Richard Gaskin analyses what is distinctive about sentences and the propositions they express--what marks them off from mere aggregates of words and meanings respectively. Since he identifies the world with all the true and false propositions, his account has significant implications for our understanding of the nature of reality.

Richard Gaskin presents a work in the philosophy of language.

In 1910 Bertrand Russell abandoned the theory of propositions that he advocated in 1903 in The Principles of Mathematics because of the problem of the unity of the proposition. This is the problem of explaining how the constituents of a proposition are bound together into a unified, representational whole. This problem has largely been ignored by contemporary advocates of Russellian propositions. I argue that this problem is the result of the Fregean distinction between content and force, the (...) arguments for the content-force distinction are inconclusive and there is independent linguistic evidence against the content-force distinction. For these reasons I think that Frege's distinction between content and force should be abandoned; this is the main negative conclusion of the dissertation. ;The main positive component of the dissertation is an account of structured propositions that individuates propositions in terms of force or mood. On the account I defend, propositions are types of representational actions or states. The proposition expressed by 'Jones loves Smith', for example, is a type of representational state, tokens of which are particular assertions that Jones loves Smith or belief-states that Jones loves Smith. The proposition expressed by 'Does Jones love Smith?', on the other hand, is a type of interrogative action or state, tokens of which are instances of asking whether Jones loves Smith or wondering whether Jones loves Smith. This account solves the problem of the unity of proposition by locating the unifying element in a proposition in an act of the subject. In addition, this account provides an empirical foundation for semantics. A semantic theory, on this view, is a theory of what speakers do with sentences in contexts. To say that a sentence S expresses a proposition p in context c is to record a fact about the literal and direct illocutionary act that could be performed using S in c. (shrink)

In this paper I do four things. (1) I explain one clear thing that ‘the problem of the unity of the proposition’ might mean. (2) I lay out a few different versions of the theory of propositions as cognitive acts, and explain why this problem arises for the version of that theory which has been defended in different forms by Peter Hanks and Scott Soames. (3) I argue that the natural ways in which the act theorist might try (...) to solve the problem fail to solve it; (4) I propose a way to fix the problem, and then I explain how the problem re-emerges in the act theorist’s treatment of propositional attitude relations. (shrink)

We present here the papers selected for the volume on the Unity of Propositions problems. After summarizing what the problems are, we locate them in a spectrum from those aiming to provide substantive, reductive explanations, to those with a more deflationary take on the problems.

The paper replies to some criticisms by Nicholas Denyer of my recent book on the unity of the proposition.

‘The Unity of the Proposition’ is a label for a problem which has intermittently intrigued philosophers but which for much of the last century lay neglected in the sad, lightless room under the stairs of philosophical progress, along with other casualties and bugaboos of early analytic philosophy such as the doctrine of internal relations, the identity theory of truth, and Harold Joachim. Yet it was while struggling with this problem (among others), that Bertrand Russell built one of the first (...) steps on the staircase by creating what came later to be called the theory of descriptions.1 According to that theory, statements containing definite descriptions are true only if there exists a unique thing satisfying the description. So nothing one says about ‘The Problem of the Unity of the Proposition’, for example, can be true unless there is one and only one such problem. Yet, as we shall explain below (§1), on the one hand it is unclear that there is any such problem at all, while, on the other, if there is a problem, there seem to be several. One might conclude, then, that everything we say in this paper is likely to be false. But perhaps the paper could be, in the context, appropriately treated as a ladder, to be kicked away after climbing. For Wittgenstein, too, was concerned with the problem: ‘At the centre of Wittgenstein’s project was the task of explaining the unity of the proposition’, says Michael Potter, for example.2 Wittgenstein had inherited the task from two of his philosophical mentors, Russell and Frege. Yet while Russell’s series of failed accounts of propositions, and then judgments, each of which was meant to resolve the problem, seemed ultimately to serve only as a sort of negative inspiration for him,3 Frege’s response to the problem proved a deep influence. We will outline Frege’s position as a backdrop to Wittgenstein’s below (§§2 and 3). As we will argue, one of the most important ways in which Wittgenstein’s position resembles Frege’s is precisely that his (Wittgenstein’s) solution to the problem of unity required treating his own book as an attempt to say the unsayable.. (shrink)

(2013). Two aspects of propositional unity. Canadian Journal of Philosophy: Vol. 43, Essays on the Nature of Propositions, pp. 518-533.

This volume advances discussions between critics and defenders of the force-content distinction and opens new ways of thinking about force and speech acts in relation to the unity problem. -/- The force-content dichotomy has shaped the philosophy of language and mind since the time of Frege and Russell. Isn’t it obvious that, for example, the clauses of a conditional are not asserted and must therefore be propositions and propositions the forceless contents of forceful acts? But, others have (...) recently asked in response, how can a proposition be a truth value bearer if it is not unified through the forceful act of a subject that takes a position regarding how things are? Can we not instead think of propositions as being inherently forceful, but of force as being cancelled in certain contexts? And what do indicators of assertoric, but also of directive and interrogative force mean? (shrink)

The problem of the unity of the proposition—what distinguishes a proposition from a mere list of constituents, so that the former is able to say something while the latter is not?—is as old as philosophy. It is evoked at the end of Plato’s Sophist, where the Stranger affirms that when one makes a statement “he does not merely give names, but he reaches a conclusion by combining verbs with nouns” ; and it is discussed by Aristotle in De Interpretatione, (...) where it is said that since “falsity and truth have to do with combination and separation”, then “the first single statementmaking sentence is the affirmation, next is the negation” ; but “every statement-making.. (shrink)

The paper contains a general argument for linguistic idealism, which it approaches by way of some considerations relating to the unity of the proposition and Tractarian metaphysics. Language exhibits a function–argument structure, but does it do so because it is reflecting how things are in the world, or does the relation of dependence run in the other direction? The paper argues that the general structure of the world is asymmetrically dependent on a metaphysically prior fact about language, namely that (...) it exhibits subject–predicate structure. (shrink)

Richard Gaskin’s work on the problem of the unity of the proposition (“the problem”, henceforth) has sometimes been called magisterial due to its vast historical and conceptual scope. Indeed, the author engages in lengthy discussions of the conceptions of propositions that have been overlooked by most previous investigations on the problem. Not only aspects of Frege’s and Russell’s theories of propositions that appear most problematic are subject to Gaskin’s investigation, it also includes Prabhākara semantics, the approach of (...) Gregory of Rimini and various parts of Abelard’s thought just to name a few. Furthermore, the purpose of this work is manifold. It is not solely a historical account of the importance of the problem, nor is it only a critique of the views explored. Gaskin ventures to provide his own original solution to the problem of the unity of the proposition. The solution and the arguments in its favour, however, are deeply embedded in the structure of the extensive historical treatment of the problem and, hence, cannot be precisely depicted due to limited space. In what follows I shall hope to provide a general idea of how Gaskin approaches the topic historically and what is his endorsement towards it. Lastly, I discuss the proposed solution to the problem. (shrink)

John Collins presents a new analysis of the problem of the unity of the proposition-how propositions can be both single things and complexes at the same time. He surveys previous investigations of the problem and offers his own novel and uniquely satisfying solution, which is defended from both philosophical and linguistic perspectives.

Originally motivated by a sophism, Pardo's discussion about the unity of mental propositions allows him to elaborate on his ideas about the nature of propositions. His option for a non-composite character of mental propositions is grounded in an original view about syncategorems: propositions have a syncategorematic signification, which allows them to signify aliquid aliqualiter, just by virtue of the mental copula, without the need of any added categorematic element. Pardo's general claim about the simplicity of (...) mental propositions is developed into several specific thesis about mental propositions: a) it is not judgement which gives its unity to mental propositions, but judicative acts always follow some previous apprehensive act that is simple in its own right; b) this simplicity is compatible with a certain kind of complexity, that can be explained in terms of the "causal history" of the acts of knowing; c) traditional conceptions about subject and predicate must be recast, while keeping their usual explicative power concerning logical properties; d) of course, the traditional conception about the copula has been modified, giving rise to a fully innovative conception of the nature of mental propositions. Nevertheless, this innovative conception of mental language seems still infected by certain "common sense" prejudices, which lead Pardo to propose also a provocative conception of vocal language, which I consider unnecessary. (shrink)

In this article I present a summary of Bertrand Russell's protracted attempts to solve the problem of the unity of the proposition, and explain the significance of the problem for Russell's philosophy. Unlike many other accounts which take the problem to be confined to Russell's early theories of propositional content, I argue that the problem (or variants of it) is a recurring theme throughout the whole of Russell's career.

ABSTRACT Although in recent years Christine Ladd-Franklin has received recognition for her contributions to logic and psychology, her role in late nineteenth- and early twentieth-century philosophy, as well as her relationship with American pragmatism, has yet to be fully appreciated. My goal here is to attempt to better understand Ladd-Franklin’s place in the pragmatist tradition by drawing attention to her work on the nature and unity of the proposition. The question concerning the unity of the proposition – namely, (...) the problem of how to determine what differentiates a mere collection of terms from a unified and meaningful proposition – received substantial attention in Ladd-Franklin’s time, and would continue to interest analytic philosophers well into the twentieth century. I argue that Ladd-Franklin had a distinct theory of the proposition and solution to the problem of the unity of the proposition that she developed over the course of her writings on logic and philosophy. In spelling out her views, I will also show her work interacted with and influenced that of the pragmatist who was her greatest influence, C.S. Peirce. (shrink)

What makes a visually appealing landscape? How can the design and use of a landscape be harmonised? In this significantly revised and updated third edition of Simon Bell's seminal text, he further explores the answers to these questions by interrogating a range of design principles, applications and ideas. Written for students, instructors and professionals, the book unveils a visual design vocabulary for anyone involved with landscape aesthetics including landscape architects, architects, planners, urban designers, landscape managers, foresters, geographers and ecologists. Structured (...) around key design terms, which are explained and illustrated using an extensive range of examples from around the world, including North America, Europe and Asia, this book enables you to describe, debate and design the visual landscape. It starts with basic elements, before moving onto variable design components, and then the ways these elements are organized into compositions, in order to demonstrate how landscapes are created and how meanings and patterns are perceived within them. This new full colour edition contains over 240 images; an updated introduction; examples from China, Vietnam, and central Asia; a chapter on how to read and understand visual design elements in the landscape; a teaching model for instructors; and expanded appendix materials including a glossary, references and further reading. (shrink)

_From Naming to Saying_ explores the classicquestion of the unity of the proposition, combining an historical approach with contemporary causal theories to offer a unique and novel solution. Presents compelling and sophisticated answers to questions about how language represents the world. Defends a novel approach to the classical question about the unity of the proposition. Examines three key historical theories: Frege’s doctrine of concept and object, Russell’s analysis of the sentence, and Wittgenstein’s picture theory of meaning. Combines an (...) historical approach with discussion and defense of a contemporary causal theory of the unity of the proposition. Establishes a view compatible with, though not dependent on, a causal theory of meaning. (shrink)

If we make the basic assumption that the components of a proposition have reference on the model of proper name and bearer, we face the problem of distinguishing the proposition from a mere list' of names. We neutralize the problem posed by that assumption of we first of all follow Wiggins and distinguish, in every predicate, a strictly predicative element (the copula), and a strictly non-predicative conceptual component (available to be quantified over). If we further allow the copula itself to (...) conform to the basic assumption, a regress ( Bradley's regress') arises: the referent of the copula will be instantiation, the instantiation of instantiation etc. To avert the regress, Wiggins simply legislates that the basic assumption is to fail for the copula. But we are entitled to regard the regress as constitution not a difficulty, but the solution: the infinitism it imports (capturable in a finitistic theory of meaning) is just what the unity of the proposition "is". (edited). (shrink)

In recent work, Scott Soames (2010, 2013, 2015, 2019) and Peter Hanks (2011, 2013, 2015) have developed a theory of propositions on which these are constituted by complexes of intellectual acts. In this article, I adapt this type of theory to provide an account of perceptual content. After introducing terminology in section 1, I detail the approach proffered by Soames and Hanks in section 2, focusing on Hanks’s version. In section 3, I introduce a problem that these theories face, (...) namely, how to account for the unity among the relevant intellectual acts. Section 4 provides an answer to this problem of unity, while section 5 explicates the relation to Soames and Hanks. In section 6, I extend the model to a theory of the unity of experiential consciousness. Finally, in section 7, I apply the preceding considerations to debates about the nature of perceptual representation. The upshot will be that experiential unity is not simply a phenomenal feature of consciousness, but central to an account of the role perceptual representation plays in perceptual cognition. (shrink)

_From Naming to Saying _explores the classicquestion of the unity of the proposition, combining an historical approach with contemporary causal theories to offer a unique and novel solution. Presents compelling and sophisticated answers to questions about how language represents the world. Defends a novel approach to the classical question about the unity of the proposition. Examines three key historical theories: Frege’s doctrine of concept and object, Russell’s analysis of the sentence, and Wittgenstein’s picture theory of meaning. Combines an (...) historical approach with discussion and defense of a contemporary causal theory of the unity of the proposition. Establishes a view compatible with, though not dependent on, a causal theory of meaning. (shrink)

This monograph reappraises the role of Bertrand Russell's philosophical works in establishing the analytical tradition in philosophy. It's main aims are to: * improve our understanding of the history of analytical philosophy * engage in the important disputes surrounding the interpretation of Russell's philosophy * make a contribution to central issues in current analytical philosophy. Drawing extensively from Russell's less well known and unpublished works, this book is a welcome addition to the literature and will undoubtedly find a place on (...) the bookshelves of philosophers around the world. (shrink)

I shall examine Quine’s conception of logic, of propositional attitudes, and of the unity of knowledge in order to show that there are some tensions in Quine’s system. I first propose a conception of the use or application of logic, stating that logic strictly speaking applies to intentional phenomena or to things that presuppose the existence of intentional phenomena. Then, I consider briefly Quine’s philosophy of logic and discuss some issues. In Quine’s philosophy, logic stays at the very center (...) of the web of our beliefs; it is central in science and ordinary knowledge as well. Then I examine Quine’s tendency to “quine” the mental, given his own maxim of minimum mutilation. Finally, I consider Quine’s thesis of the unity of knowledge, the thesis that there is continuity from ordinary to scientific knowledge. If I am right about the use of logic and the presence of the propositional attitude idiom in ordinary knowledge and social sciences and humanities, I think there is a problem of consistency in Quine’s system, and that Quine himself pointed to a part of the solution. (shrink)

Between 1903 and 1918 Russell made a number of attempts to understand the unity of the proposition, but his attempts all foundered on his failure clearly to distinguish between different senses in which the relation R might be said to relate a and b in the proposition aRb: he failed to distinguish between the relation as truth-maker and the relation as unifier, and consequently committed himself again and again to the unacceptable consequence that only true propositions are genuinely (...) unified. There is an anticipation of this confusion in the writings of the fourteenth-century philosopher Richard Brinkley. (shrink)

It is fairly well known that Wittgenstein's criticisms of Russell's multiple‐relation theory of judgement had a devastating effect on the latter's philosophical enterprise. The exact nature of those criticisms however, and the explanation for the severity of their consequences, has been a source of confusion and disagreement amongst both Russell and Wittgenstein scholars. In this paper, I offer an interpretation of those criticisms which shows them to be consonant with Wittgenstein's general critique of Russell's conception of logic and which serves (...) to elucidate some of the notoriously enigmatic passages of the Tractatus. In particular, I seek to show the continuity of Wittgenstein's criticisms of the theory of judgement with his remarks on Russell's paradox and the theory of types. In addition, I place these issues in the context of Russell's own philosophical ambitions in order to reveal the deep divisions between the two over the nature of logical form and the analysis of propositional content. (shrink)

At least since Russell’s influential discussion in The Principles of Mathematics, many philosophers have held there is a problem that they call the problem of the unity of the proposition. In a recent paper, I argued that there is no single problem that alone deserves the epithet the problem of the unity of the proposition. I there distinguished three problems or questions, each of which had some right to be called a problem regarding the unity of the (...) proposition; and I showed how the account of propositions formulated in my book The Nature and Structure of Content [2007 Oxford University Press] solves each of these problems. In the present paper, I take up two of these problems/questions yet again. For I want to consider other accounts of propositions and compare their solutions to these problems, or lack thereof, to mine. I argue that my account provides the best solutions to the unity problems. (shrink)

The purpose of this paper is to offer an interpretation of the Tractatus’ proof of the unity of logic and language. The kernel of the proof is the thesis that the sole logical constant is the general propositional form. I argue that the Grundgedanke, the existence of the sole fundamental operation N and the analyticity thesis, together with the fact that the operation NN can always be seen as having no specific formal difference between its result and its base, (...) imply that NN is intrinsic to every elementary proposition. I also argue that the picture theory of proposition is an account of the generation of propositions via naming, and that its crucial idea is that naming is the instantiation of the form of a name, which consists in arbitrarily picking out an object as the meaning of the name from those objects sorted out by the form of the name. It follows that the existential quantifier, that is, NN, is intrinsic to naming (and therefore to every elementary proposition). It is then proven that the sole logical constant is the general propositional form. This, together with the truth‐functionality of logical necessity, implies that logic and language are unified via a general rule – logical syntax. (shrink)

Please send me an email ([email protected]) if you wish to receive a copy of the book. — 'In this highly ambitious, wide ranging, immensely impressive and ground-breaking work Fabian Dorsch surveys just about every account of the imagination that has ever been proposed. He identifies five central types of imagining that any unifying theory must accommodate and sets himself the task of determining whether any theory of what imagining consists in covers these five paradigms. Focussing on what he takes to (...) be the three main theories, and giving them each equal consideration, he faults the first two and embraces the third, which argues that imagining is a special form of mental agency. The scholarship is immaculate, the writing crystal clear and the argumentation always powerful.' (Malcolm Budd). (shrink)

The paper presents an interpretation of the thinking behind the early Wittgenstein's “general form of the proposition.” It argues that a central role is played by the assumption that all domains of discourse are governed by the same laws of logic. The interpretation is presented partly through a comparison with ideas presented recently by Michael Potter and Peter Sullivan; the paper argues that the above assumption explains more of the key characteristics of the “general form of the proposition” than Potter (...) and Sullivan suppose, including, in particular, its claim that the bases from which all other propositions are derived must be elementary propositions. (shrink)

What is the relationship between unconscious and conscious intentionality? Contemporary philosophy of mind treats the contents of conscious 10 intentional mental states as the same kind of thing as the contents of un- conscious mental states. According to the standard view that beliefs and desires are propositional attitudes, for example, the contents of these states are propositions, whether or not the states are conscious or unconscious. I dispute this way of thinking of conscious and unconscious content, and propose an (...) alternative, which helps to explain why the various mental things that are called unconscious deserve that label. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Notes and Discussions THE UNITY OF WISDOM AND TEMPERANCE The attempt of Socrates to establish the unity of the virtues has long been an object of philosophic suspicion. Particular attention has been directed to the argument at Protagoras 332a-333b, in which Socrates seeks to demonstrate the unity of wisdom and temperance, by showing that they must be identified as the contrary of folly. The argument proceeds (...) on the assumption that wisdom and temperance are distinct, and so terminates in a contradiction between 'Whatever admits of a contrary admits of one only' and 'Folly, which is one thing, has two contraries, wisdom and temperance.' Scholars have generally rejected Socrates' proof of the second of the contradictory propositions. However, in a recent paper, Professor David Savan has claimed that the contradiction is derived in a formally valid way from premisses which either need no argument or are accepted by Protagoras} My intent in this paper is to cast doubt on this claim, and so to restore the status quo. According to the critics, the weak point in the argument is Socrates' defence of the statement 'Foolish acts and temperate acts are contraries.' But Saran holds that this statement follows from three conditionals, all stated by Socrates and accepted by Protagoras. These conditionals, referred to by Savan as F G and H, will here be termed P1-3. P1 If an act is right and advantageous, then it is temperate. Symbolically: (r.a) D t P2 If an act is wrong, then it is foolish. w~f P3 If an act is foolish, then it is not temperate. fD -t From these, Saran argues, "it follows that 'An act is right' and 'An act is temperate ' are truth functionallyequivalent, as are 'An act is wrong' and 'An act is foolish'.... Since right and wrong are either contraries or contradictories, temperate and foolish acts must also be either contraries or contradictories" (p. 24). Savan then argues that Protagoras, on the basis of his remarks earlier in the dialogue, must take right and wrong to be contradictories, so that temperate and foolish actions are also contradictories, rather than contraries. I want first to consider whether the material equivalences alleged to follow from P1-3 do in fact follow. They may be symbolized: Clr~t C2w =f It is immediately clear that neither C1 nor C2 follows from P1-3. But this is not surprising, because some expression of the relation between 'right' and 'wrong' 1D. Saran, "Socrates'Logic and the Unity of Wisdom and Temperance," in R. J. Butler (ed.), AnalyticalPhilosophy,2nd series(Oxford: BasilBlackwell,1965),pp. 20-26. [157] 158 HISTORY OF PHILOSOPHY must appear among the premises. Since Savan raises the question whether 'right' and 'wrong' are contraries or contradictories only after he claims to establish C1 and C2, we may suppose that the argument should hold whichever relation we assume. If 'right' and 'wrong' are contraries we add the premise: P4A r D --w And ifthey are contradictorieswe add the stronger premise: P4B r -- --w But neither C1 nor C2 follows from PI-3 together with either P4A or P4B. However, it may be urged, the argument fails only because 'advantageous' has been treated as an independent term. P1 should be replaced by: PI* r ~ t We now find that PI*-3 and P4B are sufficientto derive C1 and C2. But PI*-3 and P4A do not suffice.The most we can establish, using only P4A, is the disjunction : C1 vC2r -- t-v-w -- f And this is useless for the purposes of Socrates' argument. One further way of strengthening the premises might be suggested. Instead of dropping 'advantageous' from the argument, one might take it as related both to 'right' and to 'foolish.' That is, one might introduce additional premises from the pairs: P5A r ~ a P5B r - a and: P6A a D -f P6B a ---- -f P5A, added to P1-3 and P4B, will suffice for the derivation of C1 and C2. But no combination of premises not including P4B will do; from P1-3, P4A, P5B and P6B, one can derive C1 but not C2. It is, therefore, necessary to modify Savan's claim in two fairly important... (shrink)

ABSTRACTWhat explains the difference between a proposition and a mere list of the words it contains, presented in the same order? What unites the parts of a proposition to form a whole? José Zalaba...

What binds the constituents of a state of affairs together and provides unity to the fact they constitute? I argue that the fact that they are related is basic and fundamental. This is the thesis of Factualism: the world is a world of facts. I draw three corollaries: first, that the Identity of truth is mistaken, in conflating what represents (the proposition) with what is represented (the fact). Secondly, a popular interpretation of Wittgenstein's Tractatus, due to Steinus, whereby false (...) propositions are taken to picture non-existent state of affairs, cannot be right. For Wittgenstein, propositions had two poles, and a proposition and its negation picture the same fact. Finally, the metaphysics of modal realism must be wrong, for there are no non-actual states of affairs to constitute any world other than the actual world. (Published Online October 13 2005). (shrink)

Several philosophers advance substantive theories of propositions, to deal with several issues they raise in connection with a concern with a long pedigree in philosophy, the problem of the unity of propositions. The qualification ‘substantive’ is meant to contrast with ‘minimal’ or ‘deflationary’ – roughly, views that reject that propositions have a hidden nature, worth investigating. Substantive views appear to create spurious problems by characterizing propositions in ways that make them unfit to perform their theoretical (...) jobs. I will present in this light some critical points against Hanks’ (2015, 2019) act-theoretic view, and Recanati’s (2019) recent elaboration of Hanks’ notion of cancellation. Both Hanks and Recanati, I’ll argue, rely on problematic conceptions of fiction and pretense. (shrink)

The paper presents an interpretation of the thinking behind the early Wittgenstein's "general form of the proposition." It argues that a central role is played by the assumption that all domains of discourse are governed by the same laws of logic. The interpretation is presented partly through a comparison with ideas presented recently by Michael Potter and Peter Sullivan; the paper argues that the above assumption explains more of the key characteristics of the "general form of the proposition" than Potter (...) and Sullivan suppose, including, in particular, its claim that the bases from which all other propositions are derived must be elementary propositions. (shrink)

There is one truth, but many truths: i.e., one unambiguous, non-relative truth-concept, but many and various propositions that are true. One truth-concept: to say that a proposition is true is to say (not that anyone, or everyone, believes it, but) that things are as it says; but many truths: particular empirical claims, scientific theories, historical propositions, mathematical theorems, logical principles, textual interpretations, statements about what a person wants or believes or intends, about grammatical and legal rules, etc., etc. (...) But, as Frank Ramsey once said, “There is no platitude so obvious that eminent philosophers have not denied it”; and as soon as you ask why anyone would deny that there is one truth-concept, or that there are many true propositions, it becomes apparent that my initial, simple formula disguises many complexities. (shrink)