Pure Modal Logic of Names and Tableau Systems

. By a pure modal logic of names (PMLN) we mean a quantiﬁer-free formulation of such a logic which includes not only traditional categorical, but also modal categorical sentences with modalities de re and which is an extension of Propositional Logic. For categorical sentences we use two interpretations: a “natural” one; and Johnson and Thomason’s interpretation, which is suitable for some reconstructions of Aristotelian modal syllogistic (Johnson in Notre Dame J Form Logic 30(2):271–284, 1989; Thomason in J Philos Logic 22(2):111–128, 1993 and J Philos Logic 26:129–141, 1997. In both cases we use Johnson-like models (1989). We also analyze diﬀerent kinds of versions of PMLN, for both general and singular names. We present complete tableau systems for the diﬀerent versions of PMLN. These systems enable us to present some decidability methods. It yields “strong decidabil-ity” in the following sense: for every inference starting with a ﬁnite set of premises (resp. every syllogism, every formula) we can specify a ﬁnite number of steps to check whether it is logically valid. This method gives the upper bound of the cardinality of models needed for the examination of the validity of a given inference (resp. syllogism, formula).


Introduction
By a pure modal logic of names (PMLN) we mean a quantifier-free formulation of such a logic, in which to traditional categorical sentences we add also modal categorial sentences with modalities de re and which is an extension of Propositional Logic. In this paper we present complete tableau systems for this logic as well as for modal syllogisms in Johnson-like models for categorial sentences. Johnson [4] and Thomason [14,15] used a natural interpretation for general sentences of the form 'Every a is necessarily b' and 'No a is possibly b'. In Section 8 we describe pure modal logics for both general and singular names and some tableau systems for the logics.
We will in this paper take a natural interpretation of these four kinds of sentence, since we do not aim to reconstruct Aristotelian syllogistic. However, to reconstruct Aristotelian modal syllogistic, in [4,14,15] the authors have assumed unnatural interpretations of particular sentences of the form: 'Some a is necessarily b' and 'Some a is not possibly b', and as a consequence also of general sentences of the form: 'Every a is possibly b' and 'No a is necessarily b' (see Section 9). In the last part of the paper the unnatural interpretation is discussed and may be compared with the former results. Summing up, the general aim of the paper is to examine natural as well as unnatural interpretations of modal syllogism in the light of modern tableau tools.
In this paper we present a special kind of tableau, which we call a "minimax" tableau. Mini-max tableaus involve a very low number of proof steps. They give us a decidability method for PMLN (as well as for modal syllogistic) which yields what we call "strong decidability". It means that for every inference starting with a finite set of premises (resp. every syllogism, every formula) we can specify a finite number of steps to check whether it is logically valid. This method gives also the upper bound of the cardinality of models needed for the examination of the validity of a given inference (resp. syllogism, formula).
For syllogisms, as in [3], we can also provide another decidability method which estimates a limit of the length of a tableau for a given syllogism. This paper is both a significant extension of [3] and a modification of it.

Syntax
Formulas of modal syllogisms. We assume a countably infinite set GL which contains schematic general name letters: a 0 , a 1 , a 2 , a 3 , . . . (for the first three of these letters we will use abbreviations: 'a', 'b', and 'c', respectively). We assume that the letters from GL represent various general names.
Following Thomason [14,15] a modal syllogistic formula is an expression of any of the following forms: Aαβ, Iαβ, Eαβ, Oαβ, A αβ, A ♦ αβ, I αβ, I ♦ αβ, E αβ, E ♦ αβ, O αβ, O ♦ αβ, for α, β ∈ GL. The notation reflects a way to read non-modal and apodeictic propositions, respectively: Aab: b belongs to every a, Eab: b belongs to no a, Iab: b belongs to some a, Oab: b does not belong to some a; A ab: b of necessity belongs to every a; E ab: b of necessity belongs to no a; I ab: b belongs of necessity to some a; O ab: b of necessity does not belong to some a (see, e.g., [1] and [7, p. 10]). Similar readings hold for formulas with '♦', where 'of necessity' is replaced with 'of possibility'.
Let For syll be the set of all syllogistic formulas. All formulas with symbols A, I, E, or O we will call non-modal. For any • ∈ {blank, , ♦} all formulas with I • or O • we will call particular (for short: p-formulas), and all formulas with A • or E • we will call general (for short: g-formulas).
We present English glosses for modal syllogistic formulas under a formal interpretation which reflects a natural way of understanding them [see, e.g., 7, pp. . Moreover, the interpretations are expressed by translations into formulas of monadic first-order modal logic: 1 Syllogisms. By syllogism we mean any pair Ψ, ϕ , where Ψ is a non-empty finite subset 2 of For syll (for short: Ψ ∈ P fin (For syll ) \ {∅}) and ϕ ∈ For syll . 3

Formulas of PMLN. All formulas of PMLN are built with members of
For syll by the use of classical propositional connectives and parentheses. These formulas are members of For which is the smallest set such that: 1 Formulas of monadic first-order modal logic are defined in a standard way: 'ax' and 'bx' stands for 'x is a' and 'x is b', respectively; ' ' and '♦' stand for the de dicto operators 'it is necessary that' and 'it is possible that', respectively.
2 For any set S let P(S) (resp. P fin (S)) be the family of all (resp. finite) subsets of S. 3 The denotation of the term 'syllogism' is in our paper wider than in [1].
For any Φ ∈ P(For) let GL(Φ) be the set of letters from GL which occur in at least one formula of Φ.
If M ϕ then we say that the formula ϕ is true in M. Otherwise, i.e. M ϕ, we say that it is false in M. If for any ψ from Ψ ⊆ For we have M ψ, then we write M Ψ .
Note that for categorical sentences we obtain:

Compactness of Entailment in Pure Modal Logic of Names
We can prove that the relation |= is compact just by embedding it into the consequence relation of Classical Quantified Logic (CQL), which is compact.
To do this, we use a transformation of formulas from For into a set of formulas of a monadic first order language and then we will use the compactness property of CQL.
A transformation For into some monadic first-order language. For For we use a monadic first-order language L that has the set of variables Var := {x 0 , x 1 , x 2 , . . .} (for 'x 0 ' we will use 'x') and the set of monadic predicates P L := {α • : α ∈ GL and • is either blank, or , or ♦}. Moreover, L does not have any other non-logical constants. Finally, let For L be the set of all formulas of L which are obtained in the standard way. We use the following transformation t from For into For L , where for all α, β ∈ GL, ϕ, ψ ∈ For, and • ∈ {blank, , ♦} we put: where * = blank, ♦, , for • = blank, , ♦, respectively, where * = blank, ♦, , for • = blank, , ♦, respectively, Moreover, for any Ψ ⊆ For we put t(Ψ ) := {t(ψ) : ψ ∈ Ψ }. As we can see the transformation is an injection. For all Ψ ∈ P(For L ) and ϕ ∈ For L , also as usual, we say that in CQL Ψ logically entails ϕ (we write: Ψ |= CQL ϕ) iff there is no model M L ∈ Mod L such that M L Ψ and M L ϕ.

Auxiliary Singular Formulas for Tableaus
Auxiliary singular names. Let AI be the infinite, countable set of the following auxiliary indexes: i 0 , i 1 , i 2 , . . . (for the first two of these indexes we will use 'i' and 'j', respectively). These indexes are auxiliary singular names that are needed for the construction of tableaus.
Auxiliary singular formulas and their negations. An auxiliary singular formula is any expression which has one of the following forms: ıεα , ıε α , and ıε ♦ α , for some i ∈ AI and α ∈ GL. 6 We read, respectively, the auxiliary singular formulas 'iεa', 'iε a', and 'iε ♦ a' as follows: "i is a", "i is necessarily a", and "i is possibly a".
If σ is an auxiliary singular formula, then ¬ σ is its negation. Let Σ be the set of all auxiliary singular formulas and their negations. Moreover, let Σ ¬ be the set of all negations of singular formulas; so Σ ¬ Σ.
Notion of truth for the set For∪Σ. By Claim 1.2, the empty model M ∅ does not have an influence on the relation |=. Thus, we will use only non-empty models from Mod. Moreover, for any non-empty domain D we introduce a function v : AI → D which is a valuation of indexes. A valuation of indexes in a model is a valuation into its domain.
For any non-empty model M = D, d, d , d ♦ and any valuation of indexes v : AI → D we can extend the notion of truth to the set For ∪ Σ. For all ı ∈ AI, α ∈ GL, σ ∈ Σ, and ϕ ∈ For we put: Logical entailment for For ∪ Σ. For all Ψ ∈ P(For ∪ Σ) and ϕ ∈ For ∪ Σ, we say that Ψ logically entails ϕ (we write: Ψ |= ϕ) iff for any non-empty model M and any valuation v in its domain

Tableaus
To examine Ψ |= ϕ we will apply tableaus that are a kind of finite trees constructed from formulas of Ψ ∪ {¬ϕ} ∪ Σ by rules we introduce below.

Tableau Rules
Tableaus are build by so-called tableau rules. We will first outline the rules. Afterwards, we will provide a precise definition of a tableau that will formally specify their behaviour. We have seven groups of rules. Only the first three kinds decompose sentences with classical propositional connectives. The first group consists of a single rule (E¬¬) of elimination of double negation. Formally, this rule is the set { ¬ ¬ ϕ, ϕ : ϕ ∈ For} (cf. Table 1).
The second group has four members (E∧), E¬∨), (E¬⊃), (E≡) and they do not lead to branches in tableaus either. We will denote these rules by (NBE) for "non-branching elimination". Formally, each rule in (NBE) is a set of triples of the form ϕ, ψ, χ , where ϕ, ψ, χ ∈ For have a particular form given by the relevant schemes from Table 1. The ordered pair ψ, χ plays the role of a conjunction of conclusions.
The rules of the other four types are divided into two categories: "rules of inference" and "rules of choice".   Table 4. Rules of inference for general formulas (RIg) The fourth group of rules consists of twelve rules of inference; one rule for one type of categorical formulas. Each of these rules transforms the negation of a given categorical formula into a logically contradictory formula (cf. Table 3). So these twelve rules we denote by (RI¬). Formally, each rule in (RI¬) is a set of ordered pairs of the form ϕ, ψ , where ϕ, ψ ∈ For have a particular form given by the relevant schemes.
The fifth group of rules consists of six rules of inference; one rule for one type of general formulas (cf. Table 4). So these six rules we denote by (RIg). Formally, each rule in (RIg) is a set of ordered pairs of the form {ϕ, σ}, ς , where ϕ ∈ For and σ, ς ∈ Σ have a particular form given by the relevant schemes. In this case {ϕ, σ} plays the role of a set of premises.
The sixth group of rules consists of two rules of inference for auxiliary singular formulas (cf. Table 5). So these two rules we denote by (RIa). Formally, each rule in (RIa) is a set of ordered pairs of the form σ, ς , where σ, ς ∈ Σ \ Σ ¬ have a particular form given by the relevant schemes.
The seventh group of rules consists of six rules of choice for particular formulas; one rule for one each of particular formula (cf. Table 6). These six where * = blank, ♦, , for • = blank, , ♦, respectively. Table 6. Rules (RC) for particular formulas rules we denote by (RC). Formally, each of rules of (RC) is a triple of the form ϕ, σ, ς , where ϕ ∈ For and σ, ς ∈ Σ have a specific form given by the relevant schemes. In this case σ, ς plays the role of a conjunction of conclusions.

Tableaus. Closed Tableaus. Open Tableaus
Let Φ be a non-empty (maybe infinite) subset of For. A tableau for Φ is any non-empty finite tree of formulas from For ∪ Σ which meets all of the following four conditions: 1. The tree T has only one root (i.e., the node at the top) belonging to Φ.
2. From each of the nodes of T at most two edges (arrows) start.
3. If from a given node in T two edges start with formulas ψ 1 and ψ 2 , respectively, then in the initial path from the root to this node there is a node with the formula ϕ such that the pair ϕ, {ψ l , ψ 2 } belongs to some rule of (BE).
4. For any initial path P in T which has the form (ϕ 1 , . . . , ϕ n ) (i.e., ϕ 1 is the root of T ), for any l n at least one of the following conditions is met: (ii) There is a j < l such that ϕ j , ϕ l belongs to some rule of (E¬¬), or of (RIa), or of (RN). (iii) There is a j < l such that for some ψ ∈ For the pair ϕ j , {ϕ l , ψ} belongs to some rule of (BE) and for the initial subpath P of P which has the form (ϕ 1 , . . . , ϕ j , . . . , ϕ l−1 ) the path (P , ψ) belongs to T .
(iv) There are j, k < l such that {ϕ j , ϕ k }, ϕ l belongs to some rule of (RIg). (v) There are j < l and ψ ∈ For ∪ Σ such that for the initial subpath P of P which has the form (ϕ 1 , . . . , ϕ j , . . . , ϕ l ) the path (P , ψ) belongs to T and ϕ j , ϕ l , ψ belongs to some rule either of (NBE) or of (RC), where in the second case the index ı from ϕ l and ψ does not occur in the initial subpath P of P which has the form (ϕ 1 , . . . , ϕ l−1 ); (vi) There is a j < l − 1 such that ϕ j , ϕ l−1 , ϕ l belongs to some rule either of (NBE) or of (RC), where in the second case the index ı from ϕ l−1 and ϕ l does not occur in the initial subpath P of P which has the form (ϕ 1 , . . . , ϕ l−2 ).
Clearly, in a tableau for some Φ only a finite subset of formulas in Φ that occur in the tableau forms a proof. So any tableau for Φ is also a tableau for each set of formulas which is a superset of Φ.
Any path from a root to a bottom is called a branch. A branch in a given tableau is closed iff there are formulas from For ∪ Σ of the form ϕ and ¬ ϕ on two of its nodes; otherwise it is open. A tableau is closed iff every its branch is closed; otherwise is open.

Tableau Proofs. Soundness
For all Ψ ∈ P(For) and ϕ ∈ For, we say that ϕ has a tableau proof from Ψ (we write: Ψ ϕ) iff there is a closed tableau for the set Ψ ∪ {¬ ϕ}.
Suppose towards a contradiction that Ψ |= ϕ. Then there is a non-empty Note that all tableau rules of the types (NBE) and (RN) preserve truth and moreover at least one conclusion derived from the application of a rule of type (BE) preserves truth. Hence for some closed branch B in T all formulas from If in branch B no rules of (RC) were used, then in B there would be only formulas from For B . But this contradicts the fact that B is closed. So we suppose that on B some rule of (RC) is used at least once. Let Σ B be the set of all auxiliary formulas on B which we obtain by rules of (RC) and let I B be the set of all indexes which appear in the formulas from Σ B . We have I B = ∅. Since rules from the group (RC) were used only to formulas from For B and M For B , there is a valuation v : Since all other tableau rules preserve truth for M and v , so by induction on the length of B, we show that all formulas on B are also true in M. But this contradicts the fact that B is closed.

Completeness
Since the logical consequence relation |= is compact, we can focus on tableaus for non-empty finite sets. Let Φ be any non-empty finite subset of For.
In the light of Claim 1.3 we see that the rules of (RC) do not have to be applied to all p-formulas in a given branch of a given tableau. It is enough to apply the rules to the so-called essential p-formulas in a given branch. For any tableau T and any branch B of T a p-formula ϕ is essential in B iff there is no p-formula ψ on B such that ψ |= ϕ and ϕ |= ψ. Now we introduce a special kind of tableaus, called mini-max. They are maximal since all relevant expressions are decomposed, but minimal since only relevant expressions are decomposed.
We say that a tableau T for Φ is "mini-max " iff T satisfies the following conditions: 1. T begins with a single path in which there occur all members from Φ (we will call this path the initial list).
2. In each branch B of T, any rule that could be applied to a formula was applied exactly once, with the exception of the following cases: • rules of (RC) were applied only to essential p-formulas, • rules of (RC) were applied only to one of two equivalent formulas (see points 3 and 4 of Claim 1.3), • in B we have a pair of formulas χ and ¬ χ .
Of course, for any non-empty finite subset of For there is a "mini-max" tableau. Note that formulas of the form O αα and Oαα cannot appear in an open branch in "mini-max" tableaus (cf. Claim 1.3.6 and the rules of (RC) and (RIa)). All "mini-max" tableaus are complete in the standard sense [see, e.g., 13, p. 9].
In Sections 5-8 for a given branch B, let p B e be the number of all essential non-equivalent p-formulas in B. For the Completeness Theorem we use the following Completeness Lemma: Because T is "mini-max", we have: Moreover, Card(D) = p B e and for any ı ∈ AI we have: Second, for all α ∈ GL and • ∈ {blank, , ♦} we put: Now we show that M T is a model from Mod, i.e., for any α in GL: Now we prove that for any ϕ ∈ For we obtain: if ϕ is in B, then M T ϕ. The proof is by induction on the complexity of formulas.
(a) We consider the case of formulas of For syll . Suppose that for some • ∈ {blank, , ♦}, I • αβ is in B. Then, if I • αβ is essential in B, then by some rule of (RC), for some ı ∈ AI we have: ıεα and ıε • β are in B. 7 If I • αβ is not essential in B then • = . So if • = blank then I αβ is in B. Hence, by some rule of (RC), for some ı ∈ AI we have: ıεα and ıε β are in B. So, by some rule of (RIa), ıεβ is also in B. Moreover, if • = ♦ then I * αβ is in B, for some * ∈ {blank, }. Hence, by some rule of (RC), for some ı ∈ AI we have: ıεα and ıε * β are in B. So, by some rule of (RIa), ıε ♦ β is also in B. Thus, in all cases, So if • = blank then O αβ is in B. Therefore, by some rule of (RC), for some ı ∈ AI we have: ıεα and ¬ ıε ♦ β are in B. Hence ı ∈ d(α) and ıε ♦ β is not in B, since B is open. So, by some rule of (RIa), also ıεβ is not in B. Moreover, if • = ♦ then O * αβ is in B, for some * ∈ {blank, }. Therefore, by some rule of (RC), for some ı ∈ AI we have: ıεα and ¬ ıε β are in B, where = blank, ♦, for * = blank, , respectively. Hence ı ∈ d(α) and ıε β is not in B, since B is open. So, by some rule of (RIa), also ıε ♦ β is not in B. Thus, in all cases, ı / ∈ d * (β), where * = blank, ♦, , for • = blank, , ♦, respectively. (c) For other complex formulas we naturally consider inductive steps. We use the elimination rules and the fact that T is "mini-max".
Since all formulas from Φ are in B, so M T Φ.
By Lemma 5.1 and Theorem 1.5 (compactness of the relation |=) we obtain that the tableau approach is complete.

Decidability and Cardinalities of Models
The decidability of Ψ |= ϕ for the pure modal logic of names is considered in the case when Ψ is finite. Thus, let Ψ ∈ P fin (For). The decidability of Ψ |= ϕ can be tested in the following two ways.
First, we have already established decidability on the basis of Lemma 1.4, since the relation |= CQL is decidable. However, we dare for more! We want to estimate the cardinalities of models for the logic under research that are less than the cardinalities of models for CQL. It is known that, for Ψ ∈ P(For L ) and ϕ ∈ For L , in order to check whether Ψ |= CQL ϕ we have to examine models with a cardinality not bigger than 2 k , where k is the number of monadic predicates of P L that occur in the formulas from the set Ψ ∪ {ϕ}. Let us note that we mean not only letters from GL, but also all instances of the form α • , where α ∈ GL and • ∈ {blank, , ♦}.
Either by Lemma 1.4 and the decidability of monadic first-order logic, or by Lemma 5.1 and Theorem 4.1 we obtain: Theorem 6.1 (Decidability). For all Ψ ∈ P fin (For) and ϕ ∈ For, the problem whether Ψ |= ϕ is decidable.
As we declared, we would like to estimate the cardinality of a given model to check if Ψ |= ϕ. 8 First, we obtain the following theorem for formulas that have a disjunctive normal form. Note that each formula χ from For can be written in a disjunctive normal form χ d . Moreover, let Ψ d := {ψ d : ψ ∈ Ψ }. Theorem 6.2. For all Ψ ∈ P fin (For) and ϕ ∈ For the following conditions are equivalent:  For any i from {1, . . . , k χ }, let C i χ be the set of all non-negated p-formulas and all negated g-formulas that occur in the i-th conjunction in χ d . 9 We put the members of Ψ as a sequence (ψ 1 , . . . , ψ n ) (for Ψ = ∅ we put n := 0). Let Λ be the set of all sequences (l 1 , . . . , l n , l n+1 ) of numbers such that 1 l i k ψ i , for i = 1, . . . , n, and 1 l n+1 k ¬ϕ . Then for any λ = (l 1 , . . . , l n , l n+1 ) ∈ Λ we define the following set: Clearly, any set B λ , for λ ∈ Λ, corresponds to a branch in a tableau build For any λ ∈ Λ we define the essential p-formulas in the set B λ , as for a given branch of a given tableau (see p. 13). Let p B λ e be the number of all essential, non-equivalent p-formulas in B λ .
Once again let us note that formulas having one of the forms O αα , Oαα , ¬A ♦ αα , and ¬Aαα cannot appear in any open branch in "minimax" tableaus. So let Λ * be the set of all λ ∈ Λ such that no such formula is a member of set B λ . We put Max(Ψ, ϕ) := max{p B λ e : λ ∈ Λ * }. 10 From Theorem 6.2 we obtain: Proof. "⇒" Obvious. "⇐" Suppose that Ψ |= ϕ. Then, by Theorem 6.2, Let S = Ψ, ϕ be such a syllogism, where Ψ ∈ P fin (For syll ) \ {∅} and ϕ ∈ For syll (see p. 3). Then the set Λ has one element and any tableau for Ψ ∪ {¬ ϕ} has got only one branch. Of course, if either some formula of the form O αα or Oαα belongs to Ψ , or ϕ has the form A ♦ αα or Aαα , then Ψ |= ϕ. In the other case we obtain that an upper estimation of the cardinality of models is the number p S e of essential, but non-equivalent pformulas in Ψ ∪ {ϕ}, where ϕ is the formula of For syll which we obtain from ¬ϕ , by the suitable rule of (RN).
Of course, if S = Ψ, ϕ is non-modal then, in the second case, p S e is the number of non-modal particular non-equivalent formulas in the set Ψ ∪ {ϕ}. In [2,3] some larger estimations were made. An upper estimation of the cardinality of models was the number of non-modal particular formulas in Ψ ∪ {ϕ}. However, in those papers two various approaches to the presented problem were outlined.
In [9,11] the cardinalities of models for non-modal pure logic of names were also examined. For any non-modal formula ϕ from For the upper estimation of cardinalities of its models is 1 2 n(n + 1), where n is the number of letters from GL occurring in ϕ. It is difficult to compare that number with number Max(∅, ϕ) as it does not make sense to try to count the number of non-modal particular non-equivalent formulas built with letters occuring in ϕ.
Moreover, even for the non-modal pure logic of names, it would be a mistake to compare our results with the results presented by Kulicki [5,6]. First, Kulicki used a so-called strong interpretation for formulas of the form Aαβ : M Aαβ iff ∅ = d(α) ⊆ d(β). So we have: Aαβ |= Iαβ. Second, Kulicki examined the cardinalities of models only for Horn formulas which have the form (χ 1 ∧ · · · ∧ χ n ) → χ n+1 , where χ i is Aαβ or Iα β , for some α, β, α , β ∈ GL. These Horn formulas correspond only to syllogisms that are limited to those two kind of categorical sentences.

Tableau Proofs with Other Version of Tableau Contradiction
To make tableau proofs shorter we may use an extended version of the notion of tableau contradiction and apply it to our earlier notions of the closed tableau and the tableau proof. Hence, to the pairs of tableau contradictory formulas we defined earlier we add all pairs of logically contradictory syllogistic formulas from For syll (see p. 5). So, a branch in a given tableau is closed, if at some of its node there is a formula either of the from Oαα Iab, Aab ∨ Eab Aab (in both versions):

Aab Eab
Closed tableaus for {Iab, Aab ∨ Eab, ¬ Aab}: (i) in the first version (ii) in the second version Table 7. Example of proofs with regular and extended notion of tableau contradiction.
or of the from O αα (cf. Claim 1.3.6). Due to that we have the following fact: Proposition 7.1. Ψ ϕ in the first version iff Ψ ϕ in the second version.
"⇐" Suppose that Ψ ϕ in the second version and for Ψ ∪ {¬ϕ} there is a closed tableau T such that T has a branch B which is not closed in the first version of tableau contradiction. Then on B we have either Oαα , or O αα , or both A • αβ and O * αβ , or both E • αβ and I * αβ , where * = blank, ♦, for • = blank, , ♦, respectively. Hence, by use of appropriate rules of the types (RC), (RIa), and (RIg) we obtain, in the first case: ıεα and ¬ ıεα ; in the second case: ıεα , ¬ ıε ♦ α , and ıε ♦ α ; in the third case: ıεα , ¬ ıε • β , and ıε • β ; in the fourth case: ıεα , ıε * β , and ¬ ıε * β . So we obtain a branch B which is closed in the first version (see the example in Table 7).

Pure Modal Logic for Both General and Singular Names
So far we have considered the pure modal logic of names only for general names. It includes indexes for auxiliary singular names only in tableau proofs. Now-beside the letters from GL representing general names-we would like to use also schematic letters representing «real» singular names.
We shall show that our examination of the logic of general names from the former sections can be successfully repeated for the logic of both kinds of names.

Formulas
We fix a countably infinite set SL which contains schematic letters: n 0 , n 1 , n 2 , . . . (for the first two schematic letters we will use 'n' and 'm', respectively). We assume that these letters represent various singular names.
A singular formula is any expression which has one of the following forms: νεα , νε α , and νε ♦ α , where ν ∈ SL and α ∈ GL. Let For sin be the set of all singular formulas. Now, formulas are made of formulas For syll and For sin by the use of Boolean connectives and brackets. Let For + be the smallest set such that The notion of truth for For sin is similar to that which we used for Σ (here denotation function d s replaces valuation v ). We use the following interpretation of formulas from For + in all models M = D, d s , d, d , d ♦ ∈ Mod + , ν ∈ SL, α ∈ GL, and • ∈ {blank, , ♦}:

Models and the notion of truth for For
The remaining conditions have been left unchanged (the are on p. 4).
Logical entailment, equivalence, and contradiction. These three notions are introduced for For + in the same way as for For on p. 5. We obtain: Compactness of logical entailment. We obtain this result in a way similar to how obtained it for For in Section 1.3. However, we use a monadic first-order logic with individual constants [see, e.g., 12]. Now we use the monadic first-order language L + with monadic predicates from P L and individual constants from SL. So we have also atomic formulas of the form πν , where π ∈ P L and ν ∈ SL. Let For L + be the set of all formulas of L + which we obtain in a standard way.
The transformation t from p. 6 is enriched with the following condition for all ν ∈ SL and α ∈ GL: Ordinarily, a model for L + is any pair M L + = D, I , where D is a nonempty set and I is a function from SL into D and from P L into P(D).
We obtain the relevant counterpart of Lemma 1.4. Hence, by the compactness of Classical Quantified Logic, we also obtain the relevant version of Theorem 1.5, which is that the relation |= for For + is also compact.
Soundness, completeness and decidability. We use the same notion of tableau and the same tableau rules as in Section 3. The only difference is that now in the description of the rules a variable 'ı' takes values from the union of sets AI and SL, whereas in the former case it only took values from the set AI. Using the above rules we obtain the following facts that correspond to those in Claim 8.1: For all α, β ∈ GL and ν ∈ SL: If p B e + l B > 0, then we put: Because T is "mini-max", Card(D) = p B e + l B + 1 and for any ı ∈ SL ∪ AI we have: Moreover, for all α ∈ GL and • ∈ {blank, , ♦} we put: . Moreover, for any ν ∈ SL we put: The rest is as in the proof of Lemma 5.1.
We also have soundness theorem (a counterpart of Theorem 4.1), completeness theorem (a counterpart of Theorem 5.2), and decidability (a counterpart of theorems 6.2 and 6.3). In the last case we replace merely a number p B e with a number p B e + l B + 1; and also in the calculation of the number Max(Ψ, ϕ).
Its decidability is a result of the fact that relation |= CQL is decidable. In addition, we would like to estimate cardinalities of models for For + that are less than the cardinalities of models for CQL. It is known that for Ψ ∈ P(For L + ) and ϕ ∈ For L + to check whether Ψ |= CQL ϕ it is enough to examine all models whose cardinalities are not greater than 2 k , where k is a number of all monadic predicates from P L occurring in the set Ψ ∪ {ϕ}. So the number of individual constants occurring in this set is not essential [see 12, Corollary 2]. 11

Semantics
As we mentioned in the introduction, in [4,14,15] the aim was to reconstruct Aristotelian modal syllogistic [1], where the pairs of formulas I αβ and I βα (and so also E ♦ αβ and E ♦ βα) are equivalent. So in [4,14,15] for models of the form D, d, d , d ♦ , satisfying among others condition ( ) from p. 4, a different interpretation of I and E ♦ we use: Moreover, for some reasons given in [4,14,15] another interpretation of O and so of A ♦ were used: . 11 The latter becomes important if we consider a monadic predicate logic with equality. Then an examination of whether Ψ |=CQL ϕ requires only the examination of all models with a cardinality not greater than l + 2 k · max{n, 1}, where k is a number of all monadic predicates occurring in the set Ψ ∪ {ϕ}, l (resp. n) is a number of all individual constants (resp. variables) occurring in the non-tautological equations from this set, and the tautological equations have the form A = A . If in the set Ψ ∪ {ϕ} there are no variables then the above estimation can be reduced to max{l, 1} + min{2 k − 1, q − max{l, 1}}, where q is the number of individual constants occurring in this set [see 12, p. 52].
In [4,14,15] various conditions on a denotation for models are imposed. For example, in [14,15] ( ) is assumed plus for all α, β ∈ GL: Let Mod Ar be the class of all models satisfying conditions ( ) and (1 Ar )-(4 Ar ). For this class we construct in the standard way the relation |= Ar .
Using the revised interpretation and the conditions ( ), (1 Ar ), and (2 Ar ) we obtain: Claim 9.1. For all α, β ∈ GL: where ı is any index where j is a new index in a tableau Table 8. Rules (RI a 1 ) and (RC a ) where ı is any index, but j is a new index in a tableau Table 9. Additional rules Using the above rules we obtain the following facts corresponding to those in Claim 9.1: Under the unnatural interpretations of I , E ♦ , O and A ♦ which we have assumed here, which satisfy conditions ( ), (1 Ar )-(4 Ar ) we can reformulate and prove counterparts of the facts given in Sections 3-6.

Soundness
We prove that the revised relation Ar is sound, i.e.: if Ψ Ar ϕ then Ψ |= Ar ϕ.
Proof. Assume that Ψ Ar ϕ, i.e., there is a closed tableau T for Ψ ∪{¬ ϕ}. Suppose towards contradiction that Ψ |= Ar ϕ. Then there is a non-empty Note that all tableau rules of the types (NBE) and (RN) preserve truth and moreover at least one conclusion derived from the application of a rule of type (BE) preserves truth. Hence for some closed branch B in T all formulas from If in B no rule of type (RC) was used, then in B there would be only formulas of For B . But this contradicts the fact that B is closed. So we suppose that in B at least once some rule of type (RC) or (RC Ar ) was used. Let Σ 1 B be the set of auxiliary formulas in B which we obtain by rules of (RC) and (RC Ar ). Let I 1 B be the set of all indexes which appear in formulas from Σ 1 B . We have I 1 B = ∅. Since we apply the rules of (RC) and (RC Ar ) only to formulas from For B and M For B , there is a valuation v 1 : I 1 B → D such that M, v 1 Σ 1 B . Then we suppose we could use also a rule of type (1 Ar r )-(3 Ar r ). Let Σ 2 B be the set of auxiliary formulas in B which we obtain by rules of (1 Ar r )-(3 Ar r ). Let I 2 B be the set of all indexes which appear in formulas from Σ 2 B . We have I 2 B = ∅. Since we apply the rules of (1 Ar r )-(3 Ar r ) only to formulas from Σ 1 B and M Σ 1 B , by interpretation and the conditions (1 Ar )-(3 Ar ), there is a valuation v 2 : I 2 B → D such that M, v 2 Σ 2 B . Moreover, by interpretation and the condition (4 Ar ), all other tableau rules preserve truth for M and v 1 ∪ v 2 . Hence, by induction on the length of B, we show that all formulas in B are true in M, too. But this contradicts the fact that B is closed.

Compactness, Completeness, and Decidability
By appropriate modification to the relevant proofs, we can obtain the counterparts of the facts from Sections 3-6.
We can easily specify the notion of a mini-max tableau for any non-empty finite subset Φ of For, where the rule (1 Ar r ) we use only for some α that occurs in some formula from Φ. For each such α we use this rule exactly once and the formula of the form jε α we put in the initial list. Moreover, we do not apply rules (RC) to formulas I • αα , for any • ∈ {blank, , ♦}.
For Completeness Theorem we use the following Completeness Lemma: 13 For (1 Ar ): By rule (1 Ar r ), for any α that occurs in some formula from Φ on the initial list we have jε α . Therefore d (α) = ∅.
For (4 Ar ): Suppose that we have d(α) ⊆ d (β). Then, by (1 Ar r ) and (RI 2 ), a formula A αβ is in B. So we can use the rule (4 Ar r ) to get d ♦ (α) ⊆ d ♦ (β), if d ♦ (α) = ∅, and in consequence some ıε ♦ α is in the branch. Now we show that for any ϕ ∈ For we obtain: if ϕ is in B, then M T ϕ. The proof is by induction on the complexity of formulas as in the proof of Lemma 5.1. We use the rules of (RI Ar 1 ) and of (RC Ar ) only for A ♦ , E ♦ , I , and O .
Finally, we obtain a theorem corresponding to Theorem 6.1: Theorem 9.8 (Decidability). For all Ψ ∈ P fin (For) and ϕ ∈ For, the problem whether Ψ |= Ar ϕ is decidable.
Both theorems just give an alternative decision procedure for determining validity in the Thomason-Johnson semantics of Aristotelian modal syllogisms.