Outline. The argument from so many arguments. Framework. Royall s case. Ted Poston

Size: px

Start display at page:

Download "Outline. The argument from so many arguments. Framework. Royall s case. Ted Poston"

Clement Parsons
6 years ago
Views:

1 Outline The argument from so many arguments Ted Poston University of South Alabama Plantinga Workshop Baylor University Nov 6-8, Measuring confirmation Framework Log likelihood measure 2 The cumulative e ect of the arguments Weak independent evidence Mixed independent evidence 3 Independence Independence 4 Conclusions Conclusions Framework We can make probabilistic comparisons which respect to evidence and hypotheses.(keynes; 1921) The notion of probability is inductive probability. (Maher; 2006, 2010) The only hypotheses that concern us are theism (T) and naturalism ( T) Odds form of Bayes s Theorem: Pr(T e i ) Pr( T e i ) = Pr(e i T ) Pr(e i T ) Pr(T ) Pr( T ) Urn 1 (all white balls) and Urn 2 (equal proportion of white and black balls). An urn is selected at the flip of a fair coin, then a ball is selected at random, its color is recorded and it is placed back in the urn, and the contents are thoroughly mixed.(royall; 1997) Pr(white i Urn1) = 1 Pr(white i Urn2) = 0.5 Pr(Urn1) = Pr(Urn2) = 0.5

2 Pr(white 1 )= 3 4 [Pr(white 1 Urn1) Pr(Urn1)] + [Pr(white 1 Urn2) Pr(Urn2)] Pr(Urn1 White 1)= Pr(White 1 Urn1) Pr(Urn1) (Pr(White 1 Urn1) Pr(Urn1))+Pr(White 1 Urn2) Pr(Urn2)) = 1.5 (1.5)+(.5.5) = 2 3 Pr(Urn2 White 1)= 1 3 White 1 confirms that Urn 1 was selected. Suppose all you learn is that white i and white j. Pr(white i urn1) Pr(white i urn2) =2& Pr(white j urn1) Pr(white j urn2) =2 White i &White j independently confirm that Urn 1 was selected. Pr(White (1,2,...,n) Urn1) Pr(White (1,2,...,n) Urn2) = 1.5 =2 n n LR: Pr(E H) Pr(E H) = n Table: Number of successive white balls (b) corresponding to values of a likelihood ratio (LR) LR b Each draw of white is confirmationally independent regarding an urn hypothesis, but the draws are not unconditionally independent. E.g., Pr(white 2 white 1 ) = Pr(white 2 ). What probabilistic feature of is responsible for the intuitive judgement that w 1 and w 2 are confirmationally independent regarding the hypothesis that Urn 1 was selected? The selection of an urn screens o white on 1 from white on 2. 1 Pr(w 2 Urn1&w 1 )=Pr(w 2 Urn1) 2 Pr(w 2 Urn2&w 1 )=Pr(w 2 Urn2)

3 Log likelihood measure Key features 1 Given a likelihood ratio we can find a corresponding number of white balls. 2 The individual selections are not unconditionally probabilistically independent. 3 The individual selections are probabilistically independent regarding H (and H). 4 A clear case of the evidential value of evidential diversity. For reasons I won t discuss this uniquely picks up the log likelihood measure of confirmation. See (Fitelson; 2001) c(h, e) =log[ Pr(e h) Pr(e h) ] The log-likelihood measure is also additive. Two arguments which are entirely independent, neither weakening nor strengthening the other, ought, when they concur, to produce a[n intensity of] belief equal to the sum of the intensities of belief which either would produce separately. (Peirce; 1878) Likelihood ratios compared to log likelihood ratio 1 # of successive white balls.5 Likelihood ratio (LR) log likelihood ratio (LLR) Positive evidence for theism is an order of magnitude greater on theism than naturalism. For each e i, log[ Pr(e i T ) Pr(e i T ) ]=1 Is each item of evidence for theism individually this strong? A LLR=1 is equivalent to the selection of 3.3 white balls in a row. It s not unreasonable that some items of evidence e.g., fine-tuning, the existence of rational creatures, etc individually has a LLR of 1.(Swinburne; 2004, ) The useful feature of a LLR of 1 is that it s easy to add the cumulative e ect of multiple items of evidence.

4 Successive white balls related to LLRs # of successive white balls log likelihood ratio Table: Comparison of # of independent arguments for theism with # of successive white balls # of independent arguments for theism # of successive white balls Weak independent evidence How powerful is the evidence against theism? Does it take away all the white balls or just some? How much of a di erence is there between the priors? Is the surprise factor that theism is true like the selection 16.6 white balls from Urn 2? (e.g., LLR of 5) Nonetheless, we can say that if a LLR of 1 is correct and there are multiple arguments for theism then the evidential case for theism is strong. Suppose the evidence for theism is independent regarding T but it is only slightly more likely given T than T. Let the likelihood ratio is this: Pr(e i T ) Pr(e i T ) = a = a 9 A LR of 10 9 is very weak evidence. In this case the LLR would be It would take approximately 22 independent items of evidence to equal the confirmatory power of 1 white ball.

5 Other Models Independence This approach to thinking about confirmation makes it easy to sum the strength of independent evidence. Is the individual evidence for theism like the selection of 1 ball, 2 balls, 3 balls, etc. Then just find the corresponding LLR and add to find the cumulative e ect. If there are di erent LLR values for di erent lines of evidence then we can still just add to get a cumulative value and then find the equivalent number of white balls. Finally, we can add in the evidence against theism to get a measure of overall evidential strength. Are the arguments / evidences conditionally independent regarding theism? Definition e 1 and e 2 are (mutually) confirmationally independent regarding H according to c if and only if both c(h, e 1 e 2 )=c(h, e 1 )and c(h, e 2 e 1 )=c(h, e 2 ) (Fitelson; 2001) For each e i, e j does this condition hold? (S) Pr(e i T ) Pr(e i T ) = Pr(e i T &e j ) Pr(e i T &e j ) Independence Conclusions (S) Pr(e i T ) Pr(e i T ) = Pr(e i T &e j ) Pr(e i T &e j ) Are there families of theistic arguments / evidences that meet this condition? The ontological, cosmological, teleological, and moral arguments plausibly meet (S). Multiple design arguments do not meet condition (S) (Dougherty and Poston; 2008) Several of the two dozen or so arguments presuppose intentionality; they won t met condition (S). 1 On relatively weak assumptions theism has an evidential case as strong as the selection of 16 white balls in a row (a LLR of 5). 2 If Plantinga s two dozen arguments are probabilistically independent regarding theism then the evidential case for theism is like the selection of over 80 white balls in a row. 3 The evidential case against theism is easily taken account to yield a final verdict.

6 References Dougherty, T. and Poston, T. (2008). A user s guide to teleological arguments, Religious Studies 44: Fitelson, B. (2001). A bayesian account of independent evidence with applications, Philosophy of Science 68(3): Friedman, M. (1974). Explanation and scientific understanding, Journal of Philosophy 71(1): Keynes, J. M. (1921). ATreatiseonProbability,Macmillan. Maher, P. (2006). The concept of inductive probability, Erkenntnis 65: Maher, P. (2010). Bayesian probability, Synthese 172: Peirce, C. S. (1878). The probability of induction, Popular Science Monthly 12: Plantinga, A. (2007). Two dozen (or so) theistic arguments, in B. Deane-Peter (ed.), Alvin Plantinga, CambridgeUniversityPress,pp Royall, R. (1997). Statistical Evidence: A likelihood paradigm, Chapman and Hall. Swinburne, R. (2004). The Existence of God, 2ndedn,OxfordUniversityPress.

The argument from so many arguments

The argument from so many arguments Ted Poston May 6, 2015 There probably is a God. Many things are easier to explain if there is than if there isn t. John Von Neumann My goal in this paper is to offer