The objective of Gaskin's book is to develop and defend a solution of the problem of the unity of the proposition (the unity problem, for short). Gaskin's assessments of the views of Bradley, Frege and Russell prepare his own proposal for the solution of the unity problem. Further sources are ancient, medieval, modern and contemporary discussions of the problem.

The unity problem is the lack of a satisfactory answer to the question ‘What distinguishes the list of items a, R, b from the state of affairs that aRb?’ If we assume that there is a unifying object X distinct from a, R, b, we face the question what unifies X with a, R, and b. The original question re-occurs and we start an infinite regress. According to one reading, Frege and Russell argued that R is unsaturated and can therefore be saturated by a and b into aRb. But unsaturatedness seems just as mysterious as the problem it is supposed to solve and it even raises further problems.

Gaskin's discussion of the problem proceeds in six chapters. The chapters are dense and full of detailed argument. The following can only be a very rough summary of their contents:

Chapter 1 gives a thorough discussion of the assumptions that shape the unity problem. It introduces the main concepts (list, sentence, proposition etc.) and the doctrine of Linguistic Idealism. (42ff. See also 241ff). Linguistic Idealism takes sentences and their properties to be the primary data and derives sense and reference when trying to account for the semantic and logical relations between sentences. Interestingly, Gaskin identifies Bentham as the father of Linguistic Idealism. (188)

The main part of Chapter 2 proposes for a revision of Frege's semantic map. Frege held that an assertoric sentence in whose truth we are interested expresses a thought and purports to name a truth-value: either the True or the False. Gaskin makes a good case for the view that sentences refer to states of affairs, worldly propositions that mirror on the level of reference Fregean thoughts. The world is the totality of states of affairs.

Chapters 3 and 4 jointly argue against the Frege/Russell strategy to solve the unity problem by appealing to the unsaturatedness of concepts and concept-words. Gaskin argues that there is no convincing way to establish that singular terms are saturated, while predicates are unsaturated. The further discussion proceeds on the assumption that unsaturatedness cannot be used to solve the unity problem.

In chapter 5 Gaskin takes an important step towards his solution. Consider a language in which statements are made by configurations of names. What distinguishes configurations of names that say something, sentences, from those that do not? A sentence contains a logical copula that expresses the copulative structure of the sentence (297). For example, the logical copula in ‘Socrates is wise’ is ‘ξ is ϕ’. The logical copula unifies the different elements, which can all be (un)saturated, into a sentence. Gaskin holds that the logical copula refers to the instantiation relation. This assumption lands us in a regress.

Chapter 6 develops Gaskin's solution of the unity problem. One of his analogies helps us to understand the proposal (356). A denumerably infinite set can be a proper subset of another denumerably infinite set. Leibniz held that this shows that the infinite is paradoxical. Cantor uses the same fact to define when a set is infinite. Similarly, Gaskin argues that the infinite regress does not show that propositional unity is problematic. On the contrary: ‘My position is that one secures unity by producing and insofar as one produces a structure that generates Bradley's regress.’ (371)

The Unity of the Proposition is rich in philosophical and historical content. It is a challenging and rewarding text for an advanced graduate seminar and every philosopher who is interested in the unity problem will need to take it into account. In this review I cannot do justice to the complex web of arguments Gaskin has crafted. Instead I will have a closer look at some ‘subplots’ in his story.

1. Linguistic Idealism. Linguistic Idealism, says Gaskin, may raise ‘a few hackles’ (vi). Here is one. According to Linguistic Idealism, reference is nothing but a relation posited by the semanticist when he systematically provides truth-conditions for sentences. Why believe this part of Linguistic Idealism? Because holding the opposite is incoherent; believing in a theory-independent relation of reference is based on the ‘ludicrous’ view that words go proxy for things (244. See also 66). However, it is even more ludicrous to take one's opponents to be committed to this view. Some reference realists argue that precursors of reference can be found in the psychology of non- or not-yet linguistic thinkers. Some higher animals are able to track objects through space and time and possess recognitional capacities. Exercising these abilities is referring to an object in thought. This notion of reference underpins the linguistic notion of reference. That ‘words are proxies for things’ plays no role in the development of this form of reference realism.

2. Sense, Reference and Understanding. Linguistic Idealism is supposed to deliver ontological insight. For example, Gaskin argues for the view that the world has propositional structure by starting from reflections on the truth and significance of sentences (118f). In doing so he uses principles that connect sense, understanding and reference. Roughly, the referent of an expression ‘a’ is the object thinking of which is necessary and sufficient to count as understanding ‘a’ (66). Gaskin's discussion of these connections contains an important criticism of Frege's view that definite descriptions refer to the objects that satisfy them (77–9). If a concept-word like ‘is green’ refers to a concept and not to the objects falling under the concept or the set comprising them, the same should hold for definite descriptions which are, after all, composed from concept-words. For example, ‘the victor of Austerlitz’ does not refer to Napoleon, but to the concept of being the unique victor of Austerlitz and it cannot fail to refer to this concept. With this in mind, Gaskin amends Frege's often-discussed principle that sense determines reference to: ‘sense necessarily presents a reference’ (74). Gaskin's amended principle does not allow for sense without reference. In contrast, the principle that Frege endorsed might be rendered as follows:

Frege's original principle allows for sense without reference. This is a reason to prefer it to Gaskin's amendment. For it will be difficult to convince people that ‘Vulcan’ and other empty names are senseless. Did the 19^th century astronomers not make assertions when they uttered ‘Vulcan is a planet’? Unfortunately, Gaskin does not address this and related worries.

Gaskin uses his methodology to argue that concept-words refer to Carnapian intensions and assertoric sentences to states of affairs (worldly propositions). Let's focus on concept-expressions (85–91). Gaskin's argument assumes Linguistic Idealism and two further premises:

Result: ‘is green’ refers to an intension.

However, both assumptions in Gaskin's argument are controversial.

Against 1: Some philosophers reject sets for philosophical reasons. Does this prevent them from understanding ‘is green’? Or can reflection on understanding predicates discredit set-theoretical scepticism?

Against 2: 2 strengthens a sufficient condition for acquaintance with a set to a necessary and sufficient condition. But already the sufficient condition, namely:

is open to counter-examples. Replace ‘F’ with ‘object of acquaintance’ and you get:

A set cannot be a member of itself. Hence, S can only be acquainted with members of the set, not with the set itself. Knowledge of the membership conditions of a set cannot consist in acquaintance with the set itself. (Thanks to Keith Hossack for the example.)

3. Propositions as worldly entities. Gaskin argues that assertoric sentences refer to states of affairs. When it comes to the composition of states of affairs, philosophers are torn between a part-whole model and a function-argument model. The part-whole model underwrites the idea that the state of affairs that Socrates is wise contains Socrates and Wisdom, but it seems not to capture the unity of state of affairs (97). The function-argument model in turn has no room for constituents of states of affairs. If the state of affairs that Socrates is wise is the value of the function ξ is wise for Socrates, it does not contain Socrates as a part. Different functions may yield the same value for different arguments. Hence, the value cannot contain the arguments and functions as parts. Gaskin argues that one can combine both models:

What stops us, among other things, is that functions cannot be conceived of as parts of wholes. Assume for reductio that the plus function is a part of the whole referred to by ‘7 + 5’. How can the plus function compose with 7 and 5 this whole and compose with other numbers infinitely many further distinct wholes? Do all numbers overlap each other? Such questions are better avoided than answered.

4. Frege Exegesis. Gaskin argues against the Fregean strategy to solve the unity problem that invokes unsaturated ‘things’, especially Fregean concepts. One of the reasons Gaskin gives is that there is no viable distinction between (complete) objects and (incomplete) concepts. Gaskin's arguments for the latter thesis ignore Frege's view that the relation of identity only hold between objects, not between concepts. (For instance, Frege, G. Posthumous Writings, 120). Gaskin takes Frege to waver between an intensionalist understanding of concepts and an extensionalist one according to which ‘concepts are identical provided that they have the same extension’ (37). But for Frege concepts are not identical by any criterion. And this may be an important reason to distinguish between concepts and objects.