Simplified Kripke-Style Semantics for Some Normal Modal Logics

Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009. https://doi.org/10.12775/LLP.2009.013) proved that the normal logics K45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {K45}$$\end{document}, KB4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {KB4}$$\end{document} (=KB5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=\mathrm {KB5}$$\end{document}), KD45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {KD45}$$\end{document} are determined by suitable classes of simplified Kripke frames of the form ⟨W,A⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle W,A\rangle $$\end{document}, where A⊆W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\subseteq W$$\end{document}. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of K45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {K45}$$\end{document}. Furthermore, a modal logic is a normal extension of K45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {K45}$$\end{document} (resp. KD45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {KD45}$$\end{document}; KB4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {KB4}$$\end{document}; S5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {S5}$$\end{document}) if and only if it is determined by a set consisting of finite simplified frames (resp. such frames with A≠∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\ne \varnothing $$\end{document}; such frames with A=W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=W$$\end{document} or A=∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=\varnothing $$\end{document}; such frames with A=W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=W$$\end{document}). Secondly, for all normal extensions of K45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {K45}$$\end{document}, KB4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {KB4}$$\end{document}, KD45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {KD45}$$\end{document} and S5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {S5}$$\end{document}, in particular for extensions obtained by adding the so-called “verum” axiom, Segerberg’s formulas and/or their T-versions, we prove certain versions of Nagle’s Fact (J Symbol Log 46(2):319–328, 1981. https://doi.org/10.2307/2273624) (which concerned normal extensions of K5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {K}5$$\end{document}). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}$$\end{document} of natural numbers. In the case of extensions with Segerberg’s formulas and/or their T-versions these classes are generated by certain finite subsets of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}$$\end{document}.


Introduction
Semi-universal frames introduced in [5] for some normal logics are Kripke frames of the form W, R , where W is a non-empty set of possible worlds and R is an accessibility relation such that R = W × A, for some subset A of W (so A is a set of common alternatives for all worlds). 1 Instead of semiuniversal frames we can use simplified frames of the form W, A , where W and A are as above.In [5] it is proved that the logics K45, KB4 (= KB5) and KD45 are determined, respectively by (a) the class of all simplified frames; (b) the class of simplified frames such that A = ∅ or A = W ; (c) the class of simplified frames with A = ∅. 2 In this paper, we focus on extensions of these logics by means of Segerberg's formulas (Alt n ) and their T-versions (Talt n ), for any n > 0.
The structure of the paper is as follows.In Section 1 we introduce a modal language, basic notions and facts about normal logics, consider Kripke semantics, and recall determination theorems for normal logics.
In Section 2 we develop the notion of semi-universal frames and so simplified Kripke-style semantics for the logics K45, KD45, KB4, S5 and their normal extensions by formulas selected from (Alt n ) and (Talt m ).For such logics, we introduce a special type of frames generated by certain finite subsets of N.
In Section 3 we present Nagle's Fact from [2] and its versions for K45, KB4, KD45, S5 and for these logics with additional axioms selected from (Alt n ) and (Talt n ).We obtain that a modal logic is a normal extension of K45 if and only if it is determined by a subclass of the class of finite semiuniversal frames. 3Furthermore, we obtain that a modal logic is a normal extension of KD45 (resp.KB4; S5) if and only if it is determined by a set consisting of finite semi-universal frames with A = ∅ (resp.with A = W or A = ∅; A = W ). For the logics with an additional axiom (Alt n ) and/or (Talt m ) we obtain that a modal logic is a normal extension of one of these logics if and only if it is determined by a suitable class of semi-universal frames whose cardinalities are suitable limited using numbers n and/or m.Also in this case, we will use frames generated by certain finite subsets of N.
Let Taut be the set of classical tautologies, i.e., all truth-functional tautologies.Moreover, let PL be the set of formulas which are instances of classical tautologies.A subset Λ of For is a modal logic iff Taut ⊆ Λ and Λ is closed under two rules: detachment for material implication (modus ponens) and uniform substitution.Thus, by uniform substitution, all modal logics include the set PL.Moreover, this set is the smallest modal logic.
A modal logic Λ is normal iff Λ contains the following formula: and is closed under the necessity rule: Any normal logic Λ is closed under the monotonicity and regularity rules: Thus, for any normal logic Λ and any k 0 we obtain: We recall that K is the smallest normal modal logic.To simplify the naming of normal logics, for any formulas (X 1 ), . . ., (X k ), the smallest normal logic including all of these formulas will be denoted by KX 1 . . .X k , i.e., KX 1 . . .X k := K ⊕ {X 1 , . . ., X k }.
Remark 1.1.The formulas (Q), (Alt n ), (T q ) and (Talt n ) are connected with the following formulas for any n 0: ]. Therefore, it is unnecessary to consider formulas (C n ).

Kripke Semantics for Normal Logics
For the semantical analysis of normal logics we may use standard frames of the form W, R , where W is a non-empty set of worlds and R is a binary accessibility relation on W .For any frame W, R , a model is any triple W, R, V , where V is a function which for any pair consists of a formula and a world assigns a truth-value with respect to R.More precisely, V : For× W → {0, 1} preserves classical conditions for truth-value operators and for any ϕ ∈ For and x ∈ W we have: where for any x ∈ W we put R[x] := {y ∈ W : x R y}.
As usual, we say that a formula ϕ is true in a world x of a model W, R, V iff V (ϕ, x) = 1.We say that a formula is true in a model iff it is true in all worlds of this model.Next we say that a formula is true in a frame iff it is true in every model which is based on this frame.A formula is valid in a class of frames (resp.models) iff it is true in all frames (resp.models) from this class.Moreover, for any modal logic Λ and any class C of frames (resp.models) we say that: Λ is sound wrt C iff all formulas from Λ are valid in C ; Λ is complete wrt C iff all valid formulas in C are members of Λ; Λ is determined by C iff Λ is sound and complete wrt C . 6 binary relation R on W is called, respectively: (i) We will transfer this terminology for properties of accessibility relations to the frames with those relations.
Notice that for any binary relation R we have:

Determination Theorems for Some Normal Logics
We can assign appropriate kinds of frames to individual formulas.We have the following pairs: emptiness to (Q); reflexivity to (T); quasi-reflexivity to (T q ); vacuity to (T c ); seriality to (D); symmetry to (B); transitivity to (4); Euclideanness to (5); the condition (xi) n to (Alt n ); the condition (xii) n to (Talt n ), for any n > 0; the condition (xiii) n to (C n ), for any n 0. Determination theorems for the logic K and its normal extensions by some of the formulas (T), (T c ), (D), (B), ( 4) and ( 5) are standard [cf., e.g., 1,8,9].For normal extensions of K ⊕ (Alt n ), see [8, pp.52-53].Moreover, for normal extensions of K ⊕ (T q ), K ⊕ (Talt n ) or K ⊕ (C n ) we will adopt Segerberg's proof of Lemma 5.3 given for normal extensions of K ⊕ (Alt n ).

3.
Let Λ be a normal extension of K ⊕ (C n ), where n 0, and M Λ = W Λ , R Λ , V Λ be a canonical model for Λ.Then for any x ∈ W Λ we have Card(R Λ [x]\{x}) n.
In the standard way, we get: 2. S5 is determined by the class U of all universal frames, as well by the class U fin of all finite universal frames.
3. Triv is determined by the class of frames with R = { x, x : x ∈ W }, as well by the single universal frame Ver is determined by the class of empty frames, as well by the single empty frame F ∅ := {1}, ∅ .

Semi-Universal Frames
We say that a relation R in a frame W, R is semi-universal (and we call the frame semi-universal ) iff R = W ×A, where A is a subset of W . Furthermore, if A W , then R and W, R we call properly semi-universal.Let sU and psU be the classes of all semi-universal and properly semi-universal frames, respectively.Note that all empty frames belong to psU fin .Lemma 2.1 (5, Lemma 2.2).For any semi-universal frame W, R : 1. R is transitive and Euclidean.
Moreover, for any n 0, if R = W × A: In [5] the following was proved [cf.5, Lemma 2.1]: Lemma 2.2.Let W, R be a frame.Firstly, for arbitrary x, y ∈ W we put x R 1 y := x R y and for any n > 1 let: x R n y iff there are y 1 , . . ., Then for any x ∈ W : 6.If R is symmetric and Euclidean, then 7. If R is transitive and Euclidean, then both: Proof.Ad 4. Let R be symmetric and A x = ∅.Then for some y ∈ A x we have x R y and y R x.So x R 2 x and x ∈ A x .Ad 5. Let R be Euclidean.Suppose that x R x and y, z ∈ W x , i.e., there are n, m > 0, y 1 , . . ., Then, by assumption, we obtain: Now suppose that y, z ∈ W x \{x}.Then, firstly, there are n, m > 0, y 1 , . . ., y n−1 , z 1 , . . ., z m−1 ∈ W x such that x R y 1 , . . ., y n−1 R y n = y and xRz 1 , . . ., z n−1 Rz n = z.Then y 1 Rz 1 and, by induction, y i Rz j .So yRz.
Ad 6.Let R be symmetric and Euclidean.If A x = ∅, then W x = {x} and so either R x = {x} × {x} or R x ⊆ {x} × ∅ = ∅, by (5).If A x = ∅, then-as in the proof of (4)-we have x R 2 x.So also x R x, since R is also transitive.Hence R x = W x × W x , by (5).
Ad 7. Let R be transitive and Euclidean.For (a) ∈ R, by (5).Now suppose that y, z ∈ W x and z = x.Then, by (3), x R z and either x = y or x R y. Hence y R z, since R is Euclidean.Thus, 2. If R is transitive and Euclidean, then W x , R x ∈ sU.
3. If R is symmetric and transitive (so also Euclidean), then Given the above facts we notice that classes of semi-universal models for K45, KB4, KD45 and these logics extended by (Q), (T q ), (Alt n ) and/or (Talt m ) are connected with some classes of generated models.We make use of models generated from relational models [cf. 1, p. 97].Let M = W, R, V and x ∈ W . Then the model generated by x ∈ W is the model x in which W x and R x are as in Lemma 2.2 and for all α ∈ At and y ∈ W x we have V x (α, y) = V (α, y).Of course, V x preserves classical conditions for truth-value operators and satisfies condition (V R ) for R x .So also for all ϕ ∈ For and y ∈ W x we have For any class M of models we put the following class of generated models G(M) := {M x : M ∈ M and x is in M }.We have: Fact 2.4 (cf. 1, Theorem 3.12).For any ϕ ∈ For: The following two lemmas will be used later.
Lemma 2.5.For arbitrary non-empty sets W and S such that W ∩ S = ∅: if ϕ is not true in a universal frame on W , then ϕ is not true in the properly semi-universal frame W ∪ S, (W ∪ S) × W . 8 As a consequence we get: if ϕ is true in a non-empty semi-universal frame W, W ×A then ϕ is also true in the universal frame on A, i.e., in the frame A, A × A .
Proof.Assume that for a model M = W, W × W, V and for some x ∈ W we have V (ϕ, x) = 0. Then there is a model Lemma 2.6.For arbitrary non-empty sets W and S such that W ∩ S = ∅: if ϕ is not true in a universal frame on W then ϕ is not true in the universal frame on W ∪ S.
As a consequence we get: if ϕ is true in a universal frame on W and X W , then ϕ is true in the universal frame on W \X.
Proof.As in the proof of Lemma 2.5, we construct a model M * , but now we put for any α ∈ At Let M * be the model W ∪S, (W ∪S)×W, V * , where V * is the extension of v.
It is easy to see that for any subformula ψ of ϕ we have: We have a counterpart of the above lemma for semi-universal frames.
Lemma 2.7.For all non-empty sets W , A and S such that A W and As a consequence of we get: if ϕ is true in a properly semi-universal frame W, W × A and X A, then ϕ is true in the non-empty properly semi-universal frame W \X, (W \X) × (A\X).
Proof.Assume that for a model M = W, W × A, V and for some x ∈ W we have V (ϕ, x) = 0. Then there is a model M * = W ∪ S, (W ∪ S) × (A ∪ S), V * such that V * (ϕ, x) = 0. We consider two cases.
Firstly, if x ∈ A, we construct v : At × (W ∪ S) → {0, 1} as in the proof of Lemma 2.6.Let M * be the model W ∪ S, (W ∪ S) × (A ∪ S), V * , where V * is the extension of v.It is easy to see that for any subformula ψ of ϕ we have: Secondly, if x ∈ W \A, for a certain x 0 ∈ A we construct v as above; the only change is that we take x 0 instead of x.It is easy to see that for any subformula ψ of ϕ we have:

Semi-Universal Frames for Normal Extension of K45
For a shorter formulation of theorems we accept the following convention.Let sU w be the class of semi-universal frames with R = W × (W \{w}), for some w ∈ W . Instead of sU w we can take {F ∅ } ∪ sU w+ .Obviously, W, R ∈ sU w+ iff W, R ∈ sU w and CardW > 1. Obviously, all frames from sU w are properly semi-universal.
Note that for any k 2 the frame {1, . . ., k}, {1, . . ., k} × {2, . . ., k} belongs to sU w+ fin .Let sU wN fin be the set of all such frames extended by the single frame F ∅ .Furthermore, let U N fin be the set of universal frames based on {1, . . ., k}, for any k 1. Obviously, by Theorem 1.3 (5), the logic S5 is determined by the set U N fin .In the light of the facts form above point and Theorem 1.3 (5)  For a shorter formulation of theorems, for any n > 0, let U n and U n be the sets of universal frames with cardinality equal to n and less than or equal to n (i.e., CardW = n and CardW n), respectively.Now let U N n be the set of universal frames based on {1, . . ., k}, for any k ∈ {1, . . ., n}.
We have U N n U n and U N n U N fin .Furthermore, for any n > 0, let sU n and sU n be the set of semi-universal frames having cardinality equal to n and less than or equal to n, respectively.By analogy, we define the appropriate classes of properly semi-universal frames psU n and psU n .We put sU w n := sU w ∩ sU n , sU w n := sU w ∩ sU n and sU wN n := sU wN fin ∩ sU w n , i.e., sU wN 1 = {F ∅ } and for n 2, sU wN n is the set of frames of the form {1, . . ., k}, {1, . . ., k} × {2, . . ., k} , for any k ∈ {2, . . ., n}.
Theorem 2.9.For arbitrary n, m > 0: 9 1.S5 ⊕ (Alt n ) is determined by the following classes: U n , U N n and U n , as well by the single universal frame based on {1, . . ., n}.

Simplified Frames for Normal Extensions of K45
In the light of the following lemma, any semi-universal frame W, R may be identified with a simplified frame of the form W, A , where W is a non-empty set and A is a subset of W .If A = W then we call W, A universal.If A = ∅ then we call W, A non-empty.If A = ∅ we call W, ∅ empty.As already mentioned in footnote , empty semi-universal frames and empty frames are identical.Instead of empty frames we can use the empty frame F ∅ .
On any simplified frame W, A we construct a simplified model W, A, V , where V is a function which to any pair built out of a formula and a world from W assigns a truth-value with respect to A. More precisely, V : For×W → {0, 1} preserves classical conditions for truth-value operators and for any ϕ ∈ For and x ∈ W : Lemma 2.10 (5, Lemma 2.6).Let W be a non-empty set, A ⊆ W and v : At × W → {0, 1}.Moreover, • let W, W × A, V be a semi-universal model in which V is the extension of v by conditions for truth-value operators and (V R ) for R = W × A; • let W, A, V be a simplified model in which V is the extension of v by classical conditions for truth-value operators and (V A ) for A.
Then V = V .Thus, the semi-universal model W, W × A, V may be identified with the simplified model W, A, V .
In the light of Theorem 2.8 and Lemma 2.10 we obtain that the logics K45, KB4 (= KB5) and KD45 are determined by suitable special classes of simplified frames [see 5, Theorem 2.5].Simply, in Theorem 2.8 we replace a given class C of semi-universal frames with the following class Moreover, in virtue of Theorem 2.9 and Lemma 2.10 we obtain that also logics from the theorem are determined by special classes of simplified frames.Again it is enough to replace the term 'semi-universal' with the term 'simplified' and the class C of semi-universal frames with the class S C of suitable simplified frames.If C is a class composed of universal frames, then as a name of S C we can take the same name as for C .In other cases, if C has one of the names used in Theorems 2.8 and 2.9, then the name of S C can be obtained by replacing 'sU' (resp.'psU') with 'S' (resp.'pS').
Obviously, the logics S5, Triv and Ver are also determined by special classes of simplified frames (cf.Theorem 1.3): S5 is determined by the class of finite universal simplified frames; Triv and Ver are determined by the single universal simplified frame F 1 and the single empty frame F ∅ , respectively.

Versions of Nagle's Fact for the Remaining
In [5] to each of logics K45, KB4 (= KB5) and KD45 is assigned a suitable class consisting of finite semi-universal frames which satisfy conditions for normal extensions of K5 presented by Nagle in [2].
Nagle's Fact (2, p. 325).Every normal logic containing (5) is determined by a set consisting of finite Euclidean frames W, R which satisfy one and only one of the following conditions: (a) W is a singleton and R = ∅, is included in R and w R x, for some x ∈ W \{w}.
For all normal extensions of the logic K45 condition (c) can be replaced by the following: (c ) W is not a singleton and there is a w ∈ W such that R = W ×(W \{w}).Lemma 3.1.1.Every frame satisfying (c ) also satisfies (c).
Proof.Ad 1.Suppose that W, R satisfies (c ).Then W \{w} = ∅, the product (W \{w}) × (W \{w}) is included in the product W × (W \{w}) and w R x, for any x ∈ W \{w}.So W, R satisfies (c).Ad 2. Suppose that W, R satisfies (c) and R = W ×A, for some A W . Then W \{w} = ∅; and so W is not a singleton.Moreover, W \{w} ⊆ A W .So also A ⊆ W \{w}.
Notice that only one of conditions (a), (b), (c ) can be met.Therefore, instead of 'satisfy one and only one of conditions (a), (b) and (c )' we may just write 'satisfy one of conditions (a), (b) and (c )'.Obviously, instead of empty frames satisfying condition (a) we can use the single frame F ∅ , the frames satisfying (b) are universal and the properly semi-universal frames satisfying (c ) are the frames from sU w+ .In the light of Nagle's Fact we obtain that a normal logic is a normal extension of K45 iff it is determined by a subclass of the class of all semiuniversal frames.Moreover, also for KB4, KD45 and S5 we obtain similar results with respect to suitable classes of semi-universal or universal frames.Proof.Ad 1.If a logic is determined by a subclass of sU fin , then, it is normal and contains (4) and ( 5), since these formulas are valid in this subclass, by Lemma 2.1(1).
Ad 2. If a logic is determined by a subclass of F ∅ ∪ U fin , then it is normal and contains (4), ( 5) and (B), since these formulas are valid in this subclass, by Lemma 2.1(1,3).
Ad 3.If a logic is determined by a subclass of sU + fin , then it is normal and contains (4), ( 5) and (D), since these formulas are valid in this subclass, by Lemma 2.1 (1,4).
Ad 4. If a logic is determined by a subclass of U fin , then it is normal and contains (T) and ( 5), since these formulas are valid in this subclass, by Lemma 2.1(1,2).
Ad 5.If a logic is determined by a subset of U n , then it is normal and contains (T), ( 5) and (Alt n ), since these formulas are valid in this subset, by Lemma 2.1(1,2).Ad 6.If a logic is determined by a subclass of the class of finite semiuniversal frames with CardA n, then it is normal and contains (4), ( 5) and (Alt n ), since these formulas are valid in this subclass, by Lemma 2.1(1).
Ad 7.If a logic is determined by a subclass of the class of finite frames which are empty or universal with CardW n, then it is normal and contains (B), ( 4) and (Alt n ), since these formulas are valid in this subclass, by Lemma 2.1(1,3) and the assumption.
Ad 8.If a logic is determined by a finite subclass of the class of semiuniversal frames with 0 < CardA n, then it is normal and contains (D), ( 4), ( 5) and (Alt n ), since these formulas are valid in this subclass, by Lemma 2.1(1,4) and the assumption.
Ad 9.If a logic is determined by a subclass of the class of finite semiuniversal frames which are universal or have CardA n, then it is normal and contains (4), ( 5) and (Talt n ), since these formulas are valid in this subclass, by Lemma 2.1(1) and the assumption.
Ad 10.If a is determined by a subclass of the class of finite semiframes which are universal or have 0 < CardA n, then it is normal and contains (D), ( 4), ( 5) and (Talt n ), since these formulas are valid in this subclass, by Lemma 2.1 (1,4) and the assumption.
Ad 11.If a logic is determined by a subclass of the class of finite semiuniversal frames which are either universal with CardW n or have CardA m, then it is normal and contains (4) and ( 5), since these formulas are valid in this subclass, by Lemma 2.1 (1).Moreover, by the assumption, in any frame of this subclass either (Alt m ) is valid or both (T) and (Alt n ) are valid.So both (T) ∨ (Alt m ) and (Alt n ) ∨ (Alt m ) are valid in all frames of this subclass.Hence also both (Talt m ) and (Alt n ) are valid in this subclass.So (Talt m ) and (Alt n ) belong to Λ.
Ad 12.If a logic is determined by a subclass of the class of finite semiuniversal frames which are either universal with CardW n or have 0 < CardA m, then it is normal and contains (D), ( 4) and ( 5), since these formulas are valid in this subclass, by Lemma 2.1 (1,4).
As already mentioned in the introduction and point 2.3, instead of a semi-universal frame W, R × A we can use the simplified frame W, A .So instead of finite frames satisfying condition (c) we can use simplified frames which satisfy the following condition corresponding to (c): (c ) W is not a singleton and there is a w ∈ W such that A = W \{w}.
It is easy to show that, as in point 2.3, also in Theorems 3.2 and 3.3 it is enough to replace the term 'semi-universal' with the term 'simplified' and the class C of semi-universal frames with the class S C of suitable simplified frames.Furthermore, for these simplified frameworks we can use the names proposed on Sect.2.3.

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The authors would like to thank anonymous referees whose remarks helped to improve quality of the paper.The research of Andrzej Pietruszczak and Mateusz Klonowski presented in this paper was supported by grants from the National Science Centre, Poland: 2016/23/B/ HS1/00344 and 2015/19/N/HS1/02401.Yaroslav Petrukhin was supported by the leading scientific school of Lomonosov Moscow State University "Transformations of culture, society and history: a philosophical and theoretical understanding".Open Access.This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in any medium, provided you give Y. Petrukhin Department of Lomonosov Moscow University Moscow yaroslav.petrukhin@mail.ru by (a).
says that S5 ⊕ (Alt n ) is determined by the class U n .This class includes the classes U N n and U n .Therefore, by Lemma 2.6, this logic is determined by n , as well by the single universal frame based on the set {1, . . ., n}.Ad 2. In virtue of Theorem 1.3(5), K45 ⊕ (Alt n ) is determined by the class of transitive Euclidean frames such that for any x ∈ W , CardR[x] n.Hence, by virtue of Lemmas 2.1(5), 2.2(7), 2.5 and 2.7, Corollary 2.3(2) and Fact 2.4, this logic is determined by the listed classes.Ad 3.In virtue of Theorem 1.3(5), KB4 ⊕ (Alt n ) is determined, respectively, by the class of symmetric Euclidean frames such that for any x ∈ W , CardR[x] n.Hence, by Lemmas 2.1(5), 2.2(6) and 2.6, Corollary 2.3(3) and Fact 2.4, this logic is determined by the listed classes.Ad 4. In virtue of Theorem 1.3(5), KD45 ⊕ (Alt n ) is determined by the class of serial transitive Euclidean frames such that for any x ∈ W , CardR[x] n.Hence, by Lemmas 2.1(5), 2.2(7), 2.5 and 2.7, Corollary 2.3(4) and Fact 2.4, this logic is determined by the listed classes.Ad 5.In virtue of Theorem 1.3(5), K45 ⊕ (Talt m ) is determined by the class of transitive Euclidean frames, where for any x ∈ W either x R x or CardR[x] m.Hence, by Lemmas 2.1(6), 2.2(7), 2.5 and 2.7, Corollary 2.3(2) and Fact 2.4, this logic is determined by the listed classes.Ad 6.In virtue of Theorem 1.3(5), KD45 ⊕ (Talt m ) is determined by the class of serial transitive Euclidean frames, where for any x ∈ W either x R x or CardR[x] m.Hence, by virtue of Lemmas 2.1(6), 2.2(7), 2.5 and 2.7, Corollary 2.3(4) and Fact 2.4, this logic is determined by the listed classes.Ad 7.In virtue of Theorem 1.3(5), K45 ⊕ {Alt n , Talt m } is determined by the class of transitive Euclidean frames, where ∀ x∈W CardR[x] n and ∀ x∈W (x R x or CardR[x] m).So, by Lemma 2.1(5,6), Corollary 2.3(2) and Fact 2.4, this logic is determined by the class of semi-universal frames such that either CardA m or both W = A and CardW n.Hence, by virtue of Lemmas 2.2(7), 2.6 and 2.7, this logic is determined by the listed classes.Ad 8.In virtue of Theorem 1.3(5), KD45 ⊕ {Alt n , Talt m } is determined by the class of serial transitive Euclidean frames, where ∀ x∈W CardR[x] n and ∀ x∈W (x R x or CardR[x] m).So, by Lemma 2.1(5,6), Corollary 2.3(4) and Fact 2.4, this logic is determined by the class of non-empty semiuniversal frames where either CardA m or both W = A and CardW n.
Theorem 3.2.1.A normal extension of K45 is determined by a subclass of sU fin .Furthermore, a normal extension of K45 is determined by suitable subclasses of the classes: {F ∅ } ∪ U fin ∪ sU w+ fin and U N fin ∪ sU wN fin .Ad 4. Let be a normal extension of S5.Then, by point 3, Λ is determined by a subset C of U fin ∪ sU w+ fin .We show that C ⊆ U fin .In fact, if W, R ∈ C , then R is symmetric, because (B) is valid in W, R .Hence W, R does not satisfy condition (c ).Theorem 3.3.For arbitrary n, m > 0: 1.A normal extension of S5 ⊕ (Alt n ) is determined by a subset of U n .2.A normal extension of K45 ⊕ (Alt n ) is determined by a subclass of the class of finite semi-universal frames with CardA n.Furthermore, a normal extension of K45 ⊕ (Alt n ) is determined by suitable subsets of the following sets: {F ∅ } ∪ U n ∪ sU w+ n+1 and U N n ∪ sU wN n+1 .3.A normal extension of KB4 ⊕ (Alt n ) is determined by a subclass of the class of finite frames which are empty or universal with CardW n.Furthermore, a normal extension of K45 ⊕ (Alt n ) is determined by suitable subsets of the following sets: {F ∅ } ∪ U n and {F ∅ } ∪ U N n .4.A normal extension of KD45 ⊕ (Alt n ) is determined by a subclass of the class of finite semi-universal frames with 0 < CardA n.Furthermore, a normal extension of KD45 ⊕ (Alt n ) is determined by suitable subsets of the sets: U n ∪ sU w+ n+1 and U N n ∪ sU wN n+1 \{F ∅ }. 5.A normal extension of K45 ⊕ (Talt m ) is determined by a subclass of the class of finite semi-universal frames which are either universal or have CardA m.Furthermore, a normal extension of K45⊕(Talt m ) is determined by suitable subsets of the following sets: {F ∅ }∪U fin ∪sU w+ K45⊕{Alt n , Talt m } is determined by a subclass of the class of finite semi-universal frames which are either universal with CardW n or have CardA m.Furthermore, a normal extension of K45⊕{Alt n , Talt m } is determined by suitable subsets of the following sets: {F ∅ } ∪ U n ∪ sU w+ m+1 and U N n ∪ sU wN m+1 .8.A normal extension KD45 ⊕ {Alt n , Talt m } is determined by a subclass of the class of finite semi-universal frames which are either universal with CardW n or have 0 < CardA m.Furthermore, a normal extension of KD45 ⊕ {Alt n , Talt m } is determined by suitable subsets of the following sets: U n ∪ sU w+ m+1 and U N n ∪ sU wN m+1 \{F ∅ }.Proof.Ad 1.Let Λ be a normal extension of S5 ⊕ (Alt n ).Then Λ is also a normal extension of KB4.So, by Theorem 3.2(2), Λ is determined by a subset of {F ∅ } ∪ U fin .Let W, R be a member of this subset.However,R = ∅, because (D) is valid in W, R .Moreover, CardW n, because (Alt n ) is valid in W, R .Ad 2. Let Λ be a normal extension of K45 ⊕ (Alt n ).Then Λ is also a normal extension of K45.So, by Theorem 3.2(1), Λ is determined by a subset of {F ∅ } ∪ U fin ∪ sU w+ fin .Let W, R be a member of this subset.Then R = W × A, where either A = ∅, or A = W , or A = W \{w}, for some w ∈ W .However, CardA n, because (Alt n ) is valid in W, R .Ad 3. Let Λ be a normal extension of KB4 ⊕ (Alt n ).Then Λ is also a normal extension of KB4.So, by Theorem 3.2(2), Λ is determined by a subset of {F ∅ } and U fin .Let W, R be a member of this subset.However, CardW n, because (Alt n ) is valid in W, R .Ad 4. Let Λ be a normal extension of KD45 ⊕ (Alt n ).Then Λ is also a normal extension of KD45.So, by Theorem 3.2(3), Λ is determined by a subset of U fin ∪ sU w+ fin .The rest as in the proof of point 2. Ad 5. Let Λ be a normal extension of K45 ⊕ (Talt n ).Then Λ is also a normal extension of K45.So, by Theorem 3.2(1), Λ is determined by a subset of {F ∅ } ∪ U fin ∪ sU w+ fin .Let W, R be a member of this subset, whereR = W × A, for some A ⊆ W .Because (Talt n ) is valid in W, R , we have ∀ x∈W (x R x or CardR[x] n).Hence, A = W or CardA n, by Lemma 2.1(6).Moreover, either A = ∅, or A = W , or A = W \{w}, for some w ∈ W .So either A = ∅, or A = W , or CardW n + 1.Ad 6.Let Λ be a normal extension of KD45 ⊕ (Talt n ).Then Λ is also a normal extension of KD45.So, by Theorem 3.2(3), Λ is determined by a subset of U fin ∪ sU w+ fin .The rest as in the proof of point 5. Ad 7. Let Λ be a normal extension of K45 ⊕ {Alt n , Talt m }.Then Λ is also a normal extension of K45.So, by Theorem 3.2(1), Λ is determined by a subset of {F ∅ } ∪ U fin ∪ sU w+ fin .Let W, R be a member of this subset, where R = W × A, for some A ⊆ W .Because (Alt n ) is valid in W, R , we have CardA n.Because (Talt m ) is valid in W, R , we have ∀ x∈W (xRx or CardR[x] m).Hence, A = W or CardA m, by Lemma 2.1(6).Moreover, either A = ∅, or A = W , or A = W \{w}, for some w ∈ W .So either A = ∅, or both A = and CardW n, or both A = W and CardW m, or both A = W \{w} and CardA m.Hence either U, R = F ∅ , or U, R ∈ U n , or U, R ∈ sU w+ m+1 .Ad 8. Let Λ be a normal extension of KD45 ⊕ {Alt n , Talt m }.Then Λ is also a normal extension of KD45.So, by virtue of Theorem 3.2(3), Λ is determined by a subset of the set U fin ∪ sU w+ fin .The rest as in the proof of point 7. Furthermore, the following fact also occurs.If a logic is determined by a subclass of U fin then it is a normal extension of S5.If a logic is determined by a subclass of the class of finite semi-universal frames which are either universal with CardW n or have CardA m, then it is a normal extension of K45 ⊕ {Alt n , Talt m }.12.If a logic is determined by a subclass of the class of finite semi-universal frames which are universal with CardW n or have 0 < CardA m, then it is a normal extension of KD45 ⊕ {Alt n , Talt m }.
5.If a logic is determined by a subclass of U n then it is a normal extension of S5 ⊕ (Alt n ).6.If a logic is determined by a subclass of the class of finite semi-universal frames withCardA n, then it is a normal extension of K45 ⊕ (Alt n ).