To My Critics with Appreciation Responses to Taliaferro, Swinburne, and Koons

JORDAN HOWARD SOBEL Department of Philosophy University of Toronto

R. Douglas Geivett organized a session of the Philosophy of Religion Group at the meeting of the Pacific Division of the American Philosophical Association in San Francisco in March 2005 in which he, Robert Koons, Richard Swinburne, and Charles Taliaferro commented on Logic and Theism: Arguments For and Against Beliefs in God.1 I am grateful for their interest and the resourceful attention they have paid to the book. What follow are tokens of my appreciation in responses to scripts of comments I then had in hand.2 Each response revises and substantially elaborates scripts on which my responses in San Francisco were based.

On Explanations of the Cosmos, Cumulative Arguments, and God as a Necessary Being

1. “Sufficient-Reason” Arguments

Of explanations of the cosmos that would satisfy several “principles of sufficient reason,”3 Charles Taliaferro agrees, I think, that they must run in

1. Jordan Howard Sobel, Logic and Theism: Arguments For and Against Beliefs in God (Cambridge: Cambridge University Press, 2004). 2. I am sorry that I did not respond to Geivett’s comments in San Francisco, and that I cannot respond now. I remember that they were interesting and well-presented, but I do not remember what they were! Also, I have not had time to attend to new material in current scripts of comments by Koons and Taliaferro provided by the editor. For the reader’s ease, in this article most references to Richard Swinburne, “Sobel on Arguments from Design,” and Robert Koons, “Sobel on Gödel’s Ontological Argument,” are to the articles in Philosophia Christi 8 (2006): 227–34 and 235–47, respectively. In cases where those articles differ from the remarks presented in March 2005, the reference is to those earlier versions. 3. Namely, Leibnizian principles that would call for complete explanations of the existence and of every detail of the cosmos, and be final explanations “with which one can stop,” and so be noncircular explanations than ran “in the end” entirely in terms of necessities. cf. Principles

249 250 PHILOSOPHIA CHRISTI terms of thoroughly necessary reasons that make the cosmos necessary. “[T]he best move for the theist,” he says—the best move in response to this problem of necessity, I assume he means—“rests on articulating and defend- ing the coherence of God as a necessarily existing being, but not as a being all of whose actions are necessary.”4 A contingent cosmos, he suggests, could be due to such a being. Perhaps, but if so, it seems that it could be due as well to a similar being who did not exist of necessity, in which case it seems that a best move for the theist who would respond to the problem of demanding principles of sufficient reason would be to frame a less demanding principle of reason that can be satisfied for contingencies by capable contingently existing beings. This move would allow him to leave the problem of the sense of a necessary being whose actions are not all necessary for another day. I return to these things in section 4.

2. Cumulative Arguments for Theism and Atheism5

Taliaferro says that his “preferred approach . . . involves building a cumulative case on one side or the other” for theism or atheism. He writes, “I prefer running a version of the cosmological argument along with a design argument, and [an] argument from miracles,” and, I think he might add, versions of moral arguments, as well as arguments from common consent and religious experience, and perhaps versions of ontological arguments. It is, however, difficult to see the logic of a cumulative case built from such a mixed lot of deductive and inductive arguments.

2.1.

The idea of cumulative case-building comes up in the philosophy of religion. Sometimes it may amount to the common sense that every intelligent voice for and against should be given a hearing before one makes up one’s mind on the important issue of this philosophy. More often, at least of late, it means cumulative nondeductive reasoning of some kind in which originally deductive arguments are mined for evidence for and against. The logic intended for this cumulative reasoning is sometimes suspect. Thus Loren Meierding writes: 7 and 8 of Leibniz’s Principles of Nature and Grace (2nd ed., trans. and ed. Leroy E. Loemker [Dordrecht: D. Reidel, 1976], 638–9), and Sobel, Logic and Theism, sections 3.3 and 3.4, chapter 5. 4. Charles Taliaferro, “Cumulative Argument, Sustaining Causes, and Miracles,” paper presented at the 79th annual meeting of the Pacific Division of the American Philosophical Association, San Francisco, March 23, 2005. Cf. Taliaferro, “Cumulative Argument, Sustaining Causes, and Miracles,” Philosophia Christi 8 (2006): 219–26. 5. This section draws from the revised appendix to chapter 7 of Logic and Theism that is linked to from the Web page, “On Logic and Theism,” http://www.scar.utoronto.ca/~sobel/OnL_T. JORDAN HOWARD SOBEL 251

With the exception of the ontological argument, arguments for God’s existence are essentially inductive arguments claiming that belief in God is justified based on various kinds of available evidence. . . . [The evidence of the Consensus Gentium argument] may, when combined with other evidence . . . provide sufficient support for rational belief in God’s existence. With the addition of the evidence of common consent to other available evidence, the scale may be tipped in the favour of theism.6 Relevant to this text is that Meierding claims in his article only that the evidence of the argument he discusses “provide[s] support for God’s existence.”7 He claims only that this evidence incrementally confirms the existence of God, P(God exists|the evidence of common consent) > P(God exists), not that it alone absolutely confirms the existence of God, P(God exists) > 1/2. He is not in a position to make this claim since he neither argues nor assumes a value for the “prior probability” of theism. John Earman (somewhat sur- prisingly) writes similarly of cumulative confirmation, namely: Mere incremental confirmation may not be what theists want for their doctrines, but it is a start. And once the start is made, there does not seem to be any principled road block to achieving a substantial degree of confirmation. For example, testimonies to a number of New Testament miracles can each give bits of incremental confirmation to [some tenet . . . of Christianity] that together add up to substantial confirmation. Or the evidence of miracles can combine with the evidence of prophecy and design to provide grounds for the credibility, or even moral certainty of religious doctrines.8 There is double-trouble for the cumulative logic implicit in Meierding’s text, and nearly explicit in Earman’s. For one thing, incremental confirmations do not necessarily “accumulate”: it is not a valid principle that, ([P(h|e) > P(h)] & [P(h|e') > P(h)]) → (P[h|(e&e')] > P(h)): confirming evidence for a hypothesis when combined into one “body of evidence” can disconfirm it.9 Wesley Salmon tells a story of radioactive decay to make the point that

6. Loren Meierding, “The Consensus Gentium Argument,” Faith and Philosophy 15 (1998): 272–3. 7. Ibid., 291. 8. John Earman, Hume’s Abject Failure: The Argument against Miracles (Oxford, University Press, 2000), 66–7. 9. Suppose, for a false instance, that a fair die has been cast, and that I have no idea which number came up. Assume the abbreviations, A: either 1 or 3 came up; B: either 1 or 2 came up; C: either 2 or 3, and observe that the conjunction (B&C) entails ~A. It can be seen that 252 PHILOSOPHIA CHRISTI

[e]ven if each set of measurements [of different dimensions of an experimental result] confirms [an] hypothesis . . . [that] the conjunction of the findings . . . confirm[s] the hypothesis . . . does not follow automatically. . . . [Whether this conjunction confirms it] depends on more circumstances, including . . . that the conjunction itself is one of the predictions of the theory. . . . there are broad and basic questions about the legitimacy of . . . accumulation of many confirming test results.10 For a second, and if anything more serious trouble for the logic suggested by Meierding’s and Earman’s words, even if there is a valid restricted principle for accumulating confirmations and disconfirmations that legiti- mates the accumulation of various theistic confirmations and disconfirmations into a cumulative incremental confirmation (or disconfirmation), P(theism|cumulated evidence) > P(theism) [or <] that is not yet the result envisioned in their texts of absolute confirmation of theism (or atheism), that is, of the credibility, if not the moral certainty of theism (or atheism), P(theism) > 1/2 [or <].

2.2.

So much for bad cumulative logic. Richard Swinburne has me helped me to see that— given the objective character of his probabilities—he has a perfectly sound way of, in the first stage of his argument, accumulating incremental confirmations and disconfirmations of his theism on parts of one’s evidence to reach an “input” for the second stage of his argument, which is a Bayesian assessment that, taking into account “initial priors,” can yield an absolute confirmation or disconfirmation of this hypothesis on evidence (potential or “in hand”) taken all together. In the first accumulation stage the “background information” for the assessment of the bearing of a piece of evidence is all previous considered evidence.11 With k “tautological evidence,” and en the evidence of chapter number n (7: the existence of the world; 8: order in the world; 9: matters of consciousness and morality; 10: provisions for opportunities for us to do good; 11: evil; 12: history and miracles) he maintains that,

[P(A|B) > P(A)] & [P(A|C) > P(A)] → (P[A|(B&C)] > P(A)) 1/2 1/3 1/2 1/3 0 1/3 TT T F F Confirming evidence B and C for A, when combined, disconfirms: P[A|(B&C)] = 0 < 1/3 = P(A). 10. Wesley Salmon, “Confirmation,” Scientific American, May 1973, 80. 11. Cf. Richard Swinburne, The Existence of God, rev. ed. (Oxford: Oxford University Press, 1991), 144, 173, and 277; and Richard Swinburne, The Existence of God, 2nd ed. (Oxford: Oxford University Press, 2004), 17. JORDAN HOWARD SOBEL 253

2.3.

Swinburne’s argument has two stages. In the first stage of his argument he accumulates incremental confirmations and disconfirmations of his theism on several “pieces” of one evidence, with background information for each piece being the evidence previously considered for its bearing on theism, to reach an “input” for the second stage of his argument, which is a Bayesian assessment that, taking into account “initial priors,” can yield an absolute confirmation or disconfirmation of this hypothesis on evidence

12. Swinburne, The Existence of God, rev. ed. However, recalling that e11 is evil, Swinburne confesses in second edition of The Existence of God that he has changed his mind to the judgment that P(G|e7&e8&e9&e10&e11&k) < P(G|e7&e8&e9&e10&k) (266). This spoils the cumulating-inference to P(G|E&k) > P(G|k). But Swinburne says that “[a] diligent student of the earlier editions [1979, 1991] will . . . detect marginally more sympathy for the argument from evil against the existence of God, [is] balanced by marginally greater confidence in the force of the argument from moral awareness for the existence of God. . .” (vi). And so to restore his “cumulating-inference” to P(G|E&k) > P(G|k), he can move the section on morality and moral awareness in chapter 9 forward to chapter 11, and, after recording in this chapter his judgements that P(G|evil & k') < P(G|k') and P(G|moral awareness & k') > P(G|k'), with k' evidence taken up in chapters 7 through 9 and now diminished 10, and e the evidence of evil and moral awareness make the judgment for chapter 11, that P(G|e&k') ≥ P(G|k'). 13. Ibid., 290–1. 14. Swinburne, The Existence of God, 2nd ed., 341. 15. Swinburne, The Existence of God, rev. ed., 291, and The Existence of God, 2nd ed., 342. 254 PHILOSOPHIA CHRISTI

(potential or “in hand”) taken all together. A. P. Dawid writes of one-stage cumulative way that: The application of [the “odds” form of Bayes’s theorem: P(G|E)/P(~G|E) = P(G)/P(~G) × P(E|G)/P(E|~G); “G” for “guilty”] . . . can be done either ‘wholesale’, with E representing the totality of the evidence in the case; or ‘piecemeal’, with sequential incorporation of several items of evidence. In the latter case, the probabilities on the right-hand side [both those of the “prior odds” and those of the “likelihood ratio”] should be regarded as evaluated conditional on all previously incorporated evidence.16 It may be that Dawid intends to recommend this approach only when pieces of the total evidence are, conditional on the hypothesis, independent, since only then will their “cumulative effect” on probabilities in the likelihood ratio be the product of their “discrete effects”17: this (I add), is by the theorem that, [P(h&e)>0]→([P[(e&e')|h]=P(e|h)×P(e'|h)]≡(P(e'|h)=P[e'|(h&e)])).18

2.4.

While these ways of accumulation can work for me for potential evidence which is not of probability 1, I doubt that anything like them works for me for “old evidence,” given my way of managing “the problem of this evidence.” I propose, for assessing the confirmatory force for a person of “old evidence,” to go to what would have been his probabilities and conditional probabilities had he no inkling19 of the evidence up for explanation. Consider 16. A. P. Dawid, “Bayes’s Theorem and Weighing Evidence by Juries,” in Bayes’s Theorem, ed. Richard Swinburne (Oxford: Oxford University Press, 2002), 74. 17. Ibid., 75, 84. 18. G. E. Moore maintained that compound wholes are “organic unities” relative to their “intrinsic values,” the intrinsic value of a whole (the value of a whole “on the whole”) having no necessary relation the sum of the intrinsic values of its (“simplest”) parts. Analogously, “conjunctive body of evidence” are “organic unities” relative to their bearings on hypotheses, the evidential bearing of a conjunctive body of evidence on a proposition having no necessary relation to the product of the bearings of its conjuncts on this proposition. 19. The idea of a time when a person has no inkling of some evidence has a formal characterization in a theory that assumes, for representations of the “credal states” of its ideal subjects at times, not unique probability functions in which point-values are assigned to all propositions, but sets of such probability functions invariances across the members of which correspond to “[d]eterminate facts about the person’s beliefs. . . . For example, the person can only be said to determinately believe X to degree x [at a time] when c(X) = x for every c [in the set of probability functions that represents her credal state at this time], and she is only more confident in X than in Y if c(X) > c(Y) for every c [in this family]” (James Joyce, “How Probabilities Reflect Evidence,” in Epistemology, Philosophical Perspectives 19, ed. John Hawthorne [Malden, MA: Blackwell, 2005], 156). Each function “assigns a unique . . . number 0 ≤ c(X|Y) ≤ 1 to each proposition and X and condition Y . . . . [and] defines unconditional credences via the rule c(X) = [conditional credence] c(X|T), for T is any logical truth” (Joyce, “How Probabilities Reflect JORDAN HOWARD SOBEL 255 also the following: “Is the world considered in general and as it appears to us in this life, different from what a man . . . would, beforehand, expect from a very powerful, wise, and benevolent Deity?”20 When considering old evidence “piecemeal” a person will, as he brings bits of his evidence forward into the position of evidence to be explained, be engaged in a series of coun- Evidence,” 156; cf. Hosiasson-Lindenbaum/Kolmogorov functions in Jordan Howard Sobel, “On the Significance of Conditional Probabilities,” Synthese 109 [1997]: 311–44, and the appendix to Sobel, “Modus Ponens and Modus Tollens for Conditional Probabilities, and Updating on Uncertain Evidence,” http://www.scar.utoronto.ca/~sobel/uncertainevid.pdf). “Learning experiences can be modeled as shifts from one credal state to another that proceed in accordance with Bayes’s Rule [i.e., by conditionalization]” (Joyce, “How Probabilities Reflect Evidence,” 153) applied to each probability function in the set that represents a person’s credal state at the time of a learning experience, “certain cases aside.” There are cases in which, though a person’s new found certainty is a bona fide case of learning, he suspects that it is not, and that it is, for example, a product of dyspepsia, drugs, hypnosis, or of subliminal influences, or of deception and misplaced trust, and so on. In these cases he may, reflecting on his credences, be prepared to allow them to be “incoherent” and not to “update” on this new certainty. Cf.: “ A person’s credence for X conditional on Y [need not] coincide with her unconditional credence for X after learning Y when the learning induced belief change is . . . driven by . . . arational processes that ignore Y’s content” (Joyce, “How Probabilities Reflect Evidence,” 155). In these terms a person has “no inkling of evidence e at a time” if and only if his credal state at this time is represented by a set C of probability functions such that, for every n such that 0 ≤ n ≤ 1, there is a probability function c in C such that c(e) = n. In this case the person’s doxas- tic attitude towards p is maximally indeterminate. If representations of credal states include “weights,” 0 ≤ W ≤ 1, for functions c in the representing sets of probability functions, then a person’s “singular probabilities” can be defined as weighted averages of their probabilities in C, except in the case in which a proposition has in C every probability and these are weighted equally, the weight of a probability P for p being the sum of weights for c in C such that c(p) = P. This should be the case when he has no inkling of this proposition, in which case we should say he does not have a “singular probability” for it. “Singular probabilities,” so defined, though equal in “quantity” could differ in ways that reflected “degrees of ambiguity” it is suggested in chapter 8, section 13.2, of Logic and Theism. When a person has “no inkling of evidence e,” determinate Bayesian assessments of support that would be provided by evidence e to a member h of a partition of hypotheses, compared with the support e would provide to other members of the partition, are possible without going into details of the probability functions in the set that represents his credal state at this time, if either (i) his degrees of confidence in the probabilities for these hypotheses and for the likelihoods of the evidence on them at this time are all determinate; or (ii) for every other hypothesis h' in this set, either P(h) > P(h') and P(e|h) ≥ P(e|h'), or P(h) ≥ P(h') and P(e|h) > P(e|h')—that is, h exceeds h' in one of the dimensions of intrinsic plausibility and likelihood for the evidence, and at least matches in the other, every other hypothesis in the partition; or (iii) P(h) < P(h') and P(e|h) ≤ P(e|h'), or P(h) ≤ P(h') and P(e|h) < P(e|h')—that is, every other hypothesis in the partition exceeds h in one of those dimensions and at least matches it in the other. Evidence of colors and textures of marbles for barrel hypotheses in the Marble Problems in chapter 7, sections 3 and 4, in Logic and Theism is meant to be, for every mind, of type (i). The situation, when one had no inkling of the evidence of appearances of design and unnecessary evil and misery, of Cleanthes’s religious hypothesis in the partition of hypotheses that can be gathered from the Dialogues, would be to many minds of type (iii): Hume’s arguments for this in the Dialogues are collected in section 5 of chapter 7 of Logic and Theism. 20. David Hume, Dialogues Concerning Natural Religion, ed. Stanley Tweyman (London: Routledge, 1991), Part 11 (163). 256 PHILOSOPHIA CHRISTI terfactual shifts. The decisive shift, I think, should be a “wholesale” shift on all the evidence to be explained. When considering the evidence “piecemeal,” this shift would never be made.

3. On a Difference between the Theisms of Taliaferro and Swinburne

Guessing, I suspect that the arguments Taliaferro has in mind to accumulate—including the version of the cosmological argument that he has in mind—are all to be for incremental confirmations that culminate in Swinburne-fashion in a “likelihood,” conditional on Taliaferro’s necessarily existing being theism, of the evidence drawn from various theistic and athe- istic arguments, thence to be entered, along with “likelihoods” of alternative hypotheses, and “prior probabilities” for it and these alternative hypotheses, in a Bayesian determination of its absolute confirmation or disconfirmation on this evidence. If that is right, then I am back to the problem of how the necessity of God’s existence could be supported by Taliaferro’s cumulative arguments. Swinburne, it may be noted, makes his cumulative argument for a “diminished Anselmian theism”: it is for the existence of a being who, while in an Anselmian manner would be necessarily or essentially perfectly free, omnipotent, omniscient, perfectly good, everlasting, bodiless creator of all things, would explicitly not be, as Anselm would have Him be, a logically necessary existent.21 Swinburne gives reasons why the target of his cumulative inductive argument is not a logically necessary being.22 There are reasons of another sort why God, since He would be worshipful, needs not to be a necessary being (please see sections 1.2 and 5 of “On Gödel’s Ontological Proof” below). There are good things for us to talk about here. And there is much else of interest in Taliaferro’s wide-ranging comments which I regret not having had time to digest and to prepare responses that they merit.23

21. Swinburne, The Existence of God, 2nd ed., 7, 79. 22. Ibid., 79. 23. Of relevance to differences between William Craig and me concerning actual infinities and first cause arguments are his critical discussion of, and revisions I have made to, chapter 5 of Logic and Theism. Please see Sobel, “First Causes,” http://www.scar.utoronto.ca/~sobel/OnL _T/L_T5enRv25Mr.pdf. JORDAN HOWARD SOBEL 257

On Subjective and Objective Probabilities, Dividing the Evidence, and Fine-tuning

Richard Swinburne and I differ in our understandings of the probabilities we consider relevant for the assessments of hypotheses in the light of evidence. After drawing this difference, I relate it to strategies for dividing evidence in confirmation exercises between what is to be explained, and what is left as background information. Then I say a bit about possible explanations of the recondite evidence of fine-tuning. An appendix goes to “logical probability” according to John Maynard Keynes who began the line on “evidential (or epistemic) probability” that Swinburne follows, and who soon abandoned it.24

1. On Our “Probabilities”

1.1. Mine

Richard Swinburne is an “objective Bayesian.” He works with objective probabilities of a sort. I am a subjective Bayesian who works exclusively with subjective probabilities. 1.1.1. The idea of probabilities, in part only implicit in Logic and Theism, is that an ideally rational and highly opinionated person’s has exact credences or degrees of confidence in all possible worlds (that is, in this or that possible world’s being the actually instantiated possible world) that are “measured” by numbers25 bounded by 0 and 1 that sum to 1, and that his credences for propositions are “measured” by sums of his credences for worlds at which they are true. Terming these measures “probabilities,” it follows that this person’s probabilities for propositions satisfy the Kolmogorov axioms, (i) P(p) ≥ 0; (ii) □p → P(p) = 1; and (iii) ~◊(p&q) → P(p∨q) = P(p) + P(q). I assume that what are in this ideal intellect’s view “possible evidential bearings” of positively probable propositions are measured by conditional probabilities defined as ratios of his probabilities thus,

24. Richard Swinburne, “Introduction,” in Bayes’s Theorem, ed. Richard Swinburne (Oxford: University Press, 2002), 4. An appendix, now revised, to chapter 7 of Logic and Theism includes responses to Swinburne’s just criticisms of the appendix in the book. Section A5 contains a friendly articulation of Swinburne’s cumulative argument. Sections A1 through A3 include qualms concerning the simplicity of theism and its relevance to the “intrinsic probability” of this hypothesis. The revised appendix is linked to the Web page, “On Logic and Theism,” http://www.scar.utoronto.ca/~sobel/OnL_T. 25. Real numbers serve, except for purposes of chapter 8, appendix B3, and the “hyperreal subtext” in notes to chapter 13 buttressed by its appendix. 258 PHILOSOPHIA CHRISTI

P(q|p) = P(p&q)/P(q).26 It is a consequence in this scheme that, upon learning for sure that p at t, if t+ is a time later than t at which he is still sure of p, and t– is a time earlier than t such, for every time t' such that t– ≤ t' < t, and every proposition r, Pt–(r) = Pt'(r), then it is necessary that his probability for q at a later time t+ equals what was his conditional probability at t– (that is, it is necessary that his probability for q at t+ “updates by conditionalization” whatever was his probability for q at t–), provided that his view of the potential evidential bearing of p on q has not been perturbed since t– (that is, provided that the “rigidity condition” is satisfied to t– and t+). This conditional proposition is demonstrable for “coherent” probabilities at both t– and t+: it is a theorem that,

[Pt+(p) = 1] → ([Pt+(q) = Pt–(q|p)] ↔ [Pt+(q|p) = Pt–(q|p)]).

update by conditionalization “rigidity condition”27 for it is a theorem that, for any probability function P, [P(p) = 1] → [P(q) = P(q|p)], since it is a theorem that, [P(p) = 1] → [P(q) = P(p&q)].28 In my view, not only are probabilities at a time of an ideally rational subject constrained by Kolmogorov’s axioms, but they are rational in the further subjective sense that they are as they would be had he made full use of his rational faculties to realize the potential for affecting his credences of reflection on his evidence. That is, I assume that an ideally rational and highly

26. This scheme is elaborated in files on probability linked to a Web page for Logic and Theism, http://www.scar.utoronto.ca/%7Esobel/L_T. “I beg limitations here be remarked when I say” that the scheme is for highly opinionated subjects and the “possible evidential bearings” measured of propositions that are in his view positively probable. A scheme free of these limitations is contemplated in note 19. In it the “credal state” of a ideally rational subject is represented not by a single Kolmogorov probability function, but by a set of Hosiasson-Lindenbaum/Kolmogorov probability functions (for which please see an appendix to Logic and Theism): cf. representations of “credal states” in Joyce, “How Probabilities Reflect Evidence,” 2005. 27. Is this rigidity condition necessarily satisfied for every proposition r and condition p that an ideal intellect A could learn for sure? No. For A could learn for sure at t that it is raining at t. Let that be proposition p, and let r be the proposition that that at time t A learns p for sure. Assume that at t– A does not expect it to rain at t,Pt–(p) < 1, and that he then thinks that if it rains at t, he may not learn at t that is raining, so that for him Pt–(r&p) < Pt–(p). Then for him Pt–(r|p) < 1. However, presumably ideal intellect A remembers at t+ that he learned p at t, so that for him Pt+(r) = 1, and thus Pt+(r|p) = 1

29. Cf. David Falk’s idea of “the purely formal motivational ought”: “a willing, or impulse to action, which a person would have if he both acquainted himself with the facts . . . and test- ed his reactions to them, and which he would have necessarily, i.e., unalterably by any repeti- tion of these mental operations [the ‘testing-operations’] . . . the test [is that] of all the operations capable of affecting his attitude . . . having been carried to an ideal limit thus exhibiting to him an ‘objective’ attitude . . . most adapted to the realities of his situation and his own dispositions” (Falk, “Morality, Self, and Others,” in Ethics, ed. J. J. Thomson and G. Dworkin [Cambridge, MA: MIT Press, 1968], 38.) Presupposed by the existence of such “oughts” for a person, is that such unalterable for a person reactions would be reached eventually during such testing. Similarly for my conception of personally rationall credences. “But what can be the point of such operations, and of beliefs and attitudes adapted to one’s own dispositions?” The point, glossed in Jordan Howard Sobel, “Self-Doubts and Dutch Strategies,” Australasian Journal of Philosophy 65 (1987): 72–3, Falk says is to be an integrated subject of one mind, in which state, he plausibly maintains, everyone has a considerable stake. There is for everyone this deeply personal reason rational opining and willing: it is to opine and will authentically. 30. For example, noticing the importance to a person of her trusting her partner, though it could not, I think, directly enhance (causal notion—and so see note 65 below) her credence for his trustworthiness, could motivate care on her part lest she reflect on matters reflection on which might diminish this confidence. “Take holy water, hear mass, and give up philosophy,” Pascal could have recommended to those who would be secure in their devotion. 31. Cf. Thomas Kuhn, The Copernican Revolution (New York: Random House, 1957), 5. Differences are to be expected regarding “cosmic matters unrelated to the concerns of everyday life” (Peter van Inwagen, “The Problem of Evil, the Problem of Air, and the Problem of Silence,” in God, Knowledge and Mystery: Essays in Philosophical Theology (Ithaca, NY: Cornell University Press, 1995), 83): regarding such matters there would be little if any evolutionary pressure that in practical matters results in extensive agreements in probabilities on reflection and discussion (and, Hume with prescience could have said, explains why we expect the future to resemble the past in our daily lives, since it always has). Where van Inwagen, who believes in the possibility of objectively correct probabilities, sees reasons for thinking our incli- nations to probabilities in “cosmic matters” are unreliable, I see reasons for thinking that our 260 PHILOSOPHIA CHRISTI sity study” wherein students often find that evidence that, for others, makes hypotheses more probable than others, does so for them as well.

1.2. His

Swinburne observes that I confess “‘I do not believe in objectively reasonable levels of confidence.’”32 He does. He works with “inductive probabilities” or “logical probabilities” which he contrasts with “measures of evidential support that [are functions of] measures subjects’ degree[s] of confidence.”33 We . . . think that there are . . . right and wrong ways to assess how probable one proposition . . . makes another . . . , and the philosopher of induction tries to codify the right ways. . . . If we do not think that there are such criteria, then we must hold that no one makes any error if he regards any scientific theorem compatible with observations. . . . On the basis of our evidence . . . it would be as sensible to believe that cyanide will nourish and bread poison tomorrow. . . . Given the evi- dent implausibility of thinking thus . . . I conclude that there are correct inductive criteria that often give clear results, at any rate for claims of comparative probability. . . . Of course . . . we may misap- ply our criteria through failure to realize what are the relevant logical possibilities and necessities . . . or, through failure to realize what are the deductive or inductive consequences of our evidence. But that measure of inductive support that would be reached by a logically omniscient being . . . is what I call logical probability. It has a value determined by the content of q and r, which measures the total force of r with respect to q; to which we try to conform our judgments of inductive probability on evidence but about the value of which we may make mistakes.34

fully reflective subjective probabilities, the only kind of probabilities in which I believe, may well not agree. This could be part of an explanation of why there are perfectly reasonable uncon- vertible theists and atheists, as there seem to be. 32. Richard Swinburne, “Sobel on Arguments from Design,” Philosophia Christi 8 (2006): 229, and Sobel, Logic and Theism, 609n24. 33. Swinburne, The Existence of God, 2nd ed., 15–16. 34. Richard Swinburne, Epistemic Justification (Oxford: University Press, 2001), 64. Apropos this quotation: I “hold that no one [need make] any error if he regards any scientific theorem compatible with observations [with which it is logically compatible] . . . [and that on] the basis of our evidence . . . it [can] be as sensible to believe that cyanide will nourish and bread poison tomorrow . . . . [and that the] measure of inductive support that would be reached by [one] logically omniscient being” can differ from that that would be reached by another logically omniscient being. I do not believe in the reality of relations of “inductive support” or of “total force” of one proposition r for another q that are “determined by the content of q and r” alone. JORDAN HOWARD SOBEL 261

Those who hold that there are no such objectively correct criteria will have no use for [the concept] of logical . . . probability. . . . I . . . find it difficult to take such thinkers seriously.35 I gather from the following that in his view a person’s degree of confidence in a proposition is reasonable if and only if its subjective measure equals the inductive probability of this proposition on this person’s total evidence: “It is the probability of h on the total evidence available at some time e, by which it is rational to be guided in our actions at that time.”36 And “in so far as the probability of a hypothesis is relevant to whether or not we ought to act on it, we ought to act on a hypothesis in so far as it is rendered probable by the total evidence available to us—all we know about the world, not just some limited piece of knowledge.”37 “[T]he logical probability of p on q . . . is clearly,” Swinburne writes, “an a priori matter.”38 These probabilities are said to be determined by contents of proposition,39 without depend- ence on contingencies of human psychology, would be timeless and necessary. And yet they are to “make a difference”: we ought to act on them; they

35. Swinburne, Epistemic Justification, 69. Again: “Now in a footnote Sobel writes ‘I do not believe in objectively reasonable levels of confidence’ (609n26)” (Swinburne, “Sobel on Arguments from Design,” 229). Swinburne continues: “If that is really [Sobel’s] view, then he has ruled himself out of serious debate about the probability of the existence of God, as entered into by Cleanthes or myself or most other people. . . . So let us suppose that Sobel really believes that there are correct a priori criteria for determining intrinsic probabilities” (Swinburne, “Sobel on Arguments from Design,” 229–30). In fact, however, that really is my view. I do not believe in objectively reasonable levels of confidence. I most certainly do not believe that there are correct a priori criteria for determining “intrinsic probabilities” for “every proposition [prominently including every contingent proposition] . . . in advance of empirical discoveries” (Swinburne, The Existence of God, rev. ed., 110) and, as Hume might say, “acquaintance with the universe” (cf. Hume, Dialogues Concerning Natural Religion, Part 11, (161)]). I do not believe in correct “intrinsic probabilities” for all propositions, let alone criteria for determining such. The latter clause is an allusion to the theoretical possibility of “particularist objectivism” regarding “intrinsic probabilities.” Swinburne’s contrasting view is “that all propositions have intrinsic probabilities . . . determined solely by the a priori criteria of scope and simplicity. [Though] most propositions have only very rough intrinsic probabilities . . . the criteria . . . do often give clear verdicts on the relative intrinsic probabilities of different propositions” (Swinburne, Epistemic Justification, 147). 36. Swinburne, “Introduction,” in Bayes’s Theorem, 4. 37. Swinburne, The Existence of God, 2nd ed., 13. Cf. Swinburne, “Introduction,” in Bayes’s Theorem, 4: “It is the probability of h on the total evidence available at some time e, by which it is rational to be guided in our actions at that time.” 38. Swinburne, The Existence of God, 2nd ed., 16. “. . . we have access . . . to . . . necessary truths that are a priori in the sense that we can discover them by mere reflection. . . . The [necessary] a priori truths include . . . truths about what is evidence for what” (Swinburne, Epistemic Justification, 10). A logical probability P(q|p) exists for every pair of propositions (p, q) (Swinburne, Epistemic Justification, 103). The logical probability P(q|p) is, again, determined by the contents of q and p alone. 39. Swinburne, The Existence of God, rev. ed., 64. 262 PHILOSOPHIA CHRISTI ought to guide our actions, as well, of course, as our credal responses to observations and evidence. A person’s subjective probability for a proposition is, I take Swinburne to “say,” irrational and unreasonable, if it does not equal the logical probability of this proposition on his evidence, in which case this person ought not to act on it. This proscription applies, even if this person’s subjective probability for the proposition is subjectively reasonable in the sense recently explained (that is, it is as it would be eventually and evermore in an unend- ing process in which his evidence is given every possible of changing his mind), and even if this person has good reasons for thinking that his probability for this proposition is in this sense reasonable.40 Even in this he ought to act on the “intrinsic probability” updated by conditionalization on his evidence—he ought to act on a probability for the proposition that he does not have, and would not have eventually and then evermore were he to reflect on his evidence, tapping every capacity thereby to affect his probability for this proposition. Surely not! As John Mackie might say: Though I can find no a priori contradiction in this conception of objective probabilities, as timeless, strictly necessary, relations of “partial entailment” upon which, when their conditions match our total evidence, we ought to act on regardless of what would be our subjectively reasonable probabilities. And though this compound conception may be the conception implicit in current in ordinary unschooled talk of probabilities of things, especially argumentative talk, as said in note 34, I cannot believe in the reality of the relations of support based on contents of

40. Cf.: For Bayesians probabilistic judgment is a matter of drawing out the consequences of what one has learned (. . . evidence) in the light of one’s credal commitments (. . . prior belief state). For rationalistic Bayesians such as Laplace, Keynes, Carnap, and Harsanyi, prior belief states are . . . subject to norms of epistemic rationality, aimed for the most part at ridding the mind of bias of one kind or another. For them, belief states are justified just in case they are achieved by valid inference from a privileged initial state [cf. “every contingent proposition has an intrinsic probability . . . in advance of empirical discoveries that may increase or decrease that probability” (Swinburne, The Existence of God, rev. ed., 110)] and the totality of evidence yield- ed by experience. Subjective Bayesians, on the other hand, reject the idea of a [unique] rational prior belief state and so give up on the idea that there is a unique [rational] system of partial belief associated with any [given] evidence base. (Richard Bradley, “Radical Probabilism and Bayesian Conditioning,” Philosophy of Science 72 [2005]: 344) Objective Bayesians such as John Maynard Keynes (A Treatise on Probability [1921; London: Macmillan, 1973]) and Swinburne consider “the privileged initial state” to be not only uniquely rational, but also objectively correct or true. Subjectivists believe that “there is no truth in probability,” as well as that there is not a privileged initial state . They can believe in, or be more confident than not of, natural initial “credal states” represented by sets of probability functions. JORDAN HOWARD SOBEL 263 propositions that it presupposes.41 The reality of talk and thought of “probabilities” is, I think, exhausted by facts of credences and subjective probabilities, actual and potential on reflection.42

41. Swinburne’s idea of “intrinsic probabilities” of proposition runs into further trouble. The intrinsic probability of a proposition p would be for a tautology T the inductive probability P(p|T). And an inductive probability P(q|r) is said to measure “the total force of r with respect to q” and to be “determined by the content of q and r” (Swinburne, Epistemic Justification, 64). The idea is presumably that this “force” of r with respect to q—this “bearing” of r on q—is determined by how the contents of r and q are related. The mentioned trouble is that tautologies are contentless. “Without intrinsic probabilities you cannot even assess the probability of the most down-to-earth theory” (Swinburne, “Sobel on Arguments from Design,” 229). “[W]ithout [intrinsic probabilities] we cannot assess the force of contingent evidence in support of theories” (Swinburne, Epistemic Justification, 110). It is, as John Earman would, say “Bayes or bust” for Swinburne when it comes to assessing theories on evidence (possible or actual). Without intrinsic probabilities, Bayes’s theorem as stated in Epistemic Justification (103), P(h|e&k) = P(e|h&k) × P(h|k)/P(e|k) (see below for more on this) instantiated for his “cumulative argument for Theism,” P(E|G&k) × P(G|k) P(G|E&k) = P(E|k) wherein k is to be “tautological evidence” and E is the conjunction of evidence considered taken up, for him, patently a “bust.” Therefore, pending clarification or emendation, . . . . Incidentally, “tautological evidence” is redundant in all but “inductive probabilities,” given that tautologies are logical necessities and—this is Swinburne’s Axiom (2): (Swinburne, Epistemic Justification, 103)—given that if □(p ↔ q), P(r|p) = P(r|q). The displayed instance of Bayes’s theorem can be simplified to, P(E|G) × P(G|k) P(G|E) = P(E|k) Concerning P(h|e&k) = P(e|h&k) × P(h|k)/P(e|k), which was stated above, not explicitly stated is the antecedent, P(e|k) > 0, of the displayed theorem. Not explicitly noticed in the argument of the second edition of Swinburne’s The Existence of God is that it needs the “intrinsic probabilities” of the evidence of chapters 7 through 13—that is, the evidence of the existence of anything at all, of its orderliness, of evil, and so on—need all to be positive. It is of course no argument for the satisfaction of this requirement that our subjective probabilities for the evidence taken up are positive, though this may explain the absence of explicit reasons why their “intrinsic probabilities” are positive. What might be said for the “intrinsic probability” of there being something rather than nothing? Would not there being nothing be simpler, “infinitely simpler,” than there being something or other? Is there an argument for “intrinsic probabilities” of all contingent propositions without exception being positive? 42. I do not, however, think that the error of Swinburne’s intentions when he thinks and speaks of probabilities “has ruled [him] out of serious [discussions of evidential theistic arguments from design and evil, and regarding the evidence of testimony for miracles] . . . , as entered into by Cleanthes or myself or most other people” (Swinburne, “Sobel on Arguments from Design,” 229). After all, it is a common error, if I am right about the disparity between the concepts of ordinary thought and talk of probabilities that is not qualified by disclaimers of objective intent, and the reality of this thought and talk. Nor, of course, do I suppose that he does not really believe “that there are correct a priori criteria for determining intrinsic probabilities” (Swinburne, “Sobel on Arguments from Design,” 230). I am sure that he does believe that. 264 PHILOSOPHIA CHRISTI

1.3.

I agree with the implication of Richard Jeffrey’s title for his last book, Subjective Probability: The Real Thing.43 Swinburne agrees on the whole with John Maynard Keynes’s views of probability in A Treatise on Probability, the hard-core objectivity of which Braithwaite tells us Keynes took back in his 1931 review of “Ramsey’s 1926 paper.”44

2. On Dividing the Evidence 45

2.1.

Hume, Mackie, and I find evidence against the “prior probability” of theism when we think about the barely comprehensible character of an infinitely powerful and knowledgeable, perfectly free and good, everlasting, and bodiless creator of all things, and when we think about how—unlike all cre- ators of which we have uncontentious experience, the intentions of whom are immediately efficacious no further than the extremities of their bodies—This One would operate with intentions that were immediately efficacious every- and anywhere. But such qualms, Swinburne says, if addressed to his evidential argument for theism, would display a misunderstanding. This is because he casts his theism as an explanation of “all our empirical data,”46 of “every contingent proposition [in evidence],”47 so that there is left no contingent background evidence to make trouble for its “prior probability.” In considering the arguments for the existence of God, we shall begin with a situation of tautological background knowledge, and so dissimilarities between human persons and the postulated God will not as such affect the prior probability of theism.48 Mackie has not taken seriously my intention . . . to start without any factual background knowledge (and to feed all factual knowledge gradually into the evidence of observation [to be explained]), and so to judge the prior probability of theism solely by a priori considerations, namely, in effect, simplicity.49 43. Richard Jeffrey, Subjective Probability: The Real Thing (Cambridge: University Press, 2004). 44. R. B. Braithwaite, foreword to A Treatise on Probability by John Maynard Keynes (1921; London: Macmillan, 1973), xxii, quoted more extensively at the end of the appendix below. 45. This section draws from the revised appendix referenced in note 24 above. 46. Swinburne, The Existence of God, 2nd ed., 93. 47. Ibid., 66. 48. Swinburne, The Existence of God, rev. ed., 91, 63; cf. The Existence of God, 2nd ed., 66. 49. Swinburne, The Existence of God, rev. ed., 294. JORDAN HOWARD SOBEL 265

2.2.

Perhaps, however, Mackie felt free to ignore this arrangement of the evidence in Swinburne’s argument, realizing that, for Swinburne, as he was eventually to make explicit: The division between . . . evidence [to be explained] and background evidence can be made where you like—[though] often it is convenient to include all evidence derived from experience in e [evidence to be explained] and to regard k [background information] as being what is called in confirmation theory mere ‘tautological evidence’. . . .50 Here is a theorem that makes Swinburne’s point for a Bayesian who operates as he does with timeless “inductive probabilities.” For total contingent evidence (e&k), and hypothesis h, P(e|h&k) × P(h|k) P(e&k|h) × P(h) P(h|e&k) = = P(e&k) P(e|k)

Accepting the license of that theorem, Mackie might say: Right, the division of evidence is demonstrably in principle completely arbitrary, though I find it convenient to divide it in this case as it is natural to do, leav- ing the evidence that “wants” to trouble the “prior probability” of theism in the background, while observing that the ways this evidence would, if brought forward, challenge the predictive powers of theism, can be made to challenge instead their “prior probabilities.” For there is not only to conjure with our wonder at the image of a bodiless spirit with intentions of unbound- ed immediate efficaciousness, but also our wonder why, supposing such a spirit, the immediate efficaciousness of our intentions is so severely cur- tailed. It would have been a small blessing to have found ourselves equipped with “natural remotes” when, in early days of television, we wished the sound away from commercials.51

2.3.

However, even if that response were open to Mackie on the basis of the recently displayed theorems, it is not open to me on that basis, given that I work with subjective probabilities, and so am challenged by the problem of 50. Swinburne, The Existence of God, 2nd ed., 17 (emphasis added); cf. “[A]ny division of evidence between e and k will be a somewhat arbitrary one” (Swinburne, The Existence of God, 2nd ed., 67 [emphasis added]) and Swinburne, “Introduction,” in Bayes’s Theorem, 10. 51. Cf. Hume: “There seem to be four circumstances on which depend all or the greatest part of the ills that molest sensible creatures. . . . None of them appear to human reason in the least degree necessary [for a ‘very perfect’ being]. . . . [T]he third circumstance [is] . . . the great fru- gality with which all powers and faculties are distributed to every particular being” (Hume, Dialogues Concerning Natural Religion, Part 11 [163–5]). 266 PHILOSOPHIA CHRISTI old evidence.52 If P is a Kolmogorov probability function that measures my credences at a time, and e is certain evidence for me at this time, so that P(e) = 1, then, for every hypothesis h, P(h|e) = P(h). My way with this problem is to assess hypotheses by consulting what— “beforehand,” that is, before one had any inkling of the evidence they would explain (for which condition please see note 19 above)—would have been one’s “likelihoods” of this evidence on them,53 and what would have been one’s “prior probabilities” for them. That, in my subjective Bayesian view, means that it cannot matter in principle, how evidence is divided between what is to be explained and background information, only if the following is true: it is necessary, at least for an ideally rational and highly opinionated person whose total evidence is (e&k), that if P(e∨k) would be his probability function if he had no inkling “either way” of either e or k (and so, interme- diate confidence in each), and Pe would be his probability function if he had no inkling “either way” of e, then P(e∨k)(e&k|h) × P(e∨k)(h) P(e∨k)(e&k|h) × P(e∨k)(h) P(e∨k)[h|(e&k)]= = =Pe(h|e) P(e∨k)(e&k) P(e∨k)(e&k)

52. Cf. John Earman, Bayes or Bust: A Critical Examination of Bayesian Confirmation Theory (Cambridge, MA: MIT Press, 1992), chapter 5. Objective Bayesians do not have a problem with “old evidence.” This is an important part of James Hawthorne’s argument for Bayesians needing, in addition to measures of subjective degrees-of-belief, measures of objective degrees-of-support. “Bayesians have tried to apply [the former] to the realm of hypothesis confirmation. . . . This . . . leads to the problem of old evidence. . . . [Objective] degree of support [probabilities] . . . supplements [subjective] degree of belief [probabilities] in a way that resolves the problems of old evidence” (James Hawthorne, “Degree-of-Belief and Degree-of-Support: Why Bayesians Need Both Notions,” Mind 114 [2005]: 277). What makes Hawthorne an “objective Bayesian” is not that he advocates conditional probabilities to measure degrees-of-support that is not defined in terms of an unconditional measure of degrees of belief. So far “subjective Bayesians” are prepared to go, as in Jordan Howard Sobel, “On the Significance of Conditional Probabilitie,” Synthese 109 (1996): 311–44, and Richard Jeffrey, Subjective Probability: The Real Thing (Cambridge: Cambridge University Press, 2004). What distinguishes Hawthorne is that he maintains that Bayesians need to presume an objective measure of degrees-of-support. Objective Bayesian Swinburne does not have a problem with “old evidence.” He observes that “On the account that I have given . . . of whether and how far evidence e makes logically probable a hypothesis h, it is irrelevant whether e was known before he was formulated and perhaps taken account of in the formulation of h, or whether h was formulated first and then e was discovered. . . . the timeless view which I endorse [makes] such considerations . . . irrelevant” (Swinburne, Epistemic Justification, 220). 53. “Is the world considered in general and as it appears to us in this life, different from what a man or such a limited being would, beforehand, expect from a very powerful, wise, and benevolent Deity?” (Hume, Dialogues Concerning Natural Religion, Part 11 [163]). The “limited being” is a being of “limited intelligence” who “is not antecedently convinced of a supreme intelligence, benevolent, and powerful” (Hume, Dialogues Concerning Natural Religion, 162). JORDAN HOWARD SOBEL 267

I do not know that this is necessary, and that it is for me demonstrable that the division of evidence is in principle quite arbitrary. The following suggests that it may be. Suppose there was a time tk when a person “learned” k in the sense of becoming, to remain, certain that k, and a later time te when he “learned” e; that this person had exactly these two “learning-experiences”; and that his probability functions for times before tk, from tk to “just (e∨k) before” te, and from te to “now” are, respectively, those functions P and Pe, and the function P.

–––––––––––– tk ––––––––––– te ––––––––––– “now” P(e∨k) Pe P Then 0 < P(e∨k)(e&k) < 1, Pe(k) = 1 and 0 < Pe(e) < 1, and, assuming “rigidity” of the probability of h on e, and of h on (e&k) throughout the period of this timeline, it follows that P(h) = Pe(h|e) = P(e∨k)[h|(e&k)].

3. On What to Make of “Evidence of Fine-tuning”54

3.1.

Swinburne writes that some have argued that the fine-tuning of our universe can be explained by a theory that there are many universes . . . and each generates many other universes with universes with different laws of nature and boundary conditions from its parent, in consequence of which it is to be expected that at least one universe would be fine-tuned.55 He adds that I acknowledge that this theory would need very special characteristics. That was not my intention. What I say about theories like this one is that as explanations of why our universe is fine-tuned they have hardly gotten started, since they predict only that there should be at least one fine- tuned universe amongst very many universes the rest of which, for all these theories say, we should expect not to be fine-tuned. While making likely that there is at least one fine-tuned universe, these theories do not make especially likely that our universe should be fine-tuned. It is as with hands of bridge.

54. The issue for most of us is what to make evidence of metaphorical reports of observations and calculations of frontline physical cosmologists, reports framed in terms of “as if fine- tuning” of something or other for life. Millennial editions of the argument from design are pro- foundly different from classical arguments in the recondite character of their evidence, and in the obscurity of exactly what it is that a great designer would do. Fashioning eyes, even no- handed, makes easy sense in a way in which “establishing the fundamental constants of the standard quantum model” does not (and in the way in which “arranging for the boiling point of water at sea level to be 105° centigrade” does not). 55. Richard Swinburne, “Sobel on Arguments from Design,” 233–4 (emphasis added). 268 PHILOSOPHIA CHRISTI

On the hypothesis that there have been bridge-hands from time immemorial it is (if we get the numbers right) in every one’s reflective view likely that the hand I hold should have been dealt a number of times, but it is not especially likely that it should have been dealt to me this time.56

3.2.

The trouble with such theories of universes out of universes as an explanation of the evidence of fine-tuning, is that they are not an evolutionary for this evidence. What are needed for that are posits of processes of relevant random variation and selection, and, if universes fine-tuned for life are to be likely, long lines of descent. Lee Smolin has made a theory with these ele- ments. According to it, “universes” or “regions of space-time”57 begin in big bangs that are black holes of previous universes,58 and, Smolin postulates, while the basic laws of universes are constant and “the standard model of particle physics describes the world both before and after” these super-cosmic events,59 the values of the fundamental constants can change, though “these changes are small and random.”60 Smolin brings to that speculation argued propositions concerning black holes, light, and life. 56. It is also, as said on page 260 of Logic and Theism, as with the “old Epicurean hypothesis” somewhat revised: Instead of supposing matter infinite, as Epicurus did; let us suppose it finite. A finite number of particles is only susceptible of finite transpositions [since this is not plain- ly true, finite transpositions is best cast as part of the hypothesis]: And it must hap- pen, in an eternal duration [of random-chance trials from all possible order or position], that every possible order or position [and every sequence of these] must be tried an infinite number of times. (Hume, Dialogues Concerning Natural Religion, Part 8 [143]) This theory makes [mathematically] certain that at infinitely many times there should be appearances of design, but it does not make appearances of design especially likely: It entails “that every possible order of position must be tried an infinite number of times” [Hume, Dialogues Concerning Natural Religion, Part 8 (143)]. . . . it does not make it especially likely that there should be order and appearances of design throughout the interval of time that happens to be our time. The likelihood of appearances of design at a time on this hypothesis, including our time, is presumably (I am guessing at details of the hypothesis) 1/f, where f is the finite number of transpositions of which the finitely many particles are susceptible. (Sobel, Logic and Theism, 260) That likelihood could not be greater than the likelihood beforehand of appearances of design on the hypothesis of a finite duration that is our time. 57. Lee Smolin, The Life of the Cosmos (Oxford: Oxford University Press, 1997), 93. 58. “Previous” according to the theory, which includes that “time does not end in the cen- ters of black holes, but continues into some new region of space-time” (Smolin, The Life of the Cosmos, 63). 59. There can, it seems, be no evidence against this postulate. 60. Smolin, The Life of the Cosmos, 94. So the matter that falls into a black hole is not com- pressed to a completely lawless “state” from which it “bangs.” The “trajectory” of its “bang” is, JORDAN HOWARD SOBEL 269

The centre of [Smolin’s case] . . . is the sketch of a demonstration that the range of the nineteen parameters which determine universes with high fitness [that is, propensity for producing black holes] is extreme- ly narrow . . . the distribution of values of the parameters is highly peaked. . . . [Smolin] argues further . . . that the values of the parameters that are conducive to black holes are also conducive to life.61 Add to this the posit that ancestral lines of the vast majority of universes are enormously long, and the result, for any given universe (including ours), is a high probability, conditional on this informed and imaginative evolutionary multiverse hypothesis, that that universe should be finely tuned for black holes and thus, incidentally, for living things, to the good fortune of its living things if any.

3.3.

Now comes, as promised, to context a “dumb-luck” remark of mine, and to wrap up, a paragraph, lightly edited and tweaked, from Logic and Theism, chapter 7, section 8.7. Some say that the appearance of fine-tuning that makes life possible testifies to a pro-life Tuner: FT. Others say that it testifies to this cosmos’s being just one amongst many each of which is “tuned” by a chance mechanism that makes every possible tuning equally probable: MU. Given a choice between just those possibilities, with Earman I would accept neither, for I place hardly any credence in either. My probabilities for them sum to very little. Given especially the How- problems of FT that would beforehand severely curtail its inherent plausibility [“prior probability”], and given the easy consistency of MU with “the tumbling spectacle” that would make a mystery of a pro-life Tuner’s motivation (supposing it benign) and depreciate the likelihood of this life on MU, I think that I would place beforehand find MU more likely on the then only possible evidence of fine tuning. But I am not sure, and it does not matter whether I am right about that this soul-searching, since, to adapt Philo’s words, “it is [to my mind] a thousand, a million to one, if either . . . be the true system (Hume, Dialogues Concerning Natural Religion, Part 8, [143]). Were I persuaded that the apparent fine-tuning of which I hear talk cannot be grounded in a “deeper theory,” I would put down the appearances of fine-tuning to our dumb luck. Otherwise, supposing a worked out deep theory of fine-tuning of this cosmos, say a theory when it comes to the random determination of the fundamental constants, somewhat constrained. 61. Abner Shimony, “Can the Fundamental Laws of Nature Be the Results of Evolution?” in From Physics to Philosophy, ed. J. Butterfield and C. Pagonis (Cambridge: Cambridge University Press, 1999), 216–8. 270 PHILOSOPHIA CHRISTI

along the lines of Smolin’s evolutionary many cosmoi theory, were not only more plausible than FT and MU, as in my view Smolin’s theory is by far, but actually acceptable, which is to say, more probable than not, as to in my present, perhaps insufficiently informed, view his theory is not; I would accept its explanation of the facts of fine-tuning of this place, and put that its fundamental assumptions down to that luck. After all, not everything can be explained, not everything can have a reason,62 and a good “deep theory” of “parameters for life,” could very well be a place at which comprehensible and credible reasons and explanations for them run out, and we have only luck to thank for its fundamentals.63

Appendix: J. M. Keynes For and Against “Logical Probability”

Most of us in our unphilosophical moments and unmathematical moments think that there are objective truths about whether such-and- such evidence makes such-and-such a hypothesis very probable, or only fairly probable or very improbable. . . . When this evidential (or epistemic) probability is understood as measuring the objectively correct degree of evidential support, I shall call it logical (or inductive) probability. Detailed modern explications of evidential probability began with J. M. Keynes’s A Treatise on Probability (1921). Keynes supposed that (at any rate often and approximately) there are true values for the extent to which one proposition makes another one probable, and so his account is . . . of logical probability.64

A.1. Useful Quotations and Comments by Donald Gillies on Keynes’s Ideas

Probability as a logical relation: “Inasmuch as it is always assumed that we can sometimes judge directly that a conclusion follows from a premiss, it is no great exten-

62. This is, by the argument of Logic and Theism, chapter 6, if not everything is metaphysically necessary. 63. Sobel, Logic and Theism, 284–5. 64. Swinburne, “Introduction,” in Bayes’s Theorem, 4. In a footnote, Swinburne continues: “‘Logical probability’ is not a fully satisfactory name. . . . I do not assume that the value of every such probability . . . is a ‘truth of logic.’ For there may be a priori truths which are not truths of logic.” “Keynes started work on probability in 1906 when he was in the India office, and devoted most of his intellectual energy to it for the next five years until the book was nearly completed” (R. B. Braithwaite, foreword to A Treatise on Probability, xv). JORDAN HOWARD SOBEL 271

sion of this assumption to suppose that we can sometimes recognise that a conclusion partially follows from, or stands in a relation of probability to a premises.” (Keynes, [A Treatise on Probability,] 52) . . . “We are claiming . . . to cognise correctly a logical connection between . . . propositions which we call our evidence . . . and . . . suppose ourselves to know . . . , and . . . conclusions . . . to which we attach more or less weight according to the grounds supplied . . . . It is not straining . . . words to speak of this as the relation of probability.” ([Keynes, A Treatise on Probability,] 5–6). So a probability is the degree of a partial entailment. One immediate consequence of this approach is that it makes all probabilities conditional. “No proposition is in itself either probable or improbable . . . .” ([Keynes, A Treatise on Probability,] 7) “. . . if a knowledge of h justifies a rational belief in a of degree α, we say that there is a probability-relation of degree α between a and h.” ([Keynes, A Treatise on Probability,] 4) Here Keynes makes the assumption that if h partially entails a to degree α, then given h [that is, given h as one’s “total knowledge”] it is rational to believe a to degree α.65

When once the facts are given . . . which determine our knowledge, what is probable or improbable [and what degree of belief . . . it is rational to entertain] . . . has been fixed objectively, and is independent of our opinion.” (Keynes, [A Treatise on Probability,] 4) . . . He means objective in the Platonic sense . . . . He [Keynes] . . . writes: “The perception of some relations of probability may be outside the powers of some or all of us.” ([Keynes, A Treatise on Probability,] 18) . . . . We can see . . . the influence of G. E. Moore [Cambridge col- league, fellow Apostle] . . . . [who] argued that good was a non-natural property which could be known only by intuition.66 Also, as Moore says that the intrinsic value of a thing was dependent on its intrinsic nature, Swinburne and Keynes say that relations of probability between propositions and determined by their contents. And as Moore

65. Donald Gillies, Philosophical Theories of Probability (London: Routledge, 2000), 30–1. For the first quotation from Keynes (52), cf. Keynes, A Treatise on Probability, 16: “Speaking somewhat loosely we may say that, if our premisses make the conclusion certain, then it follows from the premisses; and if they make it very probable, then it very nearly follows from them.” 66. Gillies, Philosophical Theories of Probability, 33. “It may be perceived that I have been much influenced by W. E. Johnson, G. E. Moore, and Bertrand Russell, that is to say by Cambridge . . .” (John Maynard Keynes, preface to A Treatise on Probability [1921; London: Macmillan, 1973]). Keynes wrote the preface from King’s College, Cambridge, on May 1, 1920. 272 PHILOSOPHIA CHRISTI implies that intrinsic natures make things good and such as ought to exist,67 so Swinburne and Keynes say that contents of possible evidence and propositions makes these probable to a degree on that evidence and such as ought to be believed to that degree given this evidence.

A.2. “And Why Not?”

A . . . fundamental criticism of Mr. Keynes’ views . . . is the obvious one that there really do not seem to be any such things as the probability relations he describes. . . . I do not perceive them, and if I am to be persuaded that they exist it must be by argument; moreover I shrewdly suspect that others do not perceive them either, because they are able to come to so very little agreement [about them]. . . .68 A.2.1. Additional to Ramsey’s “no-see-um,” and persistent disagree- ments, objections, and arguments against Keynes’s objective probability relations, there is a Mackiean argument from what would be the queerness of these relations of partial entailment and objective support that would have constrain conditional credences. The queerness is, I think, compounded by— though Keynes and Swinburne do not say exactly so—what “we” think are necessary affects of apprehensions of these objective relations of propositions on a person’s subjective conditional credences. Apprehending that p makes q probable to degree n—or that p makes q more probable than, say, r—would of course entails believing that this relation between p and q is so. That is no more unremarkable than that seeing that it is raining entails believing that this is so. What is remarkable is the connection that would obtain between apprehensions of objective conditional probabilities and a person’s subjective conditional probabilities. Believers in objective inductive probabilities—believers in real probabilities presupposed in ordinary probability thought and talk—are apt to take this connection as unremarkable, but it most certainly is not. “We” think—we take for granted in ordinary probability thought and talk—that a view of an “inductive probability” is at once an impression of an entirely objective relation between propositions, and a reflection of this perceived relation in a conditional credence of the subject. It would be somewhat as Prichard says of “ought”-relations: “To feel [appreciate, recognize, apprehend] I ought to pay my bills [for example] is to be moved towards pay- ing them.”69 (There is no evidence in “Does Moral Philosophy Rest on a 67. G. E. Moore, Principia Ethica, section 13. 68. Frank P. Ramsey, “Truth and Probability,” in The Foundations of Mathematics and Other Logical Essays (London: Routledge and Kegan Paul, 1930), 161, quoted in Gillies, Philosophical Theories of Probability, 52. 69. H. A. Prichard, “Does Moral Philosophy Rest on a Mistake?” in Readings in Ethical Theory, 2nd ed., ed. W. Sellars and J. Hospers (Englewood Cliffs, NJ: Prentice-Hall, 1970), 90. JORDAN HOWARD SOBEL 273

Mistake?” that Prichard distinguished there between feeling,70 appreciating,71 recognizing,72 and apprehending73 obligations.) So “seeing” (apprehending, appreciating) that as a matter of objective fact q is, conditionally on p, probable to a degree would be feeling, conditionally on p, that degree of confidence in q. That would be some fact there. That would be an “irresistibly prescriptive objective for there.” That would be if anything stranger than the “objectively prescriptive fact” of “objective norms and value”!74 Perhaps Prichard’s dictum needs to be restricted to apprehensions of “oughts” by rational persons who are not clinically depressed and emotionally dead. Not so of this parallel for apprehensions of objective probability which would be right without restriction: rational or not, depressed or not, to see that q is probable on p to some degree would be to have a subjective probability for q on p to that degree, and similarly for seeing that q is more probable on p than is r. “We should hesitate to postulate that this strange concept [of objective value] has any real instantiations, provided that our inclination to use it can be explained adequately in some other way.”75 I agree in chapter 15, and say the same now about the concept of objective logical/inductive probability.76 There is Mackie contends a better Humean explanation according to which objective values are creatures of processes of projection and objectification. However: This projection or objectification is not just a trick of individual psychology . . . there is a system in which the sentiments of each person both modify and reinforce those of others; the supposedly objective moral features both aid and reflect this communication of sentiments, and the whole system of thought of which objectification, the false belief in the fictitious features, is a contributing part, flourishes partly because . . . it serves a social function.77 I agree with this too: there is, in chapter 15, section 9.4.2,78 a Bayesian assessment of objective intuitionist, and Humean projective-objectification, explanations of the evidence of common use of the concept of objective value, that finds the latter the better explanation. And I believe that a similar- 70. Ibid., 88, 90, 94. 71. Ibid., 90, 91, 92, 96. 72. Ibid., 91, 92. 73. Ibid., 91, 96. 74. Cf. J. L. Mackie, Ethics: Inventing Right and Wrong (New York: Penguin, 1977), 35. 75. J. L. Mackie, The Miracle of Theism: Arguments For and Against the Existence of God (Oxford: Oxford University Press, 1982), 239; cf. Mackie, Ethics, 42. 76. Sobel, “Modus Ponens and Modus Tollens for Conditional Probabilities, and Updating on Uncertain Evidence.” 77. J. L. Mackie, Hume’s Moral Theory (London: Routledge and Kegan Paul, 1980), 71–2 (emphasis added). 78. Sobel, “Modus Ponens and Modus Tollens for Conditional Probabilities, and Updating on Uncertain Evidence.” 274 PHILOSOPHIA CHRISTI ly superior Humean account can be given of our inclination to use the concept of objective probability, and to believe in relations between contents of propositions answering to it. “To be continued.” A.2.2. There are arguments against the reality of these objective relations of “inductive probability,” notwithstanding that ordinary talk and thought of the probable, absent disclaimers, presupposes their reality. Keynes writes: In the ordinary course of thought and argument, we are constantly assuming that knowledge of one statement, while not proving the truth of a second, yields nevertheless some ground for believing it. We assert that we ought on the evidence to prefer such and such a belief. We claim rational grounds for assertions which are not conclusively demonstrated. . . . And it does not seem on reflection that the information we convey by these expressions is wholly subjective. When we argue that Darwin gives valid grounds for our accepting the theory of natural selection, we do not simply mean [emphasis added] that we are psychologically inclined to agree with him; it is certain that we also intend [emphasis added] to convey our belief that we are acting ration- ally in regarding his theory as probable. We believe [emphasis added] that there is some objective relation between Darwin’s evidence and his conclusions, which is independent of the mere fact of our belief, and which is just as real and objective, though of a different degree, as that which exist if the argument were as demonstrative as a syllogism. We are claiming [emphasis added], in fact, to cognise correctly [to “see”] a logical connection between one set of propositions which we call our evidence and which we suppose ourselves to know, and another set which we call our conclusions, and to which we attach more or less weight according to the grounds supplied by the first.79 Keynes’s remarks here are I think largely correct “conceptual analyses” of ordinary talk and thought of the probable, especially disputational talk and thought, as in courts of law, absent disclaimers of objective intent such as “in my view” and “doesn’t it seem so to you too.” However, what Keynes says next, “It is this type of objective relation between sets of propositions . . . to which the reader’s attention must be directed,”80 presupposes a non sequitur, soon to be explicitly affirmed, namely, “[b]etween two sets of propositions, therefore, there exists a relation, in virtue of which, if we know the first, we can attach to the latter some degree of rational belief. This relation is the sub- 79. Keynes, A Treatise on Probability, 6–7. Cf.: “Imagine what . . . science would be like if the likelihoods were [perceived to be] highly subjective” (Hawthorne, “Degree-of-Belief and Degree-of-Support,” 287). “[I]magine how the discussion might go at a scientific conference where a new experimental or observational result O is first announced. . . : Great news. . . . This . . . changes my confidence in H1 quite a bit since my most recent likelihood for O on H1 ... was . . . = .9 while . . . [on its] strongest competitor, H2, was . . . = .2. . . . How about you [‘is it good for you too’]? How likely did you take O be on H2? . . . was O perhaps more likely on H2 than on H1 for you?” (Hawthorne, “Degree-of-Belief and Degree-of-Support,” 301). 80. Keynes, A Treatise on Probability, 6–7. JORDAN HOWARD SOBEL 275 ject-matter of the logic of probability.”81 In my view, the reality of matters regarding disputational talk and thought of probabilities is not this usually intended “subject matter,” and the beginning of a correct “ontological,” not “conceptual,” analysis of this talk and thought, is, to adapt the first sentence of Mackie’s Ethics,82 that there are no objective probabilities.83 A.2.3. It is salutary that [t]he only publication of Keynes after 1921 which had a bearing upon his theory of probability was his comment on Ramsey’s 1926 paper ... posthumously published in The Foundations of Mathematics and Other Logical Essays (1931). . . . Keynes [in his review of the book in 1931] . . . is prepared “as against the view which I had put forward” to agree with Ramsey that “probability is concerned not with objective relations between propositions but (in some sense) with degrees of belief.”84 No one had heard in 1931 of Mackie’s “have it both ways” error-theory. It is, I think, a fair bet that Keynes would have embraced such a theory of thought and talk probability in contexts of confirmation and explanation, especially disputatious contexts. Swinburne has heard of this way of his friend with objective values, but has not seen its analogue for objective probabilities to be an option worth mentioning, preliminary to giving reasons for considering it to be an inferior option for explaining the “play” of objective probabilities in thought and talk of evidence and confirmation, and, in a word, of “probability.”

On Gödel’s Ontological Proof

I have claimed that “the best and smallest change that would obviate [both the problem of modal collapse engendered by principles of Kurt Gödel’s axiomatic proof, and philosophic problems with the worshipfulness of necessary beings is to] stop counting necessary existence as a positive property that a ‘God-like’ being properly so termed would have, give up on the idea of ontological arguments, and concede that no worshipful being could be [logically or metaphysically necessary].”85 Robert Koons argues that a better response to the problem of “modal collapse” is to restrict the

81. Ibid. (emphasis added). 82. Mackie, Ethics, 15. 83. Similarly in my view for “objective chances” and “chance-connections,” the truth of which would have them “go away” with the “causes,” “powers,” and “necessary connections” David Hume sent off in section 8, “Of the Idea of Necessary Connection,” Enquiry Concerning the Human Understanding. 84. Braithwaite, foreword to A Treatise on Probability, xxii. 85. Sobel, Logic and Theism, 135. 276 PHILOSOPHIA CHRISTI domain of properties in Gödel’s axiomatic proof in a manner that does not detract from his axioms for “positive” properties, nor compromise lines of the necessary instantiation of “Godlikeness.” He cautions, however, that this is not to say that this simple change yields a flawless proof for the necessary instantiation of “Godlikeness,” since it leaves the serious problem that Anthony Anderson and I have overlooked that “we have no reason to accept Axiom 5 [that ‘necessary existence’ is a positive property] unless we already believe that all positive properties (including [Gödelian Godlikeness]) are necessarily instantiated.”86 Koons makes the interesting observation that replacing Axiom 5 with Axiom 6, the Anselmian principle that a property is positive only if the property of having it essentially or necessarily is positive, would preserve the validity of Gödel’s argument. However, he adds, this would not improve the argument, since Axiom 6 runs into the same problem: it too “presupposes . . . that every positive property . . . is instantiated of necessity, [and so, amongst other things] . . . just what [Gödel’s] ontological argument was supposed to establish.”87 I offer, in section 1, elaboration of the business of “the collapse,” including Gödel’s personal relation to it, and then, in sections 2 through 4, responses to the problem Koons has with Axiom 5 in the proof, comments on his alternative Gödelian proof that replaces Axiom 5 with Axiom 6 in an alternative Gödelian proof, and responses to the problem he has with Axiom 6 in this proof.

1. On the Modal Collapse in the System of Gödel’s Ontological Proof

“Photocopies of three handwritten pages titled ‘Gödel’s Ontological Proof’. . . began to circulate in the early 1980s. The handwriting is Dana Scott’s; the ideas are Kurt Gödel’s. They agree with ideas conveyed in two pages of notes in Gödel’s own hand dated 10 February 1970. . . .”88 Here, for ready reference, are axioms, definitions, and theorems in the notation of Dana Scott’s notes, in order of their appearances in these notes. (Axiom 1) P(~φ) ↔ ~P(φ): “good half,” P(~φ) → ~P(φ); “bad half,” ~P(φ) → P(~φ) 89 (Axiom 2) P(φ) & □∀x[φ(x) → ψ(x)] → P(ψ) (Theorem 1) P(φ) → ◊∃xφ(x)

86. Robert C. Koons, “Sobel on Gödel’s Ontological Proof,” Philosophia Christi 8 (2006): 242. 87. Ibid., 243. 88. Sobel, Logic and Theism, 115. 89. “Bad” according to Anthony C. Anderson, “Some Emendations of Gödel’s Ontological Proof,” Faith and Philosophy 7 (1990): 291, on a “moral/aesthetic” interpretation; for example, it is plausible that neither being not-red-all-over, nor being not-not-red-all over (that is, being JORDAN HOWARD SOBEL 277

(Def. G) G(x) ↔ ∀φ[P(φ) → φ(x)] (Axiom 3) P(G) (Axiom 4) P(φ) → □P(φ) (Def. Ess) φEssx ↔ φ(x)& ∀ψ[ψ(x) → □∀y[φ(y) → ψ(y)]] (Theorem 2) G(x) → GEssx (Def. NE) NE(x) ↔ ∀φ[φEssx → □∃xφ(x)] (Axiom 5) P(NE) (Theorem 3) □∃xG(x) Each principle is intended as short for the necessitation of a universal closure of it. Primitives “P,” “G,” “Ess,” and “NE” are for (1) positiveness, a property of properties—abbreviation, P: χ is a positive property; (2) Godlikeness, a property of individuals—abbreviation(s), G: a is Godlike (and Godlikeness); (3) being an essence of, a specially defined relation of a property to an individual—abbreviation, Ess: χ is an essence of a; and (4) necessary existence, a specially defined property of individuals—abbreviation(s), NE: a has the property of necessary existence (and Necessary Existence).90

1.1.

Given the generous interpretation of properties in evidence in the system of Gödel’s ontological proof of 1970, and elsewhere in his work (particular- ly, in “Russell’s Mathematical Logic,” more of which below), according to which “anything is counted as a property which can be defined by ‘abstraction on a formula’ [in which no more than one variable is free],”91 it is a further theorem of the system, derived in Logic and Theism, that every truth is a necessary truth. (Theorem 9) (Q → □Q) red-all-over) is positive in this sense, though the “bad” half of Axiom 1 says that one or the other of these properties in positive. The interested reader can find a problem for the “good” half of Axiom 1 and 2 (which entail Theorem 1), Def. Ess, Axiom 3, the “bad” half of Axiom 1, and any interpretation of “positive” such that neither being not-red-all-over, nor being not-yellow- all-over is positive in the sense of this interpretation. A related problem can be found for any interpretation of “positive” such that in its sense tautological properties (e.g., being either red or not red) are not positive: it is a consequence of the corollary of Axiom 1 (whole) that there is at least one property that is positive, and Axiom 2, that tautological properties are all positive. 90. “G” occurs in Gödel’s principles in both predicate or adjectival positions, and term of nominal positions. Similarly for “NE.” 91. Anderson, “Some Emendations of Gödel’s Ontological Proof,” 292. What is more clearly in evidence in Gödel’s writings is the idea that for any formula φα in which exactly one vari- 278 PHILOSOPHIA CHRISTI

This theorem is rigorously derived in section A1 of the appendix to an extended online version of this92 from, (Theorem 2) ∀x[G(x) → GEssx], (derivable from the “bad half” of Axiom 1 and Axiom 4, using Def. G, and Def. Ess) and (Theorem 3) □∃xG(x), (derivable from Theorem 1 [which is derivable from “good half” of Axiom 1 and Axiom 2], Theorem 2, and Axiom 5, using Def. G, and Def. NE) using the principle for property-abstractions, Properties. Every formula, □∀β(Πα[F](β) ↔ F') wherein α is an individual variable, β a term, F a formula, and F' is a formula that comes from F by proper substitution of β for α. Theorem 9 entails a “collapse of modalities.” It entails that propositions can be divided into two kinds: ones that are possible, true, and necessary, and ones that are impossible, false, and not necessary.

1.2.

My reaction in Logic and Theism at once to this logical problem of modal collapse, and to a certain philosophical problem, is to delete Axiom 5, though I mention in a footnote that a “solution that is specific [to the logical problem] would consist in confining the essence of a thing to its ‘intrinsic’ properties.”93 I see Axiom 5 as bearing primary responsibility only for the philosophical problem, which is that of the possibility of a properly-termed- “Godlike” being’s existing of necessity in the way of numbers, Platonic Forms, and propositions do, for a properly-termed-“Godlike” being would be “properly worshipful,” and a necessary condition for that is, surely, being reachable by words and gestures of worship.94 It would be, for the reason sug-

┌ ┐ able α is free, Πα[φα] names a property, the property that is had at a world w by something named by β if and only if the sentence φβ that comes from φ by proper substitution of β for α is true at w. The greater liberality of “my” property-conception is not exploited in my derivation of Theorem 9. 92. This version is linked to “On Logic and Theism,” http://www.scar.utoronto.ca/~sobel/On L_T, which is linked to my home page. 93. Sobel, Logic and Theism, 561n20. 94. Cf. J. N. Findlay: The religious frame of mind . . . desires the Divine . . . both to have an inescapable character . . . and also the character of “making a real difference.”. . . if God is to satisfy religious claims and needs, he must be a being in every way inescapable, One whose existence and whose possession of certain excellences we cannot possibly conceive away. . . . It was indeed an ill day for Anselm when he hit upon his famous proof. For on that day he not only laid bare something that is of the essence of an JORDAN HOWARD SOBEL 279 gested, absurd to worship a number or a merely Ideal Person,95 and similarly, one may philosophically fear, for any necessary being. It has been said that “in logic one needs a robust sense of reality” (Bertrand Russell). I don’t know about that. But I do think that in philosophy, especially in the philosophy of religion, it is good to have a robust sense of the absurd, even if, for broadly Pascalian reasons, this is not necessarily so in life.

1.3.

To avoid its engendering a modal collapse while preserving its validity, Koons confines the range of second order quantifiers in Gödel’s argument to “intrinsic properties,” “properties that are qualitative and nonrelational.”96 and adds axioms for the property of properties of being intrinsic sufficient, he says, “to make his proof work.”97 A simpler and more conservative solution designed by Srećko Kovač would restrict the axiom-schema Properties to properties abstracted from formulas in which no subformulas are closed or contain free variables: this prevents Pickwickian extrinsic properties such as being self-identical and such that snow is white, that is, Πx[x = x&P], from doing their nasties in deductions from the axioms and definitions used in Gödel’s ontological

adequate religious object, but also something that entails its necessary non-existence. (J. N. Findlay, “Can God’s Existence Be Disproved?” Mind 57 [1948]: 182) I am comfortable speaking for myself of necessary conditions for being properly worshipful, but not, with Findlay and others with conditions that would tend to make an object properly worshipful, since being properly worshipful would be being an object that was objectively worthy of worship, a being that objectively ought to be worshipped by all, and, for Mackiean reasons I do not believe in “the possibility of a being who would be objectively worthy of worship” (Sobel, Logic and Theism, 25; cf. 404). 95. Cf. Sobel, Logic and Theism, 137. For an illustration of “the absurd,” Thomas Nagel has offered “[declaring] your love over the telephone to a recorded announcement” (Nagel, “The Absurd,” Journal of Philosophy 68 [1971]: 718.) I know what he means. I think I may love Melanie who helps me add money to my Pay-as-You-Go plan. I mean, “What a voice! And such patience!” For another illustration, there is Jack Benny’s, “I’m thinking,” said in response to hold-up guy’s, “Well?” There has been a long pause since the guy’s, “Your money or your life!” 96. Robert C. Koons, “Sobel on Gödel’s Ontological Proof,” 239. 97. Ibid. Koons writes: “There is nothing in Gödel’s argument that rules out this interpretation of his second-order variables” (Koons, “Sobel on Gödel’s Ontological Proof,” 239). In fact, Gödel himself uses the “property” of being non-self-identical, a property that nothing has, in his deduction of Theorem 1, though there is another way, limned by Koons, that does not. Koons’s way with this small problem is to stipulate that impossible properties count as intrinsic. He continues: 280 PHILOSOPHIA CHRISTI proof.98 He writes that restricting Properties in this way (that is, restricting the λ-introduction rule in this manner) is an “(easy) way to block the unrestricted necessitations of sentences of the system [Theorem 9], since Sobel’s proof (the only one known) . . . is dependent . . . on a λ introduction that leaves a closed subformula within the scope of [the] λ operator.”99

1.4.

Kovač’s nice syntactical restriction, though it targets for exclusion not all predicates for “nonintrinsic” properties (for relational or nonqualitative properties, the properties explicitly excluded by Koons’s metaphysical restriction),100 but only predicates for Pickwickian properties, would not have recommended itself to Gödel as a solution to the modal collapse of his 1970

To make his proof work under this interpretation, we need only the following properties of the set of intrinsic properties: (IN1) If F is intrinsic, so is ~F. (IN2) The conjunction of a set of intrinsic properties is itself intrinsic. (IN3) Everything has at least one intrinsic property in every world (satisfied if the property of being self-identical counts as intrinsic), and an impossible (such as being non-self-identical) counts as intrinsic.... These are quite plausible assumptions. Furthermore, Gödel’s axioms make perfect sense under this new interpretation [of properties they are about; it is in this sense that they are sufficient “to make his proof work under this interpretation”: these axioms for intrinsicness play no roles in deductions of its theorems]. . . . Under this interpretation, Sobel’s modal collapse proof does not go through, since being such that snow is white is not a property (under the intended interpretation) [it is not among “properties that are qualitative and nonrelational”]. (Koons, “Sobel on Gödel’s Ontological Proof,” 239–40 [emphasis added]) I suppose that being such that snow is white is not an “intrinsic” property of mine, say, though it is less obvious that it is neither a qualitative nor a nonrelational property of mine. More needs to be said, I think, for it to be quite clear that Koons’s metaphysical restriction on properties rules out the peculiar “properties” that collapse modalities in Gödel’s system given his presumably unrestricted syntactical characterization of property-abstraction terms. 98. A Godlike being would have such a property for every truth P. A Godlike being would have all and only positive properties: Definition of Godlikeness, and Theorem 4 of Logic and Theism. Every positive property, since Godlikeness is necessarily instantiated (Theorem 3), is itself necessarily instantiated. So, if such properties are “covered” by Properties, it follows that, Theorem 9, if it is true that P, then it is true that it is necessary that P. 99. Srećko Kovač, “Some Weakened Gödelian Ontological Systems,” Journal of Philosophical Logic 32 (2003): 569. 100. That would be an advantage of Kovač’s restriction for traditionalists who would like being the creator of the world to be part of Gödelian “Godlikeness.” Another advantage is that it clearly rules out of Gödel’s system the “properties” I use to prove that modalities collapse in it. JORDAN HOWARD SOBEL 281 proof, unless he had by then either changed his mind about kinds of properties to which he was committed in 1944, or he was prepared to move away from the idea of “essences of individuals” as “total properties of individuals” (compare to Leibniz’s “complete individual concepts”).101 In his contribution to the Schilpp volume for Bertrand Russell, Gödel indicates assumptions he considers available for a “collapsing argument” that says that there is exactly one fact or true proposition, and exactly one falsehood (“Gödel’s slingshot” [discussed in section 14 of chapter 8, “Bertrand Russell’s Theory of Descriptions”]). One “assumption” he makes is that “every proposition ‘speaks about something,’ i.e., can be brought to the form φ(a).”102 Stephen Neale suggests, to illustrate a general device, that Gödel could say that the proposition expressed by “all men snore” can be brought to the form of “Clinton is an x such that all men snore.”103 One would expect Gödel to say that the proposition expressed by that sentence is about Clinton, and that it ascribes a property to Clinton, perhaps the property of being self-identical while all man snore (!): this would-be property could be named by “Πx[x is self-identical & all men snore].” Kovač’s restriction on Properties would make this property-abstraction “inert” in derivations.

1.5.

At least in 1944, Gödel, if he had wanted a solution to the modal collapse that we see as plaguing his axiomatic proof, would have wanted a different one from Koons’s or Kovač’s. “If he had wanted a solution to that collapse?” Well, yes, for there is strong evidence that Gödel thought his principles entailed this collapse, and was not bothered by this, that he indeed considered it a welcomed feature of the Leibnizian metaphysics he was develop- ing for himself.104

101. Koons’s “emendation” would, in Gödel’s view, include such rethinking of “essence,” if, in Gödel’s view, the Koonsian intrinsic properties of God would not be all of God’s properties, as they would not be on any ordinary understanding of His properties. God would, on a common view, have the properties of creating (a relation) these (demonstrative pronoun) things around us, and of loving (another relation) these creations, which properties would presumably not fall into “[t]he class of intrinsic properties” which, Koons explains, is “the class of properties that are qualitative and nonrelational: that pertain or fail to pertain to a thing because of its internal constitution” (Koons, “Sobel on Gödel’s Ontological Proof,” 239 [emphasis added]). 102. Kurt Gödel, “Russell’s Mathematical Logic,” in The Philosophy of Bertrand Russell, ed. P. A. Schilpp (Evanston, IL: Northwestern University Press, 1944), 129. 103. Stephen Neale, “Grammatical Form, Logical Form, and Incomplete Symbols,” in Definite Descriptions: A Reader, ed. Gary Ostertag (Cambridge, MA: MIT Press, 1998), 778. 104. Cf. Robert Adams, “Introductory Note to *1970,” in Kurt Gödel, Collected Works, vol. 3, Unpublished Essays and Lectures (Oxford: Oxford University Press, 1995), 400–1. 282 PHILOSOPHIA CHRISTI

1.5.1. To this evidence. On an undated page in a notebook begun it seems after April 1946 and before 1955105 titled “Ontological Proof” he writes, “If φ(x) ⊃ Nφ(x) is assumed [as following from the essence of x], then it is eas- ily provable that for every compatible system of properties there is a thing, but that is the inferior way. Rather φ(x) ⊃ Nφ(x) should follow first from the existence of God.”106 Gödel seems to have thought that it follows from things’ essences being their “complete properties,” that if a thing has a property, then it is necessary that it has it, □∀φ∀x[φ(x) → □φ(x)]. If he thought that, he was mistaken: all that follows in this direction from the identification of a thing’s essence with its “complete property” is that if a thing has a property, then it is necessary that, if it exists, it has this property. □∀φ∀x(φ(x) → □[E!x → φ(x)]) follows from, (1) (Def.Ess) □∀φ∀x(φEssx↔φ(x)& □∀ψ[ψ(x)→□∀y[φ(y)→ψ(y)]), given axioms for essences, (2) (Essences [everything has at least one]) □∀x∃φ φEssx, and (3) (Essentiality of Essences) □∀φ∀x[φEssx → □[E!x → φ(x)]), of which Gödel would approve.107 1.5.2. Join that mistake which Gödel may have made with the thought “that every proposition ‘speaks about something,’ i.e., can be brought to the form φ(a),”108 expressed thus, that □∀P∃x∃φ□(P ↔ φ(x)), and one has a short way to Theorem 9, □∀Q(Q → □Q),109 that does not traffic in any principles of the ontological proof of 1970 other than Def. Ess, and that makes no use of property-abstractions. Furthermore, given Theorem 9, there is a short way, with Theorem 1 (that comes from the “good half” of Axiom 1 and Axiom 2), and Axiom 3, to Theorem 3, □∃xGx. Neither the way to Theorem 9 of the previous paragraph, nor this way on to Theorem 3, uses any of Axiom 4, Axiom 5, Def. NE, and Def. G. (A4 of the appendix to the extended online version of this, for which please see note 92, contains a confirming derivation.) Gödel would have said this is an “inferior way.” He would prove first the existence of God, and then somehow infer from it the collapsing principle [φ(x) → □φ(x)]. (Strange!)

105. Kurt Gödel, Collected Works, vol. 3, Unpublished Essays and Lectures (Oxford: Oxford University Press, 1995), 429. 106. Ibid., 435 (brackets and emphasis in original). 107. A2 of the appendix to the extended online version of this, for which please see note 92 above, contains a confirming derivation. 108. Gödel, “Russell’s Mathematical Logic,” 129n, commented on above. 109. “Q” is here a proposition-variable; propositions are here functions from possible worlds to truth values; UD, UI, and EG are valid for proposition generalizations. Section A3 of the appendix cited in note 92 contains a confirming derivation. JORDAN HOWARD SOBEL 283

We can be generous to Leibniz and not say that he made the mistake of thinking that from his doctrine of the complete individual concepts of things that if a thing has a property, then it is necessary that this thing has property, that is, that there is in every possible world a counterpart of this thing that has this property. We can say that he saw, and was bothered by, that this doctrine entails that individuals are specific to possible worlds and that when human beings act could not logically have acted oth1erwise. The main present point, however, is not that Gödel seems to have made a mistake of “the logic of essences” that we should not say that Leibniz made, but that Gödel was comfortable, as Leibniz most certainly would not have been, with the universal necessity that that mistake would have entailed for him given his idea of a thing’s essence, and view that every proposition “can be brought to the form φ(a).”110

2. Koons against Axiom 5

2.1.

[T]here is a fatal flaw in Gödel’s argument. . . . The flaw concerns Axiom A5, the positivity of necessary existence. . . . Gödel’s necessary existence [defined thus, (Def. NE) NE(x) ↔ ∀φ(φEssx → □∃xφ(x)),] is provably equivalent to . . . the condition of being “contingency free” [defined thus, (Def. CF) CF(x) ↔ ∀φ(φ(x) → □∃xF(x)),] . . . . on the assumption, which Sobel rightly endorses, that everything necessarily has at least one essence.111 There is a derivation in section A5 of the appendix to the extended online version of this (for which please see note 92 above) of, (Logical Equivalence112 of NE and CF) □∀x[NE(x) ↔ CF(x)].

110. Gödel, “Russell’s Mathematical Logic,” 129n. Appendix A of chapter 6 of Logic and Theism includes a note on page 231 and following, on Leibniz and his doctrine of complete individual concepts. 111. Koons, “Sobel on Gödel’s Ontological Proof,” 241–2. 112. A discrepancy between my modal semantics and what we should say is the semantics of Gödel for this “ontological proof ” prevents me from saying with Koons that these properties are “necessarily coextensive” (Koons, “Sobel on Gödel’s Ontological Argument,” 242). I identify properties in an interpretation with functions that assign to worlds subsets of its universal domain, and make names rigid designators, while allowing a name’s designatum at a world not to exist in this world or be a member of its domain: whether or not the designatum of a name 284 PHILOSOPHIA CHRISTI

The derivation uses necessitations of universal closures of Def. NE and Def. CF, and axioms for essence (Essences) □∀x∃φ φEssx, and, (Logical Equivalence of Essences of a Thing) □∀φ∀ψ∀x(φEssx & ψEssx → □∀x[φ(x) ↔ ψ(x)]), of which Gödel would approve. “So NE is positive if and only if CF is.”113 The necessitation of that, □[P(NE) ↔ P(CF)] is an easy consequence of the Logical Equivalence NE and CF and (Axiom 2) □∀φ∀ψ[P(φ) & □∀x[φ(x) → ψ(x)] → P(ψ). (Section A6 of the appendix to the extended online version of this, for which please see note 92 above, contains a confirming derivation.) Now, edging closer to the problem Koons sees for Axiom 5, P(NE), we find that Koons writes: “whether CF . . . [is] positive depends on whether it is true that all positive properties are necessarily instantiated.”114 Why is that? It is because (backing up in Koons’s text): CF is the property of having only necessarily instantiated properties. ... CF is equivalent to the infinite conjunction of . . . the . . . comple- ments of those properties that are not necessarily instantiated (that is, that are possibly uninstantiated). CF is positive if and only if none of the properties that are possibly uninstantiated are themselves positive. If, instead, there is a positive property F that is not necessarily instantiated, then CF entails not having F, which would make CF a negative property (any property that entails not having some positive property must itself be negative).”115 These emphasized things are true, in Gödel’s system. exists in a world of every property and its negation this designatum will have one of them at this world. Gödel should be said to identify properties with functions that assign to worlds subsets of their domains of existents. There is no evidence in the texts directly relevant to Gödel’s thoughts about ontological proofs concerning his policy for names in modal logic. We agree in the range we assign to individual quantifiers at a world: we make their range the domain of existents at this world. One consequence of this difference is that their properties are necessarily “existence- entailing,” while mine need not be. Under my semantics something can have a property at a world, though this property is not instantiated in this world: that “F(A)” is true at a world does not entail that “∃xFx” is true at this world. Another consequence is that logically equivalent properties are for them, but not for me, necessarily coextensive. 113. Koons, “Sobel on Gödel’s Ontological Proof,” 242. 114. Ibid. 115. Ibid. (emphasis added). JORDAN HOWARD SOBEL 285

That “any property that entails not having some positive property must itself be negative,” (P(ψ) & □∀x[φ(x) → ~(ψx)]) → ~P(φ), follows rather obviously from Axiom 1 (the good half), ∀φ[P(~φ) → ~P(φ)], and Axiom 2, ∀φ∀ψ[P(φ) & □∀x[φ(x) → ψ(x)] → P(ψ)], and from nothing less in Gödel’s system. There is a derivation in A7 of the appendix to the extended online version of this, for which please see note 92 above, for the principle that “whether CF . . . [is] positive depends on whether it is true that all positive properties are necessarily instantiated,”116 P(CF) → ∀φ[P(φ) → □∃xφ(x)], from the same two axioms of Gödel’s system. I think that no derivation of it from less of Gödel’s system as explicitly developed in Scott’s notes is possible. Are these things true “outside” Gödel’s system and its axioms that partially define “positive”? And now to the problem Koons sees for Axiom 5, P(NE), given that “whether CF (and NE) are positive depends on whether it is true that all positive properties are necessarily instantiated.” The problem, given that, is that “[i]f some positive property is possibly uninstantiated, then CF and NE are clearly themselves negative. Thus, we have no reason to accept Axiom 5, unless we already believe that all the positive properties (including of course G) are necessarily instantiated. We have no reason to accept Axiom 5 unless we [already] know that God exists necessarily.”117

2.2.

To respond on Gödel’s behalf, first, that “all positive properties are necessarily instantiated”118 is not a consequence of Axiom 5 together with definitions and natural assumptions concerning Gödelian essences, and nothing substantial about, or partially definitive of, positive properties. That all positive properties are necessarily instantiated has only been implicitly shown by Koons to be a consequence of the conjunction of Axiom 5 with that of the good half of Axiom 1 and Axiom 2 (see the remark headed, “These emphasized things are true, in Gödel’s system,” in the previous section). However, Koons’s result is not without interest, since it means that Theorem 3, □∃xGx, can be reached with less axiomatic baggage than Gödel, as constructed by Scott, uses: Theorem 3 is evidently derivable from the conjunction of the good half of Axiom 1, Axiom 2, Axiom 3, and Axiom 5. Second, even if Axiom 5 did, without other Gödelian assumptions about positive properties, “presuppose” or entail that all positive properties are nec- 116. Ibid. 117. Ibid. 118. Ibid. 286 PHILOSOPHIA CHRISTI essarily instantiated, it would not follow that no one could have a reason for accepting this axiom unless he already believed or knew that God exists necessarily. John Findlay, in an “ontological disproof of the existence” of God, wrote: we are led irresistibly, by the demands inherent in religious reverence; to hold that an adequate object of our worship must possess its various qualities in some necessary sense. . . . we can’t help feeling that a worthy object of our worship can never be a thing that merely happens to exist. . . . his own non-existence must be wholly unthinkable.119 Axiom 5, that Gödelian necessary existence is a positive property, entails that a Godlike being who possesses all and only positive properties, satisfies this necessary condition for worshipfulness of Findlay’s. It is a condition for worshipfulness endorsed at least implicitly by all Anselmian perfect-being theists. If we understand by positiveness in the present context, being a property that either tends to make worshipful or is necessary to being worshipful,120 then Findlay had this Anselmian reason for counting necessary existence as positive: he had this reason when he did not “already believe or know that God exists necessarily,” for he had it when he was engaged in a disproof of God’s existence: he had it when writing an essay titled, “Can God’s Existence Be Disproved?” in which essay he argues yes to its question. And, third and last, though the good half of Axiom 1 is secure on any interpretation of “positive,” the same cannot be said of Axiom 2. It is a consequence of this axiom that, if there is at least one positive property, then every property that is necessarily instantiated by everything, such as the tautological properties of being either Godlike or not Godlike, and being either red or not red, is a positive. That strikes me as “seriously wrong” on a “moral/aesthetic” interpretation, and, more importantly, on a “spiritual/religious” interpretation such as the one recently floated.121 This negative opinion of mine is contrary to Koons’s attitude towards Axiom 2 (“not to men-

119. J. N. Findlay, “Can God’s Existence Be Disproved?” 51, 52. 120. In a context improved for theological purposes, there would be axioms for “negative” properties as well as “positive” properties, that did not equate being negative (positive) with being not positive (negative), and “Godlikeness” would be defined as having every positive, and no negative, property. In this context, “positiveness” (“negativeness”) could be interpreted as tending to make “worshipful” (“unworshipful”). Under these interpretations, absences of properties necessary for being worshipful (or equivalently presences of properties sufficient for being unworshipfulness) would be limiting cases of negative properties. 121. Having some property can be “necessary,” in the sense intended, to being worshipful only if it is a property that not everything of which one can speak necessarily has. Being either red or not red is a property that everything of which one can speak has. It is a property that even impossible things such as, an opponent of Anselm’s argument might say, a being that which no greater being can be conceived. Intuitively, tautological properties should be neither “positive” nor “negative” in a context such as described in note 111 above in which being positive (negative) is not the same as being not negative (positive). JORDAN HOWARD SOBEL 287 tion” the bad half of Axiom 1): he writes that “there is nothing seriously wrong with Axioms 1–4.”122 It is true that such “disjunctive properties” that featured elementary properties would be positive on the logical/ontological interpretation that Gödel floats according to which a property is positive if its “disjunctive nor- mal in terms of elementary properties contains a member without negation.”123 This, however, tells against the gloss that Gödel offers on “pure ‘attribution’ as opposed to . . . containing privation,”124 if as he implies, “being and goodness are [in his view] convertible.”125

3. A6 to the Rescue for a “Minimalist Gödelian Proof”

3.1.

Persuaded that Axiom 5 is, on pain of begging the question, not available to Gödel, Koons writes: . . . without Axiom 5, Gödel’s ontological proof is unsuccessful. There is, however, a simple repair that might do the job: replace Axiom 5 with Axiom 6: (A6) P(F) → P(□F) If a property F is positive, then so is the property of being F in every possible world. [Proceeding with Axiom 6 in place instead of Axiom 5] since Godlikeness is positive, it follows that being Godlike in every possible world is also positive.126 Positive properties are always possibly instantiated, so being necessarily Godlike is possibly instantiated. In S5, it follows that Godlikeness is necessarily, and thus also actually, instantiated.127 There is a derivation in section A8 of the appendix to the extended online version of this, for which please see note 92, that spells out this infor- mal deduction of □∃xG(x), with the aid of an axiom regarding positive properties that is not made explicit by Gödel, or for him by Scott. The derivation is from,

122. Koons, “Sobel on Gödel’s Ontological Proof,” 242. 123. Gödel, Collected Works, 3:404. 124. Ibid. (italics in original). 125. Koons, “Sobel on Gödel’s Ontological Proof,” 236. 126. Let being necessarily φ be being φ at every possible world. If φ is existence-entailing for possible things (things that are instantiated in some worlds), so that something has φ at a world only if it exists at this world, then being necessarily φ is equivalent to existing necessarily and being essentially φ,(□E!x & □[E!x → φ(x)]). 127. Koons, “Sobel on Gödel’s Ontological Proof,” 242–3. 288 PHILOSOPHIA CHRISTI

(Axiom 6) P(φ) → P[Πx(□[φ(x)])],128 (Axiom 3) P(G), (Theorem 1) □[P(φ) → ◊∃xφ(x)] (which is derivable from Axiom 2, together with the “good half” Axiom 1, or instead of this, David Johnson has observed, the principle that not every property is positive,129 ~∀φP(φ), which is a corollary of the “good half” of Axiom 1), and the unstated until now axiom, (Godlikeness Is for Possible Things Existence-Entailing [G/Poss/EE]) □∀x□(Gx → E!x), of which Gödel would approve (for my semantics). Why not simply □∀x(Gx → E!x)? Because that can go without saying in my semantics, since in it individual universal quantifiers range, at a world, over precisely the domain of this world’s existents: for every property φ, □∀x(φx → E!x). In my semantics, properties of an interpretation are functions that assign to worlds subsets of its “universal domain” which need not be the union of its world domains. In this scheme things can have properties at worlds in which they do not exist; indeed things that exist in no worlds—impossible things— can have properties at worlds.130 In my semantics, (G/Poss/EE) says of every x that exists in any world—that is, of every possible thing x—that at every world w, x has G at w only if x exists in x.131 “Not every property can make that claim” (in my semantics). The derivation, which is in section A8 of the appendix to the extended online version of this, for which please see note 92 above, uses, (Properties) Every formula, □∀β(Πα[F](β) ↔ F') wherein α is an individual variable, β a term, F a formula, and F' is a formula that comes from F by proper substitution of β for α.

128. While Axiom 6 plays no role in Gödel’s ontological proof of February 10, 1970, or in Scott’s notes, in entries in a notebook on the project of ontological proof Gödel did more than merely endorse it with reference to that project: he wrote that “[t]hat the necessity of a positive property is positive is the essential presupposition for the ontological proof ” (Gödel, Collected Works, 3:404 [emphasis added]). 129. David Johnson, Hume, Holism, and Miracles (Ithaca, NY: Cornell University Press, 1999), 99. 130. The case is otherwise for Gödel, given what we should say is his implicit semantics. If, as I suggest in note 110 above, he make properties functions that assign to worlds subsets of their domains of existents, and also (for a new observation) identify universal domains of interpretations with the union of the domains of existents of its worlds, then the possibility envisioned of something that does not exist at any world’s having G at some world cannot arise. 131. More explicitly, “□∀x□(Gx → E!x)” says that, for every possible world, w, and for each thing, x, that exists in w, at every possible world, w', it is true at w' that Gx, only if it is true at w' that x exists. JORDAN HOWARD SOBEL 289

It could use a version of Properties restricted to formulas in which no subformulas are closed (Kovač’s restriction).

3.2.

That is, in the resources it draws from Gödel’s system, a very economi- cal Gödelian proof. As observed, it can get Theorem 1 from Gödel’s Axiom 2, and Johnson’s Principle that not every property is positive, ~∀φP(φ): a derivation is given in section A9 of the appendix cited in note 92. It does not need either half of Gödel’s Axiom 1, P(~φ) ↔ ~P(φ). Nor does it need either his Axiom 4, P(φ) → □P(φ), or his Axiom 5, P(NE). It does not presuppose his definitions—it does not presuppose any definitions—of “essence” or of “necessary existence.” And, perhaps most remarkable, it makes the question of the nature of “Godlikeness” open to detailing according to one’s idea of what would make a being most worshipful, subject only to the conditions on “positiveness” of Johnson’s principle, and Axioms 2 and 6. “But surely that ‘minimalist’ proof is too good to be true. Evidence of at least that can be found in the possibility of a parallel demonstration that Devil-likeness, D, is necessarily instantiated from analogues of Johnson’s Principle and the rest for analogues of the polar opposite, N, of ‘positivity’ whatever one’s idea of positivity is.” I think this is right, and recommend for the reader’s consideration that the principal fly in the ointment of this “minimalist” proof for Gods and Devils is Gödel’s appealing-to-the-logically- minded Axiom 2, at which a finger was pointed towards the end of section 2.2 above.132

4. Koons against A6

Does A6 suffer from exactly the same flaw as A5? Unfortunately, the answer is Yes. If there is a positive property F that is possibly uninstantiated, then A6 will fail in that case, since [(i)] in that case □F or, more precisely, xˆ□F(x), (being F in every possible world), will be an impossible property, and so [(ii)] negative rather than positive. Thus, A6 presupposes that every positive property (including the conjunction of all of them) is instantiated of necessity, but this is just what the ontological argument was supposed to establish.133

132. Suppose one’s idea of a “positive” property is that of property that would tend to make a thing worthy of worship. Then to conjure with in the “minimalist” system of this section, aug- mented by analogues of its principles for “N” standing for the polar opposite of this “positiveness,” and “D” standing for Devil-likeness, would be the disjunctive property (G∨D) and the theorems thereof, P(G∨D) and N(G∨D). 133. Koons, “Sobel on Gödel’s Ontological Proof,” http://www.scar.utoronto.ca/~sobel/OnL _T/Koons_SobelonGoedel.pdf. Cp. Koons, “Sobel on Gödel’s Ontological Proof,” Philosophia Christi, 243. 290 PHILOSOPHIA CHRISTI

There is in section A10 of the appendix cited in note 92 a derivation of Koons’s (i), that if there is a positive property φ that is possibly uninstantiated, [P(φ) & ◊~∃xφ(x)], then the property Πx[□φ(x)] of being “necessarily φ” (that is, of being φ at every possible world) is “an impossible property” (that is, not possibly instantiated property), ∀φ([P(φ) & ◊~∃xφ(x)] → ~◊∃yΠx[□φ(x)]y), from the unstated until now axiom, (Positive Properties Are for Possible Things Existence-Entailing) □∀φ(P(φ) → □∀x□[φ(x) → E!x]), of which Gödel would approve for my semantics. See the paragraph in the previous section on (G/Poss/EE). However, it is plain that Koons’s (ii), according to which from the property Πx[□φ(x)]’s being not possibly instantiated it follows that this property is not positive, depends on Theorem 1. So it has not been shown that Axiom 6 presupposes or entails, without aid of other principles for positiveness, that every positive property is instantiated of necessity, ∀φ[P(φ) → □∃xφ(x)]. If it did, then it, Axiom 6, would all by itself entail Theorem 1, □∀φ[P(φ) → ◊∃xφ(x)]. The most that the argument stated establishes is that the conjunction of Axiom 6 and Theorem 1 entails that ∀φ[P(φ)→□∃xφ(x)], and that thus the conjunction of Axiom 6, Theorem 1, and Axiom 3, P(G), entails the object of Gödel’s exercise, Theorem 3, □∃xGx.

5. Weighing A5 and A6 If, as seems, Axiom 6 needs less help from Gödel’s axioms and theorems than does Axiom 5 for a proof of Theorem 3, Axiom 6 may be felt to be closer to something like question-begging than Axiom 5.134 That could encourage an ontological reasoner to go back to Axiom 5, and the idea that Necessary Existence as defined by Gödel is a positive property in a moral/aesthetic sense, or better for the object of the exercise, in the spiritual sense of tending to make worshipful. Though we may expect an ontological reasoner to hold on to the substance of Axiom 6, whether or not he uses in his proof. Axiom 6 says, for example, that if being good is a positive property then being essentially good, or necessarily good, is if anything “more so.” The substance of Axiom 6 has been prominent in the thinking of ontological reason-

134. Axiom 6 works with Theorem 1 and Axiom 3 for a derivation of Theorem 3. Axiom 5 works with Theorem 1, Theorem 2, and Axiom 3 in an articulation of Scott’s sketch of Gödel’s deduction of Theorem 3, in Logic and Theism on page 150, and, it has been noted recently, works with Axioms 1, 2, and 3. JORDAN HOWARD SOBEL 291 ers from the first one through Gödel.135 To call up again Findlay as my wit- ness, he writes: “we are led on irresistibly, by the demands inherent in religious reverence, to hold that an adequate object of our worship must possess its various qualities in some necessary manner.”136 The substance of Axiom 6 should be a “keeper” for every child of Anselm. With Axiom 5 returned to its proper place in Gödel’s scheme con- servatively emended to avoid modal collapse, Axiom 6 should be securable as a theorem, a confirmatory theorem for anyone of an Anselmian mind.137 Such a person would have all that he sought for worship, he would have a necessarily existing being that had every “positive” property essentially, “a

135. A possible problem for Axiom 6 assuming a “moral/aesthetic” interpretation of positiveness concerns presumptively positive moral properties such as veracity and generosity, and their “essential enhancements.” Could it contribute to a person’s moral perfection that he was not merely perfectly good in acts and intentions, but essentially so, with the consequence that he could not be bad and lie or “under tip,” not if he wanted to, that is, if he had no choice but to be in his conduct perfectly good? Would his being essentially honest and kind be consistent with his being praiseworthy for his honesty and kindness? It would be for a person who was essentially good not only in actions but in intentions, sentiments, and so on, absolutely impossible that he should want to lie or cheat, let alone do these bad deeds. Peter Geach was scandalized by the suggestion that he found in Richard Price’s writing that “it must not be absolutely impossible for God to do something wicked . . . or [else] God isn’t free and isn’t therefore laudable for his goodness” (Geach, “Omnipotence,” Philosophy 48 [1973]: 16). There lurks for Axiom 6 substantially the same problem assuming a “spiritual/religious” interpretation of positiveness according to which it is “worshipful, adorable, deserving-of-gratitude making or enabling,” the question being whether anything could contribute to a being’s worshipfulness, and so on that was for this being logically inescapable. For example, could a being be worshipful and deserving of “cosmic gratitude” thanks in part to the goodness and bounty of its creation, if it is not logically possible that its creation should fail to be good and bountiful, that is, if, as a matter of logical necessity, it had no choice in this matter? Price would presumably say, “No, in that case the being would be neither laudable nor worshipful nor deserving of gratitude for the goodness and bounty of his creation.” But if not for that, then for what could a being be worshipful and so on? 136. Findlay, “Can God’s Existence Be Disproved?” 53. Which, Findlay would have been happy to clarify, is not to say that “we” are led to an object whose acts are essential to it. “We” (I do not include myself) are presumably led to an object that while “inescapable in its existence and qualities” is both actively and passively contingently related to “us.” That is the problem for Findlay. “We” are led to the idea of an object, that though it does not obviously entail a priori a contradiction, he in 1957 (as in 1948) considers to be of a metaphysically or logically impossible object. 137. The main idea for a derivation, that I have not written down for a well-emended system, is that a Godlike being would not only have each positive property φ, but would necessarily have it, and would thus have the property of having it necessarily, Πx[□φ(x)], and have this property necessarily. Since a Godlike being would have only positive properties necessarily or essentially, it should be provable in some number of steps in a well-emended system that therefore, P(φ) → P(Πx[□φ(x)]), which is what we have labelled “Axiom 6.” I think I see roles in such a derivation for additional “just right” axioms for Gödelian essences (I stress Gödelian essences), namely: (Exclusive Essences) □∀φ∀x∀y(φ Ess x & ψ Ess y → x = y); and (Essentiality of Essences) □∀φ∀x[φ Ess x → □(E!x → φ Ess x)]. 292 PHILOSOPHIA CHRISTI being [who was] in every way inescapable.”138 As for me, I have no interest in worshipping anything, and if I did, would find in “a being who was in every way inescapable”—that is, in a necessarily existent being who was essentially everything it was—the last thing I was looking for.

138. Findlay, “Can the Existence of God Be Disproved?” 54; see note 91 above.