Branden Fitelson

In this paper, the authors describe their initial investigations in computational metaphysics. Our method is to implement axiomatic metaphysics in an automated reasoning system. In this paper, we describe what we have discovered when the theory of abstract objects is implemented in PROVER9 (a first-order automated reasoning system which is the successor to OTTER). After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in PROVER9's first-order syntax, and (2) how PROVER9 then finds proofs of interesting theorems of metaphysics, such as that every possible world is maximal. We conclude the paper by discussing some issues for further research.

This (brief) note is about the (evidential) “favoring” relation. Pre-theoretically, favoring is a three-place (epistemic) relation, between an evidential proposition E and two hypotheses H1 and H2. Favoring relations are expressed via locutions of the form: E favors H1 over H2. Strictly speaking, favoring should really be thought of as a four-place relation, between E, H1, H2, and a corpus of background evidence K. But, for present purposes (which won't address issues involving K), I will suppress the background corpus, so as to simplify our discussion. Moreover, the favoring relation is meant to be a propositional epistemic relation, as opposed to a doxastic epistemic relation. That is, the favoring relation is not meant to be restricted to bodies of evidence that are possessed (as evidence) by some actual agent(s), or to hypotheses that are (in fact) entertained by some actual agent(s). In this sense, favoring is analogous to the relation of propositional justification — as opposed to doxastic justification (Conee 1980). In order to facilitate a comparison of Likelihoodist vs Bayesian explications of favoring, I will presuppose the following bridge principle, linking favoring and evidential support: • E favors H1 over H2 iff E supports H1 more strongly than E supports H2.1 Finally, I will only be discussing instances of the favoring relation involving contingent, empirical claims. So, it is to be understood that “favoring” will not apply if any of E, H1, or H2 are non-contingent (and/or non-empirical). With this background in place, we're ready to begin

Naive deductive accounts of confirmation have the undesirable consequence that if E confirms H, then E also confirms the conjunction H & X, for any X—even if X is utterly irrelevant to H (and E). Bayesian accounts of confirmation also have this property (in the case of deductive evidence). Several Bayesians have attempted to soften the impact of this fact by arguing that—according to Bayesian accounts of confirmation— E will confirm the conjunction H & X less strongly than E confirms H (again, in the case of deductive evidence). I argue that existing Bayesian “resolutions” of this problem are inadequate in several important respects. In the end, I suggest a new‐and‐improved Bayesian account (and understanding) of the problem of irrelevant conjunction.

Naive deductivist accounts of confirmation have the undesirable consequence that if E confirms H, then E also confirms the conjunction H·X, for any X—even if X is completely irrelevant to E and H. Bayesian accounts of confirmation may appear to have the same problem. In a recent article in this journal Fitelson (2002) argued that existing Bayesian attempts to resolve of this problem are inadequate in several important respects. Fitelson then proposes a new‐and‐improved Bayesian account that overcomes the problem of irrelevant conjunction, and does so in a more general setting than past attempts. We will show how to simplify and improve upon Fitelson's solution.

According to Bayesian confirmation theory, evidence E (incrementally) confirms (or supports) a hypothesis H (roughly) just in case E and H are positively probabilistically correlated (under an appropriate probability function Pr). There are many logically equivalent ways of saying that E and H are correlated under Pr. Surprisingly, this leads to a plethora of non-equivalent quantitative measures of the degree to which E confirms H (under Pr). In fact, many non-equivalent Bayesian measures of the degree to which E confirms (or supports) H have been proposed and defended in the literature on inductive logic. I provide a thorough historical survey of the various proposals, and a detailed discussion of the philosophical ramifications of the differences between them. I argue that the set of candidate measures can be narrowed drastically by just a few intuitive and simple desiderata. In the end, I provide some novel and compelling reasons to think that the correct measure of degree of evidential support (within a Bayesian framework) is the (log) likelihood ratio. The central analyses of this research have had some useful and interesting byproducts, including: (i ) a new Bayesian account of (confirmationally) independent evidence, which has applications to several important problems in con- firmation theory, including the problem of the (confirmational) value of evidential diversity, and (ii ) novel resolutions of several problems in Bayesian confirmation theory, motivated by the use of the (log) likelihood ratio measure, including a reply to the Popper-Miller critique of probabilistic induction, and a new analysis and resolution of the problem of irrelevant conjunction (a.k.a., the tacking problem).

Bayesian epistemology suggests various ways of measuring the support that a piece of evidence provides a hypothesis. Such measures are defined in terms of a subjective probability assignment, pr, over propositions entertained by an agent. The most standard measure (where “H” stands for “hypothesis” and “E” stands for “evidence”) is: the difference measure: d(H,E) = pr(H/E) - pr(H).0 This may be called a “positive (probabilistic) relevance measure” of confirmation, since, according to it, a piece of evidence E qualitatively confirms a hypothesis H if and only if pr(H/E) > pr(H), where qualitative disconfirmation is characterized by replacing “>” with “ “ with “=”. Other more or less standard positive relevance measures that have been proposed are: the log-ratio measure: r(H,E) = log[pr(H/E)/pr(H)] and the log-likelihood-ratio measure: l(H,E) = log[pr(E/H)/pr(E/~H)].

In applying Bayes’s theorem to the history of science, Bayesians sometimes assume – often without argument – that they can safely ignore very implausible theories. This assumption is false, both in that it can seriously distort the history of science as well as the mathematics and the applicability of Bayes’s theorem. There are intuitively very plausible counter-examples. In fact, one can ignore very implausible or unknown theories only if at least one of two conditions is satisfied: (i) one is certain that there are no unknown theories which explain the phenomenon in question, or (ii) the likelihood of at least one of the known theories used in the calculation of the posterior is reasonably large. Often in the history of science, a very surprising phenomenon is observed, and neither of these criteria is satisfied.

mathematicians for over 60 years. Amazingly, the Argonne team's automated theorem-proving program EQP took only 8 days to find a proof of it. Unfortunately, the proof found by EQP is quite complex and difficult to follow. Some of the steps of the EQP proof require highly complex and unintuitive substitution strategies. As a result, it is nearly impossible to reconstruct or verify the computer proof of the Robbins conjecture entirely by hand. This is where the unique symbolic capabilities of Mathematica 3 come in handy. With the help of Mathematica, it is relatively easy to work out and explain each step of the dense EQP proof in detail. In this paper, I use Mathematica to provide a detailed, step-by-step reconstruction of the highly complex EQP proof of the Robbins conjecture.

Let L be a sentential (object) language containing atoms ‘A’, ‘B’, . . . , and two logical connectives ‘&’ and ‘→’. In addition to these two logical connectives, L will also contain another binary connective ‘ ’, which is intended to be interpreted as the English indicative. In the meta-language for L , we will have two meta-linguistic operations: ‘ ’ and ‘ ’. ‘ ’ is a binary relation between individual sentences in L . It will be interpreted as “single premise entailment” (or “single premise deducibility in L ”). ‘ ’ is a monadic predicate on sentences of L . It will be interpreted as “logical truth of the logic of L ” (or “theorem of the logic of L ”). We will not presuppose anything about the relationship between ‘ ’ and ‘ ’. Rather, we will state explicitly all assumptions about these meta-theoretic relations that will be required for Gibbard’s Theorem.