A Formal Explication of Blanchette's Conception of Fregean Consequence

Over the past decades, Patricia Blanchette has developed a sophisticated account of Frege's conception of logic and his views on logical consequence. One of the central components of her interpretation is the idea that Frege's conception of logical consequence is ‘semantically laden’ and not purely formal. The aim of the present paper is to provide precise explications of this as well as related ideas that inform her account, and to discuss their significance for the philosophy of logic in general and Frege's philosophy in particular.


Introduction
Frege is widely considered to be one of the founding fathers of modern logic. Yet, according to a challenging interpretation proposed by Patricia Blanchette, Frege's conception of logic is, in an important sense, not that modern after all because according to Blanchette, Frege's conception of logical consequence is not purely formal. Blanchette originally proposed her interpretation in the context of a discussion regarding Frege's controversy with Hilbert to explain Frege's dismissive attitude towards Hilbert's independence and consistency proofs contained in his celebrated Grundlagen der Geometrie (Hilbert 1903). Since then, she has expanded on her central interpretive claims and put them into a more broad perspective of Frege's conception of logic and mathematics. 1 I cannot do full justice to the nuanced position that Blanchette has developed over the years, but the simplest version of her interpretation is based on a combination of three theses about language and logic that Frege seems to be committed to. The first is the idea that a sentence of a proper language expresses a content that is not itself linguistic in nature. The second is the idea that one and the same content can be expressed by different sentences; and the third is the idea that logical consequence is primarily a relation between contents expressed by sentences, not sentences themselves. In the terminology of mature Frege, these theses may be put as follows: (F1) Each sentence S of a proper language expresses a thought T.
(F2) One and the same thought T may be expressed by different sentences S and S . (F3) Logical consequence is primarily a relation between thoughts expressed by sentences (not sentences themselves).
An important consequence of these assumptions is that unlike formal provability in some calculus, the question whether or not a thought T is a logical consequence of a set of thoughts T 1 , . . . T n is not solely a matter of the logical form of particular sentences S and S 1 , . . . S n that express those thoughts. While the existence of a formal proof of S from S 1 , . . . S n in a reasonable calculus is a reliable means to establish that T is a logical consequence of T 1 , . . . T n , the non-existence of such a formal proof does not reliably indicate that T is not a logical consequence of T 1 , . . . T n . Since formal provability only pertains to the specific syntactic structures of S and S 1 , . . . S n , the non-existence of a formal proof of S from S 1 , . . . S n does not rule out the existence of a formal proof of S from S 1 , . . . S n , where these sentences express the thoughts T and T 1 , . . . , T n in some alternative way. If such a proof exists, then T will be a logical consequence of T 1 , . . . T n , despite the fact that S is not formally provable from S 1 , . . . S n . The scenario just described can arise if we specifically consider the possibility of conceptual analysis. Here, by 'conceptual analysis' we roughly mean the result of a process whereby the sense of a simple expression e is expressed in terms of a complex expression e , and where the senses of the component-expressions of e are somehow simpler or more basic than the sense expressed by e itself. The following criterion of adequacy for conceptual analysis suggests itself: (F4) If e and e are expressions of a proper language and e is an analysis of e, then e and e express the same sense.
Thus, successful analysis preserves sense, a position that Frege seems to endorse, at least on some occasions. 2 If, in addition, we assume the compositionality of Fregean senses, i.e. the principle (F5) If e and e are expressions of a proper language and e results from e by replacing subexpressions in e by expressions with the same sense, then e and e express the same sense.
then we get the type of cases that Blanchette is mainly interested in as follows. Suppose that S is not formally provable from S 1 , . . . S n , but the senses of some of the simple terms e 1 , . . . e k in S and S 1 , . . . S n can be analysed in terms of complex expressions e 1 , . . . e k . The sentence S that results from S by replacing the primitive terms e i with their corresponding analyses e i may then become formally provable in some reasonable calculus from the analysed sentences S 1 , . . . S n because additional syntactic structure may be introduced by the analyses, which in turn may be exploited in formal proofs. By (F4) and (F5), the thoughts expressed by S 1 , . . . S n , S and S 1 , . . . S n , S respectively, are identical. So, assuming that our formal proof procedure is reasonable, this means that the thought expressed by S might, after all, be a logical consequence, in Frege's sense, of the thoughts expressed by S 1 , . . . S n . As emphasised by Blanchette, conceptual analysis may thus reveal logical connections between thoughts that are not immediately reflected in a particular expression of those thoughts. As a simple illustration, here is an example that is adapted from one of the examples discussed by Blanchette (though not by Frege). 3 Consider the argument (P) Jones had a nightmare. (C) Jones had a dream.
Clearly, the conclusion-sentence (C) is not a formal consequence of the premise-sentence (P). However, assuming that the concept 'having a nightmare' is adequately analysed as something like 'having a disturbing dream', the thought expressed by (P) is also expressed by (P') Jones had a disturbing dream.
Since (C) is a formal consequence of (P'), according to our simple version of the view Blanchette attributes to Frege, the thought expressed by (C) is, after all, a logical consequence of the thought expressed by (P).
I take it that some version of the account just described is a plausible interpretation of how Frege at some point conceptualised logical consequence, and that a version of this account is the basis of an interesting conception of consequence in its own right (although one might be hesitant to speak of 'logical consequence'). As it stands, however, the account has several problems, both as an interpretation of Frege and as a plausible account of consequence, independently of Frege. First and foremost, it relies heavily on the intelligibility of Fregean 'senses', and in particular, Fregean 'thoughts'. Since it is difficult to extract a precise notion of 'sense' from Frege's writings that squares with his main reasons for postulating such entities in the first place, the problem of the intelligibility of Fregean senses also affects the proposed account. Also, central analyses that Frege himself proposed in pursuit of his logicist project do not seem to preserve sense, regardless of how senses are construed. 4 In any case, in what follows I will be neither concerned with the simplified account of Fregean consequence described above, nor with the refined version that Blanchette has developed over the years. Instead, I will focus on a question that is raised by Blanchette towards the end of her monograph on Frege's conception of logic. There, she notes that the problems with Fregean thoughts just mentioned, along with additional problems I have not mentioned, give rise to the question of whether the central differences between (her understanding of) Frege's conception of consequence and modern conceptions, rely on the commitment to these 'mysterious entities'. While Blanchette maintains that it is convenient to present these differences in terms of thoughts, she notes in Blanchette 2012, p. 179, that [. . . ] it is important to see that none of the central views stressed in this volume, those that separate Fregean from later conceptions of logic, turn on the commitment to thoughts. Precisely the same points can be made by focusing on the idea of logical entailment and consistency as applying to (sets of) fully-interpreted sentences, independently of whether sentential interpretation is a matter of the expression of thoughts. [. . . ] [T]he idea that logical entailment applies just to fully-interpreted sentences, and that it does so in a way that's sensitive to the meaning of non-logical terms, provides the heart of the Fregean picture of entailment.
In her book, Blanchette does not follow up on these suggestions, and from what she says, it is not entirely clear why 'precisely the same points' can be made when fully-interpreted sentences are taken to be the objects of entailment rather than thoughts. The main objective of this article is to expand on this question by developing a notion of consequence that applies to sentences of a fully-interpreted language L -in the sense of modern logic -that meets the following criteria: (BF 1) Whether a sentence S of L is a consequence of sentences S 1 , . . . S n of L is in part determined by the meanings of the non-logical terms of L. (BF 2) Analysis of some of the non-logical terms of L may reveal that S is a consequence of S 1 , . . . S n despite the fact that S is not formally entailed by S 1 , . . . S n .
We will refer to notions of consequence that satisfy these two criteria as 'broadly Fregean consequence relations'. According to the explication we will develop in what follows, we will take the 'meaning' of a non-logical term of L to be its extension in L and we will take a primitive non-logical term of L to be 'analysable' in L if it is explicitly definable in terms of the remaining primitives of L. The basic idea, then, is to define broadly Fregean consequence relations for an interpreted language L by stipulating that a sentence S of L is a consequence of S 1 , . . . S n if and only if there are sentences S , S 1 , . . . S n of L that arise from S, S 1 , . . . S n by replacing some of the primitive terms e 1 , . . . e k in S, S 1 , . . . S n by expressions e 1 , . . . e k that define e 1 , . . . e k respectively, and S is formally provable from S 1 , . . . S n . On the proposed explication, for S to be a consequence of a given set of premises, it is therefore sufficient and necessary that some 'definitional variant' of S is formally provable from definitional variants of the premises. In the remainder of this article, we will make this idea precise and discuss some of the basic properties of the resulting concept of consequence. We will get back to the limits of this approach in Section 3 and discuss some of its philosophical implications.

Broadly Fregean Consequence Relations for First-Order Languages
In what follows, we will define broadly Fregean consequence relations for a restricted, but sufficiently interesting class of languages; namely, first-order languages. In Section 2.1, we will fix some terminology and review some basic facts about definability that will be needed in Section 2.2, where we define precise concepts of consequence and related notions. We will look at some of the fundamental properties of these notions in Sections 2.3 and 2.4.

Preliminaries
As announced, throughout this section we will be exclusively concerned with first-order languages. 5 Much of what follows can be found in any competent textbook on logic and is mainly presented here in order to fix some notation. We will assume the usual logical apparatus, including truth functional connectives, quantifiers, and individual variables. We will assume that the only logical symbols are negation ¬, conjunction &, the universal quantifier ∀, and identity = . Other logical symbols may be introduced by definition as usual. To simplify the setting even further, we will only consider finite, relational languages; that is, languages that only contain a finite number of relation symbols, but no individual constants and no function symbols. A finite set {R 1 , . . . , R n } of relation symbols will be called a signature. For each signature L, the L-formulas are defined inductively as usual.
where D is a set of objects and each R M i is a set of k i -tuples of elements in D (for R i a k i -ary relation symbol) that serves as interpretation of the respective relation symbol in L. We will identify an interpreted language with a signature L together with an L-structure and a variableassignment, where a variable assignment is a function that assigns to each variable some value in D. For convenience, we will use L, L , . . . both for signatures and for interpreted languages. We will use boldface letters (e.g. 'x') to refer to finite sequences of variables x 1 , . . . x k (similarly, for tuples of objects from D) and we will use ∀xϕ as an abbreviation for ∀x 1 . . . ∀x k ϕ. If s is a variable assignment, then s(x) is short for (s(x 1 ), . . . , s(x k )) and s[x/d] stands for the variant of s where each variable x i is assigned the value d i . If it is clear from the context that L, L , . . . are interpreted languages, then it should be understood that M, M , . . . are the structures, and s, s , . . . the variable assignments associated with these languages respectively. We say that L is a sublanguage of L if the signature of L is contained in that of L, M is the restriction of M to L , and the variable assignments for both languages coincide. For each interpreted language L we can define a satisfaction relation M, s ϕ in the usual way by using the semantic stipulations regarding the logical constants as clauses of an inductive definition. If M, s ϕ, we also say that ϕ is true in L.
The notion at the centre of the remainder of this subsection is that of explicit definability in an interpreted language. Roughly, a primitive relation of some language L is explicitly definable in terms of other relations in L if there exists a formula of L that only involves the latter and which is co-extensive with the former. More precisely, suppose that L and L are interpreted languages, L is a sublanguage of L, and R is a k-ary relation symbol in L\L . Then an L -formula ξ(x) is said to explicitly define R in terms of L just in case the quantified biconditional is true in L. Note that this is the case if and only if for all d ∈ D k We will also refer to a sentence like D(R, ξ) as a D-biconditional over L . A relation symbol R in L\L is explicitly definable in terms of L if there exists a formula that explicitly defines R in terms of L . If every relation symbol in L\L is explicitly definable in terms of L , then we say that L is a definitional expansion of L or, equivalently, that L is a definitional basis for L. Note that each language is trivially a definitional basis for itself. We call a definitional basis L for L non-trivial if L is a proper sublanguage of L, and we call an interpreted language L fully analysed if L contains no non-trivial definitional basis L as a sublanguage; that is, if none of its primitives are explicitly definable in terms of the others.
Let us now look at the concept of substitution. Suppose R 1 , . . . , R n is a sequence of relation symbols of a given language L, and ξ 1 , . . . , ξ n is a sequence of L-formulas. We say that the two sequences are matching if the number of free variables of each formula ξ i matches the arity of the corresponding relation symbol R i . For each pair of matching sequences we can define a translation that maps each L-formula ϕ to another L-formula; namely, the result of replacing each atomic formula R i x by the corresponding formula ξ i . 6 To avoid clutter, we simply write ϕ * for this formula. It should be clear from the context, which relation symbols are replaced by which formulas. More precisely, suppose L is an interpreted language and R 1 , . . . , R n and ξ 1 , . . . , ξ n are matching sequences. Then a substitution translation for R 1 , . . . , R n and ξ 1 , . . . , ξ n is a function * from L-formulas to L-formulas such that If is a set of formulas, then * = {ϕ * : ϕ ∈ }. Note that ϕ * = ϕ whenever ϕ contains no non-logical vocabulary.
Substitution translations are important for a variety of reasons. For our purposes, they are useful mainly because they enable us to characterise our central concept for what follows, the notion of a definitional translation. So, suppose that L and L are interpreted languages and L is a definitional basis for L. Then, a function * from L-formulas to L -formulas is a definitional translation over L if there are relation symbols R 1 , . . . , R n in L\L and L -formulas ξ 1 , . . . , ξ n such that * is a substitution translation for R 1 , . . . , R n and ξ 1 , . . . ξ n and each ξ i explicitly defines R i in terms of L . For convenience, we also count the function that maps each formula to itself as a (trivial) definitional translation for any given language.
Just like substitution translations in general, definitional translations preserve logical form. But, in addition, definitional translations also preserve truth.
Lemma 2.1: Suppose L and L are interpreted languages, L is a definitional basis for L, and * is a definitional translation over L . Then for all L-formulas ϕ: In what follows we assume that some mechanism is adopted to manage free variables to avoid potential variable clashes.

Proof: By induction on ϕ.
We continue to return to further properties of definitional translations as we go along. For now, this is sufficient.

Defining F-Consequence
To define a precise version of Fregean consequence along the lines sketched earlier, we assume some fixed formal proof-procedure for first-order logic. For the sake of definiteness, we assume some standard axiomatic calculus which is determined by a set of logical axiom schemes and a couple of inference rules, say, modus ponens and generalisation. (As we shall see, the details will not matter much.) A correct formal proof of an L-formula ϕ from assumptions is a finite sequence ϕ 1 , . . . , ϕ n of L-formulas, where ϕ = ϕ n and each formula in the sequence is either an assumption, an instance of a logical axiom scheme, or the result of an application of one of the inference rules to earlier formulas in the sequence. We say that ϕ is formally provable or derivable from and write ϕ if there exists a correct formal proof of ϕ from the assumptions . A set of formulas is formally consistent if no contradiction ϕ & ¬ϕ is formally provable from , and a formula is formally valid if it is formally provable from the empty set of assumptions. We assume that our calculus is sound with respect to the usual concept of semantic consequence: If ϕ is formally provable from , then every model of is also a model of ϕ. Although this will not be strictly needed in what follows, it will be useful to assume completeness as well, i.e. if every model of is a model of ϕ, then ϕ is formally provable from .
Formal provability is a relation between formulas of a particular language L that holds solely in virtue of the syntactical properties of the formulas involved. The interpretations that may be associated with the primitive relation symbols of L have no effect on whether a formula is formally provable from a given set of assumptions. Following Blanchette, we assume that Fregean consequence cannot be equated with formal provability because the interpretations associated with the primitive non-logical terms do actually matter. In particular, some of them may turn out to be definable in terms of others, which in turn might reveal previously hidden syntactical structure that may be exploited in formal proofs. The idea, then, is this: while formal provability of a given sentence from given premise-sentences might not be necessary for that sentence to be a consequence of the premise-sentences in Frege's sense, it does indeed seem to be necessary that the conclusion sentence is formally provable from the premise-sentences on some level of analysis of the primitive terms involved. The suggestion, then, is to define a broadly Fregean consequence relation by stipulating that an interpreted sentence S is a consequence of a given set of assumptions if and only if the relevant sentences can be transformed, via explicit definitions, into sentences such that the transformed sentence S is formally provable from the transformed assumptions. Based on this explication, we can also introduce explications of Blanchette's understanding of Fregean consistency and other concepts in the obvious way.
Definition 2.1: Suppose L is an interpreted language, a set of L-formulas, and ϕ an L-formula. Then (i) ϕ is an F-consequence of if there exists a definitional basis L for L and a definitional translation * over L such that * ϕ * .
If ϕ is an F-consequence of , we also say that ϕ F-follows from and write F ϕ.
A couple of comments to clarify. First of all, since our definition of F-consequence depends on a fixed language L, we should really write L F ϕ to make this dependence explicit. To avoid clutter, however, we will suppress this in what follows. (We will come back to this point later on.) Next, if we look at the definition of F-consequence, we can see that it has a 'syntactic' component in that formal provability plays a central role. On the other hand, F-consequence also has a 'semantic' component, because definitional translations enter the picture, which were defined in terms of the semantic notion of explicit definability in an interpreted language. Both components are important. The first ensures that consequence is characterised in terms of inferences (and, indeed, formal inferences of the kind we are used to since 1879); the second ensures that the intended interpretation of the language in question and the possibility of 'analysis' are taken into account. In particular, the notion of F-consequence is defined in such a way that information about the intended interpretation of a language, which is not reflected in a given set of assumptions, may become relevant. As an analogy, one may compare the concept of F-consequence with the semantics of modal idioms in possible worlds semantics. In order to determine whether a statement of the form 'It is possible that S' is true, it is not enough to check if S is true in the actual world. We also have to consider alternative possible worlds in which S might be true. The situation is analogous in the case of F-consequence. In order to determine whether an interpreted sentence F-follows from a given set of assumptions, it is not enough to check if the actual conclusion-sentence is derivable from the actual premise-sentences. We also have to consider alternative representations of the contents determined by those sentences (speaking somewhat loosely), and check if an alternative representation of the content associated with the conclusion-sentence is derivable from alternative representations of the contents associated with the premise-sentences.

F-Consequence: Basic Properties
Clearly, the concept of F-consequence is non-standard, and as far as I am aware, has not been studied in the literature in detail. At least in part, this may be due to considerations of purity. As we saw, the concept of F-consequence is the result of syntax as well as semantics. So, some might consider F-consequence to be the result of an unholy alliance between notions that should be kept separate. Still, in many respects, F-consequence is a well-behaved consequence relation with standard properties. In what follows, we will establish a couple of basic properties of F-consequence and its related concepts. We start with the following basic result: Theorem 2.1: Suppose L is an interpreted language, a set of L-formulas, and ϕ an L-formula. Then, Proof: For (i), note that L is a definitional basis for itself. Consider the trivial definitional translation * over L. Then * = and ϕ * = ϕ, and the claim follows from the assumption that ϕ. For (ii), suppose is formally inconsistent, i.e. ψ & ¬ψ for some ψ. Then by i), F ψ & ¬ψ and so is F-inconsistent. (iii) follows from i) for = ∅.
Theorem 2.1 ensures that formal provability in a reasonable calculus entails F-consequence. Note though, that the converses of (i), (ii), and (iii) are not generally true.
To illustrate, consider the interpreted language that contains two binary relation symbols S and P, which represent the successor and predecessor-relation over the natural numbers respectively; and suppose that s is a variable-assignment that assigns the numbers 0 and 1 to x and y respectively. Clearly, (1) Sxy Pyx It is straightforward to construct an alternative interpretation in which the premise is true and the conclusion is false. However, since the successor relation is explicitly defined (in the intended interpretation) in terms of {P} by the formula Pyx, we have (2) Sxy F Pyx Moreover, note that the formula Pyx & x = y also explicitly defines the successor relation S in terms of {P}. So, once again, while So, from the assumption that the number 1 succeeds 0, it F-follows that these numbers are distinct. Similar examples show that formal consistency does not generally imply F-consistency and that F-validity does not generally imply formal validity. A basic property we expect from any plausible notion of consequence is that it preserves truth. As the following theorem shows, this is indeed the case for F-consequence. Proof: Follows from the definition of F-consequence and Lemma 2.1.
The following theorem shows that F-consequence has several further properties one would expect from a well-behaved consequence relation. All of the items in Theorem 2.3 are immediate consequences of their counterparts for formal provability:

Theorem 2.3 (Consequence Properties):
Suppose L is an interpreted language, , are sets of L-formulas and ϕ, ϕ are L-formulas. Then, Proof: Since any interpreted language L is a definitional basis for itself, (i) follows from the existence of the trivial definitional translation over L and the fact that is reflexive. For (ii), suppose that there exists a definitional translation * over some definitional basis L of L such that * ϕ * . Since ⊆ , we have * ⊆ * . By the monotonicity of , we have * ϕ * , and so F ϕ. The other items can be proved in a similar way. 7 F-consequence is defined in terms of the notion of a definitional translation, which in turn relies on the somewhat opaque notion of substitution. The following theorem shows that we can do away with substitution and characterise F-consequence in terms of canonical extensions of a given set of assumptions. For this, recall that a D-biconditional over some definitional basis L of L is a formula of the form where R is a relation symbol in L\L and ξ(x) is an L -formula. Call a set of D-biconditionals over some basis L of L appropriate if it contains at most one D-biconditional for each relation symbol R in L\L . We have the following: Theorem 2.4: Suppose L is an interpreted language, a set of L-formulas, and ϕ an L-formula. Then the following are equivalent: There exists a definitional basis L of L and an appropriate set of D-biconditionals over L such that ∪ ϕ.
Proof: For the direction from (i) to (ii), suppose there exists a definitional translation * over some definitional basis L of L such that * ϕ * and consider the set of D-biconditionals over L that correspond to the translation * . First, note that is appropriate. Also, using basic properties of our (complete) proof procedure, it is straightforward to show by induction that The reader will have noticed that the important Cut-property, i.e. the property that F ϕ and ∪ {ϕ} F ψ implies ∪ F ψ, is missing in Theorem 2.3. I have not been able to determine whether F-consequence has the Cut-property.
for all L-formulas χ. Since each ϕ in is trivially derivable from and ⊆ ∪ , it then follows from (1) that the translation (ϕ ) * of each formula ϕ in is derivable from ∪ . Given our assumption that * ϕ * , it follows by the transitivity of formal provability that ∪ ϕ * . Hence, using (1) again, we have (2) ∪ ϕ For the direction from (ii) to (i), suppose that L is a basis for L, that is an appropriate set of D-biconditionals over L for (some or all of) the vocabulary in L\L , and that there exists a correct derivation of ϕ from ∪ , i.e. a finite sequence of formulas ϕ 1 , . . . ϕ n such that ϕ n = ϕ, and for each ϕ i we either have (1) ϕ i ∈ , or (2) ϕ i is an instance of a logical axiom schema, or (3) ϕ i results from earlier formulas by modus ponens or universal generalisation, or (4) ϕ i ∈ . Now, consider the definitional translation * over L such that for each D(R, ξ) in , Rx is mapped to ξ(x). Then it is easy to see that the sequence ϕ 1 , . . . ϕ n can be transformed into a correct derivation of ϕ * from * as follows: First, we replace each ϕ i with its translation ϕ * i and note that (1) if ϕ i ∈ , then ϕ * i ∈ * , (2) if ϕ i is an instance of a logical axiom schema, then so is ϕ * i , (3) if ϕ i results from earlier formulas in the original sequence by modus ponens or universal generalisation, then so does ϕ * i with respect to the corresponding formulas in the new sequence. Finally, if (4) ϕ i ∈ , then its translation ϕ * i is of the form for some ξ(x) that explicitly defines some R in L\L in terms of L . Since each of these formulas is formally valid, we can include appropriate formal proofs before each of them. This concludes our transformation procedure, and we thereby obtain a formal proof of ϕ * from * . Hence, F ϕ as desired.
So, F-consequence from can be characterised in terms of formal provability from plus additional assumptions that make explicit the equivalences between the definiendumexpressions and the corresponding definiens-expressions.

Non-F-Consequence and Formality
While F-consequence has several standard properties, it is unusual in other respects. In what follows we want to look at some of the more uncommon properties of F-consequence and its cognates.
First of all, one of the features of F-consequence that some might find problematic is that, due to the interplay between semantics and syntax in its definition, F-consequence is a fairly complex notion. To illustrate, compare the notions of F-validity and formal validity. It is well-known that formal validity (i.e. formal provability from no assumptions) is not decidable. That is, there is no mechanical procedure that decides whether a given formula ϕ of a given first-order language L is formally valid. On the other hand, formal validity is at least semi-decidable. So there exists an effective procedure that enumerates all and only the formally valid formulas. A brute force method would run as follows. First, we make a list of all finite sequences of formulas of a given language L. We then go through the list of sequences one by one. Since it is decidable whether a formula is a logical axiom, and it is decidable whether a formula is correctly derived from others by one of the inference rules, for each sequence it is effectively decidable whether it is a correct derivation of its last formula. If it is, we write down its last formula on a separate list and go to the next sequence, decide whether it is a correct derivation, and so on. This procedure ensures that each formally valid formula will eventually appear on the separate list.
What about F-validity? First of all, since F-validity is defined in terms of formal provability, F-validity is undecidable just like formal validity. But unlike formal validity, F-validity is not even semi-decidable. Informally, this can be seen by noticing that it is not enough to merely make a list of all correct derivations from the empty set of assumptions. In light of Theorem 2.4, we also have to consider, for each definitional basis L of L and each appropriate set of D-biconditionals over L , all the correct derivations from . In particular, this involves determining which formulas explicitly define relation symbols in L in terms of certain sublanguages of L. Obviously, there is no effective way to do this in general for any given interpreted language. So, the F-valid formulas cannot even be effectively enumerated.
(Similar results apply to F-consequence and F-consistency.) There is another peculiarity of F-consequence and its cognates. By definition, to show that a formula of an interpreted language L is an F-consequence of a given set of assumptions, we have to find a definitional translation over some basis of L as well as a formal proof of the translated formula from the translated assumptions. But what about negative F-consequence claims, i.e. claims to the effect that a certain formula is not F-valid, that a certain formula is not an F-consequence of a given set of assumptions, or that no contradiction F-follows from a given set of formulas (F-consistency)?
It follows from Theorem 2.2 that no false sentence F-follows from a given set of assumptions that are true in their built-in interpretation. But what about truths? First of all, we have the following Lemma 2.2: Suppose L is an interpreted language, a set of L-formulas, and ϕ an L-formula. If and ϕ contain no non-logical terms, then

is formally consistent, then is F-consistent. (iii) if ϕ is F-valid, then ϕ is formally valid.
Proof: (i) Suppose that * ϕ * for some definitional translation * over some definitional basis L of L. Since neither nor ϕ contain any non-logical terms, we have * = and ϕ * = ϕ, and so the claim follows from the assumption that * ϕ * . (ii) and (iii) follow from (i).
Lemma 2.2 ensures that F-consequence collapses into formal provability for formulas that contain no non-logical vocabulary. In this case, we can show that ϕ does not F-follow from just as we do for non-derivability; namely, by constructing a model (i.e. a set together with a variable assignment) in which the premises are true, but the conclusion is false. Similarly, we can show that ϕ is F-invalid by constructing a model in which ϕ is false, and we can show that is F-consistent by constructing a model in which each formula in is true. Another consequence of Lemma 2.2 is that the notion of F-validity for a given interpreted language L does not simply collapse into truth in L. For example, ∃x∃y(x = y) is true in every interpreted language whose domain contains at least two elements. However, since ∃x∃y(x = y) is not formally valid and contains no non-logical terms, it follows from Lemma 2.2 that it is not F-valid either.
Beyond these cases, the situation becomes more difficult. Indeed, one of the central themes of Blanchette's reading of Frege is that, in general, it is much harder to establish negative consequence claims in Frege's sense than non-derivability claims. This can also be observed for F-consequence. Remember that the key result that enables semantic non-derivability proofs is that derivability is sound with respect to model-theoretic consequence. So, whenever a sentence is formally provable from a set of assumptions, then every model of those assumptions is a model of the conclusion. This enables us to establish that a sentence is not formally provable from a given set of assumptions by constructing a model in which the latter are true, but the former is false. Obviously, this method cannot be used to show that a sentence is not an F-consequence of a given set of assumptions. Formal provability is tied to a particular syntactic representation of the assumptions and the conclusion, and the interpretations of the primitive non-logical terms have no effect on (non-)derivability. However, in the case of F-consequence, we have to consider alternative syntactic representations -and the interpretations do matter.
Fortunately, the situation is not as dire as it may seem, and there are methods to establish negative F-consequence claims, even if we consider formulas that contain non-logical vocabulary. First, remember that we have called an interpreted language L 'fully analysed' if no non-trivial definitional basis L of L exists; that is, if none of the primitives of L are explicitly definable in terms of others. We have:

is formally consistent, then is F-consistent. (iii) if ϕ is F-valid, then ϕ is formally valid.
Proof: For (i), suppose that * ϕ * for some definitional translation * over some definitional basis L of L. Since L is fully analysed, there is no non-trivial definitional basis L of L. Hence, * must be the trivial definitional translation and so L = L. So, * = , ϕ * = ϕ, and the claim follows from the assumption that ϕ. (ii) and (iii) follow from (i).
Thus, for fully analysed languages, F-consequence collapses into formal provability, and we can use the usual model-theoretic methods to establish non-derivability. The question is: how do we establish that a language is fully analysed? As mentioned, there is no general method to decide whether a non-logical term of some language L is explicitly definable in terms of others. However, there are methods, albeit incomplete ones, to show that a given relation is not explicitly definable in terms of others. One simple method relies on the notion of an automorphism. Suppose L is an interpreted language. Then, a bijective function α : D −→ D is an L-automorphism if for each k-ary relation symbol R in L, and all d ∈ D k : Hence, an L-automorphism is simply an isomorphism between the intended interpretation of L and itself. Now, according to a basic model-theoretic result, sometimes called the 'isomorphism lemma', isomorphic models satisfy the same formulas. Since an automorphism is a special kind of isomorphism, we can use the isomorphism lemma to prove the following: Lemma 2.4: Suppose that L and L are interpreted languages, L is a sublanguage of L, and R is a k-ary relation symbol in L\L that is explicitly definable in terms of L . Then for each L -automorphism α and all d ∈ D k : Proof: Suppose that R is explicitly defined by the L -formula ξ(x) and that α is an L -automorphism. Then for all d ∈ D M : where we use the isomorphism lemma in the third equivalence and the fact that ξ(x) explicitly defines R in the first and fifth equivalence.
Thus, relations that are explicitly definable in terms of a sublanguage of a given language are preserved by automorphisms of the sublanguage. 8 The significance of Lemma 2.4 is that it gives us a simple method to establish that a primitive relation symbol R of some interpreted language L is not explicitly definable in terms of the remaining primitives. One must simply find an automorphism α that 'moves' the intended interpretation R M of R; that is, an α such that α(R M ) = R M . 9 To sum up: in order to show that a given L-formula ϕ is not an F-consequence of a set of L-formulas , it is sufficient to show that, first, ϕ is not formally provable from , and second, that L is fully analysed, which in turn can be established by finding appropriate automorphisms that show that none of the primitives of L are definable in terms of the remaining ones. 10 To conclude this section, let me emphasise a point which should be clear from what has been said in this section, but which is worth making explicit. In the study of abstract consequence relations, a basic feature of consequence relations is the requirement of structurality or substitution invariance. In this context (using our terminology from earlier), a consequence relation is said to be 'structural' if for any set of L-formulas , any L-formula ϕ, and any substitution translation * (STR) If ϕ, then * ϕ * .
Many consequence relations are structural. Indeed, some logicians consider the structurality constraint (STR) to be a defining characteristic of a consequence relation. Clearly, (STR) fails for F-consequence. To see this, consider the example from earlier, where we have shown that (3) Sxy F x = y Now, suppose that * is a substitution translation that replaces atomic formulas Sxy by identities x = y. We then get an argument with the premise x = y and the conclusion x = y.
Since the premise and the conclusion no longer contain any non-logical terms, there is nothing left to be 'analysed' and so by Lemma 2.2, F-consequence collapses into formal provability. Obviously, x = y is not formally provable from x = y. Hence, Therefore, F-consequence is not structural. Since the structurality requirement is just a precise way to express that a consequence relation is 'formal', there is a precise sense in which F-consequence is not formal. 11 Of course, given our motivation for considering broadly Fregean consequence relations in the first place, this should not come as a surprise.

Discussion
In the previous section, we have seen that it is straightforward to define broadly Fregean consequence relations for first-order languages, i.e. ones that apply to sentences of an interpreted first-order language and respect the requirements (BF 1) and (BF 2). At least to this extent, F-consequence can be understood as a 'good' explication of the informal conception of consequence that Blanchette attributes to Frege. In what follows, we want to take a closer look at some of the presuppositions of our approach, how these presuppositions align with Frege's own views, and what our explication may teach us about Blanchette's take on these views as far as the 'heart of the Fregean picture of entailment' is concerned.

Fregean Consequence and Logical Consequence
Blanchette often presents her account of Fregean consequence as a conception of logical consequence that is opposed to modern notions of logical consequence. 12 Of course, it is partly a terminological matter whether we want to refer to a certain notion of consequence as 'logical consequence'. According to modern nomenclature, logical consequence is usually understood as a relation that holds between sentences (interpreted or not) in virtue of their logical form, which in turn is determined by a set of logical constants and how they are arranged in the relevant sentences. Relative to some fixed choice of logical constants, logical consequence can then be defined either proof-theoretically or model-theoretically. In either case, logical consequence is a purely formal relation between sentences that is independent of the meanings of the non-logical terms that occur in the relevant sentences. In what follows we will refer to conceptions of this sort as conceptions of 'logical consequence in the narrow sense'. Following Blanchette, the basic idea governing our approach to Fregean consequence was that the meanings of the non-logical terms are not irrelevant. Our explication was based on the idea that an interpreted sentence is a Fregean consequence of a set of sentences, just in case some definitional alternative of the former is a logical consequence in the narrow sense of definitional alternatives of the latter. Since, in the previous section, we have been restricting ourselves to first-order languages, we have naturally settled for first-order consequence (in its proof-theoretic guise) as a basis for our definition of Fregean consequence. But the general template underlying our definition determines an entire family of broadly Fregean consequence relations, depending on the choice of logical consequence in the narrow sense, and our particular choice is neither forced nor suggested by anything Frege says. It is well-known that many definitions that Frege came up with in the course of pursuing his logicist project rely on the presence of higher-order quantification; for example, his definition of numerical equivalence between concepts, his analogue of the membership-relation, or his definition of the ancestral of a relation. Of course, nothing prevents us from adapting our definitions from Section 2 to account for higher-order languages. 13 However, in order to obtain a determinate notion of consequence along the lines of the previous section we have to settle for some notion of logical consequence in the narrow sense, and it is not clear, from Frege's point of view, what this should be. While Frege was aware of the problem of determining 'what counts as a logical inference and what is proper to logic' (Frege 1984, p. 339), he never actually provides such an account. This point has been emphasised by Tom Ricketts (among others), who notes that Frege had no 'precise demarcation of the logical' and no 'precise overarching conception of logic' (Ricketts 1997, p. 184); a point that's also acknowledged by Blanchette. 14 While I disagree with some of the general conclusions regarding Frege's stance towards 'metalogic' that Ricketts and others have drawn from this observation, I think it is a fair point. What is important for our purposes is that even though Fregean consequence, as conceived by Blanchette, cannot be identified with any particular concept of logical consequence in the narrow sense, if the general template underlying our explication of her account is on the right track, then it is still based on one. Thus, in the absence of a determinate conception of what, according to Frege, is ultimately the 'correct' notion of 13 However, unlike in the case of first-order logic, the proof-theoretic and model-theoretic concepts of logical consequence in the narrow sense do not coincide for higher-order languages. Thus, in the case of higher-order languages, the concept of Fregean consequence as based on the proof-theoretic notion of logical consequence in the narrow sense does not coincide with the concept of Fregean consequence as based on the model-theoretic notion of logical consequence in the narrow sense either. 14 See Blanchette 2012, pp. 147, 171. It should be noted that Frege is in good company with this omission and questions about what is 'proper to logic' persist to this day. Also, as Ricketts is aware, since Frege's project of vindicating logicism did not require a 'precise demarcation of the logical', it is understandable why he was not too bothered by this problem.
logical consequence in the narrow sense, we are still left with the question: which of the infinitely many broadly Fregean consequence relations corresponds to the conception of consequence that Frege actually endorsed?

Non-Creativity and Informativeness of Definitions
One of the central ideas we were aiming to capture in our characterisation of F-consequence was the idea that sometimes the 'right' definition may enable us to see that a proposition, despite appearances, is a Fregean consequence of a given set of assumptions after all because it may provide new syntactic information that can be exploited in formal proofs. 15 But how does this square with the view, often stressed by Frege himself, that proper definitions are purely stipulative, and hence, must not enable us to prove something that is not provable without them? 16 To answer this question from the point of view of our explication, we first need to remind ourselves of the distinction between explicit definability in a language and explicit definability in a theory. We have introduced the former notion in Section 2, where it formed the basis of our explication of F-consequence. To repeat, a primitive relation symbol R of an interpreted language L is explicitly definable in a sublanguage of L, just in case there exists a formula ξ of the sublanguage whose extension in the intended interpretation of L coincides with that of R; that is, if the corresponding D-biconditional D is true in the intended interpretation of L. If this is the case, let us say that ξ explicitly defines R in the expressive sense. By contrast, R is explicitly definable in an L-theory if the corresponding D-biconditional D is not merely true in the intended interpretation of L, but -provably so. If this is the case, let us say that ξ explicitly defines R in the deductive sense (relative to ). It can easily be shown that if R is explicitly defined by ξ in the deductive sense in the theory , then the corresponding D-biconditional D adds nothing to in terms of proof-theoretic strength. That is, every R-free formula that is formally provable from together with D can be proved from without D (non-creativity or conservativeness). Moreover, for every formula that contains the relation symbol R there is a -provably equivalent formula that does not contain R (eliminability). Given these two properties, D can be understood as a 'stipulation' relative to the theory , because the definition D gives us no more and no less information about the meaning of the defined term R than is already contained in . 17 The obvious but important point here is that a primitive expression may be explicitly definable in an interpreted language without being explicitly definable in a given theory of that language. So there is no contradiction in claiming that some definitions may reveal new consequences, but at the same time requiring that definitions should be non-creative. We just have to bear in mind that the term 'definition' is used in the expressive sense in the former context, but in the deductive sense in the latter. Obviously, the distinction between definitions in the expressive sense, and definitions in the deductive sense, was not made by Frege. But it does enable us, perhaps in some non-intended way, to account for the central aspect of Frege's conception that is emphasised by Blanchette; namely, that sometimes the 'right' definition may enable us to prove new theorems.
The concept of F-consequence developed in the previous sections also throws new light on a related debate between Wilfrid Hodges and Blanchette about the role of analysis in Frege's conception of consequence. 18 The background of the debate is a passage from Frege's 'Logik in der Mathematik' where he introduces a distinction between 'analytic' and 'constructive' definitions. Constructive definitions are characterised by Frege as arbitrary stipulations where a previously meaningless symbol is assigned a sense in virtue of a definition. By contrast, analytic definitions are supposed to capture the sense of a symbol already in use, and thus, are not stipulative. Hodges points out that in the subsequent discussion of analytic definitions, Frege says that, at least in certain cases, an analytic definition is 'really to be regarded as an axiom.' 19 According to Hodges, this passage suggests that analysis cannot play the role in Frege's conception of consequence that Blanchette thinks it does, because in his interpretation, analysis does not enable us to prove new theorems from a given set of assumptions, but rather, from these assumptions together with additional axioms that express the relevant analytic definitions in the form of 'analysis biconditionals' (Hodges 2004, p. 138). In her response, Blanchette maintains that Hodges misconstrues the role of analytic definitions in Frege, noting that 'no Fregean proof includes such a biconditional' (Blanchette 2007, p. 336).
Blanchette is certainly right that analytic definitions never appear in Fregean derivations in the form of axioms (of course, they constantly appear in the form of formulas that express stipulative definitions). However, the appearance of a contradiction here is based on an equivocation. When Hodges denies that in the relevant cases the conclusion 'follows from' the premises alone, he is merely pointing out that the conclusion-sentence is not a formal consequence of the relevant premise-sentences alone. Instead, additional analysisbiconditionals must be added to make the conclusion-sentence a formal consequence of the premise-sentences. But that does not contradict Blanchette's claim that the thought expressed by the conclusion-sentence is nonetheless a 'Fregean consequence' in her sense of the thoughts expressed by the premise-sentences. Indeed, in our setting, explications of Hodges' and Blanchette's interpretations are provably equivalent. As we saw in Section 2.3, an interpreted sentence S is an F-consequence of assumptions S 1 , . . . S n (and only those assumptions) just in case S is formally provable from S 1 , . . . , S n together with the appropriate 'analysis-biconditionals' (Theorem 2.4). This suggests that Blanchette's and Hodges' readings are really just different ways of saying essentially the same thing: that an analysis can be informative in that it may provide new information that is not encoded in the syntactic structures of the original assumptions S 1 , . . . , S n . It is just that the idea of 'informative analysis' is implemented in different ways in Hodges' and Blanchette's readings.

Analysis and Reduction
In our formal model of Fregean consequence from Section 2, we have been using explicit definability in a language as an explication of the informal concept of 'analysability'. According to this explication, it is both necessary and sufficient for an analysis to capture the right extension to be correct. This, of course, may be hard to swallow. It clearly seems that something more than extensional adequacy is required for an analysis to be correct, although it is notoriously hard to pin down what this 'more' could be. While I think that some of the unease with this feature of our explication can be mitigated, the fact remains that our choice to proceed in a purely extensional framework puts limits on potential explications. In any case, capturing the right extension seems at least necessary as long as we are concerned with the analysis of particular expressions of a fixed interpreted language that have a definite extension relative to the domain of the intended interpretation of that language. In her discussions of Fregean consequence, Blanchette frequently refers to analyses of that kind; for example, when she discusses a passage in Frege's 'Boole's rechnende Logik und die Begriffsschrift'. 20 To illustrate the power of his Begriffsschrift, Frege there shows how to formally prove the proposition that the sum of two multiples of a number is a multiple of that number, based on an analysis of the multiple-relation in terms of the ancestral, using only the associativity law for addition, the proposition that for all n, n + 0 = n, and second-order logic. 21 In this case, the analysed expression ('being a multiple of') is assumed to have a determinate extension over the natural numbers, which is easily seen to be captured by Frege's definition.
However, not all 'analyses' in Frege's work are of this kind. Although Frege does not do so, it will be useful to distinguish non-foundational analyses of the sort just mentioned from foundational or reductive analyses, where a fundamental domain of mathematical objects is somehow reduced to another domain of objects. The prime example of such an 'analysis' is Frege's reduction of cardinal numbers to classes. 22 While the requirement that reference is to be preserved is plausible as a necessary condition in the case of non-foundational analyses, by definition, it makes no sense in the case of reductive analyses. In order to account for analyses of this kind, a natural generalisation of the approach from Section 2 would be to consider relative interpretability instead of definability. Similar to definability, relative interpretability comes in both an expressive and a deductive version. Roughly, an interpreted language L is relatively interpretable in another language L in the expressive sense if an isomorphic copy of the intended interpretation of L is definable in L , and an L-theory is relatively interpretable in the deductive sense in an L -theory if there is a translation of L-sentences to L -sentences such that the translation of each -theorem is a -theorem. Using either of these notions to re-define F-consequence, Frege's reduction of numbers to classes can then be understood as an interpretability-result where arithmetic 20 See Blanchette 2007;Blanchette 2007, pp. 323 ff.;and Blanchette 2012, pp. 18 ff. Frege's manuscript is reprinted in Frege 1979, pp. 27 ff. 21 The proof of the theorem uses a form of the principle of induction that is a consequence of Frege's general definition of the ancestral from § 26 of his Begriffsschrift (Frege 1879). 22 In Frege 1884, § § 68-69, Frege defines the cardinals as extensions of certain higher-level concepts. Specifically, the number of F's is defined as the extension of the concept 'being equinumerous with F', where equinumerosity between two concepts is defined in terms of the existence of a bijective mapping between the two concepts. In his Grundgesetze (Frege 1893(Frege /1903, Frege introduces 'value-ranges' and defines the numbers in terms of certain value-ranges. is shown to be relatively interpretable (either in the expressive or deductive sense) in what Frege at some point believed to be 'pure logic'. The problems with an approach such as this to reductive analyses as an explication of Frege's actual views are well-known. First, interpretability (in either the expressive or deductive sense) does not seem to be sufficient to achieve successful reduction. For example, it is straightforward to see that geometry can be interpreted in arithmetic via the link to analytic geometry; a fact that Frege was obviously aware of (naturally, he would not use this terminology). But in spite of this, Frege strongly believed that geometry cannot be reduced to arithmetic in any philosophically significant sense. 23 Secondly, multiple reductions of one structure (or theory) to another are possible. The structure of the natural numbers, for example, can be interpreted in set theory in various ways. The upshot of these considerations is that from Frege's point of view, the requirement that analysis needs to preserve reference is too restrictive for reductive analyses, but the requirement that only structural (or deductive) relationships need to be preserved is too liberal. Blanchette, being well aware of this problem, provides an account of what she believes Frege thinks is preserved in the particular case of his logicist reduction of numbers (Blanchette 2012, pp. 89 ff.). However, like Frege, she does not give us any general adequacy-conditions for reductive analyses. Since, on the view Blanchette attributes to Frege, there is no 'readily-verifiable criterion for the adequacy of analysis,' and it is therefore 'open to philosophical debate' whether a proposed analysis is adequate (Blanchette 2012, p. 104), it is equally 'open to philosophical debate' whether some interpreted sentence (or the thought it expresses) is a Fregean consequence of a given set of assumptions. I am not convinced that Frege would be happy with such a state of affairs, but it may well be that he is committed to such a view.
The main takeaway here is that Blanchette's understanding of Fregean consequence can be explicated in an extensional setting along the lines proposed in Section 2 for nonfoundational analyses in a way that arguably preserves the spirit of her interpretation. However, reductive analyses, like the ones that Frege is appealing to in his logicist reduction, present us with additional difficulties. These difficulties make it seem unlikely that Blanchette's conception of Fregean consequence (or what she takes to be 'at the heart' of Frege's conception of consequence) can be explicated along the lines discussed in this article.

Independence Proofs and Clearly Delineated Languages
As mentioned, Blanchette initially developed her account of Fregean consequence in the context of a discussion of Frege's views on independence and consistency proofs as they emerged in his controversy with Hilbert. Obviously, I cannot discuss all the relevant aspects of this controversy here in detail. 24 So, let me make a couple of observations that are connected to our explication from Section 2.
One of Blanchette's central claims is that while nothing in Frege's conception of logic prevents him from acknowledging the success of model-theoretic techniques in establishing that a sentence is not formally provable in some calculus from a given set of assumptions, the situation is radically different when it comes to proofs of independence in Frege's sense, i.e. proofs to the effect that a certain thought is not a Fregean consequence of a given set of thoughts. The standard technique of constructing counter-interpretations is not suitable to establish that a thought is not a consequence, in Frege's sense, from a given set of thoughts because analysis may reveal consequence relations between thoughts that are not reflected in the syntactic structures of the particular sentences that express them. Also, as Blanchette makes clear, the same is true if we are dealing with interpreted sentences instead of Fregean thoughts. However, Blanchette notes that model-theoretic techniques can be used to establish independence in Frege's sense for languages that are 'fully analysed'; that is, languages whose primitives cannot be further analysed. The problem, according to Blanchette, is that 'there is from Frege's point of view never a guarantee that the language in question is in fact 'fully analysed' in the sense in question' (Blanchette 2012, p. 129).
Restricting ourselves to non-foundational analyses, we have seen that in the framework we have been adopting in Section 2 there is no difficulty in explicating the condition that a language is 'fully analysed' (see Section 2.1). A language L is fully analysed just in case none of its non-logical terms are explicitly definable in terms of a sublanguage of L. Furthermore, standard techniques can be used to establish independence results with respect to F-consequence for such languages (see section 2.4, Lemma 2.3). However, in contrast to what Blanchette claims for her informal conception of Fregean consequence, in the framework we have been adopting here, nothing prevents us from showing that a language is in fact fully analysed, at least in certain cases (see section 2.4, Lemma 2.4). I am not suggesting that Frege was aware of these methods, but assuming that our framework models Blanchette's views on Fregean consequence in central respects, their existence seems to indicate that Frege is not committed to rejecting the possibility of independence proofs with respect to Fregean consequence after all.
Does this mean that Blanchette is wrong with her claim that 'there is from Frege's point of view never a guarantee that the language in question is in fact 'fully analysed' in the sense in question'? Not quite. Remember that we have assumed that F-consequence applies to sentences of a clearly delineated, interpreted language in the modern sense. Such a language is determined by a fixed stock of non-logical symbols (among other things). One important consequence of this is that the choice of the particular language for which F-consequence is defined determines whether a non-logical relation is definable in terms of the remaining ones. Add some conceptual resources to an interpreted language, and a primitive term that was undefinable in the original language may turn out to be definable in the expanded language. As a result, one and the same interpreted sentence may be an F-consequence of a set of assumptions if all these sentences are considered to be part of a language L, but it may fail to be an F-consequence of these assumptions if the relevant sentences are considered to be part of a language L that lacks certain conceptual resources of L that are required to express certain analyses. Thus, F-consequence is highly sensitive to the ambient language for which it is defined.
I do not think that the foregoing considerations disqualify F-consequence as a reasonable conception of consequence. But the underlying assumption of a clearly delineated language does seem to be at odds with views that can plausibly be ascribed to Frege -views that also seem to be presupposed by Blanchette's reading of Frege. In particular, it is not clear that Frege thought about consequence in terms of 'languages' (plural) whose basic vocabulary is fixed, and the fact that Frege takes thoughts to be the real objects of consequence is just one indication of this. Since one and the same thought can be expressed in a variety of different ways that may not be available in any fixed clearly delineated language, it is not clear why we should focus on any one of them in particular. The question whether a concept is 'analysable' should, therefore, not be understood as pertaining to any fixed language, but in some 'absolute' sense. In other words, the implicit existential quantifier in 'analysability', according to this reading of Frege, should be understood as ranging over all potential analyses, not just ones that can be expressed in some fixed language.
The situation is complicated, but it is plausible to assume that in most contexts, Frege did in fact think about 'analysability' or 'definability' as an absolute notion. As we saw, Blanchette's conception of Fregean consequence also seems to presuppose such a conception. Otherwise, her claim about the impossibility of making sure that a language is in fact 'fully analysed' would be problematic, if not plainly wrong. Studying such a notion of 'absolute definability' and how it relates to the informal notion of mathematical consequence may be a fruitful endeavour. 25 However, our formal model of Fregean consequence is tied to a relative conception of definability. Therefore, whatever its merits may be in other respects, in this particular respect it fails to be faithful to Blanchette's conception of Fregean consequence.

Conclusion
The results of the previous sections can be summarised as follows. First, we have seen that it is straightforward to define broadly Fregean consequence relations, i.e. consequence relations that apply to sentences of a fully interpreted language and that meet the requirements (BF 1) and (BF 2), which seem to be the central characteristics of Blanchette's understanding of Fregean consequence. Second, however, we have also seen that some of her claims about Frege's conceptions of consequence and analysis are based on additional assumptions about Frege's views on logic, language, and mathematics. Finally, I also hope to have shown that independently of the question whether the proposed explication is faithful to Blanchette's account, and independently of the question of whether her account is faithful to Frege's actual views, her interpretation touches upon questions that are central to the philosophy of logic and mathematics. 26 Of course, the particular explication of Blanchette's account presented here is the result of several idealisations. I do not think that there is anything wrong with this, as long as the precise concept is still explanatory with respect to paradigmatic cases and gives rise to interesting further questions, both technical and philosophical. And while a lot of work remains to be done, I hope to have shown that the explication presented in this article meets this requirement.