Overgeneration in the Higher Infinite SALVATORE FLORIO University of Birmingham s.florio@bham.ac.uk LUCA INCURVATI ILLC, University of Amsterdam l.incurvati@uva.nl This is a preprint of an article due to appear in Gil Sagi and Jack Woods (eds.), The Semantic Conception of Logic, Cambridge University Press. Abstract. The Overgeneration Argument is a prominent objection against the modeltheoretic account of logical consequence for second-order languages. In previous work we have offered a reconstruction of this argument which locates its source in the conflict between the neutrality of second-order logic and its alleged entanglement with mathematics. Some cases of this conflict concern small large cardinals. In this article, we show that in these cases the conflict can be resolved by moving from a settheoretic implementation of the model-theoretic account to one which uses higherorder resources. George Boolos once remarked that first-order Zermelo-Fraenkel set theory is 'a satisfactory if not ideal theory of infinite numbers' (Boolos 1971, p. 229). An ideal theory, he added, 'would decide the continuum hypothesis, at least'. Despite its success in capturing facts about the cardinality of infinite collections, ZF, even when augmented with the Axiom of Choice, fails to settle many natural questions about which sizes there are.1 In particular, ZFC cannot decide whether a number of interesting cardinality notions expressible in its language are instantiated. In a first-order setting, these notions are expressed by using the non-logical symbol for the membership relation. However, the language of second-order logic affords us the resources to characterize analogues of these notions without resorting to nonlogical vocabulary. One can then formulate counterparts of the cardinality questions left open by ZFC in the pure language of second-order logic. On a Tarskian account, also known as semantic account, logical validity is identified with truth in every model. Given this account, a special relation appears to exist between certain cardinality statements in set theory and the logical validity 1This, of course, holds modulo relevant consistency assumptions. Overgeneration in the Higher Infinite 2 of their counterparts in second-order logic. More specifically, there are cardinality statements that are true if and only if their second-order counterpart is true in every set-theoretic model. A number of authors have claimed this entanglement of secondorder logic with mathematics to be problematic (Etchemendy 1990; Gómez-Torrente 1996; Gómez-Torrente 1998; Hanson 1997; Hanson 1999; Priest 1995; Ray 1996; Etchemendy 2008). Others have recently defended the opposite view (Parsons 2013; Paseau 2013; Griffiths and Paseau 2016). In previous work (Florio and Incurvati forthcoming), we have exhibited a conflict between the entanglement and the view that logic should be neutral. This can be seen as vindicating the claim that the entanglement is problematic and provides the basis for a plausible reconstruction of John Etchemendy's (1990) Overgeneration Argument. Etchemendy concluded that the entanglement undermines the semantic account of logical validity. Other writers have retained the semantic account, taking the entanglement to cast doubt on the view that second-order logic is pure logic (Parsons 2013; Koellner 2010). Both reactions are premature in that they assume a specific implementation of the semantic account, one in which models are set-theoretic constructions. After all, the conflict between the entanglement and the neutrality of logic might be due to this assumption. Indeed, earlier results of ours suggest that adopting a different semantics for second-order logic might provide an alternative way of resisting our reconstruction of the Overgeneration Argument. In particular, if we embrace higher-order resources in the metatheory and take models to be higher-order entities, the paradigmatic example of the conflict either disappears or is no longer problematic. This example involves the Continuum Hypothesis (CH), which states that every set has cardinality aleph-1 if and only if it has the cardinality of the continuum. In this article, we extend this approach to other well-known cases of apparent entanglement involving the existence of certain small large cardinals, i.e. cardinals whose existence is consistent with the Axiom of Constructibility. We consider what happens to these cases of entanglement when second-order logic is given a higherorder semantics. Our findings are consistent with the optimistic outlook suggested by our earlier results. The adoption of a higher-order semantics for second-order logic helps avoid the conflict between the entanglement and the neutrality of logic. 1 Overgeneration Using a set-theoretic semantics, i.e. a semantics that construes models as sets, it is provable in ZFC that CH is true if and only if the following sentence of second-order Overgeneration in the Higher Infinite 3 logic is valid:2 (CH2) ∀X(ALEPH-1(X) ↔ CONTINUUM(X)) The cardinality notions involved in CH2 can be defined by pure formulas of secondorder logic (see, e.g., Shapiro 1991, pp. 101–5). As our discussion will make clear, what matters is the provability in a given theory of the biconditional stating that CH is true if and only if CH2 is valid. We will refer to this provability as the entanglement of second-order logic with CH. Note that one can also formulate a sentence NCH2 of pure second-order logic which is entangled with the negation of CH when ZFC is used as the background theory and the semantics is construed set-theoretically (Shapiro 1991, p. 105). In our discussion, it will be important to distinguish between a sentence of a formal language and the English (or mathematical English) sentence it formalizes. To this end, we shall adopt the following typographical convention: boldface indicates that a sentence belongs to a formal language, whereas its informal counterpart is indicated using roman characters. So, for example, CH2 stands for the English sentence that CH2 formalizes.3 If we read the second-order quantifier as ranging over properties, CH2 states that every property is ALEPH-1 if and only if it is CONTINUUM, where being ALEPH-1 and being CONTINUUM are characterized in terms of properties and their relations. Note that CH2 is different from CH, which states that every set is aleph-1 if and only if it is continuum, where these notions of size are characterized using the membership relation. Another point related to the distinction between formal and informal sentences: we use 'validity' to denote the property a formal sentence has just in case it is true in every model. Validity is meant to correspond to the notion of logical truth, which applies to informal sentences. As noted above, the claim that the entanglement of second-order logic with CH is problematic has recently been disputed. Against this, we have identified two arguments in support of the claim, both of which take the entanglement to be problematic insofar as it is in tension with the neutrality of second-order logic (Florio and Incurvati forthcoming). Their difference lies in the way in which they articulate the notion of neutrality. In both cases, the intended conclusion is that second-order logic fails 2As usual, we indicate second-order variables by means of upper-case letters. 3We are assuming that for every formal sentence there exists a unique informal counterpart, and that for every informal sentence we consider there exists a unique formalization. The uniqueness assumption is made for convenience but is not strictly needed for the purposes of the paper. What is needed is, roughly put, that the formalization operation and its inverse respect the informal relation of logical equivalence. The existence assumption is a presupposition of the debate. It may be justified by appealing to natural language sentences that seem to require variable binding of predicate positions (Higginbotham 1998, p. 3). Alternatively, it may be justified by appealing to a second-order translation of sentences involving plural quantification (Boolos 1984; Boolos 1985) or non-nominal quantification of the kind explored by Rayo and Yablo (2001). Overgeneration in the Higher Infinite 4 to be sound with respect to logical truth, i.e. there are second-order validities whose informal counterpart is not a logical truth. In this sense, the arguments purport to show that second-order logic overgenerates. The first argument is based on the idea that logic is dialectically neutral: it should be able to serve as a neutral arbiter in disputes. The argument uses the entanglement of second-order logic with CH, soundness of second-order logic with respect to logical truth and completeness of second-order logic with respect to logical truth, i.e. the thesis that there are no logical truths whose second-order formalization is not valid. Let us spell out the argument. If second-order logic is to serve as a neutral arbiter in a legitimate dispute, settling the question of whether a statement is a logical truth should not close that dispute. But now consider a dispute over CH in the context of ZFC. Suppose, on the one hand, that we agree that CH2 is a logical truth. By completeness, we have that CH2 is a higher-order validity. Using the entanglement of second-order logic with CH, we conclude that CH is true. Hence, if we agree that CH2 is a logical truth, we also have to agree that CH is true. Suppose, on the other hand, that we agree that CH2 is not a logical truth. By soundness, we have that CH2 is not a higher-order validity. Using the entanglement of second-order logic with CH again, we conclude that CH is false. Hence, if we agree that CH2 is not a logical truth, we also have to agree that CH is false. Therefore, settling the question of whether CH2 is a logical truth settles the question of whether CH is true, thereby violating dialectical neutrality.4 The proponent of the Overgeneration Argument concludes that second-order logic is not sound with respect to logical truth.5 The second argument is based on the idea that logic is informationally neutral: it should not be a source of new information. This idea licenses the following principle of informational neutrality: (IN) if a theory T does not informationally contain p and p is a consequence of T together with q, then q is not a logical truth. For had q been a logical truth, it could not have brought about such an increase in informational content. With the principle (IN) in mind, we can now turn to the 4There is an alternative version of the argument which involves NCH2. This version uses the entanglement of second-order logic with the negation of CH together with soundness and completeness of second-order logic with respect to logical truth. Below we will see that there are statements which are not entangled with second-order logic whereas their negation is. 5To save soundness, the defender of second-order logic might be tempted to deny that second-order logic is complete with respect to logical truth. In fact, the argument uses only a particular instance of completeness, namely the one which states that if CH2 is a logical truth, then CH2 is valid. Hence, denying completeness is acceptable only if one can point to features of CH2 which are not captured by the formalization. Compare with the case of propositional logic: it is usually agreed that this logic is not complete with respect to logical truth, but this is acceptable only because we can point to features of certain natural language sentences (e.g. the presence of predicates and quantifiers) which are not captured by a propositional language. Overgeneration in the Higher Infinite 5 argument. The argument is more complex than the first one, and we divide it into to steps. The first step is as follows. Suppose that CH is true. By the entanglement of second-order logic with CH, it follows that CH2 is valid. By soundness, we conclude that CH2 is a logical truth. The second step relies on this key assumption: ZFC together with the thesis that second-order logic is complete with respect to logical truth does not informationally contain CH. However, CH can be derived from ZFC plus completeness if we assume that CH2 is a logical truth. Using the key assumption, we derive from (IN) that it is not a logical truth that CH2 is a logical truth. On the supposition that logical truth can be iterated (i.e. if φ is a logical truth, then it is a logical truth that φ is a logical truth), we conclude that CH2 is not a logical truth. Therefore, the conclusion of the second step of the argument contradicts the conclusion of the first step. The argument supposes that CH is true but a parallel argument can be formulated using the assumption that CH is false. This parallel argument relies on the entanglement of second-order order logic with the negation of CH. The proponent of the Overgeneration Argument concludes, once again, that second-order logic is not sound with respect to logical truth. Both arguments make essential use of the entanglement of second-order logic with CH. The second argument, in addition, makes use of the entanglement of second-order order logic with the negation of CH. Both cases of entanglement, recall, are provable in ZFC when second-order logic is given a set-theoretic semantics. But this is just one implementation of the semantic conception of logic. So the question arises of what happens when second-order logic is given a different semantics. 2 Higher-Order Semantics Higher-order semantics is an alternative way of providing a semantics for secondorder logic which has gained popularity. On this semantics, one adopts higher-order resources in the metatheory and uses second-order entities, rather than sets, as the values of object-language second-order variables (Boolos 1984; Boolos 1985; Rayo and Uzquiano 1999; Rayo 2002; Rayo and Williamson 2003; Yi 2005; Yi 2006; McKay 2006; Oliver and Smiley 2013). This is compatible with various interpretations of the second-order resources. For example, one may interpret these resources in terms of properties, plurals, or Fregean concepts. Alternatively, one may take the second-order resources to be sui generis and not reducible to more familiar linguistic constructions (Williamson 2003, p. 459). For convenience, we will read the second-order quantifier as ranging over properties. Higher-order semantics allows us to define validity for second-order logic without ascending beyond second-order resources (see McGee 1997). Let us focus on a language whose sole non-logical symbol is the membership predicate of set theOvergeneration in the Higher Infinite 6 ory (∈). For such a language, the definition of validity amounts to truth under any reinterpretation of ∈ in any second-order domain. More formally, let φ[E/ ∈] be the result of replacing all occurrences of ∈ in φ with E and the superscript indicates the restriction of the formula's quantifiers to U . Then we say that φ is a higher-order validity if for every non-empty property U and every binary relation E, φ[E/ ∈]U holds. For instance, ∃x∃y x ∈ y is a higher-order validity just in case the following statement is true: for every non-empty property U and every binary relation E, there are x and y having U such that Exy. But the latter statement can be refuted in pure second-order logic and hence ∃x∃y x ∈ y is not a higher-order validity. If we are concerned only with arguments with finitely many premises, entailment can be defined in terms of validity. In particular, we can say that γ1, . . . ,γn entail φ just in case (γ1 ∧ . . . ∧ . . .γn) → φ is a higher-order validity. A key issue concerns the resources used in the metatheory. As we will see, the strength of the resources allowed has an impact on whether second-order logic is entangled with mathematics. In terms of logical resources, we will work with an axiomazation of second-order logic obtained by adding to first-order logic the rules for the second-order quantifiers, full second-order Comprehension and the second-order principle of Property Choice, stating that there is a choice function corresponding to every relation. As for the non-logical resources, we will consider two options. For the first option, we need the notion of a second-order closure of a theory T. This is denoted by T∗ and defined as the the set of sentences derivable from T in the axiomatization of second-order logic just described. The first option is then to use the informal counterpart of ZFC∗. The second option is to use ZFC2, the informal counterpart of the theory obtained by replacing ZFC's Replacement Schema with the corresponding second-order axiom. Since this theory has finitely many axioms, we can take it to be axiomatized by their conjunction. Thus, unlike the first option, the second option involves a theory whose non-logical axioms are formulated using distinctively higher-order resources. It is worth noting that second-order logic with higher-order semantics retains its expressive power. For example, work by Shapiro (unpublished manuscript) and Väänänen and Wang (2015) shows that an internal version of Zermelo's (n.d.) categoricity theorem is provable in pure second-order logic: there is a quasi-isomorphism between any two higher-order models of ZFC2.6 This implies that ZFC2 semantically decides any statement concerning the hierarchy up to Vκ , where κ is the first inaccessible. Since CH concerns the first few infinite levels of the hierarchy, it follows that ZFC2 semantically decides CH. That is, it is provable in pure second-order logic that ZFC2 entails either CH or it entails ¬CH. 6A function F is a quasi-isomorphism between two higher-order models 〈U1,E1〉 and 〈U2,E2〉 if F is a bijection between U1 and a subproperty of U2 or between U2 and a subproperty of U1 such that F preserves the relations E1 and E2. Overgeneration in the Higher Infinite 7 3 The Continuum Hypothesis We are now in a position to answer the question left open: 'What happens to the entanglement of second-order logic with CH when a higher-order semantics is adopted?'. As we have shown in Florio and Incurvati forthcoming, the answer turns on the metatheory used. In particular, assume ZFC and suppose that ZFC has an ω-model, i.e. a model which is correct about the natural numbers. Then we have: First Negative Theorem. ZFC∗ does not prove that if CH2, then CH. Second Negative Theorem. ZFC∗ does not prove that if CH, then CH2. In addition, the following lemma is provable in pure second-order logic. Equivalence Lemma. CH2 is true if and only if CH2 is a higher-order validity. Combining these results, one can conclude that ZFC∗ proves neither direction of the following biconditional: CH if and only if CH2 is a higher-order validity. Does the informal counterpart of this result hold? That is, can we prove in ZFC∗ that CH if and only if CH2 is a logical truth? We think not. As an analogy, consider the case of standard independence results. The independence of CH from ZFC (Gödel 1939; Cohen 1963) is usually taken to show that CH is independent of ZFC. For, presumably, any purported informal proof of CH or its negation from ZFC could be turned into a corresponding formal proof, contradicting the Gödel-Cohen result. Similarly, suppose one could give an informal proof in ZFC∗ that CH if and only if CH2 is a logical truth. Such an informal proof could then be turned into a formal proof in ZFC∗ that CH if and only if CH2 is a higher-order validity. This point extends to the relation between formal and informal statements in the case of small large cardinals considered below. While the entanglement of second-order logic with CH is not provable in ZFC∗, matters are different if we strengthen the set-theoretic background. In particular, ZFC2 proves: First Positive Theorem. If CH2, then CH. Second Positive Theorem. If CH, then CH2. Thus, given the Equivalence Lemma, we have that ZFC2 proves that CH if and only if CH2 is a higher-order validity. It follows that the informal counterpart of this biconditional may be established in ZFC2. Recall that the argument from dialectical neutrality and the argument from informational neutrality make essential use of the entanglement of second-order logic with Overgeneration in the Higher Infinite 8 CH. Thus, if we use ZFC∗ as our background theory, both arguments are blocked. What about the case in which we use ZFC2? Luckily for the defender of secondorder logic, the arguments can be blocked in that case too. The issue turns on a property of ZFC2 mentioned above, namely that it semantically decides CH. Consider the argument from dialectical neutrality first. The idea was that secondorder logic could not serve as a neutral arbiter in a dispute over CH in the context of ZFC. That is because, in the presence of the entanglement, settling whether CH2 is a logical truth settles whether CH is true. One might reformulate the argument so that it applies to ZFC2, given that this theory proves the entanglement. However, note that the original version of the argument assumed that we need to allow for a dispute over CH in the context of ZFC. This assumption is made plausible by the fact that ZFC, if consistent, entails neither CH nor ¬CH. The alternative version of the argument would replace this assumption with the new assumption that we need to allow for the same dispute in the context of ZFC2. But the fact that ZFC2 semantically decides CH gives us reason to reject the new assumption. For the fact that either ZFC2 entails CH or it entails its negation means that one of ZFC2→CH and ZFC2→ ¬CH is a higher-order validity. Therefore, in the context of ZFC2, it is not possible to agree on the logic while disagreeing on CH. Once it is agreed whether ZFC2→ CH or ZFC2→¬CH is a higher-order validity, the dispute over CH can be settled by an appeal to soundness of second-order logic with respect to logical truth followed by a simple modus ponens. Hence, the alternative version of the argument is undermined.7 Similar considerations apply to the argument from informational neutrality. The argument relied on the assumption that neither CH nor its negation is implicitly contained in ZFC. A reformulation of the argument would therefore rest on the assumption that neither CH nor its negation is implicitly contained in ZFC2. But, again, the assumption is undermined by the fact that ZFC2 semantically decides CH. The defender of second-order logic has thus reasons to switch to higher-order semantics. By adopting such a semantics, she can respond to both arguments against the soundness of second-order logic with respect to logical truth, no matter whether she chooses ZFC∗ or ZFC2 as her background theory. The viability of this response turns on two facts. First, the entanglement of second-order logic with CH is not provable in ZFC∗. Second, even on a higher-order semantics, ZFC2 semantically decides any statement concerning sets occurring below Vκ (with κ the first inaccessible) and hence CH. However, there are statements other than CH whose truth is equivalent over ZFC 7It may be observed that the alternative version of the argument from dialectical neutrality uses the entanglement of second-order logic with the negation of CH. However, analogues of the results described in this section can be proved for the case of NCH2 and ¬CH (Florio and Incurvati forthcoming). Thus, the alternative version of the argument is also blocked when moving to a higher-order semantics. Overgeneration in the Higher Infinite 9 to the set-theoretic validity of their second-order counterparts. The entanglement of these statements with second-order logic has also been deemed problematic. What's more, some of these statements concern Vκ or higher levels of the hierarchy. Therefore, despite Zermelo's categoricity theorem and its internal version mentioned in Section 2, these statements are not semantically decided by ZFC2 in either settheoretic or higher-order semantics (modulo the appropriate consistency assumptions). Can the move to higher-order semantics help with these cases of entanglement beyond categoricity? In what follows, we focus on one notable set of examples concerning the higher infinite, namely statements asserting the existence of certain small large cardinals.8 4 Inaccessible cardinals As usual, we say that a cardinal is inaccessible if it is uncountable, regular and strong limit. We use IC to denote the assertion that there is such a cardinal. As is well known, ZFC proves that κ is inaccessible if and only if Vκ is a set-theoretic model of ZFC2. This fact suggests the standard second-order counterpart of IC, namely the pure statement that there is a domain and a binary relation that satisfies ZFC2. In symbols: (IC2) ∃U∃R ZFC2[R/∈]U Now, there is no entanglement of second-order logic with IC in set-theoretic semantics (as well as in higher-order semantics). Indeed, IC2 is false in any model with a finite domain and hence the semantics easily refutes the validity of IC2. Therefore, IC does not imply that IC2 is valid. However, the other direction of the entanglement holds so we have what we might call semi-entanglement. For the fact that IC2 is not valid trivially implies that if IC2 is valid, then IC. Things change if we consider the negation of IC. In particular, let NIC2 be the negation of IC2. Then, given a set-theoretic semantics, it is provable in ZFC that ¬IC if and only if NIC2 is valid. The argument from informational neutrality as stated requires both directions of the entanglement for a statement and its negation. So it is not applicable in the 8To see that ZFC2 does not semantically decide these statements given the appropriate consistency assumption, suppose that a small large cardinal κ exists. Then, one can obtain higher-order models of ZFC2 which disagree over the existence of κ by truncating at κ . The existence of the corresponding set-theoretic models can be established by assuming that there is an inaccessible above κ . Overgeneration in the Higher Infinite 10 present case. The argument from dialectical neutrality, on the other hand, only needs that a statement or its negation be entangled with second-order logic. Thus it can be formulated using the entanglement of the negation of IC. Does this case of entanglement carry over to higher-order semantics? The answer is negative, no matter whether the background theory is ZFC∗ or ZFC2. Let us begin with ZFC2. The following two theorems hold. Theorem 1. ZFC2 proves that if NIC2 is a higher-order validity, then ¬IC. Theorem 2. ZFC2 does not prove that if ¬IC, then NIC2 is a higher-order validity. In both cases, the key observation is that, by a simple existential introduction, ZFC2 proves that IC2 holds and hence that NIC2 is not a higher-order validity. This statement is the negation of the antecedent of the conditional in Theorem 1. Thus the theorem follows trivially. For Theorem 2, suppose towards a contradiction that ZFC2 proved that if ¬IC, then NIC2 is a higher-order validity. Since ZFC2 proves that NIC2 is not a higher-order validity, it would follow that ZFC2 proved IC and hence its own consistency. But this cannot be for standard consistency reasons. Let us now turn to ZFC∗. To start with, note that ZFC∗ is a subtheory of ZFC2. Therefore, Theorem 2 yields immediately: Theorem 3. ZFC∗ does not prove that if ¬IC, then NIC2 is a higher-order validity. Unlike ZFC2, ZFC∗ does not prove the other direction of the conditional in Theorem 3. This follows from the next two results, namely Lemma 4 and Theorem 5. Lemma 4 is a version of the Equivalence Lemma above and is provable in pure second-order logic. Lemma 4. NIC2 if and only if NIC2 is a higher-order validity. The right-to-left direction is straightforward. For the other direction, assume NIC2 and suppose towards a contradiction that NIC2 is not a higher-order validity. This means that there is a non-empty U∗ and there are U and R such that ( ZFC2[R/ ∈ ]U )U∗ . By second-order Comprehension we have a property Z which something has if and only it has both U and U∗. It is easy to verify that ZFC2[R/ ∈]Z , which contradicts NIC2. The proof of Theorem 5 uses the assumption that there is an ω-model of the theory consisting of ZFC and the assertion that there are two inaccessibles. Note that the theorem is an underivability result. So, in the course of the proof, it is permissible to give the higher-order metalanguage a set-theoretic interpretation and use countermodels constructed according to the set-theoretic semantics. Theorem 5. ZFC∗ does not prove that if NIC2, then ¬IC. Overgeneration in the Higher Infinite 11 Proof. Let us abbreviate the statement that there are two inaccessibles in the constructible hierarchy L as 2ICL. First, we use the consistency assumption to construct an ω-model m of ZFC+¬IC+ 2ICL. Start from an ω-model with at least two inaccessibles and obtain by truncation a model with exactly two inaccessibles. Using forcing, make the two inaccessible cardinals of the model only weakly inaccessible (thus destroying their strong inaccessibility). This gives the desired model, since weakly inaccessible cardinals are strongly inaccessibles in L. Next, we use m to establish the consistency of the theory ZFC∗+NIC2+IC. Let κ be the second cardinal of m which is inaccessible in L. From the perspective of m, Lκ is a model of ZFC+ IC. Hence, again from the perspective of m, 〈Lκ ,P(Lκ)〉 is a model of ZFC∗+ IC. Now since m satisfies ZFC, it thinks that every set-theoretic model satisfies NIC2 if and only if ¬IC is true. But m is a model of ¬IC and therefore it thinks that 〈Lκ ,P(Lκ)〉 is also a model of NIC2. Hence m thinks that the theory ZFC∗+NIC2+ IC is consistent. Since m is an ω-model, it is correct about consistency facts. Therefore the theory ZFC∗+NIC2+ IC really is consistent. The situation, therefore, is the following. If the background theory is ZFC∗, neither direction of the entanglement of second-order logic with ¬IC is provable in higher-order semantics. If we strengthen the background theory to ZFC2, one direction of the entanglement becomes provable. Either way, the argument from dialectical neutrality is blocked. However, one might worry that the provability of semi-entanglement for the case of IC and ¬IC still poses a threat to the neutrality of logic. We address this issue in Section 6. Of course, there are natural strengthenings of ZFC2 which do prove both directions of the entanglement of second-order logic with ¬IC in higher-order semantics. For instance, the result of adding to ZFC2 the assertion that there inaccessible cardinals trivially proves that if there are no inaccessibles, then NIC2 is a higher-order validity. However, the dispute over the existence of an inaccessible is closed by such a theory quite independently of the entanglement. Therefore, its availability poses no problems for the dialectical neutrality of second-order logic. The same holds for other theories which imply the existence of inaccessibles, e.g. the theory obtained by adding a form of second-order reflection to ZFC2. 5 Mahlos and weakly compacts The fact that ZFC2 proves exactly one direction of the entanglement of ¬IC with second-order logic is a direct consequence of the definition of NIC2. That is, ZFC2 proves that there is an inaccessible property, contradicting the higher-order validity of NIC2. Thus one may wonder what the situation is with respect to other small large cardinals, such as Mahlos and weakly compacts. For as well as not proving the existence of Mahlo and weakly compact sets, ZFC2 does not prove the existence of Overgeneration in the Higher Infinite 12 Mahlo and weakly compact properties. As it turns out, perfect analogues of the results obtained for ¬IC can be obtained for the case of Mahlos and weakly compacts. Let us consider these cases. Recall that a cardinal κ is Mahlo if every normal function on κ has an inaccessible fixed point.9 We denote the set-theoretic statement that there is a Mahlo cardinal with M. Now, it can be shown in ZFC that κ is Mahlo if and only if Vκ is a set-theoretic model of ZFCM2, i.e. ZFC2 plus the following axiom: ∀F(normal(F) → ∃κ(Inacc(κ) & F(κ,κ))) Here normal(F) and Inacc(κ) formalize the statements that F is a normal functional property and κ is an inaccessible set. As before, this suggests the standard second-order counterpart of M, namely the pure statement that there is a domain and a binary relation that satisfies ZFCM2: (M2) ∃U∃R ZFCM2[R/∈]U Again, there is only semi-entanglement of second-order logic with M in set-theoretic semantics (as well as in higher-order semantics). On the other hand, if we let NM2 be the negation of M2, the following is provable in ZFC given the set-theoretic semantics: ¬M if and only if NM2 is valid. We now switch to higher-order semantics. Again, we first examine what happens when the metatheory is ZFC2. In that case, we have: Theorem 6. ZFC2 proves that if NM2 is a higher-order validity, then ¬M. Theorem 7. ZFC2 does not prove that if ¬M, then NM2 is a higher-order validity. For Theorem 6, we reason in ZFC2 and show that if there is a Mahlo cardinal, then NM2 is not a higher-order validity. Let κ be Mahlo. It can be verified that the axioms of ZFCM2 hold when restricted to Vκ . So the property of being a set in Vκ and the relation of membership restricted to Vκ provide witnesses to the existential quantifiers of M2. So M2 is true and hence NM2 is not a higher-order validity. For Theorem 7, we reason about provability in ZFC2 and establish that the relevant conditional is unprovable by providing a set-theoretic countermodel. We make 9A function on the ordinals is normal if it is strictly increasing and continuous. Since in the definition of a Mahlo cardinal one restricts attention to normal functions on a set of ordinals κ , the notion of being Mahlo is first-order definable. Note that the definition of a Mahlo cardinal given above is equivalent over the ZFC axioms to the following definition, which is perhaps more standard: a cardinal κ is Mahlo if it is regular and the set of inaccessibles below κ is stationary in κ . A set a⊆ κ is stationary in κ if a∩ c 6= /0 for each closed unbounded subset c of κ . Overgeneration in the Higher Infinite 13 the consistency assumption that there is a set-theoretic model of ZFCM2. This model must be a Vκ with κ a Mahlo cardinal. Now truncate this model at the first Mahlo cardinal μ . (If there are none, let μ simply be κ .) Clearly, Vμ is a model of ¬M. On the other hand, since μ is Mahlo, Vμ is a model of ZFCM2 and hence of M2. Therefore Vμ is a model of the statement that NM2 is not a higher-order validity. The perfect parallel between the case of inaccessibles and that of Mahlos holds even if we weaken the background theory to ZFC∗. Obviously, the direction of the entanglement unprovable in ZFC2 is also unprovable in ZFC∗: Theorem 8. ZFC∗ does not prove that if ¬M, then NM2 is a higher-order validity. Moreover, the following two results can be established by adapting the proofs of Lemma 4 and Theorem 5. Lemma 9. NM2 if and only if NM2 is a higher-order validity. Theorem 10. ZFC∗ does not prove that if NM2, then ¬M. Thus, ZFC∗ also fails to prove the other direction of the entanglement. How about weakly compact cardinals? One can formulate a second-order statement which, when added to ZFC2, yields a theory whose set-theoretic models are all and only the Vκ with κ a weakly compact cardinal (Hellman 1989, 86 ff.). Proceeding as before, one can then find a pure second-order statement (WKC2) whose settheoretic validity implies the assertion that there is a weakly compact (WKC). As in the previous cases, ZFC proves that the negation of this statement (NWKC2) is true in every set-theoretic model if and only if there are no weakly compacts (¬WKC). When moving to a higher-order semantics, one can prove analogues of the theorems obtained above. This means that we have no entanglement when the background theory is ZFC∗ and only semi-entanglement when the background theory is ZFC2. These results can be established using the techniques employed in the the case of Mahlo cardinals. 6 Semi-entanglement We now want to reassess the arguments from dialectical and information neutrality in the light of the results proved in the previous section. As mentioned, the argument from informational neutrality requires the entanglement for a statement and its negation. Therefore it does not apply in the case of small large cardinals, even on a set-theoretic semantics. In contrast, the argument from dialectical neutrality requires the entanglement of a statement or its negation. So it continues to apply with respect to small large cardinals. In these cases, higher-order semantics provides a clear way out: none of the relevant statements which are entangled in set-theoretic semantics Overgeneration in the Higher Infinite 14 continues to be entangled in higher-order semantics. So the new instances of the argument from dialectical neutrality are blocked. If the background theory is ZFC2 rather than ZFC∗, we still have semi-entanglement. Thus one might wonder whether semi-entanglement suffices to run a modified version of the arguments we have articulated. In the case of informational neutrality, it is hard to see how the argument could be modified so as to rely only on semientanglement. The case of dialectical neutrality requires closer scrutiny. Let S be any of the statements involving small large cardinals which are entangled with second-order logic in set-theoretic semantics (i.e. ¬IC, ¬M and ¬WKC) and let S2 be its second-order counterpart. Then the argument from dialectical neutrality concerning S goes as follows. By logic, either S2 is a logical truth or it isn't. If it is, using completeness of second-order logic with respect to logical truth and one direction of the entanglement, we derive that S is true. It follows that S is settled. If it isn't a logical truth, using soundness of second-order logic with respect to logical truth and the other direction of the entanglement, we derive that S is false. It follows, again, that S is settled. Therefore, no matter what logical status S2 has, S is settled. This means that second-order logic cannot be a neutral arbiter in a dispute over S. Since in the context of higher-order semantics we only have semi-entanglement when the background theory is ZFC2, it is not the case that S is settled independently of the logical status of S2. For each of S and ¬S is consistent with the assumption that S2 is not a logical truth. So the conclusion that second-order logic cannot be a neutral arbiter in a dispute over S is not available in general. However, one may point out that the conclusion is available if it is assumed that S2 is a logical truth. For, in the presence of semi-entanglement, this assumption together with completeness of second-order logic with respect to logical truth entail that S is true. So it follows that S is settled. To start with, note that this modified version of the argument leads at best to denying completeness, not soundness. Thus it cannot be considered an overgeneration argument. Rather, the denial of completeness results in the claim that second-order logic undergenerates: there are logical truths whose second-order formalization is not valid. In any case, the question we should ask is: what reasons do we have to assume that S2 is true and hence a logical truth? To answer this question, it is useful to consider first another case, namely that of CH2 and NCH2. As Michael Potter remarks: [W]e do not seem to have any intuitions about whether these secondorder principles [i.e. CH2 and NCH2] that could settle the continuum hypothesis are themselves true or false. So this observation [that there is entanglement] does not seem especially likely to be a route to an argument that will actually settle the continuum hypothesis one way or the other. (Potter 2004, p. 271) The same could be said about S2. Of course, one might deny this and insist that Overgeneration in the Higher Infinite 15 either we have no intuitions about whether S2 is true or we have intuitions that it is false. But suppose that we did think that S2 is true. Presumably this is because we have certain views about the extent of the set-theoretic hierarchy. For example, it is because we think that the hierarchy does not have Mahlo height that we think that there is no Mahlo property and hence that NM2 is true. Clearly, this is also a reason for thinking that the hierarchy contains no Mahlo set: if there was such a set, one could obtain a Mahlo property by simply considering the property of being a member of that set. Therefore the reason to accept NM2 ipso facto closes the dispute over the existence of a Mahlo. Note that this response does not carry over to the situation in which we have full entanglement. For not every reason to deny the existence of a Mahlo is ipso facto a reason to accept NM2. For instance, one might think that the universe is Vκ with κ the first Mahlo cardinal. In that case, one would hold that the hierarchy has Mahlo height without thereby thinking that there is a Mahlo set. So, in all these cases of semi-entanglement, if we have reasons to take the secondorder statement to be true, we also have reasons to take its set-theoretic counterpart to be true independently of semi-entanglement. Thus there appears to be no conflict between the relevant cases of semi-entanglement and dialectical neutrality. 7 Conclusion The semantic account of logic identifies validity with truth in every model. When models are construed set-theoretically, this account gives rise to cases of entanglement which are in conflict with neutrality of second-order logic. This conflict is captured by the arguments from dialectical neutrality and informational neutrality. These arguments make use of the assumption that second-order logic is dialectically neutral and the assumption that second-order logic is informationally neutral. Thus, one could block them by rejecting those assumptions. We have shown that the defender of second-order logic has an alternative way of blocking the arguments, namely by adopting a higher semantics. This semantics embodies a different implementation of the semantic account of logic in that it takes models to be higher-order constructions rather than sets. Of course, one might challenge the legitimacy of higher-order semantics. But this would require a separate argument targeting the use of second-order resources in semantic theorizing. Pending such an argument, the defender of second-order logic may resort to higher-order semantics in response to the Overgeneration Argument. In this article, we have focused on some cases of entanglement in the higher infinite. We have shown that these cases of entanglement, previously thought to be problematic, disappear when a higher-order semantics is adopted. In the absence of entanglement, both of our reconstructions of the Overgeneration Argument are blocked. Moreover, a modified version of the argument from dialectical neutrality Overgeneration in the Higher Infinite 16 based on semi-entanglement fails to establish its intended conclusion. This provides additional evidence that the proponent of the semantic account can hold on to the neutrality of second-order logic by employing a higher-order semantics. When one takes into account models that are not set-sized, denying the existence of small large cardinals does not translate into a commitment to the validity of certain second-order sentences. There remains one well-known case of entanglement in set-theoretic semantics which we have not considered. This is the one involving the Generalized Continuum Hypothesis, and we plan to explore it in future work. Together with the results obtained in this article, this new investigation will provide a clearer picture of the extent to which higher-order semantics can help vindicate the neutrality of secondorder logic.10 10This work has received funding from a Leverhulme Research Fellowship held by Salvatore Florio and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 758540) within Luca Incurvati's project From the Expression of Disagreement to New Foundations for Expressivist Semantics. For comments and discussion, we would like to thank Michael Glanzberg, Mario Gómez-Torrente, Graham Leach-Krouse, Beau Madison Mount, Ian Rumfitt, Gil Sagi, Kevin Scharp, Gila Sher, Jack Woods and Elia Zardini. Earlier versions of this material were presented at the workshop The Semantic Conception of Logic in Munich and the conference Disagreement within Philosophy in Bonn. 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