D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . On Logical and Scientic Strength Luca Incurvati & Carlo Nicolai 24th July 2020 Abstract The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following TimothyWilliamson and Gillian Russell, we distinguish between logical and scientic strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientic strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories. 1 Introduction Scientic theories are standardly thought to be selected on the basis of adequacy to the data and how well they fare with respect to a number of theoretical virtues (van Fraassen 1980; Lipton 2004; Keas 2017). One such virtue is strength, which has been discussed in the philosophy of science. This paper provides an account of the notions of logical and scientic strength. Our focus will be on logical and mathematical theories. However, our account is suciently exible to be applicable also in more general contexts, such as scientic theories. Our study is prompted by the recent interest in logical abductivism. This is the view that logical theories should be selected in the same way as scientic theories. Logical abductivism was famously advocated by Quine (1951), Goodman (1955), and Putnam (1968). It has received much attention in the recent literature as a way to navigate the wide array of non-classical solutions to the logical, set-theoretic and semantic paradoxes (Priest 2005; 2016; Williamson 2013; 2017). Logical abductivism promises to provide a way of resolving in a principled manner disputes between rival logics which would otherwise appear hard to settle. Abductivism, so the story goes, replaces clashes of intuition with 1 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . appeal to criteria for theory choice that are accepted by the broader scientic community. For instance, rather than debating the status of paradoxical sentences, one would determine which semantical theory scores better with respect to those criteria. According to the logical abductivist, then, theory choice in logic is no dierent from theory choice in the natural sciences. But the recent revival of interest in abductivism has been associated with the idea that logic is similar to the natural sciences in other respects. This is known as anti-exceptionalism about logic. The anti-exceptionalist may hold, for instance, that logical principles are not analytic or (metaphysically) necessary or a priori (Hjortland 2017). As Gillian Russell (2018) and Stephen Read (2018) have pointed out, however, some form of exceptionalism is compatible with abductivism. Although our focus is on abductivism, our discussion is clearly relevant for any anti-exceptionalist position which embraces an abductive methodology. Abductive methodology has been employed also for theory choice in mathematics. BertrandRussell (1973) advocated the adoption of the regressivemethod to justify mathematical axioms. An abductivist-friendly account is famously given by Gödel (1947), who suggested that set-theoretic axiomsmay be extrinsically justied. More recently, Priest (2006) defended naïve set theory against iterative set theory on the grounds of alleged greater simplicity. A thoroughgoing abductivist approach to the philosophy of set theory has been advanced by Quine (1990: 95), who argues that considerations of simplicity, economy and naturalness sanction the Axiom of Constructibility. Against this, Maddy (1997) uses the maxims Unify and Maximize to instead reject Constructibility as a candidate axiom for a foundation of mathematics. In the philosophy of science, van Fraassen (1980: 67–68) distinguishes between logical and empirical strength. A similar distinction is made byWilliamson (2017) and Russell (2018) under the labels of logical and scientic strength. Roughly speaking, the notion of logical strength of a theory takes into account only its deductive power, whereas the notion of scientic strength has mostly to do with its informational content. There has been some controversy about the status of the criterion of strength in the recent abductivist literature. Williamson thinks that logical and scientic strength are both virtues and that the former entails the latter. Russell accepts that scientic strength is a virtue but criticizes the view that logical strength should be regarded as one. A more radical position, adumbrated by Hjortland 2 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . (2017), holds that logical weakness – and therefore the capability of a logic of drawing more distinctions – is a virtue in a theory. We examine these accounts of the notions of logical and scientic strength and nd them wanting. We suggest understanding logical strength in terms of interpretability strength and scientic strength as a special case of logical strength. The emerging picture contrasts with Russell's analysis in that logical and scientic strength may still be considered to be theoretical virtues, and with Williamson's, in that scientic strength is a special case of logical strength. 2 Logical Strength The aim of this section is to oer a novel account of logical strength. To clear the ground for our account, we rst rebut arguments against the status of logical strength as a theoretical virtue and identify problems with extant accounts of logical strength. 2.1 Logical strength as a theoretical virtue Williamson characterizes logical strength in terms of deductive power. On his account, a theory T is logically stronger than a theory T ∗ just in case every theorem ofT ∗ is a theorem ofT but not vice versa. This can be extended to consequence relations by saying that a consequence relation ` is stronger than a consequence relation `∗ just in case whenever `∗ holds so does ` but not vice versa. Williamson's characterization of logical strength makes it sound as if the comparison of logical theories is a metalinguistic aair. However, Williamson aims to vindicate the idea that it is not. To this end, he considers two strategies for comparing logical theories in a non-metalinguistic way. The rst strategy consists in comparing logics by encoding a logic's consequence relation as a special set of the logic's theorems. First one reduces logically valid arguments to logical truths by replacing the entailment sign by a suitable conditional. Next, one replaces all its non-logical constants with variables of the corresponding type and universally binds them with quantiers of that type. With this reduction in place, comparing logical theories is tantamount to comparing the sets of universal generalizations corresponding to their logical consequence relations. The reduction of logical validity to logical truth makes use of the standard structural rules and of the standard rules for implication (conditional proof and 3 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . modus ponens). However, when evaluating logical theories, we want to consider substructural logics or logics which do not have a suciently strong conditional. Not to prejudge any issue against the non-classical logician, Williamson therefore considers a second strategy for comparing logical theories. According to this strategy, we compare logics by encoding their consequence relation via an operator which takes a set of premises as argument and returns the set of its consequences. Thus, if Γ is a set of sentences, Cn(Γ) is {i | Γ ` i}.1 Comparison of logical theories then proceeds by comparing the dierent Cn(Γ)s to which the logical theories give rise for dierent choices of well-conrmed Γ. The second strategy, however, appears to make theory comparison a metalinguistic aair again, contra Williamson's intentions. For the set Cn(Γ) is individuated via the relation Γ ` i, which is metalinguistic: the elements of Γ and i are mentioned rather than used.2 Williamson claims that logical strength is a theoretical virtue and that this, together with the fact that simplicity too is a virtue, amounts to a prima facie case for classical logic: Once we assess logics abductively, it is obvious that classical logic has a head start on its rivals, none of which can match its combination of simplicity and strength. Its strength is particularly clear in propositional logic, since PC is Post-complete, in the sense that the only consequence relation properly extending the classical one is trivial (everything follows from anything). Recently, Gillian Russell (2018) has challengedWilliamson's claims. She agrees with Williamson's characterization of logical strength but argues that logical strength is neither a theoretical virtue nor a theoretical vice. According to her, if logical strength were a virtue, then, ceteris paribus, if theoryT is logically stronger than theory S,T is better than S. Similarly, if logical strength were a vice, then, ceteris paribus, if theoryT is logically stronger than theory S,T is worse than S. But, she continues, is plainly not the case that, ceteris paribus, a theory is always better o (worse o) by having more (less) of logical strength: a theory can have too much or too little logical strength. Triv, the trivial logic in which any sentence follows from any set of premisses, is too strong: snow is white just does not 1In the current context it does not matter whether we characterize Cn in terms of logical consequence or derivability. Clearly, this matters when one considers logics that are not complete. 2We thank Tim Button for rasing this interesting point. 4 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . entail grass is purple. Ni, the empty logic in which nothing follows from any set of premisses, is too weak: snow is white and grass is green do entail snow is white. This argument will not persuade the defender of logical strength as a theoretical virtue. She can happily grant that if a theory is logically stronger than another theory then, ceteris paribus, it is better; but she will insist that in the case considered by Russell ceteris are not paribus. In particular, Triv is plainly not adequate to the data: by entailing everything, the theory sanctions entailments which contradict our intuitions about, say, grass is green not following from snow is white. Thus, this is just a case, where logical strength is trumped by the fact that the theory is not adequate to the data. As Williamson (2017: 335) puts it: 'First comes t with the evidence'. A similar response is available to the defender of logical strength as a vice: by entailing nothing, Ni fails to be adequate to the data. Russell presents an analogy (which she credits to Dan Marshall). She suggests that saying that logical strength always makes a theory better would be like saying that theories of love on which more people love each other are always better than ones on which fewer people do. But even granting that this analogy works, the defender of strength can accept that such a theory of love will be better other things being equal. However, she will deny that in several specic cases things are equal: the hard reality of romantic life tells us, for instance, that a theory of love on which everybody loves everybody else is hardly adequate to the data. Similar considerations apply to Read's response to Williamson abductivist argument for classical logic. Read begins by observing that classical logic is not the only logic to be Post-complete, as witnessed by the case of Abelian logic. He then writes: A good argument would still ask which logic was the right one: information is not everything, if some of that information is wrong. In the case of Abelian logic, some is indeed wrong: e.g. ((p → q) → q) → p (∗∗) is valid in Abelian logic, but is simply false (as an account of conditionals). But we take it that Williamson would agree with much of this: logical strength is not everything and the case for classical logic is to be understood with the 5 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . proviso that the logic we want ought to also be data adequate. And classical logic's t with the evidence can and has been challenged, e.g. by relevant logicians such as Read. One may consider logical strength a virtue whilst taking t with evidence as another criterion for theory of choice. Indeed, considering logical strength as a virtue is compatible with thinking that this virtue is always trumped by adequacy to the data. In the mathematical context, Maddy comes close to claiming as much. She is arguing in favour of the maxim Maximize, which tells us that we should strive for set theories which are as generous as possible. Maddy is very clear that subscribing to Maximize as a maxim in no way commits one to choosing the most generous of theories- the trivial theory. For, she says, this maxim can be trumped or at least curtailed by other maxims. In particular, she says, 'consistency is an overriding maxim' (Maddy 1997: p. 216). Thus, the idea that logical strength is a virtue remains unscathed. Nonetheless, we do not stake a stance on whether logical and also scientic strength should ultimately be considered as virtues, vices or neither. Instead, our aim is to provide a framework for comparing the strength of theories which can be used by all parties in this dispute. Even so, there are a number of issues with Williamson's characterization of logical strength as inclusion between sets of consequences. First, Williamson's characterization is not immediately applicable to cases in which one deals with dierent languages. In general, on Williamson's characterization, all we can say about the relative strength of two logics featuring disjoint sets of logical constants – such as intuitionistic propositional logic and S4 – is that they are incomparable. On the other hand, our proposal will take into account translations between languages. This will enable us, for instance, to say something informative about the relation between intuitionistic propositional logic and S4, thanks to the so-called Gödel-McKinsey-Tarski translation. Another issue with Williamson's characterization of logical strength concerns its use of the notion of a well-conrmed sentence. The idea is that we can assess a logic by considering Cn(Γ) where Γ is a set of well-conrmed sentences, such as well-established principles of physics. However, in typical cases, whether the members of Γ are well-conrmed or not depends on the background logic of the relevant theory. For instance, whether certain principles of physics can be taken to be well-conrmed depends on whether their consequences t with the data. But what these consequences are, in turn, may de6 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . pend on the background logic. Thus, it is not clear that we can nd adequate Γs which we can take to be well-conrmed independently of the background logic. Finally, Williamson claims that logical strength entails a 'looser notion' of scientic strength, but he does not provide a detailed account of scientic strength and of why such an entailment should obtain. In fact, in what follows we will provide a detailed account of scientic strength and of its relationship with logical strength in which such an entailment will fail. More specically, we will oer a characterization of logical strength based on the notion of translation. This notion will apply to theories formalized in dierent signatures, and so will bemore encompassing thanWilliamson's characterization in terms of inclusion. Second, our characterization will extend more naturally so as to apply beyond the purely logical part of a theory. Finally, our characterization will form the basis of a detailed account of scientic strength. 2.2 Characterizing logical strength In our view, the strength of a theory has to do solely with its deductive power, and in particular with the structure of its derivations. A theory is as strong as another if the former can mimic the inferential structure of the latter. As mentioned, translations allow us to compare theories in dierent languages. We propose to compare the strength of theories in terms of the existence of suitable translations between them. In order to be suitable, translations need to preserve the structure of derivations. In comparing theories with respect with their logical strength, we allow logical and non-logical primitives to be reinterpreted as long as the basic structure of derivations is preserved. Any other notion of strength which demands preservation of information in the logical or non-logical component of theories would not count as logical strength, because it would not abstract away as much as possible from specic content. For instance, our characterization entails that Peano Arithmetic (PA) and PA +¬Con(PA) – where Con(T ) is a canonical consistency statement forT – have equal logical strength. This is essentially because ¬Con(PA) can be translated in PA in a way that preserves its role in derivations while re-interpreting the notion of provability involved in the consistency statement.3 As we will see later on, what distinguishes logical and scientic strength 3For the relevant facts concerning interpretations of inconsistency in reasonable theories containing a modicum of arithmetic, we refer to (Lindström 2003: Ch 7), and in particular (Lindström 2003: Thm. 8). 7 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . of theories is how specic information contained in their logical or non-logical primitives is handled. In particular, in comparing the scientic strength of theories we will impose stricter conditions on how specic information contained in the primitive concepts of theories is preserved under suitable translations. We now implement these ideas into a formal framework for logical strength. We take mathematical theories to be axioms closed under some rules of inference. Logics are then identied with the closure of the empty set of axioms under specic rules of inference. When comparing mathematical theories in classical logic, it is customary to say that a translation from a language L1 to a language L2 consists of an ordered pair g = 〈X , F〉 where X is the domain of the translation and F is a recursive mapping associating each n-ary relation symbol R(y1 , . . . , yn) of L1 with an L2-formula F (R) (y1 , . . . , yn). The translation g commutes with the connectives and X relativizes the quantiers so that, e.g. (∀xi)g := ∀x(X (x) → ig ). A translation g from the language L1 of a theoryT1 to the language L2 of a theoryT2 is then an interpretation ofT1 intoT2 if for every set ofL1-sentences Γ andL1-sentence i, if Γ `T1 i, then Γg `T2 ig (where, as usual, Γ `T i is a shorthand for Γ ∪T ` i, and Γg is {ig |i ∈ Γ}). Finally,T1 andT2 are mutually interpretable ifT1 is interpretable inT2 and vice versa. Given these denitions, we could then characterize logical strength for theories in classical logic by saying that a theoryT1 has greater or equal logical strength than a theoryT2 just in case there is an interpretation ofT1 inT2, and that they have the same logical strength just in case they are mutually interpretable. However, since we aim to deal with mathematical theories formulated in a given non-classical logic as well, we generalize the notion of interpretation above, and call a translation fromL1 toL2 any recursivemapping that associates formulas of L2 with primitive concepts of L1 and that is recursively extended to more complex formulas by suitably commuting with the logical constants. An interpretation is then a translation that preserves provability in a such given logic. Let us consider a few examples that will be relevant also for our later discussion. The proposed characterization entails that the following pairs of theories have the same logical strength: PA + ¬Con(PA) and PA as mentioned, ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) and ZF, PA and ZFFin (ZF with the Axiom of Innity replaced by its negation).4 To mention examples 4The interpretation of nite set theory in arithmetic is due to Ackermann (1937). For the interpretation of the axiom of choice, the classical references areGödel (1948) andCohen (1963). 8 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . of non-classical theories, the theory PAk3(P) – that is Peano arithmetic formulated in the three-valued Strong-Kleene logic K3 and in the language with an additional predicate P , whose interpretation may not be classical – is mutually interpretable (relative to the logic K3) with PAk3(P)+¬Con(PA(P)).5 Hence, the proposed characterization entails that the two theories have the same logical strength. The characterization of logical strength in terms of mutual interpretability provides a precise formal counterpart to the idea that logical strength resides in a theory's capability of mimicking inferential structures, possibly via translations that reinterpret primitive concepts. However, the characterization is not sucient to deal with all cases of comparison of logical strength. For instance, we want to be able to compare pure logics, and in that case we want to reinterpret the logical vocabulary itself, whereas the standard notion of interpretation is designed so as to leave the logical vocabulary alone. Whilst we cannot hope to preserve the meanings of the connectives when translating between logics, it seems that a translation between logics, besides the basic requirement of being recursive, ought at least to (i) be uniform so that, e.g., it is not the case that p∧q is translated as p∨q but r∧s is translated as r → s and (ii) allow going beyond translating each operator with a single operator, e.g. we want to be able to translate, say, p∧q as ¬(¬p∨¬q). A suitable notion of translation is the notion of a schematic translation (Prawitz andMalmnäs 1968;Wojcicki 1988; Pellettier and Urquhart 2003). The general idea is that a translation is schematic if the translation of a complex formula is a xed schema of the translation of its parts. As a result, formulae instantiating the same schema are translated in the same way. So, for instance, if p ∧ q is translated as p ∨ q, then r ∧ s must be translated as r ∨ s. But it is possible to translate p ∨ q as ¬(¬p ∧ ¬q). To dene the notion of a schematic translation, we rst dene the notion of a schema. A schema is a map from formulae (and possibly variables) to the formulae instantiating a schema-string, i.e. an expression featuring metalinguistic variables such as i ∨ k or ∀Ui (Dewar 2018). We say that a translation from the language L1 of a logic L1 to the language L2 of a logic L2 is schematic if it is a recursive mapping g such that (i) each atom p of L1 is assigned a L2 formula, and (ii) for each piece ♠ of logical vocabulary inL1 there is anL2-schema T such that for all sequences i1 , . . . , iW of L1-formulae (♠i1 , . . . , iW )g := 5In particular, in PAk3 (P) P can appear in induction, and the induction principle of PA needs to be formulated as a rule to preserve soundness (Halbach and Nicolai 2018). 9 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . T(ig1 , . . . , i g W ). A schematic translation g from L1 to L2 is sound if for every Γ and i in the language of L1, we have that if Γ `L1 i then Γg `L2 ig . A schematic translation g from L1 to L2 is exact if for every Γ and i in the language of L1, we have that Γ `L1 i if and only if Γg `L2 ig . Schematic translations played a prominent role in the history of logic. Gödel, via the so-called negative translation, showed that there is a (exact) schematic translation of classical logic into intuitionistic logic. In doing so, he established the consistency of classical logic and classical arithmetic (Peano Arithmetic) relative to their intuitionistic counterparts. He also provided the basis of provability logic, justication logic, and Kripke semantics for intuitionistic logic by providing a schematic translation of the latter logic into the modal logic S4. We take sound schematic translatability to be a core component of our account of logical strength. In fact, if we were dealing just with logics, we could simply characterize logical strength by saying that a logic L1 has greater or equal logical strength than a logic L2 just in case there is a sound schematic translation of L2 in L1, and that they have the same logical strength if this holds mutually. One would obtain a dierent notion of logical strength with exact translations instead of sound ones. Although we believe this to be an alternative worth exploring, we here focus on sound schematic translations in order to preserve the intuitive idea that S being a sublogic ofT implies thatT is at least as (logically) strong as S. So far we have only aorded the means of comparing either dierent logics or mathematical theories cast in the same background logic. However, we also want to be able to compare mathematical theories cast in dierent logics. For instance, we want to compare the logical strength of ZF and Heyting Arithmetic (HA), the theory whose axioms are those of PA but whose logic is intuitionistic logic rather than classical logic. This kind of case leads us to our full characterization of the notion of logical strength, which is obtained via a two-stage process and subsumes the characterizations of logical strength that would be suitable in the case of logics or in the case of theories cast in the same logic. Given a theory T1 with logic L1 and a theory T2 with logic L2, the idea is that to determine whether T1 is at least as strong asT2 one rst schematically interprets L2 into L1 and then interpretsT2 (under the logic L1) intoT1. Logical strength T1 is at least as logically strong as T2 i there is a sound schematic translation g of the logic L2 of T2 in the logic L1 10 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . ofT1, and there is an interpretation (relative to the logic L1) ofT g2 inT1. We say that T1 is logically stronger than T2 if T1 is at least as logically strong as T2 but not vice versa. Our denitions entail that, for mathematical theories formulated in classical logic, logical strength coincides with the familiar notion of interpretability strength. More generally, for mathematical theories in a given logic, logical strength coincides with the notion of interpretability strength relative to that logic. Similarly, when comparing purely logical systems, our characterization of logical strength reduces to the existence of a schematic translation, since we are taking logics to be theories with the empty set of non-logical principles. We now discuss some applications of our characterization. We begin by considering cases of comparison between logics. Since schematic interpretability preserves undecidability, it is clear that classical predicate logic is logically stronger than classical propositional logic.6 The Gödel-Gentzen translation (Troelstra and Schwichtenberg 2003: §2.3) is an exact schematic translation of classical logic into intuitionistic logic. Therefore, intuitionist logic can mimic the structure of classical derivations – modulo reinterpreting some logical vocabulary. As a consequence, intuitionistic logic is as strong as classical logic. Moreover, intuitionistic logic is a sublogic of classical logic, and hence it can be trivially (schematically) translated in a sound way into classical logic. Hence, intuitionistic logic and classical logic have equal logical strength. The Gödel-McKinsey-Tarski translation is an exact schematic translation of intuitionistic logic into S4. Hence, S4 is at least as logically strong as intuitionistic logic. Similarly to the previous case, we can also reproduce the structure of S4derivations into intuitionistic logic.7 Thus, S4 and intuitionistic logic have the same logical strength. In the context of comparison between modal logics, by translating A with A ∧ A, one can show that the modal logics K and T have the same logical strength. We now turn to applications of our notions to non-logical axioms. The full power of our characterization of logical strength comes into play when we consider theories formulated in dierent logics. For instance, our notion enables 6If one allows non-eective translations, one obtains the unsound result that classic rst-order logic and classical propositional logic have the same logical strength (Kocurec 2017). 7A sound translation of S4 into intuitionistic logic can be dened as follows. One can employ the 'erasure' translation schema to translate S4 in classical logic, and then employ the GödelGentzen translation. Transitivity of sound translations then gives us the claim. 11 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . us compare ZF to HA. Clearly, there is a sound translation of intuitionistic logic into classical logic such that: HA `IL i⇒ HA `CL i. Then one simply interprets HA in ZF by means of the interpretation that relativizes quantication over natural numbers as quantication over nite ordinals. Since there is no interpretation of PA in ZF, this establishes that ZF is logically stronger than HA. Similar reasoning establishes that intuitionistic ZF (IZF for short) is logically stronger than PA.8 One rst employs the Gödel-Gentzen translation gg to obtain: PA `CL i⇔ PAgg `IL igg Then one would need to show that PAgg, qua subtheory of HA, is interpretable in IZF.9 Since IZF has (much) higher consistency strength than HA, there is no interpretation of the former in the latter theory. The examples just discussed lend support to the adequacy of our characterization of logical strength based on preservation of inferential structure. One of the main advantages of our characterization is its generality. We are able to compare both logics and theories, and various combination thereof. We believe this generality is essential to the abductive comparison of logics and theories. Without the possibility of comparing theories with dierent logical and non-logical primitive vocabulary, there is little hope for logical abductivism to succeed. Yet another advantage of our characterization is that it leads naturally to a precise characterization of scientic strength. It is to this issue that we now turn. 3 Scientic strength In this section we rst discuss Williamson's and Russell's accounts of scientic strength. We then propose our own account. 8IZF is obtained by taking the background logic to be intuitionistic and replacing ZF's Axioms of Foundation and Replacement with the Axioms of ∈-induction and Collection. 9This point requires extra care in dening a suitable notion of interpretation for intuitionistic theories. Our claim is true for reasonable notions of interpretation for intuitionistic theories (Visser 1999). 12 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . 3.1 Williamson and Russell on scientic strength Williamson (2017) holds that logical strength entails a 'looser' notion of scientic strength. For instance, since classical logic proves all instances of i ∨¬i and intuitionistic logic doesn't, the former is logically stronger, but also scientically stronger than the latter: according toWilliamson, a general claim – all instances of excluded middle are valid – is scientically more informative than its negation. Similarly, 'the time between 3:14 and 3:16' is more informative than 'the time between 4:00 and 12:00'. So, although Williamson does not provide a detailed account of scientic strength, both logical form and a certain degree of accuracy are relevant for his view. Russell (2018) rejects Williamson's claim that logical strength implies scientic strength. She does so by distinguishing between two senses of scientic strength. According to the rst, a logic L is scientically strong if it is able to decide, for each argument form in a given language, whether the argument is L-valid or not. In this rst sense, each logic is as strong as another, no matter how dierent they are in logical strength: each logic partitions the set of all argument forms into valid and invalid. Russell describes her second sense of scientic strength as follows: If our question is 'which instances of LL can we use?' (where LL is some disputed logical law) then the logically stronger logic tells us 'all of them' whereas the weaker logic says 'not all of them' – and this tells us nothing further about which particular instances are untarnished (Russell 2018: p. 12). In this second sense the trivial logic Triv is the strongest logic, because to the question 'How many instances of the argument form (Γ, i) can we use?' it answers 'All of them'. Classical logic would then seem to be scientically weaker than Triv, but stronger than, say, its logically weaker sublogics K3, the Logic of Paradox LP, and First Degree Entailment FDE. There are in fact some argument forms (Γ, i) of which, unlike Triv, classical logic can accept only some instances. Similarly, there are familiar argument forms, such as (Γ, i ∨ ¬i) or ({i, ¬i}, k), whose instances are uniformly licensed by classical logic but fail to be so in K3, LP, or FDE. Therefore, it would seem that there is a sense of scientic strength that is entailed by logical strength. However, Russell claims that this conclusion would be hasty: any sublogic ofTriv can be extended to a logic that decides which instances of an argument form are acceptable, and which 13 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . aren't. In other words, each logic can be extended in such a way that, to the question 'How many instances of the argument form (Γ, i) can we use?', it no longer provides the uninformative answer 'Not all of them'. Instead, the question is answered by providing a list of acceptable and unacceptable instances. Russell calls this process 'Triv recapture'. Now any logic that is subject to the procedure of 'Triv recapture' ends up being as informative as another. Since this equally applies to logic with substantially dierent logical strength, Russell concludes that there is no sense of scientic strength that is implied by logical strength. We believe that both accounts of scientic strength oered by Russell are unsatisfactory. We start with Russell's rst account: on this view, all logics are on a par with respect to scientic strength because either Γ L i or Γ 2L i: according to Russell, a well-dened consequence relation is cast in a set-theoretic (classical) metatheory (Russell 2018: p. 557). However, it's clear that under this characterization the specic properties of consequence relations are not relevant at all for their scientic strength. In fact, it is simply a feature of Russell's classical metatheory that excluded middle holds for logical consequence claims. It follows that, as long as a notion of consequence is well-dened, any logic is as strong as it could be. But if the notion of scientic strength is to play any role in abductive methodology, then it should be capable of discriminating at least between some logics. To avoid such an essential dependence on classical metatheory, one might try to generalize Russell's rst denition of scientic strength by requiring that each logic L is as strong as another one by its own light. On this reading, however, Russell's claims cannot be true in general. There is nothing that guarantees that the notion of logical consequence we are employing satises bivalence. For instance, if our metatheory is formulated in a paracomplete setting governed by the logic K3, it won't in general be the case that 'i follows from Γ or it's not the case that i follows from Γ', because the very notion of consequence may be partial (Nicolai and Rossi 2018). Moreover, in such scenario, it would seem that logical strength does indeed in many cases entail scientic strength. For instance, classical logic is able, for each Γ, i, to determine whether Γ  i or Γ 2 i, whereas FDE and K3 cannot. Russell's second sense of scientic strength is based on the notion of Triv recapture: any logic L can be consistently extended to a logic that decides which instances of a given argument form are valid or not. This understanding of sci14 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . entic strength faces serious diculties too. First, it is worth noticing that Russell's Triv recapture is substantially dierent from standard recapture strategies found in the literature on semantic paradoxes. Let us consider the case-study discussed by Russell. If one's language amounts to a formal syntax plus a truth predicateTr , one can provide models of transparent truth –Tr pAq is intersubstitutable with A in every context – that satisfy classical logic for all sentences without Tr . In other words, ifLTr := L∪{Tr } is the language under consideration, one can consistently formulate a logic that satises all classical principles for L and the nonclassical principles for LTr . This is what is often called 'classical recapture' (Field 2008; Beall 2013). However, this form of recapture is not sucient for Russell's purposes. She requires something much stronger – what she calls Small Square Completeness: for any argument form in a given language, one has to be able to decide which instances are licensed and which aren't. For instance, each specic instance of the form Tr piq∨¬Tr piqmust be decided one way or another. This is a hugely complex task. If Tr piq is interpreted via xed-point semantics in the style of Kripke (1975), the problem at hand reduces to a decision procedure for the set of paradoxical, or ungrounded sentences. Unlike the simple syntactic decision problem underlying recapture strategies, already in the simplest Kripkean setting (the minimal xed point) this problem is highly non-eective (Burgess 1986). And these problems become much more complex for more sophisticated constructions such as other Kripkean xed points, the revision extensions inGupta and Belnap (1993), the theory of Field (2008), just tomention a few. Moreover, the complexity of the procedure envisaged by Russell is only going to increase if we move from the specic languageLTr to less rareed languages closer to English. Therefore, the procedure of Triv recapture is simply unmanageable; it is not the case that any logic can be consistently extended to a Small-Square Complete logic, unless by logic we mean extensions of highly non-eective innitary logics whose set of validities is muchmore complex than the provable sentences of any recursively axiomatised theory. 3.2 Characterizing Scientic Strength We now come to our approach to scientic strength. Our proposal shares with Williamson's the idea that scientic strength is more closely related to the informativeness of a theory than logical strength is. Our proposal goes further in that scientic strength is obtained by placing extra conditions on the relation of 15 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . being logically stronger. Thus, scientic strength entails logical strength. Intuitively, logical strength is a coarser grained relation than scientic strength in that it mainly deals with preserving the deductive structure of theories, and hence allows for radical re-interpretation of logical and non-logical vocabulary in derivations. Scientic strength is then obtained by supplementing logical strength with stricter conditions so as to preserve some structural information contained in the theories' primitives. In particular, we no longer allow radical re-interpretations of primitives, but we impose conditions on the preservation, in derivations, of some structural aspects of logical and non-logical constants of theories. For instance, we have seen that PA and PA + ¬Con(PA) have equal logical strength, because the arithmetical primitives used to dene provability in ¬Con(PA) can be re-interpreted by PA in a way that does not entail its inconsistency. However, they will not have the same scientic strength, because our extra conditions on interpretations will require the role in derivations of ¬Con(PA) to be preserved in a much more accurate way. We formally render these ideas by means of the notion of intertranslatability. Intertranslatability is also known as denitional equivalence (Glymour 1970) and synonymy (DeBouvère 1965; Pellettier andUrquhart 2003). Earlier we distinguished between interpretations, which relate theories with non-logical axioms in the same logic, and schematic translations, which relate logics. Analogously, we now dene intertranslatibility as applied to both cases. Logics L1 and L2 are intertranslatable if and only if there are sound schematic translations f from the language L1 of L1 to the language L2 of L2 and g from L2 to L1 such that10 i àL1 (if)g for any formula i of L1; (ig )f àL2 i for any formula i of L2. Similarly, one says that theories S andT in a given logic are intertranslatable if there are interpretations f from S toT , and g fromT to S (with both f and g relative to the given logic) such that i àS (if)g for any formula i of LS; (ig )f àT i for any formula i of LT . 10For an excellent overview of various notions of translations between logics extending sound and schematic translations, including original contributions, we refer to French (2010). 16 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . Since we are dealing both with pure logics and theories featuring non-logical axioms, we again need to characterize scientic strength in terms of a two-step process. Intuitively, the idea behind our characterization is that a theory T (where, recall, logics are limiting cases of theories) is scientically stronger than another theory S if there is some subtheory ofT that can faithfully reproduce the logical and non-logical information contained in the inferential structure of S. The idea of 'faithfully reproducing' is captured in the strict requirement imposed to the translation by the notion of intertranslatibility. In particular, intertranslatability requires that both theories recognize (via provability) that the translations that relate them are 'companion' to each other in the way they process the original information: when the two translations are suitably combined, they return the original information. Scientific strength A theory T1 is scientically as strong as T2 if (i) T1 is at least as logically strong as T2, (ii) the logic L2 of T2 is intertranslatable with a sublogic of L1 – say, with g : L2 → L1 –, and (iii) there is a subtheoryT0 ofT1 such thatT g2 is intertranslatable withT0 (with respect to the logic L1). Condition (i) in the characterization of scientic strength may be dropped in certain, well behaved cases, for instance when we deal with mathematical theories cast in classical logic. However, we chose to keep it in the general case because we aim to provide a template to deal with a large class of logics, for which the notion of interpretation may be underspecied. This makes it dicult to prove that condition (i) is redundant in full generality. We now show that the denition delivers intuitively acceptable verdicts on the comparative scientic strength of theories. We start with examples of theories formulated in the same logic. Since scientic strength entails logical strength, it obviously follows that any theories that do not have the same logical strength do not have the same scientic strength either. For instance, ZFC plus the assertion that there exists a inaccessible cardinal is scientically stronger than ZFC which, in turn, is scientically stronger than PA. For T a reasonable classical theory containing a modicum of arithmetic,T + Con(T ) is logically stronger thanT , and properly so, since T + Con(T ) is not interpretable in T (Lindström 2003: Ch. 7). It is worth noticing thatCon(T ) is aΠ01-sentence of the language of arithmetic, i.e. a 17 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . purely universal claim. In general, the addition of an independent Π01-sentence results in a scientically stronger theory. This last example obviously extends to theories in dierent languages that interpret a suciently strong arithmetical theory. So our characterization of scientic strength vindicates Williamson's claim that a universally quantied sentence adds informativeness to a theory. More generally, our characterization entails that a theory is always scientically as strong as any of its subtheories. However, our notion of scientic strength is also exible enough to accommodate cases of theories that prima facie deal with dierent mathematical domains. One example concerns set theory with and without urelemente. By a result of Löwe (2006), ZF and ZF plus urelemente are intertranslatable. Therefore, they have equal scientic strength. A similar phenomenon concerns ZFC and ZFA (ZFC without Foundation plus Aczel's (1988) Anti-Foundation Axiom). ZFC-sets can be interpreted in ZFA as well-founded sets. ZFA-sets can be interpreted in ZFC as equivalence classes of graphs with lowest rank. Such interpretations yield the intertranslatability of the two theories (Visser and Friedman 2014). This example shows that sameness of scientic strength does not amount to sameness of meaning of the theories' primitives, but only to equivalence with respect to salient aspects of a theory's primitives that are relevant in the structure of derivations. Our characterization also applies to theories formulated in dierent signatures. Consider, for instance, the theory ZFFin. Although this theory is not intertranslatable with PA (Enayat et al. 2011: Thm. 5.1), it becomes so once one adds to it the claim that every set has a transitive closure (Kaye and Wong 2007). Crucially, our analysis of scientic strength yields natural counterexamples to Williamson's implication from logical to scientic strength. A striking example concerns set theory with and without the axiom of choice. ZF and ZFC have the same logical strength but not the same scientic strength. In particular, ZFC is not intertranslatable with ZF. Therefore, ZFC is scientically stronger than ZF (Enayat 2016).11 This nicely ts with the intuition that the addition of the axiom of choice to ZF, although innocent from the point of view of mere consistency strength, results in an increase of informativeness of the axioms. Similarly, although adding the Continuum Hypothesis or its negation to ZFC 11Enayat shows that, for extensions of ZF in the language L∈ of set theory, the relation of bi-interpretability – a slight weakening of the notion of intertranslatability – reduces to the subtheory relation. This yields that the two theories cannot be bi-interpretable, and therefore not intertranslatable. 18 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . does not increase its logical strength, it does increase its scientic strength. As anticipated, canonical consistency statements display a similar behaviour: although PA + ¬Con(PA) has the same logical strength as PA, it is scientically stronger than PA. There is in fact a subtheory of PA + ¬Con(PA) that is intertranslatable with PA, but the converse does not hold (Visser 2006: Cor. 9.4). A similar phenomenon holds for ZF(C) and ZF(C) + ¬Con(ZF(C)), as well as full second-order arithmetic Z2 and Z2 + ¬Con(Z2). We now turn to the comparison of logics. We said in §2.2 that classical predicate logic is logically stronger than classical propositional logic. Since classical propositional logic is a subtheory of classical predicate logic, it follows that classical predicate logic is also scientically stronger than classical propositional logic. We can also show that classical propositional logic is scientically stronger than themany-valued propositional logicsK3, LP and FDE. That classical propositional logic is as scientically strong as K3, LP and FDE obtains because of the sublogic relation. For the other direction, we can show that none of K3, LP andFDE can dene the classical connectives. Since translational equivalence for logics entails that the connectives of one logic can be dened in the other without reinterpreting propositional letters (Pellettier and Urquhart 2003: Thm. 2.8), this establishes the failure of intertranslatability. Here is our proof for K3. If K3 were intertranslatable with classical propositional logic, then it would feature formulas N (*) and O(*, *) dening in K3 classical negation and disjunction. However, in K3, one can prove by induction on its complexity that for any formula i containing only one propositional letter p, i and N (p) are K3-logically equivalent, where N (p) can be one of: p, ¬p, p ∨ ¬p, ¬(p ∨ ¬p). By employing the explosion law for p and ¬p, and excluded middle for p ∨ ¬p and ¬(p ∨ ¬p), one can see that none of these alternatives are possible.12 12In more detail: since p, ¬p classically entails q, we would have p, N (p) K3 q However, this cannot be the case ifN (p) ≡ p, ifN (p) ≡ ¬(p∧¬p), ifN (p) ≡ p∨¬p. IfN (p) ≡ ¬p, we can use O(p, q). In K3, there are only the following forms O(p, q) can take: p ∨ q , p ∨ ¬q , ¬p ∨ q , ¬p ∨ ¬q , ¬(p ∨ q) , ¬(¬p ∨ q) ∨ ¬(¬q ∨ p) But K3 does not entail: p ∨ ¬p, p ∨ ¬¬p, ¬p ∨ ¬p, ¬p ∨ p, ¬(p ∨ ¬p) , ¬(¬p ∨ ¬p) ∨ ¬(p ∨ p) , 19 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . There are also logics that despite having the same logical strength have different scientic strength. One notable example is given by the modal logics K and T. We have seen that they have equal logical strength. However, a result of Pellettier and Urquhart (2003: Th. 4.5) entails that T is scientically stronger than K (and vice versa) because they are not intertranslatable. The reason for this is that, since both logics have the nite model property, translational equivalence requires isomorphism of classes of nite models. However, since K is a sublogic of T, there are models of K of size n that are not models of T. The same result entails that the logics B, S4, S5, T, B all dier in scientic strength. There are nonetheless logics that have equal scientic strength. For instance, by a result of Lenzen (1979), the modal logics S4.4 and KD45 are intertranslatable. Finally, what has been said so far also enables us to compare theories in dierent logics by means of scientic strength. In general, if S is a subtheory of T and formulated in a sublogic of the logic of T then S will be scientically weaker than any extension of T which is not scientically as strong as T . For instance, HA is a subtheory of PA, and therefore, by Visser's result on extensions of PA, HA is scientically weaker than any proper extensions of PA. Similarly, HA is scientically weaker than any theory that is properly logically stronger than ZFFin plus the assertion that every set has a transitive closure. As we have seen in the case of logical strength, the examples just presented show that our notion of scientic strength gives intuitive verdicts for an interesting class of combinations of logics and mathematical theories. The core insight behind our notion of scientic strength is to preserve some structural information of the logical and non-logical primitives of a theory in derivations. This is realized by requiring that the re-interpretations of such primitives need to be, so to say, inverse to each other. 4 Abductivism and its strengths We have presented a framework to analyze the notions of logical and scientic strength. By employing translations between theories, the framework allows one to compare the logical and scientic strength of theories in a formally presince ¬(¬p ∨ ¬p) ∨ ¬(p ∨ p) K3 ¬¬p ∨ ¬p K3 p ∨ ¬p. 20 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . cise way. The framework is directly applicable to the debate on logical and mathematical abductivism. Williamson (2017) and Russell (2018) analyzed logical strength essentially in terms of the subtheory relation. This fails to capture many interesting cases of theory comparison. Our framework allows theory comparison between theories that are not cast in the same language. Nonetheless, it also claries how the subtheory relation ts into a more general account of logical strength. In particular, being a proper subtheory of another theory implies being logically weaker than it. One important question for the abductivist concerns the relation between logical and scientic strength. According to Williamson, logical strength entails scientic strength, essentially because more deductive power yields more information. If this is perhaps a plausible picture when comparing theories cast in the same language, it becomes harder to defend when one must translate between theories. For, if not suitably regimented, translations may compromise the information contained in theorems, and this is not compatible with theories having the same scientic strength. For instance, facts such as the interpretation of PA+'PA is inconsistent' in PA rely essentially on distorting the information contained in 'PA is inconsistent'. It then follows that logical strength cannot entail scientic strength. By ensuring that the consequences of a theory are translated in accordance to suitable information-preserving constraints, our proposal maintains the generality given by understanding logical strength in terms of translations, while providing a notion of scientic strength as a renement of the logical one. As a result, scientic strength implies logical strength but not vice versa: not all translations involved in the relation of logical strength are adequate for scientic strength. For instance, for PA to be scientically as strong as PA+'PA is inconsistent', the structural role played by 'PA is inconsistent' in derivations should be preserved, and PA has to be inconsistent after all. Hence, our notion of scientic strength gives its due to the intuitive idea that scientic strength has to do with the information contained in a theory. Our framework combines notions of reducibility and equivalence that are usually employed in dierent domains. Interpretability strength is the standard tool to comparemathematical theories, schematic translations are generally employed to compare pure logics, and intertranslatability is a standard measure of theoretical equivalence for scientic theories. Therefore, our framework paves 21 D ra ft :P le as e em ai lu s at l.i nc ur va ti@ uv a. nl ;c ar lo .n ic ol ai @ kc l.u k. ac if yo u w ou ld lik e to ci te . the way to a unied approach to the comparison of formal theories. The specic combination of notions of reducibility employed in our characterization of logical and scientic strength delivers especially intuitive verdicts when applied to canonical examples. However, several alternatives are possible. For instance, faithful interpretability – in which not only provability, but also unprovability is preserved via the translation – may replace the looser notion of interpretability. 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