TABLEAU SYSTEM FOR LOGIC OF CATEGORIAL PROPOSITIONS AND DECIDABILITY

Transcription

1 Bulletin of the Section of Logic Volume 37:3/4 (2008), pp Tomasz Jarmużek TABLEAU SYSTEM FOR LOGIC OF CATEGORIAL PROPOSITIONS AND DECIDABILITY Abstract In the article we present an application of some tableaux method presented in [1]. It is used to define a certain system of syllogistic. We consider only the basic system with Aristotelian sentences but without the assumption on non-emptiness. The presented approach can be effectively extended to enriched syllogistic in some of many ways. However, here we limit ourselves to the most basic and general perspective. Thanks to it and the method we use, we can consider a problem of decidability. In that matter we are inspired by articles [2], [3], where the problem of the cardinality of a domain for syllogistic is researched. Since we consider a pure syllogistic language, without boolean connectives, so we come to slightly different results. At the end we show a very simple connection between the formal structures of reasoning and a cardinality of domain, which is sufficient to decide whether a given reasoning is valid or not. 1. Bases: system TS Language and grammar. The alphabet of TS (in short: ALF(TS)) can be described as the union of separate sets: terms T erm = {P 1, Q 1, R 1, P 2, Q 2, R 2,... } and auxiliary symbols {a, e, i, o}. Now, we define notions of a TS formula and an auxiliary expression: This paper was supported by founds for Polish science , as a part of a research project, nr. NN Author would like to thank to Andrzej Pietruszczak for his support and suggestion to some ideas appearing in the article.

2 224 Tomasz Jarmużek Definition 2.1. To the set of formulas For belong all and only such expressions that have one of the following forms: 1. AaB (All A are B.) 2. AiB (At least one A is B.) 3. AeB (No A is B.) 4. AoB (At least one A is not B.) To the set of auxiliary expressions Ae belong all and only such expressions that have one of the following forms: 1. A +j (The object j is A.) 2. A j (The object j is not A.) where A, B T erm and j I, i.e. to the set of indexes (so to any kind of objects, in particular natural numbers). The members of For are called formulas. The set For Ae will be called set of expressions, designated by EX, and its members will be called expressions. Semantics. In the further parts we use symbols of predicate logic, logical connectives:,,, quantifiers,, and standard set-theoretical notions. Definition 2.2. [Interpretation] An interpretation I of the formulas is any pair P (D), f, where D is a set of any kind of objects, P (D) is the powerset of D, and f is a function from T erm into P (D), i.e. f : T erm P (D). Definition 2.3. [Truth] Let I be an interpretation. We define the following conditions that should be satisfied by a formula to be called true in I (in short: I = φ, where φ For): 1. I = AaB iff f(a) f(b) 2. I = AiB iff f(a) f(b) 3. I = AeB iff f(a) f(b) = 4. I = AoB iff f(a) f(b) where A, B T erm and j I. Definition 2.4. [Satisfaction, semantic consequence] Let Φ be a set of formulas. (a) Let I be an interpretation. We say that I satisfies Φ (in short: I = Φ) iff for each φ Φ: I = φ. (b) We say that Φ is satisfiable

3 Tableau System for Logic of Categorial Propositions and Decidability 225 iff there is an interpretation I such that I = Φ. (c) Let ψ be a formula. We say that ψ follows from Φ (is a semantic consequence of ) iff for any interpretation I: I = ψ, whenever I = Φ. Definition 2.5. Let Φ be a set of expressions and I = P (D), f be an interpretation. I is faithful to Φ iff 1. I = Φ For 2. there is a function γ : I D, such that for any A, B T erm and j I: (a) if A +j Φ, then γ(j) f(a) (b) if A j Φ, then γ(j) f(a). Tableaux approach We define tableaux rules of inference for TS. Definition 2.6. [Tableaux rules] Let Φ be a set of expressions, A, B T erm, and j I. We call tableaux rules for TS the following rules: Ra + Φ {AaB,A +j } Φ {AaB,A +j,b +j }, where B+j Φ Ra Φ {AaB,B j } Φ {AaB,B j,a j }, where A j Φ Re Φ {AeB} Φ {BeA}, where BeA Φ Re + Φ {AeB,A +j } Φ {AeB,A +j,b j }, where B j Φ Φ {AiB} Ri Φ {A +j,b +j }, where j does not occur in any expression in Φ, and for any k I {A +k, B +k } Φ. Φ {AoB} Ro Φ {A +j,b j }, where j does not occur in any expression in Φ, and for any k I {A +k, B k } Φ We call the initial set of any of the rules a body of a rule, whilst the result set a conclusion of a rule. For convenience we write these rules as fractions but, of course, we may also sometimes use ordered pairs, since the rules are collections of pairs defined on EX.

4 226 Tomasz Jarmużek Definition 2.7. Let Φ be a set of expressions and R a tableaux rule. We say that a body and a conclusion of R are contained in Φ iff there are such sets of expressions 1, 2, that: 1, 2 in R and 1, 2 Φ. Definition 2.8. [Inconsistent expressions] The following sets of expressions are called inconsistent: {AaB, AoB} {AeB, AiB} {A +j, A j } where A, B are terms and j I. By : For For we mean the function defined with the condition: (x) = y iff x and y are inconsistent. Let Φ be a set of expressions, containing some inconsistent expressions. Then Φ is called inconsistent. Otherwise, we call it consistent. Lemma 2.1. Let I = P (D), f be any interpretation. if φ and ψ are inconsistent formulas, then I is faithful to {φ} iff I is not faithful to {ψ} if Φ is an inconsistent set of expressions, then I is not faithful to Φ. Proof: Obvious, by the Definitions 2.3, 2.4, and 2.5. Tableaux proof Definition 2.9. [Tableaux proof] Let Φ {ψ} F or. Formula ψ has a proof on the ground of Φ (in short, we write: Φ ψ or Φ ψ, when it has no proof) iff for some finite Φ Φ there is a set of expressions Ψ = { (ψ), φ 1, φ 2,..., φ n } such that: 1. for any k n, if φ k Ψ, then one of the following conditions is satisfied: (a) φ k Φ (b) there is a tableaux rule such that its body is contained in Ψ and φ k belongs to its conclusion. 2. {A +j, A j } Ψ, for some term A, and some j I. Example 2.1. Let us prove that {P 1 aq 8, R 3 ip 2, Q 8 ar 1 } P 1 ar 1. We take the set Φ = {P 1 aq 8, Q 8 ar 1 } and build the following sequence of sets:

5 Tableau System for Logic of Categorial Propositions and Decidability 227 { (P 1 ar 1 ), P 1 aq 8, Q 8 ar 1 } = {P 1 or 1, P 1 aq 8, Q 8 ar 1 } {P 1 or 1, P 1 +1, R1 1, P 1aQ 8, Q 8 ar 1 } {P 1 or 1, P 1 +1, R1 1, P 1aQ 8, Q +1 8, Q 8aR 1 } {P 1 or 1, P 1 +1, R1 1, P 1aQ 8, Q +1 8, Q 8aR 1, R 1 +1 } = Ψ Since Ψ satisfies the conditions of the Definition 2.9, so {P 1 aq 8, R 3 ip 2, Q 8 ar 1 } P 1 ar 1. Soundness We start with a certain lemma. Lemma 2.2. For any interpretation I, any Φ EX and any expressions AaB, AeB, AoB, AiB, A +j, A j, B +j, B j : 1. if I is faithful to Φ {AaB, A +j }, then I is faithful to Φ {AaB, A +j, B +j } 2. if I is faithful to Φ {AaB, B j }, then I is faithful to Φ {AaB, A j, B j } 3. if I is faithful to Φ {AeB, A +j }, then I is faithful to Φ {AaB, A +j, B j } 4. if I is faithful to Φ {AeB}, then I is faithful to Φ {AeB, BeA} 5. if I is faithful to Φ {AiB}, then I is faithful to Φ {AiB, A +j, B +j } 6. if I is faithful to Φ {AoB}, then I is faithful to {AoB, A +j, B j }. Proof: In the proof we apply Definitions 2.3 and 2.5. Let I = P (D), f. We consider only the first case because other ones are similar and easy. Case 1. Let I be faithful to Φ {AaB, A +j }. So, I = AaB and there is a function γ : I D with γ(j) f(a), by Definition 2.5. As a consequence, by Definition 2.3, x D (x f(a) x f(b)) and γ(j) f(b). Hence, again by 2.5, I is faithful to B +j. Now, we will show that: Theorem 2.1. [Soundness] For any Φ {ψ} For: if Φ ψ, then Φ = ψ. Proof: Let Φ {ψ} F or. We assume that Φ ψ. Hence, by the definition of (2.9) for some finite Φ Φ there is a set of expressions Ψ = { (ψ), φ 1, φ 2,..., φ n } such that: 1. for any k n, if φ k Ψ, then one of the following conditions is satisfied:

6 228 Tomasz Jarmużek (a) φ k Φ (b) there is a tableaux rule such that its body is contained in Ψ and φ k belongs to its conclusion. 2. {A +j, A j } Ψ, for some terms A, and some j I. Let I be an arbitrary interpretation that satisfies Φ. We assume indirectly that I = ψ. Hence, by Lemma 2.1, I = (ψ). Therefore, by the hypothesis, I = Φ { (ψ)} ( ). Now, consider Ψ. Since Ψ = { (ψ), φ 1, φ 2,..., φ n } = n + 1, so it has been constructed by the application of the tableaux rules at most n-times in some order of inclusions: Ψ 1 Ψ 2 Ψ l = Ψ, where l n + 1. Since Ψ 1 = Φ { (ψ)}, I is faithful to Ψ 1, by ( ). Let 1 < l, i l 1 and I be faithful to Ψ i. Then Φ i+1 is a consequence of application of some tableaux rule and by the Lemma 2.2 I is faithful to Ψ i+1. As a consequence, I is faithful to Ψ l = Ψ, therefore, by Lemma 2.1, Ψ is consistent, which contradicts the consequence of the main hypothesis. Finally, I = ψ. Completeness Again, we start with some auxiliary notions. Definition [Induced interpretation] Let Φ be a consistent set of expressions and I = P (D), f be an interpretation. We say that I is induced by Φ iff 1. D = {j : A +j or A j belongs to Φ} 2. f(a) = {j : A +j belongs to Φ} where A is a term and j I. Definition [Closure under tableaux rules] Let Φ be a set of expressions. 1. We say that Φ is closed under tableaux rules iff for any tableaux rule, its conclusions is contained in Φ, whenever Φ contains the body of the rule. More precisely, iff for any sets of expresions Ψ, and a substitution of any tableaux rule Ψ, Φ, whenever Ψ Φ and conditions of the rule are satisfied. 2. Let Γ be a set of expressions. We say that Φ is a closure of Γ iff Φ is minimal among sets that are closed under tableaux rules and supersets of Γ, i.e.

7 Tableau System for Logic of Categorial Propositions and Decidability 229 (a) Φ is closed under tableaux rules (b) Γ Φ (c) there is no set EX that satisfies the conditions: i. is closed under tableaux rules ii. Γ iii. Φ. Example 2.2. Let Φ = {A 1 ob 3, A +2 1, B 2 2 }. We see that a body of only one rule (the rule Ro) is contained in Φ. But we cannot apply it to Φ because in Φ there has already been some conclusions of the rule. So, Φ is closed under tableaux rules and, since it is a minimal one, is a closure of the set {A 1 ob 3 }. Fact 2.1. Let Φ be a closure of Ψ. If some expression ψ belongs to Φ, then there is a finite Ψ 0 Ψ and a finite Φ 0, a closure of Ψ 0, such that ψ belongs to Φ 0. Clearly, by induction on the tableaux rules and forms of expres- Proof: sions. Lemma 2.3. [Extension of induced interpretation] Let Φ be consistent and closed under tableaux rules set of expressions. Let I = P (D), f be an interpretation induced by Φ. Then I = Φ For. Proof: We assume that a consistent and closed under tableaux rules set of expressions Φ induces some interpretation I. We must check all cases of formulas, showing that I = φ, if φ Φ. We shall check only one case. Assume that A, B T erm and j I. 1. Let AaB belong to Φ. Consider some j D, and let j f(a). Then, by the Definition 2.10, A +j is in Φ. Since Φ is closed under the tableaux rules, so, by the rule Ra +, also B +j belongs to Φ. Again by 2.10, j f(b). Hence, x D (x f(a) x f(b)), which by Definition 2.3 implies that I = AaB. The remaining cases are similar, as well. Now, we will show that: Theorem 2.2. [Completeness] Φ ψ. For any Φ {ψ} For: if Φ = ψ, then

8 230 Tomasz Jarmużek Proof: We assume Φ ψ, for some Φ {ψ} For. Since Φ ψ, then by Definition 2.9 for any finite Φ Φ, for all sets of expressions Ψ = { (ψ), φ 1, φ 2,..., φ n } it holds that: if ( ) for any k n, if φ k Ψ, where one of the following conditions is satisfied: 1. φ k Φ 2. there is a tableaux rule such that its body is contained in Ψ and φ k belongs to its conclusion, then for any term A and any j I, {A +j, A j } Ψ. Consider the set Φ { (ψ)}. Let be an inconsistent closure of Φ { (ψ)}. By the Fact 2.1 there is some finite set Φ 0 Φ and its closure, finite set 0, which is inconsistent. As 0 is a set satisfying the condition ( ) for some set Φ = Φ 0, so it contradicts the main hypothesis. Hence, any is consistent and closed under tableaux rules. Let I be the interpretation that some induces. By Lemma 2.3, I = For. However, since Φ { (ψ)}, I = Φ { (ψ)}. So, for some I, I = Φ, but I = ψ, by Lemma 2.1. Hence, Φ = ψ. Decidability By existential formula we understand a formula of the form: AiB or AoB, where A, B are any terms. Now, we define function λ : 2 For 2 For with the condition λ(φ) = {x Φ : x is an existential formula}. Next, we define function σ : {Φ 2 For : Φ is finite} I with the condition: σ(φ) = λ(φ). Theorem 2.3. Let Φ {ψ} be a finite subset of For. Then: Φ = ψ iff I= P (D),f (D = σ(φ { (ψ)}) (I = Φ I = ψ)) Proof: Let Φ {ψ} be a finite subset of For. The proof from left to right is by Definition 2.4. For the other side we assume that Φ = ψ. Therefore, by soundness Theorem 2.1, Φ ψ. By def. of tableaux Proof 2.9 any closure of Φ { (ψ)} is consistent. Let I = P (D), f be an interpretation induced by some closure of Φ { (ψ)}, called. From the definition of induced interpretation, 2.10, it follows that D = {j : A +j or A j belongs to }. Simultaneously, every i D was introduced by the application

9 Tableau System for Logic of Categorial Propositions and Decidability 231 of a rule to an existential formula, in each time being new. Hence, D = σ(φ { (ψ)}). As a final consequence, I = Φ, but I = ψ, since I is faithful to. References [1] T. Jarmużek, Construction of tableaux for classical logic: tableaux as combinations of branches, branches as chains of sets, Logic and Logical Philosophy 1(16) (2007), pp [2] A. Pietruszczak, Cardinalities of models for pure calculi of names, Reports on Mathematical Logic, 28(1994), pp [3] A. Pietruszczak, Cardinalities of models and the expressive power of monadic predicate logic, Reports on Mathematical Logic, 30(1996), pp Departament of Logic Nicolas Copernicus University Toruń, Poland jarmuzek@umk.pl