1. 1_1.

The atomist variation of Cartesianism was introduced by Gerauld de Cordemoi (sometimes Cordemoy). See Leibniz, «A New System of the Nature and the Communication of substances, as well as the Union between the Soul and the Body», 1695 in Philosophical Papers and Letters, Edited and translated by Leroy E. Loemker, 2nd edition, 1969, Dordrecht, Reidel, p.456. (This collection and edition hereafter referred to as Loemk.)




2. 1_2.

C.D. Broad, Leibniz (Cambridge 1975), p. 73.




3. 1_3.

Brown for example accuses Leibniz of fusing two questions, (1) explaining `the nature of organic unities' and (2) explaining how true beings must be both indestructible and indivisible. Brown acknowledges that `Leibniz took the unusual course of attempting to unite in the same theory of substance, change, activity, and final causes' and himself attempts a unified view, but he offers no explanation of why only a self-subsistent substance can act. In fact he does not distinguish between substances which are self-subsistent and those that are not. A house is a substance but not capable of action, and thus not a self-subsistent substance in Leibniz' conception. Stuart Brown, Leibniz (Brighton, 1984) p.138.




4. 1_4.

Leibniz, Correspondence with Arnauld 1686-87, in Loemker, p. 454. Also: «...After trying to explore the principles of mechanics itself in order to account for the laws of nature which we learn from experience, I perceived that the sole consideration of extended mass was not enough but that it was necessary, in addition, to use the concept of force, which is fully intelligible, although it falls within the sphere of metaphysics.» (Leibniz, «A New System of the Nature and the Communication of substances, as well as the Union between the Soul and the Body», 1695, in Loemker, p.454.)




5. 1_5.

-- to which he subscribed in the years before his trip to Paris in 1672. See Daniel Garber, `Motion and metaphysics in the young Leibniz' in Leibniz: Critical and Interpretive Essays, ed. M. Hooker (Manchester, 1982) pp. 160-184.




6. 1_6.

Leibniz, «The Confession of Nature against Atheists» 1669, in Loemker, p. 111.




7. 1_7.

Leibniz, Correspondence with Arnauld 1686-87, in Loemker, p.338.




8. 1_8.

Leibniz, «Theological Writings related to the Catholic Demonstrations», 1668-70, in Loemker, p.112.




9. 1_9.

Leibniz, «Theological Writings related to the Catholic Demonstrations», 1668-70, in Loemker, p.112.




10. 1_10.

Cf the incorporeal glue of which Garber speaks: Daniel Garber, «Leibniz and the Foundations of Physics: the Middle Years» in The Natural Philosophy of Leibniz, ed. K. Okruhlik & J.R. Brown (eds) (Dordrecht, 1986) p.35.




11. 1_11.

Leibniz, «Theological Writings related to the Catholic Demonstrations», 1668-70, in Loemker, p.112.




12. 1_12.

Garber speaks of it as the philosophy of corporeal substance, but this is as contentious a way of characterising Leibniz' thrust during this middle period as what I suggest here. Daniel Garber, «Leibniz and the Foundations of Physics: the Middle Years» in The Natural Philosophy of Leibniz, ed. K. Okruhlik & J.R. Brown (eds) (Dordrecht, 1986), pp.27-130.




13. 1_13.

Leibniz, «Discourse on Metaphysics», 1686, in Loemker, p.308. Also: `A substance cannot come into being except by creation, or perish except by annihilation', Leibniz, «Discourse on Metaphysics», 1686, in Loemker, p.308.




14. 1_14.

Leibniz, New Essays on Human Understanding (1st published 1765), this edition translated and edited by P. Remnant and J. Bennett (Cambridge, 1981) p.328/9. (Hereafter this work and edition is referred to as `New Essays ...')




15. 1_15.

There is a glancing comment in J.E. McGuire, «Labyrinthus Continui», in Motion and Time Space and Matter, ed. by P.K. Machamer & R.G. Turnbull (Ohio, 1976), p.295, footnote 22. Note though that according to McGuire Leibniz identified substantial form with suppositum. In fact the identification in the paper cited is between self-subsistent substance and suppositum.




16. 1_16.

Leibniz, «On Transubstantiation», 1668(?), in Loemker, p. 115.




17. 1_17.

For an authoritative modern account see Kneale, W. & Kneale, M., The Development of Knowledge (Oxford, 1962) pp.246-274.

Loemker (p. 119 footnote 11) refers to E.A. Moody, Truth and Consequence in Medieval Logic, Amsterdam, 1953, North-Holland.




18. 1_18.

Thus Leibniz' idea that the truly fundamental must partake of the nature of an animal -- ie share with an animal the capacity for originating action -- is at the heart of his philosophy. Catherine Wilson's suggestion that he was forced into such a thesis in manoevering with Arnauld would on this account be mistaken. See Catherine Wilson Leibniz's Metaphysics, (Manchester 1989), p.103-4.




19. 1_19.

McGuire explains the dissatisfaction Leibniz felt with both Cartesians and Newtonians in the matter of their explication of the origin of change. J.E. McGuire, «Labyrinthus Continui», in Motion and Time Space and Matter ed. P.K. Machamer & R.G. Turnbull (Ohio, 1976) p.290-1.




20. 1_20.

`The Mechanical Philosophers, of course, had denied that activity could in any sense truly exist in nature. In their explanatory program final causes were restricted to volitional acts.' McGuire, Ibid, p.299.




21. 1_21.

Leibniz, in Loemker, «A New System of the Nature and the Communication of substances, as well as the Union between the Soul and the Body», 1695, in Loemker, p.454.




22. 1_22.

Daniel Garber, «Leibniz and the Foundations of Physics: the Middle Years» in The Natural Philosophy of Leibniz, ed. K. Okruhlik & J.R. Brown (eds) (Dordrecht, 1986) pp. 38ff.




23. 1_23.

`Perfect unity should be reserved for animate bodies, or bodies endowed with primary entelechies ...' Leibniz, New Essays ..., p.328/9.




24. 1_24.

Leibniz, «On the Correction of Metaphysics and the Conception of Substance», 1694, in Loemker, p. 433.




25. 1_25.

Leibniz, «A New System of the Nature and the Communication of substances, as well as the Union between the Soul and the Body», 1695, in Loemker, p. 456.




26. 1_26.

Leibniz, Ibid, p. 456-7.




27. 2_1.

This paper benefited from the comments of Scott Lehmann and John Troyer.




28. 2_2.

Yuval Steinitz, «Necessary Beings», American Philosophical Quarterly vol. 31 (1994): 177-181.




29. 2_3.

Richard Swinburne, The Coherence of Theism (Oxford: Clarendon Press, 1977).




30. 2_4.

I adopt the convention of creating a name for an expression by placing that expression within single quotes, but do so only when the lack of single quotes might result in confusion.




31. 2_5.

A remark made by Dr. Uta Ranke-Heinemann in a PBS special on the 1992 Global Forum of Women, «Not a Bedroom War» (Dec. 1994).




32. 2_6.

Steinitz 177.




33. 2_7.

(A') is valid in an S5 modal semantics. But this suffices to establish the validity of the argument only if S5 semantics is the correct formal account of the logic of logical necessity and possibility in first-order logic. In order to assess the latter, a background theory on the nature of logical necessity and possibility is required. This is the concern of what immediately follows.




34. 2_8.

This view is held by, among others, Alfred Tarski, «On the Concept of Logical Consequence,» Logic, Semantics, and Metamathematics trans. J.H. Woodger (Oxford: Clarendon Press, 1956) 416-420; W.V. Quine, Philosophy of Logic 2nd ed. (Cambridge, Mass: Harvard UP, 1986) 47-56; Richard Jeffrey, Formal Logic: Its Scope and Limits 3rd ed. (New York: McGraw-Hill, 1991) 17-18; and Hartry Field, Realism, Mathematics, and Modality (Oxford: Basil Blackwell, 1989) 82-88.




35. 2_9.

Scott Lehmann, «Slightly Non-standard Logic», Logique Et Analyse 92 (1980): 379-392.




36. 2_10.

The L”wenheim-Skolem theorem tells us that if there exists a meaning assignment which makes a sentence p true at a possible world with a domain of n individuals (for some natural number n), then there exists a meaning assignment which makes p true in a world with no more than a denumerably infinite number of individuals. Accordingly, if there doesn't exist a meaning assignment which makes a sentence p false at a world with a denumerably infinite domain, then p is a logical truth. So, the supposition that the cardinality of the world's contents is denumerably infinite allows us to appeal solely to individuals of this world in representing logically possible situations (i.e., the domains of logically possible worlds will be subsets of the domain of the real world).




37. 2_11.

See John Etchemendy, The Concept of Logical Consequence (Cambridge: Harvard U.P., 1990) Chapters 6 and 7; and John Etchemendy, «Models, Semantics, and Logical Truth» Linguistics and Philosophy 11 (1988): 91-106.




38. 2_12.

My presentation of this totality is motivated by the combinatorialesque approach to possible worlds suggested by, among others, W.V. Quine, «Propositional Objects», Ontological Relativity and Other Essays (New York: Columbia University Press, 1968). For a detailed exposition of the idea of taking possible worlds to be set-theoretic combinatorial rearrangements of the basic atoms of which our world is composed see, M.J. Cresswell, «The World is Everything That is the Case» Australasian Journal of Philosophy 50 (1972): 1-13.




39. 2_13.

Compare this view of logical truth with a structuralist view of mathematical truth: mathematics is about the actual structures of possible worlds. `9>7' may not be true at all possible worlds, because its truth is relative to a conception of the natural numbers. If there are things in this world with the structure of the natural number sequence, then `9>7' is true of this world. In the latter case `9' and `7' would refer to some individuals and `>' to some relation between individuals. But, it may be false that 9 is necessarily greater than 7. Consider the parmenidean world w: `9>7' is false at w, since the natural number sequence is not realized in w (because the successor of a number must be different from it, and here there is nothing but «The One»). So, even though the Peano Axioms (on the standard interpretation) are mathematically necessary in the sense that they are true in all worlds which exemplify the structure of the natural numbers, they are not true at each possible world.




40. 2_14.

See, Etchemendy.




41. 2_15.

Steinitz, note 1 181.




42. 2_16.

So in determining whether or not 1=3 is coherent `1', `3', and `=' must have their ordinary meanings.




43. 2_17.

Swinburne 14.




44. 2_18.

Swinburne 12 and 14.




45. 2_19.

However, there are other senses of coherence which do not make what is coherent invariant from one world to another. For example, what is coherent depends on an ability to conceive which is determined, in great part, by the constitution of the human mind and body, the language of the conceiver, and other facts. Perhaps then in order to ascertain what is coherent in a possible world, we must imagine the powers of conceiving which would exist in that world. It may be that these powers of conceiving will vary from one world to another. For example, I can conceive of a world w in which stellar evolution is different and there are no black holes. But if there had been no black holes, it might have been the case that the inhabitants of such w would not know what is a black hole, and so no one could conceive of one. Defining logical truth in terms of this sense of coherence results in making a world possible relative to another iff the former is conceivable to someone in the latter. Then w would be possible relative to our world but not vice versa. So the validity of (A') requires a stronger sense of coherence: to say that a statement is coherent is to say something about it without reference to the ability of conceiving which may or may not exist in other states of affairs. See G.E. Hughes and M.J. Cresswell, An Introduction to Modal Logic (London: Metheun, 1968) 77-80, where Hughes and Cresswell motivate the assignment of different properties to the relative possibility relation on the basis of different senses of conceivable.




46. 2_20.

See, Paul Tidman, «Conceivability as a Test For Possibility», The American Philosophical Quarterly vol. 31 (1994): 297-309, and Stephen Yablo, «Is Conceivability a Guide to Possibility?», Philosophy and Phenomenological Research vol.53, 1993: 1-42.




47. 2_21.

Tidman 305.




48. 2_22.

Tidman 305.




49. 2_23.

Etchemendy Models, Semantics, and Logical Truth, 95 and 102.




50. 2_24.

The term is Loux's. See Michael Loux, «Modality and Metaphysics» in The Possible and the Actual ed. Loux (Ithaca: Cornell UP, 1979) 15-64, 28.




51. 2_25.

For example, Etchemendy suggests that the modal notion in (ii) may be epistemological (On the Concept of Logical Consequence 88-89): if p is logically true, then there is no way the world might, for all I know, be, which would make p false. In order to secure the invariance of logical necessity and possibility from world to world on this approach, we need to argue that from the point of view of each world, all worlds are (epistemically) possible. The challenge is to spell out the relevant sense of epistemic possibility which secures this invariance. One way is derived from the fact that we do not want logic limited by a possibly radical misconception of the world. Perhaps what is logically true, should remain true on all views about the nature of the world. Very quickly, a possible world represents a view about the facts (both modal and non-modal), and from the perspective of each view, other views are doxastic alternatives. So in ascertaining what is logically possible, we must take into account the epistemological fact that any one theory about the nature of reality could be wrong. For example, Swinburne believes that (a) positrons are electrons travelling backwards in time is logically impossible because (a) is incoherent (41). But it is possible that Swinburne is wrong, and the proponents of the truth of (a) are correct (Swinburne cites the physicist Richard Feynman as one such proponent). So, on this approach to the modal notion in (ii), it is logically possible that (a) be true. The world at which (a) is true represents the case in which Swinburne is wrong. But if the fact that I might be wrong about what I claim to know is relevant to establishing the possible falsehood of a given sentence p, then it seems that very little, if anything, turns out to be logically true.




52. 2_26.

Steinitz 180. If correct, this would be unfortunate given the role that the perception of coherence plays in the determination of logical truth (on Swinburne's account). Steinitz cites Swinburne (Swinburne 39-41) as the source for his belief that there is no conclusive demonstration for the coherence of any concept. But this implausibly strong claim is not held by Swinburne. For example, Swinburne takes the coherence of `John has red hair' as self-evident, as well as the entailment of `Someone has red hair'. Since entailment preserves coherence, this is conclusive proof that the latter is coherent according to Swinburne. The difficulty in proving coherence applies to those concepts whose coherence is dubious and which are not obviously derivable from concepts that are coherent.




53. 2_27.

Steinitz 180.




54. 2_28.

For example, suppose I believe that nobody loves Jane, and in order to convey my certainty of this I say: «If somebody loves Jane, then I am a monkey's uncle». Aren't I using the quantifier to range over the empty set?




55. 2_29.

On the view sketched in note 7, this sentence is not necessarily true.




56. 2_30.

For example, Mates tells us that «Nobody has yet been able to make the discovery needed for deciding whether the one premised argument -- The number of stars is even and greater than four; therefore, the number of stars is the sum of two primes -- is valid. Benson Mates, Elementary Logic 2nd. ed. (New York: Oxford: UP, 1972) 4. As Mates admits, the validity of this argument is an open question not because its logical structure is unknown, but rather because the truth of Goldbach's Conjecture has not been established. But why construe logical possibility so that logic becomes a hostage to the resolution of this conjecture?




57. 2_31.

See Steinitz 177.




58. 2_32.

For example, Plantinga's argument appeals to what he calls broadly logical possibility which is more restrictive than logical possibility. According to him, it is logically possible, but not broadly logically possible that `Red is a color' and `No numbers are human beings' be false. The standard approach to logical possibility sketched in section 3 also makes it the case that these sentences could logically be false.




59. 3_1.

Tarski, A. Introduction to Logic and the Methodology of Deductive Sciences. Oxford. University Press, Inc. Revised ed., 1946. p. 19.




60. 3_2.

bid, p. 76.




61. 3_3.

See Kneale, W. and M., The Development of Logic. Oxford University Press, 1962.




62. 3_4.

Geach, P. T., Logic Matters, Univ. of California Press, Berkeley-Los Angeles 1972. p. 61.




63. 4_1.

My use of `rigid designation', etc., is intended to be in keeping with Saul Kripke's use. See his Naming and Necessity (Cambridge: Harvard University Press, 1980), p. 48.




64. 4_2.

G. E. Moore, «Hume's Theory Examined», in Some Main Problems of Philosophy (New York: Collier, 1962), p. 137. Moore enumerates «Hume's Principles» as «(1) That in order absolutely to know that B must have been preceded by A, I must have observed in the past that things like B were constantly preceded by things like A; and (2) That in order to know that B was probably preceded by A, I must have observed in the past that things like B were generally preceded by things like A.» In «Hume's Theory», ibid., p. 114. (Some Main Problems now abbreviated `SMPP')




65. 4_3.

In «Material Things», SMPP, p. 160.




66. 4_4.

See BonJour's The Structure of Empirical Knowledge (Cambridge: Harvard University Press, 1985).




67. 4_5.

Bayesian analysis gives us the following. Pr (D/P) = [Pr (P/D) X Pr (D)] / Pr(P); also, Pr (S/P) = [Pr (P/S) X Pr (S)] / Pr (P). Since denumerators are equivalent, we cancel them out. Next note that since both D and S entail P (seeing the pencil), then both Pr (P/D) and Pr (P/S) are equivalent and thus drop out also. That leaves us with the prior probabilities of D and S. The weight of intuition falls on the side of S's superiority over D's here.




68. 4_6.

For Moore on «direct apprehension» see Moore, «Sense Data», in SMPP, pp. 61-66.




69. 4_7.

In SMPP, pp. 143-161.




70. 4_8.

«Hume's Philosophy», originally in The New Quarterly, 1909; reprinted in Philosophical Studies (New York: Harcourt, Brace & Company, 1922), p. 161.




71. 4_9.

In «Hume's Theory Examined», SMPP, p. 132.




72. 4_10.

The word «reliable» here poses a problem, for I examine the reliability of our belief-producing mechanisms and how that reliability is related to knowledge of the objects of perception. If I define «reliable» simple as «producing true beliefs most of the time», then by definition, even if Moore did have reliable belief-producing mechanisms he might still not know that the objects he perceived really do exist. So, by reliable I mean «producing true beliefs about the existence of the objects of perception, when those objects are both (a) readily macroscopic, and (b) well-textured substances with visible properties, plus a lot more». I would say that clouds, fog, mist, 3-D holographic images, and mirages are not well-textured substances, whereas tables, chairs, and pencils are. The «plus a lot more» clause covers the function of belief-forming mechanisms to form non-existential questions. I think these notes are sufficient to show that Moore is in no position to verify whether he has reliable (as defined) belief-producing mechanisms.




73. 4_11.

See Moore's famous paper, «The Defense of Common Sense», reprinted in his Philosophical Papers (London: Allen & Unwin, 1959).




74. 4_12.

Charles Darwin, letter to William Graham Down, July 3, 1881, in Francis Darwin, ed., The Life of Charles Darwin Including an Autobiographical Chapter (London: John Murray, 1887), Vol. 1, pp. 315-16. Quoted in Alvin Plantinga, «Is Naturalism Irrational?», in Warrant and Proper Function (Oxford, 1993), p. 219. See also Darwin's discussion of pleiotropy in On the Origin of Species (Cambridge, MA and London: Harvard University Press, 1964), p. 143.




75. 4_13.

Patricia Churchland, «Epistemology in the Age of Neuroscience», Journal of Philosophy 84 (1987), 544-553. Alvin Plantinga, «Is Naturalism Irrational?», ibid., pp. 216-237. For some of the ideas in this section, I am indebted to Alvin Plantinga and his critique of naturalism in Warrant and Proper Function.




76. 4_14.

W. V. O. Quine, «Natural Kinds», in Ontological Relativity and Other Essays (New York: Columbia University, 1969), p. 126.




77. 4_15.

Alvin Plantinga, «An Evolutionary Argument against Naturalism», Logos 12 (1991), 27-47; p. 32. This article is very similar to the chapter in Warrant and Proper Function referred to above.




78. 4_16.

Proverbs 28:1.




79. 4_17.

See William Alston, «Level Confusions in Epistemology», and «Epistemic Circularity», in Epistemic Justification: Essays in the Theory of Knowledge (Ithaca, NY: Cornell University Press, 1989).




80. 4_18.

SMPP, p. 99.




81. 5_4.

I do think that there is a correct theory, and I discuss it in my own courses. I just do not feel that arguing for it is the best way to promote the solution of important social problems.




82. 5_5.

Trans. B. Jowett, The Dialogues of Plato, vol. 1 (New York: Random House, 1920), 807.




83. 5_6.

Republic 420. Op. cit. 683.




84. 5_7.

A classic example is Karl Popper. V. his The Open Society and Its Enemies, vol. 1, The Spell of Plato (London: George Routledge & Sons, 1945).




85. 6_1.

Unfortunately we cannot yet handle TeX or LaTeX files. The convertors we've tried have proved useless.




86. 6_2.

At our home site, ftp.csic.es, there is -- hanging from our main directory /pub/sorites -- a subdirectory, WWW, which, among other files, contains one called `HTML.howto', wherein the interested reader can find some useful information on HTML editors and convertors.




87. 6_3.

Mike Albert's address is P. O. Box 535, Bedford, MA 01730, USA.




88. 6_4.

For the time being, and as a service to our readers and contributors, we have a directory called `soft' hanging from our home directory /pub/sorites at the node ftp.csic.es. The directory contains some of the non-commercial software we are referring to, such as archivers or 8-to-7 encoders (or 7-to-8 decoders).




89. 6_5.

In the case of WordPerfect 5.1, the procedure is as follows. Suppose you have a file called `dilemmas.wp5' in your directory c:\articles, and you want to submit it to SORITES. At your DOS prompt you change to your directory c:\articles. We assume your WordPerfect files are in directory c:\WP51. At the DOS prompt you give the command `\wp51\convert'; when prompted you reply `dilemmas.wp5' as your input file whatever you want as the output file -- suppose your answer is `dilemmas.ker'; when prompted for a kind of conversion you choose 1, then 6. Then you launch you communications program, log into your local host, upload your file c:\articles\dilemmas.ker using any available transmission protocol (such as Kermit, e.g.). And, last, you enter your e_mail service, start an e_mail to to sorites@ifs.csic.es and include your just uploaded dilemmas.ker file into the body of the message. (What command serves to that effect depends on the e_mail software available; consult your local host administrators.)

With WordPerfect 6 the conversion to kermit format is simple and straightforward: you only have to save your paper as a `kermit (7 bits transfer)' file.




90. 6_6.

Those devices are temporary only. Later on we'll strongly advise and encourage those of our contributors who can use neither WordPerfect format nor one of the other word-processor formats our convertors can handle automatically to resort to HTML, with certain conventions in order to represent Greek characters as well as logical and set-theoretic symbols.




91. 7_1.

The reader may find an excellent discussion of copyright-related issues in a FAQ paper (available for anonymous FTP from rtfm.mit.edu [18.70.0.209] /pub/usenet/news.answers/law/Copyright-FAQ). The paper is entitled «Frequently Asked Questions about Copyright (V. 1.1.3)», 1994, by Terry Carroll. We have borrowed a number of considerations from that helpful document.