Inferences and metainferences in ST

Inferences and metainferences in ST February 15, 2019 The logic ST has been proposed to deal with paradoxes of vagueness and with the semantic paradoxes. There is something very distinctive about ST: namely, it is classical logic for a classical language, but it provides ways of strengthening classical logic to deal with paradoxes. For example, the logic ST+ (ST for a language with a transparent truth predicate and self-referential sentences) is a conservative extension of classical logic. That is, ST+ is not only non-trivial, but it has exactly the same valid inferences as classical logic for the T -free fragment. How is this possible? Well, because ST+ preserves all classically valid inferences but not some classical metainferences. The question then arises of exactly which are the metainferences of ST+ . In a recent paper, Eduardo Barrio, Lucas Rosenblatt and Diego Tajer show that ST+ metainferences are closely related to LP inferences. In this paper we review their result and put the connection in a broader context. 1 3-valued logics based on Strong Kleene Let L be a propositional language with the usual connectives: ∧, ∨, ⊃, ¬. Let an interpretation I be a function from propositional letters to {1, 21 , 0}. Interpretations extend to formulas according to the following Strong Kleene scheme: • I(γ ∧ δ) = min(I(γ), I(δ)) • I(γ ∨ δ) = max(I(γ), I(δ)) • I(¬γ) = 1 − I(γ) • I(γ ⊃ δ) = max(1 − I(γ), I(δ)) If we think of logical consequence as necessary preservation of truth, there are two standard ways in which we can define logical consequence over this 1 semantics. First, if being true means taking the value 1 (strict truth), then an argument is valid just in case no interpretation gives all premises the value 1 and all conclusions a value less than 1. Second, if being true means taking a value greater than 0 (tolerant truth), then an argument is valid just in case no interpretation makes all premises greater than 0 and all conclusions equal to 0. Each definition leads to a familiar 3-valued logic. The first to the Strong Kleene logic K3 and the second to Priest’s Logic of Paradox LP. K3 is paracomplete in the sense that 2K3 A ∨ ¬A and LP is paraconsistent in the sense that A ∧ ¬A 2LP . Furthermore, K3 is “classical on the left” but empty on the right in the sense that for any sets of formulas Γ and ∆:1 Γ K3 ∅ iff Γ CL ∅ and ∅ 2K3 ∆. In words: Γ is K3-unsatisfiable just in case it is classically unsatisfiable and no set ∆ is K3-valid. Similarly, LP is empty on the left but classical on the right in the sense that, Γ 2LP ∅ and ∅ LP ∆ iff ∅ CL ∆. In words: not set Γ is LP-unsatisfiable but for any ∆, it is LP-valid just in case it is classically valid. The definitions of logical consequence for K3 and LP are based on the idea that logical consequence should preserve designated values from premises to conclusions. By liberalizing the notion of logical consequence – from the idea of preserving the same set of designated values from premises to conclusions to the idea of going from one designated set to another set from premises to conclusions – we arrive, in the present context, at the following two alternative definitions of logical consequence (see Cobreros et al. 2015): • An argument is ST-valid if there is no interpretation giving all premises the value 1 and all conclusions the value 0. • An argument is TS-valid if there is no interpretation giving all premises a value greater than 0 and all conclusions a value less than 1. The logics are ordered by inclusion (see Figure 2): when the language contains no constants for the values 1, 21 and 0, TS is empty and ST is classical 1 Assuming the language does not contain constant propositions like ⊤ or ⊥. 2 ST K3 LP TS Figure 1: Four 3-valued logics logic. K3 and LP are both weaker than ST and stronger than TS. K3 and LP are each other’s duals while TS and ST are self-dual.2 The logics K3, LP, ST and TS have been vindicated as suitable logics to deal with different paradoxes.3 For the Liar paradox, in particular, let L+ be L extended with the constant propositions ⊤ and ⊥, a truth-predicate T and a name-forming device hi. We assume L+ is capable of self-reference so that there are Liar-like sentences such as, λ ¬T hλi = A Kripke-construction (see Kripke 1975) shows how to find 3-valued models for L+ where the truth predicate T is transparent (that is, I(A) = I(T hAi) for any formula A) and where Liar-like sentences such as λ get value 12 . Call X+ the logic defined over these models in the way we defined the corresponding X logic above. The logics are ordered by inclusion. Double bars indicate conservative extensions, that is, the logic X+ is the logic X for the T -free fragment. This means that ST+ is a conservative extension of classical logic with a transparent truth-predicate and self-reference. This means that ST+ does not add or subtract valid inferences for the language without T . Further, ST+ is an inference extension of classical logic, in the sense that if an inference is valid in the language without T , it is valid for any uniform substitution on the language containing T .4 These facts ground our claim that ST+ preserves classical logic (Cobreros et al., 2013, 853). Two logics X and Y are duals just in case Γ X ∆ iff ¬∆ Y ¬Γ (where ¬Γ is shorthand for {¬x | x ∈ Γ}). Duality will play a role in section 3. 3 See, for example, Kripke 1975; Tye 1994 for K3, Priest 1979 for LP, Cobreros et al. 2012 for ST and French 2016; Nicolai and Rossi 2016 for TS. 4 See Ripley (2012) ,Cobreros et al. (2013), and Cobreros et al. (2015) for more on ST+ . 2 3 ST+ K3+ ST LP+ K3 TS+ LP TS Figure 2: Four 3-valued logics and their T -extensions 2 Metainferences We can think of a consequence relation extensionally as a set of pairs of sets of formulas. As with any other relation, we may be interested in different properties the relation might exhibit. We can wonder whether the relation is, say, Euclidean. That is, we can wonder whether the statement, ∀x∀y∀z((xRy ∧ xRz) ⊃ yRz) is true for the consequence relation R. As an example, take R to be LP’s consequence relation. Then the statement above is certainly false since, for example, h{A ∧ B}, {A}i ∈ LP and h{A ∧ B}, {B}i ∈ LP but h{A}, {B}i ∈ / LP  . Instead of a universally quantified statement like the one above, we will use propositional letters, under the understanding that a metainference holds just in case it is true for any uniform substitution of the propositional letters. Thus, the metainference above will be written, A ⊢ B; A ⊢ C =⇒ B ⊢ C.5 In addition to the above kind of properties, which can appropriately be called ‘structural’, we may be interested in more specific questions about our consequence relation R. We can ask whether some properties hold when we consider specific kinds of formulas (we will omit set notation in the evident way). For example, we can ask, given a formula A entailing a formula of the form B ∧ ¬B, whether A entails an arbitrary formula C (a non-structural property we might call ‘explosion’): A ⊢ B ∧ ¬B =⇒ A ⊢ C 5 Note that, assuming we are talking about finite sets of formulas, nothing is lost since an argument with multiple premises is equivalent to the argument conjoining the premises, and an argument with multiple conclusions is equivalent to the argument disjoining the conclusions. 4 The metainference holds just in case it is true for any uniform substitution of its propositional letters (this way we capture the idea that we are talking about kinds of formulas). If we take again LP’s consequence relation, the property above won’t hold, since, for example, A ∧ ¬A LP A ∧ ¬A but A ∧ ¬A 2LP B. The result in Barrio et al. (2015) shows a correspondence between ST’s consequence exhibiting a metainferential property and LP validating a specific argument. We introduce some definitions that will be used below. We restrict our attention to the propositional fragment of L+ with propositional letters: A, B, C, D...; the usual constants ∧, ∨, ⊃, ¬; plus the constant propositions ⊤, ⊥ and λ. We leave out propositions involving truth and quantifiers (compare Barrio et al. 2015, 558 Definition 6). A metainference is a conditional statement of the following form: (MI) Γ1 ⊢ ∆1 ; . . . Γn ⊢ ∆n =⇒ Γ−1 ⊢ ∆−1 ; . . . Γ−k ⊢ ∆−k Where the Γ’s and ∆’s are sets of formulas of L+ . An ST+ -instance of a metainference is a uniform substitution of propositional letters in the metainference by formulas of L+ and all turnstile symbols by the double turnstile symbol ST+ superscripted. We will say that a metainference holds for ST+ when all its ST+ instances are true. For example, A ⊢ B ∧ ¬B =⇒ A⊢C =⇒ λ ST ⊥, is a metainference and, + λ ST A ∧ ¬A + is an ST+ -instance of it. This particular instance is true since it is true that + λ ST ⊥. The metainference, however, does not hold for ST+ (as shown in Example 1 below). 3 ST+ -metainferences and LP-inferences The first collapse result of Barrio et al. shows the following short of connection between ST+ -metainferences and LP-inferences, 5 Γ1 ⊢ ∆1 ; . . . Γn ⊢ ∆n =⇒ Γ−1 ⊢ ∆−1 ; . . . Γ−k ⊢ ∆−k holds for ST+ just in case V Γ1 ⊃ W ∆1 , . . . , V Γn ⊃ W ∆n LP V Γ−1 ⊃ W ∆−1 , . . . , V Γ−k ⊃ W ∆−k (cf. Barrio et al. 2015, 557) Intuitively: a metainference holds for ST+ exactly when the result of “lowering” arguments to conditionals and substituting the metalinguistic conditional by a consequence relation symbol renders an argument that is LP-valid. For readability we will give the proof for the simple case with a single inference on either side of the conditional (see Theorem 1). Before that, though, we introduce a definition and a lemma. Definition [Fixation] Let γ be a L formula and I a 3-valued interpretation. The fixation of γ and I, written fI (γ), is the result of substituting each propositional variable A in γ according to the following rule: • if I(A) = 1 then fI (A) = ⊤ • if I(A) = 1 2 then fI (A) = λ • if I(A) = 0 then fI (A) = ⊥ Since a fixation is made up of constant propositions, it’s invariant across interpretations. We will write [[fI (γ)]] to talk about the truth-value of fixation fI (γ). Naturally, I(γ) = [[fI (γ)]]. + Lemma. I(γ ⊃ δ) > 0 iff fI (γ) ST fI (δ) I(γ ⊃ δ) = 0 iff [[fI (γ ⊃ δ)]] = 0 iff [[fI (γ)]] = 1 and [[fI (δ)]] = 0 iff + fI (γ) 2ST fI (δ) 6 In words, a conditional is LP-true if and only if the fixation of its antecedent ST+ -entails the fixation of its consequent (see Cobreros et al. 2015).6 Theorem 1. The metainference Γ ⊢ ∆ =⇒ Γ′ ⊢ ∆′ holds for ST+ V if and only if W V W Γ ⊃ ∆ LP Γ′ ⊃ ∆′ Proof. (Right to left) Suppose Γ ⊢ ∆ =⇒ Γ′ ⊢ ∆′ does not hold for ST+ . Then there is some + + ST+ -instance such that Γi ST ∆i and Γ′i 2ST ∆′i . There is, therefore, some interpretationVI for which I(γ) W = 1 for all γ ∈ Γ′i and I(δ) = 0 for all δ ∈ ∆′i . Thus, I( Γ′i ) = 1 and I( ∆′i ) = 0 and so V W + I( Γ′i ⊃ ∆′i ) = 0. On the other hand, since Γi ST ∆i , V either I(γ) W < 1 for some γ ∈ Γi or I(δ) = 0 for some δ ∈ ∆ Vi ; in either W case I( V Γi ⊃W ∆i ) > 0. The interpretation I then shows that Γi ⊃ ∆i 2LP Γ′i ⊃ ∆′i . Since Γi , ∆i , Γ′i and ∆i are instances of Γ, ∆, Γ′ and ∆ respectively, V substitution W V W we have also that Γ ⊃ ∆ 2LP Γ′ ⊃ ∆′ Proof. (Left to right) V W V ′ W ′ LP Suppose Γ ⊃ ∆ 2 Γ ⊃ there is an interpretation I such V W V ′ ∆ .WThen ′ that I( Γ ⊃ ∆) > 0 andVI( Γ ⊃ ∆W) = 0. By our V Lemma+above, W the + fixation fI is such that, fI ( Γ) ST fI ( ∆) and fI ( Γ′ ) 2ST fI ( ∆′ ). Note finally that the fixation is a uniform substitution of propositional letters in Γ, ∆, Γ′ and ∆′ so that the last is an ST+ -instance showing that the metainference Γ ⊢ ∆ =⇒ Γ′ ⊢ ∆′ does not hold for ST+ . The connection gives us a decision procedure to check whether a metainference is ST+ valid out of a decision procedure to test whether an inference is LP-valid. We use here the trees in Cobreros et al. (2012). Example 1. A ⊢ B ∧ ¬B =⇒ A ⊢ C does not hold in ST+ + A similar reasoning establishes that I(γ ⊃ δ) = 1 iff fI (γ) TS fI (δ), that is, a conditional is K3-true if and only if the fixation of its antecedent TS+ -entails the fixation of its consequent. This way, the relations between ST+ , LP and K3 below can be straightforwardly extended to TS+ . 6 7 A ⊃ (B ∧ ¬B), t ¬(A ⊃ C), s A, s ¬C, s Proof. ¬A, t ⊗ B ∧ ¬B, t B, t ¬B, t ⇑ The countermodel for the LP-inference is I(p) = 1, I(q) = 12 and I(r) = 0. Thus, the fixation showing that the metainference does not hold is: + ⊤ ST λ ∧ ¬λ 4 BUT + ⊤ 2ST ⊥ ST+ meta-anti-inferences and K3 inferences The key ingredient in the connection above between LP inferences and ST+ metainferences is the observation in the Lemma that a conditional is LP-true in an interpretation just in case the fixation of its antecedent attending to that interpretation ST+ -entails the fixation of its consequent attending to the interpretation: I(γ ⊃ δ) > 0 + fI (γ) ST fI (δ). iff This statement is equivalent to the statement that a conditional is LP-untrue in an interpretation just in case the fixation of its antecedent attending to that interpretation does not ST+ -entail the fixation of its consequent attending to the interpretation: I(γ ⊃ δ) = 0 + fI (γ) 2ST fI (δ). iff which in turn, informally reads: a conditional is K3 false in an interpretation (i.e., its negation is K3-true) just in case the fixation of its antecedent attending to that interpretation does not ST+ -entail the fixation of its consequent attending to the interpretation. So there is a connection between a conditional being K3-false and an argument being ST+ -invalid. A meta-anti-inference is a conditional statement relating kinds of antiinferences to kinds of anti-inferences, that is, a meta-anti-inference is a statement of the following form (simple case): 8 Γ 0 ∆ =⇒ Γ′ 0 ∆′ Where the Γ’s and ∆’s are sets of formulas of L+ . The following corollary follows from Theorem 1 with the observation that K3 and LP are duals. Corollary. The meta-anti-inference Γ 0 ∆ =⇒ Γ′ 0 ∆′ holds for ST+ if and only V W V W Γ ∧ ¬ ∆ K3 Γ′ ∧ ¬ ∆′ . + . Then, contrapositively, Proof. Suppose Γ 0 ∆ =⇒ Γ′ 0 ∆′ holds for ST V W V W Γ′ ⊢ ∆′ =⇒ ΓV⊢ ∆ holds ST+ .VBy Th.W1, Γ′ ⊃ ∆′VLP ΓW⊃ ∆. W forK3 By Γ ⊃ ∆)  ¬( Γ′ ⊃ ∆′ ), which is Γ ∧ ¬ ∆ K3 V duality, W ¬( ′ ′ Γ ∧¬ ∆. Like in the previous case, a procedure to decide for K3-validity can be used as a procedure to decide whether a given meta-anti-inference holds for ST+ . Example 2. A 0 C =⇒ A 0 B ∧ ¬B does not hold in ST+ A ∧ ¬C, s ¬(A ∧ ¬(B ∧ ¬B)), t A, s ¬C, s Proof. ¬A, t ⊗ ¬¬(B ∧ ¬B), t B ∧ ¬B, t B, t ¬B, t ⇑ The countermodel for the K3-inference is I(p) = 1, I(q) = 21 and I(r) = 0. Thus, the fixation showing that the meta-anti-inference does not hold is: + ⊤ 2ST ⊥ BUT + ⊤ ST λ ∧ ¬λ. Our corollary from Theorem 1 shows that ST+ is connected to K3 in much the same way it is connected to LP; at least under the assumption that 9 metainferences and meta-anti-inferences are important alike. This will be discussed in section 7. 5 Hybrid metainferences We saw that there is a connection between an inference being ST+ -valid and a conditional being LP-true, and an analogous connection between an inference being ST+ -invalid and a conditional being K3-false. Suppose we are interested in testing a hybrid metainference, that is, a conditional where the antecedent and consequent might contain inferences and anti-inferences alike, (HMI) Γ ⊢ ∆; Γ′ 0 ∆′ =⇒ Γ† 0 ∆† ; Γ∗ ⊢ ∆∗ . We want to know whether there is a uniform substitution of the propositional letters appearing in the hybrid metainference such that: the inference in the premise is ST+ -valid, the anti-inference in the premise is ST+ -invalid, the anti-inference in the conclusion is ST+ -valid, and the inference in the conclusion ST+ - invalid (if any such substitution exists, the metainference does not hold for ST+ ). Here is a natural way to go,7 construct a tree with the following initial list: V W Γ ⊃ V ′ W∆,′t V Γ† ∧ ¬ W ∆†, s VΓ ∗∧ ¬W ∆∗ , s Γ ⊃ ∆ ,t Example 3. A ⊢ B, C ; A 0 B =⇒ C, A 0 B does not hold for ST+ . 7 Notice that the same procedure can be used to decide about TS+ -metainferences by relabeling the nodes of the tree. 10 A ⊃ (C ∨ B), t A ∧ ¬B, s (C ∧ A) ⊃ B, t A, s ¬B, s Proof. ¬A, t ⊗ C ∨ B, t B, t ⊗ C, t ¬(C ∧ A), t ¬C, t ⇑ ¬A, t ⊗ B, t ⊗ The falsifying instance to show that this metainference does not hold for ST+ is A = ⊤, B = ⊥ and C = λ, which renders the well-know counterexample to transitivity in ST+ . 6 Internal vs external logic We briefly discuss in this section a second connection highlighted by Barrio et al. between ST+ and LP. These logics can be presented in proof theoretic terms via 3-sided sequents. From a semantic point of view, the sequent Γ|∆|Σ holds just in case either some element of Γ takes value 0, or some in Σ takes value 12 , or some in ∆ value 1. Barrio et al. describe a proof system S for 3-sided sequents (Barrio et al., 2015, 562) (the details about S don’t matter now). Given the system S, we can give alternative characterizations of what it takes for an argument to be valid. In particular, Definition [ST+ proof theoretic validity] An argument with premises Γ and + conclusions ∆ is proof theoretically valid in ST+ , written Γ ⊢ST ∆, just in case the sequent Γ|Γ, ∆|∆ can be proved in S. Definition [LP proof theoretic validity] An argument with premises Γ and conclusions ∆ is proof theoretically valid in LP, written Γ ⊢LP∆, just in case the sequent Γ|∆|∆ can be proved in S. In its semantic reading, that the sequent Γ|Γ, ∆|∆ can be proved in S means that either some element in Γ takes value less than 1 or some element in ∆ takes value more than 0 (as it should be: this corresponds to the above 11 definition of ST-validity). Similarly, that the sequent Γ|∆|∆ can be proved in S means that either some element in Γ takes value less than 12 or some element in ∆ takes value more than 0. The definitions above correspond to ST+ and LP’s internal logic. It is possible, however, to connect a proof theory like S to a consequence relation in an alternative (“external”) way.8 In the context of a two-sided sequent calculus, Γ externally entails ∆ iff the sequent ∅ ⇒ ∆ can be proved in the system S once we add the sequents ∅ ⇒ γ (for each γ in Γ) as axioms. The general idea is this: Γ externally entails ∆ when the set ∆ can be proved to be valid assuming each element in Γ is valid. The generalization of this idea for LP within our system S of 3-sided sequents is the following, Definition [LP external validity] An argument with premises Γ and conclusions ∆ is externally valid in LP, written Γ ⊢LP e ∆, just in case the sequent ∅|∆|∆ can be proved in S once we add to S the sequents ∅|γ|γ (for each γ in Γ) as axioms. The generalization is natural since it captures the idea that ∆ is valid (LPvalid!) once we assume each element in Γ is. As it turns out, there is no difference between LP’s internal and external logic. In order to assess external validity for ST let us review all options in the first place. Definition [External validities] 1. An argument with premises Γ and conclusions ∆ is externally valid 1 just in case the sequent ∅|∆|∆ can be proved in S once we add to S the sequents ∅|γ|γ (for each γ in Γ) as axioms. 2. An argument with premises Γ and conclusions ∆ is externally valid 2 just in case the sequent ∅|∅|∆ can be proved in S once we add to S the sequents ∅|∅|γ (for each γ in Γ) as axioms. 3. An argument with premises Γ and conclusions ∆ is externally valid 3 just in case the sequent ∅|∆|∆ can be proved in S once we add to S the sequents ∅|∅|γ (for each γ in Γ) as axioms. 4. An argument with premises Γ and conclusions ∆ is externally valid 4 just in case the sequent ∅|∅|∆ can be proved in S once we add to S the sequents ∅|γ|γ (for each γ in Γ) as axioms. Definition 1 corresponds to LP’s external validity, definition 2 to K3. Defini8 The distinction between internal and external logic appear in (Avron, 1988, 163) in the context of Linear Logic and in (Avron, 1991, 110-111) in a broader context under the names “truth” consequence relation and “validity” consequence relation. In the case of propositional modal logic, the “validity” consequence relation turns out to be global validity (see Tranchini and Cobreros (2017) for more on this last). 12 tions 3 and 4 are hybrid versions, the third ST-style and the fourth TS-style. In fact, these two last definitions turn out to be equivalent to (internal) ST and TS respectively. Now Barrio et al. give the following reason to insist that ST’s external consequence should be defined in the style of 1 (and not in the style of 3): Nevertheless, we think that none of these alternative characterizations of external validity [2 to 4] are adequate for ST+ . The reason is that they do not capture what it is for a formula to be valid in ST+ . Recall that the notion of external validity works with formulas, not with arguments, and the valid formulas of ST+ are simply the ones that have a value greater than 0 in every model.(Barrio et al., 2015, 566) Leaving aside the question of whether and how much external validity is relevant in order to assess the merits of ST+ , we think the definition favored by Barrio et al. cannot give the external logic for ST+ . It captures, at best, just half of the story about ST+ ’s external validity. External validity for a logic X is supposed to tell us something relevant about X, that is, about X’s inferences. The following constraint, then, is natural: that anything deemed equivalent by X is also treated as equivalent by X’s external logic. Γ ⊢X e∆ Γ ⊢X ∆ m =⇒ m ′ Γ′ ⊢X e∆ Γ′ ⊢X ∆′ Figure 3: e − constraint The constraint is fully met in the case of, for example, classical first-order logic for closed formulas, since in this case internal and external validity coincide. For let Γ ⊢CL ∆ be logically equivalent to Γ′ ⊢CL ∆′ . Since for classical logic, external and internal logic coincide we have that both Γ ⊢CL ∆ ′ CL ∆′ iff Γ′ ⊢CL ∆′ . Collecting all these ‘iff’s’ we get iff Γ ⊢CL e ∆ and Γ ⊢ e CL ′ CL Γ ⊢e ∆ iff Γ ⊢e ∆′ . LP The constraint, however, doesn’t hold if we assume ⊢ST e is ⊢ , since ST is self-dual while LP is not. Take, as an example, Γ = ⊤ and ∆ = A ∨ ¬A. We have that ⊤ ⊢ST A ∨ ¬A and also that ⊤ ⊢LP A ∨ ¬A but while A ∧ ¬A ⊢ST⊥, we have that A ∧ ¬A 0LP⊥. 13 Γ ⊢LP ∆ Γ ⊢ST ∆ m but 6m ¬(∆) ⊢LP ¬(Γ) ¬(∆) ⊢ST ¬(Γ) Figure 4: e − constraint failure in ST taking LP as external validity In order to restore the equivalence we would need to substitute in one of the sides LP by K3. This, we think, makes K3 as good a candidate as LP to qualify as ST’s external logic. 7 Discussion Barrio, Rosenblatt and Tajer’s result provides an insightful response to the question of which metainferences hold for the logic ST+ . Their result allows us to translate the question of whether some metainference holds in ST+ into a question about whether some corresponding inference holds in LP. In addition to this, Barrio, Rosenblatt and Tajer take the results in their paper to argue for two claims (these claims appear intertwined in the paper): (1) That, contrary to what it is claimed in Cobreros et al. (2013), ST is not classical logic. (2) That ST is no more illuminating than LP as a solution to paradoxes (in slogan: ST is LP in sheep’s clothes). In order to sustain these claims, Barrio et al. take two routes. The first, from a semantic perspective, showing a link between ST’s metainferences and LP’s inferences. The second, from a proof theoretic perspective, claiming that ST’s external logic leads to LP. Note that the second claim is stronger, LP being the paradigm of a non-classical theory. Thus, evidence for the second claim is evidence for the first. It might be, as we will comment in a moment, that no evidence is able to establish anything like claim 1, in this discussion we concentrate on a rebuttal of claim 2. The expression ‘classical logic’ seems to express an unambiguous and precise concept when restricted to a language containing only the usual connectives. When we consider a language equipped with a truth predicate, things become murky. Is it right to claim that ST+ is classical logic for such a language? There is a sense in which the claim is plainly false, since classical logic plus transparent truth leads to the trivial universal consequence relation (and we 14 know ST+ is not that logic). There is another sense in which the claim is right, since ST+ is a conservative extension and an inference extension of classical logic over the language containing the truth predicate (is there a better offer?). In their paper, Barrio, Rosenblatt and Tajer aim to show that more than this is needed to justify the claim that ST+ is classical logic. The fact that some metainferences fail for ST+ is not in itself enough reason to claim that ST+ is somehow weaker than classical logic (or that it fails to preserve something important about classical logic). As we discuss already in (Cobreros et al., 2013, 852), just every time a logic extends another, some metainferences will be lost (unless the logic is extended to the trivial universal logic). If ST+ misses something important from classical logic in the failure of some metainferences, then it must be due to specific metainferences. This is the place where the second claim (“ST+ is LP in sheep’s clothes”) enters the scene. If ST+ ’s metainferences can be characterized as inferences of LP, then ST+ ’s metainferences have indeed a non-classical flavor. While we don’t want to deny that there is something non-classical about ST+ (and this is, actually, much of its attractiveness) we believe Barrio et al. overstate in claiming that ST+ is no more illuminating than LP itself and this claim in turn discredits ST+ ’s classicality which can be understood, in the context of transparent truth, as a golden mean between a paracomplete and a paraconsistent approach. In connection to ST+ ’s external logic (second route towards second claim) our response draws on the plausibility of the e-constraint above about external logic: anything deemed as equivalent by a given logic, should also be deemed as equivalent by its external logic. If this is the case, LP cannot be the external logic of ST+ : either ST+ is systematically ambiguous about its external logic, and then both K3 and LP qualify as ST+ ’s external logic, or neither K3 or LP are ST+ ’s external logic. The argument above shows, we believe, the equidistance of ST+ among LP and K3. In connection to ST+ ’s metainferences (first route towards second claim) our response draws on the connection between ST+ ’s meta(anti)inferences and K3. A conditional A ⊃ B can be read in two directions: as expressing that A is a sufficient condition for B (forwards) or as expressing that B is a necessary condition for A (backwards). The backwards reading can be explicitly represented by contraposing the conditional: ¬B ⊃ ¬A. As with any other conditional, metainferences can also be read in two directions. In the case of metainferences the forwards reading tells us: if such and such inferences are valid, then so are some of these inferences. The backwards reading tells us: if such and such inferences are not valid, then some of these are not valid either. In conversations, we often move fluidly in both directions and some- 15 times which direction is meant can be opaque. However, either direction often highlights an interesting feature of the connection established by the conditional. Consider the following metainference which is a close relative to the Cut rule: A, B ⊢ C ; A, ¬B ⊢ C =⇒ A ⊢ C In its forwards reading it expresses the idea that if we can prove C from A and B and we can prove C from A and ¬B then we can as well prove C from A alone. In its backwards reading it shows like, A 0 C =⇒ A, B 0 C ; A, ¬B 0 C which points out to the idea that if A is consistent with ¬C, then either A + B is consistent with ¬C, or A + ¬B is. The property expressed by each reading is the same by ST+ ’s lights (since ST+ is self-dual and the metalinguistic conditional is contraposable) but each reading highlights something interesting about it. 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