1 Yablo on Subject-Matter Yablo’s book is one of the most impressive contributions to the philosophy of language in recent decades and a remarkable achievement in its own right. In his book, Yablo develops in rigorous fashion a highly original theory of subject matter and shows how it has important applications to a wide variety of different fields. There is not a topic upon which he touches which is not illuminated by his incisive and insightful observations; and the pages of the book are suffused throughout with a warmth and wit, which one would have thought impossible to achieve on such a dry and abstract topic. I am in large agreement with a great deal of what Yablo has to say and, indeed, in recent work (Fine ([2014], [2015a,b,c,d]) have pursued a similar approach. However, there are a number of differences between us; and, for this reason, I have thought it helpful to compare our views on a number of different topics. This has inevitably meant pointing out where we disagree and why I have thought it advisable to go in one direction rather than another. But the focus on our differences should not blind the reader to the underlying similarities, both in general orientation and in the treatment of particular topics. For the purposes of the comparison, I have, to a large extent, adopted my own terminology and notation, rather than Yablo’s. This is more from habit than preference; and I hope it is not too confusing to the reader. I have also suppressed most technical detail. The reader who wants more can consult my previously listed papers and Yablo’s “Aboutness Theory” (available from his website). I begin by presenting the general framework within which we both operate (§1). This involves taking seriously the idea that statements are made true by states or situations, rather than whole worlds. I then discuss the behavior, nature and role of subject matter, quite apart from any particular account of the notion (§§2-3). I turn next to particular accounts, first presenting a general framework within which they might be located (§4) and then considering the connection between positive/ negative subject matter and overall subject matter (§5) and the connection between relational and partition-like accounts of subject matter (§6). I turn finally to three central applications of the theory of subject matter: the question of what it is for one proposition to be part of another (§7); the question of what it is for a proposition to be true of some subject matter or to be the restriction of some other proposition to that subject matter (§8), and the question of how one might interpret conditional ‘incrementally’ in terms of what must be added to the antecedent to get to the consequent (§9-10). I conclude with some brief methodological remarks concerning the relationship between the possible worlds and truthmaker approaches to semantics (§11).1 §1 A General Framework 1 I am grateful for the comments of the participants in a seminar on Yablo’s book, held in the Fall of 2014 at NYU, and for the comments of the participants in a workshop on the book, held in the Summer of 2015 at the University of Hamburg, under the auspices of an Annalieser Maier Award from the Humboldt Foundation. I am especially grateful to Steve Yablo, both for his comments and for his inspiring work. 2 Yablo and I are both interested in a notion of subject matter that has to do not only with the objects that a sentence is about but also with what it says about those objects. This is amusingly illustrated in his contrast between the two headlines MAN BITES DOG and DOG BITES MAN (p. 24). Each is about the same objects (man, biting, dog), but the subject matter is different. Yablo and I both agree that: ‘a sentence’s subject matters depends upon what it says because subject matter is to do with ways of being true’ (p. 24) (and, it goes without saying for both of us, with ways of being false). We also agree in taking such talk seriously. This involves at least two elements. The first is that we recognize ‘ways’ of being true as objects in their own right. They are facts or fact-like entities that might reasonably regarded as parts or aspects of a world. The second is that they stand in a distinctive relation of ‘making true’ to the sentences of which they are the ways of being true. This relation is no mere entailment; it is not simply a matter of the way’s obtaining being sufficient for the truth of the sentence. Rather, the way’s obtaining must be responsible for its truth, so that each part of the way’s obtaining can be seen to play its part in accounting for the truth of the sentence. Yablo has talked, in this context, of reasons - of why the sentence is true - and I have talked of exact verification. At a more formal level, we may suppose that there are relations, ||- and ||-, of making true and making false, so that s ||- A indicates that s is an exact verifier (or reason or truthmaker) for the sentence A and s -|| A that s is an exact falsifier (or counter-reason or falsitymaker) for A. Given these two relations, we might then define the positive content |A|+ of the sentence A to be the set {s: s ||- A} of its verifiers, the negative content |A|- of A to be the set {s: s -|| A} of its falsifiers, and the full or bilateral content |A| of A to be the ordered pair <|A|+, |A|-> consisting of its positive and negative content. Referring to candidate verifiers or falsifiers of a sentence as ‘states, we might say, speaking more abstractly, that a set of states P (intuitively, the verifiers) is a unilateral content or proposition and that a pair of unilateral contents <P, PN> (intuitively, the verifiers and falsifiers) is a (bilateral) content or proposition. Let us use sm(A) for the subject matter of a sentence A. Then the point on which Yablo and I both agree is that the subject matter sm(A) of A depends merely upon its (bilateral) content |A|. This means that there must be a notion ó of subject matter applicable to contents for which: (1) sm(A) = ó(|A|). The subject-matter of a sentence is the subject-matter of its bilateral content, i.e. of what it says. Strictly speaking, the existence of the function ó depends upon no more than the truth of the principle that the subject matters of two sentences are the same when their contents are the same: (2) |A| = |B| implies sm(A) = sm(B). But the thought behind (1) is that we can intrinsically determine the subject matter of a content from the content itself and hence define the subject matter of a sentence in terms of the subject matter of its content. Thus we both agreed, not merely on (1), but upon the general framework within which the notion of subject matter is to be defined. The present conception of subject-matter suggests that we may differentiate the subjectmatter of a sentence into a positive and a negative component, where the positive subject matter (Yablo’s subject matter) has to do with the ways the sentence is true, the negative subject matter (Yablo’s “anti-matter” (p. 43)) with the ways the sentence is false, and where what Yablo (p. 43) calls the overall subject matter has to do with both with the ways the sentence is true and with 3 the ways it is false. Let us use sm+(A) for the positive subject matter of the sentence A and sm-(A) for its negative subject matter; and similarly, where P is a bilateral proposition, for ó+(P) and ó-(P). We may, in analogy to (1) above, suppose: (1)+ sm+(A) = ó+(|A|); (1)- sm-(A) = ó-(|A|). The positive (negative) subject-matter of a sentence is the positive (negative) subject-matter of its bilateral content. Indeed, we may go further. For presumably, ó+, in application to a bilateral proposition <P, PN>, is simply a function of the positive content P (as given by the verifiers) and ó-, in application to <P, PN>, is the same function of the negative content PN (as given by the falsifiers). Let us use ó for this function (thus ó now has application to both bilateral and unilateral content). Then we will have: [1]+ sm+(A) = ó(|A|+); [1]- sm-(A) = ó(|A|-).2 The positive subject matter of a sentence is the subject matter of its positive content and the negative subject matter of a sentence is the subject matter, in the very same sense, of its negative content. It is also natural to suppose that overall subject matter of a bilateral proposition (P, PN) is itself a function ‘+’ of its positive and negative subject matter. That is: (2) ó(<P, PN>) = ó(P) + ó(PN). Thus the task of defining the subject matter of a sentence A reduces to the task of defining the subject matter of its bilateral proposition, which reduces to the task (a) of determining the positive and negative content |A|+ and |A|- of the sentence and the task (b) of defining the subject matter ó(P) of a unilateral proposition P. I do not wish to underestimate the interest or difficulty of the first question. But the second question can, to a large extent, be pursued independently of the first; for we can simply take the verifiers and falsifiers of a sentence to be given. And in what follows, we shall largely focus on the second question, of defining the subject matter of a unilateral proposition.3 Yablo 2 Let us grant [1]+. Then [1]- can be derived from the additional assumptions that sm-(A) =sm+(¬A) and that |¬A| = <PN, P> when |A| = <P, PN>. For then sm-(A) = sm+(¬A) = ó(|¬A|+) = ó(PN) = ó(|A|-). 3 However, I should perhaps mention that Yablo and I do not altogether see eye to eye on the first question, although this may be more a matter of disagreement in emphasis than principle. I much prefer the recursive approach to truthmaking to the reductive approach (in contrast to §§4.3, 4.6) and I am inclined to regard the truthmaker content of a sentence as principally a semantic rather than a pragmatic matter (in contrast to §4.7). This calls for detailed discussion but let me simply mention how we might deal with one of his concerns, relating to the conditional (p. 60). Yablo wants p 6 p v q to be true because of the verifier q for q, whereas on the standard recursive account p 6 p v q is equivalent to ¬p w (p v q) and is either verified by a falsifier for p or a verifier for p v q. But, on the recursive account, we do not need to accept this equivalence and, as we shall see (§8), there is a recursive account which appears to deliver the 4 usually talks of the subject matter of a sentence but, since the subject matter of a sentence is always taken to be a function of its bilateral content, we may achieve greater generality by talking of the subject matter of a content, whether or not it is the content of any particular sentence. Despite this substantive area of agreement on the general framework, there are differences of detail in how we conceive the framework. For Yablo, states - the candidate verifiers and falsifiers - are, in effect, non-empty sets of possible worlds.4 This means that they are subject to two modal requirements (which hold regardless of how we represent them): Possibility Each state possibly obtains Intensionality States which necessarily co-obtain are identical. I do not insist on either requirement. Thus a state may be impossible and states which necessarily obtain, or necessarily fail to obtain, may be distinct. For me, a state space might be taken to consist of (i) a set of states, (ii) a distinguished subset of possible states and (iii) a relation of part-whole on the states. No assumption is made as to the inner structure of the states, they are simply taken as given (although one might reasonably require that every set of states has a fusion and that parts of possible states are also possible states). For Yablo, a state space can simply be identified with a family of non-empty sets of worlds.5 Each such state will be possible (obtain in some world); and one such state is naturally taken to be a part of another if it set theoretically contains the other (so that, necessarily, it obtains when the other obtains). Thus a Yablo state space may be regarded as a state space in my sense by taking of all its states to be possible and by taking the relation of part-whole to be modal entailment. But the converse does not hold and there is no reason why a state space in my sense should even be isomorphic to a Yablo-type space. Thus my approach is much more general than his. But it is not merely more general; the generality, in my opinion, pays huge dividends in terms of the naturalness, flexibility and systematicity of the resulting theory. Nothing is lost by adopting the more general framework; and a great deal is gained. Let me give one example here, although we shall later give many others. Both Yablo and I want it to be a general condition for the proposition or content P to contain the proposition Q that every verifier of Q should be contained in (or implied by) a verifier of P. But suppose that P is a contradictory proposition of the form P0 v ¬P0. Then, if we insist that every candidate verifier should be possible, then P will have no verifiers and hence will contain no proposition (even itself!). But surely we will want, in general, that the proposition P v Q should contain P. (As Yablo himself writes, ‘a paradigm of inclusion ...is the results he wants. 4 The identification is implicit in his remark that ‘No new machinery is required ...S’s reasons, or ways, for being true are just additional [truth-conditional] propositions’ (p. 2) and his ‘cellular’ treatment of subject matter (p. 45), under which any state can be regarded as the cell, i.e. non-empty set of worlds, of some subject matter. However, the reader should also take note of Yablo’s attempt to accommodate the impossible in the appendix to chapter 5 (pp. 92-4). 5 61). The worlds may, of course, be further identified - as models or valuations or the like (p. 5 relation that simple conjunctions bear to their conjuncts’ (p. 11), and one might well be perplexed by his requirement that the conjunctions to be simple.) So we should allow P0 v ¬P0 to have an impossible verifier. However, a single impossible state, of which every other state is a part, will not properly serve our purpose, since P = P0 v ¬P0 will then contain every proposition. Thus the only satisfactory solution is to admit a diversity of impossible states, each verifying their own different impossible proposition. Another difference, perhaps not as significant as the first, concerns our different conceptions of content. With each sentence A, I have associated two kinds of content, the unilateral content, the set of verifiers, and the bilateral content, the set of verifiers paired with the set of falsifiers. Yablo also associates two kinds of content with each sentence, the truthconditional - or what I prefer to call the intensional - content and the directed content (p. 49 and fn. 28 of p. 21).6 Like mine, one of them is single-barreled and the other double-barreled. But they are very different. The first is the familiar intensional content, which might be identified with the set of worlds in which the sentence is true. The second is the intensional content paired with the subject matter. Although we do not officially know this yet, the subject matter is, in effect, the unordered pair {P, PN} of the set P of verifiers and the set PN of falsifiers. Although the two “thick”, or double-barreled , notions of content seem very different, it is easy to pass (in one-one fashion) from the one to the other.7 Suppose <P, PN> is the bilateral content of the sentence A. Then the intensional content of A is the set of worlds which contains a verifier from P; and the Yablo subject matter of A is the unordered pair {P, PN}. Suppose now that <X, m> is the directed content of A. Thus m is a pair {P, Q}. If X is empty, then exactly one of P or Q will contain no possible states and it is this member of m which is the positive content of A. If X is non-empty, then exactly one of P or Q will contain a state compatible with a member of X and it is this member of m which is then the positive content of A.8 Given this correspondence between the two accounts, one might think that the difference between them is of no great significance, that whatever can be done with the one can equally well be done with the other. But I do not think this is quite right. In the first place, each account is informed by a very different picture of what thick content is. For Yablo, thick content is an enhancement of intensional content; it is intensional content plus subject matter (fn. 28, p. 21). 6 I do not like the term ‘truth-conditional’, since I think the truth-conditions of a sentence are more plausibly taken to be its exact verifiers. Calling the intensional content ‘truthconditional’ is to give it a false sheen. But the terminology is, at this state, probably too well established to be relinquished. 7 For present purposes, I am ignoring Yablo’s intensional conception of states and I am also presupposing that the sentence A is true or false in each possible world and that the state space contains the full panoply of possible worlds (is a ‘W-space’ in a terminology that I shall later introduce). One can also set up the correspondence without the use of this latter assumption but, in this case, the intensional content X of A must be identified with a suitable set of states from the worlds which, if they were to exist, would verify the sentence. 8 Two states are said to be compatible when their fusion is a possible state. 6 For me, thick content is not so much an enhancement as a modification of intensional content. Instead of looking at the worlds in which a sentence is true or false, we look at the states which make it true or false. Worlds simply drop out of view. But this difference in the picture also makes for a difference in the development of the theory. Suppose Yablo wants to construct a thick content P* from another thick content P - for example, given some subject-matter m, the part of P that is about m. Then he will attempt to determine the intensional content and subject matter of P* on the basis of the intensional content and subject matter of P. I, on the other hand, will attempt to determine the verifiers and falsifiers of P* on the basis of the verifiers and falsifiers of P. And, in practice, this will make for a difference in how one might proceed. One may speculate as to why Yablo adopts the particular conception of thick content that he does. Perhaps it is because he wants to adopt the “conservative choice” (p. 5) of building upon the standard possible worlds semantics. (I have sometimes joked that it was to my advantage to have left UCLA while it was to his disadvantage to have stayed at MIT). Perhaps it is because he thinks that the intensional content should be singled out at as the strictly semantic component of the content. But whatever the reasons, there is no doubt, in my mind, that his conception of content is unnecessarily cumbersome and leads to unnecessary complications and difficulties. This is already apparent from the way Yablo divides up the thick content. For the second component, the subject matter, already gives one almost everything that one might want. For it is the unordered pair {X, Y} of the positive and the negative content. But then all one needs to do to get the full content is to say which is which. But his first component does much more than that. For, thinking of X and Y as sets of sets of worlds, it specifies the union of whichever of X or Y is the positive content. But this information was already built into X and Y and there was no need to specify it twice over. This redundancy in the specification of the thick content leads to further redundancies down the line. For consider again, how one might attempt to define one thick content P* in terms of another thick content P. One will then need to specify the subject matter {X, Y} of P*. But in practice it will be impossible to do this without already being in a position to say which of X or Y is the positive content and which the negative content. The definition of the intensional component of P* will therefore be otiose, since it can be ascertained, if not from the subject matter of P* itself, then from the way in which it is specified. To make matters worse, one will also find that in order to define the subject matter of P*, the information that one will most directly need from P is not its subject matter, but its positive and negative content, which can itself only be indirectly determined from P. Thus the way Yablo divides up the thick content is not the way that is most convenient for the purposes of definition. It therefore seems to me - for theoretical purposes, at least - that, instead of regarding thick content as the pairing of intensional content with subject matter, we should regard it as the pairing of positive and negative content and that, instead of attempting separately to specify the intensional content and subject matter of a defined proposition on the basis of the intensional content and subject matter of given propositions, we should simply attempt to specify the positive and negative content of a defined proposition in terms of the positive and negative content of the given propositions. We thereby obtain a simpler and more 7 elegant formulation of the theory. But these differences should not blind one to the underlying commonality in our views, which is that the subject matter of a sentence will depend, not upon the worlds in which it is true or false, but upon the ways in which it is made true or false. §2 The Behavior of Subject Matter I have so far considered the question of how subject matter relates to truth making (and to falsity making). But there are questions concerning the behavior of subject matter which do not relate to truthmaking as such and upon which one might expect some kind of consensus, even among those who do not wish to relate subject matter to truthmaking or who might wish to relate it to truthmaking in some other way. One batch of such questions concerns compositionality. How does the subject matter of a complex sentence or proposition relate to the subject matter of its parts?9 It would be generally agreed (and is endorsed by Yablo, remark 3 of p. 42) that the subject matter of a proposition and its negation are the same: Negation Neutrality ó(¬P) = ó(P). The subject matter of Toby is tall, for example, is the same as that of Toby is not tall; each concerns Toby’s height. Another plausible principle of the same general sort is that the subject matter of a conjunction should be the same as that of the corresponding disjunction: Junction ó(P w Q) = ó(P v Q) The subject matter of Toby is tall and handsome, for example, is the same as that of Toby is tall or handsome; each concerns the height and looks of Toby. We might also want to accept that the subject matter of a complex proposition in these cases is a function of the subject matter of its parts: Congruence (i) ó(P) = ó(PN) implies ó(¬P) = ó(¬PN) (ii) ó(P) = ó(PN) and ó(Q) = ó(QN) implies ó(P w Q) = ó(PN w QN) (iii) ó(P) = ó(PN) and ó(Q) = ó(QN) implies ó(P v Q) = ó(PN v QN) So, for example, given that Toby is tall has the same subject matter as Toby is not tall, Toby is tall or handsome has the same subject matter as Toby is not tall or handsome by the second of these three principles. Congruence does not imply Junction since the subject matters of P w Q and of P v Q might be different functions of the subject matters of P and Q. But Congruence, in conjunction with some other plausible assumptions, will imply Junction. Let us assume: De Morgan ¬(P v Q) = (¬P w ¬Q) according to which the proposition that it is not the case that P and Q is the same as the proposition that it is not the case that P or it is not the case that Q. Then: ó(P w Q) = ó(¬¬P w ¬¬Q) by Negation Neutrality and Congruence (ii) = ó(¬(¬P v ¬Q)) by De Morgan 9 I have taken the principles to follow to be about the subject matter of propositions but there are, of course, analogous principles (and points to be made) about the subject matter of sentences. 8 = ó((¬P v ¬Q)) by Negation Neutrality = ó(P v Q) by Negation Neutrality and Congruence (iii). Given that neither Negation Neutrality nor De Morgan can reasonably be challenged, this means that Junction can only be rejected by giving up Congruence. In addition to principles of compositionality for subject matter, there are mereological principles one might wish to endorse (Yablo dubs the topic mereotopicology (p. 29)). We may plausibly suppose that given any two subject matters s and sN, there is a subject matter s + sN, which is their sum or fusion. Thus the fusion of the subject matters that concern the height of Toby and the looks of Toby will be the subject matter that concerns his height and looks. It is plausible that the operation of fusion on subject matters will be subject to standard mereological principles. So, for example, we will have the identities s + s = s and s + t = t + s; and we might even allow that the difference s - t of two subject matters s and t always exists, where s - t is disjoint from t and (s - t ) + t = s + t. But there should also be mereological principles relating the subject matter of complex propositions to their parts: Junctive Fusion (i) ó(P w Q) = ó(P) + ó(Q) (ii) ó(P v Q) = ó(P) + ó(Q). Thus the subject matter of Toby is tall and (or) handsome will be the fusion of the subject matters of Toby is tall and Toby is handsome. Such mereological principles might plausibly be taken to underlie the corresponding compositional principles. Thus Junction, ó(P w Q) = ó(P v Q), will directly follow from Junctive Fusion, since ó(P w Q) = ó(P) + ó(Q) = ó(P v Q). We therefore have another reason for thinking that Junction should hold. Junctive Fusion plays a powerful role. It provide us with an interplay between mereological and logical identities. Thus given that the mereological identity s + s = s, ó(P w P) = ó(P) + ó(P) = ó(P); and, conversely, given the logical identity ó(P w P) == ó(P) and making the modest assumption that any subject matter s is the subject matter ó(P) of some proposition P, we have s + s = ó(P) + ó(P) = ó(P w P) = ó(P) = s. It also provides us with an account of the relation of inclusion, or part-whole, between subject matters in terms of identity and the logical connectives. For in general x will be a part of y iff x £ y = y. Given two subject matters s and t, we may then suppose that there are propositions P and Q such that s = ó(P) and t = ó(Q). So s is a part of t iff s £ t = t, i.e. iff ó(P) £ ó(Q) = ó(Q). But by Sum, ó(P) £ ó(Q) = ó(P w Q) = ó(P v Q); and so s being a part of t may be alternatively defined either as ó(P w Q) = ó(Q) or as ó(P v Q) = ó(Q). Thus as long as we have the means of characterizing the subject matter of a proposition and the disjunction or conjunction of two propositions, we are in a position to provide a purely logical definition of subject matter inclusion. I have so far focused on overall subject matter but there are corresponding principles for positive and negative subject matter. Thus in the case of Compositionality, we might plausibly require: Neutrality ó+(¬P) = ó-(P) ó-(¬P) = ó+(P) Junction ó+(P w Q) = ó+(P v Q) ó-(P w Q) = ó-(P v Q) 9 Congruence ó+ (P) = ó+(PN) implies ó+(P w Q) = ó+(PN w Q) etc. ó-(P) = ó-(PN) implies ó-(P w Q) = ó-(PN w Q) etc. In the case of Mereology, we might plausibly require: Junctive Fusion óð(P w Q) = óð(P) £ óð(Q) óð (P v Q) = óð(P) £ óð(Q), where ð is either + or -. We might also want: Overall Fusion ó(P) = ó+(P) + ó-(Q) according to which the overall subject matter of a proposition is simply the fusion of its positive and negative subject matter. Granted that ó-(Q) = ó+(¬Q) (Neutrality), this would then enable us to define overall subject matter in terms of positive subject matter. Without being too precise, we might call the theory that endorses these various compositional, logical and mereological principles the standard theory of subject matter. I am not suggesting that we should necessarily endorse such a theory. But it does provide us with a standard by which we may judge deviations from the norm; and it seems to me that we should not deviate from this norm without strong and compelling reason. §3 The Nature and Role of Subject Matter Yablo’s book covers a wide range of questions concerning the nature and role of subject matter. His answers to these questions serve to define the theory and show how it might be applied; and it is largely on the basis of these answers that the success and interest of the theory is to be judged. There are four main topics - or sets of questions - from Yablo’s book that I wish to take up. There are a number of other issues I might also have considered, but I hope my discussion of the present topics is to some extent emblematic as to how the discussion of the other issues might proceed. The four sets of questions are as follows: Identity What are subject matters, considered as entities in their own right? What is the subject matter of a sentence or proposition? When is one subject matter included in - or otherwise mereologically related to - another? Containment When is the content of one sentence part of the content of another? And when, in general, is one proposition part of another? Restriction When is a sentence or proposition true about a given subject matter? What is the restriction of the proposition to the subject matter, i.e. what is the part of the proposition true about the subject matter? The Incremental Conditional How might we interpret the conditional incrementally in terms of what must be added to the antecedent to get to the consequent? The various questions, within each set and across different sets, are not unrelated. In regard to the questions under Identity, we may be able to say what a subject matter is without being able to define the subject matter of a given sentence (as Yablo points out, p. 38). However, if we can define the subject matter of a sentence, or of a proposition, then we will presumably be in a good position to say what a subject matter is, since it will be the kind of thing that is the subject matter of a sentence or a proposition. A warning here is in order. Yablo takes a thick or “directed” proposition, as I have said, 10 to be an intentional content paired with its subject matter. It is therefore trivial for him to define the subject matter of such a proposition; it is the second component. However, I take a thick proposition to be a bilateral content, i.e. a pairing of the positive and negative content; and, on such a conception, which I shall henceforth adopt, it is not at all trivial to say what its subject matter will be. Similarly, in regard to the questions under Restriction, one may be able to say when a sentence or proposition is true about a given subject matter without being able to define the restriction of the proposition to the subject matter. But it seems plausible that if one can define the restriction of the proposition to the subject matter one can say when the proposition is true about the subject matter, since the proposition will be true about the subject matter when its restriction is true simpliciter. In regard to the questions under Containment, we would like to know how the notion of subject matter might be involved in the definition of containment. Yablo’s idea here (already implicit in the earlier work of Parry [33] and Angell [77]) is that containment is the combination of some kind of implication (such as preservation of truth) with the preservation of subject matter (p. 3). Thus for A to contain B is for A to imply B and for the subject matter of A to contain the subject matter of B. Thus on this view, containment of two sentences or propositions should be understood, at least in part, in terms of the containment of their subject matter. In regard to the last set of questions, one natural thought, endorsed by Yablo, is that, at least in certain cases, the connection between the antecedent p and consequent q in the incremental conditional is given by the difference Q - P in their content, where this notion is itself related in an obvious way to the notion of propositional containment. Thus the incremental connection between p 6 p v q is given by q, which signifies the difference in content between p v q and p. Four key questions therefore emerge. What is the subject matter of a proposition? What is the restriction of a proposition to a given subject matter? What is it for one proposition to contain another and how, in particular, does the notion of subject matter figure in an account of propositional containment? And what is the difference between two contents? If we can answer these questions, then we are well on our way to answering the others. §4 The Subjugation of Subject Matter The most important of our questions is: what is the subject matter of a proposition? Taking overall subject matter to be a function of positive and negative subject matter and taking negative subject matter to be the positive subject matter of the negation, this then becomes the question: what is the positive subject matter of a proposition? What, in other words, is the subject matter of a proposition that relates to how it is true? Given P (the set of verifiers), then what is its subject matter p? In time-honored tradition, we might reduce this question to the question: when do two (unilateral) propositions P and Q have the same subject matter (are subject-matter identical)? Once we have answered this other question, we might then identify the subject matter of P with the corresponding equivalence class of propositions or with some suitable surrogate for the equivalence class. This question might be further reduced. For we might ask the question in the special case 11 in which P is a subset of Q (i.e. when every verifier of P is a verifier of Q). For suppose P and Q are any two propositions. It is then plausible that P is subject-matter identical to Q iff P and Q are both subject matter equivalent to P c Q. For clearly, the right to left direction holds. Now suppose P is subject matter equivalent to Q. Then presumably P is subject-matter identical to P w P, which is subject matter equivalent to P w Q by Congruence, which is identical and hence subject matter equivalent to P c Q. Thus we can reduce the question of whether P is subject matter equivalent to Q to the question of whether P and Q are each subject-matter identical to the superset P c Q.10 We can argue in similar manner that, among the propositions containing P to which P is subject-matter identical, there is a greatest. For P will be subject matter equivalent to the union of all such propositions ; and this union will be the greatest such proposition. Call this proposition, the closure Pc of P. Then propositions P and Q will be subject matter equivalent just in case their closures Pc and Qc are identical. Thus the question of subject matter equivalence reduces to the question of determining the closure Pc of any given proposition P. We might say that a state s bears upon (the truth of) the proposition P if it belongs to its closure. We might then attempt to define the closure of a proposition by giving conditions for when a state bears upon a proposition. These conditions will take the form: if such and such states bear upon the truth of a given proposition, then such and such related state will bear upon its truth. By giving an exhaustive account of such conditions, we might then define the closure of a proposition to be the smallest set containing the given proposition and conforming to the conditions. So our task is to give conditions for when a state bears upon the truth of a proposition. What conditions might plausibly be imposed on when a state bears upon the truth of a proposition? There are three conditions that stand out, although there are some others that might also be considered: Identity (Closure) If s is a verifier of P (i.e. if s 0 P) then s bears upon P. Part (Closure) Any part of a state that bears upon P also bears upon P. Fusion (Closure) The fusion of any states that bear upon P also bears upon P. Identity is axiomatic and does not call for further discussion. According to Part, Toby’s being tall will bear upon the truth of the proposition that Toby is tall and handsome, given that Toby’s being tall and handsome bears upon the truth of this proposition. Part (like Fusion) is intimately related to the compositional principles for subject matter. For given Part, P v Q will be subject-matter-identical to P w Q w (P v Q), at least granted that the verifiers of P v Q are of the form p £ q, for p and q respective verifiers of P and Q, and that the verifiers of P w Q w P v Q are of the form p, q and p £ q, since then the verifiers of P w Q w (P v Q) can be obtained by adding parts of verifiers to P v Q. Part is also intimately related to the notion of partial truth. For suppose we close a 10 I have here, for simplicity, assumed a certain semantics for disjunction, under which the positive content of A w B will be the union of the positive contents of A and B. There are other semantic clauses that might be given for disjunction, and for the other Boolean connectives. Most of what I say will apply, mutatis mutandis, to these other cases, although I have not aimed for full generality. 12 proposition P under part. Then the resulting proposition PN will consist of the verifiers of P and all of their parts. Now the proposition PB that P is partially true (something I have called the partial of P) may plausibly be identified with the proposition that is verified by any part of a verifier of P other than the “null” state ~ (which is the fusion of the empty set of states and which automatically obtains, no matter what).11 Thus PB = PN- {~}. Given Part, it will then plausibly follow that any proposition is subject-matter-identical to its partial, i.e. that the subject matter of the proposition P will be the same as the subject matter of the proposition that P is partially true (the converse implication is also plausible).12 According to Fusion, Toby’s being tall and handsome will bear upon the truth of the proposition that Toby is tall or handsome, given that Toby’s being tall and Toby’s being handsome both bear upon the truth of the proposition. Given Fusion, the proposition P w Q will be subject-matter-identical to the proposition P w Q w (P v Q). For we will be able to obtain any verifier p £ q of the third disjunct P v Q of the second proposition by fusing the verifiers p and q from P w Q. Thus given both Part and Fusion, the propositions P w Q and P v Q will be subjectmatter-identical - the principle of Junction above. Given the plausibility of Junction, this strikes me as a strong reason for adopting these two closure conditions. Three accounts of subject matter naturally emerge from the above discussion: the minimal account, according to which Identity is the only closure condition; the intermediate account, according to which Part is the only closure condition; and the maximal account, according to which Part and Fusion are the only closure conditions. Of course, other accounts are also possible: one might just insist upon Identity and Fusion and drop Part; or one might consider various restrictions on these conditions or other conditions altogether. But these three accounts strike me as the most natural combination of conditions. And they provide us, of course, with successively weaker notions of subject matter (where one notion is weaker than another if it results in more propositions being subject matter identical), with the intermediate account weaker than the minimal account and the maximal account weaker than the minimal or intermediate accounts. Let us say something about each of these accounts in turn: The Minimal Account According to this account, the closure Pc of a proposition P is simply the proposition P; and two propositions are subject matter identical just in case they are the same. It is the strictest of all accounts, putting the subject matter of a proposition at minimum, indeed zero, distance from the proposition itself; and it is the account of subject matter endorsed by Yablo. 11 This is not the only way of defining the partial (see Fine [2015]). But any reasonable definition will yield similar results. 12 If P contains some verifier other than ~, then PN can be obtained from PB by adding the part ~ and so the subject matters of P, PB and PN are the same. If P only contains ~, then PB = i and PN = {~} and so we only need to be assured that the null proposition i and the null state proposition {~} are subject-matter-identical. But this will presumably be so, given that the null state ~ is “contentless”. Finally, if P is empty then so are PB = i and PN. 13 It appears to be counter-intuitive. Do we really want to say that P w Q does not, in general, have the same subject matter as P v Q or as P w Q w (P v Q)? And do we want to say, in general, that intensionally distinct propositions cannot have the same subject matter, so that subject matter must cut at least as finely and, indeed, more finely, that intensional identity? There are also some counter-intuitive consequences when it comes to subject matter inclusion. We presumably want the subject matter of each of P and of Q to be included in the subject matter of each of P w Q and P v Q. But the only plausible account of subject matter inclusion which will deliver this result is one according to which the subject matter P is included in the subject matter Q just in case every verifier of P is contained in a verifier of Q.13 But it will then follow, given that s is the subject matter of P and t of Q, that their combined subject matter s + t will be the subject matter of P w Q rather than of P v Q and, in general, the subject matter of P w Q will be a proper part of the subject matter of P v Q. Even if one does not want the subject matter of P w Q and of P v Q to be the same, these further results on their mereological relationship seem completely artificial. The account also seems to go against a neutrality requirement on subject matter, according to which it should not in general be possible to determine from the subject matter of a proposition (and the facts) whether or not the proposition is true. As Yablo himself puts it (p. 42), ‘one should be able to understand what S is about while remaining ignorant of its truthvalue’. But, as he goes on to point out, the present account of subject matter violates this requirement, for the subject matter of P, i.e. P itself, will be true just in case one of its verifiers is actual, i.e. belongs to the actual world. Thus positive subject matter, on the identity account, is not, properly speaking, subject matter at all. This would not be so bad if it were not treated as such but only as a stepping stone to subject matter proper, which might appear to be Yablo’s attitude on pp. 42-3 (where positive subject matter is described merely as the ‘better candidate’ and overall subject matter is defined in terms of positive and negative subject matter). But one of Yablo’s central contentions is that subject matter is relevant to the definition of containment; and yet, when we examine his definition of containment (p. 45), we see that it appeals separately to positive and negative subject matter, which are not properly subject matters at all, rather than to overall subject matter. It would therefore be desirable to have an account of positive (and of negative) subject matter which was as equally capable of being neutral as is his account of overall subject matter. It might appear, however, that Yablo has a strong argument in favor of the Identity Account in terms of upper and lower bounds (p. 41). The crucial step in the argument is thesis 9: Something has changed, between one world and another, where S’s subject matter is concerned, only if S is differently true in the two worlds [i.e. does not have the same verifiers in the two worlds] . For ‘what could these beneath-S’s-notice tweaks have in common, if not their irrelevance to how 13 Yablo also requires each verifier of Q to contain a verifier of P (p. 30). But this would then prevent the subject matter of P being, in general, part of the subject matter of P w Q and so it seems to me that this further requirement should be dropped. 14 S obtains or is true’. But take the proposition G that God or Mind and Matter exist (of the form P w (Q v R)) and consider a world in which only God and Mind exist and a world in which only God and Matter exist. Then it may well be thought that something has changed between the two world’s where G’s subject matter is concerned and yet G has the same verifiers in the two worlds, viz. the existence of God. A proposition’s subject matter may be capable of seeing beyond its verifiers to how they are composed. Intermediate Account According to this account, the closure P* of a proposition P simply consists of the verifiers of P and their parts and two propositions are subject matter equivalent just in case the parts of their verifiers are the same. We might effectively identify the subject matter of P with its partial PB, i.e. with the proposition that P is partially true. Thus to know the subject matter of P is not to know what would make it true, as on the Identity Account, but to know what would make it partially true. The account is not as counter-intuitive as the Identity Account. Thus P v Q will have the same subject matter as P w Q w (P v Q) and intensionally distinct propositions can have the same subject matter. But it is still counter-intuitive, since P w Q will not in general have the same subject matter as P v Q or as P w Q w (P v Q); and it might well be thought invidious to insist upon the subject matter identity of P v Q and P w Q w (P v Q) while allowing the subject matter non-identity of P w Q and P w Q w (P v Q). We should also note that the definition of subject matter inclusion can be simplified somewhat. For when s is the subject matter of P* and t of Q * then s will be included in t just in case P* is a subset of Q *. However, we will still have the counter-intuitive result that the subject matter of P w Q is, in general, a proper part of the subject matter of P v Q. The Maximal Account According to this account, the closure P** of a proposition P consists of the verifiers of P, their parts, the fusions of their parts, the parts of such fusions, and so on, and two propositions are subject matter equivalent just in case their closures in this sense are the same. The account of subject matter, in this case, is subject to a striking simplification. For given a proposition P = {p1, p2, ... }, let p be the fusion p1 £ p2 £ ... of its verifiers. Then it is readily verified that the maximal closure P** and the intermediate closure {p}N, as defined above, are one and the same, i.e. that a state belongs to the maximal closure just in case it is a part of the fusion of all verifiers. But two intermediate closures {p}N and {q}N will be the same just in case their generating states p and q are the same. Thus the subject matter of a proposition P may be identified with the ‘limit’ state p. This means that, in contrast to the two previous accounts, the subject matter of a proposition will itself be a state, rather than another proposition. It will point directly, though its parts, to those states which bear upon the truth of the proposition. And if we take a state to be given as an individual, as I myself would prefer, rather than as a set of worlds, then subject matters will be of type 1, the type of an individual, rather than being of type 3, the type of of a set of sets of individuals (the individuals, in this case, being the possible worlds). A sweeping simplification in both logical and conceptual complexity! 15 We also have a further simplication in the account of subject matter inclusion: for the subject matter of P to be included in that of Q is for p to be a part of q and, similarly, the fusion of the subject matters of P and Q will be the fusion p £ q of the corresponding states. The mereology of subject matter will simply be a special case of the mereology of states. As Yablo says of a related conception of subject matter, ‘One sees what subject-matter inclusion would be on the parts-based conception. Chunks of reality stand in inclusion relations right out of the box’ (p. 30). The maximal account yields Junction, since the verifiers of P w Q can be obtained from those of P v Q via Part and the verifiers of P v Q obtained from those of P w Q via Fusion. Indeed, Junctive Fusion will hold since the subject matter of both P w Q and P v Q will be identical to p £ q. Furthermore, the account is maximal not only in the sense of incorporating all three closure conditions but also in the sense that it is plausibly the “weakest”account of subject matter, the one that makes the fewest possible distinctions between the subject matter of different propositions. For suppose the closure Pc of P were to contain a verifier q that were not part of p. Then there is a clear sense in which q would not bear upon P since it would contain a non-null part q - p that had no relevance to the truth of P. If we are after an upper bound on the possible identification of subject matter, it lies here, in p, and not, as Yablo would have it, in P. I previously referred to the standard theory of subject matter. The present view might well be called the standard account of subject matter. It delivers the standard theory and matches it in simplicity, elegance and intuitive appeal. However, the account is subject to a peculiar difficulty which needs to be resolved before it can be considered acceptable. For a start, the definition of p requires that we be allowed to fuse any states whatever, even incompatible states; and this means that, in general, p will not be a possible state. There is perhaps no harm in allowing an impossible state. But it may now be thought that any state whatever will be part of an impossible state. But this then means, under the Maximal Account, that when two propositions have incompatible verifiers, their subject matter is the same. Thus suppose an object being red and the object being blue are verifiers of the proposition that it is colored and that the object being round and the object being square are verifiers of the proposition that it is shaped. Then the state of the object being red and blue and the state of the object being round and square will be parts of the subject matters of the respective propositions and so their subject matters will be the same, which clearly is not so. In the light of these difficulties, it might be thought that Fusion should be restricted to possible states: Restricted Fusion The fusion of states which bear on a proposition also bears upon the proposition as long as it is a possible state. But if we accept this principle then we lose the sweeping simplification whereby we may identify the subject matter of a proposition with a single state; and certain general principles will also have to be rejected. Thus we could no longer say in general that P w Q is subject matter equivalent to P v Q. For suppose P and Q had incompatible verifiers p and q. Then we could have no assurance that the verifier p £ q of P v Q belonged to the closure of P w Q. However, this difficulty can be removed by abandoning the usual intensional conception of impossible states, under which there is only one impossible state of which every other state is a part, and allowing them to be more finely individuated. I have developed what I think is a very 16 natural way of doing this in Fine [2015d]. I will not go into details but suffice it to say that the resulting conception of impossible states is one in which one cannot get more possible states out of an impossible state than were put in. To be more exact, the possible states one obtains by closing a given set of states under Part and Restricted Fusion is the same as the possible states one obtains by closing a given set of states under Part and Unrestricted Fusion. Thus the detour through impossible states does not enable one to obtain more possible states. In particular, the possible state of an object being round will not be part of the impossible state of the object being red and blue; and so the previous difficulty is removed and nothing stands in the way of allowing a subject matter to be an impossible state. This is one of a number of cases, I believe, in which appeal to impossible states, once they are properly individuated, yields huge dividends. There is a further difficulty, which I should briefly consider. Yablo follows Lewis in criticizing a parts-based conception of subject matter. On such a conception, ‘for worlds to be alike with respect to m is for corresponding parts of those worlds to be intrinsically indiscernible’ (p. 25) Of course, Lewis talks of ‘corresponding parts’ here because of his adherence to counterpart theory; the rest of us could talk instead of some common objects of the worlds being intrinsically indiscernible. The objection is that ‘this approach is not sufficiently general ... facts about how many stars there are are not stored up in particular spatiotemporal regions’(p. 25). But the present maximalist approach does appear to be parts-based in roughly Lewis’ sense. Similarity in the subject matter p would appear to be agreement on the facts belonging to p; and so one might worry that the account is not ‘sufficiently general’. Oddly, Yablo’s own account of subject matter is subject to a similar criticism. For likeness in subject matter P, for him, will be a matter of agreement of the states within P. So what has gone wrong? Is there a problem with the criticism or with the accounts? I think the answer is that the lack of generality lies, not in the accounts, but in Lewis’s conception of what it is to be parts-based. Parts of a world, for Lewis, are objects and so a partsbased account will be one in which the objects match. But a much more natural conception of worldly part - both in itself and within the framework of truthmakers - is one in which a part of a world is a “state” or a “way” something can obtain. Thus a parts-based approach will be one in which similarity among worlds is a matter of agreement on some particular states or ways of obtaining. But the above criticism will then have no force, for similarity in regard to how many stars there are will be a matter of agreement on the state of there being a certain number of stars (and similarly in regard to the example of “observables” that Yablo discusses (p. 25)). However, the new parts-based approach will still be lacking in generality compared to the relational approach. For there is no reason in general to think that an arbitrary partition or equivalence relation on worlds will be induced by some set of states. If one takes a cell of the partition and asks what state belongs to (is a part of) each world of the cell, then the answer may well be none (the null state aside). Thus the cells may well not correspond to any states.14 Is this still a problem for the parts-based approach? I think not. There is a lack of generality but we may suppose that it is a lack of generality that “cuts no ice” and is irrelevant to 14 Under an obvious representation, let the worlds be {p, q}, {p, ¬q}, {¬p, q} and {¬p, ¬q} and the cells be {{p, q}, {¬p, ¬q}} and {{¬p, q}, {p, ¬q}}. Then no state is common to the worlds of either cell. 17 the proper identification of subject matter. Thus we may turn the previous criticism on its head and say that the problem now is not that the parts-based approach is insufficiently general but that the relational approach is excessively general, allowing for distinctions in subject matter where none exist. For the fact is that not every non-empty set of worlds can properly be regarded as a candidate verifier or falsifier. This is evident on the recursive account, where all the verifiers and falsifiers must be derived from the verifiers and falsifiers for atomic sentences. But it is also true on the reductive account in which we take the verifiers or falsifiers to be minimal, i.e. to have no proper parts which are verifiers or falsifiers (§4.3). For if every non-empty set of worlds were allowed to be a verifier then the sole minimal verifier of the possibly true proposition P would be P itself and the sole minimal falsifier would be ¬P and the reductive approach would simply collapse into the intensional approach. It is only by presupposing some restriction on the candidate verifiers and hence only by restricting which equivalence or similarity relations on worlds can count as subject matters that we can form a reasonable conception of what the subject matters are.15 §5 Overall Subject Matter Our focus was previously on positive and negative subject matter. I now wish to consider overall subject matter. As previously remarked, it is natural to regard the overall subject matter ó(<P, Q>) of a bilateral proposition as a function ‘+’ of its positive subject matter ó+(<P, Q>) and its negative subject matter ó-(<P, Q>). This boils down to the following congruence principle: If ó(P) = ó(PN) and ó(Q) = ó(QN) then ó(<P, Q >) = ó(<PN, QN>), granted that ó+(<P, Q>) = ó(P) and ó-(<P, Q>) = ó(Q). But what is this function +? Again, I consider three accounts, of decreasing strictness. If our concern were merely to satisfy the congruence principle, then we might identify the overall content ó(< P, Q >) with the ordered pair <ó(P), ó(Q)> of the positive and negative content. But this would be to distinguish truth from falsity (with truth first and falsity second) and the principle of Negation Neutrality, according to which ó(<P, PN>) = ó(<PN, P>) (taking <PN, P> to be the negation of <P, P N >) would not in general hold and perhaps would never hold. If our sole concern were to satisfy Negation Neutrality (in addition to Congruence), then we might identify the neutral content of ó(P, Q) of a bilateral proposition with the unordered pair {ó(P), ó(Q)} of the positive and negative content. We still draw a line between the verifiers and falsifiers of a proposition but without saying which belongs to either side of the line. This is, in effect, Yablo’s strategy. Since ó(P), for him, is simply P, this means that the neutral subject matter of <P, PN> is taken to be {P, PN}. We unorder the ordered pair. (Conversely, the rich content for him -the pairing of intensional content with subject matter 15 In different contexts, different states may count as the states, or candidate verifiers. It is any interesting question whether every possible intensional proposition corresponds to a state in some or other context. But even if the answer is ‘yes’ (which I very much doubt), the point remains that, relative to any given context, there will usually some restriction on what the candidate verifiers and hence on what the possible subject matters might be. 18 might be regarded as an indirect way of ordering the unordered pair {P , PN}). This is a quite general strategy for determining overall content from positive content, which does not depend upon Yablo’s particular conception of positive content. Thus under the earlier Maximal Account, we could take the overall content of the bilateral proposition <P, Q> to be {p, q}. Similarly, if we imagine Lewis taking the positive content of an intensional proposition P (a subset of the set of worlds W) to be P itself, then the neutral content of the bilateral proposition <P, W - P> will be the set {P, W - P}, which is the bipartite partition, or the subject matter whether or not P, corresponding to the proposition P (for contingent P). Yablo criticizes the Lewisian account (p. 39) yet, oddly, his own account of overall content is based upon the same general idea, but with states in place of worlds. It seems to me that there is a quite general problem with this strategy: it is still too strict. For the bilateral proposition P will have the same overall subject matter as its negation ¬P. So P v Q should have the same overall subject matter as ¬P v Q. But, in general, the positive subject matter of P v Q will be identical neither to the positive nor to the negative subject matter of ¬P v Q. The more general difficulty is the failure of overall subject matter, on this account, to be “congruential”; there is no guarantee that substitution of contents with the same subject matter will preserve subject matter. This brings us to the last account, according to which the overall subject matter ó(< P, PN>) of a bilateral proposition <P, PN> is to be identified with the fusion ó(P) + ó(PN) of its positive and negative subject matter (what we previously called the principle of ‘Overall Fusion’). Of course, this requires us to have previously identified the fusion of the subject matters of unilateral propositions, but it will usually be evident how this is to be done. Thus if we were to identify the subject matter of a unilateral proposition P with its closure Pc (however exactly this is defined), then we would take the fusion of two subject matters Pc and Qc to be (Pc c Qc)c. Under the Identity Account, the overall subject matter of <P, PN> would then be, not {P, PN}, as on Yablo’s approach, but P c PN. The verifiers and falsifiers would be ‘merged’, so that there would no longer even be a line between the verifiers and the falsifiers. Under the maximal account, on the other hand, the overall subject matter <P, PN> would be p £ pN, the fusion of the verifiers and falsifiers of the proposition, again without discrimination between them. I believe that the maximal account provides us with what is, in many ways, the most satisfactory account of overall subject matter. It yields a natural mereological analysis of overall subject matter in terms of positive and negative subject matter (the principle of Overall Fusion). It is congruential; subject matter equivalence is preserved under substitution. It yields the full complement of compositional principles; not only is the subject matter of a proposition the same as that of its negation, the subject matter of a disjunction and the corresponding conjunction are also the same. And we should note that the account is satisfyingly maximal in that we could not reasonably take the subject matter of <P, PN> to go beyond p £ pN. We have what may well be considered the most natural extension of the standard account of unilateral subject matter to the bilateral case. One final note. If we adopt any of the weaker conceptions of overall subject matter then we can no longer follow Yablo in identifying the thick content of a sentence with the pairing of intensional content and subject matter. Subject matter may rightly be regarded as an element - 19 and, indeed, a critical element - in the meaning of a sentence. But subject matter is inseparable from another aspect of meaning, the various ways in which the sentence is made true or false, and it is then this other aspect, the bipartite content, that is more properly regarded as constituting the thick content of the sentence. §6 The Relational Conception of Subject Matter We find two models of subject matter in Lewis [1988] - the cellular and the relational. On the cellular model, a subject matter is a partition on the set of worlds (the members of the partition being the “cells”). Intuitively, we can think of it as a collection of propositions which are individually possible and jointly exclusive and exhaustive. On the relational model , a subject matter is an equivalence relation on worlds, i.e. a reflexive, symmetric and transitive relation on worlds. Intuitively, two worlds are equivalent if they agree on the subject matter. These two models provide equivalent ways of presenting the same information and we can go in one-one fashion from either one to the other. Thus (i) two worlds will be equivalent when they belong to a common cell and (ii) each cell will be a maximal class of worlds any two members of which are equivalent. Yablo wishes to preserve the form of Lewis’ account but to relax some of its constraints. As he puts it, ‘Subject-matters will be similarity relations rather than equivalence relations; symmetry is still required, and reflexivity, but not transitivity. Alternatively, we can think of them as “divisions” of logical space - divisions being the set-theoretic whatnots standing to similarity relations as partitions stand to equivalence classes” (p. 36).16 He also wishes the form of correspondence between similarity relations and divisions ((i) and (ii) above) to remain, but without the need for the cells of a given division to be exclusive of one another. They will at best be “incomparable”, with no cell a proper subset of another (p. 5). Although he adopts both models, he seems to think of the relational model as the preferred or “official” view. Thus he writes, ‘A subject matter ... is a system of differences, a pattern of cross-world variation’ (p. 27) and, revealingly, he defines the (positive) subject matter of a sentence (p. 41) as a dissimilarity relation, i.e. as the relation that holds between two worlds when the sentence is ‘differently true in them’, i.e. does not have the same verifiers. Given that he makes appeal to the verifiers of the sentence in the definition, he could just have defined the subject matter of the sentence as the set of its verifiers (each verifier being a set of worlds). But it seems clear, both here and elsewhere in the book, that he prefers to think of subject matter in relational rather than cellular terms. I wish to press a number of objections against this way of conceiving subject matter. It seems to me that Yablo goes astray in placing so much emphasis on the relational understanding of subject matter and that it would have been better if, in developing his view, he had simply jettisoned it and stuck to the cellular approach. However, I should mention that this line of objection may not altogether be news to him, since he seems to intimate here and there that he is 16 He also wishes to allow a subject matter to be defined on a proper subset of the worlds (p. 30), though this will not concern us so much in what follows. 20 not altogether happy with the relational approach.17 One objection to the relational approach arises from taking a hyperintensional stance on states, according to which states may necessarily co-obtain and yet not be the same. If one thinks of a subject matter as being given by a set of states, then this means that certain hyperintensional differences in subject matter may be lost once one moves to the corresponding equivalence or similarity relation on worlds. One subject matter may be mathematical truth, another metaphysical truth, each constituted by certain necessary states. The subject matters are quite different and yet the corresponding relations will be same, since all worlds whatever will agree on the mathematical facts and all will agree on the metaphysical facts. But difficulties also arise under a fully intensional conception of states. One has already been noted. For a conception of subject matters as equivalence or similarity relations on worlds is one in which any equivalence or similarity relation is naturally taken to be a subject matter. But we would not naturally regard any partition or division as constituting a subject matter. The cells of the partition or division should be derived from the given states and not every cell can be so derived. A more serious difficulty concerns the lack of correspondence between the cellular and relational representations of subject matter. The correspondence works fine for equivalence classes and partitions but already begins to break down when we drop the transitivity requirement on equivalence relations and the corresponding exclusivity condition on partitions. Suppose there are three distinct worlds w1, w2, and w3 and consider the putative division D on W = {w1, w2, w3} whose cells are {w1, w2},{w1, w3}, {w2, w3}. Then it is readily shown that there is no underlying similarity relation R.18 Thus if we insist upon the correspondence with similarity relations, then D must be dropped as a genuine division. This would not be so bad if examples of this sort did not naturally arise within the framework Yablo adopts. But this is not so. Consider, for example, a block universe with three blocks a, b, and c, where a can fit on b, b on c and c on a and suppose the blocks constitute a tower. There are therefore three possible worlds, depicted below: w1 a b c w2 b c a w3 c a b 17 See especially fn 27 (p. 37), where he writes “I have decided for practical reasons to stick with divisions, leaving covers to footnotes, but “really” the whole thing should be redone with them” and he makes clear that we would then lose the one-one correspondence with similarity relations. 18 For R must relate any two distinct worlds and hence should be a relation for which {w1, w2, w3} is a maximal set of similar worlds and hence a cell. The example is from Hazen and Humberstone [2004] (Example 1.1), which is a general study of the correspondence between divisions and similarity relations. 21 Consider now the proposition that a is on b or b is on c or c is on a. This has three verifiers - a on b, b on c, and c on a, corresponding to the cells {w1, w3},{w1, w2}, {w2, w 3}, as in the example above. Thus even when we insist on minimality or require the verifying states to be “incomparable” (p. 37), there will be a problem in sustaining the correspondence with the relational model.19 Somewhat ironically, the correspondence between subjects matters and equivalence relations can be reinstated if we adopt the earlier maximal account of subject matter but under a different understanding of how the two are meant to correspond. For given a subject matter s, we may take two worlds w and v to be equivalent if w ¢ s = v ¢ s, i.e. if the common part of w and s and of v and s is the same. Moreover, given an equivalence relation defined in this way, one can, under plausible assumptions (and assuming intensionality), recover the underlying subject matter s by forming the intersection of the worlds in each equivalence class and then taking their fusion. Associated with any subject-matter s is the “cover” {t: t ¥ s}, consisting of all of the parts of s. Two worlds will then be s-equivalent just in case they agree on the associated cover, i.e. just in case the states of the cover in w and in v are the same. Thus s-equivalence is a form of agreement on subject matter, as it is for Yablo. But Yablo’s form of agreement is existential, whereas ours is universal. For two worlds to agree is, for Yablo, for them to agree on some member of the cover whereas, for us, it is for them to agree on every member of the cover.20 §7 Containment Yablo provides two kinds of account for what it is for one proposition to contain another, which I call direct and indirect. The direct account is so-called because it is stated directly in terms of the verifiers and falsifiers for the two propositions. The indirect account, by contrast, also makes appeal to the subject matter of the two propositions. It is thought that, under a suitable definition of subject matter, the two accounts will then be equivalent. So, although the direct account is perhaps more basic, the indirect account has the advantage of showing how propositional containment relates to the notion of subject matter. Each kind of account involves two clauses, which I shall call the forward clause and the reverse clause. The forward clause requires that the one proposition P imply the other Q in something like the standard sense of truth preservation. The reverse clause is then meant to exclude those implications in which the content of Q somehow goes beyond the content of P (cf. p. 3). It is the reverse clause that is thought to have a reformulation in terms of subject matter, 19 Curiously, a counter-example of this sort does not arise within a “canonical” state space, in which all states are formed from a class of independent atomic states. 20 These two forms of agreement come to the same thing for Lewis: two worlds will agree on a cell in a partition just in case they agree on every cell in the partition. Yablo is aware of these two ways of defining agreement (or difference) (fn. 40, p. 40) but goes astray, from our own point of view, in opting for the former. Note that the intermediate closure P* and the maximal closure P** will result in the same notion of equivalence (or agreement) between worlds. Thus even though it is possible to recover P** from the resulting equivalence relation, it is not possible, in general, to recover P*. 22 the idea being that it will hold when the subject matter of the second proposition is included in the subject matter of the first. There are two main ways of understanding the notion of implication in the first clause - as classical or as relevant; and we thereby arrive at a classical or relevant understanding of containment. Within our framework, classical implication is a matter of every loose verifier of P being a loose verifier of Q, where a loose verifier of a proposition is any state incompatible with a falsifier of the proposition. Relevant implication, by contrast, is a matter of every inexact verifier of P being an inexact verifier of Q, where an inexact verifier of a proposition is a state containing an exact verifier of the proposition. Alternatively, it is a matter of every exact verifier for P containing an exact verifier for Q. The two kinds of containment behave differently. Thus P v (¬P w Q) will contain Q under the classical conception though not, as a rule, under the relevant conception. Yablo seems largely to favor the classical notion (pp. 45-6), although he does discuss a version of the relevant notion (p. 59).21 The direct formulation of the reverse clause is that every verifier of Q should be contained in a verifier of P (and similarly for falsifiers, should we be working with bilateral propositions). Recall that, in the case of relevant containment, the forward clause was that every verifier of P should contain a verifier for Q. And so, with the relevant notion, we obtain a pleasing symmetry between the forward and reverse clauses. I do not wish to adjudicate between these various notions of containment here (see Fine [2015b] for further discussion). My present interest is in the relevance of subject matter to containment. To what extent can we think of the reverse clause as a requirement on subject matter under the different conceptions of subject matter? For me, the positive subject matter of a bilateral proposition P = (P+, P-) is p+, the fusion of its verifiers, its negative subject matter is p-, the fusion of its falsifiers, and its overall subject matter p is p+ £ p-.22 Suppose the direct reverse clause for P = (P+, P-) and Q = (Q+, Q-) is satisfied and let Q+ = {q1, q2, ...}. Then each qk from Q+ is contained in a pk from P+. But then q+ = q1 £ q2 £ ... is included in p1 £ p2 £ ..., which is included in p+. Thus q+ is included in p+. 21 The interpretation of the text is complicated by the fact that his three formulations of containment on pp. 45-6 ((11), (12), and (13)) are not, in fact, equivalent. For given the definition of subject matter inclusion at (5), the second clause of (11) will require that every verifier of A imply a verifier of B, where there is no such requirement in (12) and (13). I assume that (12) and (13) give Yablo’s true intent and that, as before, the definition of subject matter inclusion at (5) should be appropriately weakened. I should also note that the discussion of relevant containment on p. 59 is unduly specific in a number of ways. For there is no need (i) to tie it to a recursive conception of truthmakers, (ii) to identify states with sets of literals, or (iii) to suppose that the truthmakers of atomic sentences are themselves atomic. Here, as elsewhere, we should take ourselves to be operating within an arbitrary state space. 22 I do not discuss the role of overall subject matter in characterizing containment here, though it is relevant to the conception of containment in Parry [1933], under which P v Q, for example, will contain ¬P w Q. 23 Similarly, q- is included in p-; and, consequently, q = q+ £ q- is included in p = p+ £ p-. Thus, given the reverse clause, the natural requirement on subject matter - be it positive, negative or overall - will also be satisfied. The converse is trickier. Let us assume that the verifiers (or falsifiers) of any given proposition are closed under fusion. This means, for example, that we should adopt an “inclusive” conception of disjunction under which p £ q will be a verifier of P w Q when p is a verifier of P and q of Q. It can then be shown that the converse implication will hold: if q+ is included in p+ (q- in p-) then every verifier (falsifier) of P will be contained in a verifier (falsifier) of Q. However, the converse will not hold in the absence of this assumption. Consider, for example, the propositions (P v Q) w (P v R) and P w (Q v R). Then the positive (negative) subject matter of the second will be included in the positive (negative) subject matter of the first but the verifier q £ r of the second will not be contained in a verifier of the first. Thus subject matter inclusion will still be a necessary condition for containment but will not serve, given the forward clause, as a sufficient condition. There is no such mismatch under the intermediate and minimal accounts of subject matter; the reverse clause is simply equivalent to subject matter inclusion. This might appear to give the edge to these other accounts. I do not know if it was partly so that subject matter could exactly fulfil this role in the definition of containment that Yablo proposed the strict conception of subject matter that he did. But, as have already noted, Yablo’s notion of subject matter is not properly a notion of subject matter at all; and if we wanted a notion of subject matter to play this role, we would have done better to have adopted the intermediate account. In any case, it is not even clear that we should want the notion of subject matter to fulfil this role or that we should treat it is a mark against an account if it does not. For although we might reasonably insist that it should be necessary for P to contain Q that the subject matter of Q be included in that of P, it is not clear why containment should not amount, in the presence of the forward condition, to something more than subject matter preservation. §8 Restriction Yablo considers two ways in which propositional discourse may be restricted to a given subject matter (p. 32): the truth of the proposition may be restricted to the subject matter; and the proposition itself may be restricted to the subject matter. Let me first discuss how the “standard” theory of these two notions should go, without regard to how they might be defined, then discuss my own account, and finally Yablo’s. Suppose P is a (bilateral) proposition and s a given subject matter. Then the two notions we wish to consider are: P is true about s at the world w; and the restriction Ps of P to s, i.e. the part of P about s. The following are naturally taken to hold of the former: (i) P w Q is true about s at w iff P is true about s at w or Q is true about s at w; (ii) P v Q is true about s at w iff P is true about s at w and Q is true about s at w; (iii) If P is true in w then P is true about s in w; (iv) P is true about s and t in w iff it is true about s + t in w (v) P is true about ó(P) at w iff P is true at w. (i) and (ii) state that truth about s “commutes” with disjunction and conjunction. From 24 (iii), we obtain the following principle for negation: Either P is true about s at w or ¬P is true about s at w at least for bivalent P, since then P or ¬P is true at w and so P or ¬P is true about s at w. Note that we do not have that truth about s commutes with negation, i.e. that ¬P is true about s in w iff P is not true about s in w. Indeed, if we did then it would follow that truth about s would coincide with truth simpliciter. For if P were true in w then it would be true about s by (iii); and if P were not true in w then ¬P would be true in w, so ¬P would be true about s in w by (iv) again, and so P would not be true about s in w by Negation Commutativity. Thus it is only by allowing both a proposition and its negation to be true of some subject matter that we can draw a distinction between truth about some subject matter and truth simpliciter. (iv) states that a proposition will be true about each of two subject matters in a world iff it is true about their combined subject matters in the world; and (v) states that a proposition will be about its own subject matter at a world iff it is true at the world. We turn to propositional restriction, for which the following identity principles are plausibly taken to hold: (vi) (P w Q)s = (Ps w Qs) (vii) (P v Q)s = (Ps v Qs) (viii) Ps = Pss. (ix) Pó(P) = P. We also have the following mereological principles: (x) Ps = Ps¢ó(P) (xi) ó(Ps) = s ¢ ó(P) (xii) Ps is a part of P and hence implies P. We also have the following “bridge” principle: (xiii) P is true about s in w iff Ps is true in w according to which a proposition will be true about a given subject matter just in case the restriction of the proposition to the subject matter is true. Principle (xiii) might well be regarded as a definition of subject-related truth. Thus one might hope to derive the principles (i)-(v) from this definition and corresponding principles for propositional restriction. This can indeed be done. Thus in the case of (i), we have: P w Q is true about s at w iff (P w Q)s is true at w by (xiii) iff (Ps w Qs) is true at w by (vi) s s iff P or Q is true at w by clause for disjunction iff P is true about s at w or Q is true about s at w again by (xiii). This is by no means a complete set of principles (further principles are given in Fine [2015]) but they do give a feel for how the theory of these various notions might be developed. We can provide beautifully simple accounts of restricted truth and restricted propositions on our own approach. First a little notation. Given a subject matter (i.e. state) s, let the restriction ps of the state p to s be p ¢ s, i.e. the common part of p and s. Then: the unilateral proposition Q or the bilateral proposition P = <Q, R> will be true about the subject matter s at a world w if, for some verifier q in Q, qs is a part of w. Let the upward restriction Ps of a unilateral proposition P be {ps: p 0 P} and the downward restriction Ps be {p: p ¥ s}. We then let the 25 restriction Ps of the bilateral proposition P = <Q , R> be <Qs, Rs>. Thus, in each case, the restricted notion is obtained by appropriately restricting the verifiers (or falsifiers) in question to the given subject matter. Given these definitions and with the help of some modest assumptions, the various principles above can then be derived. Thus P v Q will be true about s at w iff P and Q are each true about s at w. We also see immediately, for example, that bilateral P relevantly contains Ps, since every verifier p of P contains the verifier p ¢ s of Ps and every verifier q of Ps is of the form p ¢ s for p a verifier (falsifier) of P and therefore contained in p and every falsifier of Ps is already a falsifier of P. Or again, the restriction of P = <P+, P-> to its subject matter p = p+ £ pwill be <Q+, Q->, where Q+ = {p ¢ p: p 0 P+} = P+ and Q- = {p: p ¥ s and p 0P-} = P-. We can also show that if P is a bivalent proposition, true or else false in each world, then so is Ps. Let us now turn to Yablo’s account of restricted truth (p. 32). Given a Lewisian subject matter s (a partition or equivalence relation on worlds), the unilateral proposition P will be true of s at a world w iff P is true in some world s-equivalent to w. Thus whereas, for me, P is true about s at w when some s-variant of P is true at w, for Yablo, P is true about s at w when P is true at some s-variant of w. I shift the proposition, he the world. An anomaly in his approach should be noted straightaway. For subject-related truth is only defined for Lewisian subject matters, whereas the overall subject matter of a proposition is not in general a Lewisian subject matter (fn. 8, p. 49).23 Yablo is aware of the anomaly but seems unperturbed by it. But surely we want to be able to say that one proposition is true about the (overall) subject matter of another proposition or that it may be restricted to the subject matter of the other proposition and, in particular, we would want to be able to assert truisms such as (v) and (ix) above, according to which a proposition is true about its subject matter at a world iff it is true at the world and the restriction of a proposition to its own subject matter is the proposition itself. Nor is it clear how the form of definition that Yablo adopts in the case of Lewisian subject matter might be extended to overall subject matter. This is not a difficulty for our own account and strikes me as a real mark against his. Although the form of the definition is rather different in the two accounts, it can be shown, under certain simplifying assumptions, that the two accounts will largely agree on particular cases. Since Yablo’s definition is stated entirely in intensional terms (no appeal is made to verifiers or falsifiers, only to possible worlds), we need to suppose, on our side, that we are working within an intensional state space in which each possible state can be identified with a set of possible worlds. We also need to show how a Lewisian subject matter X corresponds to one of our subject matters. Suppose the cells of X are the states s1, s2, ... (these are sets of worlds). Then the corresponding subject matter sX is s1 £ s2 £ ..., which will, of course, be an 23 Strictly speaking, an overall subject matter is never a Lewisian subject matter, since the former is a pair of divisions, which is not of the right set-theoretic type to be a partition. But the overall subject matter {{V}, {W - V}} might be taken to correspond to the bipartite partition {V, W - V} 26 impossible state.24 Conversely, given the subject matter s the corresponding equivalence relation //s is one which takes worlds w and v to be equivalent when they agree on s, i.e. when w ¢ s = v ¢ s. Suppose now that P is a unilateral proposition all of whose verifiers are possible states. It can then be shown that P will be true of the subject matter s at w under our account iff it is true of the Lewisian subject matter W/s at w under Yablo’s account.25 The restriction on P to possible states is important. We have already observed that a proposition P and its negation ¬P might both be true of a certain subject matter s at a world w. Now our account satisfies the requirement (principles (ii) above) that a conjunction is true of some subject matter at a world iff both of its conjuncts are; and so, on our account, the contradiction P v ¬P in such a case will be true of the subject matter in the world (a trivial case, though not the only case, is one in which P and ¬P have nothing to do with the subject matter). However, P v ¬P is never true of any subject matter at a world on Yablo’s account, since that would require it to be true at some world. Yablo is aware that of this consequence (p. 32-3). As he puts it (fn. 16), ‘truth about m is not agglomerative’. He presents the consequence merely as a feature of his account rather than as a bug. But there are a number of general principles, such as (ii) above, which are highly plausible but which do not hold under Yablo’s account; and it is surely desirable to respect these principles in full generality if we can and not encumber them with artificial restrictions. Even if we insist on working within an intensional state space and restrict the verifiers to possible states, there is still a way in which our approach is more general than Yablo’s. A subject matter in the present context is, for him, a partition of the set of worlds - in effect, a set of states s1, s2, ... which are possible and mutually exclusive and exhaustive. The corresponding subject matter, for us, is the fusion s = s1 £ s2 £ .... But there is no reason, in general, why the subject matter s should take this particular “partitive” form. A case in question relates to Yablo’s discussion of whether being restrictedly true is ““as good as true” for discussion purposes” (pp. 32-34). Yablo is especially interested in cases in which a proposition is both true and false of some subject matter but in which we only appear to be interested in asserting it as true of the subject matter. The example he discusses is the number 24 The resulting impossible states must be finely individuated and, in contrast to the possible states, cannot be identified with sets of worlds. In particular, the fusion of possible states, when the fusion is impossible, cannot be identified with the intersection of the corresponding cells. 25 The proof rests upon the not implausible assumption that s conform to the condition that if p is incompatible with q ¢ s then p ¢ s is incompatible with q ¢ s. Let us also note that, even though the mereological union p £ q can be identified, in the intensional framework, with the settheoretic intersection of p and q, the mereological intersection p ¢ q cannot be properly identified with the set-theoretic union of p and q, since the union, from an intuitive point of view, may not be a state. Thus reference to the underlying state space is required to make sense of the notion. I am not aware that Yablo ever makes reference to the mereological intersection of two states but, as we see, it does important work in justifying some of his definitions. 27 of planets > 3. We want to assert it as true (of concreta). But it is also false of concreta (since there is a numberless world concretely equivalent to our own). And so why do we not also want, in similar spirit, to asserts its negation? Yablo suggests a number of interesting responses to this question, in all of which the underlying account of truth about some subject matter is subject to modification - “we’ll want to check that a statement is true, not only in some m-equivalent world, but all m-equivalent worlds of the right type” (p. 34). But there is a treatment of the case within our own framework in which no such modification is called for. Sometimes we wish, for the purposes of discussion, to ‘bracket’ whether a certain fact obtains (this corresponds most closely to case (3b) in Yablo’s discussion of the issue). Let us treat the fact as a state s (a more general treatment is also possible). Then for the (unilateral) proposition P to be true at w bracketing (or but for) s is for p s to be a part of w for some verifier p of P. What is going on in the above example, then, is that we want the number of planets > 3 to be true but for the existence of numbers. As it stands, bracketing is not a case of subject-related truth. But note that p - s = p ¢ (€ s), where € is the “full” state, the fusion of the set of all states; and so P will be true at w but for s iff it is true at w of € - s. Thus in the example above, the subject under discussion will be everything but the existence of numbers. In this way, our interest can be seen to lie in a subjectrelated form of truth, but where the subject matter is not of simple partitive form and where only the subject under discussion, not the notion of subject-related truth itself, is in need of modification. I turn next to Yablo’s discussion of subject-restricted propositions (§3.4). Given an intensional proposition V and a Lewisian subject matter s, let Vs be the smallest /s-closed set to contain V and let Vs be the greatest /s-closed set to be included in V. Thus Vs is {w: w /s v for some v 0 V} and specifies, according to Yablo’s previous account of subject-restricted truth, when X is true about s, while Vs is {w: v 0 V whenever w /s v}. Recall that, for Yablo, each state is itself an intensional proposition; and so, given a unilateral proposition P, we may set Ps ={ps: p 0 P} and Ps ={p: p 0 P and p f (^P)s}; and, given a bilateral proposition P = <Q, R>, we may then let its restriction Ps to s be <Qs, Rs>. His account, in general form, is close to our own. The principal difference is in the definition of ps. For me, this is simply p ¢ s whereas, for him, it is {v: v /s w for some w 0 p}. I suspect that in most normal cases, the two will coincide (conceiving of p ¢ s as an intensional proposition), and so the two accounts will then yield similar results. However, as Yablo himself points out (fn. 13, p. 53), there is no general assurance that his ps could serve as a suitable verifier for Ps. But there is another difficulty. Let U be the set of worlds ^Qs in which Ps is true and V the set of worlds ^Rs in which Ps is false Then there is no no general assurance, if we start off with an arbitrary Lewisian subject matter s, that the set U c V of the worlds in which Ps is either true or false will be the total set of worlds W, for there may be a world v of V for which each r in R having v as a member overlaps with U. Such a world v will then lie in neither Qs nor Rs and the corresponding proposition Ps, in that case, will not be bivalent.26 These difficulties do not arise on our own account, for ps is defined to be a state, viz. p ¢ 26 In specifying the intensional proposition associated with A/m (clauses (1) and (3), p. 53), Yablo seems to take for granted that the proposition will be bivalent. 28 s, and it can be demonstrated that Ps will be bivalent whenever P is bivalent.27 By failing to acknowledge the existence of an underlying state space, Yablo denies himself the resources by which these various difficulties might be resolved. §9 The Incremental Conditional In his book, Yablo develops an account of logical subtraction (pp. 184-5) which, in a subsequent paper, is used to provide a semantics for the ‘incremental’ conditional A |6 C. As he puts it in the paper, ‘the idea is that |6 should be incremental, in the sense of picking up where A leaves off and stopping when it gets to C. The truth of A |6 C should turn on the truth of whatever it is that C adds to A - what is sometimes called C’s surplus content with respect to A’ (p. 3 of the paper). The account, in rough outline, is as follows. ‘A reason for C to be true-given-A [which is a reason for A |6 C to be true] is a truthmaker X for A e C that is consistent with A and makes the fullest possible use of A. ... X does that ... if it minimizes the extent to which B e C is also implied for B weaker than A’ (p. 14 of the paper and fn. 9 of p. 184 of the book). Yablo gives a somewhat complicated definition of what it is for the truthmaker to make the fullest possible use of A (or not be ‘wasteful’), but the details need not concern us in what follows. It should be noted that Yablo wishes to take the truthmakers for the incremental conditional A |6 C to be a subset of the truthmakers for the material conditional A e C. He had earlier cautioned against thinking that ‘A e C can only hold because A is false or C is true’ (p. 4 of the paper); and it is clear from his subsequent definition that he actually wishes to take every one of A #6 C’s reasons to be a truthmaker for A e C. But when is a state a truthmaker for the material conditional? Yablo does not say. And this means that we do not have a full account of the truthmakers for the conditional A #6 C but one that would appear to hinge critically on what one takes to be the truthmakers for the material conditional. This would not matter if one could adopt some pre-existing account of the material conditional. But the only accounts on offer are the recursive account, according to which A e C can only be true because A is false or C is true (p. 60), and the reductive account, according to which A e C can only be true because of a minimal truthmaker (p. 61).28 The first is clearly unsatisfactory and, as will become clear, the incremental conditional can be true without there being a minimal truthmaker for the material conditional. The appeal to truthmakers for the material conditional would also not matter if it could be dropped. But we may show that this not to be possible. For take a sentence A, such as œx(x = 27 We need to suppose that the subject matter s satisfies the condition from footnote 24. Yablo mentions another difficulty (fn. 15, p. 32), which is that ‘there is not always such a thing as a part of A about m’ for ‘it will have ... to be included in A’s subject matter a’ and ‘connect up somehow with m’, which will be impossible ‘if m and a are unrelated’. But I would have thought that the part of A about m will be the part about the common part of the subject matter of A and m, which will be the “null” subject matter when m and a are unrelated. 28 A minimal truthmaker for a sentence is a truthmaker no proper part of which is a truthmaker for the sentence (p. 61). 29 x), which is trivially true. Let E now be the sentence ‘Eve ate infinitely many of the apples a1, a2, ... ’ (this is from Alternative Eden, which contains infinitely many objects of temptation, not one). Then A #6 E is presumably made true by Eve eating any infinitude of the apples a1, a2, .... Let AN now be the sentence ‘Eve ate infinitely many of the apples a2, a3, ...’ (excluding a1). Then this is presumably made true by Eve eating any infinitude of the apples a2, a3, ... but not by her eating any infinitude of apples which includes a1. But what could explain this difference in truthmakers? When we examine Yablo’s account, we see that it is only the appeal to the truthmakers for the material conditional that could explain the difference, since all other components of the account appeal only to the intension of the consequent, which is the same for E as for EN. We may use a similar example to throw doubt on Yablo’s ‘no waste’ requirement. Let A now be the sentence ‘Eve bites into all of the apples a1, a2, ... ’ and E the sentence ‘Adam and Eve share (i.e. together bite into) infinitely many of the apples’. Then presumably the truthmakers for A #6 E are Adam’s biting into an infinitude of the apples. But such a truthmaker is always wasteful (under any reasonable definition of ‘wasteful’) since Adam’s biting into a smaller infinitude of apples will do just as well. I wish now to contrast Yablo’s account of the incremental conditional with an alternative account. Although I have given two accounts of logical subtraction in an earlier paper, I wish here to consider a third, which derives from the truthmaker semantics for intuitionistic logic developed in Fine [2014]. The reader will recall that, in the standard BHK semantics for the intuitionistic conditional, a construction for A e C is a construction which in application to a construction for A yields a construction for C. This is already reminiscent of Yablo’s intuitive idea that a truthmaker for A #6 E should be some kind of route from A to E. The reader may also recall that the corresponding Heyting algebras will be residuated in the sense that for any elements (or states) s and t there there will a least state u, which we designate as s 6 t, for which s £ u ¨ t.29 Again, we may think of s 6 t as a route, at the level of states, from s to t. In the truthmaker semantics for intuitionistic logic, these two ideas come together. We define the residual s 6 t as the least state u for which s £ u ¨ t and assume that the residual always exists. We may now project the operation of residuation upwards from states to propositions by means of the following clause: s ||- A #6 C if ›f:|A|+ 6 |C|+ for which s = ·t 0 |A|+(t - f(t)). We might here think of the state s as encoding a function (or construction) taking the verifiers of A into verifiers of C. Within intuitionistic logic, the negation ¬A might be defined as A ez and so, in particular, ¬(A e B) might be defined as (A e B) e z. But this is not possible within the current classical context and so we need to provide a separate definition for when A #6 C is falsified. Following Yablo’s general approach, we might adopt an analogous clause for falsifiers: s -|| A #6 C if ›f:|A|+ 6 |C|- for which s = ·t 0 |A|+(t - f(t)) although it should be mentioned that this allows the conditional A #6 C to be both true and 29 algebra. For the purposes of the comparison, I have reversed the usual ordering in a Heyting 30 false.30 This account is only applicable under the assumption that the residuals s 6 t always exist. But there is a straightforward way to dispense with the assumption. In effect, each would-be residual is replaced by the states that contain it. This gives: s ||- (A #6 C) if ›f, g:|A|+ 6|C|+ for which t £ g(t) ¨ f(t) for each t 0 |A|+ and s = ·t 0 |A|+ g(t) s -|| (A #6 C) if ›f, g:|A|+ 6|C|- for which t £ g(t) ¨ f(t) for each t 0 |A|+ and s = ·t 0 |A|+ g(t). It would be of interest to compare the the present account with Yablo’s - both in terms of their particular consequences and the resulting logic. But we should note that our own account may overgenerate verifiers in comparison with Yablo’s account. Consider, for example, the conditional p w q #6 (p v r) w (q v s). Identifying sentence letters with their verifiers, this will have q £ s as a verifier (using the function which takes p to p v r and q to q v s), as one might expect. But it will also have p £ r £ q £ s as a verifier (using the function which takes p to q v s and q to p v r.) Perhaps this should not be too bothersome given that |A #6 C|+ contains all the other verifiers we would like to have; and, of course, when there exist most efficient routes from A to C (in effect, minimal verifiers of (A #6 C)) then we might always remove from |(A #6 C)| any states that properly contain them. §10 Other Conditionals If I am not mistaken, Yablo’s incremental conditional is closely related to the intuitionistic conditional. But it is also related, though not so closely, to another conditional, which Yablo has introduced in his book. I will call it the ‘suppositional conditional’ since it derives from the suppositional account of the conditional in Belnap [1970]. What Yablo calls the ‘depragmatized analogue of Belnap’s conditional’ is subject to the following two clauses (p. 66): If A is true, then A_C is true (false) for the same reason(s) as C. If A is false, A_C is vacuously true - true without a truthmaker. One defect of this formulation, to my mind, is that it allows the conditional A_C to be true (viz. when A is false) without a truthmaker. But it would be desirable to be able to accept the general principle that no sentence can be true without a truthmaker; and there is, in any case, no need for Yablo to accept any lapse from this principle in the present case, since he can always take the truthmaker for A_C, when A is false, to be the null state ~ (which, within the intensional framework, may be identified with the set W of all possible worlds).31 Another feature (though not defect) of his account is that whether a truthmaker for C is a truthmaker for A_C will depend upon whether A is true; it will be a truthmaker for A_C when A 30 We might follow Yablo (p. 185) and take the conditional to be true (false) simpliciter only when it is true (false) without being false (true), although there are other means by which truth-value gluts might be avoided. 31 I do not know if it was the requirement that truthmakers should be incomparable that led Yablo to reject ~ as a truthmaker. But, as we have seen, this is not a requirement that should have been made in the first place. 31 is true but not, in general, when A is false. Thus the truthmaking relation is, in effect, relative to a world: whether a state verifies a sentence will depend not merely upon the state but upon whether the sentence is true or false in the given world.32 Let us use ‘||-w’ for world-relative truthmaking (and similarly for ‘-||w’). Thus s ||-w A indicates that s (exactly) verifies A in the world w, where it is taken for granted that this relation can only hold when s ¥ w (or, within the intensional framework, only when w 0 s). Let us also use ‘w |= A’ to indicate that A is true at the world w.33 Then the modified and more explicit version of Yablo’s clauses for ‘_’will run as follows34: (a1) s ||-w A_C iff either (i) w |= A and s ||- C or (ii) w /|= A and s = ~ (b1) s -||w A_C iff w |= A and s -|| C. One consequence of taking the verification (or falsification) of the conditional to be world-relative is that clauses for the other connectives should also be formulated in worldrelative form. Suppose, for example, that we wished to evaluate the conjunction (A_C) v (AN_CN) of two conditionals. Then since the evaluation of the component conjuncts, (A_C) and (AN_CN), is world-relative, this would appear to require that the evaluation of the conjunction should also be world-relative. However, it is pretty obvious how the corresponding worldrelative clauses should go. Thus in the case of negation and conjunction, we will have: s ||-w ¬B iff s -||w B s -||w ¬B iff s ||-w B. s ||-w B v C iff ›s1, s2: s = s1 £ s2, s1 ||-w B and s2 ||-w C s -||w B v C iff s -||w B or s -||w C I wish now to consider various modifications and refinements to these clauses. They are of some interest in their own right but will also help us to see more clearly the connection with the incremental conditional. Derelativization The present clauses for the conditional are relative to a world. But instead of thinking of 32 Perhaps somewhat oddly, Yablo in his previous discussion of the truthmakers for a universal generalization œxFx in §4.4 does not take the truthmakers to be relative to a world which contains certain individuals but takes them instead to presuppose that these are the individuals. I myself am somewhat unhappy with truthmakers carrying presuppositions in this way. Truthmakers are in the world; and it seems to me that presupposition properly belongs to language, not to the world. 33 We might define w |= A to hold when w loosely verifies A or when it inexactly verifies A (the two are equivalent in the case of worlds). 34 These are clauses I myself have adopted in some unpublished work on the semantics for imperatives. There is a question, which I shall not consider, of how clauses like the ones above can be extended to conditionals A_C in which A or C may also contain a conditional. 32 a truthmaker s for a sentence A as relative to a world w, we can think of their combination (w, s) as itself a truthmaker. We call the combination (w, s) a divided state and write it as w|s, to make vivid that the first component w is meant to serve as background and the second component s as foreground. Thus w|s is s in the context w or s given w. The clause for the conditional can now be reformulated in non-relative form, using divided states in place of undivided states. Thus in the case of the conditional A_C, we have: (a2) w|s ||- A_C iff either (i) w |= A and s ||- C or (ii) w /|= A and s = ~ (b2) w|s -|| A_C iff w |= A and s -|| C and there will be analogous clauses for the other connectives. This move is, in itself, quite trivial. It assumes a greater significance once we appreciate that the new clauses are essentially the same as the original clauses, but with a shift in the state space. This calls for further explanation. Suppose we are given two state spaces S and SN (the states from each state space need not be the same). We may then form the product space S × SN. Its states are ordered pairs of the form (s, sN), where s is a state of S and sN of SN; and one such state (s, sN) is said to be part of another (t, tN) in the product space if s is a part of t in S and sN a part of tN in SN. Yablo’s clauses are stated relative to a world. A world, for us, is a certain kind of state, what I call a “world-state”. Formally, w is a world-state if every possible state is either a part of w or incompatible with w. We may also call a space S a W-space if it contains the world-states one would expect to exist, i.e. if each possible state of S is part of a world-state in S. Suppose now that we are given a W-space S and restrict the space to the world-states of S along with the null state ~ and the full state €. Thus the world-restriction SW of S will be a “discrete” space, looking like this: « € « ¯ ¯ w1 w2 w3 w4 ¯ ¯ « « .... ~ with the worlds mereologically unrelated, the null state ~ part of every state, and the full state € containing every state as part. The world-restriction SW is effectively the plurivese W of all possible worlds, but with the null state and full state thrown in for good measure to guarantee that we still have a state space. Given a W-space S, we may then form the product space SW × S, whose states are of the form (w, s), (~, s) or (€, s), for w a world-state of S and s a state of S. We take a state (t, s) of SW × S to be possible if t is compatible with s, i.e. if t £ s is a possible state of S. Formulas can be evaluated within the product space SW × S according to the usual rules. It can then be shown that evaluating a formula in the original state space S relative to a world is effectively equivalent to evaluating the formula in the product space SW × S. Thus from this perspective, world-relative evaluation merely reflects a shift in the underlying state space and 33 does not constitute an essentially new form of evaluation.35 Two operations are involved in the passage from the original state space S to the product space SW × S: one is the operation of restriction, whereby we obtain SW from S; the other is the operation of forming a product, whereby we obtain SW × S from SW and S. These are two familiar mathematical operations on structures; and I believe that they have many varied applications within the truthmaker framework. De-Construction I have so far followed Yablo in appealing to a world in which the antecedent A of the conditional might be true. But this is quite unnecessary. For if A is true in a world, it is exactly verified by a state of the world. Thus in place of his world-relative clauses, we may substitute the following state-relative clauses: (a3) s ||-t A_C iff either (i) t ||- A and s ||- C or (ii) t -|| A and s = ~ (b3) s -||t A_C iff t ||- A and s -|| C. We might, of course, simultaneously decontextualize and deconstruct, thereby obtaining the following clauses: (a4) t|s ||- A_C iff (i) t ||- A and s ||- C or (ii) t -|| A and s = ~ (b4) t|s -|| A_C iff t ||- A and s -|| C. And it can be shown, as before, that the clauses for the divided states t|s can be seen to arise from working within the appropriate product space. Minimization According to clause (a4)(i) above, t|s will verify A_C when t verifies A and s verifies C. But what when t and s overlap? The verification of A_C may then be thought to involve overkill. Consider the conditional ‘if Eve ate the first of the two apples (a1 and a2) then she ate both of the apples’. Let us suppose that t is her eating the first apple and s is her eating both apples. Then s|t, her eating both apples given that she ate the first, will verify the conditional. But it might be thought that what properly verifies the conditional is her eating the second apple given that she eats the first. Putting the point more generally, it might be supposed that when we have a divided state t|s, in which t represents the background and s the foreground, then the two should not overlap, that what is in the foreground should be exclusive of what is in the background. We therefore arrive at the following modified version of clauses (a4) and (a5): (a5) t|s ||- A_C iff (i) t ||- A and s = t 6 u for some u for which u ||- C or (ii) t -|| A and s = ~ (b5) t|s -||t A_C iff t ||- A and s = t 6 u for some u for which u -|| C, where t 6 u is the previously defined residual. It should be noted that the operation of minimizing the verifier or falsifier for the 35 Stated more rigorously, the result is that s ||-w A in model over S iff w|s ||- A in the corresponding model over SW × S. 34 conditional A_C, i.e. using t |(t 6 u) in place of t|u makes no sense when we take the background state t to be a whole world. Thus it was only by “deconstructing” the world that we were able to come up with the present version of the semantics. Abstraction According to the previous clauses, the conditional A_C is true (or false), when A is true, in virtue of a specific connection between A and C, one that concerns the specific way in which A is true. But one might be interested in the conditional which is true (or false) in virtue of a general connection between A and C, one that does not concern any specific way in which A might be true. And this suggests that we might obtain truth (or falsity) makers for such a conditional by abstracting from the truth (or falsity) makers for A_C. Let us see how this might be done, focusing on the truthmaker case. Let the given relational content be R. Then a function f f R corresponds to a possible ‘route’ or ‘transition’ from ‘left’ to ‘right’; and we might think of the abstract content as encoding such transitions. One way is to do this is to encode f as the fusion f# of all the residuals s 6 f (s) (for s in the domain of R), so that the abstract content R# is given by: (A1) R# = { f#: f 0 R} and represents the extra steps we must take in getting from left to right. Another, simpler, way is to encode f as the fusion f*of the states in the range of f, so that the abstract content R* is given by: (A2) R* = {f*: f 0 R} and represents what is required on the right regardless of how it might have been reached on the left. The operations # and * provide us with a link between the modified semantics for the suppositional conditional and our intuitionistic-style semantics for the incremental conditional. The starting point can either be the un-minimized clauses (a4), (b4) or the minimized clauses (a5), (b5). But in each case, we should exclude the vacuous verifiers t|~ of A_C, as given by the second subclause (ii) of (a), since, in contrast to the other subclauses, they do not provide us with a way of getting from the antecedent to the consequent. If now we take the modified content |A_C|+ of the suppositional conditional under clauses (a4), (b4) as our starting point, we can obtain the content |A #6 C|+ of the incremental conditional by abstracting with # or, taking the modified content |A_C|+ of the suppositional conditional under clauses (a5), (b5) as our starting point, we can obtain the content |A #6 C|+ by abstracting with *. Both the suppositional and the incremental conditional involve some kind of connection or transition between antecedent and consequence - specific in the one case, general in the other; and this suggests that there might be a more general semantics for the conditional within the truthmaker framework which would cover these and other particular cases. Here is a brief and tentative suggestion as to how a more general semantics of this sort might go.36 We suppose that there are, in principle, two different ways in which a conditional A 6 C 36 Yablo also aims for a unified account of conditionals in his paper, but along very different lines. 35 might be verified - vacuously and non-vacuously. Vacuous verification is simply a function of the content |A| of A. Vacuous verification might simply be a matter of A’s being false. But it might also be a matter of its being impossible (as with a counterfactual); and there are perhaps other possibilities still. Non-vacuous verification is a matter of there being an appropriate transition from |A|+ to |C|+. A transition, for these purposes, might simply be identified with an appropriate subset of |A|+ ×|C|+. When a transition exists, we then take a verifier of the conditional to be a state that encodes the transition. We see that the proposed semantics involves three different elements: a relation of “vacating” that holds between a state s and a content when the conditional is vacuously true; a relation that holds between |A|+ ×|C|+ and those of its subsets that are transition relations; and a relation that holds between a transition relation and the states by which it is encoded. I would now like to suggest the following schematic clause for the conditional: s ||-A 6 C if either (i) |A 6 C| contains a transition relation encoded by s or (ii) s vacates |A|. Let us consider how various semantics for the conditional conform to this proposal. The Material Conditional s will vacate |A| just in case it falsifies A. The transition relations included in |A|+×|C|+ are the constant functions, each encoded by its constant value. The Suppositional Conditional A state will vacate |A| just in case it is of the form (t, ~) for t a falsifier of A. The transition relations included in |A|+ ×|C|+ are the individual pairs (t, s), each encoded by itself. The Incremental Conditional |A| is never vacated. The transition relations included in |A|+×|C|+ are the functions f from A|+ into |C|+, each function f encoded by f# above. The Counterfactual Conditional In Fine [2012], I provided a truthmaker semantics for the counterfactual conditional. It was supposed that, for any world-state w, there was a transition relation t 6w u holding between states. The clause for the counterfactual A < C was then: w |= A < C iff u ||- C whenever t ||- A and t 6w u. The obvious way to generalize is to make the transition relation relative to an arbitrary state: s ||- A < C iff u ||- C whenever t ||- A and t 6s u. The clause may require modification to take care of the case in which the transition t 6s u is not defined for given s. But, for simplicity, let us put this issue on one side. The above semantics for the conditional arises from the schematic semantics as follows. |A| is never vacated; the transition relations included in |A|+×|C|+ are the relations whose domain is |A|+; and each such relation R is encoded by a state s for which {(t, u): t 6s u and t 0 dm(R)} = R. §11 Methodology 36 Yablo’s approach is what one might call worlds-first, while mine is states-first. He starts off with a pluriverse of worlds (which may, for certain purposes, be identified with entities of some other sort - such as models or complete descriptions); and then, at least in the simplest case, intensional propositions are taken to be sets of propositions, states to be a special kind of intensional proposition, and hyperintensional propositions to be sets of states. I, on the other hand, start off with a space of states, which are simply taken as given; propositions are then identified with sets of states; and worlds, to the extent that I have any use for them, are taken to be a special kind of state. Thus whereas he wants to build on top of the possible worlds framework, supplementing it with truthmakers where it appears inadequate, I want to dismantle the whole framework away and start anew. In Daniel Rothschild’s nice comparison, Yablo is a Fabian while I am a revolutionary.37 I have throughout the paper given a number of reasons for preferring the states-oriented approach to the worlds-oriented approach; and it will be helpful to review these and some other reasons here. However, I should mention that it is only on the basis of experience and familiarity with each of the two approaches that one can properly adjudicate between them; and since most semanticists’ experience is almost exclusively with the possible worlds framework, it is only by familiarizing themselves with the states-first that they will be in a good position to appreciate its relative merits. We may loosely classify the ways in which it pays to be stately or does not pay to be worldly under three different heads - the theoretical, technical and philosophical, although it is sometimes hard to distinguish between them. Theoretical Gains The possible worlds approach to semantics is intensional. At the end of the day, it refuses to acknowledge distinctions in meaning which cannot be given a modal basis. There are, however, different degrees of severity to which the possible worlds theorist might be committed. The austere theorist refuses to recognize distinctions in content that are not themselves modal. Thus necessarily equivalent propositions are taken to be the same, as are necessarily co-extensive properties. However, the more relaxed theorists may admit distinctions in content that are not purely modal, as long as they are ultimately explained in modal terms. Thus a proposition may be taken to be a set of verifiers and necessarily equivalent propositions may differ in their verifiers, as long as the candidate verifiers are themselves explained in straightforward intensional terms. Most semanticists attracted to the truthmaker approach, Yablo included, have been relaxed theorists of this sort.38 They have been closet intensionalists, even when they have accepted hyperintensional distinctions of content. It is interesting to note that both the austere and the relaxed intensionalist views can be accommodated within the states-first approach. For the austere theorist’s pluriverse of possible worlds corresponds to the “discrete” state space of §10; and the truthmaker theorist’s semantical clauses, in the special case in which the verifiers or falsifiers are possible worlds, will then 37 38 Made at the Hamburger workshop on Yablo’s book. Another notable example is the Amsterdam school of inquisitive semantics. 37 correspond to the familiar clauses from the possible world semantics. On the other hand, the relaxed theorist’s framework will correspond to a state space in which the states are taken to be certain ‘privileged’ subsets of possible worlds and parthood among states is defined accordingly. The truthmaker theorist can actually go one better and make a distinction within the intensionalist approach which the standard possible worlds theorist is unable to make. For he can say what it would be for a state space to be intensional without presupposing that it is a W-space, i.e. a space in which every state obtains in some possible world (§1). Thus the distinctive aspect of the possible worlds approach, its intensionalism, can in this way be detached from its commitment to possible worlds. But the stately approach goes far beyond even the relaxed theorist’s approach in the distinctions of content it is able to accommodate. A notable example concerns impossible propositions (those which cannot be true). For the relaxed intensional approach is like the austere approach in being limited in the distinctions among impossible propositions that it can make. The austere theorist can only acknowledge one impossible proposition - in effect, the empty set of worlds. Our relaxed theorist can, at least at first pass, only admit one impossible state - again, in effect, the empty set of worlds; and so is only able to distinguish between two impossible propositions (regarded now as a set of verifiers) - either the empty set of verifiers or the set whose sole member is the impossible state. By contrast, there is no real limit on the impossible propositions that might be countenanced under the stately approach. Indeed, the introduction of a plethora of impossible states can be seen to constitute a natural rounding out of the space of possible states much as the admission of irrational or complex numbers constitutes a natural rounding out of the rational line (Fine [2015d]). To some extent, the relaxed intensionalist can make sense of these additional states in her own terms, since an impossible state can be identified with the set of its possible parts (which will themselves be intensional propositions - thus a state will now be a set of sets of worlds). But such a reconstruction of the states-first approach can only go so far; if we wish, for example, to introduce a distinct impossible proposition to the effect that x not self-identical for each object x, then there is not much the intensionalist can do. Of course, this greater generality would be for nought if it served no purpose. But it seems to me that this is far from being the case and that the admission of impossible states comes with huge advantages both in regard to the general development of the truthmaker theory and its application. It enables us, for example, to provide an adequate truthmaker semantics for intuitionistic logic, even though this would not be possible with only one or, indeed, with only a fixed finite number of impossible states (Fine [2012]). It enables us, as we have seen, to provide a more satisfactory account of propositional containment (§7) and to make sweeping simplications in the account of subject-matter (§4). It also enables us to provide a non-trivial and non-stipulative account of certain counterfactuals with counterpossible antecedents (Fine [2015d]), not to mention the application to the more familiar cases of hyperintensionality. Technical Gains To treat states as simply given, rather than as defined in terms of worlds or sets of sentences or the like, is to treat them at the right level of abstraction for mathematical purposes. To the extent that additional structure is required, then it should be imposed from above, so to 38 speak, through the positing of various external properties or relations on the states, rather than be seen as arising from the intrinsic structure of the states themselves. The point can perhaps be best appreciated by means of a comparison with the possible worlds semantics of modal logic. Before Kripke’s work, it was common to identify possible worlds with objects of some other sort - sets of sentences, perhaps, or models. But Kripke simply took the possible worlds to be given and to the extent that any further structure was required it was imposed from above and not made intrinsic to the worlds themselves. This simple move greatly facilitated the technical development of the subject; it made clear what was essential to this or that application and it meant that irrelevant detail concerning the internal structure of the worlds could be ignored. Similarly in the case of state spaces. One particular benefit in this case is that we can now freely perform various operations on state spaces without having to worry over whether the objects of the resultant space are of the right sort to be properly to be regarded as states. Thus when we took the product of two spaces in our account of the suppositional conditional (§10), the states of the product space are ordered pairs (s, t) of states from the original spaces rather than models or sets of sentences or the like. Of course, this is just one case. But I believe that there are numerous applications which call for some sort of systematic shift in the state space. Suppose, for example, that we wish to provide a semantics for sentences of the form ‘A and then B’. We can think of the verifiers of A and B as belonging to a state space S; and we form the product space S × S, where a state (s, t) from S × S intuitively represents s succeeded by t. The verifiers of A can now be taken to be of the form (s, ~) for s a verifier of A, and the verifiers of ‘then B’ to be of the form (~ , t) for t a verifier of B. The verifiers of ‘A and then B’ will be of the form (s, t) for s a verifier of A and t of B and they can be seen to be the result of applying the operation of conjunction to ‘A’ and ‘then B’ within the product space. Philosophical Gains The possible worlds approach to semantics has an unduly a limited conception of what makes a sentence true and an unduly limited conception of how a sentence is made true. The truthmakers are worlds, and truthmaking is purely modal, a matter of necessitation. To a large extent, the relaxed intensionalist is able to escape from these limitations: the truthmakers can be states, intensionally conceived, rather than worlds; and truthmaking can be relevant as well as modal, in so far as it is specified by separate recursive clauses. But relaxed intensionalism - at least in practice, if not in principle - still suffers from a lingering adherence to intensionalism. It is still regarded as a matter of building over the possible worlds semantics. Thus in Yablo, the clauses for when a state verifies a sentence are regarded as a supplement to the clauses for when a sentence is verified by a world, whereas a thoroughgoing commitment to the state-oriented approach would derive the clauses for worlds from the worlds for states, taking a world to verify a sentence when it contains a state that verifies the sentence. Often this is harmless but it can sometimes stand in the way of semantic progress. An example is provided by our treatment of the suppositional conditional in §10. Yablo, in effect, takes the verification of the conditional to be relative to a world. But by taking it to be relative to a state we can gain insight into how the suppositional conditional is related to other forms of the conditional. Another, perhaps more striking, example, is provided by the Lewis-Stalnaker 39 semantics for the counterfactual. Roughly, a counterfactual A > C is taken to be true under such an account when the closest world in which the antecedent is true is a world in which the consequent is true. But this means that when we come to evaluate an ordinary counterfactual, such as ‘If I were to strike this match it would light’, I must consider the bizarre question of how far a minor miracle at the time of striking the match would detract from the closeness of the counterfactual world to the actual world whereas no such question should be involved or would be involved under a state-oriented approach (Fine [2012]). Indeed, I have found it a useful general precept to forgo altogether the appeal to possible worlds. Even if one is convinced that the state space is a W-space (with the full panoply of possibly worlds), more is to be learned by proceeding without the assumption than with. A deeper problem with relaxed intensionalism is its continued adherence to a modal conception of meaning. Distinctions of meaning are ultimately modal distinctions. But it seems to me that this view, well entrenched as it may be, is misguided; and the truthmaker approach, once properly developed, helps one to see why. For in giving the semantic clauses for the connectives, there is no need to presuppose any distinction between possible and impossible states; and similarly in stating the clauses for other expressions (except, of course, in so far as they are themselves modal). The distinction between what is and is not possible will be involved in giving an account of classical consequence, since for C to be a classical consequence of A is for no truthmaker for A to be compatible with a falsity maker for C. But it is not at all evident that the notion of classical consequence (as opposed to the notion of propositional containment) should simply fall out of the notion of propositional content. By giving up their intensionalist ideology, the semanticists of the world have nothing to lose but their chains. References Angell, R. B., [1977] ‘Three Systems of First Degree Entailment’, Journal of Symbolic Logic, v. 47, p. 147. Belnap N., [1970] ‘Conditional Assertion and Restricted Quantification’, Nous, 4(1): 1-12. Ferguson, T., [2014] ‘Faulty Belnap Computers and Subsystems of FDE’, Journal of Logic and Computation, online at 01/2014; DOI: 10.1093/logcom/exu048. Fine K., [2012] ‘Counterfactuals without Possible Worlds’, Journal of Philosophy, 109(3), pp. 221-46 (2012). Fine K., [2014] ‘Truthmaker Semantics for Intuitionistic Logic’, Journal of Philosophical Logic 43, 549-77. Fine K., [2015a] ‘Angellic Content’, to appear in Journal of Philosophical Logic Fine K., [2015b] ‘Truthconditional Content I, II’, to appear. Fine K., [2015c] ‘A Theory of Partial Truth’, to appear Fine, K., [2015d] ‘Constructing the Impossible’, to appear in a volume for Dorothy Edgington. Lewis D., [1988] ‘Statements Partly About Observation’, in ‘Papers in Philosophical Logic’, 125-155, Cambridge: Cambridge University Press. Hazen P. A., Humberstone L., [2004] ‘Similarity Relations and the Preservation of Solidity’, Journal of Logic, Language and Information 13 (1), 24-46. 40 Parry, W. T., [1933] ‘Ein Axiomensystem fur eine neue Art von Implikation (Analytische Implikation), Ergebnisse eines Mathematischen Kolloquiums’ 4, 5-6. Yablo S., [2014] ‘Aboutness’, Princeton: Princeton University Press. Yablo S., [2015] ‘Ifs, Ands, and Buts: A Truthmaker Semantics for (Some) Indicative Conditionals’, unpublished manuscript.