Intuitionistic Euler-Venn Diagrams (extended)
aa r X i v : . [ c s . L O ] F e b Intuitionistic Euler-Venn Diagrams (extended) ⋆ Sven Linker
University of Liverpool, UK [email protected]
Abstract.
We present an intuitionistic interpretation of Euler-Venn di-agrams with respect to Heyting algebras. In contrast to classical Euler-Venn diagrams, we treat shaded and missing zones differently, to havediagrammatic representations of conjunction, disjunction and intuition-istic implication. We present a cut-free sequent calculus for this language,and prove it to be sound and complete. Furthermore, we show that therules of cut, weakening and contraction are admissible.
Keywords: intuitionistic logic · Euler-Venn diagrams · proof theory Among diagrammatic systems to reason about logic, Euler-Venn circles havea long tradition. They are known to be a well-suited visualisation of classicalpropositional logic. In previous work [11], we have presented a proof system inthe style of sequent calculus [5] to reason with Euler-Venn diagrams. There, wespeculated that, similar to sentential languages, restricting the rules and sequentsin the system would allow for intuitionistic reasoning with Euler-Venn diagrams.However, further investigation showed that such a simple change is not sufficient,due to the typical use of the syntax elements of Euler-Venn diagrams.Consider for example the diagrams in Fig. 1. In the classical interpretation,these diagrams are equivalent: the shaded zone in Fig. 1a denotes that the situa-tion that a is true and b is false is prohibited, which is exactly what the omissionof the zone included in the contour a , but not in b in Fig. 1b signifies as well. a b (a) b a (b) Fig. 1.
Euler-Venn diagrams
That is, shading a zone and omitting it is equiv-alent in classical Euler-Venn diagrams. Addition-ally, we can interpret these two diagrams in twoways: Fig. 1a may intuitively be read as ¬ ( a ∧ ¬ b ):we do not allow for the valuations satisfying a , butnot b . Fig. 1b, however, is more naturally read as a → b : whenever a valuation satisfies a , it also sat-isfies b . While in a classical interpretation, thesetwo statements are indeed equivalent, they are generally not equivalent in an in-tuitionistic interpretation. Hence, we want to treat missing zones and shaded ⋆ This work was supported by EPSRC Research Programme EP/N007565/1
Scienceof Sensor Systems Software . Sven Linker zones differently. Since typically, proof systems for Euler diagrams allow tochange missing zones into shaded zones [11,8,19], this implies a stronger de-viation from our sequent calculus rules than anticipated.Furthermore, we want to emphasise a constructive approach to reasoning. Inparticular, instead of emphasising a negative property by prohibiting interpre-tations of the diagrams, we will treat shading as a positive denotation. Whilethis would not make much of a difference in a classical system, negation in in-tuitionistic systems is much weaker, and hence not suited as a basic element forthe semantics of a language.In this paper, we present an intuitionistic interpretation of Euler-Venn di-agrams that takes the preceeding considerations into account. To that end, wewill distinguish between pure Venn , pure Euler and Euler-Venn diagrams, andpresent intuitionistic interpretations of these types of diagrams based on Heyt-ing algebras. Subsequently, we present a proof system in the style of sequentcalculus, which we prove to be sound and complete. Furthermore, we show thatthe structural rules of weakening, contraction and cut are admissible.
Related Work.
Many reasoning systems for visualisations of classical logic havebeen defined over time, for example the initial work of Venn [21] and Peirce [7]and subsequently the work of Shin [18] and Hammer [6], as well Spider diagramsby Howse et al. [8]. Most of these systems are not directly comparable to sen-tential reasoning systems, due to very different structure of the rules, with thenotable exception of the work by Mineshima et al. [14] and Takemura [20].However, the situation is different for non-classical logics. There are severalvisual reasoning systems for non-classical variants of Existential Graphs. For ex-ample, Bellucci et al. defined assertive graphs [1], including a system based onrules for iteration and deletion of graphs, among others. This logical languagereflects intuitionistic logic, but the rules manipulate only single graphs, whilesequent calculus systems manipulate sequents of diagrams. Ma and Pietarinenpresented a graphical system for intuitionistic logic [13] and proved its equiv-alence with Gentzen’s single succedent sequent calculus for propositional intu-itionistic logic. To that end, they translate the graphs into sentential formulas.They also extended their approach to existential graphs with quasi-Boolean al-gebras as their semantics [12]. Legris pointed out that structural rules of sequentcalculi can be seen as special instances of rules in the proof systems for existen-tial graphs, to analyse substructural logics [10]. de Freitas and Viana presented acalculus to reason about intuitionistic equations [4]. However, we are not awareof any intuitionistic reasoning systems using Euler-Venn-like visualisations.
Structure of the paper.
Following this introduction, we briefly recall the foun-dations of intuitionistic logic and its semantics in terms of Heyting algebras inSect. 2. In Sect. 3, we define the system of Euler-Venn diagrams, followed by thegraphical sequent calculus system, as well as soundness and completeness proofs,in Sect. 4. Section 5 contains proofs for the admissibility of the structural rules.Finally, we discuss our system and conclude the paper in Sect. 6. ntuitionistic Euler-Venn Diagrams (extended) 3
In this section, we give a very brief overview of the aspects of propositionalintuitionistic logic we will use. We start by presenting the underlying semanticalmodel we use, Heyting algebras.
Definition 1 (Heyting Algebra). A Heyting algebra H = ( H, ⊔ , ⊓ , , , isa bounded, distributive lattice, where ⊔ is the join, ⊓ the meet, the bottom and the top element of the lattice. Observe that such a bounded lattice possesses anatural partial order ≤ on its elements. The binary operation , the implication ,is defined by c ⊓ a ≤ b if, and only if, c ≤ a → b . That is, a → b is the join ofall elements c such that c ⊓ a ≤ b . We will use the abbreviation − a for a .Furthermore, we set d i ∈∅ a i = 1 and F i ∈∅ a i = 0 for any a i . We collect a few basic properties of Heyting algebras that we need in thefollowing. Proofs can be found, e.g., in the work of Rasiowa and Sikorski [17].
Lemma 1 (Properties of Heyting Algebras).
Let H be a Heyting algebra.Then for all elements a , b and c , we have a ⊓ ( a b ) ≤ b (1) ( a b ) ⊓ b = b (2) a ( b c ) = ( a ⊓ b ) c (3)The syntax of propositional intuitionistic logic is similar to classical Booleanlogic, with the difference that the operators are not interdefinable. Hence, thesigns for conjunction, disjunction, and implication are all necessary as distinctsymbols, and cannot be treated as abbreviations. We will assume a fixed, count-able set of propositional variables Vars . Definition 2 (Syntax).
An intuitionistic formula is given by the followingEBNF ϕ : = ⊥ | p | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ , where p ∈ Vars . We will treat negation as the abbreviation ¬ ϕ ≡ ϕ → ⊥ . Furthermore, welet ⊤ ≡ ⊥ → ⊥ . The semantics of a formula is based on valuations, associatingeach variable with an element of a given Heyting algebra. Definition 3 (Semantics).
Let H be a Heyting algebra and ν : Vars → H a valuation , mapping variables to elements of H . We lift valuations to formulas. ν ( ⊥ ) = 0 ν ( ϕ ∧ ψ ) = ν ( ϕ ) ⊓ ν ( ψ ) ν ( ϕ ∨ ψ ) = ν ( ϕ ) ⊔ ν ( ψ ) ν ( ϕ → ψ ) = ν ( ϕ ) ν ( ψ ) A formula ϕ holds in H , if ν ( ϕ ) = 1 . If ϕ holds for every valuation of H , wewrite H | = ϕ . If H | = ϕ for every Heyting algebra H , we say that ϕ is valid . Sven Linker
In this section, we present the syntax and semantics of Euler-Venn diagramswith an intuitionistic interpretation. Generally, a diagram can be unitary or compound . A unitary diagram consists of a set of contours dividing the spaceenclosed by a bounding rectangle into different zones . Zones may also be shaded.Depending on how the contours may be arranged, and whether zones may beshaded, we distinguish between Venn diagrams,
Euler diagrams, and
Euler-Venn diagrams. Compound diagrams are constructed recursively. Since the structureof compound diagrams is the same, regardless of the type of unitary diagrams,we present their syntax first.
Definition 4 (Compound Diagrams). A compound diagram is created ac-cording to the following syntax, D ::= d | D ∧ D | D ∨ D | D → D , where d is a unitary diagram. Definition 5 (Compound Diagram Semantics).
The semantics of com-pound diagrams for a Heyting algebra H and a valuation ν is given as follows. ν ( D ∧ D ) = ν ( D ) ⊓ ν ( D ) ν ( D → D ) = ν ( D ) ν ( D ) ν ( D ∨ D ) = ν ( D ) ⊔ ν ( D ) where D , D are compound diagrams. If ν ( D ) = 1 , for all intuitionistic models H and valuations ν then we call D valid . Observe that we did not give the semantics for unitary diagrams in the previousdefinition. While we will fill this gap in the next sections, we first present nota-tions that are used for all types of diagrams alike. Formally, a zone for a finiteset of contours L ⊂ Vars is a tuple ( in , out ), where in and out are disjoint subsetsof L such that in ∪ out = L . We will also write in ( z ) and out ( z ) to refer to thecorresponding sets of contours in z . The set of all possible zones for a given setof contours is denoted by Venn ( L ). Venn Diagrams A Venn diagram is a diagram where all possible zones fora set of contours are visible. Formally, a Venn diagram is of the shape d =( L, Venn ( L ) , Z ∗ ). Hence the only diagrammatic elements that may carry mean-ing are the presence of contours, and whether a zone is shaded. For a givendiagram d , we denote the set of shaded zones also by Z ∗ ( d ). We allow for thediagrams ⊥ = ( ∅ , { ( ∅ , ∅ ) } , ∅ ) and ⊤ = ( ∅ , { ( ∅ , ∅ ) } , { ( ∅ , ∅ ) } ). A literal is a Venndiagram for a single contour, with exactly one shaded zone. If the zone ( ∅ , { c } )is shaded in a literal, then we call it the negative literal for c , otherwise it is thepositive literal for c (see Fig. 2). Furthermore, if d is the positive literal for c ,then we call the negative literal for c the dual of d (and vice versa). Observethat our notion of literals deviates from the original definition of Stapleton andMasthoff [19] and from our previous work [11]. The main difference between ourpresentation and classical Venn diagrams is the interpretation of shaded zones. ntuitionistic Euler-Venn Diagrams (extended) 5 c c Fig. 2.
Literals
While in the traditional approach, shading denotes the emptiness of sets, we use shading as a marker of elements.That is, the semantics of a diagram consists of the join ofthe elements denoted by the shaded zones. This is more inline with the constructivist approach we want to emphasise: instead of relying ona negative aspect (emptiness), we construct the semantics out of their buildingblocks (the shaded zones).
Definition 6 (Zone Semantics).
Let H be a Heyting algebra, ν a valuation,and z a zone. The semantics of z is given by ν ( z ) = d c ∈ in ( z ) ν ( c ) ⊓ d c ∈ out ( z ) − ν ( c ) . With the semantics of single zones defined, we can now define the semantics ofa Venn diagram in general.
Definition 7 (Venn Diagram Semantics).
For a Venn diagram d , a Heytingalgebra H and a valuation ν , the semantics of d are given by ν ( d ) = F z ∈ Z ∗ ( d ) ν ( z ) . Note that we have ν ( ⊤ ) = 1 and ν ( ⊥ ) = 0, for any Heyting algebra H and valuation ν . Furthermore, for a unitary diagram with a single contour andno shaded zones, i.e. d = ( { a } , Venn ( { a } ) , ∅ ), we have ν ( d ) = 0. However, thesemantics already diverge from the classical case for a fully shaded diagram withone contour: if d = ( { a } , Venn ( { a } ) , Venn ( { a } )), then ν ( d ) = ν ( a ) ⊔− ν ( a ), whichin general is not equal to 1.Note that this semantics has one consequence in particular: we can decomposea zone into an equivalent compound diagram, and we can furthermore decomposeany unitary Venn diagram into a disjunctive normal form. Lemma 2.
Let z be a zone for the contours L . Then the semantics of the com-pound diagram d z = V c ∈ in ( z ) c ∧ V c ∈ out ( z ) c equals the semantics of z , i.e. ν ( d z ) = ν ( z ) . Furthermore, for a Venn diagram d = ( L, Venn ( L ) , Z ∗ ) , we have ν ( d ) = ν ( W z ∈ Z ∗ d z ) .Proof. Immediate by the semantics in Def. 6 and Def. 7. ⊓⊔ In particular, this implies that we cannot draw a unitary diagram that expressesintuitionistic implication.
Lemma 3.
Let a and b be propositional variables. Then there is no unitary Venndiagram d such that ν ( d ) = ν ( a → b ) for all models and valuations.Proof. By Lemma 2, every unitary diagram d can be expressed by using ∨ and ∧ only. However, → is not definable by any combination of ∨ and ∧ [17]. ⊓⊔ Observe however that we can trivially define a compound diagram a → b . Sven Linker
Pure Euler Diagrams
We need additional syntax if we want to express intuition-istic implication diagrammatically. This new syntax needs to be directed (since a → b is semantically different to b → a ). Observe that our notion of zones is already directed, and expresses topological information. So, a natural consid-eration is to allow for missing zones in the diagrams. Hence, instead of usingVenn diagrams we will now discuss pure Euler diagrams. In contrast to shadedzones, we will treat the missing zones as “restrictions on the construction” of thesemantics. First, we give the semantics of a missing zone. Definition 8 (Missing Zone Semantics).
For a Heyting algebra H , a val-uation ν and a zone z , the missing zone semantics of z is given by ν ( z ) M = (cid:16) d c ∈ in ( z ) ν ( c ) (cid:17) (cid:16)F c ∈ out ( z ) ν ( c ) (cid:17) . Definition 9 (Pure Euler Diagrams). A pure Euler diagram is a structure d = ( L, Z ) , where L is the set of contours and Z the set of visible zones of d .Furthermore, the set MZ ( d ) = Venn ( L ) \ Z is the set of missing zones of d . Thesemantics of pure Euler diagrams is that they require the constraints defined bytheir missing zones to be true. That is, for a pure Euler diagram d , we have ν ( d ) = d z ∈ MZ ( d ) ν ( z ) M . In contrast to Venn diagrams, pure Euler diagrams do not allow for any shading. To distinguish pure Euler diagrams from Venn diagrams (and Euler-Venn diagrams, see below), we draw them with dotted contours.Even with this additional syntax, we are not able to express every im-plication. A simple example would be a → a , since we cannot have a zone( { a } , { a } ). However, for this particular example, we do not lose expressivity,since a → a ≡ ⊤ for all a . But we have a diagram equivalent to a → b , b a a b Fig. 3.
Pure Euler Diagrams as shown in the left diagram of Fig. 3. The rightdiagram in Fig. 3 denotes ( a ⊓ b )
0, which is − ( a ⊓ b ). Observe that in contrast to Venn di-agrams without shaded zones, a pure Euler di-agram without missing zones denotes 1, i.e., for d = ( L, Venn ( L )), we have ν ( d ) = ν ( ⊤ ) = 1.Furthermore, the diagram without any contours and zones denotes 0, since ν (( ∅ , ∅ )) = ν (( ∅ , ∅ )) M = d c ∈∅ ν ( c ) F c ∈∅ ν ( c ) = 1 c abstractly.Intuitively such a zone is split into two zones z and z ′ that only differ insofar,as c is in in ( z ) and in out ( z ′ ). Definition 10 (Adjacent Zone).
Let z = ( in , out ) be a zone for the contoursin L and c ∈ L . The zone adjacent to z at c , denoted by z c is ( in ∪ { c } , out \ { c } ) ,if c ∈ out and ( in \ { c } , out ∪ { c } ) if c ∈ in . Now we can define a way to remove contours from a pure Euler diagram d .This contrasts to our previous work, where we allowed that the diagram to bereduced contains shading [11]. ntuitionistic Euler-Venn Diagrams (extended) 7 Definition 11 (Reduction).
Let d = ( L, Z ) be a pure Euler diagram and c ∈ L . The reduction of a zone z = ( in , out ) is z \ c = ( in \ { c } , out \ { c } ) . The reduction of d by c is defined as d \ c = ( L \{ c } , Z \ c ) , where Z \ c = { z \ c | z ∈ Z } . Lemma 4 (Properties of Reduction).
We have z \ c = z c \ c . Furthermore,for each z ′ ∈ MZ ( d \ c ) and z with z \ c = z ′ , we have z ∈ MZ ( d ) . In particular,both z ∈ MZ ( d ) and z c ∈ MZ ( d ) .Proof. Immediately from the definition of reduction. ⊓⊔ If each missing zone in a pure Euler diagram d has a missing adjacent zone,then the reduction of d by any contour is contained in the semantics of d . Inparticular, the meet of all reductions equals the semantics of d . This will allowus to show soundness of some rules of the sequent calculus in Sect. 4. Lemma 5.
Let d = ( L, Z ) be a pure Euler diagram, where for each z ∈ MZ ( d ) ,there is a contour ℓ ∈ L such that z ℓ ∈ MZ ( d ) . Furthermore, let L ′ = { c | MZ ( d \ c ) = ∅} . Then d c ∈ L ′ ν ( d \ c ) = ν ( d ) Proof.
Let c ∈ L ′ and z ′ = ( in , out ) ∈ MZ ( d \ c ). Then, let z = ( in ∪ { c } , out ).That is z \ c = z ′ and z, z c ∈ MZ ( d ) by Lemma 4 (if z = ( in , out ∪ { c } ),we can reverse the roles of z and z c in the following). Assume x ≤ ν ( z \ c ) M Then we have x ≤ d a ∈ in ν ( a ) F a ∈ out ν ( a ), if, and only if, x ⊓ d a ∈ in ν ( a ) ≤ F a ∈ out ν ( a ). This implies x ⊓ d a ∈ in ν ( a ) ≤ F a ∈ out ν ( a ) ⊔ ν ( c ), which is equivalentto x ≤ d a ∈ in ν ( a ) F a ∈ out ν ( a ) ⊔ ν ( c ) = ν ( z c ) M . Also, from x ≤ ν ( z \ c ) M ,we have x ⊓ d a ∈ in ν ( a ) ⊓ ν ( c ) ≤ d a ∈ in ν ( a ) ≤ F a ∈ out ν ( a ), which gives us x ≤ d a ∈ in ν ( a ) ⊓ ν ( c ) F a ∈ out ν ( a ) = ν ( z ) M . Hence, we have x ≤ ν ( z ) M ⊓ ν ( z c ) M .That is, for each z ′ ∈ MZ ( d \ c ), we have a z ∈ MZ ( d ) such that ν ( z ′ ) M = ν ( z \ c ) M ≤ ν ( z ) M ⊓ ν ( z c ) M . Thus, we have ν ( d \ c ) ≤ ν ( d ) for each c ∈ L ′ , thatis d c ∈ L ′ ν ( d \ c ) ≤ ν ( d ).Conversely, let x ≤ ν ( d ), i.e. x ≤ d z ∈ MZ ( d ) (cid:16) d a ∈ in ( z ) ν ( a ) F a ∈ out ( z ) ν ( a ) (cid:17) .For an arbitrary z ∈ MZ ( d ), choose c ∈ in ( z ) and c ∈ L ′ , i.e., the zone z \ c ismissing in at least one diagram (namely d \ c ). Of course, we have x ≤ ν ( z ) M , fromwhich we get by Lemma 1 (3) x ≤ d a ∈ in ( z ) \{ c } ν ( a ) (cid:16) ν ( c ) F a ∈ out ( z ) ν ( a ) (cid:17) ,which is equivalent to x ⊓ d a ∈ in ( z ) \{ c } ν ( a ) ≤ ν ( c ) F a ∈ out ( z ) ν ( a ). Further-more, from x ≤ ν ( z c ) M , we also have x ⊓ d a ∈ in ( z ) \{ c } ν ( a ) ≤ ν ( c ) ⊔ F a ∈ out ( z ) ν ( a ).By the properties of a distributive lattice, and Lemma 1 (1) and (2), we thenget x ⊓ d a ∈ in ( z ) \{ c } ν ( a ) ≤ (cid:16) ν ( c ) ⊔ F a ∈ out ( z ) ν ( a ) (cid:17) ⊓ (cid:16) ν ( c ) F a ∈ out ( z ) ν ( a ) (cid:17) ≤ (cid:16) ν ( c ) ⊓ (cid:16) ν ( c ) F a ∈ out ( z ) ν ( a ) (cid:17)(cid:17) ⊔ (cid:16)F a ∈ out ( z ) ν ( a ) ⊓ (cid:16) ν ( c ) F a ∈ out ( z ) ν ( a ) (cid:17)(cid:17) ≤ F a ∈ out ( z ) ν ( a ), which is equivalent to x ≤ d a ∈ in ( z ) \{ c } ν ( a ) F a ∈ out ( z ) ν ( a ) = ν ( z \ c ) M . Now, since z was arbitrary, this reasoning holds for all z ∈ MZ ( d )(possibly with the roles of z and z c reversed), and thus x ≤ d c ∈ L ′ ν ( d \ c ), andhence ν ( d ) ≤ d c ∈ L ′ ν ( d \ c ). ⊓⊔ For an example, consider the derivation in Sect. 5. The diagram d ∗ C as shown inTable 1 can be reduced to the three diagrams shown in the application of rule L r in derivation Π presented in Fig. 10. Sven Linker
Euler-Venn Diagrams
In this section, we combine pure Euler diagrams withthe central syntactic aspect of Venn diagrams: shading. Our main idea can besummarised as follows: We treat the information given by a pure Euler diagramas a condition for the construction of the combinations of atomic propositionsdenoted by the shading. That is, whenever we have constructions as indicated bythe spatial relations of contours in a diagram d , we also have a construction of theelements denoted by the shaded zones of the diagram. Since we use the syntacticelements of pure Euler diagrams and Venn diagrams, we will subsequently callsuch diagrams Euler-Venn diagrams .The abstract syntax of Euler-Venn diagrams is similar to Venn diagrams. Adiagram is a tuple d = ( L, Z, Z ∗ ) consisting of a set of contours L , a set of visiblezones Z over L , and a set of shaded zones Z ∗ ⊆ Z . We will often need to refer tothe pure Euler or Venn aspects of an Euler-Venn diagram separately. Hence, weintroduce some additional notation. For an Euler-Venn diagram d = ( L, Z, Z ∗ )we will write Venn ( d ) = ( L, Venn ( L ) , Z ∗ ) for the Venn diagram with the sameset of shaded zones as d , and Euler ( d ) = ( L, Z ) for the pure Euler diagram withthe same set of visible zones as d . Similarly to pure Venn and Euler diagrams, wewill refer to the missing zones of d by MZ ( d ) and to its shaded zones by Z ∗ ( d ). Definition 12 (Euler-Venn Diagram Semantics).
The semantics of a uni-tary
Euler-Venn diagram for a Heyting algebra H and a valuation ν is ν ( d ) = ν ( Euler ( d )) ν ( Venn ( d )) . Observe that with this definition, the semantics for the case MZ ( d ) = ∅ and Z ∗ ( d ) = ∅ yields ν ( d ) = 1 F z ∈ Z ∗ ( d ) ν ( z ) = F z ∈ Z ∗ ( d ) ν ( z ). Furthermore, weget ν ( ⊥ ) = 1 ν ( ⊤ ) = 1 canonical formula . Definition 13 (Canonical Formula).
The canonical formula of an Euler-Venn diagram is given by the following recursive definition. We start with thedefinition of the canonical formula of shaded and missing zones. χ z ( z ) = ^ c ∈ in ( z ) c ∧ ^ c ∈ out ( z ) − c χ m ( z ) = ^ c ∈ in ( z ) c → _ c ∈ out ( z ) c For a pure Euler diagram d e , a Venn diagram d v , an Euler-Venn diagram d andcompound diagrams D and E , the canonical formula is given as χ ( d e ) = ^ z ∈ MZ ( d e ) χ m ( z ) χ ( d v ) = _ z ∈ Z ∗ ( d v ) χ z ( z ) χ ( d ) = χ ( Euler ( d )) → χ ( Venn ( d )) χ ( D ⊗ E ) = χ ( D ) ⊗ χ ( E ) , ⊗ ∈ {∧ , ∨ , →} Remark 1.
Observe that according to Def. 13, we get χ ( c ) = c ∧ ⊤ and χ ( c ) = ⊤ ∧ − c . However, for simplicity, we will assume that the canonicalformula construction omits superfluous occurences of ⊤ and ⊥ . Hence, χ ( c ) = c and χ ( c ) = − c . Similarly, e.g., χ m (( ∅ , L )) = W c ∈ L c . ntuitionistic Euler-Venn Diagrams (extended) 9 Sequent calculus, as defined by Gentzen [5] is closely related to natural deduc-tion. It is based on sequents , which are decomposed by rule applications. In thefollowing, we will define a multi-succedent version of sequent calculus for Euler-Venn diagrams called
EDim . This version is inspired by the work of Dragalin [3],while following the more modern presentation of Negri et al. [15].
Definition 14 (Sequent). A sequent Γ ⇒ ∆ consists of multisets Γ and ∆ ofEuler diagrams. The multiset Γ is called the antecedent and ∆ the succedent .If Γ ( ∆ ) is the empty multiset, we write ⇒ ∆ ( Γ ⇒ , respectively). If a sequentis of the form p, Γ ⇒ ∆, p where p is a positive literal, then it is called an axiom .A sequent D , . . . , D k ⇒ E , . . . , E l is valid, if, and only if, ν ( D ) ⊓ . . . ⊓ ν ( D k ) ≤ ν ( E ) ⊔ . . . ⊔ ν ( E l ) for all valuations ν in all Heyting algebras. We will oftenabbreviate ν ( D ) ⊓ . . . ⊓ ν ( D k ) by ν ( Γ ) and ν ( E ) ⊔ . . . ⊔ ν ( E l ) by ν ( ∆ ) . Thatis, for the multiset Γ we always mean the meet, while for ∆ we always refer tothe join of the diagrams it consists of. A deduction for a sequent Γ ⇒ ∆ is a tree, where the root is labelled by Γ ⇒ ∆ , and the children of each node are labelled according to the rules definedbelow. If the validity of the premisses of a rule imply the validity of its conclusion,we call the rule sound . A deduction where the leaves are labelled with axioms,or instances of L ⊥ and R ⊤ , is called a proof for Γ ⇒ ∆ . We will write ⊢ Γ ⇒ ∆ to denote the existence of a proof for Γ ⇒ ∆ . In all rules, we call the diagram inthe conclusion that is being decomposed the principal diagram of the rule. Forexample, in L ∧ , the principal diagram is D ∧ E , and in the rule L s it is d . For agiven proof of Γ ⇒ ∆ , its height is the highest number of successive proof ruleapplications [15]. We will write ⊢ n Γ ⇒ ∆ if Γ ⇒ ∆ is provable with a proof ofheight at most n .We now turn to define and explain the rules of EDim . The rules to treatcompound diagrams, as shown in Fig. 4, are directly taken from sequent calculusfor intuitionistic propositional logic and are sound.
Lemma 6 (Soundness).
The rules for sentential operators are sound.Proof.
A straightforward adaptation of the proofs shown by Ono [16]. ⊓⊔ Remark 2.
If we take the placeholders D , E and F as formulas according toDef. 2 and both Γ and ∆ as multisets of such formulas, then the rules of Fig. 4together with axioms p, Γ ⇒ ∆, p form the sentential sequent calculus G3im [15].Provability in
G3im is equivalent to provability in Gentzen’s system LJ . Thesystem LJ is sound and complete [16]. Hence, G3im is sound and complete aswell. Furthermore, the structural rules of weakening, contraction and cut areadmissible [15]. Observe that we treat L ⊥ as a rule , and not as an axiom. D, E, Γ ⇒ ∆ L ∧ D ∧ E, Γ ⇒ ∆ D, Γ ⇒ ∆ E, Γ ⇒ ∆ L ∨ D ∨ E, Γ ⇒ ∆ Γ, D → E ⇒ D E, Γ ⇒ ∆ L → D → E, Γ ⇒ ∆Γ ⇒ ∆, D Γ ⇒ ∆, E R ∧ Γ ⇒ ∆, D ∧ E Γ ⇒ ∆, D, E R ∨ Γ ⇒ ∆, D ∨ E D, Γ ⇒ E R → Γ ⇒ ∆, D → E L ⊥ Γ, ⊥ ⇒ ∆ Fig. 4.
Proof Rules for Sentential Operators
Rules for Venn Diagrams.
The rules in 5a let us reduce negative to positive liter-als. Observe that we may introduce arbitrary sets of formulas into the succedent.This ensures admissability of the structural rules (cf. Lemma 13 and 14). Further-more, the rule R ⊤ lets us finish a proof similarly to L ⊥ . Let d = ( L, Venn ( L ) , Z ∗ )be a Venn diagram with | Z ∗ | >
1, and let d i = ( L, Venn ( L ) , Z ∗ i ), for i ∈ { , } ,such that Z ∗ = Z ∗ ∪ Z ∗ . Then the rules L s and R s in Fig. 5b separate d into d and d . These rules are closely related to the Combine equivalencerule for Spider diagrams [8]. For a Venn diagram d with Z ∗ ( d ) = { z } , where z = ( { n , . . . , n k } , { o , . . . , o l } ), the rules L dec and R dec of Fig. 5c decompose the single zone z into literals. c , Γ ⇒ c L neg c , Γ ⇒ ∆ c , Γ ⇒ R neg Γ ⇒ ∆, c R ⊤ Γ ⇒ ∆, (a) d , Γ ⇒ ∆ d , Γ ⇒ ∆ L s d, Γ ⇒ ∆ Γ ⇒ ∆, d , d R s Γ ⇒ ∆, d (b) n , . . . , n k , o , . . . , o l , Γ ⇒ ∆ L dec d, Γ ⇒ ∆Γ ⇒ ∆, n . . . Γ ⇒ ∆, n k Γ ⇒ ∆, o . . . Γ ⇒ ∆, o l R dec Γ ⇒ ∆, d (c) Fig. 5.
Rules for Unitary Venn Diagrams
Lemma 7.
The rules shown in Fig. 5 are sound.Proof.
In all of the following cases, let ν be an arbitrary valuation. The rule R ⊤ is clearly sound, since ν ( ⊤ ) = 1 for any valuation. For L neg , assume ν ( c ) ⊓ ntuitionistic Euler-Venn Diagrams (extended) 11 ν ( Γ ) ≤ ν ( c ). Then, we have ν ( c ) ⊓ ν ( Γ ) = ν ( c ) ⊓ ν ( Γ ) ⊓ ν ( c ) ⊓ ν ( Γ ) ≤ ν ( c ) ⊓ ν ( Γ ) ⊓ ν ( c ) ≤ ν ( Γ ) ⊓ ≤ ν ( ∆ ), where the first inequal-ity is an application of the assumption, and the second is due to Lemma 1 (1).For R neg , assume ν ( c ) ⊓ ν ( Γ ) ≤
0. Then we get, by the definition of the im-plication, the lattice properties, and the semantics of literals, ν ( Γ ) ≤ ν ( c ) ν ( c ) ≤ ν ( ∆ ) ⊔ ν ( c ).Consider R s . Assume ν ( Γ ) ≤ ν ( ∆ ) ⊔ ν ( d ) ⊔ ν ( d ), we have in particular ν ( Γ ) ≤ ν ( ∆ ) ⊔ F z ∈ Z ∗ ν ( z ) ⊔ F z ∈ Z ∗ ν ( z ). Since Z ∗ ∪ Z ∗ = Z ∗ , and since we canignore duplicate contour semantics by the lattice properties of Heyting algebras, ν ( Γ ) ≤ ν ( ∆ ) ⊔ F z ∈ Z ∗ ν ( z ), i.e., ν ( Γ ) ≤ ν ( ∆ ) ⊔ ν ( d ). Now consider L s . We haveboth ν ( d ) ⊓ ν ( Γ ) ≤ ν ( ∆ ) and ν ( d ) ⊓ ν ( Γ ) ≤ ν ( ∆ ), i.e.,( G z ∈ Z ∗ ν ( z ) ⊓ ν ( Γ )) ⊔ ( G z ∈ Z ∗ ν ( z ) ⊓ ν ( Γ )) ≤ ν ( ∆ ) ⊔ ν ( ∆ ) ⇐⇒ ( G z ∈ Z ∗ ν ( z ) ⊔ G z ∈ Z ∗ ν ( z )) ⊓ ν ( Γ ) ≤ ν ( ∆ ) ⇐⇒ ( G z ∈ Z ∗ ν ( z )) ⊓ ν ( Γ ) ≤ ν ( ∆ )which is exactly ν ( d ) ⊓ ν ( Γ ) ≤ ν ( ∆ ).Now consider L dec . By Def. 14, the premiss denotes ν ( n ) ⊓ . . . ⊓ ν ( n k ) ⊓− ν ( o ) ⊓ . . . ⊓ − ν ( o l ) ⊓ ν ( Γ ) ≤ ν ( ∆ ). But since z is the only shaded zone of d , this is exactly the semantics of d, Γ ⇒ ∆ , by Def. 6 and Def. 14. Finally,consider R dec . Then, we have ν ( Γ ) ≤ ν ( ∆ ) ⊔ ν ( n i ) and ν ( Γ ) ≤ ν ( ∆ ) ⊔ − ν ( o j )for all i ∈ { , . . . , k } and j ∈ { , . . . , l } . By the lattice properties, we get ν ( Γ ) ≤ ( ν ( ∆ ) ⊔ ν ( n )) ⊓ . . . ⊓ ( ν ( ∆ ) ⊔ ν ( n k )) ⊓ ( ν ( ∆ ) ⊔ − ν ( o )) ⊓ . . . ⊓ ( ν ( ∆ ) ⊔ − ν ( o l )),which is, by distributivity and since z is the only shaded zone in d , the same as ν ( Γ ) ≤ ν ( ∆ ) ⊔ ν ( d ). ⊓⊔ Rules for pure Euler Diagrams.
Now let d = ( L, Z ) be a pure Euler diagram,where for each z ∈ MZ ( d ) there is a contour ℓ ∈ L , such that z ℓ ∈ MZ ( d ).Furthermore, let { c , . . . , c k } ⊆ L be the maximal set of contours such that MZ ( d \ c i ) = ∅ for every i ≤ k . Then we can reduce d according to the rules L r and R r shown in Fig. 6a. Let d = ( L, Z ) be a pure Euler diagram with more thanone missing zone, i.e., | MZ ( d ) | >
1, and let d = ( L, Z ) and d = ( L, Z ) be twopure Euler diagrams such that Z ∩ Z = Z . Then the rules L MZ and R MZ ofFig. 6b separate the diagram z at its missing zones. If d is a pure Euler diagramwith a single missing zone, i.e. MZ ( d ) = { z } and z = ( { n , . . . , n k } , { o , . . . , o ℓ } ),then the rules of Fig. 6c decompose z into literals. Lemma 8.
The rules shown in Fig. 6 are sound.Proof.
The soundness of the rules L r and R r is immediate by Lemma 5. Forrules L MZ and R MZ observe that by the condition on d and d , we have MZ ( d ) ∪ MZ ( d ) = MZ ( d ). That is, ν ( d ) ⊓ ν ( d ) = ν ( d ) for all valuations and d \ c , . . . , d \ c k , Γ ⇒ ∆ L r d, Γ ⇒ ∆ Γ ⇒ ∆, d \ c . . . Γ ⇒ ∆, d \ c k R r Γ ⇒ ∆, d (a) d , d , Γ ⇒ ∆ L MZ d, Γ ⇒ ∆ Γ ⇒ ∆, d Γ ⇒ ∆, d R MZ Γ ⇒ ∆, d (b) d, Γ ⇒ n . . . d, Γ ⇒ n k o , Γ ⇒ ∆ . . . o l , Γ ⇒ ∆ L Idec d, Γ ⇒ ∆Γ, n , . . . , n k ⇒ o , . . . , o l R Idec Γ ⇒ ∆, d (c) Fig. 6.
Proof Rules for pure Euler Diagrams
Heyting algebras. The soundness of both L MZ and R MZ follows by straight-forward computations. For the rule R Idec , the proof is straightforward by thedefinition of and the lattice properties. The rule L Idec can be proven soundsimilarly to L s . ⊓⊔ Rules for Euler-Venn Diagrams.
Let d be an Euler-Venn diagram. Then therules L det and R det of Fig. 7 detach the spatial relations from the shading. d, Γ ⇒ Euler ( d ) Venn ( d ) , Γ ⇒ ∆ L det d, Γ ⇒ ∆ Euler ( d ) , Γ ⇒ Venn ( d ) R det Γ ⇒ ∆, d Fig. 7.
Proof Rules For Euler-Venn Diagrams
Lemma 9.
The rules shown in Fig. 7 are sound.Proof.
Consider R det , and assume ν ( Euler ( d )) ⊓ ν ( Γ ) ≤ ν ( Venn ( d )). Then, byDef. 1, this is equivalent to ν ( Γ ) ≤ ν ( Euler ( d )) ν ( Venn ( d )), which by Def. 12and the lattice properties implies ν ( Γ ) ≤ ν ( ∆ ) ⊔ ν ( d ). So consider L det , andassume both ν ( d ) ⊓ ν ( Γ ) ≤ ν ( Euler ( d )) and ν ( Venn ( d )) ⊓ ν ( Γ ) ≤ ν ( ∆ ). We thenhave ν ( d ) ⊓ ν ( Γ ) = ν ( d ) ⊓ ν ( Γ ) ⊓ ν ( d ) ⊓ ν ( Γ ) ≤ ν ( d ) ⊓ ν ( Γ ) ⊓ ν ( Euler ( d )) ≤ ν ( Venn ( d )) ⊓ ν ( Γ ) ≤ ν ( ∆ ). The inequalities are correct due to the first premiss,Lemma 1 (1) and the second premiss, respectively. ⊓⊔ By an induction on the height of proofs, we get the soundness theorem for
EDim , using Lemma 6, 7, 8, and 9. ntuitionistic Euler-Venn Diagrams (extended) 13
Theorem 1 (Soundness). If Γ ⇒ ∆ is provable in EDim , then Γ ⇒ ∆ isvalid. To prove completeness of the system, we first show that certain rules areinvertible. Even stronger, a rule is height-preserving invertible , if whenever wehave a proof of height n for its conclusion, its premisses are provable with a proofof at most height n . Lemma 10 (Inversions).
1. All of the rules L ∧ , R ∧ , L ∨ and R ∨ are height-preserving invertible.2. All of the rules L dec , R dec , L s , R s , L r , R r , L MZ , and R MZ are height-preserving invertible.3. If ⊢ n d, Γ ⇒ ∆ for an Euler-Venn diagram d , then also ⊢ n Venn ( d ) , Γ ⇒ ∆ .4. If ⊢ n d, Γ ⇒ ∆ for a pure Euler diagram with one missing zone z =( { n , . . . , n k } , { o , . . . , o l } ) , then also ⊢ n o i , Γ ⇒ ∆ for all ≤ i ≤ l .Proof. The propositional operator rules are height-preserving invertible as shownby Negri et al. [15] (Chap. 5, Lemma 5.3.4). For the rules L dec , R dec , L s , R s , L r , R r , L MZ and R MZ , similar arguments during an induction on the height ofthe proof yield the result. Case 3 and 4 can be shown by an induction similar tothe case of R → . ⊓⊔ That these rules can be used in an inverse manner is used in the followinglemma, where we connect provability of a sequent Γ ⇒ ∆ within EDim withthe provability of the corresponding sequent χ ( Γ ) ⇒ χ ( ∆ ) consisting of thecanonical formulas of the antecedent and the succedent. Lemma 11.
Let Γ ⇒ ∆ be a sequent of compound diagrams. Then Γ ⇒ ∆ isprovable in EDim if, and only if, χ ( Γ ) ⇒ χ ( ∆ ) is provable in G3im .Proof.
Let Γ ⇒ ∆ be provable in EDim . By Theorem 1, the sequent is valid,and hence the sequent χ ( Γ ) ⇒ χ ( ∆ ) is valid as well. Since G3im is complete (cf.Remark 2), the sequent is provable in
G3im .For the other direction, we proceed by induction on the height n of the proofof χ ( Γ ) ⇒ χ ( ∆ ). If n = 0, then χ ( Γ ) ⇒ χ ( ∆ ) is an axiom p, Γ ′ ⇒ ∆ ′ , p or aninstance of L ⊥ . In the first case, since the only diagram D with χ ( D ) = p is apositive literal, Γ ⇒ ∆ is an axiom as well. Similarly, in the second case, it is aninstance of L ⊥ of EDim . Now assume that the statement is true for all sequentswith proofs of height less than n . We proceed by a case distinction on the lastrule applied in the proof of χ ( Γ ) ⇒ χ ( ∆ ).If the last rule is R → , then the sequent is of the form χ ( Γ ) ⇒ χ ( ∆ ′ ) , χ ( D ),where D is either a compound diagram D = E → F , a pure Euler diagram D = d e with a single missing zone, an Euler-Venn diagram with missing zonesand shaded zones D = d , a single negative literal for a contour c , or D = ⊤ . Inthe first case, the premiss is then χ ( E ) , χ ( Γ ) ⇒ χ ( F ), which by the inductionhypothesis implies that E, Γ ⇒ F is provable in EDim . An application of R → then proves Γ ⇒ ∆ . Since all cases, where the principal diagram is compound are treated exactly like this, we will ignore these possibilities in the following. Forthe case where d is an Euler-Venn diagram, we have χ ( d ) = Euler ( d ) → Venn ( d ).and hence the premiss of the last step is χ ( Euler ( d )) , χ ( Γ ) ⇒ χ ( Venn ( d )). Bythe induction hypothesis, we get that Euler ( d ) , Γ ⇒ Venn ( d ) is provable, and byapplying R det , Γ ⇒ ∆, d as well. Now assume that the principal diagram is apure Euler diagram d e with a single missing zone z = ( { n , . . . , n k } , { o , . . . , o l } ).Hence, the premiss of the last step in G3im is V ≤ i ≤ k n i , χ ( Γ ) ⇒ W ≤ i ≤ l o i . Sinceboth L ∧ and R ∨ are height-preserving invertible, the provability of this sequentis equivalent to the provability of n , . . . , n k , χ ( Γ ) ⇒ o , . . . , o l , with heightless than n . Since the canonical formula is only atomic for diagram literals, wehave that n , . . . , n k , Γ ⇒ o , . . . , o l is provable by the inductionhypothesis, and hence by applying R Idec also Γ ⇒ ∆, d e . If the principal formulawas a negative literal for c , then the proven sequent is of the form χ ( Γ ) ⇒ χ ( ∆ ) , χ ( c ). Since χ ( c ) = − c = c → ⊥ , the premiss is c, χ ( Γ ) ⇒ ⊥ ,which is exactly χ ( c ) , χ ( Γ ) ⇒ ⊥ . By induction hypothesis, we get a proof for c , Γ ⇒ in EDim . Thus an application of R neg yields a proof for Γ ⇒ ∆, c .Finally, if the principal formula was ⊤ , then χ ( D ) = ⊤ , and an application of R ⊤ yields a proof for Γ ⇒ ∆ ′ , D . Observe that χ ( Γ ) ⇒ χ ( ∆ ′ ) , χ ( ⊤ ) is alsoprovable since the premiss of applying R → is an instance of L ⊥ .If the last application in the proof of χ ( Γ ) ⇒ χ ( ∆ ) was L → , the argu-ments are similar, with appropriate applications of L det , L Idec , L neg , and theinvertibility of R ∧ and L ∨ .If the last application was R ∧ , then the last sequent is of the form χ ( Γ ) ⇒ χ ( ∆ ′ ) , χ ( D ), where either D = d e is an Euler diagram with more than onemissing zone, or D = d is a Venn diagram with exactly one shaded zone. Inthe first case, this means χ ( Γ ) ⇒ χ ( ∆ ′ ) , V z ′ ∈ MZ ( d e ) χ m ( z ′ ) was proved, and thepremisses are χ ( Γ ) ⇒ χ ( ∆ ′ ) , χ m ( z ) and χ ( Γ ) ⇒ χ ( ∆ ′ ) , V z ′ ∈ MZ ( d e ) \{ z } χ m ( z ′ )for some z ∈ MZ ( d e ). Now consider the Euler diagrams d = ( L, Venn ( L ) \ { z } )and d = ( L, ( Venn ( L ) \ MZ ( d )) ∪ { z } ). Then χ ( d ) = χ m ( z ) and χ ( d ) = V z ′ ∈ MZ ( d e ) \{ z } χ m ( z ′ ). Hence, we get by the induction hypothesis that Γ ⇒ ∆ ′ , d and Γ ⇒ ∆ ′ , d are provable, and thus an application of R MZ yieldsa proof of Γ ⇒ ∆ . For the second case, assume D = d is a Venn diagramwith exactly one shaded zone z = ( { n , . . . , n k } , { o , . . . , o l } ), i.e., the sequentis in the form χ ( Γ ) ⇒ χ ( ∆ ′ ) , V ≤ i ≤ k n i ∧ V ≤ i ≤ l − o i . Assume without loss ofgenerality that n is part of the outer conjunction, i.e., the conjunction in thesuccedent is of the form n ∧ (cid:16)V ≤ i ≤ k n i ∧ V ≤ i ≤ l − o i (cid:17) . Hence, the premisses areof the form χ ( Γ ) ⇒ χ ( ∆ ′ ) , n and χ ( Γ ) ⇒ χ ( ∆ ′ ) , V ≤ i ≤ k n i ∧ V ≤ i ≤ l − o i . Since R ∧ is height-preserving invertible, all sequents of the form χ ( Γ ) ⇒ χ ( ∆ ′ ) , n i and χ ( Γ ) ⇒ χ ( ∆ ′ ) , − o i are provable with a proof of height less than n . Fromthe induction hypothesis, and Remark 1, we get that all of the sequents Γ ⇒ ∆ ′ , n i and Γ ⇒ ∆ ′ , o i are provable, and hence Γ ⇒ ∆ is provable withan application of R dec .If the last rule applied in the proof is L ∧ , the arguments are similar, withsuited applications of L MZ and L dec . ntuitionistic Euler-Venn Diagrams (extended) 15 Now, assume that the last rule applied was R ∨ . Then, the only possibility isthat the principal diagram is a Venn diagram with more than one shaded zone,i.e., the sequent is χ ( Γ ) ⇒ χ ( ∆ ′ ) , W z ∈ Z ∗ ( d ) χ z ( z ). So without loss of generalityassume that the premiss is χ ( Γ ) ⇒ χ ( ∆ ′ ) , χ z ( z i ) , W z ∈ Z ∗ ( d ) \{ z i } χ z ( z ). Considerthe Venn diagrams d = ( L, Zd, { z i } ) and d = ( L, Zd, Z ∗ d \ { z i } ), and observethat χ ( d ) = χ z ( z i ) and χ ( d ) = W z ∈ Z ∗ ( d ) \{ z i } χ z ( z ). That is, by the inductionhypothesis, we have that Γ ⇒ ∆ ′ , d , d is provable, and hence by an applicationof R s , we can prove Γ ⇒ ∆ .The case for L ∨ is similar, with an appropriate application of L s . ⊓⊔ Since every valid sequent is derivable in
G3im , we get the completeness resultfor
EDim directly from Lemma 11.
Theorem 2 (Completeness). If Γ ⇒ ∆ is valid, then Γ ⇒ ∆ is provable. Figure 8 consists of a simple proof containing only Venn diagrams with asingle contour. It shows how disjunction and shaded zones interact. That is,the presence of several shaded zones can be proven from simpler diagrams. Inparticular, this proof shows the similarity between the separation rules ( L s and R s ) and the rules for disjunction. Furthermore, we can see how the rules L neg and R neg can be used to reduce a sequent with negative literals to an axiom. a ⇒ a , a R s a ⇒ a a , a ⇒ a L neg a , a ⇒ R neg a ⇒ a , a R s a ⇒ a L ∨ a ∨ a ⇒ a Fig. 8.
Example of a Simple Proof
We show that some rules are admissible. To that end, we define the weight ofdiagrams, to order them by the number of their syntactic elements.
Definition 15.
The weight ω ( d ) of a diagram is defined inductively. The basecases are given by ω ( ⊥ ) = 0 , ω ( c ) = 0 , and ω ( c ) = 1 . Otherwise we set ω ( d ) = | Z ∗ ( d ) | + 1 , if d is a Venn diagram | MZ ( d ) | + 1 , if d is a pure Euler diagram ω ( Euler ( d )) + ω ( Venn ( d )) + 1 , if d is an Euler-Venn diagram ω ( d ) + ω ( d ) + 1 , if d = d ⊗ d for ⊗ ∈ {∧ , ∨ , →} Lemma 12.
For any diagram D , the sequent D, Γ ⇒ ∆, D is provable in EDim .Proof.
A straightforward induction on the weight of D . ⊓⊔ Lemma 13 (Admissibility of Weakening). i) If ⊢ n Γ ⇒ ∆ , then also ⊢ n D, Γ ⇒ ∆ . ii) If ⊢ n Γ ⇒ ∆ , then also ⊢ n Γ ⇒ ∆, D .Proof. By induction on the height of the proof for Γ ⇒ ∆ . For i), we can add anew diagram into the antecedent of the sequent at the inductive step, since Γ iskept from the premisses to the conclusion. In case ii), this works for most rulesas well, except, where the succecedent of the premiss is restricted (e.g. R neg ).In these cases, the weakening diagram D is simply added to the multiset ∆ inthe rule’s conclusion. ⊓⊔ Lemma 14 (Admissibility of Contraction). i) If ⊢ n D, D, Γ ⇒ ∆ , thenalso ⊢ n D, Γ ⇒ ∆ . ii) If ⊢ n Γ ⇒ ∆, D, D , then also ⊢ n Γ ⇒ ∆, D .Proof. Both cases can be proven by an induction on the height of proofs usingLemma 10 and arguments similar to Negri et al. [15]. In case ii), the only specialcase are rules with restricted right context in the premisses (e.g. R det ), wherethe contraction is done by changing the right context appropriately. ⊓⊔ Lemma 15 (Admissibility of Cut).
If both Γ ⇒ D, ∆ and
D, Γ ′ ⇒ ∆ ′ areprovable, then also Γ, Γ ′ ⇒ ∆, ∆ ′ is provable.Proof. We use a semantic proof, employing both soundness and completenessof
EDim . If both sequents are provable, they are also valid, by soundness. Sochoose an arbitrary valuation ν . Then ν ( Γ ) ≤ ν ( D ) ⊔ ν ( ∆ ) and ν ( D ) ⊓ ν ( Γ ′ ) ≤ ν ( ∆ ′ ). Now we have ν ( Γ ) ⊓ ν ( Γ ′ ) ≤ ( ν ( D ) ⊔ ν ( ∆ )) ⊓ ν ( Γ ′ ) = ( ν ( D ) ⊓ ν ( Γ ′ )) ⊔ ( ν ( ∆ ) ⊓ ν ( Γ ′ )) ≤ ν ( ∆ ′ ) ⊔ ( ν ( ∆ ) ⊓ ν ( Γ ′ )) ≤ ν ( ∆ ′ ) ⊔ ν ( ∆ ). These relations aredue to the first premiss, distributivity, the second premiss and the fact a ⊓ b ≤ a , respectively. Since ν was arbitrary, Γ, Γ ′ ⇒ ∆, ∆ ′ is valid, and due to thecompleteness of EDim , we have that
Γ, Γ ′ ⇒ ∆, ∆ ′ is provable. ⊓⊔ Remark 3.
It is also possible to prove cut admissibility with a purely syntacticargument by adapting the inductive proof for the system
G3im given by Negriet al.[15]. The proof consists of a replacement of each cut application with aderivation, where each cut either posesses a lower cut-height, or the weight ofthe cut diagram is lower. Within that proof, most cases are straightforward,where L dec , R dec , L r , R r , L MZ and R MZ are treated similarly to the rules L ∧ and R ∧ , while L s and R s play roles similar to L ∨ and R ∨ . The rules L neg , R neg , L det , R det , L Idec and R Idec need special attention, since they restrict thesuccedent in the premiss. However, the proof proceeds in these cases along thelines of the the treatment of L → and R → in G3im . While the number of cases toconsider increases, the arguments and constructions are similar. As an example,we present the case where the cut formula is principal in both premisses, and is ntuitionistic Euler-Venn Diagrams (extended) 17 a negative literal. That is, we have a derivation of the following form: c , Γ ⇒ R neg Γ ⇒ ∆, c c , Γ ′ ⇒ c L neg c , Γ ′ ⇒ ∆ ′ Cut
Γ, Γ ′ ⇒ ∆, ∆ ′ Observe that the cut-height of this cut application is m + n + 2, where m isthe height of the proof of the left premiss and n the height of the proof of theright premiss. Then, we can replace this derivation with the following. c , Γ ⇒ R neg Γ ⇒ ∆, c c , Γ ′ ⇒ c Cut
Γ, Γ ′ ⇒ ∆, c c , Γ ⇒ Cut
Γ, Γ ′ , Γ ⇒ ∆ RW , LC Γ, Γ ′ ⇒ ∆, ∆ ′ In this derivation, the uppermost cut has a lower cut-height, while the secondcut uses a cut diagram of lower weight. Here, it is crucial that the negative literalhas a higher weight than the positive literal. The last step in the derivation isa sequence of weakening and contraction. The treatment of the other cases isanalogous. Π ab c , d A ⇒ c Π ab c , d A , b ⇒ R neg ab c , d A ⇒ b Π ab c , d A , a ⇒ R neg ab c , d A ⇒ a R dec ab c , a c ⇒ b ca R det a c ⇒ ab c Fig. 9.
Proof using Euler-Venn diagrams
A derivation that uses all three types of diagrams can be found in Fig. 9.We explain parts of the proof from bottom to top. The last applied rule de-taches the pure Euler part from the Venn part of the succedent, so that wecan then decompose the single shaded zone into literals. This splits the proofinto three branches, which we treat in the sub-derivations Π , Π and Π ,respectively. For reasons of brevity, we use the abbrevations for diagrams asshown in Table 1. Now, the two right proof branches contain a negative lit-eral in the succedent, which we move to the antecedent with an application of R neg . Then, all three proof branches proceed similarly: we reduce the pure Eu-ler diagram d ∗ C into smaller diagrams. The set of missing zones is MZ ( d ∗ C ) = { ( { a } , { b, c } ) , ( { a, c } , { b } ) , ( { b, c } , { a } ) , ( { a, b, c } , ∅ ) } , and each of these missingzones has at least one adjacent missing zone. For example, ( { a } , { b, c } ) c =( { a, c } , { b } ). In particular, the reduction of d ∗ C with respect to any of the contours Table 1.
Diagram Abbreviations ab c ab c a cd C d ∗ C d A a , b and c still contains missing zones. It iseasy to check that the three diagrams shownin the derivations are indeed these reductions.Then, Π proceeds by detaching the Eulerand Venn aspects of the diagram d A , whichimmediately closes the left branch, due toLemma 12. The right branch ends in an axiom after decomposing the singleshaded zone in the antecedent. Within Π there is a similar structure, denotedby the derivation Π ′ , where the antecedent contains slightly different diagrams,but the application of rules is similar. The other branches proceed similarly.This example shows, how the reduction rules lead to smaller diagrams, and, aswe claim, better readable diagrams, due to the reduced clutter [9]. Furthermore,it shows how the admissible rules may reduce the size of the proofs, here in theform of the generalised axioms proven admissible in Lemma 12. Π b a , a c , b c , d A ⇒ a c L r d ∗ C , d A ⇒ a c d ∗ C , c , a ⇒ c L dec d ∗ C , ca ⇒ c L det d ∗ C , d A ⇒ c Π b a , a c , d A , b ⇒ b Π ′ b a , a c , d A , b ⇒ c L Idec b a , a c , b c , d A , b ⇒ L r d ∗ C , d A , b ⇒ Π b a , a c , b c , . . . ⇒ a c L r d ∗ C , d A , a ⇒ a c d ∗ C , b a , c , a , a ⇒ a L neg d ∗ C , b a , c , a , a ⇒ L dec d ∗ C , b a , ca , a ⇒ L det d ∗ C , d A , a ⇒ Fig. 10.
Auxiliary Derivations for Fig. 9ntuitionistic Euler-Venn Diagrams (extended) 19
In this paper, we presented an intuitionistic interpretation of Euler-Venn dia-grams, based on a semantics of Heyting algebras. We then defined a cut-freesequent calculus
EDim , which we have proven to be sound and complete withrespect to this semantics. Furthermore, we have shown that the structural rulesof contraction, weakening and cut are admissible.For this visualisation, we deviated from classical Euler-Venn diagrams in twoways: we did not treat missing zones and shaded zones as equivalent, and weintroduced the new syntactic element of dashed contours.The first deviation is due to the basic restrictions of intuitionistic reasoning.More specifically, intuitionistic implication cannot be treated as an abbreviationof the other operators. To have a syntax explicitly for implications, we needto increase the number of distinct syntactic elements of Euler-Venn diagrams.Hence, distinguishing these two elements is a natural choice. Of course, it can beargued that the choice we made is not the correct one, and that shading shouldbe used to reflect implications. However, we think that since the representation ofmissing zones (or rather their absence) introduces a direction into the diagram,in the form of inclusions, this choice is justified.The introduction of dashed diagrams is more debatable. Arguably, the needfor distinguishing pure Euler diagrams by dashing arises, since we interpret themissing zones of Euler-Venn diagrams as a kind of “constructive precondition”for the construction of the elements denoted by the shaded zones. That is, in theconstructive interpretation of intuitionistic reasoning, an Euler-Venn diagrammeans that, given a construction as indicated by the missing zones, we haveanother construction for the assertions given by the shaded zones. Hence, thereis an additional implication within the semantics of Euler-Venn diagrams, as canalso be seen in the rules of
EDim to detach the pure Euler aspects from the Vennaspects of a diagram. These rules behave similarly to the rules for implicationin sentential intuitionistic sequent calculus.However, the introduction of new syntactic elements is necessary, due to theindependence of the operators, and the restrictive nature of Euler-Venn diagramsmakes this need even more overt. Compare for example the intuitionistic systemsbased on Existential Graphs (EGs). While the operations in classical EGs aredenoted by juxtaposition and cuts, reflecting conjunction and negation, respec-tively, the assertive graphs [1] explicitly introduce notation for disjunction, andalso treat the “scroll” as a distinct element. Similarly, the intuitionistic EGs [13]include the notion of n -scrolls for each n > Still, there are future directions this work can be taken into. For example,our sequent calculus resembles sentential sequent calculus, while typical Euler-Venn reasoning systems work by adding syntax to single diagrams, and thenremoving unnecessary parts [2]. It is interesting to see, if we can define sucha system for intuitionistic Euler-Venn diagrams. We assume that for the rulesto introduce and remove contours, or to copy contours from one diagram intoanother, the reduction of a pure Euler diagram (cf. Def 11 and Lemma 5) willplay a significant role.
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