c,b)” is read as “c is model for modus b” D1. Mod^{1}(a) : Moda(a) bcf Mod(a,b,c,f), where “Moda(a)” is read as “a is mode” D2. Mod^{2}(b) : Modus(b) acf Mod(a,b,c,f), where “Modus(b)” is read as “b is modus” D3. Mod^{3}(c) : Model(c) abf Mod(a,b,c,f), where “Model(c)” is read as “c is model” D4. Mod^{4}(f) : Projector(f) abcMod(a,b,c,f), where “Projector(f)” is read as “f is projector” 2. Definitions of Weak Inclusions and Weak Equalities DI^{ }^{i}_{k}_{1,…,}_{km}. a ^{i}_{k}_{1,…,}_{km}_{ }b x_{k}_{1}…x_{km}(y_{p}_{1}...y_{pn}Mod(...a...) y_{p}_{1}...y_{pn}Mod(...b...)). DE^{ }^{i}_{k}_{1,…,}_{km}. a ^{i}_{k}_{1,…,}_{km}_{ }b x_{k}_{1}…x_{km}(y_{p}_{1}...y_{pn}Mod(...a...) y_{p}_{1}...y_{pn}Mod(...b...)). (in all 28 definitions of the first type and of the second) These expressions designate that variables x_{k1,}…,x_{km} stand at the places number k_{1,}…, k_{m} accordingly, variables y_{p1,}..., y_{pn} stand at the places number p_{1,}..., p_{n} accordingly in predicates Mod. Then m+n = 3, all is, k_{j}s and p_{s}s, where j=1,.., m, s=1,.., n, do not equal between each other. Terms а and b stand at place number i in the predicates Mod. Variables with index 4 (standing at the place number 4 in predicates Mod) are variables of type For example, moduses can be equale to each other on the following seven bases: a ^{2}_{1 }b x_{1}(y_{3}y_{4}Mod(x_{1},a,y_{3},y_{4}) y_{3}y_{4}Mod(x_{1},b,y_{3},y_{4})) – equality by modes a ^{2}_{3 }b x_{3}(y_{1}y_{4}Mod(y_{1},a,x_{3},y_{4}) y_{1}y_{4}Mod(y_{1},b,x_{3},y_{4})) – equality by models a ^{2}_{4 }b x_{4}(y_{1}y_{3}Mod(y_{1},a,y_{3},x_{4}) y_{1}y_{3}Mod(y_{1},b,y_{3},x_{4})) – equality by projectors a ^{2}_{1,3 }b x_{1}x_{3}(y_{4}Mod(x_{1},a,x_{3},y_{4}) y_{4}Mod(x_{1},b,x_{3},y_{4})) – equality by modes and models a ^{2}_{1,4 }b x_{1}x_{4}(y_{3}Mod(x_{1},a,y_{3},x_{4}) y_{3}Mod(x_{1},b,y_{3},x_{4})) – equality by modes and projectors a ^{2}_{3,4 }b x_{3}x_{4}(y_{1}Mod(y_{1},a,x_{3},x_{4}) y_{1}Mod(y_{1},b,x_{3},x_{4})) – equality by models and projectors a ^{2}_{1,3,4 }b x_{1}x_{3}x_{4}(Mod(x_{1},a,x_{3},x_{4}) Mod(x_{1},b,x_{3},x_{4})) – equality by modes, models and projectors ^ DSE^{ i}_{k1,…,km}. a =^{i}_{k1,…,km }b a ^{i}_{k1,…,km }b Mod(…a…) Mod(…b…), where under the designation Mod(…a…) the predicate Mod is meant, in which all the variables, besides variable a, are bounded by existential quantifiers, and variable a stands at the place number i. Index i can accept values from 1 to 4 in the expressions with the index form ^{i}_{k1,…,km}, and variable m can vary from 1 to 3 when i is fixed. For example, the following special equalities and inclusions can be distinguished: DE^{2}_{1}. a ^{2}_{1} b : a b c(Moda(c,a) Moda(c,b)), where “a b” is read as “a weakly equals b” DI^{2}_{1}. a ^{2}_{1} b : a b c(Moda(c,a) Moda(c,b)), where “a b” is read as “a is weakly included into b” DI^{2}_{3}. a ^{2}_{3} b : a * b x(Model(x,a) Model(x,b)), where “a * b” is read as “a weakly equals b by models” 4. A one special case of Equality I shall use the following equality also: DE. a = b Moda(a,b)Moda(b,a), where “a = b” is read as “a equals b” 5. Valency Definitions: DPMODA1. PModa(a) b(Moda(b,a) Moda(a,b)) Moda(a), where “PModa(a)” is read as “a is positive (not null) mode” DNMODA. NModa(a) b(Moda(b) Moda(a,b)) Moda(a), where “NModa(a)” is read as “a is negative (null) mode” DNMODUS. NModus(a) b(Model(b) Model(b,a)) c(Moda(c,a) NModa(c)) Modus(a), where “NModus(a)” is read as “a is negative (null) modus” DPMODA2. PModa(a,b) Moda(a,b) PModa(a), where “PModa(a,b)” is read as “a is positive (not null) mode for modus b” DPMODUS. PModus(a) bPModa(b,a), where “PModus(a)” is read as “a is positive (not null) modus” DIMODUS. IModus(a) Modus(a) b(Modus(b) Moda(b,a)), where “IModus(a)” is read as “a is infinite modus” DAT. At(a) PModus(a) b(PModa(b,a) b=a), here “At(a)” is read as “a is atom” DPE. a b PModa(a,b) PModa(b,a), where “a b” is read as “a is positively equivalent to b” ^ (AO1) Moda(a,b) Modus(a) d(Moda(b,d) Moda(a,d)) Moda(b,b) (AO2) Mod(a,b,c,f) (a =^{1}_{234} f(b,c)) aMod(a,b,c,f) Notation D1. a b Moda(b,a), where “a b” is read as “a is b” DP. a _{1} b PModa(b,a), where “a _{1} b” is read as “a positively is b” D*. a * b Model(b,a), where “a * b” is read as “b is model for modus a” Existential definition DEx. Ex(a) PModa(a), where “Ex(a)” is read as “a exists” ^ D^{i}_{k1,…,km}. M^{i} [x_{k1}…x_{km}(y_{p1}...y_{pn}Mod(x_{k},C,y_{p}) Mod^{k1…km}(x_{k1},…,x_{km}) (x_{k1},…,x_{km}))], where variables x_{k1,}…,x_{km} stand at the places number k_{1,}…, k_{m} accordingly, variables y_{p1,}..., y_{pn} stand at the places number p_{1,}..., p_{n} accordingly in the first predicate Mod. Then m+n = 3, all is, k_{j}s and p_{s}s, where j=1,.., m, s=1,.., n, do not equal between each other. Term C is a definable term and C stands at the place number i in the first predicate Mod. All these conditions are designated by the symbol Mod(x_{k},C,y_{p}). Variables with index 4 (standing at the place number 4 in predicates Mod) are variables of type Finally, M^{i} is 1. Expression x_{k1}…x_{km}t((x_{k1},…,x_{km}) x_{2} t ((x_{k1},…,x_{km}))_{x2}[t]) iff i=1 and there exists k_{j} = 2 between all k_{j}, where j=1,2,…,m. Expression ((x_{k1},…,x_{km}))_{x2}[t] is the result of substitution of variable t for variable x_{2} in expression (x_{k1},…,x_{km}). 2. Expression x_{k1}…x_{km}t((x_{k1},…,x_{km}) t x_{1} ((x_{k1},…,x_{km}))_{x1}[t]) iff i=2 and there exists k_{j} = 1 between all k_{j}, where j=1,2,…,m. Expression ((x_{k1},…,x_{km}))_{x1}[t] is the result of substitution of variable t for variable x_{1} in expression (x_{k1},…,x_{km}). 3. p(pp) in the other cases, where p is a propositional variable. If the definable term is a functor F of the categorial type a = b/, where type is either type N, when k_{j} < 4, or type D^{ai}_{k1,…,km}. M^{i} [x_{k1}…x_{km}(y_{p1}...y_{pn}Mod(x_{k},F(),y_{p}) Mod^{k1…km}(x_{k1},…,x_{km}) (x_{k1},…,x_{km},))], where notation is the same as earlier and, besides, is a sequence (a_{1},a_{2},…,a_{N}) of arguments of F, and symbol is a_{1}a_{2}…a_{N}. Here M^{i} is 1. Expression x_{k1}…x_{km}t((x_{k1},…,x_{km},) x_{2} t ((x_{k1},…,x_{km},))_{x2}[t]) iff i=1 and there exists k_{j} = 2 between all k_{j}, where j=1,2,…,m. Expression ((x_{k1},…,x_{km},))_{x2}[t] is the result of substitution of variable t for variable x_{2} in expression (x_{k1},…,x_{km},). 2. Expression x_{k1}…x_{km}t((x_{k1},…,x_{km},) t x_{1} ((x_{k1},…,x_{km},))_{x1}[t]) iff i=2 and there exists k_{j} = 1 between all k_{j}, where j=1,2,…,m. Expression ((x_{k1},…,x_{km},))_{x1}[t] is the result of substitution of variable t for variable x_{1} in expression (x_{k1},…,x_{km},). 3. p(pp) in the other cases, where p is a propositional variable. I shall call property (x_{k1},…,x_{km}) or (x_{k1},…,x_{km},) as modal (modus) property iff the case 2 (1) takes place. For example, we have D^{2}_{1}. ab(P(a) a b P(b)) [a(C a Moda(a) P(a))] – the case of modal definition of modus C. Condition ab(P(a) a b P(b)) defines property P as modal property here. D^{1}_{2}. ab(Q(a) b a Q(b)) [a(a C Modus(a) Q(a))] – the case of modus definition of mode C. Condition ab(Q(a) b a Q(b)) defines property Q as modus property here. D^{2}_{3}.a(C * a Model(a) Q(a)) – the case of model definition of modus C. D^{2}_{13}. abt(P(a,b) a t P(t,b)) [ab(Mod(a,C,b) Mod^{13}(a,b) P(a,b)] – the case of modalmodel definition of modus C. Condition abt(P(a,b) a t P(t,b)) defines property P as modal property here. 28 kinds of Ontological Definitions of type D^{i}_{k1,…,km} and D^{ai}_{k1,…,km} are defined in the general case. I would like to note that Moda(a) a a and Modus(a) a a. Therefore we can write consequents of D^{2}_{1} and D^{1}_{2} in the forms a(C a a a P(a)) and a(a C a a Q(a)) accordingly. ^ In general case, the following kinds of Ontological Laws of Extensionality can be accepted in Ontology: LE^{ i}_{k}_{1,…,}_{km}. ^{ i}_{k}_{1…}_{kn} {() ()}, where ^{ i}_{k}_{1…}_{kn} is one of the 28 weak equalities, and variables , have type LE^{1}_{2}. a ^{1}_{2} b {(a) (b)} LE^{2}_{1}. a ^{2}_{1} b {(a) (b)} LE^{1}_{234}. a ^{1}_{234} b {(a) (b)} LE^{2}_{134}. a ^{2}_{134} b {(a) (b)} LE^{3}_{124}. a ^{3}_{124} b {(a) (b)} LE^{4}_{123}. f ^{4}_{123} g {(f) (g)}, By analogy corresponding versions of Ontological Laws of Extensionality can be introduced for every categorial type a = b/, where b is also a categorial type and is N or LE^{a1}_{2}. (a() ^{1}_{2} b()) {(a) (b)} LE^{a2}_{1}. (a() ^{2}_{1} b()) {(a) (b)} LE^{a1}_{234}. (a() ^{1}_{234} b()) {(a) (b)} LE^{a2}_{134}. (a() ^{2}_{134} b()) {(a) (b)} LE^{a3}_{124}. (a() ^{3}_{124} b()) {(a) (b)} LE^{a4}_{123}. (f() ^{4}_{123} g()) {(f) (g)}, where is a sequence of arguments of functors a,b,f,g, while a,b have type b/N, and f,g have type b/( I shall accept the version of Ontology with LE^{2}_{1}, LE^{3}_{124}, LE^{4}_{123}, LE^{a2}_{1}, LE^{a3}_{124}, LE^{a4}_{123} below. Let us say that index form ^{i}_{k}_{1…}_{kn} is included into an index form ^{j}_{p}_{1…}_{pm} iff i=j and numbers k_{1},…, k_{n} are between numbers p_{1},…, p_{m}. One can show that Law of Extensionality LE^{j}_{p}_{1…}_{pm} is infered from the Law of Extensionality LE^{i}_{k}_{1…}_{kn} iff index form ^{i}_{k}_{1…}_{kn} is included into the index form ^{j}_{p}_{1…}_{pm}. For example, acceptane of LE^{2}_{1} (LE^{a2}_{1}) permits to prove Laws LE^{2}_{13}, LE^{2}_{14}, LE^{2}_{134} (LE^{a2}_{13}, LE^{a2}_{14}, LE^{a2}_{134}). The same is true for the Law LE^{1}_{2} (LE^{a1}_{2}). Besides, we have that equivalence between laws LE^{2}_{1} (LE^{a2}_{1}) and LE^{1}_{2} (LE^{a1}_{2}) can be proved in Ontology. ^ Theorem of mode and modus equivalence. Moda(a) Modus(a), i.e., a is mode iff a is modus. Theorem of transitivity. Moda(a,b)Moda(b,c) Moda(a,c), i.e., if a is mode of modus b and b is mode of modus c, then a is mode of modus c. Theorem of modus order.
Theorem of model unity. Modus(a) b(Model(b,a) Mod(a,a,b)), i.e., if a is modus, then for some b it is true that b is model for a and a is mode of itself in the model b. Theorem of relation of equalities. a = b a =^{2}_{1} b a =^{1}_{2} b ^ . Ex(a) Moda(a,b) Ex(b), i.e., if a exists and a is mode of b, then b exists. If we accept the following definitions and new axioms DA1. (a b) e x x e x a e a b e b [y(x e_{1} y z(y e_{1} z (a e z b e z))) NModa(x)], where “(a b)” is read as “sum of moduses a and b” DA2. (a b) e x x e x a e a b e b [y(x e_{1} y (a e y b e y)) NModa(x)], where “(a b)” is read as “product of moduses a and b” DA3. a e x x e x a e a [y(x e_{1} y (a e y)) NModa(x)], where “a” is read as “exterior of modus a” AN. aNModa(a) [^ ] AS. Moda(a) Moda(b) (a e b) x(b e_{1} x y(x e_{1} y (a e y))) [Axiom of separation], then the following important theorem can be proved Theorem of Boolean algebra on moduses. Operations , and form Boolean algebra on moduses. ^ Theorem of consistency. If Lesniewski’s protothetics is consistent, then Ontology is also consistent. To prove the theorem we should to offer an interpretation of Ontology in protothetics. I shall use the following interpretation (I) here: 1. If a is a name variable or constant, then I(a) is a propositional variable or constant accordingly. I shall use a for I(a). 2. If f is a projector, then I(f) is a functor of the type 3. I(Mod(a,b,c,f)) is the formula (af(b,c) a f(b,c) a(bc) (c b)). I shall use also the symbol Mod_{S}(a,b,c,f) for I(Mod(a,b,c,f)). In other respects formulas of Ontology do not change in the interpretation. I shall use sign “=” in the expressions “I(A) = B” (or “B = C”) for the assertion that formula B of protothetics is the interpretation of the formula A of Ontology (or interpretation B equals interpretaton C). I shall sometimes use parentheses […] for secretion of formula B or its equivalent representations. Further I(b a) = I(cfMod(a,b,c,f)) = cf(I(Mod(a,b,c,f))) = [cf(af(b,c) a f(b,c) a(bc) (c b))] [c(a(bc)) a] a b. Therefore we have: I(b a) = a b. I(Modus(b)) = [a(ab)] b I(b * c) = I(afMod(a,b,c,f)) = af(I(Mod(a,b,c,f))) = [af(af(b,c) a f(b,c) a(bc) (c b))] bc. Therefore we have: I(b * c) = bc. For Ontological Definitions D^{i}_{k1,…,km}. M^{i} [x_{k1}…x_{km}(y_{p1}...y_{pn}Mod(x_{k},C,y_{p}) Mod^{k1…km}(x_{k1},…,x_{km}) (x_{k1},…,x_{km}))], we obtain I(M^{i} [x_{k1}…x_{km}(y_{p1}...y_{pn}Mod(x_{k},C,y_{p}) Mod^{k1…km}(x_{k1},…,x_{km}) (x_{k1},…,x_{km}))]) = [I(M^{i}) (x_{k1}…x_{km}(y_{p1}...y_{pn}Mod_{S}(x_{k},C,y_{p}) Mod_{S}^{k1…km}(x_{k1},…,x_{km}) _{I}(x_{k1},…,x_{km}))) The formula x_{k1}…x_{km}(y_{p1}...y_{pn}Mod_{S}(x_{k},C,y_{p}) Mod_{S}^{k1…km}(x_{k1},…,x_{km}) _{I}(x_{k1},…,x_{km})) can be infered from the relative prothotetic definition, i.e., it is a theorem of prothotetics. Therefore all conditional is a theorem of prothotetics too. The same logic is right for the case of Ontological Definitions D^{ai}_{k1,…,km}. M^{i} [x_{k1}…x_{km}(y_{p1}...y_{pn}Mod(x_{k},F(),y_{p}) Mod^{k1…km}(x_{k1},…,x_{km}) (x_{k1},…,x_{km},))]. Further, the following equalitiy can be proved here: I(a ^{2}_{1} b) = [с(aс bс)] [ab] Therefore Ontological Law of Extentionality LE^{2}_{1} (LE^{a2}_{1}) is transformed to a Prothotetic Law of Extensionality. Another Ontological Laws of extensionality (LE^{3}_{124}, LE^{4}_{123}, LE^{a2}_{1}, LE^{a3}_{124}, LE^{a4}_{123}) are transformed to according Prothotetical Laws of Extensionality also. For axioms of Ontology we obtain: (AO1) I(b a d(d b d a) b b Modus(a)) = [a b d((b d) (a d)) (b b)]– the case of theorem of protothetics. (AO2) I(Mod(a,b,c,f) (a =^{1}_{234} f(b,c)) aMod(a,b,c,f)) = = [(af(b,c) a f(b,c) a(bc) (c b)) (a f(b,c)) a f(b,c) a(af(b,c) a f(b,c) a(bc) (c b))]  the case of theorem of protothetics too (here I used the equality I(a =^{1}_{234} b) (ab)). In other syntactical respects Ontology does not differ from LOntology (Ontology of Lesniewski) and it is known that if prothotetics is consistent, then LOntology is also consistent. Therefore, under this interpretation, all the rules of inference of Ontology become primitive or secondary rules of protothetics. This proves the theorem. 1 V.I.Moiseev “Logic of AllUnity”. Moscow: “Per Se”, 2002. – 415 p. (in Russian) 
Abstract  Е. А., 2010 об одной гипотезе космологии «Космология и внегалактическая астрофизика» опубликована статья [1] известных ученых, перевод abstract’a которой приведен ниже. Статья...  
Документы 1. /Paper.doc 2. /abstract.doc 