Modality

Joseph Melia

• Publisher: Routledge
• Year: 2014
• Print ISBN: 9781902683478
• eBook ISBN: 9781317489672
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

This introduction to modality places the emphasis on the metaphysics of modality rather than on the formal semetics of quantified modal logic. The text begins by introducing students to the "de re/de dicto" distinction, conventionalist and conceptualist theories of modality and some of the key problems in modality, particularly Quine's criticisms. It then moves on to explain how possible worlds provide a solution to many of the problems in modality and how possible worlds themselves have been used to analyse notions outside modality such as properties and propositions. Possible worlds introduce problems of their own and the book argues that to make progress with these problems a theory of possible worlds is required. The pros and cons of various theories of possible worlds are then examined in turn, including those of Lewis, Kripke, Adams, Stalnaker and Plantinga.

1 Introduction to modality

Modality

Suppose we possessed an extraordinarily comprehensive and accurate theory of the world. Suppose that the language of this theory contained a name for every object; every single thing, from the black holes hidden in the heart of the furthest galaxies, to the fine cobwebs swaying in the corner of an attic, is mentioned by this theory. Suppose also that this theory contained a predicate for every categorical property, simple or complex, that is actually instantiated. The theory says what things are like to the highest level of detail. It tells us whether something has a mass of 1.153 kg, whether it has a charge of 4.238322 coulombs, and whether it has a length of V2 metres. Suppose, finally, that everything the theory says is true. It truly reports the colours, tones and hues of each and every pixel currently appearing on my computer screen. It truly reports the shapes, sizes, masses and charges of each and every fundamental particle in my finger.

Everything in the theory is true. But does every truth appear within the theory? Would the theory account for every single matter of fact? If such a theory ever came to be written down, could thinkers and scientists finally rest, their work finished? Let us call the view that such a theory would be complete, that every truth would appear within the theory, the categorical hypothesis. Questioning the categorical hypothesis may seem absurd. By hypothesis, the theory lists all the things that exist and truly tells us what those things are like and what relations those things bear to each other. What more could one say? What else could there be to add? And yet, a number of philosophers believe that such a theory would not be a theory of everything: that there is a class of truths on which this theory is simply silent - indeed, a class of truths that this theory lacks the resources to describe.

Consider the following two sentences: "Joe is tall" and "Joe is human." Both these sentences are of a simple subject-predicate form and each ascribes a categorical property to Joe. If true, they would both appear in our theory. But, on reflection, we might note that there is an important difference in the manner or mode in which Joe possesses the two properties, a distinction ignored by our supposed theory of everything. For although Joe is tall, Joe is tall only contingently or accidentally. He could have stopped growing when he was 12. He could have lost both his legs in a terrible car accident. He could have had an unfortunately close encounter with a scythe. In brief, his being tall is an accidental property of his. By contrast, given that he is human, we might think that Joe is essentially human, that humanity is a property Joe has to have. Whereas Joe could have failed to possess the property of being tall and still be the same entity - Joe - he could not have failed to possess the property of being human and still be Joe. Such distinctions between essential and accidental properties of an object are examples of de re modality: in these cases, it is some particular thing that has a property essentially or accidentally.

As well as there being different modes in which an object may possess a property, there are also different modes in which a proposition may be true, a distinction that is again not acknowledged by the original theory. The sentences "All bachelors are unmarried" and "All emeralds are green" are both true. Both tell us something about what things there are and what categorical properties those things have. But there is a difference in kind between these two truths. On the one hand, it is necessary that all bachelors be unmarried. It is strictly impossible for there to be a married bachelor, for the trivial reason that it is part of the meaning of "bachelor" that anything that is a bachelor be unmarried. By contrast, "All emeralds are green" is merely contingently true. It is possible that there be an emerald that was red, purple or some other colour. This distinction between contingent and necessary truths is another example of a modal distinction. Here, where the modality attaches to the proposition, the modality is said to be de dicto: it is the whole truth that all bachelors are unmarried that is said to be necessary. That 2 + 2 = 4, that there are no true contradictions, that nothing can be simultaneously red and green all over are all examples of truths that are necessary: they had to be true. That there are six people in this room, that some bachelors have red hair, that London is the capital of England are all truths that are contingent: they could have been otherwise.

The de dicto and the de re distinctions are related. If Joe has the property of being human essentially then it follows that the truth "Joe is human" holds of necessity. If Joe has the property of being tall contingently then the truth "Joe is tall" holds only contingently. In general, accepting that there are some properties that are held essentially and others that are held contingently entails accepting that some truths are necessary and others contingent. The converse, however, does not hold. One can believe that there are necessary truths without believing that anything has any of its properties essentially. "All bachelors are unmarried" is a necessary truth, but this does not commit us to the existence of any object that is essentially unmarried. After all, no individual person is essentially unmarried. Although everything that is a bachelor must therefore be unmarried, nobody has to be bachelor and so nobody has to have the property of being unmarried.

The kind of modal truths that will be the focus of this book are those that go beyond the merely actual and tell us something about how things might be, or must be, or would be had things been other than they actually are. On reflection, we see that our initial theory was silent about such modal truths. It only told us how things are and what categorical properties these things actually have. Those philosophers who accept modal truths believe that the initial theory, fantastically detailed as it was, falls a long way short of the desired complete and final theory.

The modal distinctions drawn above mark distinctions about the world rather than distinctions about what we know. That there could be no true contradictions is as independent of our thoughts, beliefs and desires as is the truth that the universe is expanding. Granted, there is a use of "possibly" that is epistemic rather than metaphysical; in certain contexts, when I say that tachyons are possible I mean that the existence of these things is compatible with what I know. But the distinctions between accident and essence that were drawn above are not be understood in this way. I know very well that Joe is both bald and that he is human, yet I may also believe that he could have had hair and that he must be human. And 2 + 2 = 4 would necessarily hold whatever the state of play of my knowledge. Nor will it do to try to define possibility and necessity in terms of the a priori. For a start, the very definition of the a priori itself seems to require the modal: a sentence is a priori if it could be known without recourse to experience. But worse, it is simply an open question whether all necessary truths are knowable a priori. Goldbach's conjecture, Riemann's hypothesis and Cantor's continuum hypothesis, all presently unproved mathematical hypotheses, are all necessary truths if true at all. But whether they can be known a priori is, at the very least, an open question. It may yet be true that all necessary truths are a priori, but since this is certainly no analytic truth the concept of necessity and the a priori must be kept distinct. To accept or reject the categorical hypothesis is to take a stance on the nature of reality, not on our relation to the world.

Modality in practice

Many philosophers find the categorical hypothesis attractive. The analytic philosopher's all time favourite formal system, first-order predicate calculus, is a fine and well understood language, suited for expressing facts about what things there are and what categorical properties these things possess. It has served us well in formalizing the truths of mathematics and logic as well as the truths of science. Unfortunately, the modal truths discussed above defy the categorical hypothesis and resist formulation in first-order predicate calculus. Now, questions of essence and accident, of possibility and necessity, may seem at first to be recherche: too divorced from our everyday thought and talk to be worth much concern. Nor, one might suspect, do modal notions play a significant enough role in our philosophical theorizing to be worthy of serious attention. Perhaps we do best to preserve the categorical hypothesis, and to preserve our tried and trusted familiar formal systems by abandoning the modal altogether.¹ After all, if modal thought and talk is recherche and philosophically unimportant, then little would be lost and much would be gained by such a move.

Unfortunately, abandoning the modal is not as easy as it might at first seem. Modality is ubiquitous in both our everyday thought and talk and in our scientific and philosophical theorizing. In abandoning the modal we abandon many things that we naturally accept and think of as being trivially true or uncontentious. A philosopher who decides to abandon all talk of the temporal thereby avoids the many problems that arise in the philosophy of time, but also thereby abandons the ability to speak truly of the many uncontentious facts about time, such as that moving clocks run slow, or that John was born before Joe. Similarly, there does seem to be an abundance of uncontentious facts about the modal. Some examples are: there are many different ways the world could have been; I am unable to speak French; Joe could win his chess game in three different ways; you cannot break the laws of physics; I could have had a lot more hair than I actually do. In all these cases, we are saying things that go beyond the strictly actual and categorical. Philosophers who dare to find fault with such natural and apparently uncontentious truths had better have good reason for doing so.

Moreover, our modal thought and talk encompass far more than essence and accident - far more than possible and necessary truths. For instance, when we say that a glass is fragile, we are not saying that the glass has a certain categorical property; we are saying something about how the glass would behave in other situations, situations that may or may not actually obtain. We are saying something about how the glass would behave were it dropped, hit or treated roughly. And we take ourselves to be saying something true or false about the glass irrespective of whether or not the glass actually is dropped, hit or treated roughly. When someone says that cheap plates have a tendency to chip, again what is said seems to go beyond the strictly actual; the pronouncement is not refuted simply by showing that the object in question, as a matter of brute fact, is not chipped, for the statement says something about how cheap plates are likely to be in certain other possible situations.

Similarly, our practical reasoning involves counterfactuals: truths such as "If I had dropped the computer I would have lost a year's work" and "If Germany had invaded Britain then Germany would have won the war." Such statements are of the form "if ... then ...". But, as every philosophy undergraduate knows, these obvious and natural truths cannot be formalized by the logician's truth-functional → for A —> B is true if A is false - so any contrary-to-fact counterfactual comes out automatically and trivially true if translated in this way That can't be right! These counterfactuals aren't true simply because I didn't drop the computer or because Germany didn't invade Britain. As before, counterfactuals seem to point beyond the merely actual: they tell us something about how the world would behave were it different in certain ways. They are truths that go beyond what is actually the case and the actual categorical properties and relations of the actually existing objects.

Thought and talk about the modal is widespread and pervasive. Philosophers who abandon such talk and thought find themselves at odds with common sense. Of course, common sense is not the final arbiter of truth, but a departure from common sense is nevertheless a price to pay for one's philosophy, and the greater the departure the greater the price. Of course, if it turns out that modality is incoherent or problematic, then we will have strong reasons for revising our common-sense beliefs.² But it would be bad methodology to begin our philosophical theorizing about a discipline by departing so radically from our everyday thought and talk.

Modality in theorizing

As well as playing a major role in our everyday thought and talk, the modal also plays a major role in our scientific and philosophical theorizing. It is part of scientific practice to ascribe dispositional properties to various objects. Scientists have discovered that salt is soluble, that hydrogen is flammable and that uranium has a tendency to decay. Such truths are modal: they do not tell us just how a thing actually is, but they tell us something about the object's tendencies or capacities. Moreover, there are many who think that science uncovers a form of natural necessity. The laws of physics are universal truths, but not just universal truths. It may be a universal truth that all lumps of gold are less than a mile long but this doesn't make it a law of physics. If we wished, we could construct a lump of gold that was over a mile long, although we may never actually get around to doing it. By contrast, we cannot break the laws of physics. We can respect the intuitive idea that there is a distinction of kind between a genuine law and a mere accidental generalization easily enough by invoking modal notions, perhaps by explaining how the laws support counterfactuals while the accidental generalizations do not, or by invoking a primitive notion of natural necessity that the laws possess that the accidental generalizations do not. Whichever way we go, those who would eliminate modality have difficult work cut out for them if they wish to make sense of an apparently objective distinction between laws and accidental generalizations.³

As well as in science, modal notions also appear to be fundamental in the study of logic. One of the main concepts (some would say the most important concept) in logic is the notion of a valid argument. What is it for an argument to be valid? Typically, the definition is in modal terms: an argument is valid if it is not possible for the premises to be true and the conclusion false.

Although this is the explanation of the general notion of validity that one finds in undergraduate textbooks, those familiar with advanced logic might resist this definition. Perhaps those taken with set-theoretic semantics would prefer to say that an argument is valid if and only if (iff) there is no model M such that the premises are true-in-M and the conclusion is false-in-M. Such a definition of validity would not use modal notions (although it would require us to believe in the existence of models). But there are problems. First, this definition is restrictive: it is restricted to those languages for which logicians have already developed a model theory. Yet it seems we can talk about validity independently of whether or not a model-theoretic semantics has been developed for the theory. Moreover, if this is all there is to the notion of validity, why should we care whether or not an argument is valid? Why should the fact that there is no set-theoretical structure that bears a particular relation to a set of sentences be anything other than a purely mathematical matter? What is to stop us giving any set-theoretic definition of "true-in-all-models" and calling this validity? Arguably, what makes a particular definition of validity important to the logician is the extent to which it captures the intuitive notion of validity. Thus Mendelson, in his textbook on mathematical logic, points to the connection between models for the predicate calculus and possible worlds.⁴ Once this connection is understood we can see why the model-theoretic definition at least partly captures our intuitive notion. But the intuitive notion is there, and our formal definitions must aim to be true to it. Thus, at least at the outset, the logician must take the modal notion seriously.⁵

Modal notions have come to play a major role in philosophical theorizing. Particularly in the second half of the twentieth century, more and more philosophers have used modal concepts to solve various philosophical problems, and to provide analyses of different philosophical concepts. Of course, in philosophy, no solution and no analysis is completely uncontentious. Nevertheless, one should at least be aware of how much one has to lose by eschewing modal notions altogether. Here are some examples of how the modal can be used to help us solve problems and provide analyses.

1. The axioms of geometry appear to postulate lines that are infinitely long. Many people complain that they can find no sense in the notion of an actual infinity, but what is the alternative? Is it that lines have some arbitrary cut-off point? That if we travel far enough we shall disappear off the edge of the universe? That is even worse! There is a natural and simple solution: replace talk of the actual infinity with talk of the potential infinity. When we say that a line is infinite we don't mean that it actually stretches out for ever. Rather, for any point you might choose to travel to along a line, it is possible to have travelled a little further.
2. In logic, one expresses the limitations of formal systems by quantifying and referring to proofs. If arithmetic is consistent, then "There is no proof of the consistency of arithmetic within arithmetic" is one notorious consequence of Gödel's theorem. But for nominalists who believe only in concrete objects, this deep result is trivial. For if the nominalist is going to believe in proofs at all, they can only be concrete objects, such as marks that have actually been written down with pen on paper. But that then limits the nominalist to believing only in proofs that have actually been and (if our nominalist is a realist about the future) will be written down. That just gets Gödel wrong, for Gödel's theorem is far more interesting than the result that, as a matter of brute fact, nobody has or will prove the consistency of arithmetic within arithmetic. A Platonist has no trouble understanding Gödel because he thinks proofs are abstract entities that exist independently of our beliefs and desires; for the Platonist, all the infinitely many proofs are "out there" in the Platonic realm. But how is the nominalist to understand Gödel's result without trivializing it?
   There is an obvious and natural solution. Gödel's theorem tells us that it is not possible to prove the consistency of arithmetic within arithmetic. It is not possible to write down a series of steps that have only the axioms of arithmetic as their premises and that obey the laws of logic, and that ends with a concrete inscription asserting the consistency of arithmetic. This is surely a much more natural and intuitive way of understanding Gödel's theorem than dabbling in the Platonist's abstract realm of proofs! Yet the solution is ...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Contents
• Acknowledgements
• 1 Introduction to modality
• 2 Modal language and modal logic
• 3 Quinian scepticism
• 4 Modalism
• 5 Extreme realism
• 6 Quiet moderate realism
• 7 Possible worlds as sets of sentences
• Notes
• Further reading
• Index