Triviality Pursuit Alan Ha ´jek Published online: 12 April 2011 Ó Springer Science+Business Media B.V. 2011 Abstract The thesis that probabilities of conditionals are conditional probabilities has putatively been refuted many times by so-called ‘triviality results’, although it has also enjoyed a number of resurrections. In this paper I assault it yet again with a new such result. I begin by motivating the thesis and discussing some of the philosophical ramifica- tions of its fluctuating fortunes. I will canvas various rea- sons, old and new, why the thesis seems plausible, and why we should care about its fate. I will look at some objections to Lewis’s famous triviality results, and thus some reasons for the pursuit of further triviality results. I will generalize Lewis’s results in ways that meet the objections. I will conclude with some reflections on the demise of the the- sis—or otherwise. Keywords Probabilities of conditionals  Conditional probabilities  Stalnaker’s thesis  Triviality results  Conditionalization  Imaging  Blurred imaging  Maximum entropy  Minimum cross entropy  Boldness  Moderation  Revision rules 1 Setting the Scene Like Jason of the Friday the 13th franchise, the thesis that probabilities of conditionals are conditional probabilities has enjoyed a number of resurrections. Lewis (1976) appeared to have killed it in his famous triviality results; but van Fraassen (1976) resuscitated it. Stalnaker (1976) attacked it some more; but Rehder (1982) revived it. Further triviality results appeared to deliver fatal blows to it (e.g., Lewis 1986a, b; Ha ´jek 1989, 1994; Hall 1994; Milne 2003); yet it lives on still, albeit transmogrified, in the writings of Edgington (1995) and Bennett (2003). In this paper I will assault it yet again; but it will doubtless survive in some form in further sequels. While I think that the various negative and positive results against the thesis are interesting in their own right— and I will offer a negative result of my own—I don’t want this paper to be merely an exercise in theorem-proving. I will begin by motivating the thesis and discussing some of the philosophical ramifications of its fluctuating for- tunes. I will canvas various reasons, old and new, why the thesis seems plausible, and why we should care about its fate. There are good reasons why it keeps making come- backs. I will look at some objections to Lewis’s famous results, and thus some reasons for the pursuit of further triviality results. I will generalize Lewis’s results in ways that meet the objections. I will conclude with some reflections on the demise of the thesis—or otherwise. 2 The Thesis—or Theses ‘‘The probability that Collingwood will win if they are ahead at half time is high.’’ This seems to say that the probability of a certain conditional is high. And it seems to say exactly the same thing as: ‘‘The probability that Collingwood will win given that they are ahead at half time is high.’’ Moreover, the probability of the conditional seems to rise or fall in lockstep with the conditional probability. Symbolically: P ahead ! win ð Þ¼ Pðwin ahead j Þ; where ‘?’ represents the conditional ‘if … then’, and P(win | ahead) is given by the usual ratio formula for A. Ha ´jek (&) Research School of Social Sciences, Australian National University, Canberra, ACT 0200, Australia e-mail: alan.hajek@anu.edu.au 123 Topoi (2011) 30:3–15 DOI 10.1007/s11245-010-9083-2