Inductive logic admits a variety of semantics (Haenni et al., 2011, Part 1). This paper develops semantics based on the norms of Bayesian epistemology (Williamson, 2010, Chapter 7). §1 introduces the semantics and then, in §2, the paper explores methods for drawing inferences in the resulting logic and compares the methods of this paper with the methods of Barnett and Paris (2008) . §3 then evaluates this Bayesian inductive logic in the light of four traditional critiques of inductive logic, arguing (i) that it is language independent in a key sense, (ii) that it admits connections with the Principle of Indifference but these connections do not lead to paradox, (iii) that it can capture the phenomenon of learning from experience, and (iv) that while the logic advocates scepticism with regard to some universal hypotheses, such scepticism is not problematic from the point of view of scientific theorising. This section introduces the use of Bayesian epistemology as semantics for inductive logic. The material presented here is based on Williamson (2010) , to which the reader is referred for more details. ¶ Bayesian Epistemology: A Primer. At root, Bayesian epistemology concerns the question of how strongly one should believe the various propositions that one can express. The Bayesian theory that answers this question can be developed in a number of ways, but it is usual to base the theory on the betting interpretation of degrees of belief. According to the betting interpretation, one believes proposition θ to degree x iff, were one to offer a betting quotient for θ-a number q such that one would pay qS to receive S in return should θ turn out to be true, where unknown stake S ∈ R may depend on q-then q = x. This interpretation of degrees of belief naturally goes hand in hand with the claim that, were one to bet according to one's degrees of belief via the above betting set-up, then one shouldn't expose oneself to avoidable losses. In particular, arguably one's degrees of belief should minimise worst-case expected loss. This starting point-the betting interpretation together with the loss-avoidance claim-can then be used to motivate various rational norms that answer the main question facing Bayesian epistemology, i.e., that specify how strongly one should believe the various propositions that one can express. Since the context of this paper is inductive logic, we will be particularly concerned with propositions expressed in a logical language. Suppose then that L n is a propositional language on elementary propositions A 1 , . . . , A n , with SL n the set of propositions formed by recursively applying the usual logical connectives to the elementary propositions. Let Ω n be the set of atomic states of L n , i.e., propositions ω n of the form ±A 1 ∧ · · · ∧ ±A n , where ±A i is either A i or its negation. One norm of rational belief, which we shall call the Probability Norm, says that one's degrees of belief should satisfy the axioms of probability, for otherwise, in the worst case, stakes are chosen that ensure positive expected loss-equivalently, stakes are chosen that ensure positive loss whichever atomic state turns out to be true (a so-called Dutch book). Thus degrees of belief should be probabilities in order to minimise worst-case expected loss: Theorem 1. Define function P : SL n −→ R by P(θ) = a given agent's betting quotient for θ. The agent's bets on propositions expressible in L n avoid the possibility of a Dutch book if and only if they satisfy the axioms of probability: P1. P(ω n )≥0 for each ω n ∈ Ω n , P2. P(τ) = 1 for some tautology τ ∈ SL , P3. P(θ) = ω n |=θ P(ω n ) for each θ ∈ SL . See Williamson (2010, Theorem 3.2) for a proof. This is known as the Dutch Book Theorem, or the Ramsey-de-Finetti Theorem. P1-3 offer one way of expressing the axioms of probability over the propositional language L n : this axiomatisation makes it clear that a probability function is determined by its values on the atomic states of L n . A second norm, the Calibration Norm or Principal Principle, says that one's degrees of belief should be calibrated to known physical probabilities. In particular, if one knows just the physical probability P * (θ) of θ then one should set one's betting quotient for θ to this physical probability, P(θ) = P * (θ), for otherwise in the worst case stakes will be chosen that render the expected loss positive. 1 While in the case of the Probability Norm, minimising worst-case expected loss is equivalent to avoiding sure loss (a Dutch book), in this case the link appears in a long run of bets rather than in the single bet on θ itself: if one were repeatedly to bet on θ-like events with the same betting quotient for each bet, then one would be susceptible to sure loss in the long run (Williamson, 2010, pp. 39-42). Of course in general one might know only of certain constraints on physical probability, rather than individual physical probabilities such as P * (θ). In such a case the Calibration Norm would say that if one knows that the physical probability function P * lies within some set P * of probability functions, then one's belief function P should lie within the convex hull 〈P * 〉 of this set of probability functions. The reason being that if one remains in the convex hull one's bets do not have demonstrably greater than the minimum worst-case expected loss (equivalently, one can't be forced to lose money in the long run), but outside the convex hull one can be 1 For simplicity of exposition we take physical probability P * to be single-case here, defined over the propositions of the agent's language L . But we could instead take physical probability to be generic, defined over repeatedly instantiatable outcomes, and then consider in place of P * the single-case consequences of physical probability, i.e., the constraints that known generic physical probabilities impose on the agent's degree of belief. See Williamson (2011c) on this point. sure of sub-optimal expected loss (equivalently, one can be forced to lose money in the long run). A third norm, the Equivocation Norm, says that one should not adopt extreme degrees of belief unless forced to by the Probability Norm or the Calibration Norm: one's degrees of belief should be equivocate sufficiently between the basic possibilities that one can express (i.e., between the atomic states of L n ). Equivalently, one's belief function P should satisfy the constraints imposed by the other norms and otherwise should be sufficiently close to the equivocator function that gives the same probability to each of the 2 n atomic states, P = (ω n ) = 1/2 n for each ω n ∈ Ω n . (Distance between probability functions is measured by Kullback-Leibler divergence, d(P,Q) = ω n ∈Ω n P(ω n ) log P(ω n )/Q(ω n ).) Again this norm can be justified by minimising worst-case expected loss; the argument goes as follows (Williamson, 2010, p. 63-65). In the absence of knowledge about one's losses, one should take the loss function to be logarithmic by default, i.e., one should assume that one will lose − log P(ω n ) where ω n is the atomic state that turns out true, for such a loss function is the only one that satisfies various desiderata that are natural to impose on a default loss function. But then, under some rather general conditions, the P that satisfies constraints imposed by other norms but minimises worst-case expected loss (the robust Bayes choice of P) is the P that is closest to the equivocator. Note that we need the qualification that degrees of belief should be 'sufficiently' close to the equivocator in order to handle the case where there is no function closest to the equivocator. For example, one might know that a coin is biased in favour of heads, so that P * (H) > 1/2 where H signifies heads at the next toss. Arguably then by the Calibration Norm one ought to believe H to degree greater than 1/2, P(H) > 1/2. But there is no degree of belief greater than 1/2 that is closest to P = (H) = 1/2. Therefore the most one can expect of an agent is that P(H) is sufficiently close to 1/2, where what counts as sufficiently close depends on pragmatic considerations such as the required accuracy of calculations. In sum, the betting interpretation together with the loss-avoidance claim lead to three norms: Probability, Calibration and Equivocation. Bayesian epistemologists disagree as to whether to endorse all these norms. All accept the Probability Norm, most accept some version of Calibration Norm, but few accept the Equivocation Norm. The Probability Norm on its own leads to what is sometimes called strict subjectivism, Probability together with Calibration leads to empirically-based subjectivism and all three norms taken together lead to objectivism. Note that strict subjectivism doesn't imply that determining appropriate degrees of belief is entirely a question of subjective choice, inasmuch as the Probability Norm imposes substantial constraints on which degrees of belief count as rational. Neither does objectivism imply that rational degrees of belief are totally objective in the sense of being uniquely determined-there may remain an element of subjective choice, as in the biased coin example introduced above, where the agent is free to choose between any sufficiently equivocal probability function. In the light of the fact that the three norms have essentially the same motivation, it is hard to argue that this motivation warrants one norm but not another. Accordingly we will accept all three norms in what follows, exploring the consequences of objective Bayesian epistemology for inductive logic. ¶ Predicate Languages. Thus far we have introduced Bayesian epistemology in the context of a propositional language. But the framework extends quite naturally to a predicate language, as we shall now see. Suppose then that L is a firstorder predicate language without equality, with finitely many predicate symbols and with countably many constant symbols t 1 , t 2 , . . ., one for each element of the domain. For n≥1, let L n be the finite predicate language involving only constants t 1 , . . . , t n . Let A 1 , A 2 , . . . run through the atomic propositions of L , i.e., propositions of the form U t where U is a predicate symbol and t is a tuple of constant symbols of corresponding arity. Order the A 1 , A 2 , . . . as follows: any atomic proposition expressible in L n but not expressible in L m for m < n should occur later in the ordering than those atomic propositions expressible in L m . Let A 1 , . . . , A r n be the atomic propositions expressible in L n . An atomic n-state ω n is an atomic state ±A 1 ∧ · · · ∧ ±A r n of L n . Let Ω n be the set of atomic n-states. While the Calibration Norm remains the same as in the propositional case, the Probability Norm needs modification because the predicate language requires an extra axiom of probability: Theorem 2. Define function P : SL −→ R by P(θ) = the agent's betting quotient for θ. The agent's bets on propositions expressible in L avoid the possibility of a Dutch book if and only if P is a probability function: PP1. P(ω n )≥0 for each ω n ∈ Ω n and each n, PP2. P(τ) = 1 for some tautology τ ∈ SL , PP3. P(θ) = ω n |=θ P(ω n ) for each quantifier-free proposition θ, for any n large enough that L n contains all the atomic propositions occurring in θ, and See Williamson (2010, Theorem 5.1) for a proof. PP4, the extra probability axiom, is known as Gaifman's condition. PP1-4 imply that P(∃xθ(x)) = lim m→∞ P m i=1 θ(t i ) and P(∀xθ(x)) = lim m→∞ P m i=1 θ(t i ) . As in the propositional case, a probability function is determined by its values on the atomic n-states, but in this case n varies over the natural numbers. The Equivocation Norm carries over from the propositional case as follows. Define the equivocator by P = (ω n ) = 1/2 r n for all n and ω n . Define the n-divergence between P and Q to be d n (P,Q) = ω n ∈Ω n P(ω n ) log P(ω n )/Q(ω n ). Say that P is closer to R than Q if there is some N such that for all n ≥ N, d n (P, R) < d n (Q, R). The Equivocation Norm says that P should be, from all the functions that satisfy the constraints imposed by the Probability and Calibration Norms, one that is sufficiently close to the equivocator. As in the propositional case it is a pragmatic question as to what counts as 'sufficiently' close to the equivocator. ¶ Inductive Logic. Following Haenni et al. (2011), we will concern ourselves with inductive logics that yield entailment relationships of the form: Here ϕ 1 , . . . , ϕ k , ψ are propositions of some given logical language, and the superscripts X 1 , . . . , X k , Y denote inductive qualities that attach to these propositionscharacterising, e.g., their certainty, plausibility, reliability, weight of evidence, or probability. Such an entailment relationship can be read: if X 1 attaches to ϕ 1 , . . ., and X k attaches to ϕ k then Y attaches to ψ. As to which entailment relationships hold depends very much on the semantics for the entailment relation | ≈. Normally some given semantics will say something like, 'the entailment relationship holds if all interpretations that satisfy the premisses also satisfy the conclusion ψ Y ', and will say what an interpretation is and what it is to satisfy the premisses and satisfy the conclusion. In a probabilistic logic, or progic for short, X 1 , . . . , X k , Y are sets of probabilities and interpretations are probability functions. Still, there are a wide range of semantics one might give for a progic: the standard semantics says that P satisfies θ Z if and only if P(θ) ∈ Z, but other semantics are provided by the theories of probabilistic argumentation (which is closely related to Dempster-Shafer theory), Kyburg's evidential probability, classical statistical inference, Bayesian statistical inference and Bayesian epistemology (Haenni et al., 2011, Part I). Bayesian epistemology provides semantics in the following way. First, one interprets the logical language in which ϕ 1 , . . . , ϕ k , ψ are expressed as the language of an agent. Then one can construe the premisses ϕ as constituting the agent's evidence of physical probabilities: they say that P * (ϕ 1 ) ∈ X 1 , . . . , P * (ϕ k ) ∈ X k . Finally, interpretations are belief functions, which, by the Probability Norm, are probability functions. By the Calibration Norm a belief function P satisfies the premisses iff it lies in the convex hull of all probability functions that satisfy the relevant evidential constraints, i.e., iff P ∈ 〈P * 〉. Now if the premisses are consistent we can simply take P * = {P : But if the premisses are inconsistent we cannot identify P * = : inconsistent premisses tell us not that there is no chance function but rather that there is something wrong with the premisses. In this case, then, some consistency maintenance procedure needs to be employed. The simplest such procedure is to take P * to include any function that lies in a maximal consistent subset of {P : P(ϕ 1 ) ∈ X 1 , . . . , P(ϕ k ) ∈ X k }. Of course, if a set of constraints is consistent then the maximal consistent subset of that set is that set itself, so whether or not the premisses are consistent, we can take P * = {ϕ Y is interpreted as an assertion about rational degree of belief rather than physical probability: P(ψ) ∈ Y . The norms dictate that a belief function that satisfies the premisses also satisfies the conclusion iff P(ϕ) ∈ Y for all those P ∈ 〈 {ϕ Let E denote the set of probability functions that satisfy constraints imposed by the agent's evidence. In the above exposition of the Calibration Norm, E = 〈P * 〉. 2 Let ↓E denote the set of functions in E that are maximally equivocal (this set may be 2 This version of the Calibration Norm presumes that all constraints on degrees of belief are mediated by evidence of physical probabilities. While this is an appropriate assumption in the context of our semantics for inductive logic, where premisses are interpreted exclusively in terms of physical probabilities, this assumption may be violated in other contexts (Williamson, 2010, §3.3). empty, as in the biased coin example above) and let ⇓E denote the set of functions in E that are sufficiently equivocal. Arguably, E1 : ⇓E = . An agent is always entitled to hold some beliefs. E2: ⇓E ⊆ E. Probability functions calibrated with evidence that are sufficiently equivocal are calibrated with evidence. E3: If Q ∈ ⇓E and R ∈ E is no less equivocal than Q then R ∈ ⇓E. I.e., if Q is sufficiently equivocal then so is any function in E that is no less equivocal. If ↓E = then ⇓E = ↓E. If it is possible to be maximally equivocal then one's belief function should be maximally equivocal. E5 : ⇓⇓E = ⇓E. Any function, from those that are calibrated with evidence, that is sufficiently equivocal, is a function, from those that are calibrated with evidence and are sufficiently equivocal, that is sufficiently equivocal. In the case of a propositional language, for example, these conditions imply that: for some , where {P} = ↓〈E〉, i.e., whereP is the unique function that is maximally equivocal from all those in the convex closure 〈E〉 of E. In general, if there are no contextual factors available to determine parameters like that demarcate between sufficiently and insufficiently equivocal, the only option is to set: We shall take this as the default option in the case of the Bayesian semantics for inductive logic. We shall use the symbol | ≈ • to signify this Bayesian entailment relation, which can thus be characterised as follows: ϕ }〉, if there are any maximally equivocal members, or for every member P of 〈 {ϕ Perhaps the best way to get an intuitive feel for the above inductive logic is via a series of examples. In this section we will use examples to demonstrate how inferences can be performed in Bayesian inductive logic, and to compare this logic with a closely related logic. ¶ Categorical Premisses in Propositional Inductive Logic. Consider the following invalid argument in propositional logic: We know the argument is invalid by considering its truth table: The two left-most columns go through all the truth assignments to the atomic propositions, while the next two columns give the corresponding truth values of the premisses and the right-most column gives the corresponding truth value of the conclusion. The argument is invalid because the third truth assignment makes the premisses true but the conclusion false. Now suppose we are faced with the following question: to what extent do the premisses of the above argument entail the conclusion? This can be written: Note that the premisses are categorical in the sense that they are not qualified in any way by some inductive quality. In terms of the Bayesian semantics described above, an unqualified premiss proposition θ implies that P * (θ) = 1, where, as before P * denotes chance or physical probability. Normally, this is all that θ implies. 3 So, from the point of view of Bayesian inductive logic, the question might be written, This asks, given that a → b and b have physical probability 1, what degrees of belief would it be rational to attach to the conclusion a? The Probability Norm says that degrees of belief over the propositional language {a, b} had better be probabilities. Recall that the probability of any proposition is the sum of the probabilities of the atomic states that logically imply that proposition. In particular, This means that we can augment a truth table with the probabilities x i of the atomic states ω n to calculate the probability of any proposition: There are exceptions though, as we shall see in §3. For example, θ might say that the first hundred randomly sampled ravens have been found to be black, in which case not only does θ imply that θ has probability 1 but, arguably, that the physical probability of the next sampled raven being black is close to 1. θ is called simple with respect to P * if it only imposes the constraint P * (θ) = 1. The Calibration Norm says that degrees of belief should satisfy constraints imposed by evidence of physical probability. In the Bayesian semantics for inductive logic, the left-hand side of the entailment relation is interpreted as capturing the available information about physical probability. Since the premiss propositions have physical probability 1, P((a → b) ∧ b) = x 1 + x 3 = 1, so our augmented truth table becomes: Finally the Equivocation Norm, as applied to inductive logic, says that degrees of belief should otherwise equivocate, if possible, between the atomic states. Hence x 1 = x 3 = 1/2. The probability that attaches to the conclusion a is thus x 1 + x 2 = 1/2: Note that this tells us what probability or probabilities attach to the conclusion, given the premisses; it can be written P {a→b,b} (a) = 1/2. It does not tell us the extent to which the premisses increase the probability of the conclusion. This latter concept, degree of support rather than degree of partial entailment, can be explicated as P {a→b,b} (a) − P (a). Now the probability table for P (a) is: Clearly then P (a) = 1/4 + 1/4 = 1/2 and the degree of support is 1/2 − 1/2 = 0. Thus the premisses are irrelevant to the conclusion in this example. 4 Consider another invalid argument: either Miliband will be the next Prime Minister or Cameron will; 5 therefore, Miliband will be the next Prime Minister. To 4 Note that, as a → b and b are simple with respect to P * , P {a→b,b} (a) = P (a|(a → b) ∧ b). See Williamson (2011b) on this point. 5 We shall treat this 'or' as inclusive disjunction, under the assumption that it is possible to have two Prime Ministers in a coalition government. what extent does the premiss entail the conclusion? m ∨ c | ≈ • m ? . Here P(m ∨ c) = 1 so P(¬m ∧ ¬c) = 0 and we have the following probability table: Hence P {m∨c} (m) = 1/3 + 1/3 = 2/3. Note that P (m) = 1/2 so the premiss does support the conclusion, to degree 2/3 − 1/2 = 1/6. ¶ Qualified Premisses in Propositional Inductive Logic. We shall move to some examples of the situation in which the premisses are not categorical but have some inductive quality attached to them. Consider first the question, Here a ∧ ¬b 0.1 abbreviates a ∧ ¬b {0.1} . Suppose, for instance that a is the proposition that it rains today and b is the proposition that it will rain tomorrow. Then a ∧ ¬b 0.1 is interpreted as saying that the chance of it raining today but not tomorrow is 0.1. The question is, how strongly should one believe that it will rain tomorrow, given this information? The probability table is as follows: The probability P(a ∧ ¬b) = 0.1 follows by the Calibration Norm, while the others are due to the Equivocation Norm. Then P {a∧¬b 0.1 } (b) = 0.3 + 0.1 = 0.4. But one might also ask whether, given the chance information, getting rain today supports a forecast of rain tomorrow. One has to be a bit careful here: one cannot simply add a to the premisses because if a were the case then the chance of a ∧ ¬b would be different. (This is more obvious in the case in which the chance information is P * (a) = 0.1: then adding a as a premiss induces the constraint P * (a) = 1 which clearly supersedes the original information.) But one can ask: Here a conclusion of the form b|a Y is interpreted as an assertion about conditional probability, namely P(b|a) ∈ Y . Now P {a∧¬b 0.1 } (b|a) = 0.3/0.4 = 0.75 > 0.4, so getting rain today does indeed support the forecast of rain tomorrow. Note, in contrast, that adding a to the premisses would yield P {a∧¬b, 0.1 a} (b) = 0.9, as can be deduced from the probability table: So the use of conditional probability can yield different results to those derived by augmenting the premisses (Williamson, 2011a). Consider another example. A herder randomly sampled a hundred goats in his herd and found that 80 of them were tetchy. He noticed that 5 of the non-tetchy goats were Angora and that 2 of those had horns. To what extent should he believe that the next horned Angora goat will be tetchy? If we grant the facts of the story and that the given proportions of the sampled goats accurately reflect the proportions in the herd as a whole, then we are faced with the following question: Here we introduce conditional premisses: a premiss of the form θ|ϕ X is interpreted as saying that P * (θ|ϕ) ∈ X , while, as before, a conclusion of the form θ|ϕ Y is interpreted as saying that P(θ|ϕ) ∈ Y . We shall fill in the following probability table: To work out the probabilities of the atomic states, first we articulate the premisses as constraints on atomic states. The first constraint can be satisfied by setting since this constraint treats the atomic states that occur in it symmetrically, and those atomic states are not constrained by the other constraints (c.f., Theorem 5). The second constraint requires that 4x 2 + 4x 6 = x 2 + x 4 + x 6 + x 8 . I.e, 3x 2 + 3x 6 = x 4 + x 8 . The third constraint requires that 5x 2 = 2x 2 + 2x 6 . I.e., 3x 2 = 2x 6 . Substituting the third constraint into the second, 5x 6 = x 4 + x 8 . We can set x 4 = x 8 = x say. Then x 6 = (2/5)x and x 2 = (2/3 × 2/5)x = (4/15)x. We have probability 1/5 to share among h ∧ a ∧ ¬t, h ∧ ¬a ∧ ¬t, ¬h ∧ a ∧ ¬t, ¬h ∧ ¬a ∧ ¬t. So (4/15)x + (2/5)x + x + x = 1/5, i.e., (40/15)x = 1/5 and x = 3/40. So we have: F F F Now we can work out the degree to which the premisses partially entail the conclusion: P(t|h ∧ a) = Note that, in all these examples, the constraints imposed by the premisses are linear constraints on the atomic states. This implies that the set of probability functions satisfying these constraints is convex, and so there is no further need to take convex hulls in order to satisfy the Calibration Norm. In general, satisfying the Calibration Norm is a linear programming problem, and then satisfying the Equivocation Norm is an optimisation problem (minimisation of Kullback-Leibler divergence from the equivocator P = , or, equivalently, maximisation of entropy). As suggested in Haenni et al. (2011) , there is scope to use probabilistic networks (in particular, credal networks) as the inference machinery for probabilistic logic: a credal network can be an efficient way of representing a convex set of probability functions-in our case the set of rational belief functions that are appropriate given the premisses-and there are efficient methods for calculating the set Y of probabilities that these functions give to a conclusion proposition. Note that in the above examples, calibration with the premisses yields a set of probability functions that is closed as well as convex. Consequently there is only one function in this set that is closest to the equivocator and a single probability that attaches to the conclusion proposition (see, e.g., Paris, 1994) . ¶ Quantified Premisses in Predicate Inductive Logic. Consider a predicate language. Let us ask, ∀xθ(x) c | ≈ • θ(t 1 ) ? for some quantifier-free proposition θ and some number c in the unit interval. For instance, if the chance of all ravens being black is 0.7, then how strongly should you believe that raven t 1 is black? Let us use the notation θ n for θ(t 1 ) ∧ · · · ∧ θ(t n ). Then by the Probability Norm, the premiss implies the constraints P(θ n )≥ c for all n, and P(θ n ) −→ c as n −→ ∞. So let us write P(θ n ) = c + x n , where x n ≥ 0 and x n −→ 0 as n −→ ∞ (so P(¬θ n ) = Since the premiss is symmetric with respect to the ω n ∈ [θ n ] n , the Equivocation Norm will give them the same probability: One might think that the most equivocal probability function satisfying these constraints will set x n = 0 for all n sufficiently large (i.e., for all n such that c/|θ n | n ≥ P = (ω n ) = 1/2 r n ), since this function will be closest to the equivocator. However, if c < 1 then setting x n = 0 would yield a function that is not a probability function, in violation of the Probability Norm. Consider, for example, a language with a single unary predicate U, and let θ(x) be U x: then |θ n | n = 1 so if x n = 0 for sufficiently large n we would have for such n, in violation of axiom PP3 of §1. Similar reasoning can be used to show that if c < 1 then x n > 0 for all n. Note that if θ is tautologous then for the premiss to be consistent, c must be 1 and so x n = 0 for all n. In order to calculate x k for some fixed k, consider n≥ k and: is U x for unary predicate U then 2 r n /|θ n | n −→ ∞ and To consider another toy example, suppose that the chance of all men being mortal is 2 3 ; to what extent should one believe that Socrates is mortal given that he is human? I.e., where H stands for Human, M for Mortal and s for Socrates. We have then that P(Hs → Ms) = 11/12, so P(Hs ∧ Ms) = P(¬Hs ∧ Ms) = P(¬Hs ∧ ¬Ms) = 11/36, and P(¬(Hs → Ms)) = P(Hs∧¬Ms) = 1/12. Therefore, P(Ms|Hs) = P(Hs∧ Ms)/(P(Hs ∧ Ms) + P(Hs ∧ ¬Ms)) = 11/14: Let us consider one related example. Suppose that 80% of sampled individuals are men (i.e., human). To what extent does the proposition that all men are mortal entail the conclusion that Socrates, a randomly sampled individual, is mortal? Assuming that the frequency information warrants the estimate P * (Ht i ) = 4/5 for each sampled individual-in particular, P * (Hs) = 4/5-one might phrase this as the question: But this question cannot be answered directly since P {Hs 4/5 } (∀x(Hx → Mx)) = 0 and the conditional probability of the conclusion is undefined. In this case, though, ∀x(Hx → Mx) being the case would not in itself change the chance P * (Hs) = 4/5, so one can use the alternative formulation: The first premiss implies that P(Hs ∧ Ms) + P(Hs ∧ ¬Ms) = 4/5 while the second premiss implies that P(Hs ∧ Ms) + P(¬Hs ∧ Ms) + P(¬Hs ∧ ¬Ms) = 1. P(Hs ∧ ¬Ms) = 0 so P(Hs∧ Ms) = 4/5 and, equivocating, P(¬Hs∧ Ms) = P(¬Hs∧¬Ms) = 1/10. Hence P {Hs 4/5 ,∀x(Hx→Mx)} (Ms) = 4/5 + 1/10 = 9/10, in answer to our question. ¶ A comparison with the Barnett-Paris Method. We close this section by mentioning some examples provided in Rad (2009), a work that offers a nice comparison of the method presented here with the leading alternative method for predicate languages, the Barnett-Paris method of Barnett and Paris (2008) . Given a predicate language L , this alternative method (i) interprets the premisses of an inductive argument as constraints on rational belief over a finite predicate language L n , then (ii) calculates the rational belief function P n on this finite language by choosing from the functions that satisfy the constraints, the P n that maximises entropy (equivalently, that is closest to the equivocator in terms of n-divergence), and finally (iii) deems P ∞ , defined by P ∞ (ω m ) = lim n−→∞ P n (ω m ), to be the rational belief function over L itself. Note that the Barnett-Paris method presumes premiss constraints that delimit a closed, convex set of functions, in order that P n always be defined; hence their method is not applicable to the biased coin example mentioned in §1, where the set of constrained functions is not closed and where there is no P n with maximum entropy. But if the constraint set is closed and convex, and if the language in question contains only unary predicates, the two approaches will yield the same answers to questions in inductive logic (Rad, 2009, Theorem 29). Rad (2009, §4.1) considers the following non-unary question, phrased in a predicate language augmented with an equality symbol that satisfies the axioms of equality: 6 ∀x¬Rxx, ∀x y(¬(x = y) → (Rx y ∨ R yx)), 6 As Rad notes, this kind of example can also be phrased in a language without equality by treating some relation in that language as a surrogate for equality. Here the premisses specify that the binary relation R is a strict linear order that extends indefinitely in at least one direction. As Rad shows, the approach advocated in this paper will give an answer of 1/r! or 0 according to whether r i, j=1 is consistent with some such linear order or not (since it equivocates between all strict linear orders). On the other hand, the Barnett-Paris method will not be able to answer this question at all, since any attempt to treat the premisses as constraints on a finite sublanguage L n leads to the difficulty that the premisses have no finite model, i.e., the premisses are inconsistent if the domain is finite. 7 (This finite model problem also besets the random worlds method of Grove et al. (1994); Bacchus et al. (1996) , which provides semantics for inductive logic by considering the limit as n −→ ∞ of the proportion of all L n -models of the premisses that satisfy the conclusion.) The finite model problem is just one obstacle for the Barnett-Paris methodthere are others. Rad (2009, §3.2) shows that even where the premisses are restricted to categorical Π 2 sentences-premisses of the form ∀x∃yθ(x, y) where θ is quantifier-free-, the Barnett-Paris limit may not exist. On the other hand, the Barnett-Paris method can be used to answer all questions with unqualified premisses from the class of quantifier-free sentences, or the class of Σ 1 sentences (propositions of the form ∃xθ(x)), or the class of Π 1 sentences (∀xθ(x)), and it is known that at least in the first two cases, the Barnett-Paris method will agree with the method presented here. In more complex cases, there is no guarantee that the method of this paper will lead to a unique maximally equivocal function. For example, Rad (2009, §4.3) shows that for the question: for any function satisfying the premiss there is always a more equivocal function satisfying that premiss. In that sense, such a question is like the biased coin question of §1 and our policy is to consider any function satisfying the premiss as sufficiently equivocal (in the absence of contextual information that can offer more guidance). Hence while the approach presented here can answer such questions, it will not always attach a precise probability value to a conclusion of interest. For instance, since the premiss can be satisfied whether or not t 1 and t 2 stand in relation R. Hence the conclusions in such cases can be rather weak-they agree with the conclusions that would be generated by an empirically-based subjective Bayesian semantics ( §1), which does not appeal to the Equivocation Norm at all. But, as any subjective Bayesian will agree, weak does not imply wrong: in any particular situation there simply may be no considerations (neither evidential nor pragmatic) that further narrow down how strongly a given conclusion should be believed. 7 Recall that it is presumed that each element of the domain is picked out by a constant of the language; consequently if the language is finite, so is the domain. In this section we will examine some of the chief criticisms of inductive logic and investigate the extent to which the inductive logic of this paper-i.e., objective Bayesian inductive logic-survives these criticisms. It was suggested in the 1940s and 50s that a viable inductive logic could further our understanding of the way in which scientific hypotheses are confirmed by evidence and that it could also provide a normative foundation for reasoning under uncertainty, with application to artificial intelligence. Rudolf Carnap was the main architect of the programme to develop inductive logic (Carnap, 1945) , and, by the 1980s, a substantial body of research had been built up in this area (see, e.g., Carnap and Jeffrey, 1971; Jeffrey, 1980) . However, the view amongst philosophers at that time was that the programme had failed. This is for several reasons. Notably, Carnap did not isolate a single inductive logic, but a family of inductive logics, and an element of choice was required to pick out an appropriate logic, making inductive reasoning relative to subjective and pragmatic concerns. This was problematic from Carnap's perspective since he interpreted probability in inductive logic as a logical relation between propositions, along the lines of Keynes (1921)-an interpretation that has no obvious role for an agent or subjective choice. But as we have discussed at the end of the last section, an element of subjective choice is not in itself problematic from the perspective of the Bayesian semantics, since Bayesian probability is defined relative to an agent, and Bayesian probability need not be fully constrained by the agent's evidence and language. While the question of subjectivity is less problematic in our context, four other criticisms of the inductive logic programme deserve more discussion. First, it was thought that the inferences of an inductive logic are problematically dependent on the underlying logical language. Second, inductive logics are normally based on invariance principles, which can be thought of as applications of the principle of indifference-a principle that is known to lead to paradoxes. Third, it was recognised that Carnap's inductive logic could not simultaneously capture inductive relations between logically overlapping propositions (e.g., between The bird is black or the bird is white, and, the bird is black) and inductive relations between logically disjoint propositions (e.g., between all ravens so far observed are black, and, the next observed raven is black) (Salmon, 1967) . Fourth, attempts to develop inductive logics in which universal hypotheses (e.g., all ravens are black) receive inductive support were viewed as complex and only partially satisfactory. We shall discuss these objections in turn. While we will focus on predicate languages, most of we what will have to say will apply equally to propositional languages. ¶ Language Dependence. Since inductive logics as presented here are defined with regard to propositions of a logical language L , the question arises as to what extent inferences in an inductive logic depend on the underlying language. For the semantics presented here, there is a clear sense in which inferences do not depend on the underlying language: if one can formulate an inference in more than one language then the two formulations will agree as to whether the premisses entail the conclusion. This result, which is proved in appendix appendix §1, is important as it shows that there is normally no need to spell out the underlying language for any particular inference, and it justifies the use of the notation | ≈ • rather than | ≈ Yet this appears to confer two different levels of plausibility on what is in effect the same conclusion proposition. Which is the correct inference? The Bayesian answer to this question is that both are correct. Granting nothing at all, if your language were L 1 then you ought to believe that t 1 is Green or Blue to degree 3/4, but if your language were L 2 you ought to believe that t 1 is Grue to degree 1/2. The Bayesian can justify this stance by pointing out that, just as degrees of belief should depend on explicit evidence (because such evidence provides us with information about the world), so too should degrees of belief depend on language (because language also provides information about the world). Evidence tells us facts about the world, and one evidence base can be preferred to another if it tells us more, or if what it tells us is more accurate; language tells us about how the world can be carved up, and one language can be preferred to another if it carves up the world more effectively. 8 But the Bayesian would also say that both inferences are lacking: there is an important piece of information, namely that Grue is synonymous with Green or Blue, that has not been taken into account in the two inferences given above. Taking this information into account requires adopting a language L 3 in which one can express the proposition ∀x, Grue(x) ↔ (Green(x)∨Blue(x)). Then one can formulate the inference ∀x, Grue(x) ↔ (Green(x) ∨ Blue(x))| ≈ •3 Grue(t 1 ) 3/4 . But we also find that ∀x, Grue(x) ↔ (Green(x) ∨ Blue(x))| ≈ •3 Green(t 1 ) ∨ Blue(t 1 ) 3/4 , so there is no inconsistency. Note that this inference validates that of L 1 , the richer of the two original languages. Any dependence on the underlying language has typically been regarded as highly problematic in the literature on inductive logic and objective Bayesian methods (see, e.g., Seidenfeld, 1986) . But is such dependence as there is under the 8 Note that Bayesian epistemology provides norms that connect evidence and language on the one hand, and rational degrees of belief on the other. It says little or nothing about norms concerning evidence (e.g., one ought to gather evidence) and norms concerning language (e.g., one ought to be able to express what one wishes to reason about). Bayesian semantics really problematic? Perhaps not, given the following triviality result. Define a synonymy map between predicate languages L and L to be a consistent, countable set of propositions of the form θ i ↔ θ i where the θ i are propositions of L and the θ i are propositions of L . 9 Then demanding invariance under all possible synonymy maps is simply too tall an order, because the only entailment relations which satisfy that demand are trivial relations:
In REF Williamson provide an inference system based on Bayesian epistemology.
From Bayesian Epistemology to Inductive Logic
J. Appl. Log.
J. Appl. Log.
Computer Science