ARISTOTELIAN ASSERTORIC SYLLOGISTIC1 MOHAMED A. AMER To Raouf Doss Who introduced modern logic to Egypt Abstract. Aristotelian assertoric syllogistic, which is currently of growing interest, has attracted the attention of the founders of modern logic, who approached it in several (semantical and syntactical) ways. Further approaches were introduced later on. These approaches (with few exceptions) are here discussed, developed and interrelated. Among other things, dierent facets of soundness, completeness, decidability and independence are investigated. Specically arithmetization (Leibniz), algebraization (Leibniz and Boole), and Venn models (Euler and Venn) are closely examined. All proofs are simple. In particular there is no recourse to maximal nor minimal conditions (with only one, dispensable, exception), which makes the long awaited deciphering of the enigmatic Leibniz characteristic numbers possible. The problem was how to look at matters from the right perspective. Introduction. Aristotelian assertoric syllogistic (henceforth AAS), which is currently of growing interest (Glasho 2005), has attracted the attention of the founders of modern logic. Leibniz, Boole, De Morgan, Venn, Peirce, Frege, Hilbert, Russell and Gödel all dealt with it. For some of them it was the starting point (cf. Boole 1948). Modern treatment of AAS started closer to what may be currently called the semantical or model theoretic approach. This was threefold: arthimetical, 12010 Mathematics Subject Classication. 03A05, 03-03 (primary); 01A20, 01A45, 01A55, 01A60 (secondary). Key words and phrases. axiomatization, natural deduction, structures, models, order models, arithmetization, Venn models, soundness, completeness, decidability, sorites, independence, algebraization, inadequacy. 1 algebraic, and diagramatic (or set theoretic). The rst trend was developed by Leibniz (ukasiewicz 1998, pp. 126-9; Kneale and Kneale 1966, pp. 3378; Glasho 2002; and Sotirov 2015). The second was developed by Leibniz (Kneale and Kneale 1966, pp. 338-45; Lenzen 2004) and after about two centuries was again developed by Boole (1948), without mentioning the work of Leibniz. The last trend was developed by Euler, then by Venn (Venn 1880). With the rise of proof theory late nineteenth century, six syntactical formalizations of AAS were developed: (i) Monadic rst order formalization which goes back to Frege (1967, on p.28 the square of logical opposition may be found). This formalization is adopted by Hilbert and Ackermann (1950, pp. 44-54). (ii) Sentential formalization which goes back to Peirce (Bellucci and Pietarinen 2016, p. 226) and is adopted by Gödel (Adzic and Dosen 2016, p. 479). The most elaborate study of this formalization is that of ukasiewicz (1998). (iii) Dyadic rst order formalization which goes back to Shepherdson (1956). The novel idea of regarding categorical sentences (or propositions) as binary relational sentences (or propositions) is due to De Morgan (Valencia 2004, pp. 506-7). It is worthwile to note here that, according to Boche«ski (1968, pp. 6870), Aristotle dealt with the logic of relations among other topics which Boche«ski (1968) puts (p. 63) collectively under the title Non-analytical laws and rules, to be distinguished from syllogisms such as those considered in this article, which Boche«ski (1968) terms (p. 42) analytic. (iv) Natural deduction formalization which goes back to Corcoran (1972) and Smiley (1973). (v) First order many-sorted formalization which goes back to Smiley (1962). (vi) A recent formalization based on Hilbert's epsilon and tau quantiers (Pasquali and Retoré 2016). All of the above will be considered below with only two exceptions. The rst is the many-sorted formalization ((v) above), for it is a variant of the monadic rst order formalization mentioned in (i) above; moreover it was, 2 apparently, abandoned even by its own author (cf. Smiley 1973). The formalization based on Hilbert's epsilon and tau quantiers ((vi) above) will take us far o the current mainstream of logic. So it will be the second exception and will not be further considered here, though it may have intrinsic merit especially for those who are interested in formalizing natural languages. With only one exception, the modern syntactical formalizations of AAS degraded it to the rank of a secondary logic, a subordinate or a subsidiary sublogic of a superior fundamental or principal primary logic. In contrast, the natural deduction formalization ((iv) above, cf. Boche«ski (1968, pp. 3, 31, 42, 49, 52)) rehabilitates it to a fulledged primary logic, as was probably designed by its founder: Aristotle, and as was taken for granted for over two millennia. Accordingly, this formalization will be the focus of this article. Through completeness we shall see that as far as the basic sentences (to be dened in 1.6 below) are concerned other formalizations add nothing new. In the sequel we deal -from modern standpointswith AAS, not with medieval nor traditional syllogistic. In contrast to Boole (1948), Glasho (2007), Hilbert and Ackermann (1950), Russino (1999), Shepherdson (1956) and Sotirov (1999), term negation (or complementation, to use a modern term; cf. Boche«ski (1968, p. 50)) is not here permitted. Also, in contrast to Hilbert and Ackermann (1950), ukasiewicz (1998), Shumann (2006), Shepherdson (1956) and Sotirov (1999), Boolean combinations of categorical sentences are not here permitted. So (the extensional aspect of) AAS will be just the logic of, or the fragment of set theory which deals with, inclusion (universal armative sentences) and exclusion (universal negative sentences) and their contradictories (particular sentences). However, some exceptions may be appropriate as will be clear, or claried, at the proper places. Among other things, dierent facets of soundness, completeness, decidability, and independence are investigated. Particularly arithmetization (Leibniz), algebraization (Leibniz and Boole) and Venn models (Euler and Venn) are closely examined. All proofs given here are simple. In contrast to Corcoran (1972), Glasho (2010), Martin (1997), Shepherdson (1956), Smiley (1973) and Smith (1983), our proofs have no recourse to maximal nor minimal principles nor conditions (with only one exception, which is indirect and may be dispensed with). This makes the long awaited deciphering of the enigmatic Leibniz characteristic numbers possible. The problem was how to look at matters from the right perspective. To specify, in section 3 below we provide a polynomial time algorithm to 3 decide for any nite set of categorical sentences whether it is consistent and, if it is, to assign a Leibniz model (to be dened below) to it. This settles positively problem 2 of Glasho (2002) for nite sets. The general case is discussed in section 2. I hope that the simplicity of this exposition of AAS will help to reincorporate it into the mainstream of mathematical logic. After this introduction, the structure of the rest of the article is as follows: 1. Formalizations of AAS 2. Semantics of AAS 3. Decidability 4. Basic equivalence of the four formalizations 5. Venn soundness and completeness 6. Direct way to Venn models 7. Variations on NF (C) 8. Direct completion of direct deduction 9. Models of NF (C) revisited 10. Decidability revisited 11. Sorites 12. Independence 13. Algebraic semantics of AAS, a prelude 14. Algebraic interpretation of NF (C) 15. Annihilators: Embedding the partial into a total 16. Back to algebraic interpretation 17. Leibniz and Boole 18. Inadequacy: bounds of AAS Acknowledgements Appendix 1. Formalizations of AAS. Formalizations of AAS dier with regard to permitting the subject and the predicate of a formal symbolic categorical sentence (henceforth categorical sentence) to be the same. Smith (1983) and Glasho (2010) follow Corcoran (1972) in not permitting sameness; as accommodating sameness would entail rather more deviation from the Aristotelian text says Corcoran (1972, p. 696). On the other hand Smiley (1973) left the door open for permitting sameness, noting (p. 144) that the variables he [Aristotle] uses for the major, middle and minor terms are all distinct from one another [...]; though when it 4 comes to substituting actual terms in the resulting forms we are of course at liberty to replace dierent variables by the same term (64a1).. Consequently, it seems that Aristotle excluded sameness, for technical -not philosophicalreasons. This is, possibly, why ukasiewicz (1998) adopted sameness (see pp. 77, 88); while Martin (1997) simultaneously considered two systems, one of them is permitting sameness and the other is not. In conformity with the current mainstream of mathematical logic, sameness is here permitted. Excluding sameness, and other variations, will be considered in section 7 below. 1.1. Monadic rst order formalization of AAS. The languge here is a standard rst order language, with or without equality, whose set P of non-logical constants has at least three elements, and all of its elements are unary relational symbols. In the sequel P , Q and R will be metalinguistic variables ranging over the elements of P. With abuse of notation, P will denote this language too. ABBREVIATIONS 1.1. APQ is an abbreviation for ∀x(Px→ Qx) EPQ is an abbreviation for ∀x(Px→qQx) IPQ is an abbreviation for ∃x(Px ∧Qx) OPQ is an abbreviation for ∃x(Px∧qQx) DEFINITION 1.2. MF (P) is the theory based on P with only one non -logical axiom schema, namely, APQ → IQP (which is equivalent to the schema ∃x Px). PROPOSITION 1.3. The following are theorem schemata of MF (P): 1. EPQ↔ q IPQ 2. OPQ↔qAPQ 3. APP 4. APQ→ IQP 5. EPQ→ EQP 6. APQ ∧ AQR→ APR 7. APQ ∧ EQR→ EPR Proof. Routine.  5 1.2. Sentential formalization of AAS. The symbols A, E, I and O were made use of in section 1.1, in this section they will be made use of dierently. This abuse of notation is benign as long as the intended denotation is clear from the context, so it will be here permitted. Such abuses of notation may be permitted later on without further notice. Let J be a set (whose elements are to correspond to categorical constants) having at least three elements, let A,E, I and O be four injective functions of pairwise disjoint ranges, each of domain J×J , and let AS(J) be the union of their ranges. In the sequel i, j and k will be metalinguistic variables ranging over the elements of J . The language here is a standard sentential language whose set of sentential symbols is AS(J). With abuse of notation J will denote this language too. DEFINITION 1.4. SF (J) is the theory based on J with the following non logical axiom schemata: 1. Eij ↔qIij 2. Oij ↔qAij 3. Aii 4. Aij → Iji 5. Eij → Eji 6. Aij ∧ Ajk → Aik 7. Aij ∧ Ejk → Eik The proof machinery is modus ponens together with any standard set of sentential logical axiom schemata. REMARK 1.5. There are two kinds of substitution: sentences for sentences and indices for indices. Each may be permitted, under some conditions, as a derived rule of inference (cf. ukasiewiez 1998, p. 88; see section 1.5 below). 1.3. Dyadic rst order formalization of AAS. The language here is a standard rst order language, with or without, equality whose non-logical constants are four binary relation symbols A,E,I and O, together with a set C of individual (or categorical) constants having at least three elements. In the sequel a, b, and c will be metalinguistic variables ranging over the elements of C. With abuse of notation C will denote this language too. DEFINITION 1.6. DF (C) is the theory based on C whose non-logical axioms are the universal closures of: 6 1. Exy ↔qIxy 2. Oxy ↔qAxy 3. Axx 4. Axy → Iyx 5. Exy → Eyx 6. Axy ∧ Ayz → Axz 7. Axy ∧ Eyz → Exz 1.4. Natural deduction formalization of AAS. The language here is a sublanguage of the language dened in 1.3. The alphabet is the four binary relation symbols A,E, I and O, together with a set C of individual (or categorical) constants having at least three elements. The sentences are the equality free atomic sentences of 1.3., viz. a sentence is a string Y ab where Y ∈ {A,E, I, O} and a, b ∈ C. By abuse of notation C will denote this language too, and the set of all sentences will be denoted by S(C). In the sequel α, β, γ, δ, σ and ρ will be metalinguistic variables ranging over the elements of S(C). Sentences starting with A or E are called universal, those starting with I or O are called particular. Also, sentences starting with A or I are called armative, those starting with E or O are called negative. ForW ∈ {A,E, I, O}, sentences starting with W are called W -sentences. DEFINITION 1.7. NF (C) is the logical system based on the language C with the following deduction rules (or enrichments thereof, see sections 8 and 11 below): 0. Aaa (A-Id) 1. Aab Iba (Apc) 2. Eab Eba (Ec) 3. Aab,Abc Aac (Barbara) 4. Aab,Ebc Eac (Celarent). Barbara and Celarent are, respectively, the medieval names of the rules 3 and 4; A-Id", Apc and Ec are, respectively, abbreviations for A-identity, A-partial conversion and E-conversion. For simplicity, we may write rules instead of deduction rules. DEFINITION 1.8. A direct deduction (or d-deduction) of σ(∈ S(C)) from Γ(⊆ S(C)) is a sequence < ρi >i∈k (k ∈ N+) such that ρk−1 = σ and for each i ∈ k, σi ∈ Γ or is the consequent of some rule of NF (C) whose antecedents are previous terms of the sequence. In this case we write Γ d ` σ, and σ is said 7 to be a direct consequence (or theorem) of Γ. Also < ρi >i∈k is said to be a direct (or d-) deduction from Γ. From now on, the rules 0-4 given above will be called also d-rules. Regarding the current mainstream of mathematical logic, this denition is a typical denition. In contrast, corresponding denitions given in Corcoran (1972), Glasho (2010), Martin (1997), Smiley (1973) and Smith (1983) are atypical, each has its own peculiarity. To get closer to the Aristotelian tradition, a more restricted denition of direct deduction is presented in section 11 below, and its relationship to the above one is investigated there. As usual, the contradictory σ of σ(∈ S(C)) is dened as follows: Âab = Oab Êab = Iab Îab = Eab Ôab = Aab so σ = σ. A set Γ(⊆ S(C)) is said to be d-inconsistent (or d-contradictory) if Γ d ` σ and Γ d ` σ, for some σ ∈ S(C); otherwise Γ is said to be d-consistent. DEFINITION 1.9. The general (or g-) deduction relation g ` (⊆ ℘(S(C)) × S(C)) is dened as follows: Γ g ` σ i Γ ∪ {σ} is d-inconsistent. For Γ ⊆ S(C), Γ is g-inconsistent (or g-contradictory) and Γ is gconsistent may be dened along the above lines, replacing d by g. Obviously d ` ⊆ g `, so if Γ(⊆ S(C)) is d-inconsistent, it is g-inconsistent. For e ∈ {d, g} and Γ ⊆ S(C), Γe will denote the closure of Γ under e `, i.e. Γe= the smallest ∆ ⊆ S(C) such that Γ ⊆ ∆ and for every σ ∈ S(C), σ ∈ ∆ whenever ∆ e ` σ. LEMMA 1.10. Let Σ,Σ′ be subsets of S(C) such that for every σ ∈ Σ, Σ′ d ` σ. For every d-deduction < ρi >i∈k from Σ, there are a k′(≥ k), a d-deduction < ρ′i >i∈k′ from Σ ′ and a strictly increasing function f : k → k′ such that f(k − 1) = k′ − 1 and for every i ∈ k, ρi = ρ′f(i). Hence for every α ∈ S(C), Σ′ d ` α whenever Σ d ` α. 8 Proof. By induction on `, the number of times of making use in < ρi >i∈k of assumptions from Σ. Basis: ` = 0; take k′ = k,< ρ′i >i∈k′ = < ρi >i∈k and f the identity function on k. Induction step: assume the required for ` = m. Let ` = m + 1 and let j(∈ k) be the last line in which an assumption from Σ is made use of. The case j = 0 is easier than the case j > 0, so we shall deal only with the latter. By the induction hypothesis, there are a j′(≥ j), a deduction < αi >i∈j′ , from Σ′ and a strictly increasing function g : j → j′ such that g(j−1) = j′−1 and for every i ∈ j, ρi = αg(i). Let < βi >i∈m be a deduction of ρj from Σ ′. Put: k′ = j′ +m+ k − j − 1, γi = ρi+j+1 for i ∈ k − j − 1, < ρ′i >i∈k′=< αi >i∈j′_ < βi >i∈m_ < γi >i∈k−j−1, where _ is the concatenation operation symbol. Evidently k′ ≥ k. The completion of the proof is now easy.  Parts 3 and 4 of the next proposition are, respectively, reformulations of lemmata M1 and M2 of Corcoran (1972). PROPOSITION 1.11. Let Γ ∪ {σ} ⊆ S(C), Γ′ = {ρ ∈ S(C) : Γ g ` ρ}, U ∈ {A,E},W ∈ {I, O}, e ∈ {d, g} and a, b, c ∈ C, then: 1. Γd = {ρ ∈ S(C) : Γ d ` ρ}. 2. If Γ ∪ {Wab} d ` σ and σ 6= Wab, then Γ d ` σ. 3. If Γ is d-consistent, then Γ g ` Uab i Γ d ` Uab. 4. Γ is d-consistent i it is g-consistent. 5. If ρ ∈ Γ′ and Γ, ρ g ` σ, then Γ g ` σ. 6. Γ′g = Γ′, hence Γg = Γ′. 7. Γee = Γe; hence, for every d-rule r, if each antecedent of r belongs to Γe, then so also does its consequent. 9 Proof. 1. The required is a corollary of the above lemma. 2. Generalizing upon the metalinguistic variable σ, the resulting sentence may be proved by course of values induction on the length of the d-deduction from Γ∪{Wab}, noticing that Wab is not a premise of any rule of NF (C). 3. Let Γ g ` Uab, then for some α ∈ S(C), Γ, Ûab d ` α, α. So, by part 2, if Γ is d-consistent then Ûab ∈ {α, α}, then Uab ∈ {α, α}. So, by part 2 again, Γ d ` Uab. The other direction is obvious. 4. Let Γ be d-consistent and g-inconsistent, then there is a universal α ∈ S(C) such that Γ g ` α, α, then by part 3, Γ d ` α. Also there is β ∈ S(C) such that Γ, α d ` β, β, so, by lemma 1.10., Γ d ` β, β, hence Γ is d-inconsistent. Consequently, if Γ is g-inconsistent it is d-inconsistent. The other direction is obvious. 5. Obvious if Γ is d-inconsistent, so let Γ be d-consistent and let ρ ∈ Γ′ and Γ, ρ g ` σ. There is α ∈ S(C) such that Γ, ρ, σ d ` α, α. If ρ is universal, then by part 3 and lemma 1.10, Γ, σ d ` α, α. Also, if ρ is particular and ρ /∈ {α, α}, then by part 2, Γ, σ d ` α, α. In both cases Γ g ` σ. In the remaining case ρ must be particular and Γ, ρ, σ d ` ρ, ρ, then by part 2, Γ, σ d ` ρ. But there is β ∈ S(C) such that Γ, ρ d ` β, β, then Γ, σ d ` β, β, hence Γ g ` σ, which completes the proof. 6. By induction, part 5 may be generalized to: for every nite ∆ ⊆ Γ′, Γ g ` σ whenever Γ∪∆ g ` σ. From this it readily follows that Γ g ` σ whenever Γ′ g ` σ, hence the result. 7. By part 1 and lemma 1.10, Γdd = Γd, and by part 6, Γgg = Γ′g = Γ′ = Γg. To prove the last clause, let r be a d-rule. If each antecedent of r belongs to Γe, then its consequent belongs to Γee = Γe.  In view of part 4 of the above proposition, for e ∈ {d, g}, the prex e- may be deleted from e-consistent, e-inconsistent and e-contradictory. 10 1.5. Equality / equivalence. For Γ ⊆ S(C), e ∈ {d, g} and a, b ∈ C, Γ e ` Aab,Aba is equivalent to each of: 1. Γ e ` Aac i Γ e ` Abc all c ∈ C, 2. Γ e ` Aca i Γ e ` Acb all c ∈ C. Thus, Γ e ` Aab, Aba imply the substitutability of a, b for each other in universal positive sentences. This will be generalized below to all universal sentences, respectively all sentences, for e = d, respectively e = g. The last generalization is the essence of equality (congruence or equivalence, depending on the situation). It holds, in the respective appropriate forms, for the other formalizations as is shown in the following: THEOREM 1.12. 1. Let P,Q ∈P, then: MF (P) ` (APQ∧AQP )→ (φ↔ φ′) for every form φ of P, where φ′ is a form obtained from φ by substituting some occurrences of P (x) in φ by Q(x), or vice versa. 2. Let i, j ∈ J , then: SF (J) ` (Aij ∧ Aji) → (α ↔ α′) for every sentence α of J , where α′ is a sentence obtained from α by substituting some occurrences of i in α by j, or vice versa. 3. Let a, b ∈ C, then: DF (C) ` (Aab ∧ Aba) → (φ ↔ φ′) for every form φ of C, where φ′ is a form obtained from φ by substituting some occurrences of a in φ by b, or vice versa; provided -for languages with equalityno substitution takes place in a form or a subform of the form t = t′, where t and t′ are terms. 4. Let Γ ⊆ S(C), e ∈ {d, g} and a, b ∈ C, and let Γ e ` Aab, Aba, then for all c ∈ C: Γ e ` Y ac i Γ e ` Y bc and Γ e ` Y ca i Γ e ` Y cb, where Y ∈ {A,E}, {AE, I,O} for e = d, g respectively. Proof. The rst three parts may be proved by the standard methods developed in the respective formal systems. 11 For the last part, part 7 of proposition 1.11 secures the required for e ∈ {d, g} and Y ∈ {A,E}. It remains to consider the cases where e = g and Y ∈ {I, O}. If Γ is inconsistent the required follows by the denition of g `, so let Γ be consistent. Assume Γ g ` Aab, Aba and Γ g ` Ica, then by part 3 of proposition 1.11, Γ d ` Aab, Aba, hence Γ, Ecb d ` Eca. But there is α ∈ S(C) such that Γ, Eca d ` α, α, consequently Γ, Ecb d ` α, α and, by denition, Γ g ` Icb. The other cases are similar or easier.  1.6. Basic sentences. In each of the four formalizations MF (P), SF (J), DF (C) and NF (C) the sentences to be made use of in the Aristotelian syllogistic will be called basic (or categorical) sentences. The sets of basic sentences will be denoted, respectively, by BM(P), BS(J), BD(C) and BN(C). That is: BM(P) = {Y PQ : Y ∈ {A,E, I, O} and P,Q ∈P}. BS(J) = AS(J) (= the set of all atomic sentences of J). BD(C) = the set of all (equality free) atomic sentences of C. BN(C) = S(C) (= BD(C)). 1.7. Interpretation. Let h : C → J , with abuse of notation (no confusion will ensue) we dene another function h : BN(C) → BS(J) by h(Y ab) = Y hahb, for Y ∈ {A,E, I, O} and a, b ∈ C. As usual, for Γ ⊆ BN(C), the image of Γ under h is denoted by h(Γ); also we may write Y hab, Γh for h(Y ab), h(Γ) respectively. The function h is said to be an interpretation of BN(C) in BS(J). Similarly BM(P), BS(J) and BN(C) (= BD(C)) may be interpreted in each other. PROPOSITION 1.13. Let Γ∪{σ} ⊆ BN(C), let h and H be interpretations of BN(C) in BS(J) and BM(P) respectively, and let < Γ′, σ′, σ′, T >∈ {< Γ, σ, σ, DF (C) >, < Γh, σh, σh, SF (J) >, < ΓH , σH , σH ,MF (P) >}. Then: 1. T ` σ′ ↔qσ′, 2. T ∪ Γ′ ` σ′ whenever Γ g ` σ. Proof. 1. Easy. 12 2. By proposition 1.3 for T = MF (P), and by the denitions for the other cases.  PROPOSITION 1.14. Let T ∈ {MF (P), DF (C)}, let h be an interpretation of BS(J) in the set of basic sentences of T , and let Γ∪{σ} ⊆ BS(J). Then: T ∪ Γh ` σh whenever SF (J) ∪ Γ ` σ. Proof. The interpretations of the axioms of SF (J) are theorems of T , and the proof machinery of T is not weaker than that of SF (J).  To investigate the converses of proposition 1.14 and part 2 of proposition 1.13, we rst go to: 2. Semantics of AAS. The theoriesMF (P) and DF (C) are rst order, and the theory SF (J) is sentential; so each has its usual class of models with respect to which it is sound and complete. 2.1. Models of MF (P). A model B of MF (P) is an ordered pair < B, μ > where B is a non-empty set and μ maps P into ℘(B)− {φ}. B is called the universe, or the base, of B and may be denoted also by |B|. 2.2. Models of SF (J). A model B of SF (J) is a mapping form AS(J) into 2 (= {0, 1}), which satises all the axioms of SF (J). For σ ∈ S(J), B  σ means that σ takes the value 1 under the usual extension of B. 2.3. Models of DF (C). A structure B of the dyadic language C (or a DF (C)-structure B) is a 6-tuple < B,A∗, E∗, I∗, O∗, μ > where B is a nonempty set and A∗, E∗, I∗ and O∗ are binary relations on B corresponding to the relation symbols A,E,I and O respectively, and μ is a mapping of C into B. B is called the universe, or the base, of B and may be denoted also by |B|. B is a model of DF (C) if it satises its axioms. Since, by axioms 1 and 2, E∗ = I∗c and O∗ = A∗c (where c denotes the complement with respect to B × B) we may -by abuse of notationsay that < B,A∗, I∗, μ > is a model of DF (C) whenever the expansion < B,A∗, I∗c, I∗, A∗c, μ > is a model of DF (C). PROPOSITION 2.1. Let B 6= φ, μ : C → B and R1, R2 ⊆ B × B. Then < B,R1, R2, μ > is a model of DF (C) i: 1. R1 is reexive and transitive (i.e. R1 is a pre-ordering on B), 13 2. R2 is symmetric, 3. R1 ⊆ R2, 4. R2|R1 ⊆ R2, where R2|R1 is the relative product of R2 and R1. Proof. < B,R1, R c 2, R2, R c 1, μ > satises axioms 1-6 i conditions 1-3 above are satised. Axiom 7 is equivalent to ∀x∀z[∃y(Axy∧Eyz)→ Exz]. So axiom 7 is satised i R1|Rc2 ⊆ Rc2 which, in the presence of condition 2, is equivalent to condition 4.  2.4. Models ofNF (C). The structures in which NF (C) may be interpreted (henceforth NF (C)-structures) are exactly the DF (C)-structures. For Γ∪{σ} ⊆ BN(C) we write Γ B σ to mean that B  σ whenever B  Γ, where B is an NF (C)-structure and B  σ, B  Γ are dened as usual. Two NF (C)-structures B, B′ are said to be BN(C)-equivalent, basically equivalent, or (for short) B-eq if for every σ ∈ BN(C), B  σ i B′  σ; in this case we may say also that B is B-eq to B′. This notion may be extended in an obvious way to the other formalization of AAS. DEFINITION 2.2. An NF (C)-structure B is said to be a direct model (or, for short, a d-model) if for every Γ∪{σ} ⊆ BN(C), Γ B σ whenever Γ d ` σ. The proof of the following is straightforward. PROPOSITION 2.3. An NF (C)-structure < B,A∗, E∗, I∗, O∗, μ > is a dmodel i all of the rules of inference of NF (C) are valid in it (in the sense that if the antecedents are true in it, then so also is the consequent), i: 1. A∗μ is reexive on μ(C) and transitive (equivalently, A∗μ is a preordering on μ(C)), 2. A∗μ ⊆ ` I∗μ, 3. E∗μ is symmetric, 4. A∗μ|E∗μ ⊆ E∗μ, where, for a set X, Xμ = X∩(μ(C)×μ(C)); and for a binary relation R, ` R is its converse.  14 DEFINITION 2.4. The canonical structureBΓ corresponding to Γ(⊆ BN(C)) is the NF (C)-structure < B,A∗, E∗, I∗, O∗, μ > satisfying: 1. B = C, 2. μ = lC , where for a set X, lX is the identity function on X, 3. for every Y ∈ {A,E, I, O}, Y ∗ = {< a, b >∈ C × C : Y ab ∈ Γ}. An NF (C)-structure is said to be canonical if it is equal to BΓ, for some Γ ⊆ BN(C). The basic property of BΓ is: BΓ  σ i σ ∈ Γ all σ ∈ BN(C). Every NF (C)-structure B =< B,A∗, E∗, I∗, O∗, μ > in which μ = lC is an extension of a canonical structure; namely the canonical structure corresponding to {σ ∈ BN(C) : B  σ}. LEMMA 2.5. For Γ ⊆ BN(C), BΓd is a d-model (of Γd hence of Γ). Proof. Let ∆∪{σ} ⊆ BN(C) and let ∆ d ` σ. If BΓd  ∆ then ∆ ⊆ Γd, hence σ ∈ Γd, consequentlyBΓd  σ.  THEOREM 2.6. (Direct soundness and completeness). Direct deduction is sound and complete with respect to the class of all direct models. That is, for every Γ ∪ {σ} ⊆ BN(C), Γ d ` σ i Γ d  σ where Γ d  σ means that Γ  B σ for every d-model B. Proof. Soundness is immediate by the denition. To prove completeness assume Γ d  σ then, in particular, Γ B Γd σ. ButBΓd  Γ, thenBΓd  σ, consequently σ ∈ Γd, hence Γ d ` σ.  DEFINITION 2.7. (General models). An NF (C)-structure B is said to be a general model (or, for short, a g-model) if for every Γ ∪ {σ} ⊆ BN(C), Γ  B σ whenever Γ g ` σ. LEMMA 2.8. For Γ ⊆ BN(C), BΓg is a g-model (of Γg hence of Γ). 15 Proof. Replace d by g in the proof of lemma 2.5.  THEOREM 2.9. (General soundness and completeness). General deduction is sound and complete with respect to the class of all general models. That is, for every Γ ∪ {σ} ⊆ BN(C), Γ g ` σ i Γ g  σ where Γ g  σ means that Γ  B σ for every g-model B. Proof. Replace d by g in the proof of lemma 2.6.  THEOREM 2.10. (NF (C)-compactness). For every e ∈ {d, g}, for every Γ ∪ {σ} ⊆ BN(C), Γ e  σ i for some nite Γ1 ⊆ Γ, Γ1 e  σ. Proof. By e-soundness and e-completeness.  In theorem 9.4 below the g-models will be fully characterized. Now we conne ourselves to the following: REMARKS and denitions 2.11. 1. Every g-model is a d-model (obvious) but not vice versa. For, let Γ = {Eaa} for some a ∈ C, then BΓd is a d-model but not a g-model. 2. Every model of DF (C) is obviously a g-model (hence a d-model) but not vice versa. For let Γ = {Oaa} for some a ∈ C, then BΓg is a g-model but not a model of DF (C). 3. For every g-modelB =< B,A∗, E∗, I∗, O∗, μ > in which μ is surjective, A∗ = E∗ = I∗ = O∗ = B × B i (A∗ ∩ O∗ 6= φ or E∗ ∩ I∗ 6= φ) i for some σ ∈ BN(C), B  σ, σ. Such models are called full models. If A∗ ∪ O∗ = B × B = E∗ ∪ I∗(respectively A∗ ∩ O∗ = φ = E∗ ∩ I∗), B is said to be complete (respectively consistent). Thus B is not full i it is consistent. Bφg is an example of a g-model in which μ is bijective, while it is not complete. 16 These notions may be generalized to all NF (C)-structures, μ does not have to be surjective. 4. Let B =< B,A∗, E∗, I∗, O∗, μ > be an NF (C)-structure in which μ is surjective. Then B is not complete i for some σ ∈ BN(C), (B 2 σ and B 2 σ) i for some σ ∈ BN(C), B 2 σ and for all ρ ∈ BN(C), {σ} B ρ, ρ. 5. For Γ ⊆ BN(C), Γ is consistent i it has a consistent d-model. 6. Direct deduction is sound with respect to any class of models with respect to which general deduction is sound. 2.5. Order models and Venn models. To the best of my knowledge Shepherdson (1956) was the rst to make use of a version of order models; the ordering was pre-ordering (reexive and transitive, but not necessarily antisymmetric) and the context was the semantics of a version of DF (C). In the context of the semantics of NF (C), or versions thereof, versions of order models were made use of in Martin (1997) and in Glasho (2002). The former required a model to be some variation on a lower semi-lattice with a smallest element, the latter relaxed these conditions; none of them mentioned that Shepherdson (1956) made use of order models. Following Shepherdson (1956), let R1 be a pre-ordering on a non-empty set B; and following Glasho (2002), put: R2 = {< x, y >∈ B ×B : {x, y} has an R1-lower bound}. Let μ be a function from C to B, then < B,R1, R2, μ > is a model of DF (C), hence a g-model and a d-model. Such models are said to be order models. If R1 is a partial ordering (equivalently, antisymmetric) the order model will also be called partial (or antisymmetric). < B,R1 > is said to be the order structure underlying the order model < B,R1, R2, μ >. Notice that if < B,R1, R ′ 2, μ > is a model of DF (C), then R2 ⊆ R′2. A concrete order model (henceforth c.o.m, and c.o.ms for the plural) is an order model in which B is a collection of non-empty sets and R1 is ⊆, so the c.o.ms are partial. If R′ is dened on such a B by xR′y i , x ∩ y 6= φ then for every μ : C → B,< B,⊆, R′, μ > is a model of DF (C). Such models are said to be Venn models. In an order model B =< B,R1, R2, μ >, R2 is determined by B and R1, so we may write -for short- B =< B,R1, μ >. For similar reasons we may 17 write < B, μ > to denote the Venn model < B,⊆, R′, μ >; the c.o.m with the same B and μ is denoted by < B,⊆, μ >. Let B = {{1, 2}, {2, 3}}, then for every μ : C → B,< B,⊆, μ > is a c.o.m but not a Venn model and < B, μ > is a Venn model but not a c.o.m. This is not always the case, for if the universe is the set of all non-empty subsets of a non-empty set, then the model is both a Venn model and a c.o.m. Every Venn model is embeddable in such a model which is B-eq to it. A Venn model with universe B is a c.o.m i for every b, b′ ∈ B there is c ∈ B such that c ⊆ b ∩ b′ whenever b ∩ b′ 6= φ. Let C,C′ ∈{the class of all Venn models, the class of all c.o.ms, the class of all partial order models, the class of all order models}, then for every B ∈ C there isB′ ∈ C′ which is B-eq to it, hence C = C′ (with the usual meaning). This is a corollary of the above discussion and the following observation. Let B =< B,R1, R2, μ > be an order model. Dene the function ′ from B to ℘(B) by b′ = R1 [b], where for a binary relation ρ, ρ[y] = {x ∈ Domain ρ : xρy}. Let B′ be the range of ′ and dene the function μ′ from C to B′ by μ′(c) = μ(c)′. Then B′ =< B′,⊆, μ′ > is a c.o.m which is a Venn model and ′ is a homomorphism from B onto B′. It is an isomorphism i R1 is antisymmetric. In all cases B and B′ are B-eq. 2.6. Models and interpretations. 2.6.1. MF (P) and SF (J). Let f be an interpretation of BM(P) in BS(J) and let B be a model of SF (J). Put: μ : P→ ℘(J)− {φ}. μ(Q) = {j ∈ J : B  Ajf(Q)} then B′ =< J, μ > is a model of MF (P). It is easy to see that: 1. For every positive universal α ∈ BM(P), B′  α i B  αf . 2. For every positive particular α ∈ BM(P), B′  α only if B  αf . The other direction holds i for every i, j ∈ Range f there is k ∈ J such that both B  Aki and B  Akj whenever B  Iij. In this case: B′  α i B  αf for every α ∈ BM(P). 18 On the other hand, let h be an interpretation of BS(J) in BM(P) and let B be a model of MF (P). Put: B′ : BS(J)→ 2 B′(α) = 1 i B  αh, then B′ is a model of SF (J). 2.6.2. SF (J) andDF (C). Let f be an interpretation of BS(J) in BD(C) and let B be a model of DF (C). Put: B′ : BS(J)→ 2 B′(α) = 1 i B  αf , then B′ is a model of SF (J). On the other hand, let h be an interpretation of BD(C) in BS(J) and let B be a model of SF (J). Dene: R1 = {< i, j >∈ J × J : B(Aij) = 1}, R2 = {< i, j >∈ J × J : B(Iij) = 1}, then B′ =< J,R1, R2, h > is a model of DF (C) and for every α ∈ BD(C) B′  α i B  αh. 2.6.3. DF (C) and MF (P). Let f be an interpretation of BD(C) in BM(P) and let B =< B, μ > be a model of MF (P), then B′ =< ℘(B) − {φ}, μ ◦ f > is a Venn model of DF (C) and for every α ∈ BD(C), B′  α i B  αf . On the other hand, let h be an interpretation of BM(P) in BD(C) and let B =< B,R1, R2, μ > be a model of DF (C). Put: μ′ : P→ ℘(B)− {φ} μ′(Q) = R1 [μh(Q)], then B′ =< B, μ′ > is a model of MF (P). It is easy to see that: 19 1. For every positive universal α ∈ BM(P), B′  α i B  αh. 2. For every positive particular α ∈ BM(P), B′  α only if B  αh. The other direction holds if B is an order model, in this case: B′  α i B  αh for every α ∈ BM(P). 2.7. Leibniz models. Let η be the partial ordering dened on the set N of natural numbers by mηn i m is a multiple of n, and dene the partial ordering R on N × N by < m1, n1 > R < m2, n2 > i m1ηm2 and n1ηn2. Denote the binary operations of the greatest common divisor and the least common multiple on N by ◦ ∧ and ◦ ∨ respectively, and put: B = {< m,n >∈ N× N : m ◦ ∧ n = 1}. The restriction of R on B, to be also denoted by R, partially orders B. So, for every μ : C → B, < B,R, μ > is an order model of DF (C), hence a g-model and a d-model. Such models are called Leibniz models, for they were rst introduced -in a dierent settingby him in 1679, as may be learned from ukasiewicz (1998, pp. 126-9), Kneale and Kneale (1966, pp. 337-8) and Glasho (2002). Leibniz practically denes A∗ to be R, but he sets I∗ < m1, n1 > < m2, n2 > i m1 ◦ ∧ n2 = 1 = n1 ◦ ∧ m2. To show that this gives rise to an order model as dened in 2.5 above, notice that {< m1, n1 >,< m2, n2 >} has an R-lower bound i there is < m3, n3 >∈ B such that < m3, n3 > R< m1, n1 > and < m3, n3 > R < m2, n2 >, which is equivalent to m3η(m1 ◦ ∨ m2) and n3η(n1 ◦ ∨ n2). But m3 ◦ ∧ n3 = 1, so the condition is equivalent to (m1 ◦ ∨ m2) ◦ ∧ (n1 ◦ ∨ n2) = 1. The l.h.s. = (m1 ◦ ∧ n1) ◦ ∨ (m1 ◦ ∧ n2) ◦ ∨ (m2 ◦ ∧ n1) ◦ ∨ (m2 ◦ ∧ n2). But m1 ◦ ∧ n1 = 1 = m2 ◦ ∧ n2, so the condition is equivalent to (m1 ◦ ∧ n2) ◦ ∨ (m2 ◦ ∧ n1) = 1 which is equivalent to Leibniz condition. Via reductio ad absurdum Glasho (2002) gave a dierent proof of the same result. Every Leibniz model is isomorphic to a Venn model. The converse is not true, for B is denumerable while there are non-denumerable Venn models. 2.7.1. Assigning Leibniz models. For Γ ⊆ BN(C) put: 20 CΓ = {c ∈ C : c occurs in some element of Γ having two distinct categorical constants}. Γ will be called essentially nite if CΓ is nite. This notion may be generalized to subsets of BM(P) and BS(J). LEMMA 2.12. CΓd = CΓ. Proof. By proposition 1.11 and induction on the length of the deduction.  THEOREM 2.13. To each consistent essentially nite Γ ⊆ BN(C) a Leibniz model of Γ may be assigned (cf. Glasho 2010, Lemma 3.4). Proof. Let < B,R > be the order structure underlying the Leibniz models. Put ` = |CΓ| and let < ci >i∈` be an injective enumeration of CΓ, < pi >i∈` be an injective `-sequence of primes and b ∈ B. Dene μ : C → B as follows: μ(c) = b if c ∈ C − CΓ, and for i ∈ `, μ(ci) =< mi, ni > where mi = ∏ j ∈ ` Acicj ∈ Γd pj , ni = ∏ j ∈ ` Ecicj ∈ Γd pj; mi and ni are square free nite products (the empty product is equal to 1). By the consistency of Γ, mi ◦ ∧ ni = 1 for all i ∈ `. Therefore B =< B,R, μ > is a Leibniz model. To show that B  Γ: 1. Let i, k ∈ `, then Acick ∈ Γd only if (∀j ∈ `)[(Ackcj ∈ Γd → Acicj ∈ Γd) ∧ (Eckcj ∈ Γd → Ecicj ∈ Γd)] only if < mi, ni > R < mk, nk > only if miηpk only if Acick ∈ Γd. So Acick ∈ Γd i B  Acick. Consequently, for every c, c′ ∈ C, B  Acc′ if Acc′ ∈ Γ. Moreover if for some c, c′ ∈ C, Occ′ ∈ Γ, then by the consistency of Γ there are i, k ∈ ` such that i 6= k and c = ci, c′ = ck. Again by the consistency of Γ, Acick /∈ Γd, henceB 2 Acick, consequently B  Ocick. 21 2. Let i, k ∈ ` be such that i 6= k, then Ecick ∈ Γd only if niηpk only if ni ◦ ∧ mk 6= 1 only if B 2 Icick only if ∃j ∈ `[Acicj, Eckcj ∈ Γd ∨ Ackcj, Ecicj ∈ Γd] only if Ecick ∈ Γd. From this and the consistency of Γ it follows that for every c, c′ ∈ C, B  Ecc′, B  Icc′ whenever Ecc′ ∈ Γ, Icc′ ∈ Γ respectively.  2.7.2. Leibniz soundness and completeness. For e ∈ {d, g}, e-deduction is sound with respect to the set of all Leibniz models (to be denoted, henceforth, by L) as they are order models. Regarding completeness, for Γ ∪ {σ} ⊆ BN(C) put: Γ L σ i Γ B σ for every B ∈ L. THEOREM 2.14. If Γ is essentially nite, then Γ g ` σ whenever Γ  L σ. Proof. Obvious if Γ is inconsistent. Let Γ be consistent and Γ  L σ, then by theorem 2.13 Γ∪{σ} is inconsistent, from which the result follows.  REMARKS 2.15. 1. From ukasiewicz (1998, pp. 126-9) it follows that: SF (J) ` α i L′ α where α is any sentence (not necessarily basic) of the language J , and L′ is the obvious adaptation of L to J . Consequently, for every Γ∪{α} ⊆ BS(J): SF (J) ∪ Γ ` α only if Γ L′ α, the other direction holds if Γ is essentially nite. 2. In the above remark, as well as in theorem 2.14, only square free Leibniz models (with the obvious denition) may be taken into consideration. 2.7.3. Generalization. Theorem 2.13 cannot be unconditionally generalized to innite CΓ. For, let < ci >i∈N be an injective enumeration of some denumerable subset of C. Put: Γ = {Acici+1 : i ∈ N} ∪ {Oci+1ci : i ∈ N} 22 then Γ is consistent but has no Leibniz model, though it has a Venn model. The following theorem gives a sucient condition for Γ to have a Leibniz model if CΓ is denumerable. THEOREM 2.16. Let CΓ be denumerable and let < ci >i∈N be an injective enumeration of it. Then Γ has a Leibniz model if it is consistent and for every i ∈ N, {q ∈ N : Acicq ∈ Γd} is nite. Proof. Along the lines of the proof of theorem 2.13 with the following modications. Let < pi >i∈N be an injective enumeration of the primes, put: mi = ∏ Acicj∈Γd pj , ni = ∏ Ecicj ∈ Γd j < max{q ∈ N : Acicq ∈ Γd} pj.  To see that the condition of the above theorem is essentially necessary, dene the equivalence relation ∼Γ on CΓ by a ∼Γ b i Aab,Aba ∈ Γd (cf. section 1.5 above). THEOREM 2.17. If Γ has a Leibniz model then there is a consistent extension Γ′ of Γ such that CΓ′ = CΓ, CΓ′/ ∼Γ′ is countable and for every a ∈ CΓ′ , with at most two exceptions, Qa(= {c ∈ CΓ′ : Aac ∈ Γ′d}/ ∼Γ′) is nite. Proof. Let B(=< B,R, μ >) be a Leibniz model of Γ. Put: Γ′ = Γ ∪ {σ ∈ BN(CΓ) : B  σ}, then CΓ′ = CΓ and CΓ′/ ∼Γ′ is countable. If for some a ∈ CΓ′ , Qa is innite, then μ(a) ∈ {< 0, 1 >,< 1, 0 >} from which the last part of the theorem follows.  REMARKS 2.18. 1. In the underlying order structure of a Leibniz model B =< B,R, μ >, < 1, 1 > is the greatest element and < 0, 1 >,< 1, 0 > are the only minimal elements. Let a, c ∈ C. If μ(a) =< 1, 1 > then B  Aca. Also assuming that 23 μ(a) ∈ {< 0, 1 >,< 1, 0 >}, then B  Aca implies that μ(c) = μ(a) hence B  Aac, and B  Ica implies that B  Aac. 2. There would be no exceptions in the above theorem had N been replaced by N+ in the denition of Leibniz models, which is equivalent to excluding < 0, 1 > and < 1, 0 > from the universe of Leibniz models. 3. Noticing that c ∼Γ c′ forces c, c′ to be assigned the same value in any Leibniz model of Γ, with a slight modication of its proof, theorem 2.16 may be strengthened as follows: Γ has a Leibniz model if there is a consistent extension Γ′ of Γ such that: 1. CΓ′ = CΓ. 2. CΓ′/ ∼Γ′ is countable. 3. For every a ∈ CΓ, {c/ ∼Γ: Aac ∈ Γ′d} is nite. 4. The above strengthening is very close to be the converse of theorem 2.17. As a matter of fact, it is its converse had N been replaced by N+ in the denition of Leibniz models. 5. The completeness theorem 2.14 may be generalized in line with the above generalizations. 2.7.4. Logico-philosophical discussion of Leibniz models. It is strange that his [Leibniz's] philosophic intuitions, which guided him in his research, yielded such a sound result. says ukasiewicz (1998, p. 126). Hopefully the above reasoning would make matters less strange. Following is a further discussion taking into consideration the Liebnizian correlation between prime and composite numbers on one hand and atomic and composite sentences, propositions, concepts or attributes on the other hand (cf. Glasho 2002, 2010). If the primes p1, p2 correspond, respectively, to the atomic sentences p ′ 1, p ′ 2, it is natural to let the composite number p1p2 correspond to the composite sentence p′1 ∧ p′2 . The diculty here is that p21, which is not equal to a prime, would correspond to the sentence p′1 ∧ p′1, which is equivalent to an atomic sentence; as conjunction of sentences is idempotent, while multiplication of numbers is not. Obviously this diculty will not arise for square free numbers. Notice that in the denitions of μ : C → B given above, the values 24 assigned by μ to the elements of CΓ are always ordered pairs of square free numbers. Extending this property to all elements of C, after relaxing it to permit < 0, 1 >,< 1, 0 > also to be taken as values, gives rise to what will be called essentially square free Leibniz models. To investigate the relationship between the Leibniz models and the essentially square free Leibniz models, let < qij >i,j∈N be an injective double sequence of primes. Map the kth power of the ith prime pi on ∏ j∈k qij. This mapping may be extended in the obvious way to an injection ν from N to N such that ν(0) = 0 and for n ≥ 1, ν(n) is square free (being the empty product of primes, ν(1) = 1). The mapping ν may be further extended, in the obvious way, to N× N, the extension also will be denoted by ν. It may be easily seen that ν(B) ⊆ B and that < m1, n1 > R < m2, n2 > i ν(< m1, n1 >)Rν(< m2, n2 >) for every < m1, n1 >,< m2, n2 >∈ B. So for every Leibniz model B(=< B,R, μ >), ν is a monomorphism from B into Bν(=< B,R, νμ >) which is essentially square free and is basically equivalent to B. Moreover if in Bν , B is replaced by ν(B) and R by R∩ (ν(B)×ν(B)), then ν will be an isomorphism. Such models will be called proper Leibniz models. Since every Leibniz model is isomorphic to a proper Leibniz model, attention may be conned to the latter. Let < q′ij >i,j∈N be an injective double sequence of atomic sentences in some sentential language. For < m,n >∈ ν(B) put: λ(< m,n >) = ∧ mηqij q′ij ∧ ∧ nηqij qq′ij. As 1ηp for no prime p, ∧ 1ηqij q′ij = ∧ 1ηqij qq′ij = the empty conjunction, which is always true. So λ(< 1, 1 >) is always true. On the other hand, 0ηp for every prime p, so λ(< 0, 1 >) = ∧ i,j∈N q′ij and λ(< 1, 0 >) = ∧ i,j∈N qq′ij. These are the only innitary sentences to be considered. It may be easily seen that for every proper Leibniz modelB, B  Ac1c2 i λμc1 → λμc2 is a tautology, and B  Ec1c2 i λμc1∧λμc2 is a contradiction. As a matter of fact < 0, 1 >,< 1, 0 > and < 1, 1 > are not indispensable as elements of the universe of proper Leibniz models. To keep them or not 25 is a philosophical choice. Rejecting them is probably more compatible with the Aristotelian legacy. Following Boole (1948, p. 49), to each < m,n >∈ ν(B) the set θ(< m,n >) of all truth assignments which satisfy λ(< m,n >) may be appropriated. For every proper Leibniz model (hence for every Leibniz model) B, θ induces an isomorphism of B onto a Venn model which is a concrete order model. 3. Decidability. REMARKS 3.1. Let a, b, c ∈ C and Γ ⊆ BN(C). 1. If {Ecc,Occ} ∩ Γ 6= φ, Γ may be easily seen to be contradictory. In such a case Γ is said to be plainly contradictory. 2. Γ d ` Oab i Oab ∈ Γ. 3. If Γ d ` Ecc then Ecc ∈ Γ or c ∈ CΓ. 4. If Γ is not plainly contradictory, then Γ is contradictory i there are σ, σ ∈ (Γd ∩BN(CΓ)). 5. Γd ∩BN(CΓ) = (Γ ∩BN(CΓ))d ∩BN(CΓ). 6. In a dierent context, Glasho (2005) presents an algorithm which may be regarded as a prelude to the one given below. Roughly speaking, it amounts -in our terminologyto: For a nite Γ(⊆ BN(C)), Γd ∩ BN(CΓ) may be obtained from Γ in nitely many steps. THEOREM 3.2. There is a polynomial (of degree 8) time algorithm to decide for any essentially nite Γ(⊆ BN(C)) which is not plainly contradictory whether it is contradictory, and to assign a Leibniz model to it if it is not. Proof. Let Γ satisfy the conditions of the theorem, then BN(CΓ) is nite. Put Γ′ = Γ ∩BN(CΓ) and ∆ = Γ′d ∩BN(CΓ). The input of the algorithm is Γ′ structured as a list < γi >i∈n where n = |Γ′|, and for every i ∈ n, γi =< γij >j∈3 where γio ∈ {A,E, I, O} and 26 γi1, γi2 ∈ CΓ. CΓ may be obtained from Γ′ or supplied as a secondary input. |CΓ| ≤ 2n and |BN(CΓ)| ≤ 16n2. The next step is to extract for each Y ∈ {A,E, I, O}, Γ′Y (the set of all elements of Γ′ starting with Y ) which may be done through a simple scanning procedure in a linear time. Then construct ∆Y (with the obvious meaning) for each Y ∈ {A,E, I, O}. Notice that ∆O = Γ ′ O and ∆A is needed to construct each of ∆E and ∆I . To construct ∆A start with the list Γ ′ A. At most 16n 4 comparisons are needed to determine all the possible applicabilities of Barbara. And for each possible applicability at most 4n2 comparisons are needed to check whether the consequent is already there. If not, append it. It is needed to repeat this process at most 4n2 times to cover all the required applications of Barbara. In addition, for each c ∈ CΓ at most 4n2 comparisons are needed to check whether Acc is listed; if not, append it. It is easy to see that this completes the construction of ∆A. By simple variations on the above procedure ∆E may be constructed. Constructing ∆I is much simpler. Γ is contradictory i σ, σ ∈ ∆ for some σ, which needs at most 32n4 comparisons to check. If Γ is consistent assign to it a Leibniz model along the lines of the proof of theorem 2.13 (in the appendix a polynomial (in n of degree 6) time algorithm will be presented to generate the rst n primes). The total running time is bounded above by a polynomial (in n) of degree 8.  4. Basic equivalence of the four formalizations. Let Γ ∪ {σ} ⊆ BN(C), let h and H be bijective interpretations of BN(C) in BS(J) and BM(P) respectively, and let < Γ′, σ′, T >∈ {< Γ, σ,DF (C) >, < Γh, σh, SF (J) >, < ΓH , σH ,MF (P) >}. THEOREM 4.1. 1. Γ is consistent i Γ′ ∪ T is. 2. Γ `g σ i Γ′ ∪ T ` σ′. Proof. From proposition 1.13, if of (1) and only if of (2) follow. 27 The other two directions for < Γ′, σ′, T >=< Γh, σh, SF (J) > follow from the corresponding directions for < Γ′, σ′, T >=< Γ, σ,DF (C) >, this is a consequence of proposition 1.14. So it remains to prove these two other directions for < Γ′, σ′, T >∈ {< Γ, σ,DF (C) >, < ΓH , σH ,MF (P) >}. Only if of (1): Assume Γ is consistent. Let ∆ be a nite subset of Γ, then by theorem 2.13 it has a Leibniz model, B say. B is a model of ∆∪DF (C); from this the consistency of Γ ∪ DF (C) follows. Since ∆ may be assumed to be the inverse image of some nite ∆′ ⊆ ΓH , then by subsection 2.6.3, B induces a model of ∆′ ∪MF (P). From this the consistency of ΓH ∪MF (P) follows. If of (2): Let Γ′ ∪ T ` σ′, then Γ′ ∪ {σ′} ∪ T is inconsistent. By part (1), Γ∪{σ} is inconsistent, hence Γ `g σ.  REMARKS 4.2. 1. As far as the basic sentences are concerned, the four formalizations are equivalent in the sense expressed by part 2 of the above theorem; so it may be said, for brevity, that they are basically equivalent. 2. In the above theorem, the only if direction of (1) and the if direction of (2) may be directly proved for < Γ′, σ′, T >=< Γh, σh, SF (J) >. 5. Venn soundness and completeness. Let Γ ∪ {σ} ⊆ BN(C). DEFINITION 5.1. Γ  V σ i Γ  B σ for every Venn model B. THEOREM 5.2. (Venn soundness and completeness). General deduction is sound and complete with respect to the class of Venn models. That is Γ g ` σ i Γ  V σ. Proof. Every Venn model is a DF (C) model (subsection 2.5), then a g-model (2 of remarks 2.11). This guarantees soundness. To prove completeness, let Γ g 0 σ, then Γ∪{σ} is consistent then, by theorem 4.1, (Γ∪ {σ})H ∪MF (P) is consistent where H is a bijective interpretation of BN(C) in BM(P), for some appropriate P. By well known results in rst order logic, (Γ ∪ {σ})H ∪MF (P) has a model. By subsection 2.6.3 and 2 of remarks 2.11, Γ∪{σ} has a Venn model, hence Γ 2 V σ.  28 Alternatively theorem 6.3 below may be made use of to directly show that Γ ∪ {σ} has a Venn model. REMARKS 5.3. 1. In view of subsection 2.5, the above theorem entails that general deduction is sound and complete with respect to each of the classes of order, partial order, and concrete order models. 2. Direct ways to Venn models on one hand, and to order and partial order models on the other hand, will be presented in sections 6 and 9 respectively. 3. For the Venn soundness and completeness of ukasiewicz's system, Shepherdson (1956) may be consulted. 6. Direct way to Venn models. Let Γ ⊆ BN(C), put D = {< a, b >∈ C × C : {Iab, Iba} ∩ Γd 6= φ}, B = ℘(D)− {φ}. Dene the function μ from C to B by: μ(c) = {< a, b >∈ D : {Aac,Abc} ∩ Γd 6= φ}. Then < B, μ > is a Venn model (which is a concrete order model), denote it by BΓ. LEMMA 6.1. For every c, c′ ∈ C, the following are equivalent: 1. Acc′ ∈ Γd, 2. μ(c) ⊆ μ(c′) (which is equivalent to BΓ  Acc′), 3. < c, c >∈ μ(c′). Proof. Straightforward.  LEMMA 6.2. Let c, c′ ∈ C, consider: 1. < c, c′ >∈ D, 2. < c, c′ >∈ μ(c) ∩ μ(c′), 3. μ(c) ∩ μ(c′) 6= φ (which is equivalent to BΓ  Icc′, Ic′c), 29 4. {Ecc′, Ec′c} ∩ Γd = φ, then 1 is equivalent to 2 which implies 3 which, for consistent Γ, implies 4. Proof. The rst two parts are easy to see. For the last part assume μ(c) ∩ μ(c′) 6= φ 6= {Ecc′, Ec′c} ∩ Γd, then Ecc′ ∈ Γd and there are a, b ∈ C such that (Iab ∈ Γd or Iba ∈ Γd), (Aac ∈ Γd or Abc ∈ Γd) and (Aac′ ∈ Γd or Abc′ ∈ Γd). So there are eight cases to consider. We deal only with the case Iab, Aac, Abc′ ∈ Γd; the other cases are similar or easier. In this case Aac,Ecc′ ∈ Γd, then Eac′ ∈ Γd, then Ec′a ∈ Γd, but Abc′ ∈ Γd, then Eba ∈ Γd, then Eab ∈ Γd which contradicts that Iab ∈ Γd; from this and the consistency of Γ the result follows.  THEOREM 6.3. (Existence of Venn models). Let Γ ⊆ BN(C) be consistent, then BΓ is a Venn model (which is a concrete order model) of Γ. Proof. Let c, c′ ∈ C. By lemma 6.1, Acc′ ∈ Γd i BΓ  Acc′. From this it follows that for Y ∈ {A,O}, BΓ  Y cc′ if Y cc′ ∈ Γd. Moreover, if Icc′ ∈ Γd then < c, c′ >∈ D then, by lemma 6.2, BΓ  Icc′. Finally, if Ecc′ ∈ Γd then, by lemma 6.2, μ(c) ∩ μ(c′) = φ then BΓ  Ecc′. Lemma 6.1 syntactically characterizes {Acc′ ∈ BN(C) : BΓ  Acc′}, hence it syntactically characterizes {Occ′ ∈ BN(C) : BΓ  Occ′}. The following syntactical characterization of {Icc′ ∈ BN(C) : BΓ  Icc′}, hence of {Ecc′ ∈ BN(C) : BΓ  Ecc′}, BΓ  Icc′ i Γ d′ ` Icc′ is an immediate consequence of lemma 8.3 below; the denition of  d′ ` may be found at the beginning of section 8 below. Slightly modifying the above construction, light may be shed on the role played by the Venn models among the models of DF (C). THEOREM 6.4. For every DF (C) model B =< B,R1, R2, μ > there is a Venn model B′ =< B′, μ′ > and surjection h : B → B′ such that: 1. μ′ = hμ and for every b1, b2 ∈ B: b1R1b2 i h(b1) ⊆ h(b2) , b1R2b2 i h(b1) ∩ h(b2) 6= φ, 30 2. B and B′ are basically equivalent, 3. h is an isomorphism i R1 is antisymmetric. Proof. Put: h : B → ℘(℘(B)) h(b) = {{b1, b2} ∈ ℘(B) : (b1R2b2 or b2R2b1) and (b1R1b or b2R1b)}, B′ = h(B) , μ′ = hμ. The rest of the proof is easy.  7. Variations on NF (C). As was promised in section 1, we follow in subsection 7.1 the long standing tradition of not permitting the subject and the predicate of a categorical sentence to be the same. The resulting formalization, WF (C), and its relationship to NF (C) are discussed. In subsection 7.2 the standpoint that Acc′ requires that all c are c′ but not vice versa, will be considered. 7.1. Weak natural deduction formalization of AAS. The alphabet of the logical system WF (C), the weak natural deduction formalization of AAS, is the same as the alphabet of NF (C). The set W (C) of sentences of WF (C) is dened as follows: W (C) = S(C)− {Y cc : Y ∈ {A,E, I, O} and c ∈ C}. In accordance with subsection 1.6, the set BW (C) of basic sentences of WF (C) is W (C) itself. The rules of inference of WF (C) are those of NF (C) after dropping the rst one ( Aaa ). The weak direct and general deduction relations are respectively denoted by  wd `  and  wg `  and are dened along the lines of denitions 1.8 and 1.9 respectively. The denition of the other notions introduced in the theory of NF (C) may be modied in the obvious way to render the corresponding denitions for the theory of WF (C). The theory of WF (C) may be obtained from that of NF (C) by making the obvious modications. The key observations are the following, where 31 Γ ∪ {σ} ⊆ BW (C). PROPOSITION 7.1. Γ wd ` σ i Γ d ` σ. Proof. The only if direction is obvious. To prove the other direction let < ρi >i∈n be a d-deduction of σ from Γ. We show by induction that for every i ∈ n, Γ wd ` ρi if ρi ∈ BW (C). Distinguish between four cases: 1. ρi = Occ ′, then ρi ∈ Γ, then Γ wd ` ρi. 2. ρi = Acc ′, then the result follows by the induction hypothesis. 3. ρi = Icc ′, then Icc′ ∈ Γ or ρj = Ac′c for some j < i. From this and part 2 the result follows by the induction hypothesis. 4. σ = Ecc′, then the result follows by the induction hypothesis noting that if Ecc′ is obtained via applying Acc ′,Ec′c′ Ecc′ then the rst occurrence of Ec′c′ in the deduction must be obtained via Ac ′c′′,Ec′′c′ Ec′c′ for some c′′ 6= c′. By part 2 and the induction hypothesis Γ wd ` Acc′, Ac′c′′, Ec′′c′, hence Γ wd ` Ecc′. PROPOSITION 7.2. Γ is wd-consistent i Γ is d-consistent (hence Γ wg ` σ i Γ g ` σ). Proof. The if direction easily follows from proposition 7.1. To prove the other direction assume that Γ is d-inconsistent, then Γ d ` Y cc′, Ŷ cc′ for some Y ∈ {A,E, I, O} and some c, c′ ∈ C. If c 6= c′ the result follows by the previous proposition. Else, distinguish between two cases: 1. Occ ∈ {Y cc, Ŷ cc}, then Occ ∈ Γ which is not permitted. 2. Ecc ∈ {Y cc, Ŷ cc}. In this case there is a d-deduction of Ecc from Γ. The rule made use of to justify the rst occurrence of Ecc in this deduction must be Acc ′′,Ec′′c Ecc for some c′′ 6= c. By the previous proposition 32 Γ wd ` Ic′′c, Ec′′c, hence Γ is wd-inconsistent.  COROLLARY 7.3. Γ is wd-consistent i Γ is d-consistent i Γ is g-consistent i Γ is wg-consistent. Proof. Γ is wd-consistent only if Γ is d-consistent only if Γ is g-consistent only if Γ is wg-consistent only if Γ is wd-consistent.  REMARK 7.4. From propositions 7.1 and 7.2 it follows that the results concerning Leibniz soundness and completeness (subsection 2.7.2) and Venn and order soundness and completeness (section 5) apply to WF (C) after replacing d, g and BN(C) by wd,wg and BW (C) respectively. 7.2. Proper natural deduction formalization of AAS. AAS may be interpreted to require Acc′ to hold i all c are c′ but not vice versa. That is, extensionally, the denotation of c is required to be a proper subclass of the denotation of c′. To satisfy this requirement introduce the logical system PF (C), the proper natural deduction formalization of AAS, based on the same language as the system NF (C). So the set P (C) of sentences of PF (C) is the same as S(C). In accordance with subsection 1.6 the set BP (C) of basic sentences of PF (C) is P (C) itself. For Occ′ to remain to be the contradictory of Acc′, it must be interpreted as some c are not c′ or (all c are c′ and vice versa). The rules of inference of PF (C) are to be obtained from those of NF (C) by dropping the rst one and augmenting the remaining ones by Icc (I-Id) and Occ (O-Id). The proper direct and general deduction relations are respectively denoted by  pd ` and  pg ` and are dened along the lines of denitions 1.8 and 1.9 respectively. The denitions of the other notions introduced in the theory of NF (C) may be modied in the obvious way to render the corresponding denitions for the theory of PF (C). PROPOSITION 7.5. For Γ ∪ {σ} ⊆ BW (C): 1. Γ pd ` σ i Γ wd ` σ (i Γ d ` σ). 33 2. If Γ is pd-consistent then it is wd-consistent (equivalently d-consistent), but not always vice versa. 3. If Γ wg ` σ (equivalently Γ g ` σ) then Γ pg ` σ, but not always vice versa. Proof. 1. The proof of part 1 is similar to that of proposition 7.1. 2. That Γ is wd-consistent if it is pd-consistent easily follows from part 1. To see that the other direction does not always hold consider {Acc′, Ac′c} for some c, c′ ∈ C such that c 6= c′; this proves part 2. 3. Part 3 is a direct consequence of part 2.  PROPOSITION 7.6. ∆(⊆ BP (C)) is pd-consistent i it is pg-consistent. Proof. Along the lines of the proof of part 4 of proposition 1.11.  Order models, Leibniz models, and Venn models are not e(∈ {pd, pg})models, so it does not make sense to ask whether e is sound or complete with respect to any of these classes. However, with some modications, to be shown below, everything goes as expected. Let B =< B,A∗, E∗, I∗, O∗, μ > be a d-model of Γ(⊆ BW (C)) such that μ is injective on CΓ and A ∗μ is antisymmetric, and let Ap, Op be subsets of B ×B such that A∗μ − lμ(C) ⊆ Apμ ⊆ A∗μ and O∗μ ∪ lμ(C) ⊆ Opμ. Put: Bp =< B,Ap, E∗, I∗, Op, μ > . PROPOSITION 7.7. Bp is a pd-model of Γ. Proof. Assume B  Γ. To show that Bp  Γ, let γ ∈ Γ then γ = Y cc′ for some Y ∈ {A,E, I, O} and some c, c′ ∈ C such that c 6= c′, hence μ(c) 6= μ(c′). If Y = A then < μ(c), μ(c′) >∈ A∗μ− lμ(C) ⊆ Apμ. The other cases are obvious. To show that Bp is a pd-model, assume Bp  Acc′, Ac′c′′. If not Bp  Acc′′ then μ(c) = μ(c′′), then < μ(c), μ(c′) >,< μ(c′), μ(c) >∈ A∗μ, then μ(c′) = μ(c) = μ(c′′) which is absurd. So Barbara is valid. The other rules 34 are easier to deal with.  Accordingly, it is legitimate to adopt in the sequel the following modications: Ap = A∗ − lμ(C) , Op = O∗ ∪ lμ(C). THEOREM 7.8. Let ∆(⊆ BP (C)) be pd-consistent, then it has a modied Venn model which is a modied c.o.m and which is also a pg-model. If, in addition, ∆ is essentially nite then it has also a modied Leibniz model which is a pg-model. Proof. Put Γ = ∆∩BW (C), then Γ is pd-consistent, hence it is d-consistent, hence BΓ is a d-model of Γ in which A∗ is antisymmetric. To show that μ is injective let c, c′ ∈ C be such that c 6= c′ and μ(c) = μ(c′), then Γ d ` Acc′, Ac′c, then Γ wd ` Acc′, Ac′c, then Γ pd ` Acc′, Ac′c, then Γ pd ` Acc which contradicts that Γ is pd-consistent. Therefore BΓp is a modied Venn model (which is also a modied c.o.m) of Γ. By proposition 7.7 it is a pd-model of Γ, from which it may be easily seen that it is a pg-model of ∆. The proof of the additional result in case ∆ is essentially nite is almost the same. The only major dierence is that μ may not be injective. But its restriction to C∆ is injective, which is sucient for our purpose.  REMARK 7.9. The last theorem shows that remark 7.4 applies to PF (C) after making the obvious modications. 8. Direct completion of direct deduction. In this section the ve rules of inference given in denition 1.7 are augmented by ve more rules, in order that Γg may be directly obtained from Γ in case Γ is consistent (cf. Glasho (2005) where related problems are dealt with by brute force via a computer program). The additional ve rules are: 5. Iab Iba (Ic) 6. Iab,Abc Iac (Darii) 7. Iab,Ebc Oac (Ferio) 8. Oab,Acb Oac (Baroco) 35 9. Aba,Obc Oac (Bocardo). Taking the ten rules of inference into consideration, the d′-deduction relation  d′ ` may be dened along the lines of the denition of  d `. Likewise, all other denitions involving d may be modied in an obvious way to give corresponding denitions involving d′. PROPOSITION 8.1. 1. Γd ′ = {σ ∈ S(C) : Γ d′ ` σ}. 2. CΓd′ = CΓ. Proof. Along the lines of the proofs of the corresponding results for d: Part 1 of proposition 1.11 and lemma 2.12, respectively.  The next denition and parts 1,2 of the next lemma are essentially due to Smith (1983). DEFINITION 8.2. Let a, b, a′, b′ ∈ C. An a-b chain is a sequence < ci >i∈n∈ nC for some n ∈ N+, such that co = a and cn−1 = b. This chain is said to be a Γ-chain, or a chain in Γ, if {Acici+1 : i ∈ n − 1} ⊆ Γ; it is said to be an < a′, b′ > chain if there is i ∈ n− 1 such that ci = a′ and ci+1 = b′. LEMMA 8.3. For a, b ∈ C and Γ ⊆ BN(C): 1. Γ d ` Aab i Γ d′ ` Aab i there is an a-b chain in Γ. 2. Γ d ` Eab i Γ d′ ` Eab i there is Ea′b′ ∈ BN(C) such that {Ea′b′, Eb′a′} ∩ Γ 6= φ and Γ d ` Aaa′, Abb′. 3. Γ d ` Iab i Iab ∈ Γ or Γ d ` Aba. 3′. Γ d′ ` Iab i for some a′, b′ ∈ C, Γ d′ ` Aa′a,Aa′b or {Ia′b′, Ib′a′} ∩ Γ 6= φ and Γ d′ ` Aa′a,Ab′b. 36 4. Γ d ` Oab i Oab ∈ Γ. 4′. Γ d′ ` Oab i for some a′, b′ ∈ C, Γ d′ ` Ia′a,Ea′b or Oa′b′ ∈ Γ and Γ d′ ` Aa′a,Abb′. Proof. 1. If is easy to show that the rst statement implies the second. By induction it may be shown that the second statement implies the third. Again by induction it may be shown that the third statement implies the rst. 2. It is easy to show that the rst statement implies the second and that the third implies the rst. By induction it may be shown that the second statement implies the third. Parts 3 and 4 are easy. In each of the parts 3′ and 4′ one direction is easy, the other may be shown by induction.  PROPOSITION 8.4. For Γ ∪ {σ} ⊆ BN(C): 1. If Γ d ` σ then Γ d′ ` σ. 2. If Γ d′ ` σ then Γ g ` σ. 3. Γ is g-consistent i Γ is d′-consistent i Γ is d-consistent. (So for e ∈ {d, d′, g} the prex e- may be deleted from e-consistent, e-inconsistent and e-contradictory). Proof. Part 1 is obvious, and part 3 is an easy consequence of parts 1 and 2 above and part 4 of proposition 1.11. Part 2 is immediate if Γ is d-inconsistent. To complete the proof assume that Γ is d-consistent and proceed by course of values induction. Let Γ ` σ and let < ρi >i∈n be a d ′-deduction of σ from Γ. If the annotation of ρn−1 (= σ) is that it belongs to Γ or that it is the consequent of a d-rule whose premises are previous sentences, the result easily follows. It remains to assume that the annotation of ρn−1 is that it is the conse37 quent of a new rule. The completion of the proof depends on the specic rule in use. Following is a proof in the case of Darii. The other cases are similar or easier. Let ρn−1 = Iac and let its annotation be that it follows from Iab, Abc by Darii. By the induction hypothesis Γ g ` Iab, Abc. By part 3 of proposition 1.11, Γ d ` Abc, and by the denition of g `, there is η ∈ BN(C) such that Γ, Eab d ` η, η. Since Γ is d-consistent then, in view of parts 1 and 4 of lemma 8.3, there are c′, c′′ ∈ C such that {η, η} = {Ic′c′′, Ec′c′′}. By lemma 8.3, Γ d ` Ic′c′′ and there is Ea′b′ ∈ BN(C) such that {Ea′b′, Eb′a′} ∩ (Γ ∪ {Eab}) 6= φ and Γ ∪ {Eab} d ` Ac′a′, Ac′′b′, hence Γ d ` Ac′a′, Ac′′b′. In view of the d-consistency of Γ, lemma 8.3 implies that Eab ∈ {Ea′b′, Eb′a′}. Let Eab = Ea′b′ (the other case is similar), then Γ d ` Ac′a,Ac′′b. But Γ d ` Abc, then Γ, Eac d ` Ec′c′′, hence Γ g ` Iac.  In view of the g-deduction completeness with respect to the class of Venn models, part 2 of the above proposition is an immediate consequence of: PROPOSITION 8.5. The d′-deduction is sound with respect to the class of Venn models (hence with respect to the class of order models). Proof. Routine.  REMARK 8.6. The converse of part 2 of proposition 8.4 does not always hold. For if Γ is inconsistent then CΓg = C, while it is easy to nd an inconsistent Γ such that CΓd′ = CΓ 6= C. Also the weaker statement: Γd ′ ∩ BN(CΓ) = Γg ∩BN(CΓ), does not always hold. A counter example is Γ = {Aab,Oab}. The consistency of Γ solves the problem as the following theorem shows (cf. Smith 1983). THEOREM 8.7. For consistent Γ, Γd ′ = Γg. Proof. The inclusion of Γd ′ in Γg is guaranteed by part 2 of proposition 8.4. For the other direction assume that Γ is consistent and Γ g ` σ. If σ is universal the result follows by part 3 of proposition 1.11 and part 1 of proposition 38 8.4. So it remains to deal with the particulars. The consistency of Γ restricts what to be considered to the following: Case 1. σ is Iab for some a, b ∈ C. By the method made use of in the proof of part 2 proposition 8.4, consideration may be restricted to the following subcase only. There are c, c′ ∈ C such that Γ d ` Icc′, Aca,Ac′b, which implies that Γ d′ ` Iab. Case 2. σ is Oab for some a, b ∈ C. In this case Γ, Aab d ` ρ, ρ for some ρ ∈ BN(C). Distinguish between two subcases. Subcase 2.1. For some c, c′ ∈ C, {ρ, ρ} = {Acc′, Occ′}. Then Γ d ` Aca,Abc′, Occ′ which implies that Γ d′ ` Oab. Subcase 2.2. For some c, c′ ∈ C, {ρ, ρ} = {Ecc′, Icc′}. This subcase may be divided into the following three subsubcases. Subsubcase 2.2.1. Γ d ` Ecc′,Γ d 0 Icc′. Then Γ d ` Ac′a,Abc, Ecc′ which implies that Γ d′ ` Oab. Subsubcase 2.2.2. Γ d 0 Ecc′ and Γ d ` Icc′. Then there is Ea′b′ ∈ BN(C) such that Γ d ` Ea′b′ and Γ, Aab d ` Aca′, Ac′b′; while Γ d 0 Aca′ or Γ d 0 Ac′b′, but -by the consistency of Γnot both. This subsubcase may be further divided into two subsubsubcases. Subsubsubcase 2.2.2.1. Γ d 0 Aca′ but Γ d ` Ac′b′. Then Γ d ` Icc′, Ea′b′, Ac′b′, Aca,Aba′ from which Γ d′ ` Oab follows. Subsubsubcase 2.2.2.2. Γ d ` Aca′ but Γ d 0 Ac′b′. Similar to subsubsubcase 2.2.2.1. Subsubcase 2.2.3. Γ d 0 Ecc′ and Γ d 0 Icc′. Then there is Ea′b′ ∈ BN(C) such that Γ d ` Ac′a,Abc, Ea′b′ and Γ, Aab d ` Aca′, Ac′b′, while Γ d 0 Aca′ or Γ d 0 Ac′b′. But the consistency of Γ implies that Γ d ` Ac′b′, then Γ d 0 Aca′, then Γ d ` Aca,Aba′. In particular, Γ d ` Ac′a,Ac′b′, Aba′, Ea′b′, hence the result.  39 REMARK 8.8. In a dierent context, Smith (1983): 1. Excluded subcase 2.1 under the claim that it is impossible that Γ d ` Occ′. 2. Subsubcase 2.2.3 was deemed to be impossible. 9. Models of NF (C) revisited. An NF (C)-structure B is said to be a d′-model if for every Γ ∪ {σ} ⊆ BN(C), Γ B σ whenever Γ d′ ` σ. An immediate consequence of this denition is: PROPOSITION 9.1. An NF (C)-structure B =< B,A∗, E∗, I∗, O∗, μ > is a d′-model i it is a d-model (hence satisfying conditions 1-4 of proposition 2.3) and: 5. (I∗μ|A∗μ) ⊆ I∗μ ⊆ ` I∗μ. 6. (I∗μ|E∗μ)∪(O∗μ| ` A∗μ)∪( ` A∗μ |O∗μ) ⊆ O∗μ.  Along the lines of the proofs of lemma 2.5, theorem 2.6 and theorem 2.10, the following may be proved: THEOREM 9.2. For every Γ ∪ {σ} ⊆ BN(C): 1. BΓd′ is a d ′-model (of Γd ′ , hence of Γ). 2. d′-deduction is sound and complete with respect to the class of d′models. That is Γ d′ ` σ i Γ d′  σ. 3. Γ d′  σ i Γ1 d′  σ for some nite Γ1 ⊆ Γ. (This is called d′-compactness).  REMARK 9.3. All remarks given in remarks and denitions 2.11 hold with d′ replacing d. All proofs of the original versions essentially go through; the only exception is the rst remark, whose modied version may be proved by part 2 of proposition 8.4. THEOREM 9.4. An NF (C)-structure B(=< B,A∗, E∗, I∗, O∗, μ >) is a gmodel i it is a d′-model and: 40 1. A∗μ ∩O∗μ = φ = E∗μ ∩ I∗μ, or 2. A∗μ = E∗μ = I∗μ = O∗μ = μ(C)× μ(C). Proof. Only if: By part 2 of proposition 8.4 and an obvious generalization of part 3 of remarks and denitions 2.11. If: Every NF (C)-structure which satises condition 2 is a g-model. So, assume that B is a d′-model which satises condition 1. To see that it is a g-model, let Γ ∪ {σ} ⊆ BN(C), Γ g ` σ and B  Γ. By remark 9.3, Γ is consistent, hence by theorem 8.7, Γ d′ ` σ, hence B  σ.  Theorem 9.4 fully characterizes the class of g-models, as was promised after the proof of theorem 2.10. DEFINITIONS and remarks 9.5. 1. For an NF (C)-structure B and a relation symbol W ∈ {A,E, I, O}, dene BtWB (the basic W -theory of B), Bt+B (the basic positive theory of B), Bt−B (the basic negative theory of B) and BtB (the basic theory of B) as follows: BtWB = {Wab ∈ BN(C) : B  Wab}. Bt+B = BtAB ∪BtIB. Bt−B = BtEB ∪BtOB. BtB = Bt+B ∪Bt−B. So two NF (C)-structures are B-equivalent i they have the same basic theory. For i ∈ 2 let Bi(=< Bi, Ai, Ei, Ii, Oi, μi >) be an NF (C)-structure. 2. Bo is said to be a substructure of B1 and B1 is said to be a superstructure of Bo if Bo ⊆ B1, μo = μ1 and for every W ∈ {A,E, I, O}, 41 Wo = W1 ∩ (Bo × Bo). If, morever, Bo = Range μ1 (= Range μo), Bo is said to be a core substructure of B1. Obviously each NF (C)-structure has a unique core substructure, to be called its core substructure. Bo is a core substructure of some NF (C)-structure i it is the core substructure of itself i Bo = Range μo. In this case Bo is said to be a core structure. Obviously every canonical structure is a core structure. Bo, B1 have the same core substructure i μo = μ1 and BtBo = BtB1. 3. IfBo is a substructure ofB1 then they have the same core substructure and the three structures have the same basic theory. Hence for e ∈ {d, d′, g} if one of them is an e-model, so also are the other two. In this case Bo is said to be an e-submodel of B1, and B1 is said to be an e-supermodel of Bo; and the core substructure is said also to be the core e-submodel. If a core structure is an e-model, it is said to be a core e-model. 4. Bo is said to be a positive semisubstructure of B1 and B1 is said to be a positive semisuperstructure of Bo if Bo ⊆ B1, μo = μ1 and: Wo = W1 ∩ (Bo ×Bo) for every W ∈ {A, I}, Wo ⊆ W1 ∩ (Bo ×Bo) for every W ∈ {E,O}. In this case Bt+Bo = Bt +B1 and Bt −Bo ⊆ Bt−B1. For each e ∈ {d, d′, g} if, in addition, Bo and B1 are both e-models, it is said also that Bo is a positive e-semisubmodel of B1 and B1 is a positive e-semisupermodel of Bo. THEOREM 9.6. For each e ∈ {d′, g} if fBo (dened as above) is a consistent core e-model then there is an order model B1(=< B1, A1, μ1 >) such that: 1. B1 is a positive e-semisupermodel of Bo. 2. If Ao is a partial ordering, then so also is A1. 3. If Bo is complete, then it is the core e-submodel of B1. For e = d, the above holds after weakening part 1 to become: 1′. Bo ⊆ B1, μo = μ1, BtBo ⊆ BtB1, and Ao = A1 ∩ (Bo × Bo); hence BtABo = Bt AB1. Proof. Let e ∈ {d, d′, g} and let Bo be a consistent core e-model. Put: 42 B′ = {{ao, a1} ⊆ Bo :< ao, a1 >∈ Io or < a1, ao >∈ Io, and {ao, a1} has no Ao-lower bound}, B1 = Bo ∪B′ (Bo, B′ may be assumed disjoint), A1 = Ao∪ lB′∪{< {ao, a1}, a2 >∈ B′×Bo :< ao, a2 >∈ Ao or < a1, a2 >∈ Ao}, μ1 = μo A1 is reexive on B1 since Ao is reexive on Bo. To prove the transitivity of A1, let < bo, b1 >,< b1, b2 >∈ A1. If < bo, b1 > or < b1, b2 > belongs to lB′ then < bo, b2 >∈ A1, else < b1, b2 >∈ Ao. If < bo, b1 >∈ Ao then < bo, b2 >∈ A1. It remains to consider the case where bo = {ao, a1} for some {ao, a1} ∈ B′ such that < ao, b1 >∈ Ao or < a1, b1 >∈ Ao, in both cases < bo, b2 >∈ A1. So A1 is transitive. Hence < B1, A1, μ1 > is an order model, which is to be denoted by B1. To prove part 2 it suces to notice that if < bo, b1 >,< b1, bo >∈ A1 then they both belong to Ao or both belong to lB′ . To prove parts 1, 1′ notice that Bo ⊆ B1 and, by the disjointness of Bo, B′, Ao = A1 ∩ (Bo ×Bo). Let < ao, a1 >∈ Io. If {ao, a1} has an Ao-lower bound then it is an A1-lower bound, else the element {ao, a1} ∈ B1 is an A1-lower bound of the subset {ao, a1} ⊆ B1. In both cases < ao, a1 >∈ I1, hence Io ⊆ I1 ∩ (Bo ×Bo). At this point the proof forks into two branches: (i) Assume e ∈ {d′, g} and let < ao, a1 >∈ I1 ∩ (Bo × Bo). To show that < ao, a1 >∈ Io several cases have to be considered, following is one of them, the others are similar or easier. There is < a2, a3 >∈ Io such that < a2, ao >,< a3, a1 >∈ Ao. Since Bo = Range μo then, by theorem 9.4 and part 5 of proposition 9.1, < ao, a1 >∈ Io. So I1 ∩ (Bo ×Bo) ⊆ Io. Hence Io = I1 ∩ (Bo ×Bo). That Eo ⊆ E1 ∩ (Bo×Bo) and Oo ⊆ O1 ∩ (Bo×Bo) is guaranteed by the consistency of Bo. This completes the proof of 1. (ii) The other branch is e = d. To show that Eo ⊆ E1 assume that there is < ao, a1 >∈ (Eo −E1), then < ao, a1 >∈ I1, then {ao, a1} has an A1-lower bound. To show that this is absurd, several cases have to be considered; 43 following is one of them, the others are easier or similar. There is < a2, a3 >∈ Io such that < a3, ao >,< a2, a1 >∈ Ao. Since Bo = Range μo then, by parts 3, 4 of proposition 2.3, < a2, a3 >∈ Eo which contradicts the consistency of Bo. That Oo ⊆ O1 is guaranteed by the consistency of Bo, since Ao = A1 ∩ (Bo ×Bo). This completes the proof of 1′ and ends the forkation. For e ∈ {d, d′, g}, if Bo is complete then ⊆ may be replaced by = at the appropriate places, which proves part 3.  Taking the relationship between the e-models (e ∈ {d, d′, g}) and their respective core e-submodels into consideration, a weaker result, which holds for a wider class of e-models, immediately follows: COROLLARY 9.7. For e ∈ {d, d′, g}, if B is an e-model whose core esubmodel is consistent, then there is an order model B′ such that BtB ⊆ BtB′. Moreover, Bt+B = Bt+B′ if e ∈ {d′, g}, BtAB = BtAB′ if e = d.  In view of the last part of subsection 2.5, the above corollary may be immediately strengthened as follows: COROLLARY 9.8. In the above corollary an order model may be replaced by a partial order model which is a c.o.m and a Venn model at the same time.  Part 4 of theorem 1.12 may be extended to the case e = d′, to get a result similar to that obtained there for the case e = g; the result obtained (there) for the case e = d is weaker. Call the collection of these three results syntactical congruence. Syntactical congruence together with the denitions of core e-models (e ∈ {d, d′, g}) yield semantical congruence as formulated by parts 1 and 2 of the next theorem. Part 3 of the same theorem (whose proof is straightforward) strengthens the conclusion of part 2, under some additional condition. 44 Alternatively, semantical congruence may be directly proved by the characterizations of e-models (e ∈ {d, d′, g}) given in propositions 2.3 and 9.1 and theorem 9.4. THEOREM 9.9. Let e ∈ {d, d′, g} and let B =< B,A∗, E∗, I∗, O∗, μ > be a core e-model (consistent or not). Put ∼ = A∗∩ ` A∗, then: 1. ∼ is a congruence relation on < B,A∗, E∗, μ > and A∗/ ∼ is a partial ordering on B/ ∼. Moreover, for e ∈ {d′, g}: 2. ∼ is a congruence relation on B. The mapping b 7−→ b/ ∼ is an epimorphism from B onto B/ ∼. Hence B/ ∼ is a core e-model which is basically equivalent to B. 3. If B is, in addition, an order model, then B/ ∼ is also a partial order model.  For e ∈ {d′, g}, semantical congruence makes it possible to replace Bo in theorem 9.6 by Bo/ ∼. This provides, for e ∈ {d′, g}, an alternative proof of a weaker form of corollary 9.8, where the partial order model may be neither concrete nor Venn. The corresponding weaker result for the case e = d may likewise be obtained, but the alternative proof is a bit more involved. REMARKS and denitions 9.10. 1. Theorem 9.6 (or corollary 9.7) and corollary 9.8 (or its weaker forms) provide, respectively, direct ways to order models and partial order models for consistent Γ(⊆ BN(C)). Simply in each of them let the core e-model be the canonical structure BΓe(e ∈ {d, d′, g}). In the case of corollary 9.8 the partial order model may be required to be a concrete order model and a Venn model at the same time. 2. Let e ∈ {d, d′, g} and let C be a class of NF (C)-structures, then: 1. e is said to be C-strongly semantically complete if for every Γ ⊆ BN(C) there is B ∈ C such that BtB = Γe. 2. e is said to be C-syntactically complete if for every Γ∪{σ} ⊆ BN(C), Γ e ` σ whenever Γ C σ. 45 3. e is said to be C-consistently syntactically complete if for every e-consistent Γ ⊆ BN(C) and every σ ∈ BN(C), Γ e ` σ whenever Γ C σ. 4. e is said to be C-consistently semantically complete if every econsistent Γ ⊆ BN(C) has a model in C. For i ∈ {1, 2, 3}, the condition given in clause i implies the condition given in clause i+ 1. 3. Put: Or = the class of all order models, Po = the class of all partial order models, Le = the set of all Leibniz models, Co = the class of all concrete order models, V e = the class of all Venn models. And for e ∈ {d, d′, g} put: Be = {BΓe : Γ ⊆ BN(C)}. Also put: M = {Or, Po, Le, Co, V e} ∪ {Be : e ∈ {d, d′, g}}. 4. Le ∪ Co ⊆ Po ⊆ Or, Bg ⊆ Bd′ ⊆ Bd. 5. Every element of ⋃ M is a d-model. Every element of ( ⋃ M −Bd) ∪Bd′ is a d′-model. Every element of ( ⋃ M −Bd) ∪Bg is a g-model. 6. For e ∈ {d, d′, g}, e is Be-strongly semantically complete. For C ∈M − {Le}, d (respectively d′, g) is C-consistently semantically (respectively consistently syntactically, syntactically) complete. If C is nite, the exclusion of Le may be dropped. 7. For e ∈ {d, d′, g} and Γ ⊆ BN(C), Γ is said to be e-syntactically complete if for every σ ∈ BN(C), Γ e ` σ or Γ e ` σ. 8. For e ∈ {d, d′, g} and C ∈ M − {Le}, if Γ is consistent and esyntactically complete then there isB ∈ C such that BtB = Γe. If, moreover, CΓ is nite, the exclusion of Le may be dropped. 46 10. Decidability revisited. THEOREM 10.1. For each e ∈ {d, d′, g} there is a polynomial (of degree at most 8) time algorithm to decide for any < Γ, σ >∈ ℘(BN(C)) × BN(C) whether Γ e ` σ, provided that Γ is essentially nite and: Γ ∩ ({Ecc : c ∈ (C − CΓ)} ∪ {Occ : c ∈ (C − CΓ)}) = φ. Proof. For e = d a proof may be obtained by slightly modifying the appropriate parts of the proof of theorem 3.2. In view of lemma 8.3, a proof for the case e = d′ may be obtained along the same lines as above. In view of remarks 3.1, the rst part of this theorem may be made use of to determine whether Γ is inconsistent. If yes, Γ g ` σ; else Γ g ` σ i Γ d′ ` σ, by theorem 8.7.  11. Sorites. Soriteses are well known in Aristotelian syllogistic (see Hurley, P. J. 1982, p. 201; Rosenthal, M. and Yudin, R (eds.) 1967, p. 423; also cf. Boger, G. 1998, pp. 197-8; Smiley, T.J. 1973, pp. 139-40). The notion of a sorites may be explicated as follows. DEFINITION 11.1. Let e ∈ {d, d′} and let Γ ⊆ BN(C). An annotation of an e-deduction < σi >i∈k from Γ is said to be an e-sorites annotation if the following conditions are satised: 1. σi 6= σj whenever i 6= j (i, j ∈ k). 2. For i ∈ k − 1, σi is involved in the annotation of another sentence in the following and only in the following way. 2.1. If 1 ≤ i ≤ k − 3 then exactly one of the following holds: 2.1.1. σi+1 is annotated as the consequent of σi by some e-rule with one premise. 2.1.2. σi+1 is annotated as the consequent of σi−1, σi or σi, σi−1 by some e-rule with two premises. 2.1.3. σi+2 is annotated as the consequent of σi, σi+1 or σi+1, σi by some e-rule with two premises. 2.2. If 1 ≤ i = k − 2 then exactly one of 2.1.1 and 2.1.2 holds. 2.3. If i = 0 then exactly one of the following holds: 2.3.1. k = 2 and 2.1.1 holds. 47 2.3.2. k > 2 and exactly one of 2.1.1 and 2.1.3 holds. An e-sorites from Γ is an e-deduction from Γ which admits a sorites annotation. An e-sorites of σ(∈ BN(C)) from Γ is an e-deduction of σ from Γ which is an e-sorites. In case there is such a sorites, we write Γ es ` σ. Condition 2 of the above denition entails that, with the exception of the last sentence, every sentence occurring in an e-sorites from Γ is made use of exactly once as a premise of some application of some e-rule, and in this (hence in each) application the premise or the premises immediately precede the consequent. For e ∈ {d, d′} there is, obviously, a set Γ ⊆ BN(C) and an e-deduction from Γ which is not an e-sorites from Γ. So the best we may hope for is to nd an e-sorites of σ from Γ, for every Γ ∪ {σ} ⊆ BN(C) such that Γ e ` σ. Even this is not always attainable. Let Γo = {Aca,Ebc} and Γ1 = {Acx,Ebx, Ica}, then for i ∈ 2, Γi is consistent and Γi d′ ` Oab, but not Γi d′s ` Oab. For i = 0, adding the rule Eab Oab (E-sub) as an additional rule of inference will solve the problem. Same holds for i = 1 if, instead, Iba,Ebc Oac (Ferison) is added. 11.1. Further extension of direct deduction. Taking into consideration the following two rules of inference. 10. E-sub 11. Ferison in addition to the ten rules of inference of d′, the d′′-deduction relation  d′′ ` may be dened along the lines of the denition of  d `. Likewise all the other denitions involving d may be modied in an obvious way to give corresponding denitions involving d′′. PROPOSITION 11.2. For Γ ∪ {σ} ⊆ BN(C), Γ d′ ` σ i Γ d′′ ` σ. Proof. One direction is obvious, the other is easy.  DEFINITION/remark 11.3. The d′′-models may be dened along the lines of the denition of the d′-models. 48 By the above proposition they are the same. THEOREM 11.4. Let Γ ∪ {σ} ⊆ BN(C) and e ∈ {d, d′, d′′} then: Γ es ` σ whenever Γ e ` σ provided one of the following conditions holds: 1. σ is armative, 2. σ is universal negative and Γ is consistent, 3. σ is particular negative, e = d or Γ is consistent and e = d′′. The other direction unconditionally holds, so the two sides are equivalent if Γ is consistent and e ∈ {d, d′′}. Proof. Assume Γ e ` σ. Distinguish between the following cases. 1. σ = Aab, for some a, b ∈ C. By proposition 11.2 and lemma 8.3 there is an a-b chain in Γ, < ci >i∈n say. We may assume that this chain is injective. If n = 1, then there is an e-sorites of σ from Γ of length 1. Else n ≥ 2; dene < ρi >i∈2n−3 as follows: ρ2j = Acocj+1 = Aacj+1 j ∈ n− 1, ρ2j+1 = Acj+1cj+2 j ∈ n− 2. Then < ρi >i∈2n−3 is an e-sorites of σ from Γ. 2. σ = Iab, for some a, b ∈ C. If e = d, the result is an easy consequence of part 1 of this proof and lemma 8.3. Else, by proposition 11.2 we may assume that e = d′. By lemma 8.3 it suces to deal with the following three subcases (for some a′, b′ ∈ C): 2.1. Ia′b′ ∈ Γ and Γ d ` Aa′a,Ab′b, 2.2. Ib′a′ ∈ Γ and Γ d ` Aa′a,Ab′b, 2.3. Γ d ` Aa′a,Aa′b. Assume 2.1 (the other two subcases are not harder), then there are an a′-a chain and a b′-b chain in Γ, let them be, respectively < ci >i∈k and < c ′ j >j∈l. We may assume that the ranges of these two chains are disjoint, otherwise this subcase will be reduced to subcase 2.3. Also we may assume that each of these two chains is injective. The following is a d′ (hence a d′′)-sorites of σ from Γ: Ab′c′1, Ac ′ 1c ′ 2, Ab ′c′2, ..., Ab ′c′l−2, Ac ′ l−2b, Ab ′b, Ia′b′, Ia′b, Iba′, Aa′c1, Ibc1, ..., 49 Ibck−2, Ack−2a, Iba, Iab. 3. σ = Eab, for some a, b ∈ C and Γ is consistent. By proposition 11.2 and lemma 8.3 there are a′, b′ ∈ C such that {Ea′b′, Eb′a′} ∩ Γ 6= φ and there are an a-a′ chain and a b-b′ chain in Γ, let them be, respectively, < ci >i∈k and < c ′ j >j∈l. By the consistency of Γ, the ranges of the two chains are disjoint. Moreover, we may assume that each of them is injective. Let Ea′b′ ∈ Γ (the other case is not harder), then the following is an e-sorites of σ from Γ: Ea′b′, Ack−2a ′, Eck−2b ′, ..., Ec1b ′, Aac1, Eab ′, Eb′a,Ac′l−2b ′, Ec′l−2a, ..., Ec ′ 1a,Abc ′ 1, Eba, Eab. 4. σ = Oab, for some a, b ∈ C. If e = d, then there is a one line e-sorites of σ from Γ. Else assume that Γ is consistent and e = d′′, by proposition 11.2 and lemma 8.3 it suces to deal with the following two subcases. 4.1. There are a′, b′ ∈ C such that Oa′b′ ∈ Γ and Γ d ` Aa′a,Abb′. Making use of Bocardo and Baroco it may be shown, along the lines of part 3 of this proof, that there is a d′ (hence a d′′)-sorites of σ from Γ. 4.2. There is c ∈ C such that Γ d′ ` Ica, Ecb. As in part 3 of this proof, there are c′, b′ ∈ C such that: {Ec′b′, Eb′c′} ∩ Γ 6= φ and Γ d ` Acc′, Abb′ (∗) By lemma 8.3 it suces to deal with the following two subsubcases. 4.2.1. For some c′′ ∈ C, Γ d ` Ac′′c, Ac′′a. By this and (∗), Γ d ` Abb′, Ac′′c′, Ac′′a. So there are b-b′, c′′-c′ and c′′-a injective Γ-chains; let them be< xi >i∈k , < yi >i∈l and < zi >i∈m respectively. By the consistency of Γ, the range of < xi >i∈k and the union of the ranges of < yi >i∈l and < zi >i∈m are disjoint. Assume that the ranges of < yi >i∈l and < zi >i∈m have c ′′ only in common (the other case is similar). If Ec′b′ ∈ Γ, then there is a d′ (hence a d′′)-sorites of σ from Γ. Else Eb′c′ ∈ Γ and the following is a d′′-sorites of σ from Γ. Abx1, Ax1x2, Abx2, ..., Abxk−2, Axk−2b ′, Abb′, Eb′c′, Ebc′, Ec′b, Ayl−2c ′, Eyl−2b, ..., Ey1b, Ac ′′y1, Ec ′′b, Oc′′b (here E-sub is made use of), Ac′′z1, Oz1b, Az1z2, Oz2b, ..., Ozm−2b, Azm−2a, Oab. 4.2.2. {Ic′′a′, Ia′c′′} ∩ Γ 6= φ and Γ d ` Ac′′c, Aa′a, for some c′′, a′ ∈ C. By 50 this and (*), Γ d ` Abb′, Ac′′c′, Aa′a. So there are b-b′, c′′-c′ and a′-a injective Γ-chains; let them be < xi >i∈k, < yi >i∈l and < zi >i∈m respectively. By the consistency of Γ, the range of < xi >i∈k and the union of the ranges of < yi >i∈l and < zi >i∈m are disjoint. If the ranges of < yi >i∈l and < zi >i∈m are not disjoint, this case will be reduced to the above case; so assume that they are disjoint. If Ia′c′′ ∈ Γ then there is a d′ (hence a d′′)-sorites of σ from Γ. Else Ic′′a′ ∈ Γ, assume Eb′c′ ∈ Γ (the other case is similar), then the following is a d′′-sorites of σ from Γ. Abx1, Ax1x2, Abx2, ..., Abb ′, Eb′c′, Ebc′, Ec′b, Ayl−2c ′, Eyl−2b, ..., Ec ′′b, Ic′′a′, Oa′b (here Ferison is made use of), Aa′z1, Oz1b, ..., Oab. PROPOSITION 11.5. If Γ(⊆ BN(C)) is inconsistent then it is ds-inconsistent, in the sense that there is σ ∈ BN(C) such that Γ ds ` σ, σ. Proof. Let Γ be inconsistent, then there is a universal ρ ∈ BN(C) such that Γ d ` ρ, ρ. Distinguish between two cases: 1. ρ = Aab, for some a, b ∈ C. In this case the result is a direct consequence of theorem 11.4. 2. ρ = Eab, for some a, b ∈ C. As in part 3 of the proof of theorem 11.4, there are a′, b′ ∈ C such that {Ea′b′, Eb′a′} ∩ Γ 6= φ and there are injective a-a′, b-b′ chains in Γ; let them be, respectively, < ci >i∈k and < c ′ j >j∈l. If the ranges of these chains are disjoint, the result follows by theorem 11.4 and the methods made use of in its proof. Else there is c′′ ∈ {ci : i ∈ k}∩{c′j : j ∈ l}. Then there are injective c′′-a′, c′′-b′ chains in Γ. Along the lines of the proof of theorem 11.4 it may be shown that Γ ds ` Ea′c′′, Ia′c′′ (and Γ ds ` Eb′c′′, Ib′c′′).  To show that the consistency condition in each of the parts 2,3 of theorem 11.4 cannot be completely dispensed with, we prove: PROPOSITION 11.6. Let Γ ⊆ BN(C), e ∈ {d, d′, d′′} and a, a′, b, b′, c, c′ ∈ C; and assume that c 6= c′. 1. If every a-a′ chain in Γ is a < c, c′ > chain, then Acc′ occurs as an assumption in every e-deduction of Aaa′ from Γ; moreover it is made use of as a premise in the deduction if it is dierent from Aaa′. 51 In parts 2 and 3 below, Ebb′ is assumed to be the only universal negative sentence in Γ. 2. If every a-b chain and every a-b′ chain in Γ is a < c, c′ > chain, then Acc′ occurs as an assumption and is made use of as a premise in every ededuction of Eaa′ and every e-deduction of Ea′a from Γ. 3. If every a-b chain, every a-b′ chain, every a′-b chain and every a′-b′ chain in Γ is a < c, c′ > chain, then for every injective e-deduction < σi >i∈k of Eaa′ from Γ there is j ∈ k such that σj is made use of as a premise at least twice. In parts 4 and 5 below, Obb′ is assumed to be the only negative sentence in Γ. 4. If every b-a chain or every a′-b′ chain in Γ is a < c, c′ > chain, then Acc′ occurs as an assumption and is made use of as a premise in every e-deduction of Oaa′ from Γ. 5. If every b-a chain and every a′-b′ chain in Γ is a < c, c′ > chain, then for every injective e-deduction < σi >i∈k of Oaa ′ from Γ there is j ∈ k such that σj is made use of as a premise at least twice. Proof. Generalize the rst part to become: For every u, u′ ∈ C, if every u-u′ chain in Γ is a < c, c′ > chain, then Acc′ occurs as an assumption in every e-deduction of Auu′ from Γ; moreover, it is made use of as a premise in the deduction if it is dierent from Auu′. The stronger statement may be easily proved by course of values induction on the length of the e-deduction. Parts 2 and 4 may be proved similarly. Again generalize part 3 to become: For every u, u′ ∈ C if every u-b chain, every u-b′ chain, every u′-b chain and every u′-b′ chain in Γ is a < c, c′ > chain, then for every injective ededuction < σi >i∈k of Euu ′ from Γ there is j ∈ k such that σj is made use of as a premise at least twice. The stronger statement may be proved by course of values induction on 52 k as follows. Assume the required for r < k and let < σi >i∈k be an ededuction of Euu′ from Γ. Since {b, b′}, {u, u′} are disjoint and σk−1 = Euu′, then there are only two cases to consider: 1. For some l < k − 1, σl = Eu′u, in this case the result is immediate by the induction hypothesis. 2. For some l,m < k − 1 and some v ∈ C it is the case that l < m, {σl, σm} = {Auv,Evu′} and σk−1 is obtained from them as the conclusion of applying the rule Auv,Evu ′ Euu′ . If σl is made use of as a premise in a step whose conclusion is σj for some j < k− 1, the result is immediate. Also if there is some u-v chain in Γ which is not a < c, c′ > chain, the result follows by the induction hypothesis. So it remains to assume that every u-v chain in Γ is a < c, c′ > chain and for every j < k−1, σl is not made use of as a premise in the step which gives rise to σj. Put: ∆o = {σl}, ∆j+1 = ∆j ∪ {σi : i ∈ l and σi is made use of as a premise in a step whose conclusion is in ∆j}. Then for some n, ∆n+1 = ∆n. Hence < σi >i∈l+1,σi∈∆n is an e-deduction of σl from Γ, so by parts 1,2 above Acc ′ ∈ ∆n. Assume, towards a contradiction, that for every i ∈ k, σi is made use of as a premise at most once. Then < σi >i∈m+1,σi /∈∆n is an e-deduction of σm from Γ−∆n. Again by parts 1,2 above, Acc′ /∈ ∆n. Hence the result. Part 5 may be proved similarly.  EXAMPLES 11.7. To see that the consistency condition in each of the parts 2,3 of theorem 11.4 cannot be completely dispensed with, put: Γo = {Aac,Aa′c, Acc′, Ac′b, Ac′b′, Ebb′} , σo = Eaa′ Γ1 = {Aa′c, Abc, Acc′, Ac′a,Ac′b′, Obb′} , σ1 = Oaa′. For i ∈ 2, Γi d′ ` σi (in fact Γo d ` σo), but by the above proposition Γi d′′s 0 σi. This example still works even if d′′ is augmented by all of the Aristotelian syllogisms. To see that consistency is not always necessary, just notice that whether Γ is consistent or not, Γ ds ` σ whenever σ ∈ Γ. 53 Following are basic properties of sorites. DEFINITIONS and remarks 11.8. Let e ∈ {d, d′, d′′}, Γ ⊆ BN(C) and k ∈ N+, and let < σi >i∈k be an e-sorites of σk−1 from Γ according to some annotation. 1. Two annotations of an e-deduction from Γ are said to be essentially the same if the only dierence between them is interchanging assumption (i.e. the corresponding sentence belongs to Γ) and A-Id in some places. An e-deduction from Γ is said to have essentially one, or unique, annotation (of some sort) if all of its annotations (of this sort) are essentially the same. 2. For 1 ≤ ` ≤ k, < σi >i∈` is an e-sorites from Γ according to the restriction of the given annotation i ` = 1 or the annotation of σ`−1 is neither A-Id nor assumption. 3. Let 1 ≤ ` < k, then for at least one j ∈ {`, ` + 1}, < σi >i∈j is an e-sorites from Γ according to the restriction of the given annotation. 4. For e ∈ {d, d′}, every e-sorites from Γ has essentially one sorites annotation. This does not apply to d′′, for the d′′-deduction < Ixx,Exy,Oxy > has two d′′-sorites annotations which are not essentially the same. Only one of them is a d′-sorites annotation. 5. Ferison and E-sub are the only d′′-rules which are not d′-rules. In every d′′-sorites annotation at most one of them is made use of, at most once. 6. In each d′′-sorites at most one triple of the form < Ixx,Exy,Oxy > or < Exy, Ixx,Oxy > occurs, at most once. If no such triple occurs, the sorities will have an essentially unique d′′sorites annotation. Else all of its d′′-sorites annotations are essentially the same, with the only exception that an occurrence of Oxy may be annotated as the consequence of the preceding two sentences by Ferio (which is a d′-rule) in some of them and by Ferison (which is not) in the others. 12. Independence. DEFINITIONS 12.1. Let e be a deduction system and let r be a rule of e. The deduction system obtained from e by excluding r will be denoted by er. 1. r is said to be derivable in e if Γ er ` σ whenever Γ is a set of antecedents of an instance of r, and σ is the corresponding conclusion. Otherwise r is said to be independent in e. 54 2. e is said to be independent if each of its rules is independent in it. 3. r is said to be weakly independent in e if for some set ∆ ∪ {ρ} of sentences, ∆ e ` ρ while ∆ er 0 ρ. 4. e is said to be weakly independent if each of its rules is weakly independent in it. REMARKS 12.2. 1. Independence implies weak independence. 2. For e ∈ {d, d′, d′′}, each rule r of e is independent in e i it is weakly independent in e, hence e is independent i it is weakly independent. 3. Each of d, d′ is independent (cf. Glasho (2005) where similar results are obtained via brute force computation). 4. The independence of each of ds and d′s is an immediate consequence of the independence of each of d and d′ respectively. THEOREM 12.3. 1. E-sub, Ferio and Ferison are derivable in d′′s, hence in d′′. Each of the other rules of d′′ is independent in d′′, hence in d′′s. 2. d′′ is not independent, hence not weakly independent. 3. d′′s is weakly independent; however, it is not independent. Proof. 1. The sequence Aaa, Iaa, Eab,Oab shows that E-sub is derivable in d′′s. The corresponding proofs for Ferio and Ferison are not harder. Put r = Bocardo and let a, b, c be three pairwise distinct elements of C. Put Γ = {Aba,Obc} and σ = Oac. It is easy to see that if Γ d′′ ` ρ then ρ ∈ {Aba,Obc, Iab, Iba} ∪ {Axx : x ∈ C} ∪ {Ixx : x ∈ C}. From this the independence of r in d′′ follows. Similarly the other required results may be obtained. 2. By part 1 above and part 2 of remarks 12.2. 3. Part 1 above shows that d′′s is not independent. It shows also that to prove the weak independence of d′′s it suces to deal with E-sub, Ferio and 55 Ferison only. Put r = E-sub and let ∆ = {Eab} and ρ = Oba, for some distinct a, b ∈ C. ∆ d′′s ` Oba. To see that ∆ d′′sr 0 Oba notice that the only d′′sr rules which yield an O-sentence are Ferio, Baroco, Bocardo, and Ferison. To obtain Oba by applying Baroco or Bocardo the O-sentence occurring as one of the antecedents -in the present casewill be the same as the conclusion, which is forbidden in sorites. To apply Ferio or Ferison, the antencedents -in the present casemust be Ibb and Eba. But if ηi, ηi+1, ηi+2 is a subsequence of a d′′sr sorites deduction from ∆ and the annotation of ηi+2 is that it is obtained from ηi, ηi+1 or ηi+1, ηi by some rule, then ηi+1 ∈ ∆∪{Acc : c ∈ C}. So neither Ferio nor Ferison is applicable, hence ∆ d′′sr 0 Oba. Next, put r = Ferio (Ferison) and let ∆ = {Eab, Icb} ({Eab, Ibc}) and ρ = Oca for some pairwise distinct a, b, c ∈ C. By a slight modication of the above technique it may be shown that ∆ d′′s ` ρ, but ∆ d′′sr 0 ρ.  12.1. Independence of g and variations thereof. g and d have the same deduction rules, but the notion of g-deduction is weaker than that of d-deduction. By denition 1.9, for Γ ∪ {σ} ⊆ BN(C), Γ g ` σ i Γ ∪ {σ} is d-inconsistent. Likewise for each rule r of g (equivalently of d) dene the gr-deduction relation  gr ` by: Γ gr ` σ i Γ ∪ {σ} is dr-inconsistent. From this and remarks 12.2 it easily follows that r is independent in g i it is weakly independent in g, hence g is independent i it is weakly independent. THEOREM 12.4. g is independent. Proof. Let a, b, c be pairwise distinct elements of C, and put r = Barbara and Γ = {Aab,Abc}. Γ gr 0 Aac i Γ ∪ {Oac} is dr-consistent. But the set of all dr-consequences of Γ ∪ {Oac} is {Aab,Abc,Oac, Iba, Icb} ∪ {Axx : x ∈ C}∪{Ixx : x ∈ C}, hence Γ∪{Oac} is dr-consistent. Consequently Barbara is independent in g. The proofs of the independence of the other rules are similar or easier.  To get closer to the usual deduction systems, we introduce two new deduction systems g′, g′′ and show that each of them is equivalent to g and discuss its independence. 56 12.1.1. First variation on g. The deduction system g′ is obtained by augmenting the system d′ by the rule: ρ,ρ σ (contradiction, Co for short). Let Γ ∪ {σ} ⊆ BN(C). It is easy to see that if Γ g′ ` σ then there is a g′-deduction of σ from Γ in which Co is never made use of or it is made use of only at the last step; moreover, this applies to g′r for each rule r of d ′. THEOREM 12.5. The following are equivalent: 1. Γ g ` σ, 2. Γ g′ ` σ, 3. Γ d′ ` σ or Γ is inconsistent, 4. Γ ∪ {σ} is inconsistent. Proof. Easy if Γ is inconsistent; and in all cases parts 1 and 4 are equivalent by denition 1.9. Assume Γ is consistent. By theorem 8.7, parts 1 and 3 are equivalent, and by the denition of g′, part 3 implies part 2. Finally assume part 2, then there is a g′-deduction of σ from Γ in which Co is never made use of, this implies part 3.  The following theorem settles the indepenence of g′. THEOREM 12.6. 1. Every rule of g′ is independent in g′ i it is weakly independent in g′, hence g′ is independent i it is weakly independent. 2. Γ gco ` σ i Γ d′ ` σ. 3. For every rule r of d′, Γ g′r ` σ i Γ d′r ` σ or Γ is d′r-inconsistent. 4. g′ is independent. Proof. The proof of the rst three parts is easy. To prove the last part let a, b ∈ C, then Aab,Oab Eab is an instance of Co. But by lemma 8.3, {Aab,Oab} d′ 0 Eab. So by part 2 above {Aab,Oab} g′co 0 Eab. Therefore Co is independent in g′. To complete the proof let r be some other 57 rule of g′, then r is a rule of d′. By part 3 above the independence of r in g′ may be proved by choosing a consistent set Γ of antecedents of r such that Γ d′r 0 σ, where σ is the corresponding conclusion; which is always possible.  12.1.2. Second variation on g. Though g′ is closer than g to the contemporary deduction systems, it is not as close to the Aristotelian spirit as g. Inspired by Gentzen-type sequent systems (cf. Kleene, S.C. 1967, p. 306) we introduce a second variation g′′ on g, which will hopefully be close enough to both modern and Aristotelian traditions. The deduction rules of g′′ are: 0′. Γ`Aaa (A-Id ′) 1′. Γ`Aab Γ`Iba (Apc ′) 2′. Γ`Eab Γ`Eba (Ec ′) 3′. Γ`Aab,∆`Abc Γ∪∆`Aac (Barbara ′) 4′. Γ`Aab,∆`Ebc Γ∪∆`Eac (Celarent ′) 5′. Γ`η (Ass) 6′. Γ∪{σ}`ρ,∆∪{σ}`ρ Γ∪∆`σ (Raa) where a, b, c ∈ C, Γ ∪∆ ∪ {ρ, σ} ⊆ BN(C) and η ∈ Γ. Ass and Raa are abbreviations for Assumption and Reductio ad absurdum respectively. ` is just a symbol, instead we could have made use of ordered pairs and write, e.g. < Γ, σ > in place of Γ ` σ. DEFINITION and remarks 12.7. Let S be a set of sequents, i.e. S ⊆ {Γ ` σ : Γ ∪ {σ} ⊆BN(C)}. 1. A g′′-deduction from S is a sequence < Γi ` σi >i∈k of sequents, where k ∈ N and for each i ∈ k, Γi ` σi ∈ S or may be obtained from preceding terms of the sequence by some g′′-deduction rule. If k 6= 0, < Γi ` σi >i∈k is said to be a g′′-deduction of Γk−1 ` σk−1 from 58 S. In this case we write S g′′

Γ ` σ, g′′-deduction for g′′-deduction from φ and g′′-deduction of Γk−1 ` σk−1 (or of ∆ ` ρ) for g′′-deduction of Γk−1 ` σk−1 (or of ∆ ` ρ) from φ. 3. The above denition and remark may be generalized to subsystems of g′′. 4. The notions of derivability, independence and weak independence may be extended to g′′ in the obvious way. 5. A deduction rule of g′′ is independent in g′′ i it is weakly independent in g′′. Hence g′′ is independent i it is weakly independent. THEOREM 12.8. For every Γ ∪ {σ} ⊆ BN(C): Γ g ` σ i Γ g′′ ` σ. Proof. Let Γ g ` σ then, by denition 1.9, Γ∪{σ} d ` ρ, ρ for some ρ ∈ BN(C). Let < ξi >i∈k and < ηj >j∈l be, respectively, d-deductions of ρ and ρ from Γ∪{σ}. Then < Γ∪{σ} ` ξi >i∈k and < Γ∪{σ} ` ηj >j∈l are, respectively, g′′-deductions of Γ∪{σ} ` ρ and Γ∪{σ} ` ρ. To their concatenation (which is a g′′-deduction) add one more line to obtain Γ ` σ by Raa from lines k− 1 and k + l − 1. This proves the only if direction. To prove the other direction let Γ g′′ ` σ, let < ∆i ` ρi >i∈k be a g′′deduction of Γ ` σ, and assume that ∆i g ` ρi for each i < k − 1. To show that ∆k−1 g ` ρk−1 we deal with as many cases as there are g′′-deduction rules. Following we consider Celarent′ (4′) and Raa (6′), the other cases are similar or easier. Celarent′: There are a, b, c ∈ C and j, l ∈ N such that j < l < k − 1, ∆k−1 = ∆j ∪ ∆l, {ρj, ρl} = {Aab,Ebc} and ρk−1 = Eac. By the above assumption ∆j g ` ρj and ∆l g ` ρl. Hence ∆k−1(= ∆j ∪∆l) g ` ρj, ρl. So by part 7 of proposition 1.11, ∆k−1 g ` ρk−1. Raa: There are Σ,Σ′, η, j, l such that Σ ∪ Σ′ ∪ {η} ⊆ BN(C); j, l ∈ N; j < l < k − 1, ∆j = Σ ∪ {ρk−1}, ∆l = Σ′ ∪ {ρk−1}, {ρj, ρl} = {η, η} 59 and ∆k−1 = Σ ∪ Σ′. By the above assumption ∆j g ` ρj and ∆l g ` ρl. Hence ∆k−1∪{ρk−1}(= ∆j∪∆l) g ` ρj, ρl. So by part 4 of proposition 1.11 and the relevant denitions, ∆k−1 g ` ρk−1.  Notice that in the g′′-deductions < Γ ∪ {σ} ` ξi >i∈k and < Γ ∪ {σ} ` ηj >j∈l which occur in the proof of the only if direction of the above theorem, only the rules 0′ − 5′ are made use of. Moreover, if Γ d ` σ, then there is a g′′-deduction of Γ ` σ in which Raa is never made use of. This is essentially sucient to prove the following: COROLLARY 12.9. If Γ g′′ ` σ then there is a g′′-deduction of Γ ` σ in which Raa is never made use of or is made use of only in the last step.  This section is concluded by proving the independence of g′′. THEOREM 12.10. Let Γ∪{σ} ⊆ BN(C), r be a d-deduction rule, and r′ be the corresponding g′′-deduction rule. 1. Γ g′′ r′ ` σ i Γ gr ` σ, 2. Γ g′′Ass ` σ i φ g′′ ` σ i φ g ` σ i φ d ` σ, 3. Γ g′′Raa ` σ i Γ d ` σ, 4. g′′ is independent. Proof. The proofs of the rst and the third parts are along the lines of the proof of theorem 12.8 noting that proposition 1.11 still holds after replacing d,g by dr,gr respectively. For part 2 it is sucient to notice that each of the four statements holds i σ is of the form Y cc for some c ∈ C and some Y ∈ {A, I}. To prove the last part we consider three cases: Rules 0′ − 4′: Let r′ be one of these rules and let r be the corresponding d-rule. Since r is independent in g, there is a set Γ of antecedents of an instance of r such that Γ gr 0 σ, where σ is the corresponding conclusion. Put S = {Γ ` ρ : ρ ∈ Γ} then S is a set of antecedents of r′ and Γ ` σ is the 60 corresponding conclusion. By parts 2,3 of denition and remarks 12.7 and part 1 above: S g′′ r′

Γ ` σ i Γ g′′ r′ ` σ i Γ gr ` σ. But Γ gr 0 σ, hence r′ is independent in g′′. Rule Ass: For a, b ∈ C, φ g′′

{Oab} ` Oab while, by 2 above, φ g′′Ass 1 {Oab} ` Oab. Hence Ass is independent in g′′. Rule Raa: For distinct elements a, b of C let σ = Iab, ρ = Iba, Γ = {ρ} and S = {Γ∪{σ} ` ρ,Γ∪{σ} ` ρ} then S is a set of antecedents of Raa and Γ ` σ is the corresponding conclusion. By a slight modication of the proof given above for the rules 0′ − 4′ it may be shown that Raa is independent in g′′.  13. Algebraic semantics of AAS, a prelude. The most well known attempt to algebraically interpret Aristotelian syllogistic is that of Boole (1948, rst published 1847); however, it is not the rst. More than a century and a half earlier, this area of research was pioneered by Leibniz (Kneale and Kneale 1966, pp. 338-45; Lenzen 2004). Following is a discussion of the subject in general; the works of Leibniz and Boole will be briey discussed in section 17 below. Regarding the central role played by order models in the semantics of NF (C), they will be our starting point for algebraization. Each underlying order structure of an order model will induce an algebra which may be expanded to make the interpretation of NF (C) possible. The simplicity of order models stems from the fact that all relations are determined by only one of them, namely the interpretation of A, which is compatible with the Aristotelian view that Barbara is the essential syllogism. Likewise, algebras dened in this section will each have one (partial) binary operation and no others. DEFINITIONS and remarks 13.1. 1. Let B be a non-empty set and let ⊕ be a function from a subset of B × B to B, then ⊕ is said to be a partial binary operation on B, and < B,⊕ > is said to be a partial algebra. 61 2. Let ◦ ≤ be a binary relation on a set B, the partial binary operation + ◦ ≤ induced by ◦ ≤ on B is dened by: + ◦ ≤ : ◦ ≤ → B a+ ◦ ≤ b = a + ◦ ≤ is commutative (see 3.3 below) if ◦ ≤ is antisymmetric. < B,+ ◦ ≤ > is called the partial algebra induced by < B, ◦ ≤>. 3. Let < B, ◦ ≤> be an order structure, then < B,+ ◦ ≤ > satises: 1. Right associativity: (a+ ◦ ≤ b) + ◦ ≤ c = a+ ◦ ≤ (b+ ◦ ≤ c) in the sense that for every a, b, c ∈ B if the rhs exists, so does the lhs and they are equal. 2. Idempotence: a+ ◦ ≤ a = a all a ∈ B. If, moreover, ◦ ≤ is antisymmetric, then < B,+ ◦ ≤ > satises: 3. Commutativity: a+ ◦ ≤ b = b+ ◦ ≤ a in the sense that for every a, b ∈ B, if both sides exist they are equal. Honouring Leibniz, a partial algebra < B,⊕ > satisfying conditions 1 and 2 will be called a Leibniz algebra (LA for short). If, moreover, it satises condition 3 it will be called a commutative Leibniz algebra (CLA for short). So, < B,+ ◦ ≤ > is a LA if < B, ◦ ≤> is an order structure; moreover, it is a CLA if ◦ ≤ is antisymmetric. 4. An idempotent partial algebra < B,⊕ > will be called a weak Leibniz algebra (WLA for short) if it satises: 62 1. Weak right associativity: (a⊕ b)⊕ c = a⊕ (b⊕ c) in the sense that for every a, b, c ∈ B if (a⊕ b) and the rhs both exist, then the lhs exists and equals the rhs. If, moreover, < B,⊕ > is commutative (in the sense of condition 3.3 above), it will be called a commutative weak Leibniz algebra (CWLA for short). Obviously every LA (CLA) is a WLA (CWLA). 5. With abuse of notation, LA, CLA, WLA and CWLA will denote also the classes of all LAs, CLAs, WLAs and CWLAs respectively; what is intended will be clear from the context. Abuses of notations such as this may take place later on without further notice. 6. Let < B,⊕ > be a partial algbra, the binary relation ≤⊕ induced by ⊕ on B is dened by: ≤⊕= {< a, b >∈ B :<< a, b >, a >∈ ⊕}, so a ≤⊕ b i a⊕ b = a (in the sense that the lhs exists and equals the rhs). Obviously, ≤⊕ is antisymmetric if ⊕ is commutative. 7. Let ◦ ≤ and ⊕ be, respectively, a binary relation and a partial binary operation on a set B, and let + ◦ ≤ , ≤⊕, ≤+◦ ≤ and +≤⊕ be as dened above. Then: 1. ≤+◦ ≤ = ◦ ≤. 2. +≤⊕ ⊆ ⊕; moreover, +≤⊕ is commutative if ⊕ is. 8. Let < B,⊕ > be a WLA, then < B,≤⊕> is an order structure, called the order structure induced by < B,⊕ >. Moreover, ≤⊕ is antisymmetric if ⊕ is commutative. 63 14. Algebraic interpretation of NF (C). DEFINITION 14.1. Let B =< B,⊕ > be a WLA and let μ : C → B. The structure Bμ =< B,⊕, μ > is said to be a weak Leibniz structure (WLS for short). The reduct B shall be called the WLA base of Bμ. Leibniz structures (LS for short), commutative Leibniz structures (CLS for short) and commutative weak Leibniz structures (CWLS for short) are dened analogously. The following denition shows how NF (C) may be interpreted in these structures. So they may, and will, be considered as NF (C)-structures and will be treated like other NF (C)-structures when dealing with semantics. In particular, all semantical notions (such as Bμ is a (n algebraic) model of Γ or Γ  C σ, for Γ∪{σ} ⊆ BN(C) and C ⊆WLS) will be assumed to be known. DEFINITION 14.2. Let Bμ =< B,⊕, μ > be a WLS, and let a, b ∈ C, then: 1. Bμ  Aab i μa⊕ μb exists and equals μa (i << μa, μb >, μa >∈ ⊕). 2. Bμ  Iab i the system of equations x⊕ μa = x, x⊕ μb = x has a solution (i the equation x⊕ μa = x⊕ μb has a solution, i the equation x⊕ μa = y ⊕ μb has a solution). 3. Bμ  Eab i Bμ 2 Iab. 4. Bμ  Oab i Bμ 2 Aab. REMARKS 14.3. 1. The order (partial order) model < B, ◦ ≤, μ > is basically equivalent to the WLS (CLS), < B,+ ◦ ≤ , μ >. 2. The WLS (CLS), < B,⊕, μ >, is basically equivalent to the order (partial order) model < B,≤⊕, μ >. 3. Consequently, every WLS (hence every LS, every CWLS and every CLS) is an e-model for e ∈ {d, d′, d′′, g}. 64 4. In the light of remarks and denitions 9.10 it may be easily seen that: 1. For e ∈ {d, d′, d′′, g}, e is sound wrt WLS, hence wrt every subclass of it. 2. g is CLS-syntactically complete. 3. For e ∈ {d′, d′′, g}, e is CLS-consistently syntactically complete. 4. For e ∈ {d, d′, d′′, g}, e is CLS-consistently semantically complete. In clauses 2-4, CLS may be replaced by any class intermediate between it and WLS. 15. Annihilators: Embedding the partial into a total. An annihilator of a (partial) binary operation ∗ on a set B is an element b ∈ B such that: x ∗ b = b = b ∗ x all x ∈ B Obviously ∗ has at most one annihilator. An annihilator algebra is an ordered triple B =< B, ∗, b > such that the reduct rB =< B, ∗ > is a partial algebra, and b is an annihilator of ∗. The subreduct srB of B is the ordered pair < B′, ∗′ >, where: B′ = B − {b} , ∗′ = ∗ ∩ (B′ ×B′)×B′ Here, and in the sequel, B′ is assumed to be non-empty. DEFINITIONS 15.1. 1. An annihilator Leibniz algebra (ALA for short) is an annihilator algebra whose subreduct is a LA. Annihilator commutative Leibniz algebras (ACLA for short), annihilator weak Leibniz algebras (AWLA for short) and annihilator commutative weak Leibniz algebras (ACWLA for short) are dened analogously. 2. A Leibniz algebra with annihilator (LAA for short) is an annihilator algebra whose reduct is a LA. Commutative Leibniz algebras with annihilators (CLAA for short), weak Leibniz algebras with annihilators (WLAA for short) and commutative weak 65 Leibniz algebras with annihilators (CWLAA for short) are dened analogously. As usual, an algebra or a structure based on an algebra is said to be total if each of its operations is total. TLA will stand for total Leibniz algebra, TLS will stand for total Leibniz structure and similarly for the other cases. REMARKS 15.2. 1. The subreduct of an annihilator algebra is a LA (respectively CLA, WLA or CWLA) if the reduct is. Hence LAA ⊆ ALA, and similarly for the other cases. 2. Let B =< B, ∗, b > be a total annihilator algebra whose subreduct also is total. Then B is LAA i it is ALA; LA may be replaced by CLA, WLA or CWLA. 3. TCLAA = TCWLAA = ICSGA (idempotent commutative semigroups with annihilators) = OSLA (operational semilattices with annihilators) The order structures induced by these algebras are lower semilattices with smallest elements. In the above equations C, the last A or both, may be dropped everywhere (the corresponding parenthetic clause is to be modied accordingly). The following denition designates to each partial algebra a total annihilator algebra in which it may be embedded. DEFINITION and remarks 15.3. 1. For i ∈ 2, let Bi(=< Bi, ∗i >) be a partial algebra. A bijection f from B0 to B1 is said to be an isomorphism from B0 to B1 if for every x, y, z ∈ B0: << x, y >, z >∈ ∗0 i << fx, fy >, fz >∈ ∗1. B0 and B1 are said to be isomorphic if there is an isomorphism from one of them to the other. 66 2. If two partial algebras are isomorphic and one of the partial binary operations has an annihilator, then its image is an annihilator of the other. 3. Two annihilator algebras are said to be isomorphic if their reducts are. 4. Two total annihilator algebras are isomorphic i their subreducts are. 5. Every partial algebra B(=< B, ∗ >) is the subreduct of some total annihilator algebra. For, let 0 /∈ B and 0B = B ∪ {0}. Put: 0 ∗ : 0B × 0B → 0B x 0 ∗ y = { x ∗ y if < x, y >∈ Domain ∗, 0 otherwise Then 0B(=< 0B, 0∗, 0 >) is a total annihilator algebra, and B is its subreduct. 6. Every total annihilator algebra whose subreduct is isomorphic to B, is isomorphic to 0B. This warrants calling 0B the total annihilator algebra induced by B. The identity map on B is an embedding of B into 0B. 7. 0B is a TALA i B is a LA. LA may be replaced by CLA, WLA or CWLA. 16. Back to algebraic interpretation. Let B(=< B, ∗, 0 >) be a WLAA, then its reduct rB(=< B, ∗ >) is a WLA. So < B, ∗, μ > is a WLS, for every μ : C → B. Obviously, for all a, b ∈ C, Iab is satised in this structure. Hence none of d, d′, d′′ nor g is consistently semantically complete wrt any class of such structures, though every one of them is sound wrt each of these classes. Evidently expanding the structure to < B, ∗, 0, μ > will not solve the problem. As a matter of fact, the annihilator is the source of the diculty, and we may get around it by not permitting the annihilator to be assigned as a value corresponding to any element of C, nor accepting it as a solution of any of the relevant equations below. An additional advantage of this approach is to be able to consider the more general AWLA. 67 DEFINITION 16.1. (non-annihilator interpretation of NF (C) in annihilator algebras) 1. Let B(=< B, ∗, 0 >) be an AWLA and let μ : C → B′(= B − {0}). The structure Bμ(=< B, ∗, 0, μ >) is called an annihilator weak Leibniz structure (AWLS for short). The reduct B of Bμ is called the AWLA base of Bμ. The structures based on the other algebras (total or not) are dened, and their names are abbreviated, analogously. 2. For each a, b ∈ C: 1. Bμ  Aab i μa ∗ μb exists and equals μa (equivalently << μa, μb >, μa >∈ ∗). 2. Bμ  Iab i the system of equations x ∗ μa = x, x ∗ μb = x has a solution dierent from 0 (i the equation x ∗ μa = x ∗ μb has a solution which makes x ∗ μb 6= 0, i the equation x ∗ μa = y ∗ μb has a solution which makes y ∗ μb 6= 0). 3. Bμ  Eab i Bμ 2 Iab. 4. Bμ  Oab i Bμ 2 Aab. This shows how NF (C) may be interpreted in the structures dened in part 1. So they may, and will, be considered as NF (C)-structures and will be treated like other NF (C)-structures when dealing with semantics. In particular, all semantical notions (such as Bμ is a (total algebraic) model of Γ or Γ C σ, for Γ∪{σ} ⊆ BN(C) and C ⊆AWLS) will be assumed to be known. REMARKS 16.2. 1. Bμ and srBμ are basically equivalent, hence every AWLS is an e-model for e ∈ {d, d′, d′′, g}. 2. In part 4 of remarks 14.3, WLS and CLS may be, respectively, replaced by TAWLS and TACLS. 3. If Bμ is a TCLSA (equivalently TCWLSA), the provisions given in part 2 of denitions 16.1 may be simplied in the obvious way; in particular, the second provision will be equivalent to μa ∗ μb 6= 0. 68 To investigate the relationship between TCLSA and the Venn models we make use of a (n intermediate) subclass of TCLSA, namely the subclass of those TCLSA based on OSLA which are reducts of Boolean algebras. These reducts will be called Boolean-Leibniz algebras with annihilators, BLAA for short. As usual BLSA is a Boolean-Leibniz structure with annihilator, i.e. a LSA based on a BLAA. PROPOSITION 16.3. Every TCLAA may be embedded in a BLAA. Proof. Let B(=< B, ∗, 0 >) be a TCLAA. The mapping: f : B → ℘(B) f(b) = {x ∈ B : x ∗ b = x} − {0} is an embedding ofB in the BLAA:< ℘(B),∩, φ >.  < ℘(B),∩, φ > will be called the BLAA corresponding to B and will be denoted by Bl(B). For μ : C → B, < ℘(B),∩, φ, fμ > is a BLSA; it will be called the BLSA corresponding to, the TCLSA, Bμ and will be denoted by Bl(Bμ). The relevant denitions and part 3 of remarks 16.2 show that Bμ and Bl(Bμ) are basically equivalent. THEOREM 16.4. Every TCLSA is basically equivalent to a Venn model. And every Venn model is basically equivalent to a BLSA (hence to a TCLSA); moreover, the BLSA may be assumed to be based on a concrete BLAA whose universe is a power set. Proof. Let Bμ(=< B, ∗, 0, μ >) be a TCLSA, then Bl(Bμ) is a BLSA and B′ =< ℘(B)−{φ}, fμ > is a Venn model. They all are basically equivalent. On the other hand, letB(=< B, μ >) be a Venn model, then< ℘( ⋃ B),∩, φ, μ > is a BLSA which is basically equivalent to it.  COROLLARY 16.5. In part 4 of remarks 14.3, WLS and CLS may be, respectively, replaced by TWLSA and BLSA (either the superclass TCLSA, or the subclass consisting of those elements each of which is based on a concrete BLAA whose universe is a power set, may replace BLSA).  69 17. Leibniz and Boole. The calculus de continentibus et contentis, or the calculus of identity and inclusion -which is an algebraic treatment of conceptswas developed by Leibniz during 1679-90 (Kneale and Kneale 1966, p. 337). As may be gathered from a passage of the same reference (pp. 340-3), or from a translation of an original text of Leibniz (Lewis 1960, pp. 297-305), this calculus is the theory of operational semilattices (OSL for short) with applications to concepts; commutativity and idempotence are explicitly stated, while associativity is implicitly taken for granted (the aforementioned passage is abbreviated with some slight changes from the aforementioned translation (Kneale and Kneale 1966, p. 343); notice that the edition of Lewis' book referred to in Kneale and Kneale (1966) is earlier than the one referred to above). Kneale and Kneale (1966)'s assessment of this calculus is unfavorable. It asserts (p. 337) that Leibniz intended, no doubt, to produce something wider than traditional logic. [...]. But [...] he never succeeded in producing a calculus which covered even the whole theory of syllogism.. On p. 345 this assertion is elaborated What he [Leibniz] produced was certainly much less than he hoped to produce. For the last scheme [the calculus de continentibus et contentis], lacking as it does any provision for negation or for consideration of conjunction and disjunction together, is still a fragment. So far from including all Aristotle's syllogistic theory as a part, it contains no principle of syllogism except the rst [...]. Likewise, Lenzen (2004)'s assessment of the calculus de continentibus et contentis is unfavorable. It asserts (p. 28) that this calculus remains a very weak and uninteresting system [...]; thus it shall no longer be considered here.. On the contrary, we have shown that neither negation (of terms) nor any additional operations are needed to algebraically interpret AAS. It suces to require the OSL to possess an annihilator, i.e. to be OSLA. For the structures based on the OSLA are the TCLSA and, by corollary 16.5, AAS is both sound and complete with respect to them. According to Kneale and Kneale (1966, p. 339) it may be seen that Leibniz practically introduced annihilators when he interpreted Eab as ab (μa ∗ μb, in our terminology) is nothing. Lenzen (2004) goes even further. It (pp. 2-3) asserts that Leibniz developed stronger calculi, the most important of them (p.3) is L1, the full algebra of concepts [...], L1 is deductively equivalent or isomorphic to the ordinary algebra of sets. Since Leibniz happened to provide a complete set of axioms 70 for L1, he discovered the Boolean algebra 160 years before Boole.. Moreover, Lenzen (2004) asserts that Leibniz succeeded in making use of his logical theory to derive the basic laws of Aristotelian syllogism (p. 55). In particular, the Aristotelian inferences may be derived as theorems of L1, or the stronger calculus L2 (p. 56); a detailed discussion of the subject may be found in Lenzen (2004, 8, pp. 55-73). Indeed, as we have shown, AAS does not need all of this. Boole did more than just algebraically interpreting AAS. In addition to annihilators, which are sucient for dealing with Aristotelian syllogisms (which involve no term negation), he introduced complementation (which corresponds to term negation) and a second binary operation. This is possibly to: 1. be able to interpret all the Aristotelian categorical sentences into equations (cf. Boole 1948, pp. 20-5), 2. deal with medieval categorical sentences which may involve term negation (cf. Boole 1948, pp. 20, 27-47), or 3. deal with hypotheticals (cf. Boole 1948, pp. 48-59). In addition to establishing the Aristotelian syllogistic rules, Boole (1948) established some non-Aristotelian ones. For example (p. 37) Ezy,Oyx Ox′z , where x′ denotes not-x. Boole (1948) did not address the question of completeness, neither did he consider consequences of more than two premises. However, it discussed (pp. 76-81) a general scheme to solve arbitrarily nite systems of simultaneous equations in arbitrarily nitely many variables; applying, in particular, Lagrange's method of indeterminate multipliers. This discussion took place after making (p. 18) the confounding assertion [...] all the processes of common algebra are applicable to the present [Boolean] system.. For one more confounding assertion see below. 18. Inadequacy: bounds of AAs. Calling the symbols of its system elective symbols (p. 16), Boole (1948) makes (p. 59) another confounding assertion: Every Proposition which language can express may be represented by elective symbols, and the laws of combination of those symbols are in all cases the same; but in one class of instances the symbols have reference to collections of objects, in the other, to the truths of constituent Proposition.. This, probably, amounts -in modern languageto asserting: Every proposition which language can express is equivalent to a sentential combination of categorical sentences (SCCS for short). 71 SCCS should be taken seriously, since a stronger assertion has dominated human thought over more than two millennia: Every argument can be put in a syllogistic form. Even Bertrand Russell (1967, p. 198) asserts Of course it would be possible to re-write mathematical arguments in syllogistic form, but this would be very articial and would not make them any more cogent.. Concerning these assertions, it is worthwile to bring to the fore what Boche«ski (1968) calls attention to. On p. 63 it observes that Artistotle says explicitly that not all logical entailment is Syllogistic.. Moreover it observes on the same page that Aristotle declares that some logical entailments cannot be reduced to syllogisms. So it may be concluded that Artistotle himself contradicts the aforementioned assertions of Boole and Russell, which makes making them deeply confounding, and makes it more urgent for historians of thought to investigate the matter. Understanding SCCS depends on understanding the notion of categorical sentences. If term negation is permitted, the sentences will be called Boolean categorical sentences and the corresponding assertion will be denoted by SCBCS. Otherwise, the sentences will be called Aristotelian categorical sentences and the corresponding assertion will be denoted by SCACS. Hilbert and Ackermann (1950) formalizes the Boolean categorical sentences (pp. 44-8) and informally refutes SCBCS (pp. 55-6). To formally discuss SCACS (making use only of the methods developed above and the well known results of sentential logic) augment the alphabet of the language C of the natural deduction formalization dened in section 1.4 above, by a ternary relation symbol E ′, and add E ′abc (a, b, c ∈ C) to the set of sentences based on C. Denote the new set of sentences by BN ′(C). Intuitively, we like E ′abc to mean that no a which is b, is c. This may be formalized as follows: Interpret BN ′(C) in a WLS Bμ =< B, ∗, μ > by adding the following provision to the provisions of denition 14.2. 5. Bμ  E ′abc i the system of equations: x ∗ μa = x, x ∗ μb = x and x ∗ μc = x has no solution. In an AWLS Bμ =< B, ∗, 0, μ >, BN ′(C) is interpreted by adding the following provision to the provisions of part 2 of denitions 16.1. 5. Bμ  E ′abc i 0 is the only solution of the system of equations: x ∗ μa = x, x ∗ μb = x and x ∗ μc = x. Recall that 0 is not in the range of μ; also notice that if Bμ is a TCLSA, 72 then this provision is equivalent to 5′. μa ∗ μb ∗ μc = 0. The other syntactical and semantical notions remain the same, or to be appropriately modied in the obvious way. Let Γ0,Γ1 ⊆ BN ′(C) and let D ⊆ WLS ∪AWLS. Γ0 is said to D-imply Γ1 (symbolically Γ0  D Γ1) if for every D ∈ D, D  Γ1 whenever D  Γ0. Γ0 is said to be D-equivalent to Γ1, or Γ0,Γ1 are D-equivalent, if each of them D-implies the other. Γ0 is said to be D-valid if φ  D Γ0, it is said to be D-consistent if D  Γ0 for some D ∈ D. The above notions may be generalized, in the obvious way, to sets of sentential combinations of elements of BN ′(C). If Γ0 or Γ1 is a singleton, it may be replaced by its unique element, e.g. ρ  D σ may replace {ρ}  D {σ}. In what follows c0, c1 and c2 are assumed to be pairwise distinct elements of C. For every D ⊆ WLS∪AWLS, Ec0c1 D-implies E ′c0c1c2. The converse depends on D. In particular it does not hold for D = BLSA. As a matter of fact we have the following: THEOREM 18.1. Let σ be a sentential combination of elements of BN(C), then: 1. E ′c0c1c2 is not BLSA-equivalent to σ, hence 2. E ′c0c1c2 is not deductively equivalent to σ (i.e. one of them does not deductively entail the other), for each deductive system which is sound with respect to BLSA. To prove this, we rst prove: LEMMA 18.2. Put: Γ0 = {Ic0c1, Ic1c2, Ic2c0} and Γ1 = Γ0 ∪ {E ′c0c1c2} then: 1. Γ1 is BLSA-consistent. 2. Γ1 is not BLSA-implied by any BLSA-consistent Γ ⊆ BN(C). Proof. Part 1 is easy. To see part 2, assume that there is a subset Γ ⊆ BN(C) which is both BLSA-consistent and BLSA-implies Γ1. Then there is Bμ ∈ BLSA which is a model of Γ ∪ Γ1. By theorem 16.4 it may be assumed that Bμ =< ℘(B),∩, φ, μ > for some B. 73 Let B′ = B∪{a} for some a /∈ B and let B′μ′ =< ℘(B′),∩, φ, μ′ > where for every c ∈ C, μ′(c) = { μ(c) ∪ {a} if Bμ  Acic for some i ∈ 3, μ(c) otherwise. B′μ ′ is a BLSA which is basically equivalent to Bμ, hence it is a model of Γ; but it is not a model of Γ1. From this the required follows.  Proof of theorem 18.1. Assume that E ′c0c1c2 is BLSA-equivalent to a sentential combination of elements of BN(C), σ say. Then E ′c0c1c2 ∧ Ic0c1 ∧ Ic1c2∧ Ic2c0 (ρ for short) is BLSA-equivalent to σ∧ Ic0c1∧ Ic1c2∧ Ic2c0 (σ1 for short) which also is a sentential combination of elements of BN(C). By sentential logic, σ1 may be assumed to be a disjunction of conjunctions of elements of BN(C) and their negations. Since ρ is BLSA-consistent and the negation of any element of BN(C) is BLSA-equivalent to some element of BN(C), σ1 may further be assumed to be a non-empty disjunction of BLSA-consistent conjunctions of elements of BN(C). Consequently ρ is BLSA-implied by each of these conjunctions, which contradicts part 2 of lemma 18.2. From this the required follows.  Acknowledgements. Several friends were kind enough to provide me with references which proved to be very helpful. My deep gratitude is hereby expressed to each of them: Wak Lotfalla, Essawy Amasha, Sharon Amasha, and Fawzy Hegab. I am most indebted to two more friends: Azza Khalifa for pointing out some misprints, and Ahmed Ghaleb for patiently and carefully proofreading the manuscript and transforming its scientic Workplace le into TEX. Appendix. The following algorithm, to generate the rst n(> 0) primes, may not be ecient, but it is simple, and its running time (see below) makes it sucient for our purposes. Input: n (positive integer) Output: p (the strictly increasing list of the rst n primes) Procedure: Declare i, j, k,m natural number parameters; p0 ← 2; 74 If n = 1 go to ∗ ∗ ∗ Else p1 ← 3, i← 1, m← 2 End If; For1 i < n− 1 do k ← pi + 2,m← mpi For2 k ≤ m+ 1 do j ← 0 For3 j ≤ i do If pj|k go to ∗ Else j ← j + 1 End If; Repeat End For3; ∗ If j > i go to ∗∗ Else k ← k + 2 End If; Repeat End For2; ∗∗ i← i+ 1 pi ← k Repeat End For1; ∗ ∗ ∗ Print p; End Algorithm. The termination of this algorithm is guaranteed by the respective upper bounds stipulated at the beginnings of the three For loops. The correctness is guaranteed by the well known fact which goes back to Euclid's Elements: pi+1 ≤ 1+ i Π j=0 pj, together with the simple fact that pi+1 is the rst (odd) integer greater than pi, which is not a multiple of any of p0, ..., pi. To estimate the running time, notice that (Landau 1958, p. 91) for large n, pn < n 2. 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[33] Venn, J. (1880). On the diagrammatic and mechanical representations of propositions and reasoning. Philosophical Magazine, series 5, 10:59, pp. 1-18, DOI: 10.1080/14786448008626877. Department of Mathematics Faculty of Science Cairo University Giza Egypt m0amer@hotmail.com , amer@sci.cu.edu.eg URL: http://scholar.cu.edu.eg/?q=mohamedamer/