On Blass translation for Leśniewski's propositional ontology and modal logics
aa r X i v : . [ m a t h . L O ] J un On Blass translation for Le´sniewski’spropositional ontology and modal logics
Takao Inou´e
Abstract.
In this paper, we shall give another proof of the faithfulness of Blass transla-tion (for short, B -translation) of the propositional fragment L of Le´sniewski’s ontology inthe modal logic K by means of Hintikka formula . And we extend the result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logic. As a resultof observing the proofs we shall give general theorems on the faithfulness of B -translationwith respect to normal modal logics complete to serial or transitive or irreflexive or Eu-clidean or reflexive or symmetric Kripke frames. As an application of the theorems, forexample, B -translation is faithful for the provability logic PrL (= GL ), that is, K + ✷ ( ✷ φ ⊃ φ ) ⊃ ✷ φ . The faithfulness also holds for normal modal logics, e.g., KD , KT , K4 , KD4 , KB , KB4 , KTB , S4 and S5 . We shall conclude this paper with the sectionof comments which contains my ideas for the translations, some open problems and myconjectures. Keywords : Le´sniewski’s ontology, propositional ontology, Le´sniewski’s epsilon, modal in-terpretation, interpretation, translation, faithfulness, embedding, normal modal logic,modal logic K , Hintikka formula, tableau method, provability logic, deontic logic, se-rial frame, transitive frame, irreflexive frame, Euclidean frame, (almost) reflexive frame,(almost) symmetric frame, dense frame, convergent frame, bi-intuitionistic logic, displaylogic, bimodal logic, temporal logic.
1. Introduction
In Inou´e [19], a partial interpretation of Le´sniewski’s epsion ǫ in the modallogic K and its certain extensions was proposed: that is, Ishimoto’s propo-sitional fragment L (Ishimoto [24]) of Le´sniewski’s ontology L (refer to Ur-baniak [43]) is partially embedded in K and in the extensions, respectively,by the following translation I ( · ) from L to them:(1.i) I ( φ ∨ ψ ) = I ( φ ) ∨ I ( ψ ),(1.ii) I ( ¬ φ ) = ¬ I ( φ ),(1.iii) I ( ǫab ) = p a ∧ ✷ ( p a ≡ p b ),where p a and p b are propositional variables corresponding to the name vari-ables a and b , respectively. Here, “ L is partially embedded in K by I ( · )”1means that for any formula φ of a certain decidable nonempty set of formu-las of L (i.e. decent formulas (see § φ is a theorem of L if and only if I ( φ ) is a theorem of K . Note that I ( · ) is sound.The paper [21] also proposed similar interpretations of Le´sniewski’s ep-silon in certain von Wright-type deontic logics, that is, ten Smiley-Hansonsystems of monadic deontic logic and in provability logics (i.e., the full sys-tem PrL (= GL ) of provability logic and its subsystem BML (= K4 ),respectively. (See ˚Aqvist [1], and Smory´nski [38] or Boolos [3] for thoselogics.)The interpretation I ( · ) is however not faithful. A counterexample forthe faithfulness is, for example, ǫac ∧ ǫbc. ⊃ .ǫab ∨ ǫcc (for the details, see[21]). Blass [2] gave a modification of the interpretation and showed thathis interpretation T ( · ) is faithful, using Kripke models. In this paper, weshall call the faithful interpretation (denoted by B ( · )) Blass translation (forshort, B -translation ) or Blass interpretation (for short,
B-interpretation ).The translation B ( · ) from L to K is defined as follows:(2.i) B ( φ ∨ ψ ) = B ( φ ) ∨ B ( ψ ),(2.ii) B ( ¬ φ ) = ¬ B ( φ ),(2.iii) B ( ǫab ) = p a ∧ ✷ ( p a ⊃ p b ) ∧ .p b ⊃ ✷ ( p b ⊃ p a ),where p a and p b are propositional variables corresponding to the name vari-ables a and b , respectively.Later, I found another faithful and sound translation (embedding) offrom L to K taking(1.W) I ( ǫab ) = ⋄ p a ⊃ p a . ∧ ✷ ( p a ≡ p b )instead of (1.iii) (see Inou´e [22]).Now the purpose of this paper is to give another proof of the faithfulnessof Blass translation ( B -translation) of Le´sniewski’s propositional ontology L in the modal logic K by means of Hintikka formula . This will be done in §
4. In §
3, we shall give chain equivalence relation and so on, as importanttechnical tools for this paper. After the main result in §
4, we shall extendthe faithfulness result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logics in §
5. Observing the proof of thefaithfulness for K , we shall, in §
6, give a general theorem on the faithfulnessof B -translation with respect to normal modal logics with transitive or ir-reflexive frames. As an application of the generalization, we shall obtain thefaithfulness for provability logics (i.e., the full system PrL (= GL ) of prov-ability logic and its subsystem BML (= K4 )). PrL is characterized by K + ✷ ( ✷ φ ⊃ φ ) ⊃ ✷ φ . PrL is also characterized by K4 + ✷ ( ✷ φ ⊃ φ ) ⊃ ✷ φ .From careful consideration of the proof of the faithfulness for deontic log-ics, we shall obtain further general theorems for the faithfulness for thepropositional normal modal logics which are complete to serial or transitiveor irreflexive or Euclidean or almost reflexive or almost symmetric Kripkeframes. We shall also consider reflexive and symmetric relations, among oth-ers, i.e. dense and convergent relations As the result of the consideration,we shall show that the faithfulness also holds for normal modal logics, e.g., KD , KT , K4 , KD4 , KB , KB4 , KTB , S4 and S5 . This will be presentedin §
7. The section of comments ( §
8) contains some open problems and myconjectures. In the following §
2, we shall first collect the basic preliminariesfor this paper.
2. Propositional ontology L and its tableau method Let us recall a formulation of L , which was introduced in [24]. The Hilbert-style system of it, denoted again by L , consists of the following axiom-schemata with a formulation of classical propositional logic CP as its ax-iomatic basis:(Ax1) ǫab ⊃ ǫaa ,(Ax2) ǫab ∧ ǫbc . ⊃ ǫac ,(Ax3) ǫab ∧ ǫbc . ⊃ ǫba ,where we note that every atomic formula of L is of the form ǫab for somename variables a and b and a possible intuitive interpretation of ǫab is ‘the a is b ’.We note that (Ax1), (Ax2) and (Ax3) are theorems of Le´sniewski’s on-tology (see S lupecki [35]).The modal logic K is the smallest logic which contains all instances ofclassical tautology and all formulas of the forms ✷ ( φ ⊃ ψ ) ⊃ . ✷ φ ⊃ ✷ ψ beingclosed under modus ponens and the rule of necessitation (for K and basicsfor modal logic, see Bull and Segerberg [4], Chagrov and Zakharyaschev [6],Fitting [7], Hughes and Cresswell [14] and so on).Let us take ∨ (disjunction) and ¬ (negation) as primitive for the purepropositional logic part of L . We shall employ a useful tool, i.e. positive ( negative ) parts (for short, p . p . ’s (n . p . ’s)) of a formula due to Sch¨utte [32]and [34]. The positive and negative parts of a formula φ of L are inductivelydefined as follows: (i) φ is a p . p . of φ ; (ii) If η ∨ ξ is a p . p . of φ , then η and ξ are p . p . ’s of φ ; (iii) If ¬ ψ is a p . p . of φ , then ψ is a n . p . of φ ; (iv) If ¬ ψ isa n . p . of φ , then ψ is a p . p . of φ . By a notation F [ φ + ] ( G [ φ − ]), which also is due to [32], we mean thata formula φ occurs in a formula F [ φ + ] ( G [ φ − ]), as a p . p . (n . p . ) of F [ φ + ]( G [ φ − ]). Such expressions as F [ φ + , ψ − ] and the like are understood analo-gously under the conditions such that for example, F [ φ + , ψ − ], the specifiedformulas φ (as a p . p . ) and ψ (as a n . p . ) in F [ φ + , ψ − ] do not overlap witheach other. Definition . (Kobayashi-Ishimoto [27]) A formula φ of L is said to bea Hintikka formula of L if it satisfies all the following conditions: (1) φ isnot of the form F [ ψ + , ψ − ]; (2) If φ contains η ∨ ξ as a n . p . of φ , then itcontains η or ξ as a n . p . of it; (3) If φ contains ǫab as a n . p . of φ , then itcontains ǫaa as a n . p . of it; (4) If φ contains ǫab and ǫbc as n . p . ’s of φ , thenit contains ǫac as a n . p . of it; (5) If φ contains ǫab and ǫbc as n . p . ’s of φ ,then it contains ǫba as a n . p . of it.We shall review the tableau method TL for L . The tableau method TL for L is defined on the basis of the following four reduction rules: ∨ − G [ φ ∨ ψ − ] G [ φ ∨ ψ − ] ∨ ¬ φ | G [ φ ∨ ψ − ] ∨ ¬ ψǫ G [ ǫab − ] G [ ǫab − ] ∨ ¬ ǫaaǫ G [ ǫab − , ǫbc − ] G [ ǫab − , ǫbc − ] ∨ ¬ ǫacǫ G [ ǫab − , ǫbc − ] G [ ǫab − , ǫbc − ] ∨ ¬ ǫba where for TL , we take the same logical symbols as primitive as for L .(This tableau method TL is due to [27].)By reducing a formula by these reduction rules, we obtain a tableau ofthe formula. A branch of a tableau is closed if it ends with a formula of theform F [ φ + , φ − ], which is called an axiom of TL . A branch of a tableauis open if it is not closed. A tableau is said to be closed if every branch ofit is closed, otherwise we call it open . A tableau is said to be finite , if ithas a finite number of branches and evey branch of it is obtained by a finitenumbers of applications of reduction rules. A formula of TL is provable in TL if there exists a closed tableau of it. Theorem . ([27] (The Fundamental Theorem for TL ) Given a formula ( of L ( TL )), by reducing it with reduction rules we obtain a finite tableau,each branch of which ends either with an axiom of TL or with a Hintikkaformula, where every branch of it is extended by a reduction rule only if theformula to be reduced, say φ , is not an axiom of TL and the reduction givesrise to a formula not occurring in φ as a negative part of φ . A tableau is said to be normal , if it is constructed in compliance withthe condition of Theorem 2.2.If a normal tableau has a Hintikka formula φ at the end of some branchof it (in other words, if the formula to be reduced is not provable in TL ),then it is obvious from its normality that φ cannot be reduced further. Notethat a normal tableau of a formula is not always the shortest one of thepossible tableaux of it. Definition . Let φ be a formula of L . For any formula φ of L , we call ψ to be a Hintikka formula of φ , if one of open branches of a normal tableauof φ ends with ψ . Theorem . ([20] and [27]) For any formula φ of L , we have ⊢ TL φ ⇔ ⊢ L φ. A p . p . is said to be minimal if it is not of the form ¬ φ or φ ∨ ψ . A n . p . is said to be mininal if it is not of the form ¬ φ . (See [34, p . Theorem . ([34, p . If, under a sentential valuation v , every minimalp . p . of a formula φ takes the value falsity ) and every minimal n . p . of φ takes the value truth ) , then every p . p . of φ takes the value and everyn . p . of φ takes the value . ( In particular, v ( φ ) = 0 holds, since φ is a p . p . of φ . )
3. Chain equivalence relation, one more preliminary
As one more preliminary, we introduce the notion of chain equivalence classesand tails of a Hintikka formula, which are important for our proofs.
Definition . For any formula φ of L , the name vaiable set N V φ of φ isdefined as follows: N V φ = { a : a is a name variable of φ } . We note that for any formula φ of L , for any Hintikka formula ψ of φ , N V φ = N V ψ holds. Definition . For any Hintikka formula φ of L , the chain name variableset CN φ of φ is defined as follows: CN φ = { a ∈ N V φ : ∃ b ∈ N V φ ( φ = G [ ǫab − , ǫba − ]) } . It is possible that CN φ is empty for some Hintikka formula φ , e.g. φ = ǫaa . Definition . For any Hintikka formula φ of L , we define a relation onthe chain name vaiable set CN φ of φ is defined as follows: ∼ φ = { ( a, b ) ∈ CN φ × CN φ : φ = G [ ǫab − , ǫba − ] } . Proposition . For any Hintikka formula φ of L , the relation ∼ φ on CN φ is an equivalence relation. Proof.
Let φ be a Hintikka formula of L . For any a ∈ CN φ , G [ ǫaa − ] = φ holds by Definition 2.1.(3) and Definition 3.3. So we have the reflexivity a ∼ φ a . For any a, b ∈ CN φ , the symmetry ( a ∼ φ b ⇒ b ∼ φ a ) trivially holdsby Definition 3.3 and the definition of negative parts. For any a, b, c ∈ CN φ ,we have a ∼ φ b, b ∼ φ c ⇒ φ = G [ ǫab − , ǫba − , ǫbc − , ǫcb − ] . Apply Definition 2.1.(4) and Definition 3.3 to pairs ( ǫab, ǫbc ) and ( ǫcb, ǫba ) . Then, φ = G [ ǫac − , ǫca − ] . This means a ∼ φ c. So we obtain the transitivity. Hence the relation is anequivalence relation.
Definition . Let φ be a Hintikka formula of L . For any a ∈ CN φ , theequivalence class of a with respect to ∼ φ ,[ a ] ∼ φ = { b ∈ CN φ : a ∼ φ b } is the set of all elements of CN φ equivalent to a , which we call the chainequivalence class of a ( with respect to ∼ φ ). Definition . For any Hintikka formula φ of L , we define ” CN φ modulo ∼ φ ” to be the set ChainQ ( φ ) = CN φ / ∼ φ = { [ a ] ∼ φ : a ∈ CN φ } of all chain equivalence classes of CN φ . We call ChainQ ( φ ) the chain quo-tient set of φ . We call an element of
ChainQ ( φ ) a chain of φ . Definition . ([27]) For any Hintikka formula φ of L , a Kobayashi-Ishimoto-chain of φ is a nonempty (finite) set C of name variables, say C = { a , a , . . . , a n } ( n ≥
1) such that: (1) Every pair a i and a j (1 ≤ i ≤ n ,1 ≤ j ≤ n ) belonging to C are connected by a relation defined as ǫa i a j and ǫa j a i both of which are n . p . ’s of φ ;(2) The set is maximal with respect tothe property in (1). Definition . For any Hintikka formula φ of L , we define Kobayashi-Ishimoto-chain set
ChainKI ( φ ) of φ as follows: ChainKI ( φ ) = { C : C is a Kobayashi-Ishimoto-chain of φ } . Proposition . For any Hintikka formula φ of L , we have ChainQ ( φ ) = ChainKI ( φ ) . Proof.
Let φ be a Hintikka formula of L . We first prove ChainQ ( φ ) ⊆ ChainKI ( φ ). Let C be an element of ChainQ ( φ ). Take a name variable a such that C = [ a ] ∼ φ holds. For any x, y ∈ C , φ = G [ ǫxy − , ǫyx − ] holds, since φ = G [ ǫxa − , ǫax − , ǫya − , ǫay − ] . Apply Definition 2.1.(4) and Definition 3.3 to pairs ( ǫxa, ǫay ) and ( ǫya, ǫax ).Then, φ = G [ ǫxy − , ǫyx − ] . This satisfies Definition 3.7.(1). By Definition 3.6, C ∈ ChainQ ( φ ) is anelement of a partition of CN φ . So C satisfies Definition 3.7.(2). Hence C ∈ ChainKI ( φ ).We shall prove ChainKI ( φ ) ⊆ ChainQ ( φ ). Let C be an element of ChainKI ( φ ). By Definition 3.6, we have C ⊆ CN φ . Take an element a of C , arbitrarily. Then, by Defintion 3.7.(1), φ = G [ ǫax − , ǫxa − ] holds for any x ∈ C . We obtain C ⊆ [ a ] ∼ φ . Let y ∈ [ a ] ∼ φ . Themaximality of C follows y ∈ C . So, C = [ a ] ∼ φ . Thus C ∈ ChainQ ( φ ).Thus, a Kobayashi-Ishimoto-chain is nothing but our chain. Our Defini-tion 3.6 clarifies the notion of Kobayashi-Ishimoto-chains. Definition . ([27]) For any Hintikka formula φ of L and any chain C of φ , a tail of C is a name variable b such that ǫab is a n . p . of φ with a ∈ C and b / ∈ C . By T N φ , we denote the set of all tails of a Hintikka formula φ . We may have a chain of a given Hintikka formula φ without tails, e.g. φ = ¬ ǫaa . Proposition . For any Hintikka formula φ of L and any chain C of φ , no tail of C belongs to any other chains of φ . Proof.
Let φ be a Hintikka formula of L . Let C and C be distinct chainsof φ and b a tail of C . We may suppose that they are nonempty. Thenthere is a name variable a of C such that ǫab is a n . p . of φ . Suppose that b is a member of C . Since b is an element of C , by Definition 3.2, there issome name variable, say c such that ǫbc and ǫcb are n.p.’s of φ . Since ǫab isa n . p . of φ , by Definition 2.1.(5), φ contains ǫba as its n . p . . In other words, b is a member of C , which contradicts the definition of tails. Definition . Let φ be a Hintikka formula of L . By Rest φ , we denote N V φ − ( CN φ ∪ T N φ ).We can analyse the set of name variable occurring in the set of all minimal(atomic) p.p.’s and n.p.’s of a Hintikka formula of L . And it plays animportant role for the proof of our main theorem. Theorem . (Characterization Theorem for name variables of a Hintikkaformula) For any Hintikka formula φ of L and any name variable a ∈ Rest φ , a occurs in some minimal p.p. of φ . Further, N V φ = CN φ ∐ T N φ ∐ Rest φ ( ∗ ) holds, where ∐ means a disjoint union. Proof.
Let φ be a Hintikka formula of L . From Definition 3.10 and Propo-sition 3.11, we have CN φ ∪ T N φ = CN φ ∐ T N φ . From Definition 3.12, thisleads to ( ∗ ). Lemma . (Tail Lemma) Let φ be a Hintikka formula of L . For any tail a of a Chain of φ and any b ∈ N V φ , ǫab does not occur as a n.p. of φ . Proof.
Let φ be a Hintikka formula of L . Let a be a tail of a chain of φ and b ∈ N V φ . Suppose φ = G [ ǫab − ]. Since φ is a Hintikka formula of L ,we have φ = G [ ǫab − , ǫaa − ]. This means a ∈ [ a ] ∼ φ . That is, a is an elementof the chain [ a ] ∼ φ . This contradicts Proposition 3.11.
4. Another proof of the faithfulness of B -translation usingHintikka formula In this section, we shall prove the faithfulness of B -translation by means ofHintikka formula with respect to K , that is: Theorem . (Blass [2]) For any formula φ of L , we have ( ♣ ) ⊢ K B ( φ ) ⇒ ⊢ L φ. That is, B -translation is faithful with respect to K . Proof.
Let φ be a formula of L . For the proof, it is sufficient to showthat (not ⊢ TL φ ) ⇒ (not ⊢ K B ( φ )). Then by Theorem 2.4 we obtain thedesired meta-implication in ( ♣ ). If φ is a theorem of TL , then we triviallyhave the meta-implication. Suppose that φ is not a theorem of TL . Then,there exists an open normal tableau such that it has a branch ended with aHintikka formula, say ψ . We immediately see that φ is a p . p . of ψ (observethe reduction rules for TL ). (We can eventually prove it by induction onderivation.) That is, ψ is a Hintikka formula of φ . We may choose sucha Hintikka formula arbitrarily as a Hintikka formula of φ for our modelconstruction below.Let v be a sentential valuation such that by v , every atomic p . p . (n . p . ) of ψ is assigned falsity 0 (truth 1) (the rest of the assignment is at a person’sdisposal), where we do not need to look into the structure of atomic formulassuch as ǫab . Since ψ is a Hintikka formula, the valuation v makes everyminimal p . p . (n . p . ) false (true). Hence by Theorem 2.5, if we have v ( ψ ) = 0,then v ( φ ) = 0 holds, since φ is a p . p . of ψ .Since B ( · ) commutes disjunction and negation, B ( φ ) in the setting of K is falsified by the (adapted) valuation v such that we have v ( ψ ) = 0 in thatof L . In other words, in B ( φ ) we can regard formulas of the form B ( ǫab )as atomic formulas, when we assign truth-values to them in order to falsify B ( φ ).We shall below construct such a valuation, namely a model in which B ( φ ) is false. (If we have a model falsifying B ( φ ), then by the completenesstheorem for K , B ( φ ) is not a theorem of K .) Given a Hintikka formula ψ ,we have a finite numbers of its chains and their associated tails, observingits atomic negative and positive parts. Say ChainQ ( ψ ) = { C , . . . , C n } ( n ≥ n = 0 (that is, ψ contains no atomic negative parts), then wetake a Kripke model M = < G , R , V > such that G = {∗ , g } ∪ G ω ( ∗ 6 = g ), R = { ( ∗ , g ) ∪ { ( η, η ) : η ∈ G ω } .0 V ( p a , x ) = 0 for any name variable a and any x ∈ G , where G ω = { η j : j ∈ ω } , Card ( G ω ) = ℵ , {∗ , g } ∩ G ω = ∅ .( V ( p, x ) stands for the truth-value of a propositional variable p in a world x (by V ) in M ). Then we easily see that B ( φ ) is false in ∗ in M , since B ( ǫab ), that is, p a ∧ ✷ ( p a ⊃ p b ) ∧ .p b ⊃ ✷ ( p b ⊃ p a )is trivially false in ∗ in M regardless of the modality. So suppose n ≥ M = < G, R, V > (below we shallwrite g | = φ for | = M g φ (i.e., φ is true in a world g in M ) as follows.(M1) G = {∗ , g , . . . , g n } ∪ G ω ,where G ω = { η j : j ∈ ω } , Card ( G ω ) = ℵ , {∗ , g , . . . , g n } ∩ G ω = ∅ and ∗ , g , . . . , g n are distinct.(M2) R = { ( ∗ , g i ) : 1 ≤ i ≤ n } ∪ { ( η, η ) : η ∈ G ω } .(M3)(i) For any name variable a ∈ N V ψ , V ( p a , ∗ ) = (cid:26) a ∈ CN ψ ,0 otherwise.(ii) Take one name variable c i from each C i (1 ≤ i ≤ n ) as its representa-tive, that is, say C i = [ c i ] ∼ ψ . Assume that if the p c i is assigned a truth-valuein a world, then every propositional variable p corresponding to the othername variable belonged to C i should be assigned the same truth-value in thesame world. Under this assumption, we take the following assignment: V ( p c , g ) V ( p c , g ) . . . V ( p c , g n ) V ( p c , g ) V ( p c , g ) . . . V ( p c , g n )... ... . . . ... V ( p c n , g ) V ( p c n , g ) . . . V ( p c n , g n ) = U where U is n × n unit matrix in linear algebra. (Here we remark that for thisconstruction of Kripke model, we do not need the notion of connectednessused in [21].) We remark that the order of C , . . . , C n does not matter.(iii) If a name variable a is a tail of some chain, we take the followingassignment. First fix the assingment of (M3).(ii). Take all chains such that a is a tail of them, say ˜ C , . . . , ˜ C M (0 ≤ M ≤ n ) (if there is a tail, then M ≥ d i ∈ ˜ C i for any 1 ≤ i ≤ M , arbitrarily. Take all indices ξ , . . . , ξ M such that V ( p d i , g ξ i ) = 1 for any 1 ≤ i ≤ M (note that d i may beeventually equal to c ξ i in the notation of (M3).(ii)). Such ξ , . . . , ξ M exist1because of (M3).(ii), Proposition 3.4 and Definition 3.6. Then take, for any1 ≤ i ≤ n , V ( p a , g i ) = (cid:26) i ∈ { ξ , . . . , ξ M } ,0 otherwise.(iv) For any p of the rest of propositional variables and any 1 ≤ i ≤ n , V ( p, g i ) = 0.(v) For any propositional variable p and any η ∈ G ω , V ( p, η ) = 1.Now, let us verify that the model just constructed above actually falsifies B ( ψ ) in the world ∗ , which makes B ( ψ ) invalid. So = K B ( φ ) holds fromTheorem 2.5. Thus, K B ( φ ) by the completeness theorem for K . Let D , . . . , D s be all distinct atomic p . p . ’s of ψ and E , . . . , E t all distinct atomicn . p . ’s of ψ ( s + t ≥ t ≥
1) because of n ≥
1. In order to apply Theorem2.5 to our case, we have to show that( † ) ∗ 6| = B ( D k ) for any 1 ≤ k ≤ s ( s may possibly be 0.),( †† ) ∗ | = B ( E k ) for any 1 ≤ k ≤ t .First we remark that we do not need to consider the worlds of G ω at all,since ∗ does not relate any element of G ω . We put the G ω in order to keepa generality of models.For verifying ( † ) and ( †† ), we need to classify all the name variables (inminimal parts of ψ ) occurring in ψ , thus in φ as follows.(NV1) All distinct name variables occurring in chains, say x , . . . , x p ( p ≥ CN ψ = { x , . . . , x p } .(NV2) All distinct name variables which are tails of some chain, say y , . . . , y q ( q ≥ T N φ = { y , . . . , y q } .(NV3) All distinct name variables which are not tails, occurring in someatomic positive parts of B , but not in any of chains, say z , . . . , z r ( r ≥ Rest φ = { z , . . . , z r } .Note that x , . . . , x p , y , . . . , y q , z , . . . , z r are mutually distinct variablesbecause of Theorem 3.13.We shall first verify ( † ). There are the following cases (Case 1)–(Case 5)for D k (1 ≤ k ≤ s ). Suppose s ≥
1. If s = 0, we ignore the verification forthe cases..(Case 1): The case of D k = ǫx i x j or ǫx j x i for any i and j with i < j , suchthat x i and x j are not in the same chain. (Because of Definitions 2.1.(1) and3.5, it is not possible that x i and x j belong to the same chain, if ǫx i x j is ap . p . of ψ . We do not need to consider the case of i = j because of Definition2.1.(1).) In this case, by (M3).(ii), there are exactly two indices α and β ( α = β ) (1 ≤ α ≤ n , 1 ≤ β ≤ n ) such that(a) x i ∈ [ c α ], g α | = p x i , g α = p x j ,2 (b) x j ∈ [ c β ], g β | = p x j , g β = p x i hold. So we have g α = p x i ⊃ p x j and g β = p x j ⊃ p x i . Since ∗ | = ✷ ( p x i ⊃ p x j ) ⇔ .g | = p x i ⊃ p x j ∧ g | = p x i ⊃ p x j ∧ · · · ∧ g n | = p x i ⊃ p x j holds, we obtain ∗ 6| = ✷ ( p x i ⊃ p x j ), when D k = ǫx i x j . Similarly, we get ∗ 6| = ✷ ( p x j ⊃ p x i ), when D k = ǫx j x i . Thus we have ∗ 6| = B ( D k ) in bothcases.(Case 2): The case of D k = ǫx i y j for any i and j such that y j is not a tailof the chain to which x i belongs. Then, by (M3).(ii) and (M3).(iii), there isexactly one index 1 ≤ α ≤ n such that V ( p x i , g α ) = 1 and V ( p y j , g α ) = 0.From this, we have ∗ 6| = ✷ ( p x i ⊃ p y j ). So we get ∗ 6| = B ( D k ).(Case 3): The case of D k = ǫx i z j for any i and j . In this case we have ∗ | = p x i and ∗ | = p z j ⊃ . ✷ ( p z j ⊃ p x i ). But ∗ 6| = ✷ ( p x i ⊃ p z j ) holds as (Case1). (Case 4): The case of D k = ǫy i b for any i where b is arbitrary. From(M3).(i) and Theorem 3.13, ∗ 6| = p y i holds. This makes our case verified.(Case 5): The case of D k = ǫz i b for any i where b is arbitrary. Then, by(M3).(i), ∗ 6| = p z i , thus ∗ 6| = B ( D k ).Next, we shall verify ( †† ). In this case, there are only the following twocases for E k (1 ≤ k ≤ t ).(Case 6): The case of E k = ǫx i x j for any i and j such that x i and x j belong to the same chain (because ψ is a Hintikka formula). By (M3).(i), ∗ | = p x i . By (M3).(ii), we have ∗ | = ✷ ( p x i ⊃ p x j ) and ∗ | = p x j ⊃ ✷ ( p x j ⊃ p x i ), since g α | = p x i ≡ p x j holds for any (1 ≤ α ≤ n ). Thus, ∗ | = B ( E k ).(Case 7): The case of E k = ǫx i y j for any i and j such that y j is a tailof the chain to which x i belongs (because ψ is a Hintikka formula). By(M3).(i), ∗ | = p x i . By (M3).(ii) and (M3).(iii), we obtain ∗ | = ✷ ( p x i ⊃ p y i ).By (M3).(i), we have ∗ | = p y j ⊃ ✷ ( p y j ⊃ p x i ), since ∗ | = p y i and ∗ 6| = ✷ ( p y j ⊃ p x i )holds as (Case 6). Hence, ∗ | = B ( E k ).We can now conclude ∗ 6| = B ( ψ ). This completes the whole proof for( ♣ ). Corollary . (Blass [2]) B -translation is an embedding of L in K . Proof.
From Theorem 4.1.3
5. The faithfulness for von Wright-type deontic logics
As von Wright-type deontic logic, we shall deal with ten Smiley-Hansonsystems of monadic deontic logic after ˚Aqvist [1], that is, OK , OM , OS4 , OB , OS5 , OK + , OM + , OS4 + , OB + , OS5 + .As primitive logical connectives, we take ⊤ (verum), ⊥ (falsum), ¬ (na-gation), O (obligation), P (permission), ∧ (conjunction), ∨ (disjunction), → (implication), ↔ (material equivalence). (We may think of O and P as ✷ and ✸ , respectively.)The well-formed formulas of each system are defined as usual as those ofpropositional modal logics.The two rules of inferences, modus ponens and O -necessitation ( ⊢ A implies ⊢ OA ) are common to all the ten Smiley-Hanson systems of monadicdeontic logic.We need the following axiom schemata for the system.(A0) All classical propositional tautologies(A1) P A ↔ ¬ O ¬ A (A2) O ( A → B ) → ( OA → OB )(A3) OA → P A (A4) OA → OOA (A5)
P OA → OA (A6) O ( OA → A )(A7) O ( P OA → A )First five systems are defined as follows. OK = A0–A2 OM = A0–A2, A6 OS4 = A0–A2, A4, A6 OB = A0–A2, A6, A7 OS5 = A0–A2, A4, A5Note that A6 and A7 are derivable in
OS5 .Let X be any of these five systems. Then we define: X + = X , A3.We shall recall the definition of accessibility relations as follows.(AR1) R is serial in W ∀ x ∃ ( xRy )(AR2) R is transitive in W ∀ x ∀ y ∀ z ( xRy ∧ yRz. ⊃ xRz )(AR3) R is Euclidean in W ∀ x ∀ y ∀ z ( xRy ∧ xRz. ⊃ yRz )(AR4) R is almost reflexive in W ∀ x ∀ y ∀ z ( xRy ⊃ yRy )(AR5) R is almost symmetric in W ∀ x ∀ y ∀ z ( xRy ⊃ .yRz ⊃ zRy ),4where W is the set of possible worlds. With these relations we characterizethe systems as follows:The class of OK -models has no condition R being imposed.The class of OM -models has almost reflexive R .The class of OS4 -models has transitive and almost reflexive R .The class of OB -models has almost symmetric and almost reflexive R .The class of OS5 -models has Euclidean and transitive R .The class of OK + -models has serial R .The class of OM + -models has serial and almost reflexive R .The class of OS4 + -models has serial, transitive and almost reflexive R .The class of OB + -models has serial, almost symmetric and almost re-flexive R .The class of OS5 + -models has serial, Euclidean and transitive R . Theorem . (See [1]) The soundness and completeness theorems hold forany ten Smiley-Hanson systems.
We shall take trivial adaptation B O of B -tranlation for the Smiley-Hanson systems.(O.i) B O ( φ ∨ ψ ) = B O ( φ ) ∨ B O ( ψ ),(O.ii) B O ( ¬ φ ) = ¬ B O ( φ ),(O.iii) B O ( ǫab ) = p a ∧ O ( p a ⊃ p b ) ∧ .p b ⊃ O ( p b ⊃ p a ),where p a and p b are propositional variables corresponding to the name vari-ables a and b , respectively.We may regard OK as K . The other systems are stronger than OK .Hence the following theorem is easily proved model theoretically. Theorem . For any of ten Smiley-Hanson systems, say SH , and anyformula φ of L , we have ⊢ TL φ ⇒ ⊢ SH B O ( φ ) . With the slight modification of the accessibility relations as above, we canobtain following theorem for the faithfullness of the adapted B O -translation. Theorem . For any of ten Smiley-Hanson systems, say SH , and anyformula φ of L , we have ⊢ SH B O ( φ ) ⇒ ⊢ L φ. Proof.
Let SH be one of ten Smiley-Hanson systems. Let φ be a formulaof L . If the number of chain n ≥ G and R in the model construction of Theorem 4.1,5(M O G O = {∗ , g , . . . , g n } ∪ G ω ,where G ω = { η j : j ∈ ω } , Card ( G ω ) = ℵ , {∗ , g , . . . , g n } ∩ G ω = ∅ and ∗ , g , . . . , g n are distinct.(M O R O = BR ∪ { ( g i , g j ) : i = j, ≤ i ≤ n, ≤ j ≤ n } if SH is OS5 or OS5 + BR ∪ { ( g i , g i ) : 1 ≤ i ≤ n } otherwise.where BR = { ( ∗ , g i ) : 1 ≤ i ≤ n } ∪ { ( η, η ) : η ∈ G ω } .If n = 0 holds we take G O = {∗ , g } ∪ G ω as (M O R O = { ( ∗ , g ) } ∪ { ( η, η ) : η ∈ G ω } . if SH is OS5 or OS5 + { ( ∗ , g ) , ( g, ∗ ) , ( g, g ) } ∪ { ( η, η ) : η ∈ G ω } . otherwise.We shall take the same assignment as that for Theorem 4.1. The abovemodification of the accessibility relations does not effect the truth of ∗ 6| = B O ( φ ), since the additional parts of relations do not related to ∗ . Hence weprove this theorem with the same procedures of Theorem 4.1. We remarkthat we can take the following relation R Oc = BR ∪ { ( g i , g j ) : 1 ≤ i ≤ n, ≤ j ≤ n } in place of R O . Still R Oc is Euclidean and it satisfies all other relations foreight deontic logics.From the above theorem, Corollary . For any of ten Smiley-Hanson systems, say SH , and anyformula φ of L , B -translation is an embedding of L in SH .
6. A general theorem on the faithfulness
First, as usual, a normal modal logic X is defined as follows:(nor.1) K ⊆ X ,(nor.2) X is closed under modus ponens, substitution and the rule ofnecessitation (i.e., ⊢ φ implies ⊢ ✷ φ ).We may say that normal modal logics are extensions of K .In this section, we shall give a general theorem on the faithfulness of B -translation with respect to normal modal logics with transitive frames orirreflexive ones.6 Theorem . Let X be a normal modal logic. Let I be an index set suchthat ω ⊆ I . Suppose that X is complete with respect to a set of Kripkeframes, say F = { ( G α , R α ) : α ∈ I } where for any α ∈ I , G α is a nonemptyinfinite set and R α is an accessibility relation on G α . Let H be a set ofKripke frames such that (6 . .i ) H = { ( G ∗ i , R ∗ i ) : i ∈ ω } , (6 . .ii ) For any i ∈ ω , G ∗ i = {∗ , g , g , . . . , g i } ∪ G ωi , where G ωi = { η ij : j ∈ ω } , Card ( G ωi ) = ℵ , {∗ , g , . . . , g n } ∩ G ωi = ∅ and ∗ , g , . . . , g n are distinct and R ∗ i = { ( ∗ , g j ) : 0 ≤ j ≤ i } ∪ { ( η, η ) : η ∈ G ω } . If H ⊆ F holds, then B -translation is faithful with respect to X . Proof.
Suppose the assumption and the condition of the theorem to beproved. Then we can take the same model-construction in the proof of ( ♣ )in § Definition . By ExtK tirr , we denote the set of all normal logics suchthat they are elements of
ExtK and they are complete w.r.t. a set of tran-sitive or irreflexive Kripke frames.
Corollary . Let X ∈ ExtK tirr . B -translation is faithful with respectto X . Proof.
From Theorem 6.1.
Corollary . Let X ∈ ExtK tirr . B -translation is an embedding of L in X . Proof.
From Corollary 6.3.As a direct consequece of the above corollary, we have the following.
Corollary . B -translation is an embedding of L in K4 . Proof. K4 is stronger than K . And it is complete with respect to transitiveframes. Thus we obtain the soundness and faithfulness of B from Corollary6.4.7 K4 is a subsystem of the provability logic PrL (= K4 + ✷ ( ✷ φ ⊃ φ ) ⊃ ✷ φ ). The subsystem is also called BML . PrL is also characterized by K + ✷ ( ✷ φ ⊃ φ ) ⊃ ✷ φ (see [6]).We know that φ is a theorem of PrL iff φ is valid in all finite transitiveand irreflexive frames (see e . g . Boolos [3];
PrL is denoted by GL in [3], alsosee Carnielli and Pizzi [5]). (Frames ( W, R ) in which W is finite and R isirreflexive and transitive are called strict partial orders. ) Thus we have: Corollary . B -translation is an embedding of L in PrL . Proof.
In this case, we may take G ω = ∅ in the model construction of theproof of Therem 4.1. So we can do all in finite models. From Theorem 6.1.and Cororllary 6.4, our corollary holds, since our accessibility relations areall finte transitive and irreflexive.
7. Further general results for the faithfulness
Considering the proof of Theorems 4.1, 5.3 and 6.1, we can obtain furthergeneral theorems for the faithfulness of B -translation and their applications. Theorem . Let X be a normal modal logic. Let I be an index set suchthat ω ⊆ I . Suppose that X is complete with respect to a set of Kripkeframes, say F = { ( G α , R α ) : α ∈ I } where for any α ∈ I , G α is a nonemptyinfinite set and R α is an accessibility relation on G α . Let H be a set ofKripke frames such that (7 . .i ) H = { ( G ∗ i , R ∗ i ) : i ∈ ω } , (7 . .ii ) For any i ∈ ω , G ∗ i = {∗ , g , g , . . . , g i } ∪ G ωi , where G ωi = { η ij : j ∈ ω } , Card ( G ωi ) = ℵ , {∗ , g , . . . , g n } ∩ G ωi = ∅ and ∗ , g , . . . , g n are distinct and R ∗ i = { ( ∗ , g j ) : 0 ≤ j ≤ i } ∪ { ( g j , g k ) : j = k, ≤ j ≤ i, ≤ k ≤ i }∪{ ( η, η ) : η ∈ G ωi } . If H ⊆ F holds, then B -translation is faithful with respect to X . Proof.
From Theorem 4.1, 5.3 and 6.1.Theorem 7.1 is applicable to normal modal logics with serial or irreflexiveor Euclidean or almost symmetric Kripke frames.8
Theorem . Suppose that we take the same assumptions about X , I , F , H , G ∗ i , G ωi ( for any i ∈ ω ) in Theorem 7.1 with R ∗ i = { ( ∗ , g j ) : 0 ≤ j ≤ i } ∪ { ( g j , g j ) : 1 ≤ j ≤ i }∪{ ( η, η ) : η ∈ G ωi } . If H ⊆ F holds, then B -translation is faithful with respect to X . Proof.
From Theorems 4.1, 5.3 and 6.1.Theorem 7.2 is applicable to normal modal logics with serial or transitiveor irreflexive or almost reflexive or almost symmetric Kripke frames.
Theorem . Suppose that we take the same assumptions about X , I , F , H , G ∗ i , G ωi ( for any i ∈ ω ) in Theorem 7.1 with R ∗ i = { ( ∗ , g j ) : 0 ≤ j ≤ i } ∪ { ( g j , g k ) : 1 ≤ j ≤ i, ≤ k ≤ i }∪{ ( η, η ) : η ∈ G ωi } . If H ⊆ F holds, then B -translation is faithful with respect to X . Proof.
From Theorems 4.1, 5.3 and 6.1.Theorem 7.3 is applicable to normal modal logics with serial or transitiveor irreflexive or Euclidean or almost reflexive or almost symmetric Kripkeframes.
Definition . By ExtK d , we denote the set of propositional modal logicsstronger than K such that they are complete w.r.t. a set of serial or transitiveor irreflexive or Euclidean or almost reflexive or almost symmetric Kripkeframes. Theorem . For any logic X ∈ ExtK d , B -translation is faithful withrespect to X . Proof.
From Theorem 7.3.
Corollary . For any logic X ∈ ExtK d , B -translation is an embeddingof L in X . Proof.
From Theorem 7.5.9We recall the naming of modal logics as follows (refer to e.g. Poggiolesi[31] and Ono [30], also see Bull and Segerberg [4]): KD : K + ✷ φ ⊃ ✸ φ ( D , serial relation) KT : K + ✷ φ ⊃ φ ( T , reflexive relation) K4 : K + ✷ φ ⊃ ✷✷ φ ( , transitive relation) KD4 : K + D + (serial and transitive relation) KB : K + φ ⊃ ✷✸ φ ( B , symmetric relation) KB4 : K + B + (symmetric and transitive relation) KTB : KT + B (reflexive and symmetric relation) S4 : KT + (reflexive and transitive relation) S5 : S4 + B (reflexive, transitive and symmetric relation, i.e., equivalencerelation) or equivalently: KT + ✸ φ ⊃ ✷✸ φ ( , Euclidean relation),where + means ⊕ in the sense of [6], which the system is closed under modusponens, substitution and the rule of necessitation. We use + in the senseeveywhere in this paper. For those logics, we can give the correspondinggeneral theorems. Theorem . Suppose that we take the same assumptions about X , I , F , H , G ∗ i , G ωi ( for any i ∈ ω ) in Theorem 7.1 with R ∗ i = { ( ∗ , g j ) : 0 ≤ j ≤ i } ∪ { ( g j , ∗ ) : 0 ≤ j ≤ i }∪{ ( g j , g k ) : j = k, ≤ j ≤ i, ≤ k ≤ i }∪{ ( η, η ) : η ∈ G ωi } . If H ⊆ F holds, then B -translation is faithful with respect to X . Proof.
From Theorem 4.1, 5.3 and 6.1, we can take the same model con-struction in the proof of Theorem 4.1 and prove it. We do not need takecare of { ( g j , ∗ ) : 0 ≤ j ≤ i } , since we may only consider the worlds which ∗ can access.Theorem 7.7 is applicable to normal modal logics with serial or irreflexiveor Euclidean or symmetric Kripke frames. Theorem . Suppose that we take the same assumptions about X , I , F , H , G ∗ i , G ωi ( for any i ∈ ω ) in Theorem 7.1 with R ∗ i = { ( ∗ , g j ) : 0 ≤ j ≤ i } ∪ { ( g j , ∗ ) : 0 ≤ j ≤ i }∪{ ( g j , g k ) : 1 ≤ j ≤ i, ≤ k ≤ i } ∪ { ( ∗ , ∗ ) }∪{ ( η, η ) : η ∈ G ωi } . If H ⊆ F holds, then B -translation is faithful with respect to X . Proof.
From Theorem 4.1, 5.3 and 6.1, we can take the same model con-struction in the proof of Theorem 4.1 and verify the theorem. What we needto check is that of the cases (Case 1) and (Case 2) with the formulas as such ∗ | = ✷ ( p x i ⊃ p x j ) ⇔ . ∗ | = p x i ⊃ p x j ∧ g | = p x i ⊃ p x j ∧ g | = p x i ⊃ p x j ∧ · · · ∧ g n | = p x i ⊃ p x j . In (Case 1), ∗ | = p x i ⊃ p x j does not effect the proof of ∗ 6| = ✷ ( p x i ⊃ p x j )in the proof of Theorem 4.1. In (Case 2), ∗ 6| = p x i ⊃ p y j leads to ∗ 6| = ✷ ( p x i ⊃ p y j ). From this we have ∗ 6| = B ( D k ). We remark that we shall take G = { ( ∗ , ∗ ) } ∪ { ( η, η ) : η ∈ G ω } and the same valuation, if the number ofchain is 0.Theorem 7.8 is applicable to normal modal logics with serial or transitiveor reflexive or Euclidean or symmetric Kripke frames.From Theorem 7.8, we can obtain the following theorem. Theorem . Let X be one of logics KD , KT , K4 , KD4 , KB , KB4 , KTB , S4 and S5 . B -translation is faithful with respect to X . Proof.
From Theorem 7.8.
Corollary . Let X be one of logics KD , KT , K4 , KD4 , KB , KB4 , KTB , S4 and S5 . B -translation is an embedding of L in X . Proof.
From Theorem 7.9.We shall remark that Theorem 7.8 is applicable to normal modal logicswith the following axioms and accessibility relations: C4 : ✷✷ φ ⊃ ✷ φ ( ∀ x ∀ y ( xRy ⊃ ∃ z ( xRz ∧ zRy )), dense relation) C : ✸✷ φ ⊃ ✷✸ φ ( ∀ x ∀ y ∀ z ( xRy ∧ xRz. ⊃ ∃ w ( yRw ∧ zRw )), convergentrelation). Theorem . Let X be a normal modal logic. If X has the finite modelproperty, then ’infinite’, G ∗ i = {∗ , g , g , . . . , g i } ∪ G ωi and R ∗ i can be replacedby ’finite’, G ∗ i = {∗ , g , g , . . . , g i } and R ∗ i − { ( η, η ) : η ∈ G ωi } respectively,in Theorems 6.1, 7.1, 7.2, 7.3, 7.7 and 7.8. Proof.
Let X be a normal modal logic. Suppose that X has the finitemodel property. In the proof of Theorem 4.1, we can take G ω = ∅ . Thenall proofs of Theorems 6.1, 7.1, 7.2, 7.3, 7.7 and 7.8 follow the modificationwithout any trouble.For the finite model property, refer to e.g. Carnielli and Pizzi [5], Ono[30] (especially for algebraic models) and so on.1
8. Comments
One motive from which I wrote [19] and [21] is that I wished to understandLe´sniewski’s epsilon ǫ on the basis of my recognition that Le´sniewski’s epsilonwould be a variant of truth-functional equivalence ≡ . Namely, my originalapproach to the interpretation of ǫ was to express the deflection of ǫ from ≡ in terms of Kripke models. We may say that this is an approximationof ǫ by something simple and symmetric, that is equivalence. In analysis,the fundamental idea of calculus is the local approximation of functions bylinear functions. In mathematics, we can find a lot of such examples asthat of calculus. As an example in logic, a Gentzen-style formulation ofintuitionistic logic is nothing but an expression of the deflection from thecomplete geometrical symmetry of Gentzen’s LK . Gentzenization of a logicalso has such a sense, as far as I am concerned.It is well-known that Le´sniewski’s epsilon can be interpreted by theRussellian-type definite description in classical first-order predicate logicwith equality (see [24]). Takano [41] proposed a natural set-theoretic in-terpretation for the epsilon. The other motive for the above papers of mineis that I wanted to avoid such interpretations when we interpret Le´sniewski’sepsilon. I do not deny the interpretation using the Russellian-type definitedescription and a set-theoretic one. I was rather anxious to obtain anotherquite different interpretation of Le´sniewski’s epsilon having a more proposi-tional character.We now know the quite satisfactory B -translation of which the discoveryis, in my opinion, monumental in the study of Le´sniewski’s systems. B -translation is very natural, if one tries to understand it in an intuitive way.However I believe that there are still other interpretations which are in thespirit of my original recognition mentioned above. One fullfilment of mybelief is the (2.W) in § L can be used as examples of which the validity in K is checked (see e.g.Lagniez et al [28]).I shall give an open problem being directly related to this paper. Open problem 1 . Is the condition of Theorems 6.1, 7.1, 7.2, 7.3, 7.7and 7.8 (i . e . , as { (6.1.i) and (6.1.ii)) } and { (7.1.i) and (7.1.ii)) } ) the bestpossible one for the faithfulness of B -translation ?Other interesting open problems are: Open problem 2 . Give an algebraic proof of the faithfulness of B -translation. Open problem 3 . Search for possibilities of embedding Le´sniewski’s2propositional ontology L in bi-intuitionistic logic and the display logic forbi-intuitionistic logic (see Gor´e [13] and Wansing [45] for those logics). Open problem 4 . Search for possibilities of embedding Le´sniewski’spropositional ontology L in logics treated in Marx and Venema [29].I think that this book is very important to consider. It is also worthconsidering Gabbay, Kurucz, Wolter and Zakharyaschev [12] for temporaland computational aspects, and products for modal logics, for example.In relation to the theme of this paper, I shall give my conjectures. Conjecture 1 . L is embedded in some bimodal logic by some soundtranslation.I shall suggest a direction that, I believe, is to be proceeded for thisconjecture. Let T , T be some normal modal propositional logics withmodal operators ✷ , ✷ , respectively. Let T b be a bi-modal propositionallogic T + T + ✷ ( φ ⊃ ψ ) ⊃ ✷ ( ψ ⊃ φ ), for example. ∗ Then we define atranslation B d ( · ) from L to T b as follows:(3.i) B d ( φ ∨ ψ ) = B d ( φ ) ∨ B d ( ψ ),(3.ii) B d ( ¬ φ ) = ¬ B d ( φ ),(3.iii) B d ( ǫab ) = p a ∧ ✷ ( p a ⊃ p b ) ∧ .p b ⊃ ✷ ( p b ⊃ p a ),where p a and p b are propositional variables corresponding to the name vari-ables a and b , respectively. By this translation, I believe that L is embeddedin T b for some T b . Conjecture 2 . Le´sniewski’s propositional ontology L is embedded inmany other modal logics which are listed in, for example, Chagrov andZakharyaschev [6], Humberstone [15] and Gabbay and Guenthner [10]. Conjecture 3 . Le´sniewski’s propositional ontology L is embedded insome temporal logics with several ways.It seems to me that this conjecture 3 will give a broad possibility offuture research. Conjecture 4 . Le´sniewski’s elementary ontology L is embedded in some first-order modal predicate logic.This conjecture 4 is the most important open problem in the next stageof our direction of study. It is well-known that L with axiom ∃ S ( ǫSS ) isembedded in monadic second-order predicate logic (see Smirnov [36], [37]and Takano [40]). I believe that by introuducing a modal operator, we couldreduce second-order to first-order for the embedding. In a similar direction,a certain attempt has been carried out in Urbaniak [42] and [44]. If we findsuch an embedding in a particular formal system with a modal operator ∗ There would be many choices of relationizing ✷ and ✷ , I believe. Acknowledgements.
I would like to thank Prof. Andreas Blass, Prof.Mitio Takano, the late Prof. Stanislaw ´Swierczkowski and the late Prof.Emeritus Arata Ishimoto for their valuable comments, stimulation and en-couragement. Further, I would also like to thank, especially, the late Prof.Anne Troelstra and Prof. Dirk van Dalen for their encouragement.
References [1] ˚Aqvist , L., ‘Deontic logic’, pp. 605–714 in [9] and also pp. 147–264 (as a revisedversion) in [10].[2]
Blass , A., ‘A faithful modal interpretation of propositional ontology’,
MathematicaJaponica
Boolos , B.,
The Logic of Provability , Cambridge University Press, 1993.[4]
Bull , R. A. and K.
Segerberg , ‘Basic modal logic’, pp. 1–88 in [9] and also pp.1–82 in [10].[5]
Carnielli , W. and C.
Pizzi , Modalities and Multimodalities , Springer-Verlag, 2008.[6]
Chagrov , A. and M.
Zakharyaschev , Modal Logic , Clarendon Press, 1997.[7]
Fitting , M.,
Proof Methods for Modal and Intuitionistic Logics , D. Reidel, 1983.[8]
Fontaine , P, S.,
Schulz and J.
Urban (eds.),
Proceedings of the 5th Workshop onPractical Aspects of Automated Reasoning (PAAR 2016) , Coimbra, Portugal, 02-07-2016, 2016. published at http://ceur-ws.org[9]
Gabbay , D. and F.
Guenthner (eds.),
Handbook of Philosophical Logic, vol. II:Extensions of Classical Logic , D. Reidel, 1984.[10]
Gabbay , D. and F.
Guenthner (eds.),
Handbook of Philosophical Logic, vol. 3, 2nded. , Springer-Verlag, 2001.[11]
Gabbay , D. and F.
Guenthner (eds.),
Handbook of Philosophical Logic, vol. 8, 2nded. , Springer-Verlag, 2002.[12]
Gabbay , D. M, A.
Kurucz , F.
Wolter and M.
Zakharyaschev , Many-Dimensional Modal Logics: Theory and Application , Elsevier Science B.V., 2003.[13]
Gor´e , R., ‘Dual Intuitionistic Logic Revisited’, pp. 252–267 in: R.
Dyckhoff (ed.),
Proceedings Tableaux 2000, LNAI 1847 , Springer-Verlag, 2000, .[14]
Hughes , G. E. and M. J.
Cresswell , A Companion to Modal Logic , Methuen, 1984.[15]
Humberstone , L,
Philosophical Applications of Modal Logic , College Publications,2015.[16]
Inou´e , T., ‘On Ishimoto’s theorem in axiomatic rejection – the philosophy of unprov-ability –’, (in Japanese),
Philosophy of Science
Inou´e , T., ‘On rejected formulas – Hintikka formula and Ishimoto formula –’, (ab-stract),
The Journal of Symbolic Logic [18] Inou´e , T., ‘Cut elimination theorem, tableau method, axiomatic rejections’, (ab-stract),
Abstracts of Papers Presented to the American Mathematical Society
Inou´e , T., ‘Partial interpretation of Le´sniewski’s epsilon in modal and intensionallogics’, (abstract),
The Bulletin of Symbolic Logic
Inou´e , T., ‘Hintikka formulas as axioms of refutation calculus, a case study’,
Bulletinof the Section of Logic
Inou´e , T., ‘Partial interpretations of Le´sniewski’s epsilon in von Wright-type deonticlogics and provability logics’,
Bulletin of the Section of Logic
Inou´e , T., ‘A modal interpretation for Lesniewski-Ishimoto’s propositional ontology’,(abstract),
The Bulletin of Symbolic Logic
Inou´e , T. and A.
Ishimoto , ‘Cut elimination theorem and Hilbert- and Gentzen-style axiomatic rejections’, (abstract),
Abstracts of Papers Presented to the AmericanMathematical Society
Ishimoto , A., ‘A propositional fragment of Le´sniewski’s ontology’,
Studia Logica
Ishimoto , A., ‘Logicism revisited in the propositional fragment of Le´sniewski’s on-tology’, pp. 219–232 in
Philosophy of Mathematics Today , (Episteme vol. 22), editedby E.
Agazzi and G.
Darvas , Kluwer Academic Publishers, 1997.[26]
Iwanu´s , B., ‘On Le´sniewski’s elementary ontology’,
Studia Logica
Kobayashi , M. and A.
Ishimoto , ‘A propositional fragment of Le´sniewski’s ontologyand its formulation by the tableau method’,
Studia Logica
Lagniez , J. M., D.
Le Berre , T. de Lima and V.
Montmirail , ‘On checking Kripkemodels for modal logic K’, pp. 69–81 in [8].[29]
Marx , M. and Y.
Venema , Multi-Dimensional Modal Logic , Springer Sci-ence+Business Media, 1997.[30]
Ono , H.,
Proof Theory and Algebra in Logic , Springer-Verlag, 2019.[31]
Poggiolesi , F.,
Gentzen Calculi for Modal Propositional Logic , Springer-Verlag,2011.[32]
Sch¨utte , K.,
Beweistheorie , Springer-Verlag, 1960.[33]
Sch¨utte , K.,
Vollstandige Systeme modaler und intuitionistischer Logik , Springer-Verlag, 1968.[34]
Sch¨utte , K.,
Proof Theory , Springer-Verlag, 1977.[35]
S lupecki , J., ‘S. Le´sniewski’s calculus of names’,
Studia Logica
Smirnov , V. A., ‘Embedding the elementary ontology of Stanis law Le´sniewski intothe monadic second-order calculus of predicates’,
Studia Logica
Studia Logica
Smirnov , V. A., ‘Strict embedding of the elementary ontology into the monadicsecond-order calculus of predicates admitting the empty individual domain’,
StudiaLogica
Smory´nski , C., ‘Modal logic and self-reference’, pp. 441–495 in [9].[39]
Takano , M., ‘A semantical investigation into Le´sniewski’s axiom of his ontology’,
Studia Logica [40] Takano , M., ‘Embeddings between the elementary ontology with an atom and themonadic second-order predicate logic’,
Studia Logica
Takano , M., ‘Syntactical proof of translation and separation theorems on subsystemsof elementary ontology’,
Mathematical Logic Quaterly
Urbaniak , R.,
Le´sniewski’s Systems of Logic and Mereology; History and Re-evaluation , PhD thesis, Department of Philosophy, University of Calgary, 2008.[43]
Urbaniak , R.,
Le´sniewski’s Systems of Logic and Foundations of Mathematics ,Springer-Verlag, 2014.[44]
Urbaniak , R., ‘Plural quantifiers: a modal interpretation’,
Synthese
Wansing , H., ‘Sequent Systems for Modal Logics’, pp.61–146 in [11].
Takao Inou´e