Normality Operators and Classical Recapture in Many ...

Normality Operators and Classical Recapture in Many-valued Logic ROBERTO CIUNI, Department FISPPA, Section of Philosophy, University of Padova, Piazza Capitaniato 3, Padova, Italy. E-mail: [email protected] MASSIMILIANO CARRARA, Department FISPPA, Section of Philosophy, University of Padova, Piazza Capitaniato 3, Padova, Italy. E-mail: [email protected] Abstract In this paper, we use a ‘normality operator’ in order to generate Logics of Formal Inconsistency and Logics of Formal Undeterminedness from any subclassical many-valued logic that enjoys a truthfunctional semantics. Normality operators express, in any many-valued logic, that a given formula has a classical truth value. In the first part of the paper we provide some setup, and focus on manyvalued logics that satisfy some (or all) of three properties, namely subclassicality and two properties that we call fixed-point negation property and conservativeness. In the second part of the paper, we introduce normality operators, and explore their formal behavior. In the third and final part of the paper, we establish a number of classical recapture results for systems of Formal Inconsistency and Formal Undeterminedness that satisfy some or all the properties above. These are the main formal ⊛ results of the paper. Also, we illustrate concrete cases of recapture by discussing the logics K⊛ 3 , LP , ⊛ ⊛ w , PWK and E , respectively. Kw⊛ , PWK and E , that are in turn extensions of K , LP, K 3 fde fde 3 3 Keywords: Many-valued logic, Classical Recapture, Normality Operators, Logics of Formal Inconsistency, Logics of Formal Undeterminedness

Introduction In this paper, we provide a semantic method to build Logics of Formal Inconsistency [14, 16, 17] and Logics of Formal Undeterminedness [37] from any given many-valued logic that has a truth-functional semantics, and we establish classical recapture theorems for some families of resulting systems. A distinctive feature of the method we present is that, when fed with a paraconsistent and paracomplete many-valued logic, it will output a system that is both a logic of Formal Inconsistency and a logic of Formal Undeterminedness. This is done by introduction of a normality operator ⊛, which allows for expressing, in any many-valued logic, that a given formula has a classical truth value.1 The normality operator suggests a natural application to classical recapture, that is the problem of specifying at which conditions we can reason classically when using a weaker (typically, many-valued) reasoning tool. In particular, the operator offers a straightforward strategy for recapture, namely: to express within the object-language 1 The label ‘normality operator’ makes implicit reference to the ‘normality view’ that we introduce below in this Introduction. According to this view, situations where a formula has a classical truth value are ‘normal’, while situations where a formula has a non-classical truth value are ‘abnormal’.

L. J. of the IGPL, Vol. 0 No. 0, pp. 1–30 0000

1

© Oxford University Press

2

Normality Operators and Classical Recapture in Many-valued Logic

itself the conditions at which we can reason classically. The theorems that we establish in this paper (Section 4) provide general recapture results for a number of many-valued logics that (1) include a normality operator and (2) are defined by satisfaction of given formal properties that we introduce in Section 2. The normality operator the we define in this paper is a special case of the ‘recovery operators’ that are the focus of this volume. In particular, the normality operator is designed in order to help recover Classical Logic specifically (see Definition 3.1 in Section 3), while for every logic S and sublogic S′ , a suitably defined recovery operator helps recover S within S′ . Operators that can help recapture Classical Logic are called ‘classicality operators’ in [40]. We prefer the term ‘normality operator’ here, since it hints at the fact that recapture, far from being just a technical option, is motivated by a philosophical view on the relations between subclassical many-valued systems and Classical Logic. Also, this choice of terms allows us to be coherent with the terminology from our recent [20]. The paper proceeds as follows. In the remainder of this introduction, we provide some background on Logics of Formal Inconsistency and Logics of Formal Undeterminedness, and we discuss the relevance of the results of the paper. Section 1 introduces the general setup, notation, and basic notions that we use throughout the entire paper. Section 2 introduces many-valued logic; in particular, we focus on formalisms that have a truth-functional semantics and satisfy some (or all) of three formal properties, namely subclassicality (Definition 2.3), and two properties that we call fixed-point negation property (Definition 2.4) and conservativeness (Definition 2.8), respectively. Beside, the section zooms in on a handful of many-valued systems; these will work as reference systems for the logics that we introduce in Section 5. Section 3 introduces normality operators and details some valid principles involving them. The satisfaction of some (or all) of the properties highlighted in Section 2 proves relevant for such principles. Beside, the section explores the relation between normality operators, Logics of Formal Inconsistency, Logics of Formal Undeterminedness, and their operators. Section 4 establishes a number of recapture theorems (Theorems 4.2–4.9). Again, these theorems depend on the satisfaction of some (or all) of the properties defined in Section 2. Section 5 exemplifies the recapture strategies from the theorems in Section 4 by discussing particular cases of recapture in five different many-valued logics including the normality operator. In particular, we discuss some specific features of the systems based on Kw 3 and PWK. Finally, Section 6 sums up the results of the paper and discuss some possible topics for future research. Logics of Formal Inconsistency and Logics of Formal Undeterminedness. The Logics of Formal Inconsistency (from now on, LFIs) are paraconsistent logics that have been first introduced in [17] and further developed in [14, 16]. LFIs control the behavior of contradictions—and single them out—by internalizing the notion of consistency 2 in the language, and expressing whether a formula is consistent or not. Most LFIs lack a traditional truth-functional semantics,3 but there is increasing 2 LFIs have been introduced in order to overcome some deficiencies in the da Costa’s calculi Cn originating from [23]. The connections between da Costa’s project and the LFI project are explained in detail in [16], to which we refer the reader.The original proposal by [17] also included an inconsistency operator. See Section 3 for this. 3 For instance, the logic mbC [14], and the logic LETJ [18] lack a truth-functional semantics. mbC and some of its extensions have been given a non-deterministic many-valued semantics by [4]; LETJ is given a quasi-matrix semantics

Normality Operators and Classical Recapture in Many-valued Logic 3 interest today for LFIs that have a truth-functional semantics, like the logics that we investigate in this paper—see for instance the logics LFI1, LFI2 [17], LFI3 [39], and the systems from [40]. The Logics of Formal Undeterminedness (from now on, LFUs) are a relatively new addition to the literature in many-valued logic, although the idea of recovering excluded middle analogously to how non-contradiction and explosion are recovered in LFIs traces back to [25]. LFUs have been first introduced in [37] and have been later explored in [6]. They dualize the project of LFIs; in particular, LFUs are paracomplete logics that control—and single out—determined formulas by internalizing the notion of determinedness in the language. A system that is both a LFI and a LFU is introduced by [18] under the name of LETJ . This logic extends Nelson’s logic N4 with a normality operator (in our terminology), and it is introduced in order to provide a propositional tool for reasoning about truth, inconsistent, and incomplete evidence.4 Classical Recapture. The philosophical applications of many-valued logics usually come with a story of ‘normality’: there are a number of ‘abnormal phenomena’ for which we need many-valued reasoning—logical paradoxes, partial information, vagueness, among others—but as long as the situation is normal —that is, no abnormal phenomena is at stake—Classical Logic is perfectly in order as it is. The view seems to be justified by the following consideration: numerous as the abnormal phenomena may be, we may assume that they are not ubiquitous. For instance, take semantic and set-theoretical paradoxes; to put it with [41, p.235]: ‘paradoxical sentences seem to be a fairly small proportion of the sentences we reason with. . . It would seem plausible to claim that in our day-to-day reasoning we (quite correctly) presuppose that we are not dealing with paradoxical claims’. Thus, if one follows the ‘normality story’ above, then one may just want to resort to classical reasoning when possible—that is, in ‘normal situations.’ This in turn prompts the following recapture question: how can we recapture Classical Logic CL in manyvalued logic? That is, how can we secure inference of classical conclusions, under the assumption that we are facing no abnormality? [8, 41, 42]. There are also more specific reasons to pursue recapture of classical reasoning. As is well-known, the success of the many-valued strategy in dealing with abnormal phenomena consists in weakening Classical Logic. For instance, paracomplete approaches to paradoxes drop the Law of Excluded Middle α∨¬α, while paraconsistent approaches drop Ex Contradictione Quodlibet α, ¬α ⊧ β.5 However, every many-valued logic seem to come with undesired casualties. For instance, the Law of Identity α ⊃ α is not valid in the strong Kleene logic K3 [34], and Modus Ponens α, α ⊃ β ⊧ β is not valid in the Logic of Paradox LP [41, 44]. These seem to be very basic principles of our reasoning. Thus, ability to display classical reasoning when possible is a desideratum. Relevance of the results of this paper. Classical recapture is one of the most important problems in the philosophy of many-valued logic and non-classical logic in by [18]. Other LFIs are given a possible-translation semantics—see [14] for this device and its application to LFIs. 4 We believe that LETJ is a very interesting addition to LFIs and LFUs, but we will not discuss it further in this paper. The reason is that LETJ lacks a standard matrix-based semantics, which is the kind of semantics on which we focus here. We wish to devote future investigation on the addition of a normality operators to logics that are based on quasi-matrices [18] like LETJ . 5

See Definitions 2.1–2.2 for the definition of paracompleteness and paraconsistency, respectively.

4


general, which in turn explains the relevance of the recapture theorems that we pursue in this paper. The potential of LFIs for classical recapture has already been noticed by [14], which presents recapture results for a number of systems. However, notice that the current recapture results in the literature from LFIs are ‘system-tailored’, in the sense that, for each of the systems considered, they provide a specific result. By contrast, the results that we present in Section 4 are more general, in that they provide recapture strategies for families of logics. In our opinion, this makes an interesting progress w.r.t. (with respect to) current results. We briefly discuss the difference between our results and the theorems from [36] at the end of section 4.

1

Preliminaries

Given a similarity type ν and a countably infinitely set X of generators, the absolutely free algebra Fml over X is called the formula algebra of type ν. F ml denotes the universe of Fml. We call propositional variables—or variables, simply—the members of X, and we denote them by p, q, r, . . . . We call ν-formulas the members of F ml, and we denote them by α, β, γ, . . . . We use Σ, Γ, ∆, . . . to denote sets of formulas.6 We omit reference to the type ν when this is not necessary. The two different formula algebras that we investigate in this paper are defined as follows: • Fml1 is a formula algebra of type (2, 2, 1, 0, 0), namely, of the type containing the connectives ∨, ∧, ¬, , ⊺; • Fml2 is the formula algebra of type (2, 2, 1, 1, 0, 0), namely, of the type containing the connectives ∨, ∧, ¬, ⊛, , ⊺; A logic of type ν is a pair S = ⟨Fml, ⊢S ⟩, where Fml is a formula algebra of type ν and ⊢S ⊆ ℘(F ml) × F ml is a substitution invariant consequence relation. A νmatrix —or, simply, a matrix —is a pair M = ⟨A, D⟩ with A an algebra of type ν with universe A and D ⊂ A. D is called the filter of M. Just to make an example: Classical Logic CL is defined as ⟨Fml1 , ⊧MCL ⟩, and MCL is defined as ⟨B2 , {t}⟩, where B2 is the two-element Boolean algebra. The following notion of a submatrix will be useful: Definition 1.1 (Submatrix) A matrix M = ⟨A, D⟩ is a submatrix of a matrix M′ = ⟨A′ , D′ ⟩ (M ⊑ M′ ) iff (if and only if) A is a subalgebra of A′ and D = D′ ∩ A. Informally, we think of the members of A as truth values. Under this informal reading, the members of DM are naturally thought of as designated values. Definition 1.2 (Valuation) A valuation is a homomorphism v ∶ Fml Ð→ A from a formula algebra Fml into an algebra A of the same type. We denote by HomFml,A the set of valuations for Fml defined on A. When Fml is clear by the context and we wish to focus on the matrix rather than on the algebra, we write HomM . For every M = ⟨A, D⟩, we let HomM (Σ) be the set {v ∈ HomM ∣ v[Σ] ⊆ DM } of the models of Σ based on M. 6 Unless specified otherwise, in this paper we consider just finite sets of formulas, with the exception, of course, of F ml itself.

Normality Operators and Classical Recapture in Many-valued Logic 5 Definition 1.3 (Matrix Consequence) If M is a matrix, the relation ⊧M ⊆ ℘(F ml) × F ml defined as follows: Σ ⊧M β ⇔ for every valuation v ∈ HomM , ν[Σ] ⊆ DM implies ν(β) ∈ DM is a consequence relation. As usual, we say that α is a tautology if ∅ ⊧M α. We write α ⊧M β instead of {α} ⊧M β, and α, β ⊧M γ instead of {α, β} ⊧M γ. We also use other standard listlike shorthand; for instance, Γ, ∆ for Γ ∪ ∆, or Γ, α for Γ ∪ {α}. The following are useful notations: α ⊃ β is an abbreviation for ¬α ∨ β, and var(Σ) denotes the set of variables from Σ.7

2

Many-valued Logic

Systems of many-valued logic allow for other possible truth values beside the ‘classical’ truth values f (false) or t (true). If it takes one or more non-classical logic into account, a system of many valued logic will not satisfy the principle of bivalence, according to which any formula in Fml is assigned either f or t.8 Systems of many-valued logic may admit a matrix-based semantics, which is truth-functional, or interpretations that are not truth-functional, such as the non-deterministic semantics from [5] or the quasi-matrix semantics from [24]. In this paper, we focus on systems of many-valued logic that have a truth-functional semantics.9 Classical Logic CL itself is a ‘limit-case’ of this kind of logics, in the sense that it satisfies bivalence beyond having a truthfunctional semantics.10 In philosophical logic, the focus is usually on the following two properties that a many-valued logic can enjoy, namely paracompleteness and paraconsistency:11 Definition 2.1 A logic S = ⟨Fml, ⊧M ⟩ is paracomplete iff ∅⊧ / M α ∨ ¬α Definition 2.2 A logic S = ⟨Fml, ⊧M ⟩ is paraconsistent iff α, ¬α ⊧ /M β In this paper, we focus on many-valued logics that are subclassical and enjoy what we call the fixed-point negation property (fpn property, for short): 7

For the basic setting defined in this section, see also chapter 1 from [30].

8

A different formulation of the principle states that any formula in Fml is assigned either a designated value, or f . This version makes some paraconsistent logics bivalent. In this paper, however, we will follow the stricter version of the principle above. 9 Truth-functionality holds when the truth value of a formula is entirely determined by the truth values of its components. Some many-valued logics have a non-truth functional semantics. The logics mbC [14] and LETJ [18] provide a clear example of this. 10 In addition, here we focus on truth-functional many-valued logics whose consequence relation is the matrix consequence relation of a single matrix. The reason for this is that the truth-functional systems of many-valued logic that are most widespread in philosophical logic today belong to this family. From now on, we will be referring to those systems of many-valued logic when using the label ‘many-valued logics’. 11 The definitions of paracompleteness and paraconsistency that we give here can be generalized beyond manyvalued logic to any arbitrary logic. Since we are dealing just with many-valued logic in this paper, we feel free to restrict the definitions to this kind of systems.

6


Definition 2.3 A logic S = ⟨Fml1 , ⊧M ⟩ is subclassical iff S is a proper sublogic of CL, that is iff: Σ ⊧MS β Σ ⊧MCL β

⇒ ⇒ /

Σ ⊧MCL β Σ ⊧MS β

Definition 2.4 A logic S = ⟨Fmli , ⊧M ⟩ with i ∈ {1, 2} satisfies the fixed-point negation (fpn) property iff: ¬x = x for every x ∈ A− where A− = A ∖ {f , t} is the set of ‘non-classical’ values in A. The following observations will be relevant in Section 3: Observation 2.5 For every logic S = ⟨Fml1 , ⊧M ⟩, if ⟨B2 , {t}⟩ is a submatrix of M, then Σ ⊧MS β ⇒ Σ ⊧MCL β Observation 2.6 If S = ⟨Fml, ⊧M ⟩ satisfies the fpn property, then: 1. ∅ ⊧ / M α ∨ ¬α 2. α, ¬α ⊧ /M β

⇔ ⇔

∃x ∈ A− ∶ x ∉ DM ∃x ∈ A− ∶ x ∈ DM

The proof of the observations is elementary and we leave it to the reader. Additionally, we consider a further property, that we call conservativeness with respect to Classical Logic—from now on, conservativeness, simply. In order to define it, we first need to introduce the notion of the classical counterpart of a valuation function:12 Definition 2.7 (Classical Counterpart of a Valuation) For every S = ⟨Fml1 , ⊧M ⟩ and v, v ′ ∈ HomFml,A , v ′ is the classical counterpart of v iff: 1. 2.

v ′ (p) = t v ′ (p) = f

if if

v(p) ∈ DM v(p) ∉ DM

Definition 2.8 (Conservativeness) A logic S = ⟨Fml1 , ⊧M ⟩ is conservative with respect to Classical Logic CL iff: For every v, v ′ ∈ HomFml,A and α ∈ F ml, if v ′ is the classical counterpart of v, then: 1. 2.

v ′ (α) = t v ′ (α) = f

if v(α) = t if v(α) = f

Since the logics we consider have ⟨B2 , {t}⟩ as a submatrix, we can represent classical valuations within any of the logics we investigate in this paper. In particular, we say that v ∈ HomFml1 ,A is a classical valuation iff v(p) ∈ {f , t} for every p ∈ X, and that 12 In what follows, we abuse notation a bit and write v(p) (v(φ)) instead of vp (vφ). We believe this makes the symbolism more readable.

Normality Operators and Classical Recapture in Many-valued Logic 7 v ∈ HomFml1 ,A is a classical model of Σ iff v is a classical valuation and v[Σ] = {t}. Conservativeness of S = ⟨Fml, ⊧M ⟩ implies that, for every v ∈ HomM , if v[Σ] = {t}, then Σ has a classical model. Lack of conservativeness implies that existence of a model v ∈ HomM such that v[Σ] = {t} does not imply that Σ has a classical model. These fact will play a role in the results from Section 3. Subclassicality and the fpn property determine infinitely many systems, but more importantly, they are satisfied by the most widespread formalisms in many-valued logic.13 Below, we zoom in on some prominent examples of subclassical many-valued logics that satisfy the fpn property, such as the Strong Kleene logic K3 [34], the Logic of Paradox LP [41, 44], and the relevant logic Efde [2, 3, 11]. K3 and LP are also conservative logics in the sense specified by Definition 2.8. We also introduce the Weak Kleene logic Kw 3 [12, 34] and its paraconsistent kin PWK stemming from [33]. These are also subclassical, conservative logics satisfying the fpn property, and they are attracting increasingly more attention in the many-valued community [10, 13, 19, 22, 29, 46] and in the LFI/LFU community—see especially [6, 47]. Other subclassical logics with the fpn property include Lukasiewicz’ L3 , Deutsch’s Sfde [28], Daniels’ S∗fde [27]—this is independently introduced by Priest as FDEφ [45]— the relevant logic many-valued logic RM3 and a wide family of sublogics of Kw 3 and PWK that has been recently introduced by [6, 47] and investigated by [21]. With the exception of S∗fde and Efde , all the above logics are also conservative.

2.1

Kleene Logics

Logics based on Strong Kleene Algebra. The so-called ‘Strong Kleene algebra’ has been first introduced in [34]. The paracomplete Strong Kleene logic K3 is due to [34], while the paraconsistent Logic of Paradox LP is due to [41, 44]. Definition 2.9 (Strong Kleene Logics) K3 = ⟨Fml1 , ⊧MK3 ⟩ and LP = ⟨Fml1 , ⊧MLP ⟩, where: • MK3 = ⟨SK, {t}⟩ • MLP = ⟨SK, {e, t}⟩ • SK is the algebra whose universe is {f , e, t} and whose operations are given by Table 1.

Table 1 ¬ t e f

f e t

∨ t e f

t t t t

e t e e

f t e f

∧ t e f

t t e f

e e e f

f f f f

13 Notice that subclassicality is not trivial: many-valued connexive logics [48] are not subclassical. However, we focus on classical recapture in this paper, and this justifies our focus on subclassical logics.

8


The following observation details some validities and the most notable failures of K3 and LP: Observation 2.10 The following holds for K3 -consequence and LP-consequence: 1a 2a 3a 4a 5a 6a 7a

∅⊧ / MK3 α for every α ∈ F ml1 β⊧ / MK 3 α for α a classical tautology α, ¬α ⊧MK3 β α ⊃ (β ∧ ¬β) ⊧MK3 ¬α α, α ⊃ β ⊧MK3 β ¬β, α ⊃ β ⊧MK3 ¬α α ⊃ β, β ⊃ γ ⊧MK3 α ⊃ γ

∅ ⊧MLP α for α a classical tautology β ⊧MLP α for α a classical tautology α, ¬α ⊧ / MLP β α ⊃ (β ∧ ¬β) ⊧ / MLP ¬α α, α ⊃ β ⊧ / MLP β ¬β, α ⊃ β ⊧ / MLP ¬α α ⊃ β, β ⊃ γ ⊧ / MLP α ⊃ γ

1b 2b 3b 4b 5b 6b 7b

We refer the reader to [8, 9, 41, 44] for these failures and validities. An immediate corollary of Observation 2.10 is that: Corollary 2.11 1. K3 is paracomplete and falsifies the Law of Identity: ∅ ⊧ / MK3 α ⊃ α. 2. LP is paraconsistent and falsifies Modus Ponens (MP): α, α ⊃ β ⊧ / MLP β. Logics based on Weak Kleene Algebra. Weak Kleene algebra has been first introduced in [34]. The paracomplete Weak Kleene logic Kw 3 originates from [12] and is independently discussed in [34]. The logic PWK originates from [33] and has been recently investigated in a number of papers including [13, 19, 46].14 Definition 2.12 (Weak Kleene Logics) Kw ⟩ and PWK = ⟨Fml1 , ⊧MPWK ⟩, where: 3 = ⟨Fml1 , ⊧Kw 3 • MKw = ⟨WK, {t}⟩ 3 • MPWK = ⟨WK, {e, t}⟩ • WK is the weak Kleene algebra whose universe is {f , e, t} and whose operations are given by Table 2. Table 2 ¬ t e f

f e t

∨ t e f

t t e t

e e e e

f t e f

∧ t e f

t t e f

e e e e

f f e f

The so-called Principle of Contamination describes the behavior of the third value in Weak Kleene logics: 14 Weak Kleene logic K3 is the so-called internal fragment of the logic of meaninglessness introduced in [12]; PWK is the internal fragment of the logic of nonsense from [33]. See Section 5.

Normality Operators and Classical Recapture in Many-valued Logic 9 Observation 2.13 (Contamination) For every k ∈ {1, 2}, α1 , . . . αk ∈ F ml1 , i ∈ {1, . . . , k}, v ∈ HomFml1 ,WK , and ◇k ∈ {¬, ∨, ∧}: v(αi ) = e ⇒ v(◇k (α1 , . . . , αk ) = e Logics satisfying the Principle of Contamination for some value e are usually called infectious logics—see [6, 47]. Kw 3 and PWK share all the validities and failures of K3 and LP from Observation 2.10, respectively. However, they also have distinctive validities and failures, which follow from Observation 2.13, DMKw = {t}, and DMPWK = 3 {e, t}: Observation 2.14 The following holds for Kw 3 -consequence and PWK-consequence: α⊧ / MK w α ∨ β 3 α ∨ β ⊧MKw α ∨ ¬α

1a 2a

α∧β ⊧ / MPWK α α, ¬α ⊧MPWK α ∧ β

1b 2b

3

2.2

The relevant logic Efde

The logic Efde has been first introduced in [2] and motivated as the fde-fragment of the relevant logic E in [3]. It has been later generalized to a ‘useful four-valued logic’ by Nuel Belnap [11], and it is usually interpreted on the bilattice FOUR that has been first introduced by Matthew Ginsberg [31]. Definition 2.15 (The logic Efde of bilattices) Efde = ⟨Fml1 , ⊧MEfde ⟩, where: • MEfde = ⟨FOUR, {b, t}⟩ • FOUR is a bilattice whose universe is {f , n, b, t}; in it, operations ¬, ∨, and ∧ are given by Table 3. Table 3 ¬ t b n f

f b n t

∨ t b n f

t t t t t

b t b t b

n t t n n

f t b n f

∧ t b n f

t t b n f

b b b f f

n n f n f

f f f f f

Remark 2.16 Notice that the bilattice FOUR also includes two weak orders ≤t and ≤k —which are sometimes called the truth order and the information order, respectively—and join and meet operations for each order. In particular, ∨ and ∧ are defined as the join and meet of the truth order ≤t . The join and meet of the information order ≤k are usually denoted by ⊕ and ⊗, respectively. The operations defined on the information order have no counterpart in the language of Efde , and this is why we are skipping them in Definition 2.15.

10


Efde is both paraconsistent and paracomplete, and is a sublogic of K3 and LP. The following is a distinctive failure of the logic: Observation 2.17 Efde fails Confusion: α, ¬α ⊧ / MEfde β ∨ ¬β. Remark 2.18 It is clear from Definition 2.9 and Table 1 that ⟨B2 , {t}⟩ is a submatrix of MK3 and MLP . From this and Observation 2.5, it follows that K3 and LP are subclassical. Also, Table 1 makes it clear that the two logics satisfy the fpn property. Similar remarks apply to the Weak Kleene logics and Efde . Hence, all these logics are subclassical and satisfy the fpn property. Table 1 and Table 2 also suffice to see, by an easy induction on the complexity of the formula, that Kleene logics are conservative. By contrast, Efde is not conservative. Indeed, take a valuation v ∈ HomFml1 ,FOUR such that v(p) = n and v(q) = b. From Table 3, we have v(p ∧ ¬p) = n, v(q ∧ ¬q) = b, and v((p ∧ ¬p) ∨ (q ∧ ¬q)) = t. But of course, we have v ′ ((p ∧ ¬p) ∨ (q ∧ ¬q)) = f for the classical counterpart v ′ of v, since, by construction, v ′ (p) = f , v ′ (q) = t, and v ′ (p ∧ ¬p) = v ′ (q ∧ ¬q) = f . We will go back this feature of Efde in Section 4—see comment to Corollary 4.10.

3

Normality Operators

In this section, we introduce a semantic method to build systems that are both LFIs and LFU from any many-valued logic. In particular, if we are given a logic that is both paraconsistent and paracomplete, our method will generate a system that is both a LFI and a LFU. The method centers on a normality operator, which secures the expressive ability to state that a formula α has a classical truth value—that is, is assigned either f or t. The normality operator generalizes the truth conditions of the main operator from [40]—see end of Section 4 for this. We define normality operators semantically as follows: Definition 3.1 (Normality Operators) Given a formula algebra Fml, and valuation functions v ∶ Fml Ð→ A with B2 a subalgebra of A, a unary connective k is a normality operator iff, for every α ∈ F ml: v(kα) = t ⇔ v(α) ∈ {f , t} and v(kα) = f ⇔ v(α) ∉ {f , t} In this paper, we consider the formula algebra Fml2 from Section 1, which can be seen as an extension of Fml1 with connective ⊛. Also, from now on we interpret ⊛ as a normality operator in every logic whose formula algebra is Fml2 . Table 4 illustrates how ⊛ would behave in three-valued and four-valued logics, such as the Kleene logics and the relevant Efde from Section 2, respectively: The following terminology will be convenient in what follows: Definition 3.2 (⊛-extensions of many-valued logics) Given a logic S = ⟨Fml1 , ⊧MS ⟩, we say that S⊛ = ⟨Fml2 , ⊧MS⊛ ⟩ is the ⊛-extension of S iff MS⊛ = ⟨A⊛ , DMS⊛ ⟩ is such that: • A⊛ has the same universe as A ∈ MS and the same similarity-type as Fml2 ; • Operations ¬, ∨, ∧, ⊺, are as in A; operation ⊛ is given by: ⊛t = ⊛f = t, and ⊛x = f for every x ∉ {f , t};

Normality Operators and Classical Recapture in Many-valued Logic 11 Table 4: Normality operators (three- and four-valued cases), consistency operators and determinedness operators (four-valued case) ⊛ t e f

t f t

⊛ t b n f

t f f t

○ t b n f

☆ t b n f

t f t t

t t f t

• DMS⊛ = DMS . Thus, for instance, PWK⊛ = ⟨Fml2 , ⊧MPWK⊛ ⟩ is the ⊛-extension of PWK, and its matrix ⟨WK⊛ , {e, t}⟩ includes an algebra WK⊛ of universe {f , e, t} that extends the operations from WK by ⊛. Similarly, Efde ⊛ = ⟨Fml2 , ⊧ME ⊛ ⟩ is the ⊛-extension fde

of Efde , and its matrix ⟨FOUR⊛ , {b, t}⟩ includes an algebra FOUR⊛ of universe {f , n, b, t} that extends the operations from FOUR by ⊛. An immediate consequence of Definition 3.2 is: Observation 3.3 For every logic S and corresponding ⊛-extension S⊛ , if Σ, β ∈ F ml1 , then Σ ⊧MS⊛ β ⇔ Σ ⊧MS β which in turn implies that every ⊛-extension S⊛ is a conservative extension of S. From Observation 3.3, it follows that if S is subclassical, satisfies the fpn property, is conservative, then its ⊛-extension S⊛ has these property.

3.1

Normality Operators, Logics of Formal Inconsistency and Undeterminedness

The normality operator ⊛ is a generalization of the consistency and determinedness operators from LFIs and LFUs, respectively. The two families of systems are defined as follows:15 Definition 3.4 (Logics of Formal Inconsistency) A logic S = ⟨Fml, ⊧MS ⟩ is a Logic of Formal Inconsistency iff (1) S is paraconsistent, and (2) S satisfies the following, for some unary connective k: ● α, kα ⊧ / MS β ● ¬α, kα ⊧ / MS β ● α, ¬α, kα ⊧MS β

for some α, β ∈ F ml for some α, β ∈ F ml for every α, β ∈ F ml

(Principle of Gentle Explosion)

15 Definition 3.4 is a variation of [14, Definition 15], as the first two conditions from the former are not included in the latter. We believe that these conditions highlight a distinctive feature of the consistency operator, which we define semantically below (Definition 3.8)

12


Definition 3.5 (Logics of Formal Undeterminedness) A logic S is a Logic of Formal Undeterminedness iff (1) S is paracomplete, and (2) S satisfies the following, for some unary connective k: ● ● ●

∅⊧ / MS α, kα ∅⊧ / MS ¬α, kα ∅ ⊧MS α, ¬α, kα

for some α ∈ F ml for some α ∈ F ml for every α ∈ F ml

(Principle of Gentle Implosion)

Remark 3.6 Notice that in Definition 3.5 we are considering the multiple-conclusion version of the consequence relation over matrices from Definition 1.3; by abusing notation a bit, we also denote this with ⊧.16 This is done in order to account for the behavior of ∨ in the ⊛-extension Kw⊛ of Kw 3 . Aside from this, multiple-conclusion consequence is 3 not needed in the rest of the paper.17 Since this kind of consequence does not prove especially revealing for our focus, we prefer to consider single-conclusion consequence for our formal investigation. In the LFI tradition, any operator k satisfying the conditions from Definition 3.4 is called a consistency operator. We follow the standard LFI notation and we use the symbol ○ for the operator.18 The reason for the label ‘consistency operator’ is clear: the Principle of Gentle Explosion (PGE) implies that, if ○α is satisfied (in a given model), then at least one of α and ¬α is not satisfied (in the model), and vice versa. In a word, ○α states that α is consistent. Similarly, in the LFU tradition, any operator k satisfying the conditions from Definition 3.5 is called an undeterminedness operator. The operator is usually written as ★ in the LFU tradition—see [18, 37], for instance. Again, we believe the reason for the label is clear: the Principle of Gentle Implosion (PGI) implies that, if ★α is not satisfied (in a valuation), then at least one of α and ¬α is satisfied (in that model), and if neither α and ¬α are satisfied, ★α is satisfied. Remark 3.7 Many LFIs include an inconsistency operator ● alongside the consistency operator ○, with ●α being satisfied iff both α and ¬α are satisfied.19 In the logics that we consider in this paper, the operators ● and ○ are interdefinable—in particular, we have ○α = ¬ ● α and ●α = ¬ ○ α. The choice to introduce ○ has a primitive has now become more widespread, and we are following it. Similarly, in the logics that we consider in this paper, we can indifferently introduce a determinedness operator ☆ such that ∅ ⊧MS α, ¬α, ¬ ☆ α holds good. Indeed, in such logics we have ☆α = ¬ ★ α and ★α = ¬ ☆ α. Notice that, given this interdefinability, the rule above secures that ☆α is satisfied iff either α or ¬α are satisfied.20 Here we focus on the determinedness operator ☆, since it allows for a more straightforward comparison with the normality operator. 16 Multiple-conclusion matrix consequence is standardly defined as a relation R ⊆ ℘(F ml)×℘(F ml), with R(Γ, ∆) being the case iff some formula in ∆ is designated in any valuation where all formulas in Σ are designated. 17

Indeed, all the other logics from Section 5, verify the natural single-conclusion version of PGI above.

18

Notice, however, that [40] uses ○ for a normality operator (which is the main operator of that paper), and ○′ for a consistency operator that fails Definition 3.1. 19 The two formulas need not be equivalent, however, since ●α ∧ ¬ ● α is not satisfiable in any LFI, while (α ∧ ¬α) ∧ ¬(α ∧ ¬α) is satisfiable in some LFI. In a nutshell, ●α consistently expresses that α is inconsistent. 20

Again, ☆α ∨ ¬ ☆ α is a tautology in systems of LFU, while (α ∨ ¬α) ∨ ¬(α ∨ ¬α) typically fails in LFUs.

Normality Operators and Classical Recapture in Many-valued Logic 13 If we confine our attention to the many-valued LFIs/LFUs that are endowed with a matrix-based semantics, the consistency operator ○ and its kin the determinedness operator ☆ can be given straightforward semantic definitions: Definition 3.8 (Consistency and Determinedness Operators) Given a formula algebra Fml, a matrix M, and valuation functions v ∶ Fml Ð→ A with A a subalgebra of ⟨B2 , {t}⟩ and A ∈ M, the consistency operator ○ and the determinedness operator ☆ are defined as follows: 1. 2.

v(○α) ∈ DM ⇔ (v(α) ∈ DM ⇒ v(¬α) ∉ DM ) v(☆α) ∈ DM ⇔ (v(α) ∈ DM or v(¬α) ∈ DM )

An exemplification of these definitions is found in Table 4, where the four-valued case is considered. Definition 3.1 and Definition 3.8 imply that any normality operator is also a consistency and a determinedness operator. On the one hand, if v(⊛α) = t and v(α) = t, then v(¬α) = f , and if v(⊛α) = t and v(¬α) = t, then v(α) = f ; in both situations, the conditions for ○α are met. By contrast, if v(α) ∈ DM and v(¬α) ∈ DM , then v(α) ∉ {f , t}, which in turn implies v(⊛α) = f ; if the conditions for ○α are not met, those for ⊛α are also not met. On the other hand, if v(⊛α) = t, then either v(α) = t or v(¬α) = t, and thus the conditions for ☆α are met; by constrast, if v(α) ≠ DM and v(¬α) ≠ DM , then v(α) ∉ {f , t} and v(⊛α) = f : if the conditions for ☆α are not met, those for ⊛α are also not met. By contrast, a consistency (determinedness) operator may fail to be a normality operator. Just to get a concrete feeling of this, take Efde ⊛ . Definition 3.1 implies that the normality operator satisfies the criteria from both Definition 3.4 and Definition 3.5 in any logic that is both paraconsistent and paracomplete, such as Efde ⊛ . As in any logic, ⊛ will satisfy PGE and PGI in these systems, and we will have v(⊛α) = t iff v(α) ∈ {f , t}. By contrast, if we extend Efde with ○, the semantic clause for ○ allows v(○α) = t even in case v(α) = n, since n ∉ DMEfde . Dually, the semantic clause for ☆ allows v(☆α) = t even in case v(α) = b, since b ∈ DMEfde . Hence, in Efde ⊛ , neither ○ nor ☆ satisfy Definition 3.1. More in general, in subclassical logics that are both paraconsistent and paracomplete, consistency or determinedness operators may fail from being normality operators. There are, however, two interesting cases where the normality operator collapses on either the consistency or the determinedness operator. First, in every paraconsistent three-valued LFI satisfying the fpn property, the clause for ○α reduces to v(○α) ∈ DM ⇔ v(α) ∈ {f , t}.21 From this and the fact that our semantics is based on classical set-theory, we conclude that, in a paraconsistent three-valued logic, v(○α) = t iff v(α) ∈ {f , t}, and v(○α) = f otherwise. Second, in every paracomplete threevalued LFU S satisfying the fpn property, the clause for ☆α reduces to v(☆α) = t iff v(α) ∈ {f , t}, and v(☆α) = f otherwise.22 These remarks clarify our statement that the normality operator generalizes the operators from the LFI and LFU traditions: the conditions for a consistency (determinedness) operator from Definition 3.8 are necessary, but not sufficient, to yield a 21 Indeed, by Observation 2.6, paraconsistency and three-valuedness of S, we have that A− = A− ∩ DM = x for S some x ∈ A. From this and the fpn property, it follows that ○α is satisfied if α is not assigned x. But from three-valuedness, this implies that α is assigned either f or t. 22

The reasoning is similar to the one we summed-up for the paraconsistent case. We leave it to the reader.

14


normality operator; by contrast, a normality operator is always both a consistency and a determinedness operator. At the end of Section 4, we also discuss how exactly the normality operator defined here also generalizes the main operator from [40].

3.2

Some Principles involving the Normality Operator

Here, we present some validities involving the normality operator ⊛. Interestingly, satisfaction of fpn property or conservativeness are crucial for some of these validities. Observation 3.9 The following holds for the ⊛-extension S⊛ of any subclassical many-valued logic: 1a. ∅ ⊧MS⊛ ⊛ ⊛ α 2a. ⊛α ⊧MS⊛ α ∨ ¬α 3a. ⊛α ∧ ⊛β ⊧MS⊛ ⊛(α ∧ β)

1b. ∅ ⊧MS⊛ ⊛¬ ⊛ α 2b. α, ¬α ⊧MS⊛ ¬ ⊛ α 3b. ⊛α ∧ ⊛β ⊧MS⊛ ⊛(α ∨ β)

Proof. (1a) If v(α) ∈ {f , t}, then v(⊛α) = t by Definition 3.1. Otherwise, v(⊛α) = f . Again by Definition 3.1, both cases imply v(⊛⊛α) = t. The same applies to 1b. (2a) If v(⊛α) = t, then v(α) ∈ {f , t}. Since S is subclassical, whenever v(α) ∈ {f , t}, we have v(α ∨ ¬α) = t. This implies that, if v(⊛α) = t, then v(α ∨ ¬α) ∈ DMS⊛ . (2b) Either S is paraconsistent, or it is not. If it is not paraconsistent, then 2b trivially holds. If it is paraconsistent and v(α), v(¬α) ∈ DMS⊛ , then v(α) ∉ {f , t}, since S is subclassical. But this implies v(¬ ⊛ α) = t. Hence, if v(α), v(¬α) ∈ DMS⊛ , then v(¬ ⊛ α) ∈ DMS⊛ . (3a) and (3b) are elementary and we leave them to the reader. Observation 3.10 The following holds for the ⊛-extension S⊛ of any subclassical many-valued logic that satisfies the fpn property: 1a. ¬ ⊛ α ⊧MS⊛ ¬ ⊛ ¬α 2. α ∨ ¬α ⊧MS⊛ ⊛α 3. ¬ ⊛ α ⊧MS⊛ α ∧ ¬α

1b. ¬ ⊛ ¬α ⊧MS⊛ ¬ ⊛ α if S is paracomplete and not paraconsistent if S is paraconsistent and not paracomplete

Proof. (1a) Assume v(¬ ⊛ α) = t. This implies v(α) = x for some x ∈ A− . Since S satisfies the fpn property, v(¬α) = x. This in turn implies v(¬ ⊛ ¬α) = t. 1b is proved along the same lines. (2) Since S is paracomplete, is not paraconsistent, and satisfies the fpn property, by Observation 2.6, we have DMS = {t}. Thus, if v(α ∨ ¬α) ∈ DMS , then v(α) ∈ {f , t}, and, by Definition 3.1, v(⊛α) = t. (3) is proved along the same lines. Observation 3.11 The following holds for the ⊛-extension S⊛ of any subclassical many-valued logic that satisfies the fpn property and conservativeness: 1. ⊛(α ∧ β) ⊧MS⊛ ⊛α ∨ ⊛β 2. ⊛(α ∨ β) ⊧MS⊛ ⊛α ∨ ⊛β Proof. We start from (1). Assume v(⊛(α ∧ β)) = t. Suppose that x ∧ y = t for some x, y ∈ A− in the relevant algebra. We distinguish three cases: (a) x, y ∈ DMS ; (b) x, y ∉ DMS ; (c) x ∈ DMS , y ∉ DMS . Consider x, y ∈ DMS . Since S enjoys the fpn

Normality Operators and Classical Recapture in Many-valued Logic 15 property, we would have (x ∧ ¬x) ∧ (y ∧ ¬y) = t. But if this would be the case, we should have v(α) = x, v(β) = y, and v((α∧¬α)∧(β ∧¬β)) = t. Since S is conservative, the classical counterpart v ′ of v should be such that v ′ ((α ∧ ¬α) ∧ (β ∧ ¬β)) = t. But this is impossible. Hence, either x ∉ DMS or y ∉ DMS . Again, since S is conservative, this implies that either x ∈ {f , t} or y ∈ {f , t}. The same strategy applies to the other two cases. Hence, if v(⊛(α ∧ β)) = t, then either x ∈ {f , t} or y ∈ {f , t}. But since v is arbitrary, this equates with ⊛(α ∧ β) ⊧MS⊛ ⊛α ∨ ⊛β. The proof of (2) goes exactly along the same lines.

4

Recapture via Normality Operators

In this section, we establish recapture results for (a) any subclassical many-valued logic, (b) any subclassical many-valued logic satisfying the fpn property, (c) any logic that, additionally, satisfies conservativeness. Below, we also explain why it is worth pursuing specific recapture results beside the one that applies to all subclassical manyvalued logic. First, we need an auxiliary notion: Definition 4.1 (Normal Counterpart) Given a set Γ ⊆ F ml1 , we say that the set Γ⊛ = {⊛α ∈ F ml2 ∣ α ∈ Γ} is the normal counterpart of Γ. Informally, the normal counterpart of a set of formulas is the set obtained by replacing each α ∈ Γ with ⊛α. Thus, the normal counterpart of {α, α ⊃ β} is {⊛α, ⊛(α ⊃ β)}. Since {α}⊛ is {⊛α}, we just denote by ⊛α the normal counterpart of singleton {α}. If both Γ and Γ⊛ are designated, this implies that the formulas in Γ are classically true. We are now ready to establish our recapture results. The following theorem applies to all subclassical many-valued logics, independently from their position w.r.t. the fpn property or conservativeness: Theorem 4.2 For every subclassical many-valued logic S and Σ, β ⊆ F ml1 , the extension S⊛ of S is such that: Σ, (var(Σ))⊛ , (var(β))⊛ ⊧MS⊛ β ⇔ Σ ⊧MCL β Proof. (⇒) Assume Σ, (var(Σ))⊛ , (var(β))⊛ ⊧MS⊛ β. Since Σ, β ⊆ F ml1 , this implies that v(β) = t for every v ∈ HomMS such that (1) v[Σ] = {t}, and (2) v(p) ∈ {f , t} for every p ∈ var(Σ) ∪ var(β). Every such valuation v can be turned into a corresponding classical valuation v ′ ∈ HomMS as per Definition 2.7. By construction, we have that v ′ [Σ] = {t} and v ′ (β) = t. Since these exhaust the classical models of Σ, we can conclude that Σ ⊧MCL β. (⇐) Assume Σ ⊧MCL β. Every valuation v ∈ HomMS such that (1) v[Σ] = {t} and (2) v(p) ∈ {f , t} for every p ∈ var(Σ) ∪ var(β) can be turned into a corresponding classical valuation v ′ ∈ HomMS . This implies that v(β) = t. Otherwise, we would have v ′ (β) = f , which just contradicts the initial hypothesis. Since Σ, β ⊆ F ml1 , this implies Σ, (var(Σ))⊛ , (var(β))⊛ ⊧MS⊛ β.

16

4.1


Classical Recapture for Many-valued Logics satisfying the fpn property

Theorem 4.3 For every subclassical many-valued logic S satisfying the fpn property and for every Σ, β ⊆ F ml1 , if ∅ ⊧MS α ∨ ¬α, then the extension S⊛ of S is such that: Σ, (var(Σ))⊛ ⊧MS⊛ β ⇔ Σ ⊧MCL β Proof. (⇒) Assume Σ, (var(Σ))⊛ ⊧MS⊛ β. Since Σ, β ⊆ F ml1 , this implies that v(β) ∈ DMS for every v ∈ HomMS such that v[Σ] = {t} and v(p) ∈ {f , t} for every p ∈ var(Σ). If there is no valuation v such that v[Σ] = {t}, then Σ has no classical model, and Σ ⊧MCL β trivially follows. Suppose that there exists at least a valuation v such that v[Σ] = {t} and additionally that v(q) ∈ {f , t} for every q ∈ var(β). This implies that v(β) ∈ {f , t} and, since v(β) = f would contradict the initial assumption, from this we can conclude that v(β) = t. Every such valuation v can be extended to a valuation v ′ such that v ′ (p) = v(p) for every p ∈ var(Σ) ∪ var(β). This implies v ′ [Σ] = {t} and v ′ (β) = t. Since these exhaust the classical models of Σ, it follows that Σ ⊧MCL β. (⇐) Assume Σ ⊧MCL β. Now suppose that there is a v ∈ HomMS such that (1) v[Σ] = {t} and v(p) ∈ {f , t} for every p ∈ var(Σ), and (2) v(β) ∉ DMS . From ∅ ⊧MS α ∨ ¬α, S being subclassical and satisfying the fpn property, it follows that x ∈ DMS for every x ∈ A− —otherwise, S would be CL itself, which contradicts the fact that S is subclassical. By this, fpn property, and Observation 2.6, we have that S is paraconsistent and A ∖ DMS = {f }. As a consequence, 2 implies v(β) = {f }. Any valuation v satisfying 1 and 2 can be extended to a valuation v ′ ∈ HomMS such that (a) v ′ (p) ∈ {f , t} for every p ∈ X and (b) v ′ (r) = v(r) for every v ∈ var(Σ) ∪ var(β). By construction, we have v ′ [Σ] = {t} and v ′ (β) = f . However, since v ′ is a classical valuation, this contradicts the initial hypothesis. This in turn implies that v(β) ∈ DMS for every v ∈ HomMS such that v[Σ] ⊆ DMS and v(p) ∈ {f , t} for every p ∈ var(Σ). But since Σ, β ⊆ F ml1 , this equates with Σ, (var(Σ))⊛ ⊧MS⊛ β. Theorem 4.4 For every subclassical many-valued logic S satisfying the fpn property and for every Σ, β ⊆ F ml1 , if α, ¬α ⊧MS β, then the extension S⊛ of S is such that: Σ, (var(β))⊛ ⊧MS⊛ β ⇔ Σ ⊧MCL β Proof. (⇒) Assume Σ, (var(β))⊛ ⊧MS β. Since Σ, β ⊆ F ml1 , this implies that v(β) ∈ DMS for every v ∈ HomMS s.t. (1) v[Σ] ⊆ DMS and (2) v(p) ∈ {f , t} for every p ∈ var(β). If there is no valuation v satisfying conditions 1–2 and v(q) ∈ {f , t} for every q ∈ var(Σ), then Σ has no classical model, and Σ ⊧MCL β trivially follows. Suppose that there is a valuation v that satisfies conditions 1–2 and v(q) ∈ {f , t} for every q ∈ var(Σ). Then v(β) = {f , t}. Since v(β) = f contradicts the initial hypothesis, we conclude that v ′ (β) = t. But this implies Σ ⊧MCL β. (⇐) Assume Σ ⊧MCL β. Now suppose that there is a v ∈ HomMS such that (1) v[Σ] ⊆ DMS and v(p) ∈ {f , t} for every p ∈ var(β), and (2) v(β) ∉ DMS . From α, ¬α ⊧MS β, S being subclassical and satisfying the fpn property, it follows that x ∉ DMS for every x ∈ A− —otherwise, S would be CL itself, which contradicts the fact that S is subclassical. By this, fpn property, and Observation 2.6, we have that S is

Normality Operators and Classical Recapture in Many-valued Logic 17 paracomplete and DMS = {t}. From this and condition 1, it follows that v[Σ] = {t}. From 1 and 2, it follows that v(β) = f . Valuation v can be extended to a valuation v ′ ∈ HomMS such that (1) v ′ (p) ∈ {f , t} for every p ∈ X and (2) v ′ (r) = v(r) for every v ∈ var(Σ) ∪ var(β). By construction, we have v ′ [Σ] = {t} and v ′ (β) = f . However, since v ′ is a classical valuation, this contradicts the initial hypothesis. This in turn implies that v(β) ∈ DMS for every v ∈ HomMS such that v[Σ] ⊆ DMS and v(p) ∈ {f , t} for every p ∈ var(β). But since Σ, β ⊆ F ml1 , this equates with Σ, (var(β))⊛ ⊧MS⊛ β. A special application of Theorem 4.2 is to subclassical logics S that are both paracomplete and paraconsistent, including those that satisfy the fpn property. Indeed, it immediately follows from the theorem that: Corollary 4.5 For every subclassical many-valued logic S satisfying the fpn property and for every Σ, β ⊆ F ml1 , if ∅ ⊧ / MS α ∨ ¬α and α, ¬α ⊧ / MS β, then the extension S⊛ of S is such that: Σ, (var(Σ))⊛ , (var(β))⊛ ⊧MS⊛ β ⇔ Σ ⊧MCL β Notice that the recapture recipes from Theorem 4.3 and Theorem 4.4 do not suffice for a paracomplete and paraconsistent logic. Consideration of the ⊛-extension Efde ⊛ of Efde suffices to see this. In particular, it is easy to check that we have p∧¬p, ⊛r ⊧ / ME ⊛ fde r and p, ⊛p ⊧ / ME ⊛ q ∨ ¬q. This, together with p ∧ ¬p ⊧MCL r and p ⊧MCL q ∨ ¬q, implies fde

that Theorem 4.3 and Theorem 4.4 do not suffice to secure recapture in Efde ⊛ . Remark 4.6 Here, we briefly discuss the role played by the fpn property in Theorems 4.3–4.4. In order to see that the fpn property is indispensable for Theorem 4.3, consider a slight modification of Efde ⊛ , which we call S1 . Given the same values {f , n, b, t} as Efde ⊛ , the operations of S1 differ from those of Efde ⊛ as follows: ¬b = n, ¬n = b, t ∨ n = n. S1 does not satisfy the fpn property, and it is easy to see that S1 is not paracomplete. Now take a valuation v such that v(p) = v(⊛p) = t and v(q) = n. The valuation is such that v(p ∨ q) = n. Thus, p, ⊛p ⊧ / MS1 p ∨ q, but of course, p ⊧MCL p ∨ q. This implies that Theorem 4.3 does not hold for S1 . An example like the above is ruled out if we add the fpn property to the other conditions specified in Theorem 4.3. Indeed, the fpn property implies that, if a non-designated non-classical value is included, then the logic in question must be paracomplete, thus contradicting one of the conditions. As for Theorem 4.4, consider a further modification S2 where ¬b = n, ¬n = b, b ∧ f = b. Again, S2 does not satisfy the fpn property, and it is easy to see that S2 is not paraconsistent. Now take a valuation v such that v(p) = b and v(q) = f . The valuation is such that v(⊛q) = t. Thus, p ∧ q, ⊛q, ⊧ / MS1 q, but of course, p ∧ q ⊧MCL q. This implies that Theorem 4.4 does not hold for S2 . Again, an example like this is ruled out if we add the fpn property to the other conditions specified in Theorem 4.3.

18

4.2


Classical Recapture for Conservative Many-valued Logics satisfying the fpn property

Theorem 4.7 For every conservative subclassical many-valued logic S satisfying the fpn property and for every Σ, β ⊆ F ml1 , the extension S⊛ of S is such that: Σ, Σ⊛ , ⊛β ⊧MS⊛ β ⇔ Σ ⊧MCL β Proof. (⇒) Assume Σ, Σ⊛ , ⊛β ⊧MS⊛ β. Since Σ, β ⊆ F ml1 , this means that v(β) = t for every v ∈ HomMS such that v(β) ∈ {f , t} and v[Σ] = {t}. Since S is conservative, the classical counterparts v ′ of any such v will be such that v ′ [Σ] = {t} and v ′ (β) = t. But these exhaust the classical models of Σ. Hence, Σ ⊧MCL β. (⇐) Assume Σ ⊧MCL β. Since S is conservative, we have v(β) = t for every v ∈ HomMS such that v[Σ] = {t} and v(β) ∈ {f , t}. Otherwise, we would have a classical counterpart v ′ of some v such that v ′ (β) = f and v[Σ] = {t}, but this would contradict the initial hypothesis. From this and Σ, β ⊆ F ml1 , we conclude Σ, Σ⊛ , ⊛β ⊧MS⊛ β. Theorem 4.8 For every conservative subclassical many-valued logic S satisfying the fpn property and for every Σ, β ⊆ F ml1 , if ∅ ⊧MS α ∨ ¬α, then the extension S⊛ of S is such that: Σ, Σ⊛ ⊧MS⊛ β ⇔ Σ ⊧MCL β Proof. (⇒) Assume Σ, Σ⊛ ⊧MS⊛ β. Since Σ, β ⊆ F ml1 , this equates with {v ∈ HomMS ∣ v[Σ] = {t}} ⊆ HomMS (β). Either Σ has a classical model, or it hasn’t. If Σ has no classical model, then Σ ⊧MCL β. If Σ has a classical model, then β has a classical model too. Take any classical model v of Σ. Since v(β) = f would contradict the initial hypothesis, we have v(β) = t. Since v is an arbitrary valuation, we can conclude that Σ ⊧MCL β. (⇐) Assume Σ ⊧MCL β. Again, either Σ has a classical model, or it hasn’t. If Σ hasn’t a classical model, then by the conservativeness of S, we have {v ∈ HomMS ∣ v[Σ] = {t}} = ∅. Since Σ, β ⊆ F ml1 , this implies Σ, Σ⊛ ⊧MS⊛ β. If Σ has a classical model, then also β has a classical model. Suppose that v(β) ∉ DMS for some v ∈ HomMS s.t. v[Σ] = {t}. Since S satisfies the fpn property and is not paracomplete, it has no non-designated non-classical value: DMS = {f } (by Observation 2.6). Thus, v(β) = f . Now take the classical counterpart v ′ of v. Since S is conservative, we have (1) v ′ [var(Σ)] ⊆ {f , t}, (2) v ′ [Σ] = {t}, (3) v ′ (β) = f . 1–3 together contradict the initial hypothesis. As a consequence, v(β) ∈ DMS for every v ∈ HomMS s.t. v[Σ] = {t}. This in turn equates with Σ, Σ⊛ ⊧MS⊛ β. Theorem 4.9 For every conservative subclassical many-valued logic S satisfying the fpn property and for every Σ, β ⊆ F ml1 , if α, ¬α ⊧MS β, then the extension S⊛ of S is such that: Σ, ⊛β ⊧MS⊛ β ⇔ Σ ⊧MCL β Proof. (⇒) Assume Σ, ⊛β ⊧MS⊛ β. Since Σ, β ⊆ F ml1 , this equates with {v ∈ HomMS (Σ) ∣ v(β) ∈ {f , t}} ⊆ HomMS (β). In particular, we have {v ∈ HomMS (Σ) ∣ v(β) ∈ {f , t}} ⊆ {HomMS (β) ∣ v(β) ∈ {f , t}}. Since {v ∈ HomMS (Σ) ∣ v[Σ ∪ {β}] ⊆ {f , t}} ⊆ {v ∈ HomMS (Σ) ∣ v(β) ∈ {f , t}}, we conclude {v ∈ HomMS (Σ) ∣ v[Σ ∪ {β}] ⊆

Normality Operators and Classical Recapture in Many-valued Logic 19 {f , t}} ⊆ {HomMS (β) ∣ v(β) ∈ {f , t}}. Since S is conservative, this implies that v ′ (β) = t for every classical counterpart v ′ of the valuations v in question. These are in turn such that v ′ [Σ] = {t}. This suffices to conclude Σ ⊧MCL β. (⇐) Assume Σ ⊧MCL β. Take any valuation v ∈ HomMS such that v[Σ] ⊆ DMS and v(β) ∈ {f , t}. Since S is not paraconsistent and satisfies the fpn property, by Observation 2.6 we have DMS = {t}. This implies v[Σ] = {t} along with v(β) ∈ {f , t}. Now suppose v(β) = f . Since S is conservative, the classical counterpart v ′ of v would be such that v ′ (β) = f and v[Σ] = {t}. But this contradicts the initial hypothesis. Thus, v(β) = t for every v ∈ HomMS such that v[Σ] = t. Since Σ, β ⊆ F ml1 , this suffices to conclude Σ, ⊛β ⊧MS⊛ β. A special application of Theorem 4.7 is to subclassical logics S that satisfy the fpn property and conservativeness, and are both paracomplete and paraconsistent. Indeed, it immediately follows from Theorem 4.7 that: Corollary 4.10 For every conservative subclassical many-valued logic S satisfying the fpn property and for every Σ, β ⊆ F ml1 , if ∅ ⊧ / MS α ∨ ¬α and α, ¬α ⊧ / MS β, then the extension S⊛ of S is such that: Σ, Σ⊛ , ⊛β ⊧MS⊛ β ⇔ Σ ⊧MCL β Notice that, by contrast, the recapture recipes from Theorem 4.8 and Theorem 4.9 do not suffice here. Just to get a concrete feeling of this, consider the paracomplete and paraconsistent logic Sfde [28], which satisfies subclassicality, fpn property, and conservativeness.23 The matrix for this logic obtains by the matrix for Efde by replacing n with a non-designated contaminating value e in the style of MKw . This 3 suffices for the logic to fail adjunction (α ⊧ α ∨ β). In particular, there is a valuation v ∈ HomMSfde where v(p) = t and v(q) = v(p ∨ q) = e. This implies that, in the corresponding ⊛-extension Sfde ⊛ , α, ⊛α ⊧ / MS ⊛ α ∨ β. But α ⊧MCL α ∨ β. Thus, the fde

recipe from Theorem 4.8 does not suffice to secure recapture in Sfde ⊛ . Also, there is a valuation v ∈ HomMSfde where v(p) = b and v(q) = f . This implies that, in Sfde ⊛ , α, ¬α, ⊛β ⊧ / MS ⊛ β. But α, ¬α ⊧MCL β. Thus, the recipe from Theorem 4.9 does not fde

suffice to secure recapture in Sfde ⊛ . Remark 4.11 Here, we briefly discuss the role played by conservativeness in Theorems 4.7–4.9. In order to see that conservativeness is indispensable for Theorem 4.7, consider Efde ⊛ and a valuation v ∈ HomME ⊛ where v(r) = f , v(p) = n and v(b). By Table 3, we fde have v((p ∧ ¬p) ∨ (q ∧ ¬q)) = t. This in turn implies v(⊛((p ∧ ¬p) ∨ (q ∧ ¬q))) = t. Since v(r) = f , the valuation falsifies (p ∧ ¬p) ∨ (q ∧ ¬q), ⊛((p ∧ ¬p) ∨ (q ∧ ¬q)), ⊛r ⊧ME ⊛ r. fde But of course (p ∧ ¬p) ∨ (q ∧ ¬q) ⊧MCL r. Thus, we can have Σ ⊧MCL β and yet Σ, Σ⊛ , ⊛β ⊧ / ME ⊛ β. Theorem 4.7 may fail for some subclassical logics that are not fde conservative. Conservativeness is also indispensable for Theorem 4.8 and Theorem 4.9. Take a three-valued subclassical paraconsistent logic S that satisfies the fpn-property. Suppose that A− = A− ∩ DMS = {x} for some x ∈ A, and that t ∨ x = f . The latter 23

For more details on Sfde , we refer the reader to [29], where the system is called AL.

20


implies that S is not conservative. In this logic, we may have v(p) = t, v(q) = x and v(p ∨ q) = f . This implies that, in the corresponding ⊛-extension S⊛ , we can have v(p) = v(⊛p) = t and yet v(p ∨ q) = f , although p ⊧MCL p ∨ q. As for the paracomplete case, suppose that A− = {x} and DMS = {t} for a subclassical logic S that satisfies the fpn property. Set f ∧ x = t; this implies that the logic is not conservative. In this logic, we may have v(p) = x, v(q) = f , and v(p ∧ q) = t. This implies that, in the corresponding ⊛-extension S⊛ , we can have v(p ∧ q) = v(⊛q) = t and yet v(q) = f , although p ∧ q ⊧MCL q.

4.3

Discussion of the Results

Here, we discuss Theorems 4.2–4.9, the relevance of Theorems 4.3–4.9, and we compare our theorems with some existing results of classical recapture. Some facts about Theorems 4.2– 4.9. Theorems 4.2–4.9 provide an array of recapture results for logic in the LFI/LFU traditions, and for logics in their intersection (namely, the paracomplete and paraconsistent logics including ⊛). An interesting feature of our theorems is that they do not rely on the syntactic complexity of the conclusion or (of the formulas in) the premises. Thus, the above results also apply to richer (or poorer) languages, as soon as the resulting logics satisfy some (or all) of subclassicality, fpn property, and conservativeness. Beside, all the different results in this section rely on the very same feature of ⊛-extensions of any many-valued logic—that is, these results let us reason within the object-language itself about the conditions at which we can deploy classically valid inferences. As is clear from Theorems 4.2–4.9, the exact application of this feature depends on the formal properties of the systems in question. Also, consider that ⊛-extension S⊛ is a conservative extension of S (Observation 3.3). Thus, Theorems 4.2–4.9 seems to provide, in a broader sense, recapture strategies for S itself: they express, in a richer language, the conditions at which we can reason classically within S. Relevance of Theorems 4.3–4.9. Theorem 4.2 provides a very general result of classical recapture, applying to any subclassical (many-valued) logic. However, the more specific Theorems 4.3–4.9 are also relevant, and for a simple reason: their application requires less information than the application of Theorem 4.2. Theorems 4.3–4.4 and Theorems 4.7–4.9 prove that, depending on the position of the logic w.r.t. paracompleteness and paraconsistency, we may dispense from some of the formulas required by Theorem 4.2—namely, those expressing that the variables in the premise-set or conclusion have a classical value. As for Theorem 4.7, its conditions are less demanding: in many logics, they are verified by a wider set of valuations than the conditions from Theorem 4.2—a brief look at Table 1 and Table 3 suffices to see this. In terms of information, the requests by Theorem 4.2 are more demanding, and it may be convenient to make use of less exacting conditions when possible.

4.4

Relevance with respect to existing background

Here, we briefly discuss the relevance and originality of our approach and results w.r.t. existing recapture theorems from the LFI/LFU tradition, and w.r.t. the closely re-

Normality Operators and Classical Recapture in Many-valued Logic 21 lated work by [40].24 Existing recapture results in LFIs/LFUs. As we have mentioned in the Introduction, recapture theorems are not new in the tradition of LFI, where they are usually called Derivability Adjustment Theorems. For instance, [36, Theorem 3.11] establishes a similar classical recapture theorem for the paraconsistent logic bC [14], and [36, Theorem 3.46] proves a particular recapture that involves a satisfaction-preserving translation τ from CL to the paraconsistent logic Ci [36]. We briefly comment on the differences between these results, which are particularly significant in the LFI/LFU recapture tradition, and ours. First, Theorem 3.11 from [36] relies on adding the normal counterpart of some set ∆ of well-formed formulas, and not necessarily the normal counterpart of the premise-set Σ itself. Second, Theorem 3.46 from [36] relies on translating CL into Ci. By contrast, no translation is needed for our purposes. We have already mentioned a difference in scope between our results (Theorems 4.2–4.9) and current recapture theorems in the LFI tradition. The specific points we have discussed here helps get a more concrete feeling of these differences. Other relevant background. The present paper has close connections with [40]. Here, we briefly discuss similarities and differences. [40] is, to our knowledge, the first paper to present a systematic analysis of a normality operator (which [40] calls classicality operator ) and to focus on the distinction between this operator and the consistency operator. In particular, [40] investigates the paraconsistent and paracomplete logic BS4, which enjoys a four-valued matrix-based semantics and includes a normality operator and a detachable conditional.25 Also, in case of logics that can be interpreted on FOUR, Definition 3.1 from this paper is equivalent with the semantical definition of the normality operator from [40, §6], which is given in terms of Dunn’s ‘relational semantics’.26 Thus, the present paper and [40] clearly overlap in considering logics where the normality operator would not need collapse on the consistency (or determinedness) one, and in aiming at an operator that is able to signal ‘classicality’—that is, assignment to f or t to the formula in the scope of the operator. However, there are three significant differences. First, the scope of the present paper is somehow wider: although we zoom in on specific examples, the results of the paper apply to potentially infinite many-valued logics including a normality operator; by contrast, [40] is much more system-tailored. Second, and more important, although it acknowledges the potential of the normality operator in specifying when we can reason classically, [40] does not pursue, discuss, or hint at results of classical recapture. Contrary to our paper, these results lies completely out of the focus of [40]. Thus, the main formal contribution from our paper is completely novel and original w.r.t [40]. Third, we believe that Definition 3.1 generalizes the definition from [40, §6] in a significant way. Indeed, the latter is based on ‘relational semantics’,27 and it is not 24

We thank the editors for suggesting a discussion on the relations between the present paper and [40].

25

It also devises the paraconsistent and paracomplete logic cBS, which includes a normality operator, a Nelsonstyle constructive conditional, and is given an intensional (possible-world) semantics. This logic is less relevant to the topic of this paper, though. 26 To be precise, the equivalence between the two definitions holds for logics F ml1 -fragment can be interpreted on FOUR. The relational semantics we mention here must not be confused with possible-world semantics and its variations. 27

Here, we consider the functional version of Dunn’s relational semantics, in order to keep closer to [40]. Notice,

22


clear if the latter can give a conceptually clear semantical insight on logics based on Weak Kleene algebras and their sublogics. Take S∗fde , for instance. The relational semantics condition for disjunction in K3 is ‘t ∈ v + (α ∨ β) iff t ∈ v + (α) or t ∈ v + (β)’. + + + In a logic like Kw 3 , this must be adjusted to ‘t ∈ v (α ∨ β) iff (i) t ∈ v (α) or t ∈ v (β) + + 28 and (ii) ∅ ≠ v (α) and ∅ ≠ v (β)’. In our view, the latter blurs the intuition behind relational semantics,29 but even worse, none of them applies to S∗fde , since the interpretation of this logic comprises both a value n in the style of K3 and a value e in 30 the style of Kw 3 , thus making the above conditions neither necessary nor sufficient. So, it is not clear to generalize the definition from [40, §6] to (every) subclassical many-valued logic that has a matrix-based semantics. By contrast, Definition 3.1 applies in a straightforward way to any such logic.

5

Exemplifying Classical Recapture

In this section, we give a concrete feeling of how recapture via normality operator works. We zoom in on some convenient cases in Efde ⊛ and we briefly discuss the ⊛ ⊛ w⊛ possibility of semantically closed extensions of K⊛ 3 , LP , and K3 , PWK . Also, we ⊛ w⊛ focus on some specific features of K3 and PWK .

5.1

Classical Recapture in K⊛3 and LP⊛

The following is an immediate corollary of Theorem 4.8 and Theorem 4.9, and it gives a concrete example of how recapture via normality recovers the classical inferences or laws that fail in the logics based on Strong Kleene algebras K3 and LP—see Section 2: Corollary 5.1 ⊛ The following holds for K⊛ 3 -consequence and LP -consequence: 1. 2. 3. 4. 5. 6. 7. 8.

⊛α ⊧MK ⊛ α 3 ⊛α ⊧MK ⊛ α ∨ ¬α 3 ⊛α ⊧MK ⊛ α ⊃ α 3 α, ¬α, ⊛α, ⊛¬α ⊧MLP⊛ β α ⊃ (β ∧ ¬β), ⊛(α ⊃ (β ∧ ¬β)) ⊧MLP⊛ ¬α α, α ⊃ β, ⊛α, ⊛(α ⊃ β) ⊧MLP⊛ β ¬β, α ⊃ β, ⊛¬β, ⊛(α ⊃ β) ⊧MLP⊛ ¬α α ⊃ β, β ⊃ γ, ⊛(α ⊃ β), ⊛(β ⊃ γ) ⊧MLP⊛ α ⊃ γ

for α a classical tautology

Item 1 secures recapture of classical tautologies in K⊛ 3 (see the corresponding failure in Observation 2.10.1a). Recapture of the failures from Observation 2.10.2a to 7a follows as a special case. Items 2–3 provides recapture strategies for LEM and LI however, that this is equivalent to the original version. 28 Here, v + ∶ Fml → ℘({f , t}), which is the distinctive valuation function of the functional version of Dunn’s ’relational semantics’. When it comes to a paracomplete and non-paraconsistent logic, the function must be adjusted to v + ∶ Fml → ℘({f , t}) ∖ {{f , t}}. 29 Indeed, this semantics aims at reducing non-classical values to a co-presence or lack of f and t, while keeping a traditional understanding of how the classical values interact with the connectives. Condition (ii) seems not to match this intuition. 30 Of course, this does not imply that no relational semantics can be found for S∗ fde . The question, however, is: ‘How much conceptual insight could we gain from this semantics, in the case of S∗ fde ? ’

Normality Operators and Classical Recapture in Many-valued Logic 23 31 in K⊛ Items 4–6 provide a recapture of ECQ,32 Reductio ad 3 (see Corollary 2.11). Absurdum (RAA) and MP in LP⊛ , item 7 recaptures Modus Tollens (MT) and item 8 recaptures the transitivity of the material conditional (Observation 2.10.3b to 7b). These recaptures also apply to Kw⊛ and PWK⊛ , respectively. 3 ⊛ The ⊛-extensions K⊛ of Strong Kleene logics K3 and LP have already 3 and LP appeared, under different names, in [32] and [15, 17, 39], respectively. In particular, LP⊛ is just a variant of the well-known logic LFI1 from [17]. This is a logic of type (2,2,2,1,1,0,0)—its formula algebra contains connectives ∨, ∧, →, ¬, ●, , ⊺—and it is interpreted on a matrix ⟨A, D⟩ where A = {f , e, t} and D = {e, t} (as in LP and LP⊛ ). The operations of A are given by the following table:

Table 5

t e f

¬ f e t

● f t f

∨ t e f

t t t t

e t e e

∧ t e f

f t e f

t t e f

e e e f

f f f f

→ t e f

t t t t

e e e t

f f f t

In order to see that the connectives from Table 5 are definable in LP⊛ , consider that the consistency operator ○ from Table 6 is definable in LFI1 as ○α = ¬ ● α, and that, by Definition 3.1, ○ is the same operator as ⊛ in LP⊛ . Consider: Observation 5.2 The operators ∼ from Table 6 and → from Table 5 are defined in LFI1 as follows: 1. 2.

∼α α→β

= ¬α ∧ ○α = ∼α∨β

Definition of ∼ is just equivalent with the one noted by [15, p. 128]. For Definition of →, see [39, Corollary 12]. Table 6

t e f

∼ f f t

t e f

○ t f t

On the other hand, it is clear by the above that ⊛ is definable in LFI1 as ⊛α = ¬ ● α— or, equivalently, as ⊛α = (α ∧ ¬α) → (∼ α∧ ∼∼ α).33 This suffices to see that LFI1 and 31 32

Notice that, in K⊛ 3 , recapture of ECQ is equivalent with PGI. Notice that, in LP⊛ , recapture of ECQ is equivalent with PGE.

33 Notice that ∼ is a strong negation, according to the terminology from [17], since it verifies α, ∼ α ⊧LP⊛ β. The operator also satisfies ∅ ⊧LP⊛ α ∨ ∼ α.

24


LP⊛ are the same logic. ⊛ Semantically closed versions of K⊛ 3 and LP . In philosophy, the most widespread application of K3 and LP is to logical paradoxes, especially to semantic ones: K3 is the bedrock logic for Kripke’s fixed point construction in truth theory [35], and LP has been proposed by Priest as the basic logical frame of a dialetheic solution to the logical and set-theoretical paradoxes.34 These truth theories employ a semantically closed language, that is a language that can express its own concept of truth. This is done by expressions of the form T r(φ), where T r is a truth predicate and φ is the name of formula φ. As is well-known, semantic closure is fatal to CL: the Liar Paradox λ ≡ ¬T r(λ)—is derivable in semantically closed CL, and by ECQ, this trivializes the logic.35 By contrast, K3 and LP does not derive any contradiction from the Liar,36 ⊛ and do not become trivial with semantic closure. However, K⊛ 3 and LP do not share ⊛ ⊛ this convenient feature: semantically closed extensions of K3 and LP are trivial, as proved by [32] and [6], respectively. This has undermined the interest of the two logics in philosophy. As we shall see, the same results do not apply to the ⊛-extensions Kw⊛ 3 and PWK⊛ of Weak Kleene logics.

5.2

Classical Recapture in Kw⊛ and PWK⊛ 3

Weak Kleene logics Kw en 3 and PWK have been proposed by Bochvar [12] and Halld´ [33] in order to reason about the impact of meaningless expressions on our reasoning. Interestingly, they have both been proposed as fragments of more comprehensive formalisms, which in turn employ a normality operator or some similar expressive device. The two Weak Kleene logics are discussed by Bochvar and Halldén as the so-called ‘internal fragment’ of a full logic of meaninglessness [12] and of a full logic of nonsense [33], respectively. The full logic of nonsense from [33] is in turn PWK⊛ , and in this particular application, the normality operator from Table 3 (three-valued case) reads ‘α is meaningful’. The full logic of meaninglessness B3 employs a stricter operator that expresses that a formula is meaningful and true,37 and it is strictly related to Kw⊛ 3 . As is easy to check, all the cases of recapture from Corollary 5.1 also apply to Kw⊛ 3 w⊛ and PWK⊛ . Since Kw and PWK⊛ 3 and PWK are conservative, it follows that also K3 are. This, Theorem 4.8 and 4.9 guarantee the following recapture for the failure from Observation 2.14: Observation 5.3 ⊛ The following holds for Kw⊛ 3 -consequence and PWK -consequence: α, ⊛β ⊧MKw⊛ α ∨ β 3

α ∧ β, ⊛(α ∧ β) ⊧MPWK⊛ α

Also, Contamination has a straightforward impact of the validities concerning ⊛ in Kw⊛ and PWK⊛ : 3 34 Dialetheism is the philosophical view holding that some sentences are both true and false. It suggests a nonepistemic reading of value e in LP and b in Efde . 35 An implicit assumption here is that T r satisfies the so-called Tarski Axioms. This is in turn a widely shared assumption. 36 37

Notorioulsy, the success with this is due to failure of LEM and ECQ, respectively.

This is in turn the same as the operator ⊚ form [39], that expresses (in a three-valued logic) that a formula is assigned value t.

Normality Operators and Classical Recapture in Many-valued Logic 25 Observation 5.4 ⊛ The Principle of Contamination from Observation 2.13 implies that, for S⊛ ∈ {Kw⊛ 3 , PWK } and ◇ ∈ {∨, ∧}: ⊛(α ◇ β) ⊧S⊛ ⊛α ◇ ⊛β Indeed, by contamination, truth (or falsity) of a formula α ◇ β implies that neither α nor β have the non-classical value. The validity above is just expressing this in object-language reasoning. This has two interesting consequences. First: Corollary 5.5 For every Σ, β ⊆ F ml1 : Σ, Σ⊛ , ⊧MPWK⊛ β ⇔ Σ, (var(Σ))⊛ , (var(β))⊛ ⊧MPWK⊛ β Σ, ⊛β ⊧MKw⊛ β ⇔ Σ, (var(β))⊛ ⊧MKw⊛ β 3

3

That is, Theorem 4.3 collapses on Theorem 4.8 in PWK⊛ , and Theorem 4.4 collapses on Theorem 4.9 in Kw⊛ 3 . In the two logics, the models where a formula α as a classical value are exactly the same where all p ∈ var(α) have a classical value. This is a particular case where the recapture strategies from the different theorems ends up requiring exactly the same information. Second, notice that: 1. α ∧ ¬α ∧ ⊛α ⊧ / PWK⊛ β α ∨ ¬α ∨ ⊛α 2. ∅ ⊧ / Kw⊛ 3 Indeed, any valuation v ∈ HomMPWK⊛ where v(β) = f and v(α) = e is such that v(α ∧ ¬α ∧ ⊛α) = e, thus falsifying (1); similarly, any valuation v ∈ HomMKw⊛ where 3

v(α) = e is such that v(α ∨ ¬α ∨ ⊛) = e, thus falsifying (2)—remember that MPWK⊛ = = {t}. By contrast, it is easy to check that {α, ¬α, ⊛α} has no model {e, t} and MKw⊛ 3 in PWK⊛ , to the effect that the logic verifies principle PGE from Definition 3.4. Also, if we upgrade to the multiple-conclusion version of ⊧MKw⊛ , it is easy to check that PGI 3

is satisfied, since in any model, at least one element of {α, ¬α, ⊛α} is true. In particular, failure of 2 above explains why consideration of multiple-conclusion consequence is indispensable when defining the criteria for a Logic of Formal Undeterminedness. Semantically closed versions of Kw⊛ and PWK⊛ . Semantically closed versions 3 ⊛ w⊛ of K3 and PWK , and related formalisms have been recently explored by [6]. In particular, [6] proves that semantically closed infectious logics are non-trivial, contrary ⊛ to what happens to K⊛ 3 and LP . As a consequence, the addition of a truth predicate ⊛ w⊛ T r to K3 and PWK is harmless. Whether Kw⊛ and PWK⊛ can be suitable logics 3 for reasoning about truth, it is for a dedicated investigation to say. To the purposes of this paper, suffices it to notice that it is possible to have logics that are semantically closed, include normality operators and are non-trivial.

5.3

Classical Recapture in Efde ⊛

We close with a very brief zoom in on Efde ⊛ .38 Since Efde ⊛ is not conservative, its recapture recipe is given by Corollary 4.5. Consider (p ∧ ¬p) ∨ (q ∧ ¬q) ⊧CL r. As we 38 Efde ⊛ is not the first four-valued logic proposed at the intersection of the LFI and LFU traditions. In particular, Efde ⊛ is a sublogic of the four-valued Belnap-Dunn-style logics BS4 from [40], that we have already mentioned in Section 4.

26


have seen in commenting Corollary 4.10, the rule fails in Efde . Corollary 4.5 implies the following recapture for the rule: (p ∧ ¬p) ∨ (q ∧ ¬q), ⊛p, ⊛q, ⊛r ⊧Efde ⊛ r Corollary 4.5 also provides recapture for the classical failures from Observation 2.10— all these failures affect Efde , since the latter is a sublogic of both K3 and LP. We believe that it is clear how this recipe works, and so we will not detail the recapture here. An interesting point is that some of the inferences failing in Efde can be recaptured in Efde ⊛ without necessarily considering the values of the variables. For instance, it is easy to check that the following holds: α, ¬α, ⊛α, ⊛¬α ⊧ME

fde

⊛

β

To see this, consider that, for every α ∈ F ml1 there is no v ∈ HomMEfde such that v(α) = v(¬α) = t. One relevant question is: ‘At which condition can we relax the recipe from Corollary 4.5 when we provide recapture of given inferences in Efde ⊛ ? ’ We wish to reply this question in a future paper.

6

Conclusions

In this paper, we have provided a general way to build Logics of Formal Inconsistency (LFIs) and Logics of Formal Undeterminedness (LFUs) from any given many-valued logic, and we have established classical recapture results for families of such systems. In particular, we have extended the basic language of propositional logic with normality operator ⊛, which allows us to distinguish the valuations where α has a classical value (f or t) from those where α lacks a classical value. When applying to paraconsistent and paracomplete logics, this operator generates systems that are both LFIs and LFUs (Section 3). After presenting some properties of the operator (Section 3), we have established recapture results for a family of many-valued logics (Theorems 4.2–4.9, Section 4), which are in turn defined by formal properties introduced in Section 2. We have also discussed the rationale of these results, and their relation with respect to previous results and proposals, especially in LFIs and LFUs. We have exemplified the recapture results by illustrating some concrete cases in the ⊛ ⊛ ⊛ w⊛ logics K⊛ 3 , LP , K3 , PWK , and Efde . The present results also open interesting issues. We just mention two, to which we plan to devote future research. First, truth theory and logical paradoxes do not exhaust applications of paracomplete and paraconsistent logics: K3 and other paracomplete formalisms have been applied to partial information [1, 26], LP and other paraconsistent logics have been applied to paraconsistent belief revision [38, 43]. We aim at exploring the potential of normality operator ⊛ and related device in these particular applications, where the possibility to express that information is consistent, or that any information is held at all, seems desirable. Second, many classical recapture strategies have been presented, especially on the paraconsistent camp. Two prominent proposals here are classical collapse from [8, 9] and the non-monotonic minimal inconsistency from [42]. The tradition of adaptive logics also offers a non-monotonic way to classical recapture.39 In future work, we wish to compare our present proposal with these three approaches. 39 Adaptive logic was devised by Diderik Batens in the 1980s, and it has been developed and investigated in a huge number of papers; we refer the reader to [7] for a survey on the first twenty years of research in adaptive logic.

Normality Operators and Classical Recapture in Many-valued Logic 27

Acknowledgments We wish to thank three anonymous referees, Eduardo Barrio and Walter Carnielli for their helpful comments. Preliminary versions of this paper have been presented at the Logic and Metaphysics Workshop of the Graduate Center, CUNY, New York, and at the VI Workshop on Philosophical Logic organized by the BA Logic Group of the SADAF, Buenos Aires. We thank Graham Priest, Thomas M. Ferguson, Dave Ripley, and the other participants of the two workshops for their valuable comments. Research for this paper was carried while Roberto Ciuni was a Piscopia Fellow with the MSCA Cofund DYTEBEL project at the Department FISPPA, University of Padova (2016–2018).

Bibliography [1] A.N. Abdallah. The Logic of Partial Information. Springer, Berlin, 1995. [2] A. Anderson and N. Belnap. Tautological entailments. Philosophical Studies, 13:9–24, 1962. [3] A. Anderson and N. Belnap. Entailment. The logic of relevance and necessity. Princeton University Press, 1975. [4] A. Avron. Non-deterministic semantics for logics with a consistency operator. International Journal of Approximate Reasoning, 45:271–287, 2007. [5] A. Avron and I. Lev. Non-deterministic multiple-valued structures. Journal of Logic and Computation, 15:241–261, 2005. [6] E. Barrio, F. Pailos, and D. Szmuc. A cartography of logics of formal inconsistency and truth. manuscript, 2016. [7] D. Batens. A survey of inconsistency-adaptive logics. In D. Batens, G. Priest, and J. P. van Bendegem, editors, Frontiers of Paraconsistent Logic, pages 49–73. Kings College Publication, Baldlock, 2000. [8] Jc Beall. Multiple-conclusion LP and default classicality. Review of Symbolic Logic, 4(2):326–336, 2011. [9] Jc Beall. LP+ , K3+ , FDE+ and their classical collapse. Review of Symbolic Logic, 6(4):742–754, 2013. [10] Jc Beall. Off-topic: a new interpretation of Weak Kleene Logic. Australasian Journal of Logic, 13(6):136–142, 2016. [11] N. Belnap. A useful four-valued logic. In J. M Dunn and G. Epstein, editors, Modern uses of multiple-valued logic, pages 5–37. Reidel, Dordrecht, 1977. [12] D. Bochvar. On a three-valued calculus and its application in the analysis of the paradoxes of the extended functional calculus. Matamaticheskii Sbornik, 4:287– 308, 1938. [13] S. Bonzio, J. Gil-Ferez, F. Paoli, and L. Peruzzi. On Paraconsistent Weak Kleene logic. Studia Logica, 105(2):253–297, 2017. [14] W. Carnielli, M. Coniglio, and J. Marcos. Logics of Formal Inconsistency. In D. Gabbay and F. Guenthner, editors, Handbook of Philosphical Logic, volume 14, pages 1–93. Dordrecht: Springer-Verlag, 2007. [15] W. Carnielli, J. Marcos, and S. de Amo. Formal inconsistency and evolutionary databases. Logic and Logical Philosophy, 8:115–152, 2000. 28

BIBLIOGRAPHY

29

[16] W. A. Carnielli and M. E. Coniglio. Paraconsistent Logic: Consistency, Contradiction and Negation, volume 40 of Logic, Epistemology, and the Unity of Science series. Springer, 2016. [17] W. A. Carnielli and J. Marcos. A taxonomy of C-systems. In W. A. Carnielli, M. E. Coniglio, and I. M. L. d’Ottaviano, editors, Paraconsistency: The Logical Way to the Inconsistent, volume 228 of Lecture Notes in Pure and Applied Mathematics, pages 1–94. Marcel Dekker, 2002. [18] W. A. Carnielli and A. Rodrigues. An epistemic approach to paraconsistency: a logic of evidence and truth. Synthese, Online, https://doi.org/10.1007/s11229017-1621-7 2017. [19] R. Ciuni and M. Carrara. Characterizing logical consequence in Paraconsistent Weak Kleene. In L. Felline, A. Ledda, F. Paoli, and E. Rossanese, editors, New Developments in Logic and the Philosophy of Science, pages 165–176, London, 2016. College Publications. [20] R. Ciuni and M. Carrara. Normality operators and classical collapse. In P. Arazim and T. Lavicka, editors, Logica Yearbook 2017, London, forthcoming. College Publications. [21] R. Ciuni, T. Ferguson, and D. Szmuc. Logics based on linear orders of contaminating values. ms. [22] M. E. Coniglio and M.I. Corbalan. Sequent calculi for the classical fragment of Bochvar and Halldén’s nonsoense logic. In D. Kesner and Petrucio, V., editors, Proceedings of the 7th LSFA Workshop, Electronic Proceedings in Computer Science, pages 125–136, 2012. [23] N. C. A. da Costa. On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15:497–510, 1974. [24] N.C.A. da Costa and E. H. Alves. A semantical analysis of the calculi cn. Notre Dame Journal of Formal Logic, 18:621–630, 1977. [25] N.C.A. da Costa and D. Marconi. A note on paracomplete logic. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, 80(7–12):504–506, 1986. [26] M. D’Agostino. Semantics and proof-theory of depth-bounded boolean logics. Theoretical Computer Science, 480:43–68, 2013. [27] C. Daniels. A note on negation. Erkenntnis, 32:423–429, 1990. [28] H. Deutsch. Relevant analytic entailment. The Relevance Logic Newsletter, 2:26– 44, 1977. [29] T.M. Ferguson. Logics of nonsense and Parry systems. Journal of Philosophical Logic, 44(1):65–80, 2014. [30] J.M. Font. Abstract Algebraic Logic. College Publications, London, 2016. [31] M. Ginsberg. Multivalued logics: A uniform approach to reasoning in arti cial intelligence. Computational Intelligence, 4(3):265–316, 1988. [32] A. Gupta and N. Belnap. The Revision Theory of Truth. MIT, Cambridge, MA, 1993. [33] S. Halldén. The Logic of Nonsense. PhD thesis, Uppsala University, 1949. [34] S. Kleene. Metamathematics. North Holland, Amsterdam, 1952.

30

BIBLIOGRAPHY

[35] S. Kripke. Outline of a theory of truth. The Journal of Philosophy, 72(19):690– 716, 1975. [36] J. Marcos. Logics of Formal Inconsistency. PhD thesis, Universidade Estadual de Campinas, 2005. [37] J. Marcos. Nearly every normal modal logic is paranormal. Logique et Analyse, 48:279–300, 2005. [38] E. Mares. A paraconsistent theory of belief revision. Erkenntnis, 56(2):229–246, 2002. [39] H. Omori. Halldén’s logic of nonsense and its expansions in view of logics of formal inconsistency. In Database and Expert Systems Applications (DEXA), 27th International Workshop, pages 129–133, 2016. [40] H. Omori and K. Sano. da Costa meets Belnap and Nelson. In R. Ciuni and H. Wansing and C. Willkommen, editors, Recent Trends in Philosophical Logic, pages 145–166. Springer, Berlin, 2013. [41] G. Priest. The logic of paradox. Journal of Philosophical Logic, 8:219–241, 1979. [42] G. Priest. Minimally inconsistent LP. Studia Logica, 50(2):321–331, 1991. [43] G. Priest. Paraconsistent belief revision. Theoria, 67(3):214–228, 2001. [44] G. Priest. In Contradiction. Oxford University Press., 2 edition, 2006. [45] G. Priest. The logic of the catuskoti. Comparative Philosophy, 1(2):24–54, 2010. [46] G. Priest. Natural deduction systems for logics in the FDE family. ms. [47] D. Szmuc. Defining LFI and LFU’s in extensions of infectious logics. Journal of Applied Non-Classical Logic, 26(4):286–314, 2016. [48] H. Wansing. Connexive modal logic. In R. Schmidt et al., editor, Advances in Modal Logic, volume 5, pages 367–383. College Publications, London, 2005. Received Day Month Year.

Normality Operators and Classical Recapture in Many ...

Normality Operators and Classical Recapture in Many ...

Suggest Documents

Classical recapture - arXiv

OPERATORS IN QUANTUM AND CLASSICAL OPTICS

A Criterion for the Normality of Unbounded Operators and Applications ...

NORMALITY OF SPACES OF OPERATORS AND QUASI-LATTICES

on majorization and normality of operators (1) (d - American ...

Holonomy control operators in classical and quantum completely

Normality

Classical Kantorovich operators revisited 1 Introduction

THE BOUNDEDNESS OF CLASSICAL OPERATORS ON ... - icmat

Classical Resolution for Many-Valued Logics

Classical Solutions of the Many-time Functional

Classical foundations of many-particle quantum chaos

Almost-conserved operators in nearly many-body localized systems

Semi-classical monopole operators in Chern-Simons-matter theories

Normality in Analytical Psychology

Minimal Classical Logic and Control Operators Zena M. Ariola Hugo ...

Charged Free Fermions, Vertex Operators and Classical Theory of ...

Regular Keplerian motions in classical many-body ... - Eugene Butikov

Normality and Sharing Values

Regular Keplerian motions in classical many-body systems

Normality and Sharing Values

Mark-recapture estimates of the survival and recapture rates of ...

Dissociation and recapture dynamics in H2O

NORMALITY AND PROPERTIES RELATED TO COMPACTNESS IN ...