Functors for the Piagetian Robotics Paper

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1 Functors or the Piagetian Robotics Paper Joshua Taylor 1 and Selmer Bringsjord 1,2 tayloj@cs.rpi.edu, selmer@rpi.edu Rensselaer AI & Reasoning (RAIR) Lab Department o Cognitive Science 2 Department o Computer Science 1 Rensselaer Polytechnic Institute (RPI) Troy NY USA April 22, Introduction We describe three unctors that map proos dierent logical systems. The languages under consideration the propositional calculus (L PC ), the irst-order predicate calculus (L FOL ) and a subset thereo (L FOL ), the propositional modal logic S5 (L S5 ), and the description logic ALC, (L ALC ) (Schmidt-Schauß & Smolka 1991). Functor 1 The irst unctor maps sentences and proos o L FOL to sentences and proos o L PC. The translation o ormulae is based on the Barwise & Etchemendy s (1999) truth-unction orm algorithm. Functor 2 The second unctor maps sentences and proos o L S5 to sentences and proos o L FOL. The translation rom the ormulae o L S5 to those o L FOL is based on reiying modal ormulae and by quantiying over possible worlds [cite... ]. Functor 3 The third unctor maps sentences o L ALC and the products o a decision procedure or L ALC to the sentences and proos o L FOL. We assume no amiliarity with category theory, and deine the concepts that we use. We present speciic grammars or the logical languages that we use, but do assume a general amiliarity with logic, ormal semantics, natural deduction, and sequent calculus based reasoning. 2 Formal Preliminaries A category C consists o two disjoint collections: a collection o objects, and a collection o arrows. Associated with each arrow is a domain and a codomain each o which is an object o the category. An arrow whose domain is A and codomain is B denoted : A B. For any arrows : A B and g : B C in C, there is a composite g : A C. For each object A in C, there is an identity arrow, 1 A : A A. The arrows o a category satisy associativity, i.e., or : A B, g : B C, and h : C D, h (g ) = (h g) Furthermore, composition with identity is such that or : A B, composition with 1 A and 1 B satisies 1 A = = 1 B. TODO: Deine subcategories too. Functors are mappings between categories. A unctor F : C 1 C 2 between categories C 1 and C 2 maps the object and arrows o C 1 to the objects and arrows o C 2, respectively, satisying the ollowing constraints. The domain and codomain o the mapping o an arrow are exactly the mapping o the domain and codomain o ; that is, or arrow : A B, the domain and codomain o F ( ) are F (A) and F (B). The mapping o a composition is exactly the composition o mappings; that is, F (g ) = F (g) F ( ). Finally, the mapping o an identity arrow o an object is the identity arrow o the mapping o the object; that is, F (1 A ) = 1 F (A). 1

2 Atomic Propositions Compound Propositions Form Syntax Form Syntax True Negation φ False Conjunction φ ψ Propositional Variables P 1, P 2,... Disjunction φ ψ Implication φ ψ Co-implication φ ψ Figure 1: Syntax o the Propositional Calculus, where φ and ψ represent arbitrary propositions. In practice, propositional variables may be replaced by mnemonics, e.g., r or P 4. Deductive calculi can be expressed as categories, where objects are sentences o the language on which a calculus operates, and arrows are deductions, or proos, within the calculus, subject to several restrictions. For each sentence φ in the language there must be a proo o φ rom φ (and one such proo designated as the identity arrow, 1 φ ), proos can be composed, and proo composition is associative. The axioms and inerence rules o the deductive calculus deine what other arrows will be present in the category. 3 Four Deductive Systems We now describe our languages and their deductive calculi as categories. Some symbols are used in the ormulae o multiple logical languages. Strictly speaking, each language has its own version o the symbol; we subscribe symbols with their language when necessary. For instance PC belongs to L PC ; FOL to L FOL. When there is no danger o conusion, we simply use. 3.1 Propositional Calculus The irst treat the propositional calculus. The syntax o sentences in L PC is shown in Figure 1. These sentences are the objects o C PC. We present the arrows o C PC schematically in Figure 2 (and will continue this practice or other categories). The schematic notation indicates that should the category satisy the conditions atop the line, it contains the arrows described below. The irst two schemata address the requirement that categories contain identity arrows or each o their objects and that their arrows compose. Since the arrows o C PC are proos, we take as the identity arrow or a sentence φ the simple proo o φ rom φ by reiteration, reitφ. Proos compose, and we use the standard notation or composition. Thus, the irst two schemata ensure that C PC is a category. The remaining schemata correspond to Fitch-style inerence rules, ollowing the convention o Barwise & Etchemendy s (1999) Language, Proo, and Logic. Using the arrow schemata, we may iner the presence o arrows. For instance, Figure 3 demonstrates the existence o a proo o P R rom (P Q) (Q R), i.e., that there is an arrow to the ormer rom the latter. 3.2 First-Order Predicate Calculus The next logical language and deductive calculus we treat is the irst-order predicate calculus, L FOL. We call this category C FOL. The syntax o sentences in L FOL is given in Figure 4. These sentences are the objects o C FOL. The arrows o C FOL are generated by the schemata or C PC, given in Figure 2, with the addition o those generated by the schemata shown in Figure 5. The new schemata correspond to inerence rules or handling the quantiied ormulae and equality. We also note the subcategory C FOL whose objects are the objects o C FOL and whose arrows are only those generated by Figure Propositional S5 We now turn propositional S5. Its syntax includes that o L PC, but adds sentences φ and φ, read, respectively, necessarily φ and possibly φ. C S5 s, like C FOL, includes C PC s arrow schemata. Modal logics are characterized by 2

3 φ reit φ φ [reit] φ ψ ψ g ρ φ g ρ [ ] 1 φ1... n φn intro 1,..., n φ1... φ n [ intro] φ 1... φ n elim (i) φ i [ elim] φ i intro φ 1,...,,...,φ n φ1... φ i... φ n [ intro] 1 δ1... δ n 2 δ1 φ... elim 1,..., n+1 φ n+1 δn φ [ elim] φ intro φ [ intro] φ elim φ [ elim] φ g φ intro,g [ intro] elim φ [ elim] φ 1... φ n ψ intro (i) φ 1... φ i 1 φ i+1... φ n φ i ψ [ intro] φ ψ intro φ ψ [ intro] φ ψ g φ elim,g ψ [ elim] φ ψ g ψ φ intro,g φ ψ [ intro] φ ψ (or ψ φ) elim,g φ g ψ [ elim] Figure 2: Arrow schemata o C PC. a P (P Q) (Q R) b P (P Q) (Q R) c P (P Q) (Q R) d P (P Q) (Q R) e P (P Q) (Q R) P (P Q) (Q R) g (P Q) (Q R) reitp (P Q) (Q R) P (P Q) (Q R) elim a(1) P elim a(2) P Q elim c,b Q elim a(3) Q R elim e,d R intro (1) P R Figure 3: Demonstration o an arrow (P Q) (Q R) P R. 3

4 Terms Quantiied Formulae Type Syntax Type Syntax Variable v 1, v 2,... Universal Quantiication v i φ(v i ) Function Application i n(τ 1,...,τ n ) Existential Quantiication v i φ(v i ) Atomic Formulae Compound Formulae Type Syntax Type Syntax True Negation φ False Conjunction φ ψ Predicate Application p n i (τ 1,...,τ n ) Disjunction φ ψ Identity τ 1 = τ 2 Implication φ ψ Co-implication φ ψ Figure 4: Syntax o First-Order Predicate Calculus. φ and ψ are arbitrary ormulae, τ i an arbitrary term. Constants are denoted by applications o nullary unctions i 0 (ater which parentheses may be omitted). Sentences are precisely those ormulae in which no variable appears without the scope o a quantiier. In practice we will use mnemonics or unctions and predicate symbols, e.g., MotherO or 4 1 and Likes or p2 78. = intro n n = n [= intro] φ g n = m (or m = n) = elim,g φ[n m] [= elim] φ [ intro] intro,c,x x φ[c x] where c does not appear in x φ elim,c φ[x c] [ elim] φ intro,c,x x φ[c x] [ intro] x φ g φ[x c] ψ [ elim] elim,g ψ where c does not appear in or ψ Figure 5: Additional arrow schemata or C FOL. 4

5 φ nec φ [nec] (φ ψ) dist φ,ψ φ ψ [dist] φ T φ φ [T] φ 5 φ φ [5] Figure 6: Arrow schemata or S5 s characteristic rule s and axioms. Concept Descriptions Description Type Syntax Semantics Top (Universal Concept) I Bottom (Empty Concept) /0 Atomic Concept A A I Negation (Complement) C I \C I Intersection C D C I D I Union C D C I D I Value Restriction R.C {x y (x,y) R I y C I } Existential Restriction R.C {x y (x,y) R I y C I } Axioms Name Syntax Semantics Concept Subsumption C D C I D I Concept Equivalence C D C I = D I Concept Membership C(a) a I C I Role Membership (R(a,b)) ((a I,b I ) R I ) Figure 7: Syntax and semantics o ALC concept descriptions and axioms. A is a an atomic concept name, C and D are any concept names, R and S role names, and a and b object names. their rules and axioms; S5, speciically, by the necessitation rule, distribution axiom, T, and 5. These, which complete the schemata o C S5, are given in Figure Description Logic Description logics are named or their ocus on concept and role descriptions. A concept represents a set o individuals within the domain o an interpretation. An interpretation I is a non-empty set I, called the interpretation s domain, and a unction, ( ) I, mapping atomic concept names to subsets o the domain and role names to subsets o I I. The syntax and semantics or the language ALC are given in Figure 7. There are several techniques or reasoning within description logics, the most common o which are tableau algorithms. The structure o description logics is such that eicient tableau algorithms can be designed that decide the satisiability o a set o axioms. These algorithms determine whether or not there is an interpretation that satisies each axiom o a set, but do not produce object-level proos. We briely describe a tableau algorithm that decides ALC satisiability. The tableau algorithm determines whether a set o axioms is satisiable. Tableau rules speciy ways o augmenting the set and splitting the satisiability decision procedure recursively Each o the ollowing rules can be applied to a set o axioms A to produce one or more new sets A, A,..., with the relationship that i any o the new sets are satisiable, A is satisiable, but i every new set is unsatisiable, A is unsatisiable. 5

6 Σ reit Σ Σ [reit] Σ Γ Σ g Γ g [ ] {C(a), C(a)} Σ clash 1 C(a),Σ /0 [clash 1 ] { (a)} Σ clash 2 (a),σ /0 [clash 2 ] {C(a), D(a), (C D)(a)} Σ /0 {(C D)(a)} Σ /0 [ ] {C(a), (C D)(a)} Σ /0 {D(a), (C D)(a)} Σ g /0 {(C D)(a)} Σ,g /0 [ ] {R(a,b), C(b), R.C(a)} Σ /0 { R.C(a)} Σ /0 (b not appearing in Σ) [ ] {C(b), R.C(a), R(a,b)} Σ /0 { R.C(a), R(a,b)} Σ /0 [ ] Figure 8: Arrow schemata or C ALC. (C 1 C 2 )(x) A C 1 (x) A or C 2 (x) A A {C 1 (x),c 2 (x)} A [ -rule] (C 1 C 2 )(x) A C 1 (x) A C 2 (x) A A {C 1 (x)} A A {C 2 (x)} A [ -rule] R.C(x) A no y such that R(x,y) A,C(y) A A {C(a)} A or a new name y [ -rule] R.C(x) A R(x,y) A C(y) A A {C(y)} A [ -rule] A set is contains a clash when, or some individual a and concept description C, it contains C(a) and C(a). I a set does not contain a clash, and none o the rules can be applied to it, it is satisiable. The rules are applied nondeterministically to an initial set o interest, until the set is shown satisiable or unsatisiable. Discuss rewriting into normal orm. Discuss issues with cyclic deinitions. The decision procedure just described is not a deductive calculus, being concerned with determining the satisiability o sets o axioms rather than ormulae that ollow a set. However, we can create a deductive calculus rom the tableau algorithm, and then construct a category based on it. We develop calculus and category in parallel. The objects o the calculus are the same as the objects on which the tableau rules operate, i.e., sets o ALC axioms. Eight arrow schemata, given in Figure 8, generate the arrows o the category. The irst pair, reit and, guarantee that C ALC is a category. The next two clash rules guarantee that, or any concept C and individual a, any set containing C(a) and ( C)(a) is unsatisiable, as is any set containing (a). The remaining rules correspond to the tableau algorithm rules. Recall that the tableau algorithm produces rom a set A a number o new assertion boxes A,A,... such that i each o A,A,... is unsatisiable, then A is unsatisiable. Each o the arrow schemata captures the notion that i every set generated by a tableau rule is unsatisiable, then the input o the tableau is also unsatisiable. For instance, the -rule applies to a set A containing (C D)(a), but neither C(a) nor D(a), and generates A = {C(a)} A and A = {D(a)} A, such that i both A and A are unsatisiable, then A is unsatisiable, and this relationship is captured by the schema. Figure 9 compares tableau-based and arrow-based demonstrations o inconsistency. 4 Functors We now construct three unctors between the our categories just described. Particularly, F 1 : C FOL C PC, F 2 : C S5 C FOL, and F 3 : C ALC C FOL. For each unctor, we deine the mapping between objects, and then show that or each arrow in the source category, there is a corresponding arrow in the target category, and based on the correspondence, we deine the mapping o arrows. 4.1 F 1 : C FOL C PC F 1 maps the objects and arrows o C FOL to the objects and arrows o C PC. The objects o C FOL are exactly the objects o C FOL, i.e., the sentences o L FOL. F 1 maps each sentence o L FOL to its truth-unctional orm, speciied in 6

7 1 ( C D)(b), ( R.C R. D)(a), R(a,b) 3 ( R.C)(a) -rule 2 ( C)(b), D(b) -rule -rule 4 ( R. D)(a) 5 C(b) -rule -rule 6 ( D)(b) a b c d e {( D)(b), ( R. D)(a), C(b), D(b), ( C D)(b), ( R.C R. D)(a), R(a,b)} {( R. D)(a), C(b), D(b), ( C D)(b), ( R.C R. D)(a), R(a,b)} {C(b), ( R.C)(a), C(b), D(b), ( C D)(b), ( R.C R. D)(a), R(a,b)} {( R.C)(a), C(b), D(b), ( C D)(b), ( R.C R. D)(a), R(a,b)} { C(b), D(b), ( C D)(b), ( R.C R. D)(a), R(a,b)} {( C D)(b), ( R.C R. D)(a), R(a,b)} clash 1 D(b) /0 a /0 clash 1 C(b) /0 c /0 d,b /0 e /0 Figure 9: Comparison o tableau and arrow demonstrations o unsatisiability. In the tableau demonstration, wherein each box represents the set o its contents and those o its ancestors, boxes 5 and 6 contain clashes, and close each branch o the tableau, and showing that 1 is unsatisiable. The arrow demonstration starts with the unsatisiability o the ully expanded sets, and de-expands the sets into the original set, carrying the unsatisiability result along the way. 7

8 Object mapping o F 1 Sentence Type C FOL object C PC object Sentence Type C FOL object C PC object True (False) ( ) ( ) Negation φ F 1 (φ) Predicate Application p n i (τ 1,...,τ n ) P p n i (τ 1,...,τ n ) Conjunction φ ψ F 1 (φ) F 1 (ψ) Identity τ 1 = τ 2 P τ1 =τ 2 Disjunction φ ψ F 1 (φ) F 1 (ψ) Universal Quantiication x φ(x) P x φ(x) Implication φ ψ F 1 (φ) F 1 (ψ) Existential Quantiication x φ(x) P x φ(x) Co-implication φ ψ F 1 (φ) F 1 (ψ) Figure 10: F 1 maps sentences o L FOL to their truth-unctional orm. Each predicate application, identity, and quantiication is mapped to its own unique propositional variable. Object mapping o F 2, by F 2 (w,φ) Sentence Type C S5 object C FOL object Sentence Type C S5 object C FOL object True (False) ( ) ( ) Negation φ F 2 (w,φ) Conjunction φ ψ F 2 (w,φ) F 2 (w,ψ) Disjunction φ ψ F 2 (w,φ) F 2 (w,ψ) Implication φ ψ F 2 (w,φ) F 2 (w,ψ) Co-implication φ ψ F 2 (w,φ) F 2 (w,ψ) Necessity φ w F 2 (w,φ) Possibility φ w F 2 (w,φ) Figure 11: Object mapping o F 2. F 2 (φ) is a special case o the more general translation, F 2 (w,φ), with respect to a particular world w, where the world is w. We present here the more general F 2. Figure 10. Since, by deinition, the arrows o C FOL are generated by the same schemata as the arrows o C PC, F 1 simply maps each arrow o C FOL to its counterpart in C PC. 4.2 F 2 : C S5 C FOL F 2, which maps the sentences and proos o L S5 to sentences and proos o L FOL is based on encoding the Kripke semantics o S5 in the irst-order predicate calculus. Kripke semantics speciy a set o possible worlds, a privileged element representing the real world, and an accessibility relation over the worlds. Each world provides an interpretation or modal ormulae, subject to the constraint that φ is true in a world w when φ exactly when φ is true in every world accessible rom w. Under these semantics, the various modal logics can be classiied based on the structure o their accessibility relations. The translation o propositional modal sentences into irst-order sentences is based on taking the universe o discourse as possible worlds, designating a particular constant the real world, and parameterizing propositional variables over worlds by converting the propositional variables into singulary predicates. We let w denote the real world, and R(w 1,w 2 ) that w 2 is accessible rom w 1. As an example, the modal propositional ormula (P Q) would be translated w R(w,w) P(w) Q(w). S5 is characterized by total accessibility between worlds, so the quantiied ormulae in translations may replace the conditional whose antecedent is an accessibility claim by its consequent. The complete translation o sentences is given in Figure 11. Some o the arrows o C S5 are generated by schemata shared with C PC, and these arrows are mapped to their appropriate counterparts. That the arrows introduced to capture the modal inerence rules and axioms have counterparts in C FOL requires demonstration. We now show that nec, dist, T, and 5 have counterparts in C FOL. The knowledge generalization rule nec is derived in Figure 12, but as the only schema with preconditions, deserves some note. First, we assume that the initial arrow : S5 φ has a counterpart in C FOL. Since this counterpart necessarily has FOL as its domain, w cannot appear in its domain and so is a candidate or intro, generalizing rom w to a universally quantiied w. The counterpart to dist is a proo showing that x φ(x) ψ(x) = x φ(x) x ψ(x), whose existence is demonstrated in Figure 13. T is the simplest arrow to map. The translation o φ airms that the world-respecting translation o φ holds in every world w, and the translation o φ that is holds in the actual world, w, so T s counterpart is based on elim. Figure 14 gives the derivation. 5 s counterpart is an arrow to xφ(x) rom y x φ(x). The derivation is given in Figure 15. 8

9 S5 φ a FOL F 2 ( ) F 2 (w,φ) b FOL intro a,w w F 2 (w,φ) F 2 ( S5 ) F 2 ( ) nec F 2 ( φ) Figure 12: Demonstration that F 2 ( nec ) is well deined. a b c d e g h w F 2 (w,φ) F 2 (w,ψ) F 2 ( (φ ψ)) reit ( w F2 (w,φ)) ( w F 2 (w,φ) F 2 (w,ψ)) elim a(2) elim b,w elim a(1) elim d,w elim c,e intro,w,w intro g(1) dist φ,ψ F 2 ( ) F 2 ( φ ψ) w F 2 (w,φ) F 2 (w,ψ) F 2 (w,φ) F 2 (w,ψ) w F 2 (w,φ) F 2 (w,φ) F 2 (w,ψ) w F 2 (w,ψ) w F 2 (w,φ) w F 2 (w,ψ) Figure 13: Demonstration that F 2 ( dist ) is well deined. a b w F 2 (w,φ) w F 2 (w,φ) F 2 ( φ) reit w F 2 (w,φ) elim a(w ) F 2 ( ) T φ w F 2 (w,φ) F 2 (w,φ) F 2 (φ) Figure 14: Demonstration that F 2 ( T ) is well deined. a b c d e w F 2 (w,φ) w w F 2 (w,φ) w F 2 (w,φ) w w F 2 (w,φ) w w F 2 (w,φ) w F 2 (w,φ) w w F 2 (w,φ) w w F 2 (w,φ) F 2 ( φ) reit w F 2 (w,φ) w w F 2 (w,φ) w F 2 (w,φ) w w F 2 (w,φ) elim a(1) w F 2 (w,φ) intro b(1) w F 2 (w,φ) w F 2 (w,φ) elim a(1) w w F 2 (w,φ) elim d,c,x w F 2 (w,φ) or any x F 2 ( 5 φ ) F 2 ( φ) Figure 15: Demonstration that F 2 ( 5 ) is well deined. 9

10 Object Translation o F 3 Type C S5 object C FOL object Empty Set /0 Singleton Set {φ} F 3 (φ) General Set Σ F 3 (φ) F 3 (Σ\{φ}) where φ is the minimum element o Σ Axiom Translation F 3 Type ALC axiom C FOL sentence Type ALC axiom C FOL sentence Subsumption C D x F 3 (C,x) F 3 (D,x) Concept Membership C(a) F 3 (C,c a) Equivalence C D x F 3 (C,x) F 3 (D,x) Role Membership R(a,b) F 3 (R,c a,c b ) Concept and Role Membership Translation F 3 x Type Arguments C FOL ormula Type Arguments C FOL ormula Top/Bottom /, x / Negation C, x F 3 (C,x) Atomic A, x P A (x) Role R, x, y P R (x,y) Value R.C, x y F 3 (R,x,y) F 3 (C,y) Existential R.C, x y F 3 (R,x,y) F 3 (C,y) Intersection C D, x F 3 (C,x) F 3 (D,x) Union C D, x F 3 (C,x) F 3 (D,x) Figure 16: The object mapping o F 3. Axiom sets are mapped to conjunctions, atomic concept and role names A and R to predicate symbols P A and P R respectively, and individual name a to constant symbol c a. 4.3 F 3 : C ALC C FOL Our third unctor, F 3, maps rom the description logic ALC to a subset o irst-order logic. The correspondence o ALC, as well as o other description logics, to ragments o irst-order logic is well-established. While the expressions themselves have a well deined mapping to the ormulae o irst-order logic, the tableaux produced do not have immediate correspondents in natural deduction style proo calculi. (There are, o course, tableaux algorithms or irst-order logic; to these the tableaux produced or description logics can be more readily mapped.) The object mapping o F 3 is straightorward. It is based on translating concept names as singulary predicates, role names as binary relations, individual names as constant symbols; complex concept constructors are translated recursively by auxiliary unctions F 3 and F 3. The auxiliary conjunctions and alse symbol introduced in Section 3.4 are mapped to conjunctions and the alse symbol o C FOL. The ull translation is given schematically in Figure 16. As with the other unctors, we present derivations o the correspondents or the arrow schemata o C ALC ormally and in prose. The clash arrow s counterpart is easily derived. The translation o a conjunction containing both C(x) and C(x) will contain as conjuncts both F 3 (C(x)) and F 3 (C(x)). Arrows or elim and intro quickly produce FOL to which ALC maps. This is given ormally in Figure 17. The -rule is a bit more complex ormally, but is intuitively simple, being primarily an application o elim. The derivation is shown in Figure 18. The arrow corresponding to the -rule is a special instance o proo by cases, the method which elim captures. A derivation is shown in Figure 19 The -rule has been recast as an arrow schema similar to elim, and the correspondence in C FOL is given in Figure 20. The -rule corresponds, roughly, to C FOL s elim. A derivation is given in Figure 21. In each o these ormal derivations, the bulk o the work is in breaking down and rebuilding conjunctions in n arbitrary order. Since ALC directly is equivalent to a ragment o irst-order logic, it is not surprising that the tableau algorithms match so closely inerence rules or irst-order logic. 10

11 a F 3 ({C(a), ( C)(a)} Σ) b F 3 ({C(a), ( C)(a)} Σ) c F 3 ({C(a), ( C)(a)} Σ) d F 3 ({C(a), ( C)(a)} Σ) reit F 3 ({C(a), ( C)(a)} Σ) φ 1... φ n where φ i = F 3 (C(a)), and φ j = F 3 (( C)(a)) elim a(i) P C (c a ) elim a( j) P C (c a ) intro b,c F 3 ({C(a), ( C)(a)} Σ) F 3 ( clash 1 ) F 3 (/0) a F 3 ({ (a)} Σ) b F 3 ({ (a)} Σ) reit φ 1... φ n where φ i = F 3 ( (a)) = elim a(i) F 3 ({ (a)} Σ) F 3 ( clash 2 ) F 3 (/0) Figure 17: Demonstration that F 3 ( clash 1 ) and F 3 ( clash 2 ) are well deined. {C(a), D(a), (C D)(a)} Σ /0 a F 3 ({C(a), D(a), (C D)(a)} Σ) F 3 ( ) b F 3 ({(C D)(a)} Σ) reit ψ 1... ψ n ψ1... ψ n where ψ i = F 3 ((C D)(a)) c x=1,...,n F 3 ({(C D)(a)} Σ) elim b(x) ψ x d F 3 ({(C D)(a)} Σ) elim c i (1) F 3 (C(a)) e F 3 ({(C D)(a)} Σ) elim c i (2) F 3 (D(a)) F 3 ({(C D)(a)} Σ) intro F 3 ({C(a), D(a), (C D)(a)} Σ) g F 3 ({(C D)(a)} Σ) a F 3 ({(C D)(a)} Σ) F 3 ( ) F 3 (/0) Figure 18: Demonstration that F 3 ( ) is well deined. In, intro is parameterized by the appropriate permutation o (a subset o) c 1,..., c n, d, and e. {C(a), (C D)(a)} Σ /0 C(a), D(a) Σ {D(a), (C D)(a)} Σ g /0 a φ 1... φ n1 F 3 ( ) where φ i = F 3 (C(a)) b ψ 1... ψ n2 F 3 (g) where ψ j = F 3 (D(a)) c F 3 ({(C D)(a)} Σ) d F 3 ({(C D)(a)} Σ) e F 3 ({(C D)(a)} Σ) F 3 ({(C D)(a)} Σ) intro a(i) intro b( j) F 3 (C(a)) F 3 (D(a)) reit ρ 1... ρ n3 ρ1... ρ n3 where ρ k = F 3 ((C D)(a)) = F 3 (C(a)) F 3 (D(a)) elim e(k) elim,c,d g F 3 ({(C D)(a)} Σ) F 3 ({(C D)(a)} Σ) F 3 ( ) F 3 (/0) F 3 (C(a)) F 3 (D(a)) Figure 19: Demonstration that F 3 ( ) is well deined. 11

12 {R(a,b), C(b), R.C(a)} Σ a F 3 ({R(a,b), C(b), R.C(a)} Σ) b (F 3 (R(a,b)) F 3 (C(b))) F 3 ({ R.C(a)} Σ) c d e g x=1,...,n h i j F 3 ({ R.C(a)} Σ) k l m F 3 ({ R.C(a)} Σ) F 3 ( ) F 3 (/0) /0 b not appearing in Σ F 3 ( ) reit elim b(1) F 3 (R(a,b)) F 3 (C(b)) elim c(1) F 3 (R(a,b)) elim c(2) F 3 (C(b)) elim b(2) ψ 1... ψ n elim (x) ψ x intro F 3 ({R(a,b), C(b), R.C(a)} Σ) a i intro j(1) F 3 ({R(a,b), C(b)}) reit φ 1... φ n φ1... φ n where φ j = F 3 ( R.C(a)) elim l( j) F 3 ( R.C(a)) elim m,l Figure 20: Demonstration that F 3 ( ) is well deined. Step h is parameterized by a permutation o (a subset o) d, e, and g 1,..., g n. {C(b), R.C(a), R(a,b)} Σ a F 3 ({C(b), R.C(a), R(a,b)} Σ) b F 3 ({ R.C(a), R(a,b)} Σ) c x=1,...,n F 3 ({ R.C(a), R(a,b)} Σ) d F 3 ({ R.C(a), R(a,b)} Σ) e F 3 ({ R.C(a), R(a,b)} Σ) F 3 ({ R.C(a), R(a,b)} Σ) g F 3 ({ R.C(a), R(a,b)} Σ) h F 3 ({ R.C(a), R(a,b)} Σ) F 3 ({ R.C(a), R(a,b)} Σ) /0 F 3 ( ) reit φ 1... φ n φ1... φ n where φ i = F 3 ( R.C(a)), φ j = F 3 (R(a,b)) elim b(x) elim c i (b) elim b( j) elim d,e intro φ x F 3 (R(a,b)) F 3 (C(b)) F 3 (R(a,b)) F 3 (C(b)) F 3 ({ R.C(a), R(a,b), C(b)} Σ) a g F 3 ( ) F 3 (/0) Figure 21: Demonstration that F 3 ( ) is well deined. In g, intro is parameterized by an appropriate permutation o (a subset o) c 1,...,c n, and. 12

13 Reerences Awodey, S. (2006), Category Theory, number 49 in Oxord Logic Guides, Oxord University Press. Barr, M. & Wells, C. (1999), Category Theory or Computing Science, third edn, Les Publications CRM, Montréal. Barwise, J. & Etchemendy, J. (1999), Language, Proo, and Logic, Seven Bridges, New York, NY. Konyndyk, K. (1986), Introductory Modal Logic, University o Notre Dame Press, Notre Dame, IN. Lambek, J. & Scott, P. J. (1986/1988), Introduction to higher order categorical logic, number 7 in Cambridge studies in advanced mathematics, irst paperback edn, Cambridge University Press. Schmidt-Schauß, M. & Smolka, G. (1991), Attributive concept descriptions with complements, Artiicial Intelligence 48(1),