aa r X i v : . [ m a t h . L O ] F e b On axioms and rexpansions ∗Carlos Caleiro and Sérgio Marcelino {ccal,smarcel}@math.tecnico.ulisboa.pt
SQIG - Instituto de TelecomunicaçõesDep. Matemática - Instituto Superior TécnicoUniversidade de Lisboa, Portugal
Abstract
We study the general problem of strengthening the logic of a given(partial) (non-deterministic) matrix with a set of axioms, using the idea ofrexpansion. We obtain two characterization methods: a very general butnot very effective one, and then an effective method which only appliesunder certain restrictions on the given semantics and the shape of theaxioms. We show that this second method covers a myriad of examplesin the literature. Finally, we illustrate how to obtain analytic multiple-conclusion calculi for the resulting logics.
The work reported in this paper has three underlying aims.First, and foremost, on a higher-level reading, this paper is an acclamationof the modularization power enabled by non-deterministic matrices (Nmatri-ces) , as proposed and developed by Arnon Avron, along with his coauthors andstudents over the past 15 years [8, 2, 3, 4, 5, 9, 6, 7, 10], and used by manyothers [31, 17, 23, 11, 18, 32, 24] when seeking for a clear semantic rendering oflogics resulting from strengthening a given base logic.Secondly, in the technical developments we propose, this paper can be seenas an application of the ideas behind rexpansions [10] of Nmatrices, in the formof a generalization of the systematic method put forth in [23] for obtainingmodularly a suitable semantics for a given logic strengthened with additionalaxioms (and new unary connectives). Expectedly, the method may yield ingeneral a partial non-deterministic matrix (PNmatrix) [11], partiality being afeature that adds to the conciseness of Nmatrices but which is known to contendwith analyticity . ∗ This research was funded by FCT/MCTES through national funds and when applicableco-funded EU funds under the project UIDB/EEA/50008/2020. Work done under the scopeof the CaCTus initiative of SQIG at Instituto de Telecomunicações. Ax of thelogic of a given PNmatrix M . The first method, presented in Subsection 3.1,is completely general but unfortunately produces an infinite PNmatrix evenwhen a finite one would be available. In order to overcome this drawback, inSubsection 3.2, we present another more economic method, generalizing [23],which, under suitable requirements, always provides a finite PNmatrix whenstarting from finite M and Ax . Section 4 is devoted to illustrating the applica-tion of the method of Subsection 3.2 to some meaningful examples. Then, inSection 5, we show that (under minimal expressiveness requirements on M ) theresults of [32, 17] can be used to provide analytic multiple-conclusion calculito the strengthened logics by exploring the semantics obtained by our method,and provide illustrative examples. We close the paper in Section 6, with someconcluding remarks and topics for future work. For the sake of self-containment, and in order to fix notation and terminology,we start by recalling (or suitably adapting, or generalizing) a number of usefulnotions and results. Instead of going through this material sequentially, thereader could as well jump this section for the moment and refer back herewhenever necessary.A propositional signature Σ is a family { Σ ( k ) } k ∈ N of sets, where each Σ ( k ) contains the k -place connectives of Σ . To simplify notation, we express the factthat © ∈ Σ ( k ) for some k ∈ N by simply writing © ∈ Σ , and we write Σ ′ ∪ Σ or Σ ′ ⊆ Σ to denote the union or the inclusion, respectively, if Σ ′ is also a signature.Given a signature Σ , the language L Σ ( P ) is the carrier of the absolutely free Σ -algebra generated over a given denumerable set of sentential variables P .Elements of L Σ ( P ) are called formulas . Given a formula A ∈ L Σ ( P ) , we denoteby var ( A ) (resp. sub ( A ) ) the set of variables (resp. subformulas) of A , defined asusual; the extension of var and sub , and other similar functions, from formulasto sets thereof is defined as expected. A substitution is a member σ ∈ L Σ ( P ) P ,that is, a function σ : P → L Σ ( P ) , uniquely extendable into an endomorphism · σ : L Σ ( P ) → L Σ ( P ) . Given Γ ⊆ L Σ ( P ) , we denote by Γ σ the set { A σ : A ∈ Γ } .For A ∈ L Σ ( P ) , define A inst = { A σ : σ ∈ L Σ ( P ) P } and Γ inst = S A ∈ Γ A inst .2iven formulas A, A , . . . , A n ∈ L Σ ( P ) with var ( A ) ⊆ { p , . . . , p n } , we write A ( A , . . . , A n ) to denote the formula A σ where σ ( p i ) = A i for ≤ i ≤ n .Given a signature Σ , a Σ -PNmatrix (partial non-deterministic matrix) is astructure M = h V, D, · M i such that V is a set (of truth-values ), D ⊆ V is theset of designated values, and © M : V k → ℘ ( V ) is a function ( truth-table ) foreach k ∈ N and each k -place connective © ∈ Σ . When © M ( x , . . . , x k ) = ∅ for all x , . . . , x k ∈ V we say that the truth-table of © in M is total . When© M ( x , . . . , x k ) has at most one element for all x , . . . , x k ∈ V we say that thetruth-table of © in M is deterministic . Of course, deterministic does not implytotal. Given Σ ′ ⊆ Σ , we say that M is Σ ′ -total if the truth-tables in M of theconnectives © ∈ Σ ′ are all total. Analogously, we say that M is Σ ′ -deterministic if the truth-tables in M of the connectives © ∈ Σ ′ are all deterministic. Whenthe Σ -PNmatrix M is Σ -total, or just total, it is simply called a Σ -Nmatrix, orNmatrix (non-deterministic matrix) . When a Σ -Nmatrix M is Σ -deterministic,or just deterministic, it is simply called a Σ -matrix , or a logical matrix. For thesake of completing the picture, when a Σ -PNmatrix M is deterministic we callit a Σ -Pmatrix , or Pmatrix.Granted a Σ -PNmatrix M = h V, D, · M i , a M -valuation is a function v : L Σ ( P ) → V such that v ( © ( A , . . . , A k )) ∈ © M ( v ( A ) , . . . , v ( A k )) for every k ∈ N , every k -place connective © ∈ Σ , and every A , . . . , A k ∈ L Σ ( P ) . We denotethe set of all M -valuations by Val M . Given a formula A ∈ L Σ ( { p , . . . , p n } ) ,we extend the usual notation for connectives and use A M : V n → ℘ ( V ) todenote the function defined by A M ( x , . . . , x n ) = { v ( A ) : v ∈ Val M with v ( p i ) = x i for ≤ i ≤ n } for every x , . . . , x n ∈ V .As is well known, if M = h V, D, · M i is a matrix then every function f : Q → V with Q ⊆ P can be extended to a M -valuation (in an essentially unique wayfor all formulas A with var ( A ) ⊆ Q ). As a consequence, A M ( x , . . . , x n ) is asingleton when M is a matrix, or more generally when there is Σ ′ ⊆ Σ suchthat A ∈ L Σ ′ ( P ) and M is Σ ′ -deterministic and Σ ′ -total. If M is only knownto be Σ ′ -deterministic, we can at least guarantee that A M ( x , . . . , x n ) has atmost one element. When M is a Nmatrix, however, A M ( x , . . . , x n ) can be alarge (non-empty) set. Still, we know from [9] that a function f : Γ → V with Γ ⊆ L Σ ( P ) can be extended to a M -valuation provided that sub (Γ) ⊆ Γ and that f ( © ( A , . . . , A k )) ∈ © M ( f ( A ) , . . . , f ( A n )) whenever © ( A , . . . , A k ) ∈ Γ . Incase M is a PNmatrix, in general, one does not even have such a guarantee [11],unless f (Γ) ∈ T M = S v ∈ Val M ℘ ( v ( L Σ ( P ))) . In other words, given X ⊆ V , wehave X ∈ T M if the values in X are all together compatible in some valuationof M . Of course, A M ( x , . . . , x n ) = ∅ if { x , . . . , x n } ∈ T M .A set of valuations V ⊆
Val M characterizes a generalized (multiple conclu-sion) consequence relation ⊲ V ⊆ ℘ ( L Σ ( P )) × ℘ ( L Σ ( P )) defined by Γ ⊲ V ∆ whenfor every v ∈ V if v (Γ) ⊆ D then v (∆) ∩ D = ∅ . Of course, it also defines themore usual (single conclusion) consequence relation ⊢ V ⊆ ℘ ( L Σ ( P )) × L Σ ( P ) such that Γ ⊢ V A when Γ ⊲ V { A } . In both cases, ⊲ V and ⊢ V are substitutioninvariant , and respectively a Scott [35] and Shoesmith and Smiley [36] conse-quence relation, or else a Tarskian consequence relation, when V is closed for3ubstitutions, that is, if v ∈ V and σ ∈ L Σ ( P ) P then v ◦ ( · σ ) ∈ V .We simply write ⊲ M or ⊢ M , instead of ⊲ Val M or ⊢ Val M , respectively, and saythat the consequences are characterized by M . With respect to given conse-quence relations ⊲ or ⊢ , we say that M is sound if ⊲ ⊆ ⊲ M or ⊢ ⊆ ⊢ M , and wesay that M is complete if ⊲ M ⊆ ⊲ or ⊢ M ⊆ ⊢ .A refinement of a Σ -PNmatrix M = h V, D, · M i is any Σ -PNmatrix M ′ = h V ′ , D ′ , · M ′ i with V ′ ⊆ V , D ′ = D ∩ V ′ , and © M ′ ( x , . . . , x k ) ⊆ © M ( x , . . . , x k ) for every k ∈ N , every k -place connective © ∈ Σ , and every x , . . . , x k ∈ V ′ .It is clear, almost by definition, that Val M ′ ⊆ Val M . When it is always thecase that © M ′ ( x , . . . , x k ) = © M ( x , . . . , x k ) ∩ V ′ then the refinement is called simple and M ′ is denoted by M V ′ . Clearly, v ∈ Val M implies that v ∈ Val M V ′ with V ′ = v ( L Σ ( P )) , and also that M V ′ is a non-empty total refinement of M .This observation justifies the equivalent definition of T M put forth in [17]. E : V → ℘ ( U ) is an expansion function if E ( x ) = ∅ for every x ∈ V ,and E ( x ) ∩ E ( x ′ ) = ∅ if x ′ ∈ V is distinct from x . Given X ⊆ V , we abusenotation and use E ( X ) to denote S x ∈ X E ( x ) . One associates to E its contrac-tion e E : E ( V ) → V such that, for each y ∈ E ( V ) , e E ( y ) ∈ V is the uniquesuch that y ∈ E ( e E ( y )) . The E -expansion of a Σ -PNmatrix M = h V, D, · M i is the Σ -PNmatrix E ( M ) = hE ( V ) , E ( D ) , · E ( M ) i such that © E ( M ) ( y , . . . , y k ) = E ( © M ( e E ( y ) , . . . , e E ( y k ))) for every k ∈ N , every k -place connective © ∈ Σ ,and every y , . . . , y k ∈ E ( V ) . By construction, it is clear that e E preserves andreflects designated values, i.e., e E ( y ) ∈ D if and only if y ∈ E ( D ) . Further, givena function f : L Σ ( P ) → E ( V ) , f ∈ Val E ( M ) if and only if e E ◦ f ∈ Val M .A rexpansion of a Σ -PNmatrix M = h V, D, · M i is a refinement of some E -expansion of M . When M † = h V † , D † , · M † i is a rexpansion of M , we stillhave that if v † ∈ Val M † then e E ◦ v † ∈ Val M . Consequently, we have that e E ( A M † ( x , . . . , x n )) ⊆ A M ( e E ( x ) , . . . , e E ( x n )) , for every A ∈ L Σ ( { p , . . . , p n } ) and x , . . . , x n ∈ V † .It is easy to see that the refinement relation, the expansion relation, andthus also the rexpansion relation, are all transitive.We end this section with a very simple but useful lemma. Lemma 2.1.
Let Σ ′ ⊆ Σ and M = h V, D, · M i be a Σ ′ -deterministic Σ -PNmatrix.If M † = h V † , D † , · M † i is a rexpansion of M , A ∈ L Σ ′ ( { p , . . . , p n } ) , and y, z ∈ A M † ( x , . . . , x n ) then y ∈ D † if and only if z ∈ D † .Proof. Assume that M † is a refinement of the expansion of M with E . If y, z ∈ A M † ( x , . . . , x n ) then e E ( y ) , e E ( z ) ∈ e E ( A M † ( x , . . . , x n )) ⊆ A M ( e E ( x ) , . . . , e E ( x n )) .Since M is Σ ′ -deterministic and A ∈ L Σ ′ ( P ) it follows that A M ( e E ( x ) , . . . , e E ( x n )) has at most one element, and thus e E ( y ) = e E ( z ) . Therefore, y ∈ D † iff e E ( y ) ∈ D iff e E ( z ) ∈ D iff z ∈ D † . 4 Adding axioms
Given a signature Σ , a Tarskian consequence relation ⊢ over Σ , and Ax ⊆ L Σ ( P ) , the strengthening of ⊢ with (schema) axioms Ax is the consequence re-lation ⊢ Ax defined by Γ ⊢ Ax A if and only if Γ ∪ Ax inst ⊢ A .Our aim is to provide an adequate (and usable) semantics for ⊢ Ax , given asemantic characterization of ⊢ , a task that is well within the general effort ofcharacterizing combined logics [16, 30, 31]. The following simple result, whose(simple) proof we omit, is a corollary of Lemma 2.7 of [18]. Proposition 3.1.
Let M = h V, D, · M i be a Σ -PNmatrix and Ax ⊆ L Σ ( P ) . Theconsequence relation ⊢ Ax M is characterized by Val Ax M = { v ∈ Val M : v ( Ax inst ) ⊆ D } . Our aim in the forthcoming subsections is to design some systematic way ofusing the ideas behind rexpansions for transforming M into a PNmatrix whosevaluations somehow coincide with Val Ax M . As a first attempt, we employ a general technique from the theory of com-bining logics [16, 30, 31]. The overall idea, when starting from a given PNmatrixand a set of strengthening axioms, is to pair each formula of the logic with itspossible values but guaranteeing that instances of axioms can only be pairedwith designated values.
Theorem 3.2.
Let M = h V, D, · M i be a Σ -PNmatrix and Ax ⊆ L Σ ( P ) .The consequence ⊢ Ax M is characterized by the rexpansion M ♭ Ax = h V ♭ Ax , D ♭ Ax , · M ♭ Ax i of M defined by: • V ♭ Ax = { ( x, A ) ∈ V × L Σ ( P ) : if A ∈ Ax inst then x ∈ D } , • D ♭ Ax = D × L Σ ( P ) , • for each k ∈ N and © ∈ Σ ( k ) ,© M ♭ Ax (( x , A ) , . . . , ( x k , A k )) = { ( x, © ( A , . . . , A k )) ∈ V ♭ Ax : x ∈ © M ( x , . . . , x n ) } . Proof.
We prove, in turn, that M ♭ Ax is a rexpansion of M , and then the soundnessand completeness of M ♭ Ax with respect to ⊢ Ax M . Rexpansion.
It is easy to see that the PNmatrix M ♭ Ax is a refinement of theexpansion of M with E ( x ) = { x } × L Σ ( P ) . e E : V ♭ Ax → V is such that e E ( x, A ) = x , and clearly preserves and reflects designated values. UsingProposition 3.1, it suffices to show that { e E ◦ v ♭ : v ♭ ∈ Val M ♭ Ax } = Val Ax M .5ote that if v ♭ ∈ Val M ♭ Ax and v ♭ ( A ) = ( x, B ) then B ∈ A inst . Namely, we have B = A σ where σ ∈ L Σ ( P ) P is such that σ ( p ) = C if v ♭ ( p ) = ( y, C ) . Soundness.
Since M ♭ Ax is a rexpansion of M with E , we know that if v ♭ ∈ Val M ♭ Ax then e E ◦ v ♭ ∈ Val M . Further, if A ∈ Ax inst and v ♭ ( A ) = ( x, B ) then B ∈ ( Ax inst ) inst = Ax inst and e E ( v ♭ ( A )) = x ∈ D . We conclude that { e E ◦ v ♭ : v ♭ ∈ Val M ♭ Ax } ⊆ Val Ax M and thus that ⊢ Ax M ⊆⊢ M ♭ Ax . Completeness.
Reciprocally, if v ∈ Val M and v ( Ax inst ) ⊆ D then v = e E ◦ v ♭ with v ♭ ( A ) = ( v ( A ) , A ) for each A ∈ L Σ ( P ) . Since v ∈ Val M , the fact that v ( Ax inst ) ⊆ D guarantees that v ♭ ∈ Val M ♭ Ax . We conclude that Val Ax M ⊆{ e E ◦ v ♭ : v ♭ ∈ Val M ♭ Ax } and thus that ⊢ M ♭ Ax ⊆ ⊢ Ax M .In the definition of M ♭ Ax , if © M ( x , . . . , x k ) ∩ D = ∅ and moreover one has© ( A , . . . , A k ) ∈ Ax inst then © M ♭ Ax (( x , A ) , . . . , ( x k , A k )) = ∅ , which in gen-eral explains why the resulting PNmatrix may fail to be total. Still, M ♭ Ax isdeterministic (actually a Pmatrix) when M is a (P)matrix. These two observa-tions mean that the construction actually uses partiality in a most relevant way,but not non-determinism, which is simply imported from the starting PNma-trix. Note also that the construction, though fully illustrative of the power ofrexpansions (generalized to PNmatrices) to accomodate new axioms, has otherdrawbacks. In fact, M ♭ Ax is always infinite, even if starting from a finite M .Further, the structure of M ♭ Ax is quite syntactic, as it incorporates an obviouspattern-matching mechanism for recognizing instances of axioms into the re-ceived structure of M .In general, it is not possible to do much better, as it may happen that ⊢ Ax M cannot be characterized by a finite PNmatrix. For instance, as noted in [10],Avron and coauthors show in [4] that the logic resulting from strengtheningthe Nmatrix characterizing the basic paraconsistent logic BK of [6] with theaxiom ¬ ( p ∧ ¬ p ) → ◦ p yields a logic that cannot be characterized by a finiteNmatrix. Thus, in order to improve on our result, it can be useful to look forsuitable ways of controlling the shape of the axioms considered, as many otherexamples are known to have finite characterizations [2, 4, 6, 23, 10, 20].On the other hand, the construction of Theorem 3.2 unveils a very interestingproperty of PNmatrices: every axiomatic extension of the logic of a finite (ordenumerable) PNmatrix can be characterized by a denumerable PNmatrix. Justby itself, the result entails that intuitionistic propositional logic ( IPL ) can begiven by a single denumerable PNmatrix, sharply contrasting with the knownfact that a characteristic matrix for IPL needs to be non-denumerable (see [25,39, 38]).
Example 3.3.
Fix a suitable signature containing the two-place connective → , and use the method above for strengthening with the usual axioms Int of6ntuitionistic logic the consequence relation characterized by the Nmatrix MP = h{ , } , { } , · MP i where © MP ( x , . . . , x k ) = { , } for every k -place © ∈ Σ suchthat © = → , and → MP has the truth-table below . → MP , ,
11 0 0 , It is easy to see that ⊢ MP is precisely the consequence determined by the singlerule p p → qq of modus ponens , and so ⊢ MP ♭ Int is precisely
IPL . △ This idea applies also to propositional normal (global) modal logic K . Example 3.4.
For simplicity, take a signature containing only the -placemodality (cid:3) , and the -place connective → . The logic determined by the rulesof modus ponens and necessitation , i.e., p (cid:3) p , is easily seen to be characterizedby the Nmatrix MP (cid:3) = h{ , } , { } , · MP (cid:3) i given by the truth-tables below. → MP (cid:3) , ,
11 0 0 , (cid:3) MP (cid:3) ,
11 1
Collecting in
Norm the usual axioms of classical implication plus the normal-ization axiom (cid:3) ( p → q ) → ( (cid:3) p → (cid:3) q ) and applying Theorem 3.2, we get adenumerable PNmatrix ( MP (cid:3) ) ♭ Norm characterizing K . △ These cases suggest another possible obstacle to improving our result, namelywhen the received PNmatrix is not-deterministic and actually mixes designatedwith undesignated values in some entry of its truth-tables. When the basis isdeterministic (enough) many examples are known to be finitely characterizable.
In order to improve on the construction presented in the previous subsection,we will borrow full inspiration from the construction in [23], and try to pushthe boundaries of the scope of application of the underlying ideas.Let Σ be a signature, fix Σ d ⊆ Σ and set U ⊆ (Σ \ Σ d ) (1) to be the set of all1-place connectives not in Σ d . We shall consider the set U ∗ of all finite stringsof elements of U (the Kleene closure of U ). We shall use ε to denote the emptystring , and uw ∈ U ∗ to denote the concatenation of strings u, w ∈ U ∗ . We use prfx ( w ) to denote the set of all prefixes of string w , including ε . Given w ∈ U ∗ and A ∈ L Σ ( P ) we will use wA to denote the formula defined inductively by εA = A , and • wA = • ( wA ) if • ∈ U . For simplicity, in this and other examples, we omit the usual brackets of set notation whendescribing the truth-tables. efinition 3.5. Let © ∈ Σ be a k -place connective. Σ d -simple formulas basedon © are formulas B ∈ L Σ ( { p , . . . , p k } ) such that B = A σ for some structureformula A ∈ L Σ d ( { q , . . . , q n , r , . . . , r m } ) and some substitution σ for which:• σ ( q i ) = w i p j with w i ∈ U ∗ and ≤ j ≤ k , for each ≤ i ≤ n , and• σ ( r l ) = u l © ( p , . . . , p k ) with u l ∈ U ∗ , for each ≤ l ≤ m .For ease of notation, we will simply write A ( . . . w i p j . . . u l © ( p , . . . , p k ) . . . ) for a generic Σ d -simple formula based on ©.The look-ahead set induced by B is Θ B = ( ∪ ni =1 prfx ( w i )) ∪ ( ∪ ml =1 prfx ( u l )) .We call Σ d -simple formula to any formula which is Σ d -simple based on some connective of Σ . The look-ahead set induced by a set Γ of Σ d -simple formulasis Θ Γ = { ε } ∪ ( ∪ B ∈ Γ Θ B ) . △ Σ d -simple formulas will be the allowed shapes of our (schema) axioms. Com-paring with [23], our setup is strictly more general in that it allows for an arbi-trary base signature Σ . If we set Σ d to consist of the usual 2-place connectivesof positive logic ∧ , ∨ , → , and let Σ = Σ d ∪ U where U collects a number ofadditional 1-place connectives (e.g., ¬ , ◦ ), we recover the setup of [23].For instance, axiom B = ◦¬ ( p ∧ p ) → ( ¬ ◦ p ∨ ¬ ◦ p ) is Σ d -simplein this setting, as can be seen by taking A = r → ( q ∨ q ) , © = ∧ , and σ ( q ) = w p = ¬ ◦ p , σ ( q ) = w p = ¬ ◦ p , thus with w = w = ¬◦ , and σ ( r ) = u ( p ∧ p ) = ◦¬ ( p ∧ p ) , thus with u = ◦¬ .Easily, all axioms covered in [23] are Σ d -simple. However, p ∧ ¬ p or p → ( ¬ p → ¬ p ) fall outside the scope of [23], but are still Σ d -simple (based on anyof the -place connectives, as the r l variables are not necessary). Axioms like ¬ ( p ∧ ¬ p ) → ◦ p are not Σ d -simple, due to the interleaved nesting of ¬ and ∧ , and fall outside the scope of both methods.Having set up our syntactic restriction on the set of allowed axioms, we willstill need to match them with appropriate semantic restrictions. Before we doit, we need to shape up another crucial idea from [23]: when strengthening witha set of axioms Ax , the truth-values of the intended PNmatrix will correspondto suitable functions f : Θ Ax → V where V is the set of truth-values of the givenPNmatrix; when the value of a formula A is f this does not only settle its facevalue to f ( ε ) but also gives as look-ahead information the value f ( w ) for thevalue of formulas wA with w ∈ Θ Ax . Since not all the variables q , . . . , q n , r , . . . , r m need to occur in A , it may well happenthat the subformula © ( p , . . . , p k ) ends up not appearing in the Σ d -simple formula B basedon ©. For this reason, such a Σ d -simple formula can also be based on any available k ′ -placeconnective distinct from ©, as long as k ′ ≥ k (more precisely, k ′ needs to be at least as bigas the number of distinct variables p j occurring in B ). Note that, in our definition, Θ Γ is not simply the union of the look-ahead sets of eachformula in Γ . We not only want Θ Γ to be closed for taking prefixes, but we want ε ∈ Θ Γ evenif Γ = ∅ (a rather pathological case). efinition 3.6. Let M = h V, D, · M i be a Σ -PNmatrix and Ax a set of Σ d -simpleformulas. For each v ∈ Val M and A ∈ L Σ ( P ) , we define f Av ∈ V Θ Ax by letting f Av ( w ) = v ( wA ) for each w ∈ Θ Ax . △ It is worth noting that, by definition, f Av ( uw ) = f wAv ( u ) whenever uw ∈ Θ Ax .We can finally put forth our improved construction, taking Σ d -simple axioms.In order to make it work it will suffice to require that the given PNmatrix is Σ d -deterministic (not necessarily Σ d -total). The more general condition, though,will be to require that the PNmatrix is a rexpansion of a Σ d -deterministic PN-matrix, as the crucial necessary property is granted by Lemma 2.1. Theorem 3.7.
Let M = h V, D, · M i be a Σ -PNmatrix and Ax ⊆ L Σ ( P ) .If there exists Σ d ⊆ Σ such that M is a rexpansion of some Σ d -deterministicPNmatrix, and the formulas in Ax are all Σ d -simple, then the consequence ⊢ Ax M is characterized by the rexpansion M ♯ Ax = h V ♯ Ax , D ♯ Ax , · M ♯ Ax i of M defined by: • V ♯ Ax = S v ∈ Val Ax M { f Av : A ∈ L Σ ( P ) } , • D ♯ Ax = { f ∈ V ♯ Ax : f ( ε ) ∈ D } , • for each k ∈ N and © ∈ Σ ( k ) ,© M ♯ Ax ( f , . . . , f k ) = [ v ∈ Val Ax M { f © ( A ,...,A k ) v : A i ∈ L Σ ( P ) with f A i v = f i for ≤ i ≤ k } . Proof.
We prove that M ♯ Ax is a rexpansion of M , and then its soundness andcompleteness with respect to ⊢ Ax M . Rexpansion.
It is simple to check that the PNmatrix M ♯ Ax is a refinement ofthe expansion of M with E ( x ) = { f ∈ V Θ Ax : f ( ε ) = x } . Just note thatone has f © ( A ,...,A k ) v ( ε ) = v ( © ( A , . . . , A k )) ∈ © M ( v ( A ) , . . . , v ( A k )) = © M ( f A v ( ε ) , . . . , f A k v ( ε )) whenever it is the case that v ∈ Val M , k ∈ N ,© ∈ Σ ( k ) and A , . . . , A k ∈ L Σ ( P ) . e E : V ♯ Ax → V is such that e E ( f ) = f ( ε ) ,and clearly preserves and reflects designated values. As before, usingProposition 3.1, it suffices to show that { e E ◦ v ♯ : v ♯ ∈ Val M ♯ Ax } = Val Ax M .For a 1-place connective • ∈ U and u ∈ U ∗ such that u • ∈ Θ Ax , given avaluation v ♯ ∈ Val M ♯ Ax , we have that v ♯ ( • A )( u ) = v ♯ ( A )( u • ) , simply because v ♯ ( • A ) ∈ • M ♯ Ax ( v ♯ ( A )) and by definition of • M ♯ Ax there must exist v ∈ Val Ax M suchthat v ♯ ( • A ) = f • Bv and v ♯ ( A ) = f Bv . It easily follows, by induction, that if w ∈ U ∗ is such that uw ∈ Θ Ax then also v ♯ ( wA )( u ) = v ♯ ( A )( uw ) .9 oundness. Since M ♯ Ax is a rexpansion of M with E , if v ♯ ∈ Val M ♯ Ax then e E ◦ v ♯ ∈ Val M . Hence, when B = A ( . . . w i A j . . . u n © ( A , . . . , A k ) . . . ) ∈ Ax inst then setting y = v ♯ ( B )( ε ) we have y ∈ A M ( . . . v ♯ ( w i A j )( ε ) . . . v ♯ ( u n © ( A , . . . , A k ))( ε ) . . . ) = A M ( . . . v ♯ ( A j )( w i ) . . . v ♯ ( © ( A , . . . , A k ))( u n ) . . . ) . By definition of © M ♯ Ax , we know there exist v ∈ Val Ax M and B , . . . , B k ∈ L Σ ( P ) such that v ♯ ( © ( A , . . . , A k )) = f © ( B ,...,B k ) v and v ♯ ( A j ) = f B j v for ≤ j ≤ k . Thus, we have y ∈ A M ( . . . f B j v ( w i ) . . . f © ( B ,...,B k ) v ( u n ) . . . ) = A M ( . . . v ( w i B j ) . . . v ( u n © ( B , . . . , B k )) . . . ) . Clearly, setting z = v ( A ( . . . w i B j . . . u n © ( B , . . . , B k ) . . . )) we also have z ∈ A M ( . . . v ( w i B j ) . . . v ( u n © ( B , . . . , B k )) . . . ) . Using Lemma 2.1, since A ∈ L Σ d ( P ) and M is a rexpansion of a Σ d -deterministic PNmatrix, we conclude that y ∈ D iff z ∈ D . Now, it isalso the case that A ( . . . w i B j . . . u n © ( B , . . . , B k ) . . . ) ∈ Ax inst and weknow that v ∈ Val Ax M , so we conclude that z ∈ D . Therefore, y ∈ D and v ♯ ( B ) ∈ D ♯ Ax . We conclude { e E ◦ v ♯ : v ♯ ∈ Val M ♯ Ax } ⊆ Val Ax M and ⊢ Ax M ⊆⊢ M ♯ Ax . Completeness.
Reciprocally, if v ∈ Val Ax M then v = e E ◦ v ♯ with v ♯ ( A ) = f Av for each A ∈ L Σ ( P ) . It is immediate, by definition of © M ♯ Ax , that f © ( A ,...,A k ) v ∈ © M ♯ Ax ( f A v , . . . , f A k v ) for every k -place connective © ∈ Σ and formulas A , . . . , A k ∈ L Σ ( P ) . We conclude that v ♯ ∈ Val M ♯ Ax . There-fore, we have Val Ax M ⊆ { e E ◦ v ♭ : v ♭ ∈ Val M ♭ Ax } and ⊢ M ♭ Ax ⊆⊢ Ax M .As intended, we have pushed the boundaries of the method in [23] as muchas we could. Beyond the arbitrariness of the signature, and the more permissivesyntactic restrictions on the axioms, we also allow a more general PNmatrix tostart with. Instead of demanding it to be the two-valued Boolean matrix on the Σ d -connectives, we simply require that it be a rexpansion of any Pmatrix. Thishas the advantage of applying to a large range of non-classical base logics, butalso of making the method incremental, allowing us to add axioms one by oneand not necessarily all at once. Further, in our method, the interpretation ofthe connectives not in Σ d is completely unrestricted, which constrasts with [23],where the remaining (1-place) connectives are implicitly forced to be fully non-deterministic. This additional degree of freedom allowed by our method appliesnot only to the connectives in U , but also to any other connectives not appearingin the structure formulas of the axioms.10 Worked examples
In order to show the workings and scope of the method we have put forth inSubsection 3.2, we shall now consider a few meaningful illustrative examples.
Example 4.1.
Suppose that we want to add to the logic of classical implicationa negation connective satisfying the explosion axiom p → ( ¬ p → p ) .We consider the signature Σ with a single 2-place connective → , and asingle 1-place connective ¬ , and we start from the two-valued (P)Nmatrix B = h{ , } , { } , · B i given by the truth-tables below. → B ¬ B ,
11 0 , Clearly, → B corresponds to the usual matrix truth-table of classical impli-cation. The truth-table of ¬ B is fully non-deterministic.Setting Σ d to contain only → , and U = {¬} it is clear that B is Σ d -deterministic and that the axiom is Σ d -simple. With Exp = { p → ( ¬ p → p ) } ,we have that Θ Exp = { ε, ¬} . From Theorem 3.7, the strengthening of ⊢ B with Exp is characterized by the PNmatrix B ♯ Exp = h{ , , , } , { , } , · B ♯ Exp i where: → B ♯ Exp
00 01 10 1100 10 10 10 ∅
01 10 10 10 ∅
10 00 ,
01 00 ,
01 10 ∅ ∅ ∅ ∅ ¬ B ♯ Exp
00 00 , , Note that, for ease of notation, we are denoting a function f ∈ V ♯ Exp simply bythe string f ( ε ) f ( ¬ ) . For instance, the value corresponds to the function suchthat f ( ε ) = 0 and f ( ¬ ) = 1 . In this example, all four possibilities correspond totruth-values of the resulting PNmatrix. The reader may refer to Example 4.3below, for a situation where this does not happen.For illustration purposes, let us clarify why ¬ B ♯ Exp (10) = { , } . Easily, if xy ∈ ¬ B ♯ Exp (10) it is clear that x = 0 as this is the value of the ¬ look-aheadprovided by the value . The fact that y can be either or boils downto noting that ¬ B (0) = { , } , none of these choices being incompatible withsatisfying the axiom. Namely,
10 = f p v and
00 = f ¬ p v for any B -valuation v with v ( p ) = 1 and v ( ¬ p ) = v ( ¬¬ p ) = 0 and classical for other formulas,whereas
10 = f p v and
01 = f ¬ p v would result from any fully classical B -valuationwith v ( p ) = 1 , both valuations clearly in Val Exp B . Another interesting case is ¬ B ♯ Exp (01) = { } . Easily, the on the left of is explained by the on theright of . Once again, ¬ B (1) = { , } . However, we must exclude because
01 = f Av and
11 = f ¬ Av would jointly imply that v ( ¬ A → ( ¬¬ A → A )) = 0 and11herefore v / ∈ Val
Exp B . Similar justifications can be given, for instance, to explainwhy → B ♯ Exp
00 = ∅ .The PNmatrix B ♯ Exp obtained is slightly more complex than one could expect.Note, however, that the value is isolated from the others in the sense thata valuation that assigns to some formula must assign to all formulas.Concretely, B ♯ Exp has two maximal total refinements: the three-valued Nmatrix ( B ♯ Exp ) { , , } one would expect, plus the trivial one-valued matrix ( B ♯ Exp ) { } (whose only trivial valuation is irrelevant for the definition of ⊢ Exp B ). △ Let us now consider a slight variation on this theme.
Example 4.2.
To see the contrast with the previous example, suppose nowthat we want to add to the logic of classical implication a negation connectivesatisfying the weaker partial explosion axiom p → ( ¬ p → ¬ p ) . This is a casethat is out of the scope of the method in [23].The setting up we need to consider is the same used in Example 4.1: thesame Σ , Σ d and U , and the same starting PNmatrix B . Setting now Exp ¬ = { p → ( ¬ p → ¬ p ) } , we still have that Θ Exp ¬ = { ε, ¬} . From Theorem 3.7, thestrengthening of ⊢ B with Exp ¬ is now characterized by the PNmatrix B ♯ Exp ¬ = h{ , , , } , { , } , · B ♯ Exp ¬ i where, using the same notation convention usedin Example 4.1, we have: → B ♯ Exp ¬
00 01 10 1100 10 10 10 ∅
01 10 10 ,
11 10 1110 00 ,
01 00 ,
01 10 ∅ ∅ ∅ ¬ B ♯ Exp ¬
00 00 , , , The PNmatrix B ♯ Exp ¬ is more interesting than before. Note that it alsohas two maximal total refinements: the three-valued Nmatrix ( B ♯ Exp ¬ ) { , , } (which is precisely the same as the one obtained in Example 4.1), plus the two-valued matrix ( B ♯ Exp ¬ ) { , } (whose implication is classical but whose negationis always designated). △ Next, we will analyze a number of examples that appear scattered in the lit-erature, and show how our method can be systematically used in all of them. Westart by revisiting an example from [3], paradigmatic of many similar examplesconsidered by Avron and coauthors.
Example 4.3.
Let us consider strengthening the logic CL u N from [13, 12]with the double negation elimination axiom ¬¬ p → p . Actually, for the sakeof simplicity, we shall consider only the {¬ , →} -fragment of the logic.Let Σ d contain a single 2-place connective → , U contain a 1-place connective ¬ . The (fragment of the) logic CL u N is characterized by the Nmatrix M = h{ , } , { } , · M i with: 12 M ¬ M , It is clear that M is Σ d -deterministic and that the axiom is Σ d -simple. If welet DNe = {¬¬ p → p } , we have that Θ DNe = { ε, ¬ , ¬¬} . From Theorem 3.7,the strengthening of ⊢ M with DNe , which is well known to coincide with thelogic C min of [21, 22], is characterized by the four-valued Nmatrix M ♯ DNe = h{ , , , } , { , , } , · M ♯ DNe i where: → M ♯ DNe
010 101 110 111010 D ♯ DNe D ♯ DNe D ♯ DNe D ♯ DNe
101 010 D ♯ DNe D ♯ DNe D ♯ DNe
110 010 D ♯ DNe D ♯ DNe D ♯ DNe
111 010 D ♯ DNe D ♯ DNe D ♯ DNe ¬ M ♯ DNe
010 101101 010110 101111 110 , Above, for ease of notation, we are denoting a function f ∈ V ♯ DNe simply bythe string f ( ε ) f ( ¬ ) f ( ¬¬ ) . As this is a new feature in our row of examples, it isworth explaining why only four of the eight possible such functions appear astruth-values of the resulting Nmatrix. Namely, , , are all unattainableas f Av in the Nmatrix M since ¬ M ♯ DNe (0) = 1 . The remaining string isexcluded for more interesting reasons, as any M -valuation v with
001 = f Av makes v ( ¬¬ A → A ) = 0 and thus v / ∈ Val
DNe M .This example shows that our method, though very general, may not be astight as possible. It is a mandatory topic for further research to best under-stand how to equate the equivalence between this Nmatrix and the three-valuedNmatrix from [3].If we want to strengthen the resulting logic, C min , with the double negationintroduction axiom p → ¬¬ p , we can readily apply Theorem 3.7 to M ♯ DNe and
DNi = { p → ¬¬ p } , obtaining (up to renaming of the truth-values) thethree-valued Nmatrix N = ( M ♯ DNe ) ♯ DNi = h{ , , } , { , } , · N i where: → N
01 10 1101 10 ,
11 10 ,
11 10 , ,
11 10 , ,
11 10 , ¬ N
01 1010 0111 11
Note that, by construction, the Nmatrix N has three values g : Θ DNi → V ♯ DNe which, given that Θ DNi = { ε, ¬ , ¬¬} , can be written in string notation as g ( ε ) g ( ¬ ) g ( ¬¬ ) , corresponding to the strings , , .Clearly, each of them can be named simply by their first two symbols.It is interesting to further note that this Nmatrix is isomorphic to M ♯ DNe ∪ DNi .This is a particularly happy case as, in general, adding axioms incrementally,instead of all at once (as in [23]), will yield an equivalent PNmatrix but notnecessarily the same, often with more truth-values. △
13e now consider a more elaborate example in the family of paraconsistentlogics, as also tackled by Avron and coauthors, which is developed in detailin [23].
Example 4.4.
As in Example 5.1 of [23], we want to characterize the logicobtained by adding two additional 1-place connectives ¬ , ◦ to positive classicallogic, subject to the set of axioms Ax containing: p ∨ ¬ p p → ( ¬ p → ( ◦ p → p )) ◦ p ∨ ( p ∧ ¬ p ) ◦ p → ◦ ( p ∧ p )( ¬ p ∨ ¬ p ) → ¬ ( p ∧ p ) Let Σ d contain the three 2-place connectives ∧ , ∨ , → , and U contain the two1-place connectives ¬ , ◦ and consider the Nmatrix C = h{ , } , { } , · C i with: ∧ C ∨ C → C ¬ C ◦ C , ,
11 0 , , It is clear that C is Σ d -deterministic and that the axioms are all Σ d -simple.Further, we get Θ Ax = { ε, ¬ , ◦} . From Theorem 3.7, the strengthening ⊢ Ax C ischaracterized by the PNmatrix C ♯ Ax = h{ , , , } , { , , } , · C ♯ Ax i where: ∧ C ♯ Ax
011 101 110 111011 011 011 011 ∅
101 011 101 ∅ ∅
110 011 ∅ ∅ ∅ ∅ ∅ ∨ C ♯ Ax
011 101 110 111011 011 101 110 ∅
101 101 101 ∅ ∅
110 110 ∅ ∅ ∅ ∅ ∅ → C ♯ Ax
011 101 110 111011 101 ,
110 101 110 ∅
101 011 101 ∅ ∅
110 011 ∅ ∅ ∅ ∅ ∅ ¬ C ♯ Ax ◦ C ♯ Ax
011 101 ,
110 101 , For ease of notation, once again, we are denoting a function f ∈ V ♯ Ax simplyby the string f ( ε ) f ( ¬ ) f ( ◦ ) .Notably, the PNmatrix C ♯ Ax is slightly different from the PNmatrix obtainedusing the method in [23]. Still, it is easy to see that C ♯ Ax has two maximaltotal refinements: the three-valued PNmatrix ( C ♯ Ax ) { , , } (which is anequivalent refinement of the PNmatrix in [23] maximizing the partiality), plusthe trivial one-valued matrix ( C ♯ Ax ) { } . △ Example 4.5.
The addition of a paraconsistent Nelson-like [33, 37, 34] strongnegation ∼ to a given intermediate logic (as in [29]) can be easily captured byour construction.Let Σ d be a signature containing binary connectives ∧ , ∨ , → , and U containthe 1-place connective ∼ , and consider an Nmatrix M = h V, { } , · M i whose {∧ , ∨ , →} -reduct of M , dubbed N , is an implicative lattice [34], and such that ∼ M ( x ) = V for every x ∈ V , and let Ax contain: ∼∼ p → p p → ∼∼ p ∼ ( p ∨ p ) → ( ∼ p ∧ ∼ p ) ( ∼ p ∧ ∼ p ) → ∼ ( p ∨ p ) ∼ ( p ∧ p ) → ( ∼ p ∨ ∼ p ) ( ∼ p ∨ ∼ p ) → ∼ ( p ∧ p ) ∼ ( p → p ) → ( p ∧ ∼ p ) ( p ∧ ∼ p ) → ∼ ( p → p ) Clearly, the axioms in Ax are Σ d -simple and Θ Ax = { ε, ∼ , ∼∼} . From The-orem 3.7, ⊢ Ax M is characterized by the matrix M ♯ Ax = h V ♯ Ax , D ♯ Ax , · M ♯ Ax i isomorphicto the well known full twist-structure N ⊲⊳ over N (see [34]). Namely, we have V ♯ Ax = { f ∈ V { ε, ∼ , ∼∼} : f ( ε ) = f ( ∼∼ ) } . For simplicity, we can represent eachsuch function f ∈ V ♯ Ax simply by the pair ( f ( ε ) , f ( ∼ )) . Hence, we have:• V ♯ Ax = V × V and D ♯ Ax = { } × V ,• ( x , y ) ∧ M ♯ Ax ( x , y ) = ( x ∧ M x , y ∨ M y ) ,• ( x , y ) ∨ M ♯ Ax ( x , y ) = ( x ∨ M x , y ∧ M y ) ,• ( x , y ) → M ♯ Ax ( x , y ) = ( x → M x , x ∧ M y ) , and• ∼ M ♯ Ax ( x, y ) = ( y, x ) .When we take N to be the two-valued Boolean matrix, and using now xy insteadof ( x, y ) , we obtain, M ♯ Ax = h{ , , , } , { , } , · M ♯ Ax i where: ∧ M ♯ Ax
00 01 10 1100 00 01 00 0101 01 01 01 0110 00 01 10 1111 01 01 11 11 ∨ M ♯ Ax
00 01 10 1100 00 00 10 1001 00 01 10 1110 10 10 10 1011 10 11 10 11 → M ♯ Ax
00 01 10 1100 10 10 10 1001 10 10 10 1010 00 01 10 1111 00 01 10 11 ∼ M ♯ Ax
00 0001 1010 0111 11 true implication of Avron [1].If we further impose the axiom ∼ p → ( p → p ) we obtain corresponding explosive versions of Nelson’s construction. Making Ax ′ = Ax ∪{∼ p → ( p → p ) } , the resulting twist-structure is now a refinementresulting from isolating the truth-value (1 , , i.e., such that for ∗ ∈ {∧ , ∨ , →} we have (1 , ∗ M ♯ Ax ( x, y ) = ( x, y ) ∗ M ♯ Ax (1 ,
1) = ∅ if ( x, y ) = (1 , . Concretely, ifwe take N to be the two-valued Boolean matrix, again, we obtain the Pmatrix M ♯ Ax ′ = h{ , , , } , { , } , · M ♯ Ax ′ i where: ∧ M ♯ Ax ′
00 01 10 1100 00 01 00 ∅
01 01 01 01 ∅
10 00 01 10 ∅ ∅ ∅ ∅ ∨ M ♯ Ax ′
00 01 10 1100 00 00 10 ∅
01 00 01 10 ∅
10 10 10 10 ∅ ∅ ∅ ∅ → M ♯ Ax ′
00 01 10 1100 10 10 10 ∅
01 10 10 10 ∅
10 00 01 10 ∅ ∅ ∅ ∅ ¬ M ♯ Ax ′
00 0001 1010 0111 11
Easily, M ♯ Ax ′ has two maximal total refinements: the three-valued matrix ( M ♯ Ax ′ ) { , , } , plus the trivial one-valued matrix ( M ♯ Ax ′ ) { } . Expectedly, wehave that ( M ♯ Ax ′ ) { , , } is precisely the matrix characterizing the three-valuedlogic of Vakarelov [37, 29] (which coincides with ⊢ Ax ′ M , and is known to be trans-lationally equivalent to Łukasiewicz’s three-valued logic). △ Next, we will show, by means of an example, that our method subsumes theidea of swap-structure semantics put forth in [20, 24].
Example 4.6.
As in [24], we consider obtaining a semantic characterizationof the non-normal modal logic T of Kearns [28], which coincides with the logic S a + of Ivlev [27]. This can be done by using our method to characterize thelogic obtained by a 1-place connective (cid:3) to the {¬ , →} -fragment of classicallogic, further demanding the Tm axioms of [24], namely: (cid:3) ( p → p ) → ( (cid:3) p → (cid:3) p ) (cid:3) ( p → p ) → ( (cid:3) ¬ p → (cid:3) ¬ p ) ¬ (cid:3) ¬ ( p → p ) → ( (cid:3) p → ¬ (cid:3) ¬ p ) (cid:3) ¬ p → (cid:3) ( p → p ) p → (cid:3) ( p → p ) (cid:3) ¬ ( p → p ) → (cid:3) ¬ p (cid:3) ¬ ( p → p ) → (cid:3) p (cid:3) p → p (cid:3) p → (cid:3) ¬¬ p (cid:3) ¬¬ p → (cid:3) p Let Σ d contain → , and U = {¬ , (cid:3) } . Take the Nmatrix D = h{ , } , { } , · D i with: → D ¬ D (cid:3) D ,
11 0 0 , Clearly the axioms in Tm are Σ d -simple. Furthermore, now, we have that Θ Tm = { ε } ∪ prfx ( { (cid:3) , (cid:3) ¬ , ¬ (cid:3) ¬ , (cid:3) ¬¬} ) = { ε, ¬ , ¬ (cid:3) , ¬ (cid:3) ¬ , (cid:3) , (cid:3) ¬ , (cid:3) ¬¬} . Notethat for any f ∈ V ♯ Tm and ¬ w ∈ Θ Tm we have f ( ¬ w ) = 1 − f ( w ) . Note alsothat due to the last two axioms of Tm , it follows that f ( (cid:3) ¬¬ ) = f ( (cid:3) ) for any f ∈ V ♯ Tm . Hence, we can represent each f simply by the string f ( ε ) f ( (cid:3) ) f ( (cid:3) ¬ ) .Further, note that the antepenultimate axiom (cid:3) p → p guarantees both that f ( (cid:3) ) ≤ f ( ε ) and f ( (cid:3) ¬ ) ≤ f ( ¬ ) = 1 − f ( ε ) . Now, applying Theorem 3.7, weconclude that the strengthening ⊢ Tm D is characterized by the four-valued Nmatrixgiven by D ♯ Tm = h{ , , , } , { , } , · D ♯ Tm i where: → D ♯ Tm
000 001 100 110000 100 ,
110 100 100 ,
110 110001 110 110 110 110100 000 000 100 ,
110 110110 000 001 100 110 ¬ D ♯ Tm (cid:3) D ♯ Tm
000 100 000 , , , , It is straightforward to check that this Nmatrix is isomorphic to the Kearns andIvlev semantics [28, 27], also recovered in [24], by renaming the truth-values , , , by f, F, t, T , respectively. △ We finish this section with another example, starting from a non-classicalbase, namely, Łukasiewicz’s five-valued logic.
Example 4.7.
We start from Łukasiewicz’s logic L and strengthen it by axiom (( p → ¬ p ) → p ) → p in order to obtain Łukasiewicz’s three-valued logic L (see, for instance, [38, 26]). In this case, no new connectives are added.Let Σ d contain the 2-place connective → , and also the 1-place connective ¬ , and let U = ∅ . Let also Ax = { (( p → ¬ p ) → p ) → p } . Consider thefive-valued matrix L = h{ , , , , } , { } , · L i with:17 L
14 12 34
10 1 1 1 1 1
14 34
12 12 34
34 14 12 34
14 12 34 ¬ L
14 3412 1234 14
Clearly the axiom is Σ d -simple and Θ Ax = { ε } . Hence we represent any f ∈ V ♯ Ax simply by f ( ε ) . From Theorem 3.7, the strengthening ⊢ Ax L is characterized bythe well-known three-valued matrix ( L ) ♯ Ax = L = h{ , , } , { } , · L i where: → L
10 1 1 1
12 12 ¬ L
12 12 △ Examples 4.1, 4.4, 4.6 are also covered by the method in [23]. The two-valued based case of Example 4.5 could also be obtained using [23], but not thegeneral case we deal with, over an arbitrary implicative lattice. Example 4.3,the way it is formulated, is outside the scope of [23], not only because it startsfrom a Nmatrix where negation is not fully non-deterministic, but also becausewe are adding one axiom and then another. Examples 4.2, 4.7 are also notcovered by [23]. Namely, Example 4.2 uses an axiom which does not respecttheir syntactic criteria, and Example 4.7 uses a five-valued non-classical matrix.
In the work of Arnon Avron on Nmatrices and rexpansions, obtaining aconcise semantics for a logic (typically in the form of a Nmatrix) is not anend in itself but a means for obtaining (sequent-like) analytic calculi for thatlogic [5, 6, 7]. In other works (e.g., [23, 11]), the semantics (typically in the formof a PNmatrix) is not a basis for obtaining a calculus but it is still instrumentalin proving its analyticity (when the PNmatrix is total). In this paper, so far,we have not worried about proof-theoretic aspects. Therefore, this is a goodpoint for applying to our previous construction the techniques developed in [32,17] for obtaining analytic multiple-conclusion calculi for logics defined by finitePNmatrices, under a reasonable expressiveness proviso. This contrasts with theabove mentioned results for sequent-like calculi [7, 11, 23], for which partialityseems to devoid them of a usable (even if generalized) subformula propertycapable of guaranteeing analyticity (and elimination of non-analytic cuts).In what follows, we will consider so-called multiple-conclusion calculi , a sim-ple generalization of Hilbert-style calculi with (schematic) inference rules of the18orm Γ∆ where Γ ( premises read conjunctively, as usual) and ∆ ( conclusions read disjunctively) are sets of formulas. Such calculi were studied by Shoe-smith and Smiley in [36], and have very interesting properties. A set R of suchmultiple-conclusion rules induces a consequence relation ⊲ R by means of anadequate notion of proof, simply defined as a tree-like version of Hilbert-styleproofs. We shall show some illustrative examples later, but refer the readerto [36, 32, 31] for details. As usual, we say that R constitutes a calculus for aconsequence relation ⊲ if ⊲ R = ⊲ .A set S ⊆ L Σ ( { p } ) induces a simple notion of a generalized subformula: A isa S -subformula of B if A ∈ sub S ( B ) = sub ( B ) ∪ { S ( B ′ ) : S ∈ S , B ′ ∈ sub ( B ) } .We say that R is an S -analytic calculus if whenever Γ ⊲ R ∆ then there existsa proof of ∆ from Γ using only formulas in sub S (Γ ∪ ∆) . For finite S , we haveshown in [32, 31] that S -analyticity implies that deciding ⊲ R is in coNP , andthat proof-search can be implemented in EXPTIME .Producing analytic calculi for logics characterized by finite PNmatrices ispossible, as long as the syntax of the logic is sufficiently expressive (a notionintimately connected with the methods in [36, 5, 7, 19, 23]). Fix a Σ -PNmatrix M = h V, D, · M i . A pair of non-empty sets of elements ∅ 6 = X, Y ⊆ V are separated , X Y , if X ⊆ D and Y ⊆ V \ D , or vice versa. A formula S with var ( S ) ⊆ { p } with S M ( z ) = ∅ for every z ∈ V , and such that S M ( x ) S M ( y ) issaid to separate x and y , and called a (monadic) separator . The PNmatrix M issaid to be monadic if there is a separator for every pair of distinct truth-values.Granted a monadic PNmatrix M = h V, D, · M i and some set S = { S xy : x, y ∈ V, x = y } of monadic separators for M such that each S xy separates x and y , a discriminator for M is the V -indexed family e S = { e S x } x ∈ V , with each e S x = { S xy : y ∈ V \ { x }} . Each e S x is naturally partitioned into Ω x = { S ∈ e S x : S M ( x ) ⊆ D } and ✵ x = { S ∈ e S x : S M ( x ) ⊆ V \ D } . This partition is easily seento characterize precisely each of the truth-values of M .Given X ⊆ V , we denote by Ω ∗ X any of the possible sets built by choosingone element from each Ω x for x ∈ X , that is, Ω ∗ X ⊆ S x ∈ X Ω x is such that Ω ∗ X ∩ Ω x = ∅ for each x ∈ X . Analogously, we let ✵ ∗ X denote any of thepossible sets built by choosing one element from each ✵ x for x ∈ X , that is, ✵ ∗ X ⊆ S x ∈ X ✵ x is such that ✵ ∗ X ∩ ✵ x = ∅ for each x ∈ X . The following resultis taken from [17]. Theorem 5.1.
Let M = h V, D, · M i be a monadic PNmatrix with discriminator e S . Then, R e S M = R ∃ ∪ R D ∪ R Σ ∪ R T is an S -analytic calculus for ⊲ M , where: • R ∃ contains, for each X ⊆ V and each possible ✵ ∗ X and Ω ∗ V \ X , the rule ✵ ∗ X ( p )Ω ∗ V \ X ( p ) • R D contains, for each x ∈ V , the rule Ω x ( p ) p, ✵ x ( p ) if x ∈ D or Ω x ( p ) , p ✵ x ( p ) if x / ∈ D R Σ = S © ∈ Σ R © where, for © ∈ Σ ( k ) , R © contains, for each x , . . . , x k ∈ V and y / ∈ © M ( x , . . . , x k ) , the rule S ≤ i ≤ k Ω x i ( p i ) , Ω y ( © ( p . . . , p k )) S ≤ i ≤ k ✵ x i ( p i ) , ✵ y ( © ( p . . . , p k )) • R T contains, for each X ⊆ V with X / ∈ T M , the rule S x i ∈ X Ω x i ( p i ) S x i ∈ X ✵ x i ( p i ) . It is worth understanding the role of each of the rules proposed, as they fullycapture the behaviour of M . Namely, R ∃ allows one to exclude combinations ofseparators that do not correspond to truth-values. Actually, in examples wherethe separators S are such that, in all cases, S M ( z ) ⊆ D or S M ( z ) ⊆ V \ D , onecan always in practice set up the discriminator in a way that makes all R ∃ rulestrivial, in the sense that they will necessarily have a formula that appears bothas a premise and as a conclusion. Rules in R D distinguish those combinationsof separators that characterize designated values from those that characterizeundesignated values. Again, in practice, whenever M has both designated andundesignated values and S ( p ) = p is used to separate them, all R D rules are alsotrivial. The most operational rules are perhaps R Σ , as they completely deter-mine the interpretation of connectives in M . The rules in R = R ∃ ∪ R D ∪ R Σ already guarantee that ⊲ R = ⊲ M , but not necessarily analyticity. The rulesin R T are crucial in proving analyticity (they are already derivable from theprevious rules, but with seemingly non-analytic proofs). Indeed, rules in R T guarantee that one deals with combinations of separators that correspond tovalues taken within a total refinement of M .In order to be able to apply this general result to obtain analytic calculifor the logics characterized by the PNmatrices produced by the method wehave devised in Subsection 3.2, we need to make sure that the PNmatrices aremonadic. Of course, not every PNmatrix is monadic, but we can easily showthat our construction preserves monadicity. Proposition 5.2.
Let M = h V, D, · M i be a Σ -PNmatrix and Ax ⊆ L Σ ( P ) thatfulfill the conditions of Theorem 3.7. If M is monadic then M ♯ Ax is also monadic.Proof. Let f A v , f A v ∈ V ♯ Ax with f A v = f A v . This means that there exists w ∈ Θ Ax such that x = f A v ( w ) = f A v ( w ) = x . Given that M is monadic, weknow that there exists S ∈ L Σ ( { p } ) which separates x from x in M , that is, S M ( x ) S M ( x ) . We show that R ( p ) = S ( w p ) separates f A v from f A v in M ♯ Ax .20iven f Bv ∈ V ♯ Ax we know (from the completeness part of the proof of Theo-rem 3.7) that v ♯ ( C ) = f Cv for each C ∈ L Σ ( P ) defines a valuation v ♯ ∈ Val V ♯ Ax .Easily, then, v ♯ ( R ( B )) ∈ R M ♯ Ax ( v ♯ ( B )) = R M ♯ Ax ( f Bv ) , and therefore R M ♯ Ax ( f Bv ) = ∅ .In order to show that R M ♯ Ax ( f A v ) R M ♯ Ax ( f A v ) we just need to show that R M ♯ Ax ( f A v )( ε ) ⊆ S M ( x ) S M ( x ) ⊇ R M ♯ Ax ( f A v )( ε ) , and use the fact that in arexpansion designated values are preserved and reflected.Take i ∈ { , } and any valuation v ♯ ∈ Val M ♯ Ax with v ♯ ( p ) = f A i v i . We havethat v ♯ ( R ( p )) = v ♯ ( S ( w p )) ∈ S M ♯ Ax ( v ♯ ( w p )) . Thus, it follows that v ♯ ( R ( p ))( ε ) ∈ S M ♯ Ax ( v ♯ ( w p ))( ε ) ⊆ S M ( v ♯ ( w p )( ε )) = S M ( v ♯ ( p )( w )) = S M ( f A i v i ( w )) = S M ( x i ) .Note that this result encompasses the sufficient expressiveness preservationresult of [23], as the two-valued Boolean matrix is trivially separable using just S ( p ) = p .We now illustrate the powerful result of Theorem 5.1 by producing suitablyanalytic calculi for the resulting logics in each of the examples of Section 4. Insome cases, we also take the opportunity to illustrate the (obvious) notion ofproof in multiple-conclusion calculi. In each of the examples, rules R ∃ and R D are omitted, as they are all trivial, as discussed before. We refer the readerto [32, 17] for further details. Example 4.1, revisited.
In Example 4.1 we have obtained a four-valued PN-matrix characterizing the strengthening of the logic of classical implication withthe additional axiom p → ( ¬ p → p ) . Easily, S = { p, ¬ p } is a correspondingset of monadic separators, which yields the discriminator e S with e S x = S foreach truth-value x . This gives rise to the following partitions. x Ω x ✵ x ∅ { p, ¬ p } {¬ p } { p } { p } {¬ p } { p, ¬ p } ∅ Using Theorem 5.1, the following rules constitute an S -analytic calculus R for the logic. p , p → q r p , p → qq r qp → q r p , ¬ pq r Exp
After simplifications, the rules r – r correspond to R → , and r Exp to R T with X = { , } , X = { , } , and X = { , } .For illustration, we next depict an analytic proof of ⊲ R p → ( ¬ p → p ) .Note that rules with multiple conclusions give rise to branching in the proof-tree,which makes it necessary for the target formula p → ( ¬ p → p ) to appear inall the branches. 21 p → ( ¬ p → p ) p ¬ p → p p → ( ¬ p → p ) r ¬ p p → ( ¬ p → p ) r Exp r r △ Example 4.2, revisited.
In Example 4.2 we have obtained a four-valued PN-matrix characterizing the strengthening of the logic of classical implication withthe additional axiom p → ( ¬ p → ¬ p ) . Easily, one can reuse the set ofmonadic separators, and the discriminator, from the previous example.Using Theorem 5.1, an S -analytic calculus R for the logic can be obtainedby replacing the rule r Exp of Example 4.1 with the rule below. p , ¬ p ¬ q r Exp ¬ Expectedly, rule r Exp ¬ corresponds to R T with X = { , } , and X = { , } . △ Example 4.3, revisited.
In Example 4.3 we have obtained a four-valued Nma-trix characterizing C min , the strengthening of the logic CL u N with the additionalaxiom ¬¬ p → p . Easily, S = { p, ¬ p, ¬¬ p } is a corresponding set of monadicseparators, which allows for the discriminator e S with e S = { p } , e S = { p, ¬ p } ,and e S = e S = { p, ¬ p, ¬¬ p } , giving rise to the following partitions. x Ω x ✵ x ∅ { p } { p } {¬ p } { p, ¬ p } {¬¬ p } { p, ¬ p, ¬¬ p } ∅ Using Theorem 5.1, the following rules constitute an S -analytic calculus R for C min . p , p → q r p , p → qq r qp → q r p , ¬ p r ¬¬ pp r After simplifications, the rules r – r correspond to R → , and r , r to R ¬ .We then obtained a three-valued Nmatrix characterizing the strengtheningof C min with the axiom p → ¬¬ p . Easily, S ′ = { p, ¬ p } is a corresponding set22f monadic separators, which allows for the discriminator e S ′ with e S ′ = { p } ,and e S = e S = { p, ¬ p } , giving rise to the following partitions. x Ω x ✵ x ∅ { p } { p } {¬ p } { p, ¬ p } ∅ Using Theorem 5.1, an S ′ -analytic calculus R ′ for the logic can be obtainedby joining to the calculus R obtained above the new R ¬ rule: p ¬¬ p △ Example 4.4, revisited.
In Example 4.4 we have obtained a four-valued PN-matrix characterizing the strengthening of positive classical logic with axioms p ∨ ¬ p p → ( ¬ p → ( ◦ p → p )) ◦ p ∨ ( p ∧ ¬ p ) ◦ p → ◦ ( p ∧ p )( ¬ p ∨ ¬ p ) → ¬ ( p ∧ p ) It is easy to see that S = { p, ¬ p, ◦ p } is a corresponding set of monadicseparators, which allows for the discriminator e S with e S = { p } , e S = { p, ¬ p } ,and e S = e S = { p, ¬ p, ◦ p } . This gives rise to the following partitions. x Ω x ✵ x ∅ { p } { p } {¬ p } { p, ¬ p } {◦ p } { p, ¬ p, ◦ p } ∅ Using Theorem 5.1, the following rules constitute an S -analytic calculus R for the logic. p , qp ∧ q r p ∧ qp r p ∧ qq r ¬ p ¬ ( p ∧ q ) r pp ∨ q r qp ∨ q r p ∨ qp , q r p , p → qq r qp → q r p , p → q r p , ¬ p r p , ◦ p r p ¬ p , ◦ p r p , q , ¬ q ¬ p r p , ¬ p , ◦ pq r After simplifications, the rules r – r correspond to R ∧ , r – r to R ∨ , r – r to R → , r to R ¬ , r and r to R ◦ . Finally, r and r result from R T , with X = { , } and X = { , } , respectively.Sample proofs, namely for some of the axioms, with a very similar calculuscan be found in [17]. △ xample 4.5, revisited. In Example 4.5 we have obtained a four-valued twist-structure characterizing the addition of a paraconsistent Nelson-like strong nega-tion to positive classical logic. Easily, S = { p, ∼ p } is a corresponding set ofmonadic separators, yielding the discriminator e S with e S x = S for each truth-value x . This gives rise to the following partitions. x Ω x ✵ x ∅ { p, ∼ p } {∼ p } { p } { p } {∼ p } { p, ∼ p } ∅ Using Theorem 5.1, the following rules constitute an S -analytic calculus R for the logic. p ∧ qp r p ∧ qq r p , qp ∧ q r ∼ p ∼ ( p ∧ q ) r ∼ q ∼ ( p ∧ q ) r ∼ ( p ∧ q ) ∼ p , ∼ q r pp ∨ q r qp ∨ q r p ∨ qp , q r ∼ ( p ∨ q ) ∼ q r ∼ ( p ∨ q ) ∼ q r ∼ p , ∼ q ∼ ( p ∨ q ) r p , p → qq r qp → q r p , p → q r ∼ ( p → q ) p r ∼ ( p → q ) ∼ q r p , ∼ q ∼ ( p → q ) r p ∼∼ p r ∼∼ pp r After simplifications, the rules r – r correspond to R ∧ , r – r to R ∨ , r – r to R → , r and r to R ∼ .A strengthening with an additional (explosion) axiom ∼ p → ( p → p ) wasthen shown to be characterized by a four-valued Pmatrix. It is straightforwardto see that one can reuse the set of monadic separators, and the discriminator,from above. Using Theorem 5.1, an S -analytic calculus R ′ for the logic can beobtained by simply adding to R the new R T rule p , ∼ pq obtained by considering X = { , } , X = { , } , and X = { , } . △ Example 4.6, revisited.
In Example 4.6 we obtained a four-valued Nmatrixcharacterizing the non-normal modal logic of Kearns and Ivlev [28, 27]. It isnot difficult to check (namely, using Proposition 5.2) that S = { p, (cid:3) p, (cid:3) ¬ p } isa set of monadic separators for the Nmatrix. This allows for the discriminator e S with e S = e S = { p, (cid:3) ¬ p } , and e S = e S = { p, (cid:3) p } , which gives rise tothe following partitions. 24 Ω x ✵ x ∅ { p, (cid:3) ¬ p } { (cid:3) ¬ p } { p } { p } { (cid:3) p } { p, (cid:3) p } ∅ Using Theorem 5.1, we get an S -analytic calculus R for the logic. p , p → q r p , p → qq r qp → q r p , ¬ p r p , ¬ p r (cid:3) ( p → q ) , (cid:3) p (cid:3) q k (cid:3) ( p → q ) , (cid:3) ¬ q (cid:3) ¬ p k (cid:3) p , (cid:3) ¬ q (cid:3) ¬ ( p → q ) k (cid:3) ¬ p (cid:3) ( p → q ) m (cid:3) q (cid:3) ( p → q ) m (cid:3) ¬ ( p → q ) (cid:3) ¬ q m (cid:3) ¬ ( p → q ) (cid:3) p m (cid:3) pp T (cid:3) p (cid:3) ¬¬ p dn (cid:3) ¬¬ p (cid:3) p dn After simplifications, the rules r – r , k , k – k , m – m correspond to R → , r – r and dn – dn to R ¬ , and T to R (cid:3) . It is interesting to note that rules r – r characterize classical logic, and the remaining rules are in a one-to-onecorrespondence with the axioms considered (see [24]). The only less obviouscase is the rule k . For this reason we present below an analytic proof of thecorresponding axiom K = ¬ (cid:3) ¬ ( p → q ) → ( (cid:3) p → ¬ (cid:3) ¬ q ) , i.e., ⊲ R K . Notethat K is obtained in all the branches of the proof-tree, except for the leftmostone, which is discontinued due to rule r (as signaled by the use of ∗ ). ∅ K ¬ (cid:3) ¬ ( p → q ) (cid:3) p → ¬ (cid:3) ¬ qK r (cid:3) p ¬ (cid:3) ¬ q (cid:3) p → ¬ (cid:3) ¬ qK r r (cid:3) ¬ q (cid:3) ¬ ( p → q ) ∗ r k r r r △ xample 4.7, revisited. In Example 4.7 we have obtained the usual three-valued Łukasiewicz’s matrix (by strengthening the five-valued Łukasiewicz logicwith an additional axiom). Easily, S = { p, ¬ p } is a set of monadic separators,yielding the discriminator e S with e S = e S = { p, ¬ p } , and e S = { p } , which givesrise to the following partitions. x Ω x ✵ x {¬ p } { p } ∅ { p, ¬ p } { p } ∅ Using Theorem 5.1, the following rules constitute an S -analytic calculus R for L . p , ¬ p r p ¬¬ p r ¬¬ pp r p , p → q , ¬ q r p , p → qq r qp → q r ¬ pp → q r ¬ q , p → q ¬ p r ¬ ( p → q ) p r ¬ ( p → q ) ¬ q r p , ¬ q ¬ ( p → q ) r After simplifications, the rules r – r correspond to R ¬ , and r – r to R → .For illustration, we depict an analytic proof of the added axiom A = (( p →¬ p ) → p ) → p ) , i.e., ⊲ R A . ∅ ¬ pp → ¬ p ¬ (( p → ¬ p ) → p ) A r r r A ( p → ¬ p ) → p ¬¬ ppA r r p → ¬ ppA r r pA r r r △ Concluding remarks
In this paper we have shown that rexpansions of (P)(N)matrices are a uni-versal tool for explaining the strengthening of logics with additional axioms.This does not come as a surprise, as non-determinism and partiality are wellknown for enabling a plethora of compositionality results in logic. Our generalmethod in Theorem 3.2 is not effective, but it still brings about some inter-esting phenomena, such as the possibility of building a denumerable semanticsfor intuitionistic propositional logic (where the precise roles of non-determinismand partiality need further clarification). More practical, though, is our less gen-eral method in Theorem 3.7 as, despite the necessary restrictions on its scope, itbrings about an effective method for producing finite semantic characterizationswhenever starting from a finite basis. Our results cover a myriad of examplesin the literature, namely those motivated by the study of logics of formal incon-sistency, which played an important role in the work of Arnon Avron. Besides,our effective method, while more general and incremental, is fully inspired bythe fundamental ideas in [23]. It is also worth noting that our results apply notjust to the Tarskian notion of consequence relation, but also to the multiple-conclusion case. An obvious topic for further work is to provide a usable toolimplementing these methods.Other opportunities for further research, aimed at generalizing the resultspresented, would be to find more general syntactic conditions on the set ofallowed axioms. For instance, the number of sentential variables occurring inan axiom seems to be easy to flexibilize by artificially extending the logic withbig-arity connectives. Beyond axioms, one could think even further away, andconsider strengthening logics with fully-fledged inference rules. In any case, suchextensions will expectedly need more sophisticated techniques than the simpleidea behind look-aheads .These results reinforce the need to better understand the conditions underwhich two (P)(N)matrices characterize the same logic. This is by no means atrivial question, but we believe that the notion of rexpansion can be a usefultool in that direction.If not for its own sake, this line of research aimed at providing effectivesemantic characterizations for combined logics is quite well justified by anotherrecurring goal of many of the papers that inspired us: ultimately obtainingsuitably analytic calculi for the resulting logics.
References [1] O. Arieli and A. Avron. The value of the four values.
Artificial Intelligence ,102(1):97–141, 1998.[2] A. Avron. Non-deterministic matrices and modular semantics of rules. InJ.-Y. Béziau, editor,
Logica Universalis , pages 149–167. Birkhäuser, 2005.273] A. Avron. A non-deterministic view on non-classical negations.
StudiaLogica , 80(2-3):159—194, 2005.[4] A. Avron. Non-deterministic semantics for logics with a consistency op-erator.
International Journal of Approximate Reasoning , 45(2):271–287,2007.[5] A. Avron, J. Ben-Naim, and B. Konikowska. Cut-free ordinary sequentcalculi for logics having generalized finite-valued semantics.
Logica Univer-salis , 1(1):41–70, 2007.[6] A. Avron, B. Konikowska, and A. Zamansky. Modular construction ofcut-free sequent calculi for paraconsistent logics. In
Proceedings of the27th Annual IEEE Symposium on Logic in Computer Science (LICS 2012) ,pages 85–94, 2012.[7] A. Avron, B. Konikowska, and A. Zamansky. Cut-free sequent calculi forC-systems with generalized finite-valued semantics.
Journal of Logic andComputation , 23(3):517–540, 2013.[8] A. Avron and I. Lev. Non-deterministic multiple-valued structures.
Journalof Logic and Computation , 15(3):241–261, 2005.[9] A. Avron and A. Zamansky. Non-deterministic semantics for logical sys-tems: A survey. In D. Gabbay and F. Guenthner, editors,
Handbook ofPhilosophical Logic , volume 16, pages 227–304. Springer, 2011.[10] A. Avron and Y. Zohar. Rexpansions of non-deterministic matricesand their applications in non-classical logics.
Review of Symbolic Logic ,12(1):173–200, 2019.[11] M. Baaz, O. Lahav, and A. Zamansky. Finite-valued semantics for canonicallabelled calculi.
Journal of Automated Reasoning , 51(4):401–430, 2013.[12] D. Batens. Paraconsistent extensional propositional logics.
Logique et Anal-yse , 23(90–91):195–234, 1980.[13] D. Batens. A survey of inconsistency-adaptive logics. In
Frontiers of para-consistent logic , pages 49–73. Research Studies Press, 2000.[14] N. Belnap. How a computer should think. In G. Ryle, editor,
ContemporaryAspects of Philosophy , volume 2 of
Episteme , pages 30–55. Oriel Press,1977.[15] N. Belnap. A useful four-valued logic. In G. Epstein J.M. Dunn, editor,
Modern Uses of Multiple-Valued Logic , volume 2 of
Episteme , pages 5–37.Oriel Press, 1977. 2816] C. Caleiro, W. Carnielli, J. Rasga, and C. Sernadas. Fibring of logics as auniversal construction. In D. Gabbay and F. Guenthner, editors,
Handbookof Philosophical Logic, 2nd Edition , volume 13, pages 123–187. Springer,2005.[17] C. Caleiro and S. Marcelino. Analytic calculi for monadic PN-matrices. In
Logic, Language, Information and Computation(WoLLIC 2019) , LNCS. Springer, in print. Preprint available at http://sqig.math.ist.utl.pt/pub/CaleiroC/19-CM-axiomPNmatrices.pdf .[18] C. Caleiro, S. Marcelino, and J. Marcos. Combining fragments of clas-sical logic: When are interaction principles needed?
Soft Computing ,23(7):2213–2231, 2019.[19] C. Caleiro, J. Marcos, and M. Volpe. Bivalent semantics, generalized com-positionality and analytic classic-like tableaux for finite-valued logics.
The-oretical Computer Science , 603:84–110, 2015.[20] W. Carnielli and M. Coniglio.
Paraconsistent Logic: Consistency, Contra-diction and Negation , volume 40 of
Logic, Epistemology, and the Unity ofScience . Springer, 2016.[21] W. Carnielli and J. Marcos. Limits for paraconsistent calculi.
Notre DameJournal of Formal Logic , 40:375–390, 1999.[22] W. Carnielli and J. Marcos. A taxonomy of C-systems. In W. Carnielli,M. Coniglio, and I. D’Ottaviano, editors,
Paraconsistency: The logical wayto the inconsistent , volume 228 of
Lecture Notes in Pure and Applied Math-ematics , pages 1–94. Marcel Dekker, 2002.[23] A. Ciabattoni, O. Lahav, L. Spendier, and A. Zamansky. Taming paracon-sistent (and other) logics: An algorithmic approach.
ACM Transactions onComputational Logic , 16(1):5:1–5:23, 2014.[24] M. Coniglio and A. Golzio. Swap structures semantics for Ivlev-like modallogics.
Soft Computing , 23(7):2243–2254, 2019.[25] K. Gödel. Zum intuitionistischen aussagenkalkül. In
Mathematisch – natur-wissenschaftliche klasse , volume 69 of
Anzeiger , pages 65–66. Akademie derWissenschaften, Wien, 1932.[26] S. Gottwald.
A Treatise on Many-Valued Logics , volume 9 of
Studies inLogic and Computation . Research Studies Press, 2001.[27] J. Ivlev. A semantics for modal calculi.
Bulletin of the Section of Logic ,17(3–4):114–121, 1988.[28] J. Kearns. Modal semantics without possible worlds.
Journal of SymbolicLogic , 46(1):77–86, 1981. 2929] M. Kracht. On extensions of intermediate logics by strong negation.
Journalof Philosophical Logic , 27:49–73, 1998.[30] S. Marcelino and C. Caleiro. Decidability and complexity of fibred logicswithout shared connectives.
Logic Journal of the IGPL , 24(5):673–707,2016.[31] S. Marcelino and C. Caleiro. Disjoint fibring of non-deterministic matri-ces. In R. de Queiroz J. Kennedy, editor,
Logic, Language, Informationand Computation (WoLLIC 2017) , volume 10388 of
LNCS , pages 242–255.Springer, 2017.[32] S. Marcelino and C. Caleiro. Axiomatizing non-deterministic many-valuedgeneralized consequence relations.
Synthese , doi 10.1007/s11229-019-02142-8, 2019.[33] D. Nelson. Constructible falsity.
Journal of Symbolic Logic , 14:247–257,1948.[34] S. Odintsov.
Constructive Negations and Paraconsistency , volume 26 of
Trends in Logic . Springer Netherlands, 2008.[35] D. Scott. Completeness and axiomatizability in many-valued logic. InL. Henkin, J. Addison, C. Chang, W. Craig, D. Scott, and R. Vaught,editors,
Proceedings of the Tarski Symposium , volume XXV of
Proceedingsof Symposia in Pure Mathematics , pages 411–435. American MathematicalSociety, 1974.[36] D. Shoesmith and T. Smiley.
Multiple-Conclusion Logic . Cambridge Uni-versity Press, 1978.[37] D. Vakarelov. Notes on N-lattices and constructive logic with strongnegation.
Studia Logica: An International Journal for Symbolic Logic ,36(1/2):109–125, 1977.[38] R. Wójcicki.
Theory of Logical Calculi , volume 199 of
Synthese Library .Kluwer, 1998.[39] A. Wroński. On the cardinality of matrices strongly adequate for the in-tuitionistic propositional logic.