FABIEN SCHANG Negation and Dichotomy. Do They Refer to the Same 'Thing'? On the Identity of Logical Negation from a Semantic Perspective* Properties of classical negation '~' Reductio ad Absurdum RA1 If φ ⇒ ψ and φ ⇒ ~ ψ, then ~ φ RA3 If φ ⇒ ~ φ, then ~ φ RA2 If ~ φ ⇒ ψ and ~ φ ⇒ ~ ψ, then φ RA4 If ~φ ⇒ φ, then φ Contraposition C1 If φ ⇒ ψ, then ~ ψ ⇒ ~ φ C3 If ~ φ ⇒ ψ, then ~ ψ C2 If ~ φ ⇒ ~ ψ, then ψ ⇒ φ C4 If φ ⇒ ~ ψ, then ψ ⇒ ~ φ Double negation DN1 φ ⇒ ~~ φ DN2 ~~ φ ⇒ φ The two 'laws' LC (~ (φ ∧ ~ φ))∈{T} LEM (φ ∨ ~ φ)∈{T} De Morgan laws Conjunction DM ∧ 1 ~ (φ ∧ ψ) ⇔ ~ φ ∨ ~ ψ DM ∧ 3 ~ (φ ∧ ~ ψ) ⇔ ~ φ ∨ ψ DM ∧ 2 ~ (~ φ ∧ ψ) ⇔ φ ∨ ~ ψ DM ∧ 4 ~ (~φ ∧ ~ ψ) ⇔ φ ∨ ψ 226 FABIEN SCHANG Disjunction DM ∨ 1 ~ (φ ∨ ψ) ⇔ ~ φ ∧ ~ ψ DM ∨ 3 ~ (φ ∨ ~ ψ) ⇔ ~ φ ∧ ψ DM ∨ 2 ~ (~φ ∨ ψ) ⇔ φ ∧ ~ ψ DM ∨ 4 ~ (~ φ ∧ ~ ψ) ⇔ φ ∨ ψ Conditional DM → 1 φ → ψ ⇔ ~ φ ∨ ψ DM → 3 φ → ~ ψ ⇔ ~ φ ∨ ~ ψ DM → 2 ~ φ → ψ ⇔ φ ∨ ψ DM → 4 ~ φ → ~ ψ ⇔ φ ∨ ~ ψ 1. Negation from an algebraic viewpoint According to a Fregean or referential view of semantics, each formula from an interpreted language names and is associated with a reference, viz. a truth-value. Thus, such a sentence like 'Socrates is a philosopher' is taken to be a name for one logical object among two possible ones, namely: truth or falsity, depending upon whether Socrates is a philosopher or not. Each formula φ is interpreted algebraically by a mapping (valuation) from a set of formulas Lφ to a set of references or truth-values Vφ, to be symbolized as follows: Lφ → Vφ (see Figure 1). Each non-atomic, complex formula ⊕n with n components is interpreted by a mapping from a set of input values in Vφ to the set of output values in Vφ, to be symbolized as follows: Vφ → Vφ. For example, if the constructor-sign ⊕ stands for the unary operator of classical negation '~' then '~φ' is a complex formula, to be interpreted by a mapping from the reference of an atomic formula φ to the reference of its negated form ~φ. In classical logic, such a mapping will proceed from truth to falsity or from falsity to truth exclusively, depending upon the input value of φ. Let us use '~' as a symbol for classical negation, i.e. the operator to be attached only to sentences in a bivalent frame; any assignment of truth or falsity to ordinary sentences turns them into propositions. If we accept such a referential defi nition of semantic interpretation, each formula appears as a sort of defi nite description the reference of which may vary: it may name just one reference or truth-value (the mapping is a total function, that is a one-to-one or bijective relation between two sets), several truth-values (the mapping is a one-to-many or surjective relation between two sets), or no truth-value at all (the map is a partial function). The number of truth-values in V can be discussed and will result in different logical systems. How many truth-values are to be contained in a semantic set: two, three, or infi nitely many values? Some philosophers like Quine claimed that any 227 NEGATION AND DICHOTOMY Lφ (n = 2) Vφ Classical valuation as a one-to-one (bijective) bivaluation φ• •1 ~φ• •0 does map into does not map into Lφ (ex: n = 3) Vφ Non-Classical valuation as a one-to-one (bijective) n-valuation φ• •1 ~φ• •0 •1/2 Figure 1 228 FABIEN SCHANG semantic set must count only two elements, i.e. the subsets of true and false formulas.1 Other logicians hold some similar but more complex position, namely Roman Suszko and his plea for two-valuedness;2 we will return to Suszko's position later, since it will be used as a central argument for restoring some commonsensical logical laws as opposed to some scientifi c (many-valued) ones. As a pioneer of many-valued logics in their application to philosophical problems, Jan Łukasiewicz once suggested two different kinds of manyvalued sets, namely: a set of three truth-values or an infi nity of them, depending upon the meaning to be assigned to the third value of 'possibility' besides truth and falsity.3 The preceding quotation by Suszko has shown that he was clearly opposed to Łukasiewicz's many-valued sets, hence his distinction between algebraic and logical values. Each of these semantic sets are to be found in modern logical systems: a case for bivalence (with n = 2 truth-values) is Classical Logic (hereafter: CL); a case for trivalence (with n = 3 truth-values) is exemplifi ed by several threevalued systems with a specifi c meaning of the third truth-value as 'neither ... nor'. There is Łukasiewicz's Ł3 for contingent events with 'true', 'false', and 'indeterminate' as truth-values; Kleene's K3 for mathematical statements with 'true', 'false', and 'undecided' as truth-values; Bochvar's B3 for paradoxical statements with 'true', 'false' and 'senseless' as truth-values. In spite of Łukasiewicz's preceding plea for either three or infi nitely many truth-values, let us notice that he himself recognized afterwards (in 1953) the inappropriateness of his three-valued system and suggested instead a fourvalued system of modal logic in order to avoid some unpleasant result in Ł3, namely: the Law of Non-Contradiction didn't hold in it, whereas Łukasiewicz wanted to invalidate the Law of Excluded Middle only. Four-valued logics appear as a specifi c case of semantic sets in which each truth-value is a combined element from the powerset P(V). In a set V = {{0},{1}} with n = 2 elements, that is, {0} as the subset of only false formulas and {1} as the subset of only true formulas, the powerset P(V) = {{∅},{0},{1},{1,0}} includes 2n = 22 = 4 subsets with two additional cases, namely: {∅} as the set of neither-true-nor-false formulas, and {1,0} as the set of both-true-andfalse formulas. Three samples of four-valued logics (inter alia) are: Directional Logic (hereafter: DL), by Leonard Slawomir Rogowski (in 1961). In this logical system, the classical sets {1} and {0} are reinterpreted by 't' as strictly true and by 'f' as strictly false, respectively. In addition to these two classical values, Rogowski adds the 'subtrue' set {u} as coming to be false and the 'subfalse' set {i} as coming to be true. 229 NEGATION AND DICHOTOMY Relevance Logic, by Nuel Belnap (1977) (see Figure 2). This system is concerned with information or data bases in computer science: φ∈{N} is the set in which no information occurs about φ; φ∈{F} is the set with an information saying that φ fails; φ∈{T} is the set with an information saying that φ holds; and φ∈{B} is the set with an information saying that φ both holds and fails. Overclassical Logic, by Newton C. A. da Costa and Jean-Yves Béziau (1997) (see Figure 2). Such a system presents nonclassical diagrams with a relative complement: the pair <+,-> means that a formula φ is 'absolutely true', i.e. is located in the class of true formulas but not in the class of false formulas; <+,+> means that φ is 'relatively true', i.e. is located both in the class of true formulas and the class of false formulas; <-,-> means that φ is 'relatively false', i.e. is located neither in the class of true formulas nor in the class of false formulas; and <-,+> means that φ is 'absolutely false', i.e. is located in the class of false formulas but not in the class of true formulas. Finally, a case for indefi nitely many truth-values is some species of fuzzy logics [0,1], where an infi nity of truth-values occurs between the extreme cases of falsity {0} and truth {1}; infi nitary-valued logics are closely related to probabilitary logics or even to Jerzy Łoś's logic of assertion (in 1948), where each truth-value corresponds to a specifi c standpoint within a group of speakers. How to defi ne the logical constant of negation, given that the output value of a formula may change according to the cardinality of a semantic set? If, for instance, a given formula φ is neither true nor false in a logical system, it is pretty sure that the output value of the resulting ~φ will be neither false nor true and, thus, won't meet the algebraic defi nition of classical negation (from 1 to 0, or from 0 to 1). Now that does not mean that the main features of classical negation should be deeply revised in non-classical systems: such is the main thesis to be defended in this paper. As a minimal and necessary precondition for being a logical constant of negation in any system, it will be argued that negation operates as a dichotomy: As in Plato's Sophist, each dichotomy splits each set (say, xi for any i) into two subsets xi and not-xi of being and not-being (see Figure 3). According to Buridan's distinction, logical negation discriminates a set of elements {{x1},...,{xi}} from the remaining ones {{xi},...,{xn}} either positively (negatio infi nitans), or negatively (negatio negans). Thus, to say 230 FABIEN SCHANG N T B F Figure 2 Belnap's Relevance Logic da Costa and Béziau's Overclassical Logic <+;-> <+ ;+ > <-;+> <-;-> <+;-> absolutely true (only true) {1} <+;+> relatively true (true, false) {1,0} <-;+> absolutely false (only false) {0} <-;-> relatively false (not true, not false) {∅} B both true and false {1,0} T only true {1} F only false {0} N neither true nor false {∅} 231 NEGATION AND DICHOTOMY that 'the table is not-red' positively entails that the table is either blue, or black, or green ... i.e. possesses another defi nite colour. It follows that to be 'not-true' amounts to be another value, and not just 'not to be true'. Does it entail that the Law of Excluded Middle still holds as an excluded third with n = 2 truth-values in V, as an excluded fourth with n = 3, ..., as an excluded (xi+1)th with n = xi? If so, to defi ne negation as a dichotomy seems to result in a regressio ad infi nitum when applied to excluded middle, as argued by Church (1928) with respect to Burali-Forti's Paradox.4 But it is not so: logical negation can be properly defi ned as a dichotomy whatever V may be. Against Church's former objection to non-classical semantic sets with n > 2, Barzin and Errera (1929) insist that negation essentially proceeds as a negatio negans, i.e. as an indefi nite process of exclusion out of a class.5 The same observation will be applied in the following, thus yielding a general defi nition of negation as a dichotomy and restoring the above properties of the so-called 'classical' negation even within non-classical systems. Isn't this absurd, given that the classical rules of inference for negation V = {{x1}} V = {{x1},{x2}} (n = 1) (n = 2) V = {{x1},{x2},{x3}} (n = 3) V = {{x1},{x2},{x3},{x4}} (n = 4) V = {{x1},{x2},{x3},...,{xn}} (n → ∞) x1 x1 x2 (= not-x1) x2 x3 (= not-x2) x3 x4 (= not-x3) ...... ... xn (not-xn–1) xn xn+1 (= not-xn) Figure 3 Dichotomy 232 FABIEN SCHANG (contraposition, reductio ad absurdum, and the like) normally hold as a whole in classical logic only? A more careful attention upon 'classical laws' is in order in the following, so as to have a more comprehensive view of the notion of dichotomy. Two crucial pairs of notions are concerned here, namely: truth and falsity, on the one hand; affi rmation and negation, on the other hand. How are they combined, and do many-valued logics really entail a revision of 'classical' negation? The point is that the classical properties of negation fail in nonclassical systems when defi ned extensively, that is, in terms of truth-values. But the present paper wants to show that the process of dichotomy can be introduced within these non-classical systems in order to maintain such 'commonsensical' properties as excluded middle and non-contradiction while defi ning logical negation as a general constant (i.e. for any V). Negation can be distinguished from conjunction, disjunction, conditional and other logical constants as a peculiar process of dichotomy. For this purpose, let us look back on the past and Aristotle's Term Logic in order to support our 'intensional' defi nition of the concept of negation (beyond its extensional view in terms of output values). In Aristotle's Logic of Terms, a more fi ne-grained distinction between basic propositions was made as opposed to modern sentential logic. Such a distinction was called 'the Four' by Aristotle: given a basic predication 'S is P' for every atomic proposition, such an affi rmative form can be enriched if we introduce negation in it (see Figure 4). We thus obtain: the denial 'S is not P', the contraffi rmation 'S is not-P' and, fi nally, the contradenial 'S is not not-P'. An example of an affi rmation is 'Socrates is a philosopher', to be added with 'Socrates is not a philosopher', 'Socrates is a non-philosopher', and 'Socrates is not a non-philosopher'. Please note that a clear-cut distinction is to be made between denial and contraffi rmation: as noted by Englebretsen (1981), if Socrates doesn't exist the above denial is true whereas the contraffi rmation is not. Such a difference disappeared in modern logic, given that negation as a term predicate doesn't make sense therein and only serves as a sentential operator. But this very distinction will turn out to be crucial for the following argument. Let Lφ be a language of φ-order, φ any formula from Lφ and Vφ a set {{x1},...,{xn-1},{xn}} with n truth-values. Again, any valuation is an assignment of truth-values upon sentences, i.e. a mapping from an arbitrary φ of Lφ into a set in Vφ: φ → {xi}. Let {1} be the set of true propositions and {0} the set of false propositions. Then we can render φ's being true as φ∈{1}, and φ's being false as φ∈{0}. Now two different levels of negation occur in our ordinary speech-acts: a fi rst, linguistic negation '~' is attached to the entire sentence. If φ is read ''φ is true'', then ~φ means ''φ is not-true". 233 NEGATION AND DICHOTOMY Figure 4 'The Four' in Term Oppositions Affi rmation S is P ("Socrates is a philosopher'') Denial S is not P ("Socrates is a not-philosopher'') Contraffi rmation S is not-P ("Socrates is not a philosopher'') Contradenial S is not not-P ("Socrates is not a not-philosopher'') S is P contrariety S is not-P contradiction S is not not-P subcontrariety S is not P Oppositions as pairs of valuations Contrariety A and B cannot be both true, A and B can be both false Contradiction If A is true, then B is false; if A is false, then B is true Subcontrariety A and B cannot be both false; A and B can be both true A B A B A B A second, metalinguistic negation '∉' is attached to the truth-values of sentences. If φ∈{1} is read ''It is the case that φ is true'' (or, equivalently, ''φ is true''), then φ∉{1} means ''It is not the case that φ is true" (or "φ is not true"). It follows from the latter that if φ doesn't belong to the set {xi} of truth-values, then it belongs to any other set in Vφ than {xi}. As a case of metalinguistic negation, let V be a set of three truth-values {{x1},{x2},{x3}}; 234 FABIEN SCHANG thus according to Barzin and Errera's view of negation as exclusion out of a class, if n = 3 then φ∈{x1} means that φ∉{x2} or φ∉{x3}, φ∈{x2} means that φ∉{x1} or φ∉{x3}, and φ∈{x3} means that φ∉{x1} or φ∉{x2}. Generally speaking, if Vφ = {{x1},{x2},...,{xa},...,{xn-1},{xn}}, then φ∈{xn} if and only if (hereafter: iff) φ∈UVφ–xn , just as φ is not true iff φ∉{1} and φ is not false iff φ∉{0}. In the light of these two different levels of negation, the difference between Classical Logic (hereafter: CL) and Non-Classical Logics (hereafter: NCL) can be rephrased set-theoretically. Negation in CL concerns logics in which Vφ = {{1},{0}}, i.e. a set of n = 2 sets of truth-values, so that φ∈{1} whenever φ∈{0} and φ∈{0} whenever φ∈{1}. In a nutshell, CL is the class of bivalent semantic sets with n = 2 elements ({1} and {0}, say). Since Vφ = {{1},{0}} in CL, it follows that ~φ is true iff φ is not-true, i.e. φ is false (~φ∈{1} iff φ∉{1}, i.e. φ∈{0}), and ~φ is false iff φ is not-false, i.e. φ is true (~φ∈{0} iff φ∉{0}, i.e. φ∈{1}). According to such a bivalent relation between truth ({1}, say) and falsity ({0}, say), we thus obtain the following 'laws' in CL: Law of Bivalence (hereafter: LB): every sentence is either true or false In other terms, there are only n = 2 truth-values: for any φ, (φ∈{1} U φ∈{0}), so that φ∈{1} iff φ∉{0} and φ∈{0} iff φ∉{1}. Law of Excluded Middle (hereafter: LEM): either an affi rmation or its denial is true Either (S is P) is true or (S is not P) is true; Either (S is not-P) is true or (S is not not-P) is true, so that φ∈{1} or ~φ∈{1} for any φ. Law of (Non-)Contradiction (hereafter: LC): an affi rmation and its denial cannot be both true (S is P) and (S is not P) are not true; (S is not-P) and (S is not not-P) are not true, so that φ∉{1} or ~φ∉{1} for any φ. Negation in NCL concerns logics in which Vφ is a set of n > 2 sets of truth-values. LB notably fails in NCL: for some φ's, φ∈{1/2} means that φ∉{1} and φ∉{0}. Note that the 'intermediary' set {1/2} needn't be a unitclass: whatever is neither true nor false fi lls the bill. The linguistic negation '~' is truth-functional, that is: the output value of ~φ is uniquely determined by the input value of φ. In CL, Vφ = {{1},{0}}, so that ~φ is true iff φ is false (~φ∈{1} iff φ∈{0}); in NCL, if Vφ = 235 NEGATION AND DICHOTOMY {{1},{1/2},{0}} then ~φ is true iff φ is not-true (~φ∈{1} iff φ∉{1}), and ~φ is indeterminate iff φ is indeterminate (~φ∈{1/2} iff φ∈{1/2}). As in CL, the negation of {1} and {0} still yields {0} and {1} with so-called normal negations in NCL;6 and although the negation of {1/2} yields the same output value {1/2}, such a redundant operation doesn't constitute a counterexample against the general defi nition of negation as dichotomy or 'otherness'-operator, however; this will be argued later by means of a metalinguistic characterization of negation. The metalinguistic negation is not strictly truth-functional in NCL. That is: the value of φ is not uniquely determined by the value of φ in a bijective mapping. Thus, φ is not true iff φ is either false or indeterminate (φ∉{1} iff φ∈{0} or φ∈{1/2}); φ is not indeterminate iff φ is either true or false (φ∉{1/2} iff φ∈{1} or φ∈{0}); and φ is false iff φ is either true or indeterminate (φ∉{0} iff φ∈{1} or φ∈{1/2}). Another point to clarify is about the meaning of '1/2' as an intermediary value between 0 and 1. If φ is neither true nor false, then it could be claimed that φ doesn't have any value so that {1/2} stands for the empty set {∅}; if φ is both true and false, then φ has two values and {1/2} stands for the non-empty set {1,0}. Now any logic with {1/2} as a proper set is a 3-valued logic, insofar as {∅} and {1,0} count as two distinct subsets in Vφ in addition to {1} and {0}. The case of overdeterminacy (being both true and false) is not {{1},{0}} but {1,0}, and the case of indeterminacy (being neither true nor false) can also be seen as a three-valued system with the empty set as a proper set in its own part (see Figure 5). In the light of such defi nitions, let us turn to special cases of NCL and the purported reasons to revise the properties of classical negation. By gappy logics, we mean the class of paracomplete logics in which some sentences are neither true nor false: φ is neither true nor false whenever φ is not true and φ is not false (φ∈{1/2} whenever φ∉{1} and φ∉{0}, i.e. φ∈{∅}). LEM fails in gappy logics: if φ∈{1/2}, then φ∉{1} and φ∉{0}, i.e. ~φ∉{1}, so that (φ ∨ ~φ)∉{1} with φ∈{1/2}. By glutty logics, we mean the class of paraconsistent logics in which some sentences are both true and false: φ is both true and false whenever φ is true and φ is false (φ∈{1/2} whenever φ∈{1} and φ∈{0}, i.e. φ∈{1,0}. LC fails in glutty logics: if φ∈{1/2}, then φ∉{0} and φ∉{1}, i.e. ~φ∉{0}, so that ~(φ ∧ ~φ)∉{1} with φ∈{1/2}. Defi ning negation as a process of dichotomy generally works only if it is considered from a metalinguistic point of view. When negation is about sentences, we note this by ~φ∈{xi}; when negation is about truth-values, 236 FABIEN SCHANG Lφ (n = 2) Vφ 1/2 as the ouput value from a total function φ• •1 ~φ• •0 Lφ (n = 3) Vφ 1/2 as the output value from a total function φ• •1 ~φ• •0 •1/2 •1/0 Figure 5 237 NEGATION AND DICHOTOMY we note this by φ∉{xi}. In order to extensionalize this metaproperty, we propose to make use of internalization as follows. Lφ+1 is said to be an internalization of Lφ when some metalinguistic symbols with respect to Lφ are introduced into the object-language of Lφ+1. For instance, '∉' and '{xi}' are two metalinguistic symbols, whereas 'φ' and '~' are two linguistic symbols. The truth-values in Lφ are semantic (i.e. metalinguistic) predicates in Lφ+1, whereas valuations (upon Lφ in Vφ) are unary operators in Lφ+1. Turning back to 'the Four' (see Figure 6), we can interpret more extensively the general form 'S is P' as the semantic judgment 'φ is true' (φ∈{1}), the contraffi rmation 'S is not-P' as 'φ is not-true' (φ∉{1}, i.e. φ∈{0}), the denial 'S is not P' as 'φ is not true' (φ∉{1}, i.e. φ∈{0} or φ∈{1/2}), and the contradenial 'S is not not-P' as 'φ is not not-true' (φ∉{0}, i.e. φ∈{1} or φ∈{1/2}). Therefore, the contrary relation between φ∈{1} and φ∈{0} means that these cannot be both true and both false. Now if {1} means 'to be true', the aforementioned contrariety amounts to say that being true and being false cannot be true together. We thus iterate the notion of truth, but without entailing any antinomy with these judgments. For if I say: the judgment 'φ∈{0}' is true, that does not entail φ∈{1} because a typed distinction is made between the value of the sentence φ and the value of a judgment about it. In order to clarifi y such a distinction between sentences and judgments, let us call for a semantic distinction between plain values and designated values. Figure 6 Semantic Oppositions Affi rmation φ∈{1} ("It is true that Socrates is a philosopher'') Denial φ∉{1} ("It is not true that Socrates is wise'') Contraffi rmation φ∈{0} ("It is false that Socrates is a philosopher'') Contradenial φ∉{0} ("It is not false that Socrates is a philosopher'') φ∈{1} contrariety φ∈{0} contradiction φ∉{0} subcontrariety φ∉{1} 238 FABIEN SCHANG By a 'designated value' {T} is meant a specifi c subset of truth-values in V; such a subset may count only one or several elements. For example, if 'Bydgoszcz is in Poland' is true in Lφ, i.e. φ∈{1}, then ''that Bydgoszcz is in Poland is true'' is true in Lφ+1, i.e. (T{1}∈{T}). In Bochvar (1938), a similar use of judgment-operators was made with the so-called external operator of assertion Aφ: 'it is true that φ', to be read as 'φ∈{1}'. By this distinction between internal (sentential) and external (judgmental) operators, Bochvar made a typed distinction between affi rmations and negations on the one hand, assertions and denials on the other hand. The same rationale will be used in the following in order to internalize the normally metalinguistic notions of truth-values. As a matter of fact, the set {1} of true sentences is taken to be the only case of designated value; but some non-classical logicians supplement the subset of designated values with {1/2} (see Figure 7). When a truth-value does not belong to the subset of designated values, it is said to be a non-designated value {⊥}.7 By extension, designated values are used to defi ne the logical truth of any formula semantically, namely: for any formulas φ and ψ, ψ is a logical consequence of φ iff if ψ∈{T} whenever φ∈{T}. The several properties of logical negation can be thus understood as preserving the designated value from the premises to the conclusion: φ ⇒ ψ. Here is the core point of the paper: by means of internalization and designated values, and in accordance with Suszko's thesis, any Vφ with n > 2: {{x1},{x2},...{xn}} sets of truth-values can be reduced to a Vφ+1 with n' = 2: {{xi},{xj,...,xn}} subsets of truth-values. In rephrasing affi rmation and denial as '∈' and '∉', the properties of negation can be 'classicized' by both internalizing truth-values while following Suszko's thesis. This will be done in two successive steps, namely: in section 2, an increasing step from bivalent logics (with n = 2) to manyvalued logics (with n > 2); and in section 3, a decreasing step from manyvalued logics to their bivalued (but not bivalent!) counterparts (with n' = 2). Just recall that the step from bivalence to bivaluation does not amount to a back to CL at all: it is a transition from so-called 'algebraic' values in NCL to 'logical' values. Such a reduction has been supported by Suszko and will be exemplifi ed by internalizing several non-classical logics.8 2. From bivalence to many-valuation From Vφ = {{0},{1}} we can derive the following subsets of truth-values: true and false, i.e. {1}∩{0} = {1,0}; true and not false, i.e. {1}∩~{0} = {1}; not true and false, i.e. ~{1}∩{0} = {0}; not true and not false, i.e; ~{1}∩~{0} = {∅}. 239 NEGATION AND DICHOTOMY Figure 7 Designated and non-designated values in NCL In Ł3 (Łukasiewicz's 3-valued logic) {T} •1 •1/2 •0 {⊥} •1 •1,0 •0 {T} {⊥} In PLP (Priest's 3-valued logic of paradox) Given CL as the set of logics in which either φ∈{1} or φ∈{0} (tertium non datur) for every φ, and NCL as the set of logics in which φ∉{1} and φ∉{0}, i.e. φ∈{1,0} or φ∈{∅} for some φ's (with {{1,0},{∅}}⊆{1/2}), two sorts of many-valued logics can be discriminated according to the meaning of {1/2}, namely: gappy logics (including K3, DL), and glutty logics (including PLP). Some general rules of valuation can be afforded for both sorts of NCL (whether paracomplete or paraconsistent), provided that the same ordering 240 FABIEN SCHANG relation obtains between their truth-values. As it is the case for both K3 and PLP, let us state the following rules of valuation. Given an ordering relation {1}>{1/2}>{0}, for any two 1-0 truth-values {x1},{x2} we have the following rules for logical constants: ~{x1}∈|{1-x1}| {x1} ∨ {x2}∈max{x1,x2} {x1} ∧ {x2}∈min{x1,x2} {x1} → {x2}∈max{1-x1,x2} ⊕φ∈min(⊕{x1,y1},⊕{x2,y2}), 9 for any pairs of 1-0 truth-values {x1,x2},{y1,y2} and any 1-ary or 2-ary operator ⊕. 2.1. Gappy Logics An example of gappy logic with {1/2} = {∅} is Heyting's Intuitionistic Logic (hereafter: HIL). In accordance with Brouwer's objections to the dual opposition of truth and falsity and the realist approach of mathematical reasoning, the philosophy of intuitionism roughly consists in reading proofs as mental constructions and refuses to assign any value to a sentence so long as no proof has been constructed for it. Consequently, such classical properties of negation as LEM and Double Negation (hereafter: LDN) are cancelled by intuitionists because of their wrongly dual treatment of affi rmations and negations. Following Heyting's original notation, we'll make use of '¬' as a symbol for intuitionistic or strong negation.10 Given the truth-conditions for intuitionistic negation, LB and LEM both fail in HIL with φ∈{1/2}. But the core question rather concerns the identity of LEM from a logical system to another one: how can LEM be said to have the same meaning in both classical and intuitionistic logics, assuming that logical negation is not the same for classicists and intuitionists? Pending an answer for this question, let us note that the two following properties of classical negation still hold when interpreted intuitionistically: (H1) ¬φ ⇒ (φ → ψ) (H2) ((φ → ψ) → (φ → ¬ψ)) ⇒ ¬φ However, the following invalid formulas (*) making use of negation are not logical truths in HIL: (H1*) (¬φ ∨ ψ) ⇒ (φ → ψ) 241 NEGATION AND DICHOTOMY (H2*) (φ → ψ) ⇒ ¬(φ ∧ ¬ψ) (H3*) (φ ∨ ψ) ⇒ (¬φ → ψ) (H4*) (φ ∨ ψ) ⇒ ¬(¬φ ∧ ¬ψ) (H5*) (φ ∧ ψ) ⇒ ¬(¬φ ∨ ¬ψ) In order to give a semantic interpretation of these formulas, a language L and a 3-valued matrix V for HIL have been supplied by Kleene (1952)'s K3-system, namely: LK3 = 〈∼,∧,∨,→〉 and VK3 = 〈{1},{1/2},{0}〉, in which classical negation '~' can be used as a primitive operator defi ning intuitionistic negation, that is: ¬φ means 'φ∈{0}', i.e. '~φ∈{1}'. The truthconditions for the logical constants in LK3 are truth-functionally defi ned in matrices (see Figure 8). Another case of gappy logic is Rogowski's Directional Logic (hereafter: DL), the aim of which was to formalize Hegel's dialectical logic of change. Several kinds of modal operators11 are introduced into a 4-valued matrix in DL, namely: LDL = 〈N→,∧,∨,→〉, and VDL = 〈{1,1},{1,0},{0,1},{0,0}〉 As for most of non-classical systems, the designated value corresponds to the single value {{1,1}}⊆{T}, whereas non-designated values are the three remaining ones {{1,0},{0,1},{0,0}}⊆{⊥}. In accordance with the motivation of DL, i.e. to express changes between 'being' and 'not-being', an interpretation for the four values in DL yields φ∈{1,1} as 'it is true that φ', φ∈{1,0} as 'it comes to be false that φ', φ∈{0,0} as 'it is false that φ', and φ∈{0,1} as 'it comes to be true that φ'. Given the ordering relation {1,1}>{0,1}>{1,0}>{0,0}, the general rules of valuation for formulas in DL can be defi ned within matrices (see Figure 9) in basic terms of classical negation '~'. For any pairs of 1-0 values {x1,y1},{x2,y2}, we have the following valuations for connectives: N{x1,y1}∈{~x1,~y1}; N+{x1,y1}∈{~x1,~y1} iff x1 = y1; N +{x1,y1}∈{1,1}, otherwise; N→{x1,y1}∈{x1,~y1} iff x1 = y1; N →{x1,y1}∈{~x1,y1}, otherwise; N←{x1,y1}∈{~x1,y1} iff x1 = y1; N ←{x1,y1}∈{x1,~y1}, otherwise; 242 FABIEN SCHANG T{x1,y1} = N +N{x1,y1}; H→{x1,y1}∈{x1,y1} iff x1 = y1; H →{x1,y1}∈{~x1,y1}, otherwise; H←{x1,y1}∈{x1,y1} iff x1 = y1; H ←{x1,y1}∈{x1,~y1}, otherwise. Together with the following valuations for the set of classical connectives {∨,∧,→}, namely: {x1,x2} ∨ {y1,y2}∈max({x1,x2},{y1,y2}); Figure 8 Logical matrices for K3 φ ~φ {1} {0} {1/2} {1/2} {0} {1} φ ∧ ψ {1} {1/2} {0} {1} {1} {1/2} {0} {1/2} {1/2} {1/2} {0} {0} {0} {0} {0} φ ∨ ψ {1} {1/2} {0} {1} {1} {1} {1} {1/2} {1} {1/2} {1/2} {0} {1} {1/2} {0} φ → ψ {1} {1/2} {0} {1} {1} {1/2} {0} {1/2} {1/2} {1/2} {1/2} {0} {1} {1} {1} 243 NEGATION AND DICHOTOMY {x1,x2} ∧ {y1,y2}∈min({x1,x2},{y1,y2}); {x1,x2} → {y1,y2}∈max(N{x1,x2},{y1,y2}). LEM, LC, and the Law of Identity (φ → φ)12 fail in DL with φ∈{1,0} and φ∈{0,1}. Furthermore, the same does with all classical properties of negation that fail with the non-static truth-values {1,0} and {0,1}. Thus, such 'obvious' laws as LEM or LC are rejected within many-valued systems; just as the former are invalidated in some gappy logics, most of the properties of logical negation are also discarded in glutty logics. Figure 9 Logical matrices for DL φ Nφ N+φ N→φ N←φ Tφ H←φ H→φ {1,1} {0,0} {0,0} {1,0} {0,1} {1,1} {1,1} {1,1} {0,1} {1,0} {1,1} {1,1} {0,0} {0,0} {0,0} {1,1} {1,0} {0,1} {1,1} {0,0} {1,1} {0,0} {1,1} {0,0} {0,0} {1,1} {1,1} {0,1} {1,0} {0,0} {0,0} {0,0} φ ∨ ψ {1,1} {0,1} {1,0} {0,0} {1,1} {1,1} {1,1} {1,1} {1,1} {0,1} {1,1} {0,1} {0,1} {0,1} {1,0} {1,1} {0,1} {1,0} {1,0} {0,0} {1,1} {0,1} {1,0} {0,0} φ ∧ ψ {1,1} {0,1} {1,0} {0,0} {1,1} {1,1} {0,1} {1,0} {0,0} {0,1} {0,1} {0,1} {1,0} {0,0} {1,0} {1,0} {1,0} {1,0} {0,0} {0,0} {0,0} {0,0} {0,0} {0,0} φ → ψ {1,1} {0,1} {1,0} {0,0} {1,1} {1,1} {0,1} {1,0} {0,0} {0,1} {1,1} {0,1} {1,0} {1,0} {1,0} {1,1} {0,1} {0,1} {0,1} {0,0} {1,1} {1,1} {1,1} {1,1} 244 FABIEN SCHANG 2.2. Glutty logics In many-valued logics, the main case for gluttiness is the family of paraconsistent logics, i.e. systems interpreted by logical matrices with φ∈{1/2}: 'φ is both true and false'. A sample of glutty logic is Priest's Logic of Paradox (hereafter: PLP). Since paraconsistent negation is not attached only to a classical value {1} or {0}, let us symbolize it as '–'. Just as in gappy logics and any many-valued logics, LB and LC both fail in PLP. The main difference between PLP and the preceding gappy systems concerns the sets of designated and not-designated values: {T} and {⊥} differ in extension given that {T} = {{1/2},{1}} in PLP.13 Now just as intuitionistic negation didn't have the same meaning as classical negation, how can we say that paraconsistent negation, and the properties of logical negation, do have the same meaning in CL and NCL? Pending an answer to this matter of meaning for logical constants, a semantic interpretation for PLP was given within logical matrices by Priest (1979)'s 3-valued logic PLP. It relies on a language to be interpreted in a 3-valued matrix (see Figure 10): LPLP = 〈–,∧,∨,→〉 and VPLP = 〈{1},{1/2},{0}〉 As just observed, LC does not 'classically' hold (φ∉{1} for some φ's) but still holds in PLP with φ∈{1/2}: (1/2 ∧ ~1/2) = (1/2 ∧ 1/2) = 1/2, therefore LC∉{1} with φ∈{1/2}; now {1/2}∈{T} in PLP, then LC 'weakly' holds with {1/2}. Valuations in PLP turn on the three following truth-values: {1} as 'only true', i.e. {1,1}; {1/2} as 'both true and false', i.e. {1,0} or {0,1}; and {0} as 'only false', i.e. {0,0}, within an ordering relation {1}>{1/2}>{0}. The following formulas with logical negation still hold in PLP: (P 1) (φ → ψ) ⇒ (–ψ → –φ) (P 2) (–φ ∧ –ψ) ⇒ –(φ ∨ ψ) (P 3) (–φ → –ψ) ⇒ (ψ → φ) (P 4) –(φ ∨ ψ) ⇒ –φ (P 5) φ ⇒ – –φ 245 NEGATION AND DICHOTOMY (P 6) – –φ ⇒ φ (P 7) –φ ⇒ –(φ ∧ ψ) (P 8) –(φ → ψ) ⇒ φ (P 9) (φ ∧ –ψ) ⇒ –(φ → ψ) (P10) –φ ⇒ (φ → ψ) (P11) (φ → –φ) ⇒ –φ Some of the logical truths in CL fail in PLP, namely: (P1*) (φ ∧ –φ) ⇒ ψ (P2*) (φ ∧ (–φ ∨ ψ)) ⇒ ψ Figure 10 Logical matrices for PLP φ ~φ {1} {0} {1/2} {1/2} {0} {1} φ ∧ ψ {1} {1/2} {0} {1} {1} {1/2} {0} {1/2} {1/2} {1/2} {0} {0} {0} {0} {0} φ ∨ ψ {1} {1/2} {0} {1} {1} {1} {1} {1/2} {1} {1/2} {1/2} {0} {1} {1/2} {0} φ → ψ {1} {1/2} {0} {1} {1} {1/2} {0} {1/2} {1/2} {1/2} {1/2} {0} {1} {1} {1} 246 FABIEN SCHANG (P3*) ((φ → ψ) ∧ –ψ) ⇒ –φ (P4*) (φ → (ψ ∧ –ψ)) ⇒ –φ The invalidity of (P1*) symbolizes non-triviality in PLP and means that, from a pair of inconsistent formulas, we cannot derive anything; (P2*) is a rejection of disjunctive syllogism, normally used in order to deduce triviality in an inconsistent system; fi nally, (P3*) and (P4*) are two variants of the principle of reductio ad absurdum (hereafter: RA) and mean that any formula entailing an inconsistency is not to be always rejected as such. In the light of these three non-classical systems and their many-valued semantics, it is established as a commonplace that most of the classical properties of logical negation (especially LEM or LC) do not hold universally and fail whenever the semantic frame V includes more than the two classical truth-values {0} and {1}. Does it mean that our alleged commensensical reading of negation is incompatible with the preceding commonplace, or that it should be restricted to some current interpretations of negation? In order to concile the current (classical, bivalent) view of negation and its special (non-classical, manyvalued) uses, it will be claimed in the following that: logical negation is defi ned with respect to its arguments, i.e. the truth-values it maps onto; by this way, non-classical logics don't appear as a deviation but as an extension of negation in CL (from n = 2 to n > 2); the commensensical reading of logical negation can be preserved even in NCL, by internalizing the truth-values as supplementary unary operators; most of the classical properties of logical negation may be thus restored, depending upon the translation of LEM, LC, and the like in the internalized systems; beyond CL and NCL, logical negation can be viewed as a general operator of dichotomy, while clearly making a distinction between bivalence and bivaluation (i.e. bipartition) in V. 3. From many-valuation to bivaluation According to Suszko's thesis, every many-valued logical matrix can be reduced to a two-valued one, whenever a distinction is made between algebraic values ({x1},...,{xn}) in Lφ and logical values ({T} and {⊥}) in 247 NEGATION AND DICHOTOMY Lφ+1. Two internalized systems may exemplify this thesis, namely: K3+ as an internalization of K3 for the semantics of HIL, and PLP+ as an internalization of PLP. 3.1. Gappy logics An internalization of K3 thus yields the Logic of Truth K3+: L3+ = 〈∼,∧,∨,→,T〉 and V3+ = 〈{1},{1/2},{0}〉, in which classical negation '~' is used as a basic term for defi ning its single intuitionistic counterpart. Following logical truth in K3, the only designated value in K3+ is {{1}}⊆{T}, so that the not-designated values are {{1/2},{0}}⊆{⊥}. K3+ is a semicomplete logic, extending K3 with T as a primitive unary operator together with two other interdefi nable operators F,I. Tφ helps to internalize φ∈{1} and stands for 'It's established as true that φ'; a counterpart for Tφ in Modal Logic (hereafter: ML) is the notion of necessity, φ. Fφ internalizes φ∈{0} and stands for 'It's established as false that φ'; a counterpart for Fφ in ML is the notion of impossibility, ~φ. Iφ internalizes φ∈{1/2} and stands for 'Nothing is established about φ'; a counterpart for Iφ in ML is the notion of contingency or two-sided possibility,φ =df (~φ ∧ ~~φ). We thus get the following τ-translations of the valuations in VK3 into formulas in LK3+: τ(φ∈{1}) = Tφ; τ(φ∈{1/2}) = Iφ; and τ(φ∈{0}) = Fφ The language for K3+ helps to translate the formulas from HIL and, above all, to compare classical and intuitionistic views of negation. For instance, a τ-translation of the left-sided formulas from HIL yields the right-sided formulas from LK3+: τ(φ) = Tφ, i.e. φ∈{1}; and τ(¬φ) = T~φ, i.e. φ∈{0} According to the logical matrices in K3+ (see Figure 11), both values {1} and {0} in HIL uniquely correspond to Tφ and T~φ in K3+. Let us note however that, while Fφ (= T¬φ) corresponds to a well-formed formula (hereafter: wff) in HIL, no wff corresponds to the resulting Iφ (= ~Tφ) in K3+. It is so because of the strictly strong meanings of truth and falsity as established values in HIL; in other terms, there is no room for contingent 248 FABIEN SCHANG (unestablished) truth in Heyting's view of intuitionistic logic. Now if such a strong reading of truth and falsity is translated in K3+, it seems that the intuitionistic version of LEM, namely: (Tφ ∨ Fφ), actually corresponds to the metalinguistic LB and thus expresses a relation between contrary values, rather than contradictory ones. A comparison between the classical and intuitionistic readings of LEM can be made within Blanché's hexagon of oppositions (see Figure 12), thus yielding a contrast between the syntactic (i.e. according to their logical forms) and semantic formulations (i.e. according to their truth-values) of LEM: either the latter contains a pair affi rmation-denial and thus corresponds to the classical version only; or it is about a pair truth-falsity and thus corresponds to both classical and intuitionistic versions. By analogy with a famous paper by Slater (1995): "Paraconsistent Logics?'', who doubted about their foundations because paraconsistent negation is not a contradictory - but subcontrary-forming operator, the same question can Figure 11 Logical matrices for K3+ φ Tφ {1} {T} {1/2} {⊥} {0} {⊥} φ ¬φ Fφ {1} {0} {⊥} {1/2} {1/2} {⊥} {0} {1} {T} Iφ {1} {⊥} {1/2} {T} {0} {⊥} Wffs in HIL Wffs in K3+ 249 NEGATION AND DICHOTOMY be asked about intuitionistic negation given that it is not a contradictorybut contrary-forming operator (see Figure 12). It is a thing that not every logical negation must be a contradictory-forming operator, as witnessed in the history of logic by several forms of contrary negations within ancient and medieval logics;14 but it is another thing to say that not-contradictoryforming operators of negation occur in LEM as such. Is excluded middle strictly related to classical negation, or does it ultimately express a semantic relation between truth-values, i.e. irrespective of its logical form and displayed opposition? According to Wiredu (1975), the intuitionistic objection against LEM does miss the point: a disjunctive relation is stated therein between an affi rmation ('is true') and its contraffi rmation ('is false'), whereas LEM is stated as an Figure 12 Blanché's Hexagon of Oppositions Meta-operators in K3+ Tφ contrariety Fφ contradiction ∼Fφ subcontrariety ~Tφ A contrariety E contradiction I subcontrariety O U Y ~Iφ Classical LEM Intuitionistic LEM ~Iφ 250 FABIEN SCHANG opposition between and affi rmation ('is true') and its denial ('is not true'). In a nutshell, the identity of LEM seems to vacillate between its syntactic and semantic defi nition. Anyway, some iteration laws can be put in K3+ in order to simplify its modal formulas: Tφ ⇔ TTφ; Tφ ⇔ φ (for any ∈{T,F,I}); FTφ ⇔ ~Tφ; Fφ ⇔ TFφ. The class of theorems in K3+ are S5-valid when intuitionistic negation is translated as ~T, so that this class is larger than in Gödel (1933)'s translation of intuitionistic negation as a S4-modal system. The reason is that Gödel's translation squares with Heyting's version of '¬' as 'T~' (= 'F'), and not '~T' (the latter does not make sense in HIL, again). It can be established both (see the Appendix) that: K3+ is a translation of HIL whenever '¬' is translated by the strong negation 'T~', in accordance with Heyting's modal interpretation of intuitionistic negation; the properties of classical negation can be preserved in K3+ whenever '¬' is translated by the weak, classical negation '~T'; as for what the 'genuine' translation of LEM in K3+ is, namely: (Tφ ∨ T~φ) or (Tφ ∨ ~Tφ), the question remains open. Another case of internalization concerns DL, with a Logic of Being-theCase DL+. DL is self-internalizing, i.e. LDL already contains an operator of assertion T that may be used in order to internalize truth-values and preserve the 'classical' properties of negation. Examples of internalized assertions in DL+ are 'It is the case that it is true that φ', i.e. T{1,1}∈{T}; 'it is the case that it begins to be true that φ', i.e. T{0,1}∈{⊥}; 'it is the case that it begins to be false that φ', i.e. T{1,0}∈{⊥}; and, fi nally,'it is the case that it is false that φ', i.e. T{0,0}∈{⊥}. A proof of the validity for LEM and LC in DL+ is given in the Appendix below. 3.2. Glutty logics An internalization of PLP yields the Logic of Veridication PLP+, as suggested by Strössner and Strobach (2007): 251 NEGATION AND DICHOTOMY LPLP+ = 〈~,∧,∨,→,V〉 and VPLP+ = 〈{1},{1/2},{0}〉 in which classical negation is used as a basic constant defi ning its possible paraconsistent counterparts. Following logical truth in PLP, there are two designated values in PLP+, namely: {{1},{1/2}}⊆{T}, whereas the single not-designated value is {{0}}⊆{⊥}. PLP+ is a semiconsistent logic extending PLP, with V as a primitive unary operator together with one other interdefi nable operator W. Wφ helps to internalize φ∈{1} and stands for 'It's only true that φ'; a counterpart of Wφ in ML is φ. W~φ internalizes φ∈{0} and stands for 'It's only false that φ'; a counterpart for W~φ in ML is ~φ; Vφ internalizes either φ∈{1} or φ∈{1,0} and stands for 'It's at least true that φ'; a counterpart for Vφ in ML is φ. We thus have the following translations of the valuations in VPLP into the formulas in LPLP: τ(φ∈{1}) ⊇ {Wφ,Vφ}; τ(φ∈{1/2}) = Vφ; and τ(φ∈{0}) ⊇ {W~φ,V~φ} Note also that Wφ and Vφ are duals, i.e. Vφ ⇔ ~W~φ. The language for PPL+ helps to translate the formulas from PPL and, above all, to compare classical and paraconsistent views of negation. As the meaning of paraconsistent negation is not as strong as its intuitionistic counterpart but still differs from the classical one, we thus have the following possible translations from left-sided formulas in PPL to right-sided formulas in PLP+: τ(φ)= Vφ or Wφ; and τ(–φ)= V~φ or W~φ. According to the logical matrices in PLP+ (see Figure 13), both classical values {1} and {0} in PPL correspond to Wφ or Vφ and W~φ or V~φ in PLP+; Wφ (i.e. ~V~φ), W~φ (i.e. ~Vφ) and Vφ (i.e. ~W~φ) correspond to wffs in PPL, whereas the non-classical value {{1},{0}} uniquely corresponds to Vφ. Contrary to the translations from HIL-formulas to K3+formulas, in which the resulting τ(φ) and τ(¬φ) are defi nite, the translations from PLP-formulas to PLP+-formulas can thus vary according to the adopted translations (syntactic interpretations) of φ and ~φ in PLP+. A comparison between classical and both possible paraconsistent readings of LC can be made within Blanché's hexagon of oppositions (see Figure 14), thus yielding a contrast between the syntactic (i.e. their logical forms) and semantic formulations (i.e. their truth-values) of LC. 252 FABIEN SCHANG Some iteration laws can be put in PLP+, thus: Wφ ⇔ Wwφ; Wφ ⇔ Wφ (for any ∈{W,V,W~}); and W~φ ⇔ W~φ. Theorems in PLP+ are S4-valid when paraconsistent negation '–' is read as 'W~' (like Gödel's S4, given that F and W~ share the same valuations); whereas they are S5-valid when '–' is read as '~W', i.e. as 'V~'. It can be shown (see the Appendix) that: every logical truth of PLP+ is a theorem of PLP whenever '–' is read as thre weak negation 'V~'; every classical property of negation can be restored in PLP+ whenever '–' is read as the strong negation 'W~'. 4. Conclusion: Negation as Dichotomy In conclusion, the properties of negation in CL may be maintained depending upon the meaning of truth and falsity in NCL; and assuming that two Figure 13 Logical matrices for PLP+ φ Wφ Vφ {1} {T} {T} {1/2} {⊥} {T} {0} {⊥} {⊥} φ –φ = V~φ {1} {⊥} {1/2} {T} {0} {T} φ –φ = W~φ {1} {⊥} {1/2} {⊥} {0} {T} Wffs in PPL Wffs in PLP+ 253 NEGATION AND DICHOTOMY formulas are equivalent iff they have the same meaning, a translation of intuitionistic and paraconsistent negations '¬' and '–' by '~T' and '~W' (i.e. 'V~'), respectively, would maintain LEM, LC, LDN, and the like within non-classical (many-valued) systems. As depicted by a fi rst transition from bivalent to many-valued logics and, then, by a second transition from many-valued to bipartitioned logics, the classical properties of negation in CL can be maintained in applying Suszko's thesis and replacing bivalence with bipartition. Indeed, every property from RA to DM→ may thus be saved when logical consequence and logical truth are defi ned in Vφ+1 = {T,⊥} rather than in Vφ = {x1,...xn}; here are the main advantages and shortcomings of defi ning negation as a dichotomy: Figure 14 Blanché's Hexagon of Oppositions Meta-Operators in PLP+ Wφ contrariety W~φ contradiction Vφ subcontrariety V~φ A contrariety E contradiction I subcontrariety O U Y Wφ or W~φ Classical LC Paraconsistent LC (1) Paraconsistent LC (2) Vφ and V~φ 254 FABIEN SCHANG 4.1 Dichotomy and n-chotomies A defi nition of dichotomy is the following: 'being twofold; a classifi cation into two opposed parts or subclasses', in which it relates to bipartition of classes and don't need to be synonymous with bivalence (it is so only when each subset is a unit-class, i.e. only in CL). The conclusive claim of this paper is that, beyond the variety of logical negations from a semantic perspective, negation can be viewed as a metalinguistic process of dichotomy, i.e. as a bipartition of any two subsets of truth-values. It follows from this defi nition that: Figure 15 {1} {1} {0} {1} {0} {1/2} n = 3 n = 2 n → ∞ {0} Truth-polygons {1} {0} {0,1}{1,0} n = 4 255 NEGATION AND DICHOTOMY CL is a peculiar case in which the number of algebraic values (metalinguistic predicates: {0}, {1}, ..., {xn}) is the same as the number of logical values (designated values); - 'classical negation' is a minimal negation, in the sense that no lower Vn than n = 2 can be used in logic (if n < 2, then no consequence relation can be stated between formulas); every negation consists in dividing a set of n elements into 2 subsets of truth-values, and not always a set of n = 2 elements into 2 subsets (this latter case holds in CL only); any n-chotomy is an increasing set of algebraic values tending to infi nitely many, whereas negation as a dichotomy differs from these "referential'' assignments and only concerns logical (impredicable) values {T,⊥}. 4.2 Negations as specifi c functions Such an intensional defi nition of negation as dichotomy makes the mapping non-truthfunctional, that is: each particular (i.e. cyclic, strong, weak, external, internal, and so on) negation (see Figure 15 and Figure 16) helps to fi x one and only one value for ~{xi} by means of {xi} in Vφ, but the same does not hold in Vφ+1 since different formulas can have the same truth-value. Several specifi c functions are related to negation, namely: complementary negation (classical negation as a contradictoryforming operator) φ∈{xi} in Vφ+1, iff ~φ∈UVx-{xi} in Vφ polar negation (intuitionistic negation as a contrary-forming operator) φ∈{x1} in Vφ = {x1,...,xn} iff ~φ∈{xn} symetric negation For any scale of values in Vφ ={0,...,xi=1/n,...,1}, |{xi} + {~xi}| = 1 in Vφ cyclic negation (directional negation as a "backward-or-forward''-forming operator) If φ∈{xi} in Vφ, then ~φ∈{xi-1} or ~φ∈{xi+1} in Vφ DN1 and DN2 can be replaced by cyclic laws of n-fold negation: ~nφ ⇔ φ (for any Vφ) 256 FABIEN SCHANG Values in DL {1,1} Truth {1,0} Sub-falsity {0,1} Sub-truth {0,0} Falsity Negations in DL N+φ: strong negation refl ective (), 1-fold clockwise (), 1-fold counterclockwise () Nφ: weak negation refl ective (, / ,) N→φ: initiation n-fold clockwise cyclic (	, ,,,	,...) N← φ: fi nalization counter-clockwise cyclic ( ,	,,, ,...) n = 4 Figure 16 {1,1} {0,0} {1,0}{0,1} N← N+ N+ N N N N+ N← N← N← N→ N→ N→ N→ An example of 4-valued polygon: Directional Logic, with its cyclic (blue) and refl ective (red) negations 257 NEGATION AND DICHOTOMY In sum, the commensensical view of logical negation is closely related to the metalinguistic negation '∉' (as applied to sets of truth-values) and not to the linguistic negation '~' (as applied to sets of formulas). Our usual confusion between both negations is due to our usual employment of notions like 'affi rmation' and 'negation' from a bivalent point of view. A distinction between both views of negations helps to restore the 'classical' properties of negation even in NCL, whenever a proper translation of many-valuations is given in internalized modal systems while following Suszko's thesis about logical values. Appendix: Internalizing many-valued logics (I) From K3 to K3+ Theorem 1. Every set of theorems TK3 is TK3+. Proof: by induction upon the list of axioms in HIL (see below). Nagation in concentric spheres Figure 17 x1 ~x1=x2 ~x2=x3 ... ~xi-1=xi n = 1 n = 2 n = 3 n = i n = ... 258 FABIEN SCHANG Figure 18 φ∈{1} ~φ∈{0} CL: complete, consistent φ∈{1} φ∉{1}∩φ∉{0} ~φ∈{0} NCL: paracomplete, consistent φ∉{1}∩φ∉{0} φ∈ {1 }∩ φ∈ {0 } φ∈{1} ~φ∈{0} NC: paracomplete, paraconsistent 259 NEGATION AND DICHOTOMY Theorem 2. K3+ is equivalent with K3. Proof: Every logic L1 is equivalent with another logic L2 iff (1) LL1 = LL2 and (2) TL1 = TL2. (1) Let L1 be LHIL and L2 be LK3+. LHIL = LK3+ iff wffs in K3 are identical with wffs in K3+ or wffs in K3 are translated into wffs of K3+. LK3+ are translations of LK3, hence LK3 = LK3+ (2) TK3 = TK3+, by Theorem 1. Proof of TK3 = TK3+. Every translation of TK3 is TK3+, as can be checked in the following: (K1) Tφ ⇒ (Tφ ∧ Tφ) (K2) (Tφ ∧ Tψ) ⇒ (Tψ ∧ Tφ) (K3) (Tφ → Tψ) ⇒ ((Tφ ∧ Tχ) → (Tψ ∧ Tχ)) (K4) ((Tφ → Tψ) ∧ (Tψ → Tχ)) ⇒ (Tφ → Tχ) (K5) Tψ ⇒ (Tφ → Tψ) (K6) Tφ ⇒ ((Tφ → Tψ) → Tψ) (K7) Tφ ⇒ (Tφ ∨ Tψ) (K8) (Tφ ∨ Tψ) ⇒ (Tψ ∨ Tφ) (K9) ((Tφ → Tχ) ∧ (Tψ → Tχ)) ⇒ ((Tφ ∨ Tψ) → Tχ) (K10) Fφ ⇒ (Tφ → Tψ) (K11) ((Tφ → Tψ) ⇒ (Tφ → Fψ)) → Fφ The following HIL-logical truths (a)-(e) are also K3+-logical truths: (a) (¬φ ∨ ψ) ⇒ (φ → ψ) (Fφ ∨ Tψ) ⇒ (Tφ → Tψ) (b) (φ → ψ) ⇒ ¬(φ ∧ ¬ψ) (Tφ → Tψ) ⇒ F(Tφ ∧ Fψ) (c) (φ ∨ ψ) ⇒ (¬φ → ψ) (Tφ ∨ Tψ) ⇒ (Fφ → Tψ) (d) (φ ∨ ψ) ⇒ ¬(¬φ ∧ ¬ψ) (Tφ ∨ Tψ) ⇒ F(Fφ ∧ Fψ) (e) (φ ∧ ψ) ⇒ ¬(¬φ ∨ ¬ψ) (Tφ ∨ Tψ) ⇒ F(Fφ ∨ Fψ) 260 FABIEN SCHANG Theorem 3: Any converse of the preceding formulas is not valid in K3+. K3+ is equivalent with K3, by Theorem 2. Hence if (a)*-(b)* are not logical truths in HIL, the same does in K3+. Proof: by induction upon the converses of (a)-(e). (a)* (φ → ψ) ⇒ (¬φ ∨ ψ) (Tφ → Tψ) ⇒ (Fφ ∨ Tψ) (b)* ¬(φ ∧ ¬ψ) ⇒ (φ → ψ) F(Tφ ∧ Fψ) ⇒ (Tφ → Tψ) (c)* (¬φ → ψ) ⇒ (φ ∨ ψ) (Fφ → Tψ) ⇒ (Tφ ∨ Tψ) (d)* ¬(¬φ ∧ ¬ψ) ⇒ (φ ∨ ψ) F(Fφ ∧ Fψ) ⇒ (Tφ ∨ Tψ) (e)* ¬(¬φ ∨ ¬ψ) ⇒ (φ ∧ ψ) F(Fφ ∨ Fψ) ⇒ (Tφ ∧ Tψ) (a)*-(e)* are of the form (A ⇒ B), so that any of these is not a logical truth iff A∈{T} and B∉{T}, i.e. B∈{⊥} for some assignment(s) of their components φ and ψ. Thus we have: (a)* (Tφ → Tψ)∈{T} and (Fφ ∨ Tψ)∈{⊥} with φ∈{1/2} and ψ∈{1/2} or ψ∈{0} (b)* (F(Tφ ∧ Fψ))∈{T} and (Tφ → Tψ)∈{⊥} with φ∈{1} and ψ∈{1/2} (c)* (Fφ → Tψ)∈{T} and (Tφ ∨ Tψ)∈{⊥} with φ∈{1/2} and ψ∈{1/2} or ψ∈{0} (d)* F(Fφ ∧ Fψ)∈{T} and (Tφ ∨ Tψ)∈{⊥} with φ∈{0} and ψ∈{1/2} or φ∈{1/2} and ψ∈{0} (e)* (F(Fφ ∨ Fψ)∈{T} and (Tφ ∧ Tψ)∈{⊥} with φ∈{1} and ψ∈{1/2} or φ∈{1/2} and ψ∈{1} Theorem 4: Any converse (a)*-(e)* is valid in K3+ when F is translated as ~T. Proof: for any member of (a)*-(e)*, either A∈{T} and then B∈{T}; or B∈{⊥} and then A∈{⊥}. Therefore A ⇒ B for any value of A,B in VK3+. (II) From DL to DL+ Theorem 5: LEM and LC are theorems in DL+. Proof: τ(LEM) = T(φ) ∨ NT(φ) in DL+ and τ(LC) = N(T(φ) ∧ NT(φ)) in DL+. Any formula φ in DL is a theorem iff φ∈{1,1}; now 261 NEGATION AND DICHOTOMY (LEM)∈{1,1} and (LC)∈{1,1} for any interpretation of φ in DL. Therefore, LEM∈{T} in DL+ and LC∈{T} in DL+. Theorem 6: any property of negation is classicized in DL+. Proof: by induction on the properties of classical negation, with the translation φ =df Tφ for any modal formula φ in DL. (III) From PLP to PLP+ Theorem 7. Every TPLP is TPLP+. Proof: by induction upon the list of axioms in HIL (see below). Theorem 8. PLP+ is equivalent with PLP. Proof: Every logic L1 is equivalent with another logic L2 iff (1) LL1 = LL2 and (2) TL1 =TL2. (1) Let L1 be LPLP and L2 be LPLP+. LPLP+ = LPLP+ iff wffs in PLP are identical with wffs in PLP+ or wffs in PLP are translated into wffs of PLP+. LPLP+ are translations of LPLP, hence LPLP =LPLP+ (2) TPLP = TPLP+, by Theorem 1. Proof of TPLP = TPLP+. Every translation of TPLP is TPLP+. We assume that, for some plausible translations τ1 in PLP+ of a formula (A ⇒ B) in PLP, τ(A ⇒ B)∈{T}. It can then be verifi ed that any of (P1)- (P11) are logical truths in PLP+, especially the following ones: (P1) (φ → ψ) ⇒ (–ψ → –φ) (P2) (–φ ∧ –ψ) ⇒ –(φ ∨ ψ) (P3) (–φ → –ψ) ⇒ (ψ → φ) (P4) –(φ ∨ ψ) ⇒ –φ (P5) φ ⇒ ––φ (P6) – –φ ⇒ φ (P7) –φ ⇒ –(φ ∧ ψ) (P8) –(φ → ψ) ⇒ φ 262 FABIEN SCHANG (P9) (φ ∧ –ψ) ⇒ –(φ → ψ) (P10) –φ ⇒ (φ → ψ) (P11) (φ → –φ) ⇒ –φ Theorem 9: (f)*-(i)* are not logical truths in PLP+ for some τ1translation. PLP+ is equivalent with PLP by Theorem 2. Hence if (f)*-(i)* are not logical truths in PLP, the same does in K3+ under some τ1-translation. Proof: by induction upon the translated formulas of PLP. (f)* (φ ∧ –φ) ⇒ ψ (Vφ ∧ V~φ) ⇒ Vψ (g)* (φ ∧ (–φ ∨ ψ)) ⇒ φ (Vφ ∧ (V~φ ∨ Vψ)) ⇒ Vφ (h)* ((φ → ψ) ∧ –ψ) ⇒ –φ ((Vφ → Vψ) ∧ V~ψ) ⇒ V~φ (i)* (φ → (ψ ∧ –ψ)) ⇒ –φ (Vφ → (Vψ ∧ V~ψ)) ⇒ V~φ (f)*-(i)* are of the form (A ⇒ B), so that any of these is not a logical truth iff A∈{T} and B∉{T}, i.e. B∈{⊥} for some assignment(s) of their components φ and ψ. Thus we have: (f)* (Vφ → Vψ)∈{T} and (Vψ)∈{⊥} with φ∈{1/2} and ψ∈{1/2} or ψ∈{0} (g)* (Vφ ∧ (V~φ ∨ Vψ))∈{T} and (Vψ)∈{⊥} with φ∈{1/2} and ψ∈{0} (h)* ((Vφ → Vψ) ∧ V~ψ)∈{T} and V~φ∈{⊥} with φ∈{0} and ψ∈{1/2} (i)* (Vφ → (Vψ ∧ V~ψ))∈{T} and V~φ∈{⊥} with φ∈{0} and ψ∈{1/2} Theorem 10: Any converse (f)*-(i)* is valid in PLP+ when V~ is substituted by W~. Proof: for any member of (f)*-(i)*, either A∈{T} and then B∈{T}; or B∈{⊥} and then A∈{⊥}. Therefore A ⇒ B for any value of A,B in VPLP+. 263 NEGATION AND DICHOTOMY Notes * I am especially grateful to Konrad Turzynski for his very helpful comments on Rogowski's Directional Logic, as well as his detailed observations during private correspondences; see also his: "The temporal functors in the directional logic of Rogowski - some results.'' Bulletin of the Section of Logic. Vol. 19 (1990), pp. 80-82. 1 "The original motivation [of many-valued investigations] was an abstractly mathematical one: the pursuit of analogy and generalization. From such a perspective, many-valued logic is logic by analogy only; indeed, it is an uninterpreted theory, or abstract algebra.'' (Quine 1973, p. 124) 2 Łukasiewicz is the chief perpetrator of a magnifi cent conceptual deceit lasting out in mathematical logic to the present day.'' (Suszko 1977, p. 377) 3 "Among all these many-valued systems, just two are entitled to claim to some philosophical involvement: the three-valued, and the infi nitely-valued one. For if any other values than 0 and 1 are read as 'the possible,' we can reasonably distinguish only two cases: either we assume that the possible does not include degrees, so that we get the three-valued system; or we assume the contrary, so that it is natural to recognize, as in the calculus of probability, that there is infi nitely many degrees of the possible, what leads to the system with infi nitely many values." (Łukasiewicz 1930, p. 72) 4 If excluded middle is rephrased as the view that no third value stands besides 1 and 0 in a bivalent system with n = 2, no fourth value stands besides 1, 0 and 1/2 in a trivalent system with n = 3 etc., Church claims that such a recurrent sequence cannot avoid a paradox with transfi nite ordinal numbers of truth-values: n = ω, n = ω+1, ... . As a conclusion, Church says, "this paradox, in fact, compels us to regard as illegitimate the consideration of this sequence as a whole.'' (Church 1928, p. 78) 5 "... after characterizing a number of groups by some positive properties, we'll reject the whole residue into a last group which won't have any special properties, except that its components won't have any of the properties that would have let them introduced into one of the fi rst divisions. The same does for logical classifi cation. Classical logic acknowledges a true, which it characterizes by a positive property, e.g. correspondence with an external reality; then the non-true or false, that is, all that doesn't have such a feature. In other places, especially in mathematics, it characterizes the false by a positive property, i.e. contradiction; so that the true is the non-false according to it. Now if we defi ne the true and the false each by some positive quality, there might be a residue into our classifi cation. Such a residue will be the tiers, which could be defi ned as being the neither-true-nor-false.'' (Barzin and Errera 1929, pp. 9-10) 6 A unary operator of negation is said to be "normal" iff if ~{1/2} = {1/2}. Post's cyclic negation is not a normal negation, given that the negation of 1/2 yields 1 in it. 7 The pairs of elements {1}-{0} and subsets {T}-{⊥} are one and the same in CL: obviously, the bivalent frame of classical logics entails that whatever is true is designated, and whatever is false is not-designated. Therefore, the distinction between designated and not-designated values is relevant only in NCL (with n > 2). 8 Tsuji sketches Suszko's thesis of reduction for logical matrices as follows: "In short, according to [Suszko], many-valued logics are in essence two-valued logics with many-valued referential (semantic) correlates; these semantic correlates are not to be confused with truth and falsity, which are after all the only possible logical values in such cases'' (Tsuji 1998, p. 302) 9 Suszko meant by "true" and "false" that what is symbolized here by {T} (the designated value) and {⊥} (the non-designated). 264 FABIEN SCHANG Let us give three examples of valuation with pairs of truth-values, assuming throughout that {1,1} = {1}, {1,0} = {0,1} = {1/2}, and {0,0} = {0}: a) Let ⊕ be '~', and φ with {x1,y1} = {1/2}. Then ~φ∈min{~x1,~y1}, i.e. ~φ∈min({0,1},{1,0}), hence ~φ∈{1/2}. b) Let ⊕ be '∧', φ with {x1,y1} = {1/2} and ψ with {x2,y2} = {1}, i.e. {1,1} Then (φ ∧ ψ)∈min({1∧1,0∧1},{0∧1,1∧1}), i.e. (φ ∧ ψ)∈min({1,0},{0,1}), hence (φ ∧ ψ)∈{1/2}. c) Let ⊕ be '→', φ with {x1,y1} = {1/2}, and ψ with {x2,y2} = {1/2} Then (φ→ψ)∈ min({1→1,0→0},{1→0,0→1},{0→1,1→0},{0→1,0→1}); hence (φ→ψ)∈ min({1,1},{0,1}},{1,0},{1,1}), i.e. (φ ∧ ψ)∈{1/2}. Note that the above truth-conditions don't obtain in Łukasiewicz's three-valued system, where {1/2 →1/2} = {1} and not {1/2}. 10 See von Wright (1959)'s strong negation, the truth-conditions of which are more stringent that the classical or 'weak' negation. In a nutshell, this strong negation amounts to the modal notion of impossibility: "By saying that ~p means 'the proposition p is not true', it is meant 'it is impossible for p to be true'.'' (Heyting 1932, p. 122) Hence the intuitionistic defi nition of negation as absurdity: ¬φ =df (φ → ⊥). 11 These are "modal" in the sense that they proceed as unary operators upon sentences, but they could be said not to be "modal" insofar as that they proceed truth-functionally within a functionally complete matrix. The informal readings of modal operators in DL are the following: negations include, besides the primitive negation N→ as initiation, N← as fi nalization, N as weak negation (it is not the case that), and N+ as strong negation; the other modal operators are H→ as protention, H← as retention, and T as assertion. 12 The failure of self-identity doesn't matter for our present purposes concerning negation, but it is a special feature of DL: according with Hegel's dialectic, objects don't have any static properties and, hence, no identity as such. 13 A consequence of this extended definition of logical truth is the distinction between contradiction, inconsistency, and triviality in PLP: (φ ∧ –φ) and –(φ ∧ –φ) are logically true in PLP, but that the former contains inconsistent theorems φ and –φ does not entail that anything is true in PLP (according to the classical law ex contradictio sequitur quodlibet: (φ ∧ –φ) ⇒ ψ, for any ψ). Triviality thus fails with ψ∈{0}. 14 A case for non-classical negations is given by Dutilh-Novaes (2003), for instance. The writer rightly insists that the problem with paraconsistent logics is not so much about non-classical negation in general than about contradiction: there are some non-classical negations in logic, but how to accept a negation N with (φ ∧ Nφ)∈{1}? References Barzin, M. and Errera, A. (1929) "Sur le principe du tiers exclu.'' L'Enseignement Mathématique. Vol. 30, pp. 4-15. Belnap, N. (1977) "A useful four-valued logic.'' In Modern Uses of MultipleValued Logic. J. M. Dunn & G. Epstein (ed.). Reidel, pp. 5-37. 265 NEGATION AND DICHOTOMY Bochvar, D.A. (1938) "On a three-valued calculus and its application to analysis of paradoxes of classical extended functional calculus.'' Matématičéskij Sbornik. Vol. 4, pp. 287-308. Church, A. (1928) "On the law of excluded middle.'' Bulletin of the Mathematical Society. Vol. 34, pp. 75-78. da Costa, N.C.A. and Béziau, J.-Y. 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