properties of change to demonstrate that change is impos- sible. (a) The ‘race-course’ (also known as the ‘stadium’ or ‘dichotomy’). A runner has to run a given length. Before running the whole length, he must have run half of it. Then, before running the second half, he must have run half of that half. And so on. Since the division again never terminates, the whole stretch is composed of infinitely many successive pieces, each of some length. But the run- ner cannot finish the task of traversing infinitely many substretches in succession. (b) The ‘Achilles’. A slow run- ner is given a start by a fast runner. The fast one can never catch up: again he has to traverse infinitely many succes- sive stretches, first to the slower runner’s starting-point, then to the point the slow runner has reached by then, and so on. (c) The ‘arrow’. In any indivisible instant of its flight, is a flying arrow moving or at rest? If the former, how can it move in an instant? If the latter, it is never moving, and therefore is at rest. (d) The ‘moving rows’ (also known as the ‘stadium’) A paradox involving relative motion; the details are unclear. (3). Other arguments recorded are: (a) one about ‘place’, again constructing an infinite progression (if everything that is is in a place, and place is, then a place is in a place, and so ad infinitum); (b) possibly the first sorites argument (about the smallest *heap of grain to make an audible noise when dropped; details unreliable). e.l.h. G. E. L. Owen, ‘Zeno and the Mathematicians’, Proceedings of the Aristotelian Society (1957–8). W. C. Salmon (ed.), Zeno’s Paradoxes (Indianapolis, 1970). G. Vlastos, ‘Zeno of Elea’, in P. Edwards (ed.), The Encyclopedia of Philosophy (New York, 1967). zombies. The zombies of Haitian voodoo lore and horror movies are the ‘living dead’, but the philosopher’s zom- bies are merely the stuff of *thought experiments. A zom- bie, if there could be such a thing, would be a living creature that was indistinguishable in its physical constitu- tion and in terms of its outward appearance and behaviour from a normal human being, but in whom the light of *consciousness was completely absent: a being with no inner, conscious mental life, no first-person point of view, no *qualia—in short, a being such that there would be nothing it was like to be that being. But could there be such a being? It is not enough to point out that we can imagine there being zombies, because not everything that we can imagine is really possible (*time-travel may be an example). It may be difficult to determine whether zombies really are possible, but the issue undoubtedly has far-reaching implications for the metaphysics of mind. e.j.l. D. J. Chalmers, The Conscious Mind (Oxford, 1996). Zoroastrianism. An ancient Persian religion, most likely to be known to philosophers either in connection with Nietzsche’s naming the central character of Thus Spoke Zarathustra after its founder or because Pierre Bayle in his Dictionary (1697) presented it as a key to the problem of evil. Zarathustra is now thought to have flourished in the middle of the second millennium bc. In an audacious transvaluation, he proclaimed the gods (daevas) wor- shipped by the very ancient Persians to be evil. The leader of the daevas is the eternal opponent of the one good God, Ahura Mazda. The feature of Zoroastrianism that attracted Bayle is that the forces of good and evil are about equally matched. At the end of time Ahura Mazda will score a final victory, but until then he often fails to control events. Consequently Zoroastrianism, unlike Christian- ity, Judaism, and Islam, escapes the paradox of an all- powerful God who is responsible for what many people take to be unnecessary *evil. j.j.k. Mary Boyce, Zoroastrians: Their Religious Beliefs and Practices (London, 1979). 970 Zeno of Elea APPENDIX Logical Symbols A book like this cannot define the logical symbols precisely, both because they may have somewhat different definitions in different logical systems, and because the methods of definition used by logicians cannot be explained in a few words. The following list merely offers rough equivalents in English for symbols and letters that are used in the Companion, with a few comments. For a more generous list of some of the alternatives see the entry ‘notations, logical’. And do note that any explanation given within a particular article overrides what is said here. ~ or ¬ or – or N not · or & or and ∨ or ⊃ or → if (i.e. ‘P ⊃ Q’ and ‘P → Q’ mean ‘If P, Q’; signs for material implication) — 3 if (similarly; a sign for strict implication) ≡ or ↔ if and only if (material equivalence) = is the same as, or if and only if (strict equivalence) ∀ all (‘∀x’ etc. are sometimes written ‘(x)’ etc.) ∃ some, at least one, there exists, i.e. not ∀ not ٗ or L necessarily ◊ or M possibly, i.e. not ٗ not ∈ is a member of (a set or class) ∉ negates ∈ ≠ negates = ∩ indicates intersection ∪ indicates union Letters are very variously employed, and the following is no more than a guide to usage in the Companion.What are here called schematic letters (*schema) are sometimes brought under the general label *‘variables’. P, Q or p, q, etc. schematic letters for replacement by indicative sentences, or by names of such sentences F, G, etc. schematic letters for replacement by predicates (e.g. ‘is a swan’, ‘laughs’, ‘is to the left of’), or by terms (e.g. ‘swans’, ‘black things’) ∨ R schematic letter for replacement by two-place predicates (e.g. ‘is to the left of’), or by indicative sentences S, P schematic letters for replacement by terms a, b, etc. or X, Y schematic letters for replacement by singular names or refer- ring expressions (in predicate logic these are written after predicate letters, e.g. ‘Fa’, ‘Rba’), or by terms x, y, z, etc. individual variables α, Γ, ∆, etc. variables used informally, i.e. not as parts of a logical lan- guage, for talking about—usually generalizing over— expressions of a logical language, or sets of them n, m the same for talking about numbers t, t 1 , etc. the same for time instants w, w 1 , etc. the same for possible worlds E, F the same for events φ, ψ like α, Γ, ∆ etc. or general like x, y, z,etc. A, B, etc. very general—used either like S, P, etc. or like α, Γ, etc. or like a, b, etc. or even like P, Q, etc. Other letters are explained in their places in the book, or are self-explanatory. c.a.k. 972 Logical Symbols APPENDIX Maps of Philosophy Mapping philosophy is as difficult as mapping the world. Asia and Alaska are likely to be a whole map-width apart, despite the mere 56 miles that really sep- arates them. On one projection Africa will look like a squashed-up kidney bean, on another like a woebegone banana. On one the world itself will appear as an ellipse, on another as two circles. The world has two hemispheres (east and west) but also two other hemispheres (north and south). Equal-area and equal- population maps (where equal areas on the page represent equal areas on the ground, or equal populations, respectively) may be almost unrecognizable as Aesthetics Philosophy of education Social philosophy Moral philosophy Epistemology Logic and philosophical logic Political philosophy Philosophy of language Philosophy of history Philosophy of science Metaphysics Philosophy of mind Philosophy of law Philosophy of religion Philosophy of mathematics Inner and outer circles of philosophy referring to the same planet. Yet maps are made and are useful. Most normal maps (though not all possible ones) will show London as between Cambridge and Brighton. The situation is the same in philosophy. There is no one way of mapping it. Different, perhaps overlapping, perhaps inconsistent, maps may be used for different purposes, and will all be horribly misleading unless used merely as over-simplified rough guides. It is essential that the reader remember these points when looking at the following pages. Inner and Outer Circles of Philosophy Philosophy can be thought of as concerning what in the most general sense there is, what we can know and how, and the most general conditions that must be satisfied by any coherent thought. This gives us the three items in the central circle. The items in the outer circles are less general and concern limited areas. They also tend to depend on the central items in ways that those do not depend on them in return. For instance, moral philosophy often depends on theories of implication, which belong in logic and philosophical logic, but logic and philo- sophical logic do not themselves depend for their tools on moral philosophy. The relation between the two outer circles is somewhat similar, though less markedly so. Political philosophy, for instance, seems to presuppose moral philosophy without being presupposed by it. No doubt for these reasons philosophers have given more attention to the more central items, so that the diagram also to some extent maps popularity. However, both the circles them- selves and the regions within them should be thought of as only rather vaguely delimited. There are multiple overlaps, and in particular no attempt has been made to order the items within each ring, which are arranged alphabetically, reading clockwise from the top; no significance attaches to co-radiality. Groups of Parts of Philosophy Any grouping is bound to be somewhat arbitrary and roughshod, but the reader may find it helpful for certain similarities to be pointed out, bearing in mind 974 Maps of Philosophy I Epistemology Philosophy of science III Aesthetics Moral philosophy Political philosophy V Philosophy of education Philosophy of history Philosophy of law Social philosophy II Metaphysics Philosophy of mind Philosophy of religion IV Logic Philosophical logic Philosophy of language Philosophy of mathematics always that the grouping presented here, though it has the rationale explained below, is certainly not unique. Group I has in common a concern with the conditions under which we can know something, the justifications that we can offer for claims to know it, and the methods that may help us to come to know it. Group II asks primarily about what there is, either completely generally or in certain obviously important spheres such as that of beings as developed as our- selves or that of the ultimate power, if any, behind the universe. It then asks about the nature of these various things. Group III combines various questions concerned in one way or another with value: what sorts of value there are, what things are valuable in these various ways, and what connection there is between value and a duty to produce it, as well as the question what alternatives, if any, to value can be offered as a foun- dation for our duties. Group IV mainly concerns abstract structures, and in particular the structure of any coherent thinking and the tools that are essential for such thinking— since presumably we could not think in any effective way without language. Group V, finally, is a bit of a ragbag since it consists of philosophical problems directed at various particular subject areas. The list could be extended almost indefinitely, since there are usually at least some philosophical problems attached specifically to each of the special sciences or other major areas of human activity. Those mentioned have achieved a certain entrenchment, pre- sumably because, although each of them has connections with various items in the other four groups, they are thought to raise more problems of their own than the philosophies of, say, physics or economics. One special case deserving mention is the subject often called philosophy of action: its subject is quite gen- eral and not a particular sphere of human activity; but in fact it is usually regarded as a branch of philosophy of mind. Parts of Philosophy and Philosophical Positions and Doctrines In what follows, those diagrams headed ‘Epistemology’, ‘Metaphysics’, ‘Logic and philosophical logic’, ‘Philosophy of mind’, ‘Moral philosophy’, ‘Political philosophy’, ‘Philosophy of language’, and ‘Philosophy of science’ represent the parts of philosophy, or questions that can be asked, while the others repre- sent philosophical positions and doctrines, or answers that might be given. Solid lines represent relations in a tree diagram. Dotted lines represent connec- tions, as when in the former group the ‘main related subjects’ are listed, or else emphasize that the items they connect share a greater than usual degree of overlap, or merge into each other and cannot be sharply distinguished. For instance, in ‘Theories on mind and body’ property dualism, though presumably to be classified under dualism, is closely bound up with certain monist views. The lists of ‘main related subjects’, and items linked only by dotted lines to the main subject, are not always limited to subjects within philosophy itself. The few attributions of views to named figures should be regarded as approximate, sometimes controversial, and of course not exhaustive. There are more philosophers who might be mentioned in connection with each view. a.r.l. Maps of Philosophy 975 This page intentionally left blank Maps of Philosophy 977 Epistemology Main related subjects 1 See scepticism; fallibilism. 2 See scepticism; common sense; arguments from illusion. 3 See reliabilism. 4 See epistemological justification. 5 See epistemological justification. 6 See holism; epistemological relativism. 7 See the given. 8 See scepticism; fallibilism; epistemological relativism; epistemological justification; foundationalism. Relations of knowledge to other notions Knowledge Knowledge and certainty 1 and belief Knowledge Knowledge and doubt 2 and causation 3 Knowledge and Knowledge Knowledge and justification 4 and evidence 5 revisability 6 Types of knowledge Of objects Of concepts Propositional A priori Empirical A priori Empirical Innate Intuition Direct Indirect Philosophy of language Perception Memory Testimony Inference Abduction Philosophy of mind Philosophy of psychology Evolutionary epistemology Philosophy of biology Anthropology Naturalized epistemology Philosophy of science Sociology of knowledge Moral epistemology Moral philosophy Objects of knowledge External The The Values Abstractions Minds Philosophy world past future of mind Our own Our own Other minds experiences 7 inner states Sources of knowledge Perception Memory Reason Introspection Other alleged sources Deduction Induction Other Intuition Telepathy Clairvoyance Precog- nition Possibility of knowledge 8 978 Maps of Philosophy Empiricism As psychological doctrine: about origin of concepts or knowledge (locke, hume) As epistemological doctrine Of concepts Of propositions Extreme or Moderate or Extreme: Less extreme: Weak: strong: all weak: some all knowledge all knowledge some knowledge concepts are concepts are is of depends on is empirical empirical empirical sense-data sense-data (locke) (locke) As semantic doctrine: logical empiricism/logical positivism Re meanings Re meanings of words: of sentences: operationalism verificationism Anti-realism Conclusive Weak: experience must be relevant (ayer) In practice In theory (schlick) Constructive empiricism (van fraassen) Philosophy of science Maps of Philosophy 979 Rationalism As psychological doctrine (about the origin of our ideas and beliefs) Innate Instinctive ideas ideas Cartesian Linguistic (descartes (chomsky) held both the versions below) Strict: Loose: all ideas some ideas innate innate (hume’s ‘impressions’) As epistemological doctrine (about the justification of our beliefs) Innate Intuitions Inescapable knowledge conceptual apparatus (kant; cf. chomskyan linguistic rationalism) Extreme: Moderate: senses senses totally partly rejected accepted (eleatics)(descartes, leibniz) In moral philosophy Reasoning Intuition as basis as basis of morality of morality (kant)(kant) Intuition Intuition of general of particular principles moral facts In religion Rejection of Rejection of revelation religion as such (the usual meaning of ‘rationalism’ here) . and philosophical logic Political philosophy Philosophy of language Philosophy of history Philosophy of science Metaphysics Philosophy of mind Philosophy of law Philosophy of religion Philosophy of mathematics Inner. again he has to traverse infinitely many succes- sive stretches, first to the slower runner’s starting-point, then to the point the slow runner has reached by then, and so on. (c) The ‘arrow’ both the circles them- selves and the regions within them should be thought of as only rather vaguely delimited. There are multiple overlaps, and in particular no attempt has been made to order the