Disputing the many-valuedness of topos logic

DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC∗ LUIS ESTRADA-GONZÁLEZ NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM), MEXICO [email protected] Abstract: I study here the implications of Suszko-like reductive results for the common wisdom on the internal logic of a topos . My aim here is to question certain slogans about topos logic, and especially on its many-valuedness. The originality of this paper comes not from the observation that the internal logic of a topos is bivalent because it is in the scope of Suszko’s or any other reduction; that is easy. Rather, the originality comes from (1) recognizing the import of applying Suszko’s and similar results in a topos-theoretical setting, (2) the suggestions to give categorial content to the reduction, and (3) the extrapolation of the debate about Suszko’s thesis to the topos-theoretical framework, which gives us some insight about the scope of another theorem, namely that stating the intuitionistic character of the internal logic of a topos. Keywords: Topos logic, Suszko’s reduction, logical bivalence, logical manyvaluedness. ∗ This paper was written under the financial support from the European Union through the European Social Fund (Mobilitas grant no. MJD310), the CONACyT project CCB 2011 166502 “Aspectos filosóficos de la modalidad”, and the PAPIIT project IA401015 “Tras las consecuencias. Una visión universalista de la lógica (I)”. I thank Robert Goldblatt for his skepticism on the whole project and his helpful comments and suggestions at a very earliy stage, as well as Axel Barceló-Aspeitia, Jean-Yves Béziau, Daniel Cohnitz, Marcelo Coniglio, Thomas Ferguson, Carlos César Jiménez, Chris Mortensen, Zbigniew Oziewicz and Ivonne Pallares-Vega for their comments, suggestions or encouragement over the years I have spent working on this topic. I have presented several drafts to many conferences, from the XIII SLALM (2006) to the Contest “Scope of Logic Theorems” during UNILOG 2013 (which I could attend thanks to the support from the Mobilitas grant), and benefited greatly from the interaction with the audiences. Diagrams were drawn using Paul Taylor’s diagrams package v. 3.94. 193 194 1 LUIS ESTRADA-GONZÁLEZ Introduction A category can be thought of as a universe of objects and their transformations or connections, called morphisms, subject to some very general conditions. An example of a category is Set, whose objects are sets and its morphisms are functions between sets. In Set there is a special kind of objects, namely objects with two elements. As objects with two elements, these objects are isomorphic and each of them has all and only those mathematical properties (as expressible in categorial terms) as any other, so the sign ‘2Set ’ can be used to denote any of them and speak as if there were only one of them. We will say that an object with the property of having exactly two elements is unique up to isomorphism. 2Set act as truth values object in Set in the sense that suitable compositions with codomain 2Set serve to expresses that certain sets are part of others. Hence, the two elements of 2Set are conveniently called trueSet and f alseSet . Other logical notions besides truth values, such as zero- and higher-order connectives, can be defined in Set. It can be proved that the right logic to study the objects in Set, its internal logic, is that induced by the algebra formed by 2Set and the connectives, which turns out to be classical. This logic is called internal because it is formulated exclusively in terms of the objects and morphisms of the topos in question and it is the right to reason about the topos in question because it is determined by the definition of its objects and morphisms in a way that using a different logic for that purpose would alter the definitory properties of those objects and morphisms and thus it would not be a logic for the intended objects and morphisms; it cannot be a canon imposed “externally” to reason about the topos. As in usual axiomatic membership-based set theories like ZFC, most of mathematics can be interpreted and carried out in Set. However, a set theory developed from a category-theoretic point of view is not based on the notion of membership, but rather on those of function and composition (of functions). There are other Set-like categories, called elementary toposes or simply toposes. In a topos E there are objects object which play the role of 2Set in the particular case of Set, i.e. they serve to express that certain objects are part of others via suitable compositions of morphisms. An object that plays such a role in a topos is also unique up to isomorphism and any of them can be denoted by the sign ‘ΩE ’ and speak as if there were only one of them. Logical notions like truth values and zero- and higher-order connectives can also be defined in a topos. However, in general ΩE has more than two elements and, since ΩE has all the same universal properties as 2Set and the latter can be considered a truth values object, so can the former. In addition, the logic appropriate for dealing with the objects and morphisms in a topos, its internal logic, is in general intuitionistic, not classical. This is precisely a logic arising from objects and morphisms themselves, not from our devices to reason about them. Like Set, toposes also allow for the interpretation of set theoretical DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC195 notions and hence of significant parts of mathematics, but the reconstruction of mathematics carried out in a topos corresponds to mathematics as done in an intuitionistic set theory. If toposes can be considered universes of sets and, given that at least parts of mathematics can be reconstructed in a set theory, toposes also allow for the reconstruction of those parts of mathematics, then the universal laws of mathematics are those valid across all universes of sets, viz. the laws of intuitionistic logic. This beautiful picture of logic in a topos can be summarized in the following slogans1 : (S1) ΩE is (or at least can be seen as) a truth-values object. (Common categorial wisdom, see for example [12], [16], [17], [18].) (S2) The internal logic of a topos is in general many-valued. (Common categorial wisdom, but see [5], [3], [4], [12], [16], [17], [21].) (S3) The internal logic of a topos is in general (with a few provisos) intuitionistic. (This also is common categorial wisdom, just to name but two important texts where this is asserted see [12] and [18].) (S4) Intuitionistic logic is the objective logic of variable sets. (A powerful metaphor widely accepted. See [14], [15], [22]) (S5) The universal, invariant laws of mathematics are those of intuitionistic logic. (Cf. again [5], [3], [4].) With the exception of (S5)2 , which is a claim specifically due to Bell, these slogans are theses so widely endorsed by topos theorists as accurate readings of some definitions, results and constructions in topos theory that it is hardly worth documenting, but I have done it just to show that they appear in several major texts by leading category theorists. Recent research on so-called complement-toposes (cf. [23], [24] or [8]) runs against (S3) and (S4) showing that the internal logic of a topos does not have to be intuitionistic unless some prior, non-categorial conceptualizations have been made. Whereas in standard topos theory, the categorial reconstruction of logic starts by conceptualizing as “true” one of the elements Ω, in complement-toposes it is conceptualized as “f alse”. These authors stress the fact that the very categorial structure of toposes supports different prior non-categorial conceptualizations for some crucial morphisms, so the internal logic of a topos could be equally described as a kind of dual-intuitionistic, paraconsistent logic. The decidedly philosophical claim in (S5) proclaiming the intuitionistic character of the mathematical universal laws would not be entirely justified, either. 1 I use the word ‘slogan’ here pretty much in the sense of van Inwagen: “a vague phrase of ordinary English whose use is by no means dictated by the mathematically formulated speculations it is supposed to summarize” ([30, p. 163]), but that looks as if it was, I would add. 2 And maybe also of (S4), due mostly to the appearance of Hegelian terminology (“objective”), very frequent in Lawvere but not in other topos-theorists. Omitting that, one can add [1] and [12] as supporters of this slogan. 196 LUIS ESTRADA-GONZÁLEZ Slogans (S1) and (S2) are also in a similar, difficult position, but the challenge comes from a different source. There is a family of results initiated by Roman Suszko which states that every logic satisfying certain conditions –a Tarskian structural logic in the case of Suszko, i.e. a logic whose consequence relation is reflexive, transitive, monotonic and structural– has a bivalent semantics. A philosophical intuition behind Suszko’s and akin results is the distinction between algebraic truth values and logical truth values. Logical values are those values used to define consequence: If every premise is true, then so is (at least one of) the conclusion(s). In a contrapositive form, the other logical value can also be used to explain valid semantic consequence: If the (every) conclusion is not true, then so is at least one of the premises. Thus only the two logical truth values true and not true or, more generally, designated and antidesignated, are needed in the definition of consequence. Many-valuedness could be recovered at a different level, by taking it into account from the very characterizations of logical consequence. There are at least two other characterizations of logical consequence that do not presuppose that designated and antidesignated values form two collectively exhaustive and mutually exclusive collections, and that could introduce interesting complications in the theory of the internal logic of toposes. The first one is Malinowski’s Q-consequence (“Q” for “quasi”): If the premises are not antidesignated then the conclusion is designated (cf. [19], [20]). Another is Frankowski’s P-consequence (“P” for “plausible”): If the premises are designated then the conclusion is not antidesignated (cf. [10], [11]). The Tarskian properties are indissolubly tied to the canonical characterizations of logical consequence, but Q- and P-consequence are non-Tarskian. For example, Q-consequence is not reflexive. I study here the implications of reductive results for the picture of the internal logic of a topos described above. My aim here is to question slogans (S1), (S2) and (S3), and suggest that they cannot be jointly true if ‘manyvalued’ in (S2) means logically many-valued. The originality of this paper comes not from the observation that the internal logic of a topos is bivalent because it is in the scope of Suszko’s or any other reduction; that is easy. Rather, the originality comes from (1) recognizing the import of applying Suszko’s and similar results in a topos-theoretical setting, (2) the suggestions to give categorial content to the reduction, and (3) the extrapolation of the debate about Suszko’s thesis to the topos-theoretical framework, which gives us some insight about the scope of another theorem, namely that stating the intuitionistic character of the internal logic of a topos, embodied in (S3). The plan of the paper is as follows. Knowledge of basic category theory (as in chapters 1, 2 and 4 of [21]) is presupposed and knowledge of basic topos theory would be useful (good sources are [12] and again [21]). In the next section I present only the topos theory needed for the rest of the paper. In section 3 I expound Suszko’s reduction, following closely the presentations in [7], [29] and [32]. In section 4 I show that the internal logic of a topos can be described as algebraically many-valued but not necessarily logically DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC197 two-valued. I introduce there the notions of a Suszkian object and of Suszkian bivaluation, as opposed to a truth values object and a subobject classifier, as a means to internalize Suszkian bivalence in certain toposes. Finally, in sections 5 to 7 I suggest how logical many-valuedness could be recovered by moving to notions of consequence as those mentioned above, but at the price of letting the internal logic of a topos become variable. 2 A quick survey of the internal logic of a topos A (standard) topos is a category S E with equalizers, (binary) products, coequalizers, coproducts, exponentials, and a morphism S true : 1 −→ S Ω, called subobject classifier, has the following property: Comprehension axiom. For each S ϕ : O −→ S Ω there is an equalizer of S ϕ and S trueO , and each monic m : M ֌ O is such an equalizer for a unique S ϕ. In diagrams, S true is such that for every S ϕ and every object T and morphism o : T −→ O, if m ◦S ϕ = m ◦S trueO and x ◦S ϕ = x ◦S trueO , then there is a unique h : X −→ M that makes the diagram below commutative: M > m > Sϕ O < ∧ h > Ω > S S trueO x X Example 2.1. Let Set be the (standard) category of (abstract constant) sets as objects and functions as morphisms. S ΩSet has only two elements with the order S f alseSet < S trueSet . Hence, in this category S ΩSet = 2S Set . Thus, for every element t of O, t : 1 −→ O, t ∈ O if and only if S ϕ ◦ t = S trueSet , and t ∈ / O if and only if S ϕm ◦ t = S f alseSet , since S f alseSet is the only morphism distinct from S trueSet . Example 2.2. S S ↓↓ is the category of (standard irreflexive directed multi-) graphs and graph structure preserving maps.3 An object of S S ↓↓ is any pair s of sets equipped with a parallel pair of maps A t > >V where A is called the set arrows and V is the set of dots (or nodes or vertices). If a is an element of A (an arrow), then s(a) is called the source of a, and t(a) is called the target of a. Morphisms of S S↓↓ are also defined so as to respect the graph structure. s That is, a morphism f : (A t 3 Nice > >V s′ ) −→ (E t′ > >P) in S S↓↓ is defined to be any introductions to this category can be found in [31] and [17]. 198 LUIS ESTRADA-GONZÁLEZ pair of morphisms of Set fa : A −→ V , fv : E −→ P for which both equations fv ◦ s = s ′ ◦ fa f v ◦ t = t′ ◦ f a are valid in S Set. It is said that f preserves the structure of the graphs if it preserves the source and target relations. A terminal object in this category, 1S S ↓↓ , is any arrow such that its source and target coincide. This topos provides a simple yet good example of a truth values object with more than two elements. S ΩS ↓↓ has the form of a graph like that in Figure 1 above. There are exactly three morphisms 1S S ↓↓ −→ S ΩS ↓↓ in this category, which means that S ΩS ↓↓ has three truth values with the order S f alseS ↓↓ < S(st )S ↓↓ < StrueS ↓↓ . The Comprehension axiom enables us to define the more usual standard connectives as operations on S Ω, morphisms of the form k : (S Ω×. . .× .. .S ΩX .. SΩ −→ S Ω (with S Ω × . . . × S Ω n times and S Ω . t times, S Ω) n, t ≥ 0), in a way that they come with certain truth conditions.4 For the purposes of this paper only the following ones are needed: Negation. ¬ : S Ω −→ S Ω ¬p = S true if and only if p = S f alse, otherwise ¬p = S f alse Disjunction. ∨ : S Ω× S Ω −→ S Ω (p ∨ q) = sup(p, q) Implication. ⇒: S Ω× S Ω −→ S Ω (p ⇒ q) = S true if and only if p ≤ q, otherwise (p ⇒ q) = q 5 The internal logic of a standard topos is the algebra of operations of S Ω, that is, the algebra of operations of its object of truth values. In general, S Ω has more than two elements, and that is why it is said that the internal logic of a topos is in general many-valued. There is a theorem establishing necessary and sufficient conditions for a proposition S p being the same morphism as S true in a given standard topos S E. Let ‘|=I ’ indicate that logical consequence deploys theorems and valid inferences as in intuitionistic logic. Then the following theorem holds: SΩ 4 For details see for example [12, § 6.6]. and ≤ here are relative to the partial order formed by the elements of phisms with codomain S Ω). 5 sup SΩ (mor- DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC199 Theorem 2.3. For every proposition S p, |=S E S p for every topos S E if and only if |=I S p. i.e. S Ω is a Heyting algebra.6 Hence the commonplace that the internal logic of a topos is, in general, intuitionistic. Example 2.4. The internal logic of S Set is classical. For example, in S Set, every proposition p is the same as one and only one of S trueSet and S f alseSet . ¬ ◦S trueSet =S f alseSet and ¬ ◦S f alseSet =S trueSet . Hence, for any p, ¬¬p = p. Also, for any p (p ∨ ¬p) = ∨ ◦ hp, ¬pi = sup(p, ¬p) =S trueSet . Example 2.5. In S S ↓↓ , negation gives the following identities of morphisms: ¬ StrueS ↓↓ = Sf alseS ↓↓ , ¬ S(st )S ↓↓ = f alseS ↓↓ , ¬ Sf alseS ↓↓ = StrueS ↓↓ Since (p ⇒ q) = S true if and only if (p∧q) = p, in general (¬¬p ⇒ p) 6= S true in S ↓↓ because even though (¬ ¬ p ⇒ p) = S trueS ↓↓ either when p = S trueS ↓↓ or when p = S f alseS ↓↓ , (¬ ¬ p ∧ p) 6= ¬ ¬ p when p = S (st )S ↓↓ . Given that (¬ ¬ p ⇒ p ) 6= S trueS ↓↓ but there is no formula Φ such that Φ = true in classical logic and Φ = f alse in intuitionistic logic, (¬¬p ⇒ p) = S (st )S ↓↓ when p = S (st )S ↓↓ . Moreover, p ∨ ¬p fails to be the same morphism as S trueS ↓↓ since (p ∨ q) = S true if and only if either p = S true or q = S true. If p = S (st )S ↓↓ , ¬p = S f alseS ↓↓ , so neither p = S trueS ↓↓ nor ¬p = S trueS ↓↓ and hence (p ∨ ¬p) 6= S trueS ↓↓ . Logical consequence in a topos is assumed to be traditional, Tarskian consequence: q is a (Tarskian) logical consequence of premises Γ, in symbols Γ |=T q, if true is preserved from premises to the conclusion and is not a consequence if the premises are the same morphism as true but the conclusion is not. A theorem is a consequence of an empty set of premises, i.e. if it is a morphism which is the same morphism as true. A non-theorem is a morphism which is different from true. But the two values true and not true (or untrue, etc.) are the only values required to define (Tarskian) consequence. The internal logic of complement-toposes (cf. [23], [24]) is Tarskian, too. Even though the subobject classifier and the connectives are described in a different, dual way, the notion of consequence in the internal logic of complementtoposes is the same as that of (ordinary or standard) toposes. Therefore, the subsequent discussion can be cashed in terms of toposes simpliciter, ignoring for the rest of the paper whether they are standard or not unless otherwise indicated. 6 In rigor, p is a morphism which corresponds to a formula ( p)∗ in a possibly different S S language, but there is no harm if one identifies them, hence the abuse of notation. A proof can be found in [12, see §8.3 for the soundness part and §10.6 for the completeness part]. 200 3 LUIS ESTRADA-GONZÁLEZ Suszko’s reduction Let F be a non-empty collection of, say, formulas, and let A be a non-empty collection of truth values. Let us assume A = D+ ∪ D− for suitable disjoint collections D+ = {d1 , d2 , . . .} and D− = {a1 , a2 , . . .} of designated and antidesignated values, respectively. A semantics over F and A is said to be any collection sem of mappings σ : F −→ A, called n-valuations (where ‘n’ corresponds to a suffix, uni-/bi-/tri-, and so, according to |A|, the cardinality of A). Given some n-valued semantics sem and some formula ϕ ∈ F, it is said that one has a model for ϕ when there is some σ ∈ sem such that σ(ϕ) ∈ D+ ; when this is true for every σ ∈ sem it is said that ϕ is validated. A notion of entailment given by a consequence relation |=sem ⊆ ℘(F)XF associated to the semantics sem is then defined by saying that a formula ϕ ∈ F follows from a set of formulas Γ ⊆ F whenever all models of all formulas of Γ are also models of ϕ, that is: Γ |=sem ϕ iff for all σ ∈ sem, σ(ϕ) ∈ D+ whenever σ(Γ) ⊆ D+ I will omit the semantics in the index whenever it is clear from the context. Clauses like Γ, ϕ, ∆ |= ψ are written to denote h Γ ∪ ϕ ∪ ∆, ψ i ∈ |=; such clauses can be called inferences. It was early remarked by Tarski that the above notion of consequence may be abstractly axiomatized as follows. For every Γ, ϕ, ∆, ψ: (T1) ϕ ϕ (Reflexivity) (T2) If ∆ ϕ then Γ, ∆ ϕ (Monotonicity) (T3) If Γ, ϕ ψ and ∆ ϕ then Γ, ∆ ψ (Transitivity) A logic L can be defined simply as a structure (typically a set of propositions, formulas, sentences or something similar) together with a consequence relation defined over it. Logics respecting axioms (T1)-(T3) are called Tarskian. Notice, in particular, that when sem is a singleton, one also defines a Tarskian logic. A theory will be any subset of a logic. A logic given by some convenient structure and a consequence relation is said to be sound with respect to some given semantics sem whenever Γ ϕ implies Γ |=sem ϕ (that is, ⊆ |=sem ); and is said to have a complete semantics when Γ |=sem ϕ implies Γ ϕ (that is, |=sem ⊆ ). A semantics which is both sound and complete for a given logic is often called adequate. In case a logic L is characterized by some n-valued semantics, it will be dubbed a (general) n-valued logic, where n = |A|; L will be called finite-valued if n<ℵ0 , otherwise it will be called infinite-valued. Note that, as a logic may have different semantical presentations, the ‘n’ in n-valued is not necessarily unique for L. Given a family of logics {Li }i∈I , where each Li is ni -valued, it is also said that the logic given by L = ∩i∈I Li is Maxi∈I (ni )-valued (this means that an arbitrary intersection of Tarskian logics is still a Tarskian logic). Suszko distinguished between the logical values on the one hand, and algebraic values on the other hand. According to Suszko, many-valued logics DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC201 resulted from a purely referential phenomenon, i.e. from the fact that when one defines homomorphisms between a logic L = hF, |=i and one of its models, one can associate with an element of F = |F| any number of algebraic values, for these homomorphisms are in fact merely admissible reference assignments for F . Said otherwise, they are algebraic valuations of L over the model. Logical values would play a different role, since they are used to define logical consequence. One of them, TRUE, is used to define consequence as follows: If every premise is T RU E, then so is (at least one of) the conclusion(s). By contraposition, the other logical value also can be used to explain valid semantic consequence: If the (every) conclusion is NOT TRUE, then so is at least one of the premises. Theorem 3.1 (Suszko’s Reduction). Every Tarskian logic is logically twovalued. Proof. For any n-valuation σ of a given semantics sem (n), and every consequence relation based on An and Dn+ , define A2 = {T RU E, N OT T RU E} and D2+ = {T RU E} and set the characteristic total function bσ : F −→ A2 to be such that bσ (ϕ) = T RU E if and only if σ(ϕ) ∈ D+ 2 . Now, collect all such bivaluations bσ ’s into a new semantics sem(2), and notice that Γ |=sem(2) ϕ if and only if Γ |=sem(n) ϕ. Reductive results similar in spirit to Suszko’s were presented independently by other logicians, for example Newton da Costa (see e.g. [13]), Dana Scott (cf. [27], [28]) and Richard Routley and Robert K. Meyer [26]. Moreover, there is a family of akin results of different strengths under the label “Suszko’s reduction”. Suszko’s reduction in rigor, required from the logic not only to be reflexive, transitive and monotonic, but also structural. Suszko-da Costa’s reduction, which is closer to what I expound here, dropped the structurality requisite. Suszko-Béziau’s reduction only requires reflexivity from the logic (cf. [29]). Suszko declared that many-valuedness is “a magnificent conceptual deceit” and he claimed that “(. . . ) there are but two logical values, true and false (. . . )”. This claim is now called Suszko’s thesis and can be stated more dramatically as “All logics are bivalent” or “Many-valued logics do not exist at all”. Reductive results, especially the strongest form (Suszko-Béziau’s reduction), seem to be overwhelming evidence in favor of Suszko’s thesis because virtually all logics regarded as such are in the scope of these theorems. A possible way to resist Suszko’s thesis is by extending the scope of logics to cover non-Tarskian logics, especially to non-reflexive ones to avoid SuszkoBéziau’s reduction, and this reply is what I will discuss more extensively after discussing reductions in a categorial setting. 202 4 LUIS ESTRADA-GONZÁLEZ Suszkoing toposes. . . These reductive results can be given categorial content. The internal logic of a topos is said to be algebraically n-valued if there are n distinct morphisms 1 −→ Ω in the given topos. As reductive results have shown, an algebraically n-valued Tarskian logic in general is not also logically n-valued. Accordingly, the internal logic of a topos is said to be logically m-valued if its notion of consequence implies that there are m distinct logical values. The internal logic of a topos, whether standard or complement, is a Tarskian logic, and this means that it is in the scope of Suszko’s theorem. Such internal logic is defined, as usual, by distinguishing between those propositions that are the same morphism as true and the other ones, no matter what or how many algebraic truth values there are, i.e. what is playing the leading logical role is the bipartition true and not true, independently of the number of algebraic values (elements of Ω). So, provided that the notion of logical consequence is that usually assumed, and it is to deliver intuitionistic logic in the standard case and a dual of this in the complement case, the internal logic of a topos is logically bivalent. The logical m-valuedness of the internal logic of a topos can be itself internalized just in case it can be replicated appropriately in terms of morphisms and compositions of the topos itself. I study here first the case of internalizing m-valuedness when m = 2 and suggest a more general definition in section 6. Definition 4.1. In a non-degenerate category C with (respectively, dual) subobject classifier, a Suszkian logical truth values object, or Suszkian object for δ+ short, is an object S such that there are exactly two morphisms 1 δ− > >S and a morphism sep : Ω : C −→ S such that sep is the unique morphism which satisfies the following properties: (S1) sep ◦ p = δ + if p = trueC , and (S2) sep ◦ p = δ − if p 6= trueC The morphisms δ + and δ − can be collectively denoted by biv and are called a Suszkian bivaluation. Thus, the diagram below commutes according to the above definition of biv and the conditions (S1) and (S2): 1 p > ΩC sep biv > ∨ S From the very definition of a Suszkian object, for every proposition ϕ, either sep ◦ ϕ = δ + or sep◦ϕ = δ − . Now, for every theorem Φ, sep◦Φ = δ + , and for every non-theorem Ψ, sep ◦ Ψ = δ − . Consider the three truth values DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC203 in S S ↓↓ . Then sep ◦S trueS ↓↓ = δ + and sep ◦S f alseS ↓↓ = sep ◦S (st )S ↓↓ = δ − . Hence, for example, sep ◦ (¬ ¬ p ⇒ p) = δ − , for (¬¬p ⇒ p) =S (st )S ↓↓ when p =S (st )S ↓↓ . Something similar happens with (p ∨ ¬p). As a consequence of the definition, there is no morphism ψ : 1 −→ S such that ψ ∈ δ + and ψ ∈ δ − . However, this does not mean that ∩ ◦ (δ + , δ − ) = ∅. This implies that, in general, S is not isomorphic to 2 in spite of having exactly two morphisms 1 −→ S (nonetheless, it is easily proved that a Suszkian object and Ω are isomorphic for example in Set). Unlike a subobject classifier, a Suszkian object does not necessarily classify subobjects and it does not necessarily count them, either, for it collapses every other proposition different from S true into δ − . A Suszkian object provides a bivaluation biv = sep ◦ p for Ω, i.e. it says whether a proposition is logically true or not, full stop. This justifies the suggested definition of a Suszkian object, but the difficult part is proving the claim that every (non-degenerate) topos has a Suszkian object as defined here. Maybe not all toposes have a Suszkian object as defined here but only those with certain discreteness conditions. However, I only wanted to show that it seems possible to internalize the notions involved in the reductive results. 5 . . . and de-Suszkoing them? There are notions of logical consequence which are not Tarskian and that could introduce interesting complications in the theory of the internal logic of toposes. Consider Consider first Frankowski’s P-consequence (“P” for “Plausible”; cf. [10], [11]): P-consequence. q is a logical P-consequence from premises Γ, in symbols Γ |=P q, if and only if any case in which each premise in Γ is designated is also a case in which ϕ is not antidesignated. Or equivalently, there is no case in which each premise in Γ is designated, but in which q fails to be not antidesignated. Thus logical many-valuedness in a topos could be obtained at a different level, by taking it into account from the very characterization of logical consequence. However, this would result in a change in the description of the internal logic, for it would be no longer intuitionistic. The Tarskian properties are indissolubly tied to the canonical characterizations of consequence, but P-consequence is non-Tarskian: It is not transitive. Let me exemplify how radical the change would be in the internal logic if P-consequence is adopted instead of the Tarskian one. In general, P-consequence does affect the collection of theorems. Since theorems are those propositions which are consequences of an empty set of premises, theorems according to P-consequence are those propositions that are not antidesignated. So theorems of the internal logic are not the same as those when Tarskian consequence is assumed even if trueE is taken as the only 204 LUIS ESTRADA-GONZÁLEZ designated value. For example, let us assume as above that S trueS ↓↓ is the only designated value in S S↓↓ and that S f alseS ↓↓ is the only antidesignated value. p ∨ ¬p would be a theorem in S S↓↓ because there is no case in which it is antidesignated. P-consequence affects also the validity of inferences. Remember that unlike Tarskian consequence, P-consequence is not transitive. Suppose that p = S trueS ↓↓ , q = S (st )S ↓↓ and r = S f alseS ↓↓ . Thus p |=PS ↓↓ q and q |=PS ↓↓ r, S S but p 2PS ↓↓ r, because P-consequence requires that if premises are designated, S conclusions must be not antidesignated, which is not the case in this example. However, in being reflexive P-consequence is in the scope of Béziau’s reduction to bivalence, it does not assure logical many-valuedness. This is not the case with Malinowski’s Q-consequence (“Q” for “Quasi”; cf. [19], [20]), though: Q-consequence. q is a logical Q-consequence from premises Γ, in symbols Γ |=Q q, if and only if any case in which each premise in Γ is not antidesignated is also a case in which q is designated. Or equivalently, there is no case in which each premise in Γ is not antidesignated, but in which q fails to be designated. The changes in the internal logic would be as follows. Theorems are those propositions which are consequences of an empty set of premises, so theorems are propositions that are always designated. This is just the usual notion of theoremhood, but whether Q-consequence affects the collection of theorems depends on what are the designated values, because one has to choose by hand, as it were, what are the designated, antidesignated and neither designated nor antidesignated values. If E true is the only designated value as usual, the theorems of the internal logic are the same whether Tarskian or Q-consequence is assumed. Nonetheless, Q-consequence does affect the validity of inferences even if E true is the only designated value. Unlike Tarskian consequence, Q-consequence is not reflexive. For example, let us assume that S trueS ↓↓ is the only designated value in S S↓↓ and that S f alseS ↓↓ is the only antidesignated value. Suppose that p =S (st )S ↓↓ . Then p 2QS ↓↓ p, because Q-consequence requires S that if premises are not antidesignated, conclusions must be designated, which is not the case in this example. The above are not the only possible changes. Consider the case when designated and antidesignated values form mutually exclusive and collectively exhaustive values; for simplicity take Tarskian logical consequence |=T , which states that if premises are designated then the conclusions are also designated; equivalently, under the preceding assumption on values, if conclusions are antidesignated, premises are also antidesignated. Elaborating on [32], let me isolate and separate these forms of logical consequence: D+ -consequence: q is a logical D+ -consequence from premises Γ, in symbols + Γ |=D q, if and only if any case in which each premise in Γ is designated is DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC205 also a case in which q is designated. This is called forwards preservation (of D+ ). D− -consequence: q is a logical D− -consequence from premises Γ, in symbols − Γ |=D q, if and only if any case in which q is antidesignated is also a case in which some premise in Γ is antidesignated. This is called forwards preservation (of D− ). When the arrangement of values is such that D+ ∪ D− 6= A or D+ ∩ D− 6= ∅, D+ -consequence and D− -consequence do not coincide. Let us consider the category S S↓↓ and let us assume that S trueS ↓↓ is the only designated value and that S f alseS ↓↓ is the only antidesignated value. Then one has + − p ∧ (p ⇒ q) |=D q but p ∧ (p ⇒ q) 2D q. Again, mutatis mutandis, examples similar to the above can be given to show how each notion of consequence modifies the internal logic of a complement-topos.7 6 Is this logical consequence? An obvious worry at this point is whether these strange notions of consequence are notions of logical consequence at all. I sketch four arguments supporting the idea that they do not lead us so far off the usual business of logic.8 First, one of the non-Tarskian notions of consequence is well-known, nonmonotonic logic, but non-reflexive and non-transitive consequence relations are not so popular. However, there is no prima facie reason as to why it is possible to do without monotonicity but not without reflexivity or transitivity: As properties of a relation of logical consequence they seem to be pari passu. Thus, the first and easiest answer to the question of this section is that, at least technically, non-Tarskian notions of consequence such as Q- and P-consequence are as legitimate as non-classical logics are. More elaborate answers could be given along the lines of non-monotonic logics. For example, Q-consequence would serve to “stick” to conclusions more certain than the premises, whereas P-consequence would allow to “jump” to conclusions less certain than the premises. Beall and Restall [2] say that they are not on equal footing “because preservation of designated values (from premises to conclusions) is a reflexive and transitive relation”. This begs the question, for (and this is a second argu7 [32] is a good source of inspiration for other notions of logical consequence. Abstraction on the notions of logical consequence could go further up to a definition of a logical structure analogous to that of an algebraic structure given in Universal Algebra such that other notions of consequence and particular logics appear as specifications of that structure: That is the project of Universal Logic, see [6] for an introduction. However, I will stop generalization here because of limitations of space and because it has been enough to show that the issue of the many-valuedness of topos logic is not as neat as thought in the categorial orthodoxy. 8 This section draws upon my [9]. 206 LUIS ESTRADA-GONZÁLEZ ment for non-Tarskian logics) they are precisely asking for ways of logically connecting premises and conclusions others than preservation of designated values. Third, a good signal that we are not very far of logic is that under minimal classical constraints on the structure of truth values, these notions of consequence are indistinguishable from the traditional one. If there are only two truth values, true and f alse with their usual order, the collections of designated and antidesignated values exhaust all the possible values, designated = not antidesignated and not designated = antidesignated. But if this were a feature merely of classical logic, surely Q- and P-consequence would have arisen before they did. However, these notions of consequence collapse if the collections of designated and antidesignated values are supposed to be mutually exclusive and collectively exhaustive with respect to the total collection of values given, as is assumed in most popular logics. Fourth, these notions of logical consequence satisfy what some of their detractors9 call “the core tradition of logic”, necessary conditions that something to be called “a logic”. According to them, an account of logical consequence must be a tool useful for the analysis of the inferential relationships between premises and conclusions expressed in arguments we actually employ, and logical consequence has at least the features of necessity (certain values of the premises in a valid argument necessitates the that conclusions have certain values), formality (valid arguments are so in virtue of their logical form) and normativity (rejecting a valid argument is irrational). Though nothing is completely uncontentious, there has been hardly any disagreement about this; I do not count disagreements about how to spell out some crucial notions in the above –like formality or normativity, to mention just two– as disagreements with the spirit of these premises.10 Let me discuss first how the strongly non-Tarskian notions of consequence can serve as tools for the analysis and evaluation of arguments we actually employ. It is easy to see that, in spite of the appearances, we are still in the business of logic. If one uses cognitive states like acceptance and rejection (or their linguistic expressions, assertion and denial) to define validity, these different notions of consequence arise almost naturally. For example, Let V be a collection of “dialogical properties” so that one uses, for example, ‘accepted’ for D+ and ‘rejected’ for D− : 9 For example, Beall and Restall: “Non-transitive or non-reflexive systems of ‘entailment’ may well model interesting phenomena, but they are not accounts of logical consequence. One must draw the line somewhere and, pending further argument, we (defeasibly) draw it where we have. We require transitivity and reflexivity in logical consequence.” [2, p. 91, italics in the original] 10 As it has been noted in [25], there might be disagreement on whether normativity, formality and necessity are the only settled features of logical consequence –others might be, say, aprioricity or universality–, but not that they are indeed features of logical consequence. I think that discussing these features will be enough to illustrate my point and dialectically it is enough for the detractors do not require much more than these. DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC207 - An argument is (generalized) Tarskian-valid if and only if, if for every case the premises are accepted, then the conclusions are also accepted. Equivalently, if the conclusions are non-accepted, at least some of the premises are nonaccepted too. The relevant dialogical property is accepted, so there is only one property forwards-preserved. That property determines another property, non-accepted, which makes the collections of properties mutually exclusive and collectively exhaustive. - An argument is Q-valid if and only if, if for every case the premises are nonrejected, then the conclusions are accepted. Equivalently, if the conclusions are non-accepted in some case, at least some of the premises are rejected. The relevant dialogical properties are those that count as non-rejected, but there is only one of them that has to be forwards-preserved: accepted. Accepted and rejected are mutually exclusive, but probably not collectively exhaustive; non-accepted and non-rejected are collectively exhaustive, but probably not mutually exclusive. - An argument is P-valid if and only if, if for every case the premises are accepted, then the conclusions are non-rejected. Equivalently, if the conclusions are rejected in some case, at least some of the premises are non-accepted. The relevant dialogical properties are those that count as non-rejected, since any of them can be preserved from accepted. Again, accepted and rejected are mutually exclusive, but probably not collectively exhaustive; non-accepted and non-rejected are collectively exhaustive, but probably not mutually exclusive. Secondly, necessity. Clearly, designatedness is a case of non-antidesignatedness, so Tarskian consequence is a case of a more general notion which also encompasses Q- and P-consequence: Preservation of non-antidesignatedness. For example, Q-consequence deals with preservation of non-antidesignated values but in such a strong way that it rather forces passing from non-antidesignated values to designated values. Similarly, P-consequence is preservation of nonantidesignated values but in such a weak way that it allows passing from designated values to some non-designated values (but never from designated to antidesignated values!). Let me call this notion of consequence TMFconsequence (for Tarski, Malinowski and Frankowski) and define it as follows: TMF-consequence. q is a logical TMF-consequence from premises Γ, in symbols Γ |=T M F q, if and only if any case in which each premise in Γ is not antidesignated is also a case in which q is not antidesignated. Or equivalently, there is no case in which each premise in Γ is not antidesignated, but in which q fails to be antidesignated. Again equivalently, if there is a case in which q is antidesignated, then at least one premise in Γ is also antidesignated. In short, in the non-Tarskian notions of consequence the non-antidesignatedness of premises necessitates the non-antidesignatedness of conclusions. This does not immediately reduce to bivalence because of the further conditions impossed by P- and Q-consequence. Now, normativity. In a monist framework, with only one logic at play, normativity is not so problematic: There is but one canon saying what is 208 LUIS ESTRADA-GONZÁLEZ rational and what is irrational concerning argumentation. But even under the (generalized) Tarskian framework, there are many logics, each sanctioning as (in)valid different argument-forms as they might be operating on different specifications of cases. So normativity should not be understood from the outset as “what must be done in all cases simpliciter ” (or “in absolutely all cases”), but as “what must be done in all cases of a kind k”. Just as it could be irrational, in spite of what bold Tarskian consequence could say, to reason monotonically in cases in which information does not have the required properties to do so (i.e., it is incomplete, not known but merely probable, etc.), inferring according to Q-consequence (accepting a conclusion on the basis that there is no case of the pertinent kind in which the premises are non-rejected but the premise is non-accepted) in some contexts –whether by time pressure, resource-saving, etc.– might be the rational thing to do, instead of waiting until the information is not merely not-rejected but fully accepted. Ampliative inferences have their contexts of application and establish their own sets of norms for evaluating arguments in all the contexts of the pertinent kind. Thirdly, normativity. It is easy to see that, in spite of the appearances, we are still in the business of logic. If one uses cognitive states like acceptance and rejection (or their linguistic expressions, assertion and denial) to define validity, these different notions of consequence arise almost naturally: - An argument is TMF-valid if and only if, if the premises are not rejected, then conclusions are also not rejected. Equivalently, if the conclusions are rejected, premises are rejected too. The relevant dialogical properties are those that count as not rejected, so there is more than one property that could be forwards-preserved. The preservation of these properties is required, but the direction is not important since not rejected determines another property, whose name depends of what the components of the first property are taken to be, which makes the collections of properties mutually exclusive (but probably not collectively exhaustive). Note that TMF-consequence has a dialogical version too, which illustrates its usefulness for evaluating ordinary arguments: - An argument is TMF-valid if and only if, if the premises are not rejected, then conclusions are also not rejected. Equivalently, if the conclusions are rejected, premises are rejected too. The relevant dialogical properties are those that count as not rejected, so there is more than one property that could be forwards-preserved. The preservation of these properties is required, but the direction is not important since not rejected determines another property, whose name depends of what the components of the first property are taken to be, which makes the collections of properties mutually exclusive (but probably not collectively exhaustive). About formality: It is difficult to say exactly what constitutes the formality of a logic, but still it is fairly innocuous given that the non-Tarskian notions of consequence are generalizations of the Tarskian notion, as I have showed, so if the latter is formal, the former are too. The meanings of no DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC209 other terms than those accepted by the notion of Tarskian consequence are in play here, so there is no threat on formality if the original notion is already formal. Thus, the route of exploring notions of logical consequence other than the Tarskian one provides a way to give Suszko’s thesis up, and a fortiori a way of preserving logical many-valuedness in the internal logic of a topos. 7 Logical m-valuedness A problem at this point is to know whether the non-Tarskian notions of consequence can be internalized in a topos. Without trying to settle that question here, I will probe an idea at least for consequences based on a form of forwards-preservation. I say that a topos is algebraically n-valued if there are n morphisms from 1 to Ω. A topos is said to be logically m-valued if the assumed notion of consequence, , implies that there are m logical values. Logically m-valuedness is internalized in a topos if (1) there is an object V in the topos such that it is the codomain of exactly m morphisms with domain 1 such that to each logical value implied by corresponds one and only one morphism from 1 to V; and (2) there is a unique morphism sep : Ω −→ V such that sep satisfies the following properties: (2.1) for every δi : 1 −→ V there is a p : 1 −→ Ω such that sep ◦ p = δi (2.2) If p q implies that p and q have certain -logical values vi and vj , respectively, then if sep ◦ p = δi , sep ◦ q = δj (where δi corresponds to vi and δj corresponds to vj ). The morphisms δ1 , . . . δm can be collectively denoted by m-val and are called a logical m-valuation (based on ). Thus, the diagram below commutes according to the definition of m-val just given and the conditions (1) and (2): 1 p > Ω sep m−val > ∨ V The morphism δi such that sep ◦ trueE = δi will be called “morphism of designated values” and will be denoted “δ + ”; the morphism δj such that sep ◦f alseE = δj will be called “morphism of antidesignated values” and will be denoted “δ + ”. A similar procedure has to be followed to individuate each additional logical value, if any. Again, an open problem is to determine which kind of toposes are logically m-valued in the sense defined above and to provide a definition that could encompass all toposes. 210 8 LUIS ESTRADA-GONZÁLEZ Conclusions The conceptualization of the internal logic of a topos can be summarized in certain slogans. I have been concerned with three, claiming its manyvaluedness and its intuitionistic character: (S1) ΩE is (or at least can be seen as) a truth-values object. (S2) The internal logic of a topos is in general many-valued. (S3) The internal logic of a topos is in general (with a few provisos) intuitionistic. In this paper I have argued, against part of that common categorial wisdom, that there is no easy answer to the question of how many logical values there are in a topos. At first sight, there seems to be several of them, because of the many elements of ΩE . However, if logical values are those used to define logical consequence, every Tarskian (reflexive, transitive and monotonic) logic is bivalent, and indeed every reflexive logic is bivalent, as proved by several reductive results inspired by Suszko’s ideas on the subject. Given that the usual internal logic of a topos, whether standard or a complement-topos, is Tarskian, it follows directly that such internal logic is logically bivalent, contradicting (S2). I have showed an attempt to internalize such bivalence in a topos through the notion of a Suszkian object. However, the range of such internaliation is an open problem. More importantly, the notion of consequence is assumed externally, i.e. nothing in the categorial structure of toposes obliges us to adopt the Tarskian notion of logical consequence, although its extension is determined by the internal structure. Logical many-valuedness can be recovered from the start if in defining logical consequence we require more than two logical values. Malinowski’s Q-consequence and Frankowski’s P-consequence are examples of notions of logical consequence which can accommodate more than two logical values. However, recovering logical many-valuedness comes with a cost: In the case of standard toposes, for example, the internal logic ceases to be intuitionistic; thus, saving (S2) requires to abandon (S3). This is no surprise because we are actually converting the internal logic into a non-Tarskian logic. Further generalization came with the notions of D+ - and D− -consequence, with which preservation of values from premises to conclusions (forwardspreservation) in general does not give the same results as preservation of values from conclusions to premises (backwards-preservation). The most pressing open problem in this regard is to investigate whether it is possible to internalize P- and Q-consequence, i.e. whether there are “Malinowskian” and “Frankowskian” objects in a topos. In general, it would be desirable to know what objects of m logical values there are in a topos. This is left for further work. DISPUTING THE MANY-VALUEDNESS OF TOPOS LOGIC211 References [1] AWODEY, S. (2006). 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