2. A general account of degrees of confirmation

This section develops a formal framework for degrees of confirmation that extends beyond their standard probabilistic representation. My approach lies within the Carnapian tradition of conceptual engineering and proceeds by formulating desiderata or axioms for a precisified concept of confirmation. In contrast to Carnap and his successors, I will avoid an immediate commitment to the calculus of probabilities; instead, I will spell out the desiderata without presupposing any specific calculus. In a second step, I shall translate the desiderata into mathematical language, which will define a formal framework. The next section will situate probabilities within it.

I wish to start with a disclaimer. My ambition is not to formulate immutable axioms that are unquestionable a priori truths for all conceivable cases of confirmation. Rather, I acknowledge that instances of conceptual engineering are generally limited in scope. For example, the explication of degrees of confirmation by conditional probabilities is certainly highly fruitful in a vast array of applications; however, it is not ipso facto the right explication for exotic cases such as fair infinite lotteries. My explication will be considerably more general, but I do not claim it to be entirely unrestricted either.

Throughout, I adopt the standard notation by which c(h, e) denotes the degree of confirmation of h with respect to e (Hosiasson-Lindenbaum Reference Hosiasson-Lindenbaum1940, 133). Here, h stands for a hypothesis, and e stands for some evidence. There are two plausible ways of understanding h and e. The first considers them to be (declarative) sentences of some formal language. This is the choice made in early writings (see Hosiasson-Lindenbaum Reference Hosiasson-Lindenbaum1940; Carnap Reference Carnap1962), and it has more recently been defended by Hawthorne (Reference Hawthorne, Prasanta and Malcolm2011). The alternative is to take h and e to be propositions, possibly corresponding to subsets of an outcome space or a set of possible worlds. Carnap (Reference Carnap, Carnap and Richard1971) made and justified this choice in his later work on inductive logic.

Even if one chooses to understand h and e as sentences, orthodoxy holds that degrees of confirmation are intensional (Hosiasson-Lindenbaum Reference Hosiasson-Lindenbaum1940; Hempel Reference Hempel1945; Carnap Reference Carnap1962):

Confirmation is commonly supposed to be concerned with the factual content of statements and not with how this content is presented to us. As we shall see, this diminishes the relevance of the distinction between declarative sentences and propositions.

The second desideratum states that degrees of confirmation are a quantitative generalization of the logical entailment relation. Logical entailment provides the limiting cases of confirmation: Nothing confirms h stronger than evidence e that entails it, and nothing disconfirms h stronger than evidence that entails its negation $\overline{h}$ .

Of course, this implies a quantitative generalization of the entailment relation only if there are more than two possible values that degrees of confirmation can take. In fact, the very notion of “degrees” alludes to the possibility of nuance and comparability. This brings me to the next desideratum.

A partial order is reflexive, antisymmetric, and transitive, yet being partial, it does not guarantee that any two values can be compared. A possible strengthening would be to demand a total order. This would suggest itself if we presupposed a probabilistic representation because probabilities take values in the real unit interval, whose natural order is total. Yet if we considered imprecise probabilities represented by intervals, it would already no longer be clear how to compare two overlapping intervals with the same center, such as $[0.1, 0.9]$ and $[0.2, 0.8]$ (for further discussion, see Norton Reference Norton2007, 147–49). So if we wish not to anticipate a particular formalism, we should refrain from demanding a total order.

Norton (Reference Norton2007, 149–50) goes a step further; he not only questions totality but even transitivity. He imagines a confirmation theory in which degrees of confirmation are fundamentally evaluated along several dimensions, each corresponding to a different theoretical virtue. The evaluations are then combined by “majority vote”: A hypothesis is preferred over another if it scores higher on the majority of theoretical virtues considered. It is easy to conceive of examples where this procedure violates transitivity.Footnote ³ There are two possible replies. The first is to acknowledge, as I have done, that my desiderata do not claim entirely unrestricted applicability. The second is to state more precisely that degrees of confirmation as understood in this work are only intended to measure how “likely” a hypothesis is to be true on the evidence; this is a fundamentally one-dimensional assessment. If we aid ourselves by using distinct dimensions and they disagree among one another, then instead of settling on an incoherent evaluation, we may simply refuse the comparison. On this interpretation, Norton’s argument speaks more against totality than against transitivity.

The next and final desideratum states that it is harder to confirm logically stronger hypotheses:

This requirement again follows from the idea that degrees of confirmation measure how likely a hypothesis is to be true on the evidence. Since a logically weaker hypothesis is guaranteed to be true when a stronger hypothesis obtains, its degree of confirmation cannot be smaller.

Had we wanted to anticipate the calculus of conditional probabilities, we would have added two further desiderata. The first would have to ensure additivity, and the second would have to ensure conditionalization.Footnote ⁵ Both would presuppose algebraic operations on the set of possible values of degrees of confirmation (addition and division), essentially prefiguring the algebraic structure of a field such as the real numbers. While this may be seen as an advantage because of the mathematical power and convenience that come with it, the algebraic structure is also restrictive. For example, it is the (countably) additive structure of probabilities that conflicts with the idea of a fair infinite lottery.

My next step is to translate the desiderata into mathematical language, which will lead to a straightforward representation theorem. In short, degrees of confirmation that satisfy all the desiderata are defined by truth-value aggregation: They are obtained via an aggregation function $m_e$ , which aggregates the possible truth-values of h, under the assumption of e, into a joint evaluation.

In order to get there, some formal preparation has to be done. The first concerns the formalization of h and e. I hold that we may follow the late Carnap (Reference Carnap, Carnap and Richard1971) in taking h and e to be propositions or events corresponding to subsets of a set $\Omega$ of outcomes or possible worlds. In the familiar way, the logical connectives $h \land h'$ (conjunction), $h \lor h'$ (disjunction), and $\overline{h}$ (negation) then correspond to the set-theoretic operations $h \cap h'$ (intersection), $h \cup h'$ (union), and $\Omega \setminus h$ (complement).

Suppose that we had wanted instead to understand h and e as declarative sentences of some formal language of classical logic (as suggests Hawthorne Reference Hawthorne, Prasanta and Malcolm2011). Then, by virtue of i ntensionality, we could redefine the confirmation function so that it takes as arguments not h and e but their respective classes of logically equivalent sentences. The collection of these equivalence classes forms a Boolean algebra (Halmos Reference Halmos1956), and every Boolean algebra is isomorphic to an algebra of subsets (Stone Reference Stone1936). In other words, the class of sentences that are logically equivalent to a given declarative sentence defines a proposition in the sense of the basic possible-worlds account.

A valuation ${\unicode{x27E6}{\mkern 1mu} \cdot {\mkern 1mu} \unicode{x27E7}}$ is a function that determines the truth-values of propositions for all possible worlds. I write ${\unicode{x27E6}h\unicode{x27E7}^\omega }$ for the truth-value of h at the world $\omega$ : ${\unicode{x27E6}h\unicode{x27E7}^\omega } = 1$ if $\omega$ represents h as true, and ${\unicode{x27E6}h\unicode{x27E7}^\omega } = 0$ if $\omega$ represents h as false. We will also need restricted valuations ${\unicode{x27E6}\cdot \unicode{x27E7}_e}$ to a subset e of possible worlds, corresponding to the evidence; they are only defined on e-worlds.

I call $\mathcal{F}$ the algebraFootnote ⁶ of propositions under consideration. The set of valuations $\unicode{x27E6}h\unicode{x27E7}$ of propositions h on the algebra $\mathcal{F}$ is written as $\unicode{x27E6}{\mathcal F}\unicode{x27E7}$ ; likewise, ${\unicode{x27E6}{\mathcal F}\unicode{x27E7}_e}$ denotes the set of the restricted valuations on e-worlds.Footnote ⁷

In standard theories of confirmation built on conditional probabilities, one usually defines degrees of confirmation c(h, e) only if e has some nonzero prior probability. Similarly, I will allow for a restriction to “admissible” evidence, yet I shall not spell out any specific condition. Minimally, admissible evidence has to be logically possible.

Now I can state the representation theorem for degrees of confirmation under the chosen desiderata. Its proof is provided in the appendix.

Let me briefly unwrap the theorem. Degrees of confirmation c(h, e) that satisfy the desiderata can be represented by an aggregation function $m_e$ . This function takes in the truth-values of h at e-worlds—encoded in the restricted valuation ${\unicode{x27E6}h\unicode{x27E7}_e}$ —to produce a graded assessment between two limiting values $\mathbf{0}$ and $\mathbf{1}$ . The first value is attained if h is false at all e-worlds; the second is attained if h is true at all e-worlds. Between the two extremes, there is room for nuance, depending on how fine-grained the set X of possible values is chosen to be. The gradation has to be compatible with m onotonicity: If a hypothesis h′ is true at all e-worlds where h is true, then its aggregated value is at least as high as that of h.

It is perhaps easiest to understand this in the finite case, for example, where e only consists of two possible worlds $\omega_1$ and $\omega_2$ . Restricted valuations ${\unicode{x27E6}h\unicode{x27E7}_e}$ then are equivalent to truth-value pairs $\langle {\unicode{x27E6}h\unicode{x27E7}^{{\omega _1}}},{\unicode{x27E6}h\unicode{x27E7}^{{\omega _2}}}\rangle $ , and the aggregation function can be interpreted as a function of these two values. The theorem states first that ${\bf{0}} = {m_e}(0,0) \lt {m_e}(1,1) = {\bf{1}}$ . Truth-value aggregation respects something like idempotency: The constantly zero valuation is aggregated into the value $\bf{0}$ ; the valuation that yields 1 everywhere is aggregated into $\bf{1}$ . The second statement of the theorem translates to the inequalities $m_e(0, 0) \leq m_e(0, 1) \leq m_e(1, 1)$ and $m_e(0, 0) \leq m_e(1, 0) \leq m_e(1, 1)$ . As more truth is added, degrees of confirmation increase (or stay constant).

All this may sound very abstract, but what it describes is utterly familiar. Imagine a judge who assesses the hypothesis h that the suspect is guilty of the crime, given the accepted evidence e. She will consider several scenarios compatible with the evidence so that in each of them, the suspect’s guilt or innocence is clearly established. Ideally, all remotely plausible courses of events are taken into consideration. These scenarios correspond to the possible e-worlds. The respective determinate truth-values of the hypothesis define the restricted valuation ${\unicode{x27E6}h\unicode{x27E7}_e}$ . If the suspect has committed the crime in all scenarios, the judge will evaluate the hypothesis to be maximally confirmed by the evidence; if the suspect is innocent in all scenarios, the opposite will be the case. This is in line with (1) of the theorem. Between these extremes, the more scenarios add to the truth of the hypothesis (without subtraction of other scenarios), the more strongly the hypothesis is confirmed; this corresponds to (2). Nothing here hinges on the particular example; truth-value aggregation, as described by the theorem, seems to capture something fairly generic about confirmation.

3. Special case: The calculus of probabilities

I have argued earlier that degrees of confirmation are the result of truth-value aggregation. In this section, I show that if the aggregation function is chosen to be the weighted arithmetic mean, degrees of confirmation are probabilities in the mathematical sense. If the weights are constant, degrees of confirmation are conditional probabilities.

Truth-value aggregation (as characterized by theorem 1) can be thought of as something like a mathematical mean or average of truth-values. In a very general sense, a mathematical mean $M(x_1, x_2, \ldots, x_n)$ of values $x_1, x_2, \ldots, x_n$ is a value that is intermediate,

$$\min \{ {x_1}, \ldots, {x_n}\} \le {M}({x_1}, \ldots, {x_n}) \le \max \{ {x_1}, \ldots, {x_n}\}, $$

and increasing in each argument:

$${\rm{if}}\;{x_i} \le {x'_i},\;{\rm{then }}\ {M}({x_1}, \ldots, {x_i}, \ldots, {x_n}) \le M({x_1}, \ldots, {x'_i}, \ldots, {x_n}).$$

Truth-value aggregation satisfies precisely these two properties by (1) and (2) of theorem 1.Footnote ⁸

The most familiar mean is the arithmetic mean; it corresponds to how we ordinarily compute averages of numbers. We shall need its weighted version, where each value $x_i$ has a nonnegative weight $\alpha_i$ :

$${M}_\alpha ^{{\rm{ari}}}({x_1}, \ldots, {x_n})\mathop = \limits^{{\rm{df}}} {{{\alpha _1}{x_1} + \ldots + {\alpha _n}{x_n}} \over {{\alpha _1} + \ldots + {\alpha _n}}}.$$

Here, the weights are collected in the tuple $\alpha = \langle \alpha_1, \ldots, \alpha_n \rangle$ , and at least one of the weights has to be positive. The natural generalization to an infinite collection of values draws on measure theory.Footnote ⁹ Used for truth-value aggregation, the weighted arithmetic mean produces the probability calculus:

A brief proof is given in the appendix.

That the proposition is mathematically straightforward can be seen by a simple illustration. Imagine a fair die is thrown, and we ask for the probability that it shows a number divisible by 3. The quick reply would simply count possibilities: two favorable ones (3 and 6) out of six, so a probability of one-third. A more tedious solution could go over each outcome (1, 2, etc.), adding the outcome’s probability if it is a favorable case and nothing otherwise: $0 + 0 + {1 \over 6} + 0 + 0 + {1 \over 6} = {1 \over 3}$ . This is just the calculation of a weighted arithmetic mean of the possible truth-values of the hypothesis:

$${M}_{\langle {1 \over 6}, \ldots, {1 \over 6}\rangle }^{{\rm{ari}}}(0,0,1,0,0,1) = {{{1 \over 6} \cdot 0 + {1 \over 6} \cdot 0 + {1 \over 6} \cdot 1 + {1 \over 6} \cdot 0 + {1 \over 6} \cdot 0 + {1 \over 6} \cdot 1} \over {{1 \over 6} + \ldots + {1 \over 6}}} = {1 \over 3}.$$

We could have chosen any other constant positive number for the weights in this calculation; if the weights, like in the example, are normalized (their sum is 1), they behave like probabilities for the individual outcomes. Normalized weights thereby correspond to a probability mass function.

It may be less apparent that Bayesian conditionalization is also a weighted arithmetic average of truth-values, as can be seen by rewriting the formula of conditional probability in a slightly unusual way:

$$P(h\mid e) = {{P(h \wedge e)} \over {P(e)}} = {{P(h \wedge e) \cdot 1 + P(\bar h \wedge e) \cdot 0} \over {P(h \wedge e) + P(\bar h \wedge e)}} = M_{\langle P(h \wedge e),P(\bar h \wedge e)\rangle }^{{\rm{ari}}}(1,0).$$

Here, we average the truth-values of h given e by distinguishing $h \land e$ (where h’s truth-value is 1) and $\overline{h} \land e$ (where it is 0). The weights are the probabilities of these two cases. In turn, each of these probabilities is the sum over the probability masses (or weights) of the corresponding individual outcomes. Hence, this connects to proposition 1, leading to the following observation:

Qua truth-value aggregation, the weighted arithmetic mean produces not only the calculus of probability but specifically of conditional probability. The only additional assumption is that the weights used in the average are constant, irrespective of the evidence e. This ensures that the probability function $c(\,\cdot\,, e)$ given e remains the same, up to normalization, as the prior probability function $P = c(\,\cdot\,, \top)$ restricted to e, a property characteristic of conditionalization.Footnote ¹⁰

If the additional condition is violated and the weights assigned to the outcomes are changed when evidence is added, degrees of confirmation no longer conform to conditionalization. In orthodox Bayesian terms, this would correspond to changing the prior ex post, a move often criticized as irrational (Teller Reference Teller1973; Briggs and Pettigrew Reference Briggs and Pettigrew2020). However, alternative updating schemes upon learning new evidence have been defended in the literature. Williamson (Reference Williamson2010, chap. 4) holds that probabilities should be updated on the grounds of the principle of maximum entropy, whose application can deviate from standard conditionalization. Douven (Reference Douven2022, chap. 4) defends an alternative updating rule in which the hypothesis that best explains the evidence is more strongly confirmed than Bayesian conditionalization would have it. These accounts are compatible with proposition 1 but reject the additional premise of proposition 2.

Mathematically, the results of this section are unremarkable and, fundamentally, nothing new.Footnote ¹¹ Philosophically, however, they show how (conditional) probabilities naturally arise in the broader framework of truth-value aggregation. Truth-value aggregation is analogous to a mathematical mean, and the most straightforward mean with a fine-grained value set is the arithmetic mean. And it is the (weighted) arithmetic mean that defines the probability calculus. Section 5 will show how probabilities also arise in a more general setting via a transform.

There are even simpler aggregation methods than the arithmetic mean once we reduce the value space for degrees of confirmation. Instead of the real unit interval [0, 1] chosen for the probability calculus in this section, one could, for instance, consider degrees of confirmation with only three possible values—only one intermediate value between the minimal value ${\bf 0}$ and the maximal value ${\bf 1}$ .Footnote ¹² The corresponding aggregation methods would be simple but indiscriminating. It is not merely simplicity that distinguishes the arithmetic mean but the combination of its relative simplicity with the extraordinary richness of its set of possible values: The real unit interval is totally ordered, dense,Footnote ¹³ and complete.Footnote ¹⁴

4. Special case: Fair infinite lotteries

In this section, I show how the general framework of truth-value aggregation accommodates fair infinite lotteries, which defy the probability calculus. This illustrates the usefulness of the framework in a philosophically and scientifically relevant example. The final result will be equivalent to Norton’s (Reference Norton2021a) “infinite lottery inductive logic.”

In a fair infinite lottery, a ticket is drawn randomly out of an infinite set of tickets. The most discussed case is that of a countable set so that we may label the tickets by the natural numbers $1, 2, 3, \ldots$ In its most stringent and faithful interpretation, “fairness” here means that the chance of an outcome (a particular ticket) or an event (a subset of tickets) does not depend on the labeling. A relabeling is a permutation, a bijective map from the sample space onto itself. The following condition imposes this on the resulting degrees of confirmation on an algebra $\mathcal{F}$ of events:

Here, the propositions or events h and e again correspond to subsets of possible outcomes. For instance, h could be the proposition that the ticket has an odd number (corresponding to the set of all odd numbers), and e could be the proposition that the ticket has a number between 1 and 3 (corresponding to the set $\{1, 2, 3\}$ ). Then c(h, e) is the degree to which the ticket lying between 1 and 3 confirms the hypothesis that the ticket is odd. By label independence, a situation that only differs by labeling, such as for the hypothesis h′ (that the ticket is even) and the evidence e′ (that it is between 2 and 4), must yield the same result, $c(h', e') = c(h, e)$ .Footnote ¹⁶

It is easy to see that the standard probability calculus cannot satisfy label independence if e corresponds to an infinite set (and the algebra contains all singletons). For any two different outcomes, there is a permutation that exchanges them. So every individual outcome has to receive the same probability by label independence, which confronts the probabilist with a dilemma. Either this probability is zero, or it is nonzero. If it is zero, then (by countable additivity) the probability of the certain event (to draw any ticket at all) is also zero. If it is nonzero, then the probability of the certain event diverges to infinity, which violates normalization.

This observation famously motivated de Finetti (Reference De Finetti, Machí and Smith1974) to reject countable additivity of probabilities in favor of finite additivity, allowing him to escape the dilemma’s first horn. A zero probability of each ticket no longer contradicted the probability of 1 for the whole sample space because we would have to add up an infinity of terms to relate the former to the latter. De Finetti’s solution, however, faces two problems. First, Williamson (Reference Williamson1999) shows that de Finetti’s Dutch-book argument extends to countable additivity; thus, giving up on countable additivity threatens to undermine de Finetti’s original justification of the probability calculus. Second, a zero probability for each ticket seems to be too small, for it is perfectly sensible to suppose that it is strictly more likely to draw one of the tickets between 1 and 1,000 than it is to draw the particular ticket 466, which in turn seems more likely than the impossible event of drawing no ticket at all. Yet according to de Finetti, the corresponding probabilities are all the same—namely, zero.

There is a natural temptation to suspect that the real numbers are simply not fine-grained enough for the job. On the one hand, the probability of each ticket should not be strictly zero; on the other hand, every positive real number is too big. What we seem to need are infinitesimals, infinitely small yet nonzero numbers. Wenmackers and Horsten (Reference Wenmackers and Horsten2013) pursue this route with mathematical rigor. Their solution, however, violates label independence; in a sense, they have exchanged the original problem for another: The lottery they describe is not fair according to the original, stringent understanding.Footnote ¹⁷

In contrast to the probability calculus, the more general framework of truth-value aggregation, introduced in section 2, is compatible with label independence. The mathematical starting point is an observation that relates label independence to considerations of cardinality. The cardinality $|S|$ of a set S is a measure of how many elements it contains. If S is finite, its cardinality simply is the number of its elements. If S is infinite, cardinalities can still be compared: One has $|S| = |S'|$ if there is a bijective map (one-to-one and onto) between them, and one has $|S| \leq |S'|$ if there is an injective map (one-to-one but not necessarily onto) from S to S′. Accordingly, $|S| \lt |S'|$ is defined by $|S| \leq |S'|$ and $|S| \ne |S'|$ .

This result states that label independence forces degrees of confirmation of two hypotheses to be equal just in case the hypotheses, as well as their complements, each intersected with e, have equal cardinality; I give a proof in the appendix.

Whenever the cardinalities of two hypotheses or of their complements differ, a discrimination is possible. I shall next define the maximally fine-grained order relation that makes these discriminations.

This strict order is turned into a total order $\leq$ by inclusion of equality in accordance with proposition 3.

It is easiest to get an intuitive grasp of this order relation if we think of h and e as sets of tickets; we know that the ticket lies in e, and we ask what the “chance” is that it lies in h. In a finite fair lottery, it is clear that the chance of the ticket being in h (for fixed e) is higher the more tickets are in the intersection of h with e. This directly corresponds to the first part of cardinality order: c(h, e) increases if the number of tickets in $h \cap e$ does. The second part of cardinality order is redundant in the finite case; as the number of tickets in $h \cap e$ increases, the number of tickets in the complement $\overline{h} \cap e$ decreases accordingly because their sum is the constant number of tickets in e.

The infinite case, however, comes with a few twists. For one, an infinite set does not increase in cardinality when a finite number of elements is added. As a consequence, the cardinality of $h \cap e$ may change while the cardinality of the complement in e, $\overline{h} \cap e$ , does not, or vice versa, since one may be finite and the other infinite. Hence, cardinality order looks at the cardinality of both $h \cap e$ and its complement. For another, two sets that may appear very different in size can have equal cardinality. As an example, the set of even numbers, whose asymptotic densityFootnote ¹⁸ in the natural numbers is one-half, has the same cardinality as the comparatively sparse set of squared integers (whose asymptotic density is zero); the respective complements are also equal in cardinality so that the cardinality order, with $e = \mathbb{N}$ , does not distinguish between the hypothesis h that the ticket has an even label and h′ that the ticket’s label is a square number: $c(h, e) = c(h', e)$ .Footnote ¹⁹ While this may seem counterintuitive, it is a direct consequence of label independences: There is a relabeling of the natural numbers that maps h onto h′.Footnote ²⁰ That the set of even numbers appears bigger than the set of squares depends on how the natural numbers are arranged. If fairness of a lottery means that labels are confirmationally irrelevant, both sets have to be treated alike.

Formally, we can use the strict order to give a name to each possible value for degrees of confirmation. I write these values as $v_{\kappa, \lambda}$ , where $\kappa$ is the cardinal number of $|h \cap e|$ and $\lambda$ of $|\overline{h} \cap e|$ ; so we may write $c(h, e) = v_{|h \cap e|, |\overline{h} \cap e|}$ . Consider the case where e is countably infinite. Then its cardinality is that of the natural numbers, commonly represented by the cardinal number $\aleph_0$ (with the Hebrew letter aleph). The set of possible values for degrees of confirmation under the cardinality order is exhausted by

$${\bf 0} = v_{0, \aleph_0} \lt v_{1, \aleph_0} \lt v_{2, \aleph_0} \lt \ldots ; \lt v_{\aleph_0, \aleph_0} \lt ; \ldots \lt v_{\aleph_0, 2} \lt v_{\aleph_0, 1} \lt v_{\aleph_0, 0} = {\bf 1},$$

where a semicolon delimits an infinite sequence. An equivalent characterization is found in Norton (Reference Norton2021a). The method does not presuppose that e is countably infinite, and it directly generalizes to higher cardinalities.Footnote ²¹

To demonstrate how this fits into the framework of truth-value aggregation of section 2 is merely an exercise in formal rewriting. In this framework, the value of c(h, e) is determined by an aggregation function $m_e$ whose argument is the valuation $\unicode{x27E6}h\unicode{x27E7}_e$ of h restricted to outcomes in e. Now $h \cap e$ equals the subset of e on which the valuation $\unicode{x27E6}h\unicode{x27E7}_e$ is 1; that is, it equals the preimage $\unicode{x27E6}h\unicode{x27E7}_e^{-1}(1)$ . Likewise, $\overline{h} \cap e = \unicode{x27E6}h\unicode{x27E7}_e^{-1}(0)$ . This indicates how we can obtain the degrees of confirmation of a fair infinite lottery by means of an aggregation function: $c(h, e) = m_e(\unicode{x27E6}h\unicode{x27E7}_e) = v_{|\unicode{x27E6}h\unicode{x27E7}_e^{-1}(1)|, |\unicode{x27E6}h\unicode{x27E7}_e^{-1}(0)|}$ . With the cardinality order stated earlier, this aggregation function manifestly satisfies the statements of theorem 1.

So far, I have only defined an ordering of the possible values of degrees of confirmation without specifying these values in a quantitative sense. I shall now partially fill this lacuna and retain, as expected, that when starting from a fair infinite lottery and learning the evidence e that the ticket is in a set of N tickets (with finite N), the degrees of confirmation c(h, e) automatically turn into a standard uniform probability distribution with the numerical probability ${1 \over N}$ for each of these tickets. This is achieved as follows. Define the value $p_\kappa \mathop = \limits^{{\rm{df}}} v_{1, \kappa - 1}$ ; this is the chance of drawing an individual ticket out of a set of cardinality $\kappa$ , where subtraction is defined as usual for cardinal numbers.Footnote ²² If $\kappa$ is infinite, we may think of $p_\kappa$ as an infinitesimal number. We assume multiples of $p_\kappa$ satisfying $n p_\kappa = v_{n, \kappa - n}$ for all finite n (and if $\kappa$ is finite, $n \leq \kappa$ ). This ensures that the likelihood of an event with n tickets is n times the chance of each particular ticket. We may also assume ${\bf 1} - n p_\kappa = v_{\kappa - n, n}$ so that the chance of drawing a ticket out of a set of n tickets accords, in the usual way, with the chance of the complementary event. In the finite case $\kappa = N \in \mathbb{N}$ , we find $N p_N = v_{N, 0} = {\bf 1}$ , so after identifying the maximal degree of confirmation ${\bf 1}$ with the ordinary number 1, we get the probability of each ticket as ${p_N} = {1 \over N}$ .

5. Conditional probabilities by convention

Truth-value aggregation—which defines degrees of confirmation in the framework introduced in section 2—can be thought of as a mathematical mean of truth-values, where the notion of a mean is understood quite broadly. In section 3, I have shown that if the mathematical mean is chosen to be the familiar weighted arithmetic mean, then truth-value aggregation imposes the standard calculus of probabilities on degrees of confirmation. In the present section, I will considerably generalize this result. I will look at a standard class encompassing all popular mathematical means, the weighted generalized f-means, and show that if any of these defines truth-value aggregation, then the resulting degrees of confirmation satisfy a probability transform, whereby they can be represented by probabilities. I shall also introduce a generalized version of conditioning that holds for degrees of confirmation defined by a generalized f-mean with constant weights.

Readers well versed in mathematics will immediately think of a number of alternatives to the arithmetic mean: the geometric mean, the harmonic mean, the root mean square, the power means, and so forth. All these popular choices are examples of generalized f-means, where f is a continuous injective function defined on an interval of real numbers. In its weighted version with nonnegative weights $\alpha = \langle \alpha_1, \ldots, \alpha_n \rangle$ (see Bullen Reference Bullen2003, 266), it reads

$${M}_\alpha ^f({x_1}, \ldots, {x_n})\mathop = \limits^{{\rm{df}}} {f^{ - 1}}\left[ {{{{\alpha _1}\ f({x_1}) + \ldots + {\alpha _n}\ f({x_n})} \over {{\alpha _1} + \ldots + {\alpha _n}}}} \right].$$

In the infinite case, a finite measure takes the place of the weights.Footnote ²³ Different choices for the function f define different means: If f is a linear function, the f-mean is the arithmetic mean; if f is a quadratic function, the f-mean is the root mean square; if f is a logarithm, the f-mean is the geometric mean; if $f(x) = 1/x$ , the f-mean is the harmonic mean; and so forth.

If f is defined on the real unit interval, then the generalized f-mean can be employed for truth-value aggregation.Footnote ²⁴ This means that we aggregate the possible truth-values of a hypothesis h given the evidence e by calculating their generalized f-mean. If e corresponds to the set of outcomes $\{\omega_1, \ldots, \omega_n\}$ ,

$$c(h, e) = m_e(\unicode{x27E6}h\unicode{x27E7}_e) = {M}^f_\alpha\!\left( \unicode{x27E6}h\unicode{x27E7}^{\omega_1}, \ldots, \unicode{x27E6}h\unicode{x27E7}^{\omega_n} \right).$$

For example, if we asked—as in section 3—for the degree of confirmation c(h, e) of the hypothesis h that a fair die shows a number divisible by 3 without additional evidence, and choosing $f(x) = x^2$ (corresponding to the root mean square), we would obtain

$${M}_{\langle {1 \over 6}, \ldots, {1 \over 6}\rangle }^f(0,0,1,0,0,1) = \sqrt {{0^2} + {0^2} + {1 \over 6} \cdot {1^2} + {0^2} + {0^2} + {1 \over 6} \cdot {1^2}} = {1 \over {\sqrt 3 }},$$

where uniform weights are chosen because of the assumed fairness of the die. The arithmetic mean, corresponding to the standard probability calculus, would have given the standard answer of $1/3$ .

Although generalized f-means except the arithmetic mean produce unfamiliar degrees of confirmation that violate the axioms of probability, it is easy to transform these degrees of confirmation to standard probabilities post hoc. In the previous example, this is as straightforward as calculating the square, which transforms the result $1/\sqrt{3}$ to the standard probability $1/3$ . The following theorem assures that this observation holds generally. It is demonstrated in the appendix.

This theorem is a considerable generalization of proposition 1, which stated that the weighted arithmetic mean gives rise to the probability calculus. If the function f is a linear function (defining the arithmetic mean), then the probability transform g is the identity. All other f-means defined on the unit interval produce nonprobabilistic degrees of confirmation, which, however, can be transformed to probabilities. Hence, if truth-value aggregation is explicated by any generalized f-mean, probabilities can still be chosen to represent degrees of confirmation.

Proposition 1 had an important addendum: If the weights chosen for the arithmetic mean attributed to the outcomes are constant (that is, independent of e), then degrees of confirmation c(h, e) are conditional probabilities of the hypotheses h on the evidence e (see proposition 2). I shall next establish an analogous result for the generalized f-mean.

Beforehand, I have to explain what conditioning means in the generalized case. Since degrees of confirmation given by truth-value aggregation with some f-mean are, in general, not even probabilities, they cannot be conditional probabilities either. Nonetheless, there are two natural ways of defining conditioning for them; both will turn out to be equivalent.

The first is to use standard conditioning for the transformed degrees of confirmation (which are probabilities by virtue of theorem 2) and to transform them back using the inverse of the probability transform. So with $P(h) \mathop = \limits^{{\rm{df}}} g[c(h, \top)]$ being the probability function corresponding to the prior degrees of confirmation, we say that c(h, e) satisfies generalized conditioning if $c(h, e) = g^{-1}[P(h \mid e)]$ .

The second idea is to generalize the observation made in section 3 to the effect that standard conditionalization is itself a weighted arithmetic mean of truth-values; namely, $P(h \mid e) = M^\mathrm{ari}_{\langle P(h \land e), P(\overline{h} \land e) \rangle}(1, 0)$ . Likewise, we may say that generalized conditioning should be a weighted f-mean of truth-values: c(h, e) is said to respect generalized conditioning if $c(h, e) = M^f_{\langle g[c(h \land e, \top)], g[c(\overline{h} \land e, \top)] \rangle} (1, 0)$ . Here, the probability transform has been used in the weights because weights are additive. The following result, shown in the appendix, states that the two ideas coincide.

If the weights used in the generalized f-mean are attributed directly to the outcomes and held constant irrespective of the evidence, the resulting degrees of confirmation are guaranteed to fall in line with generalized conditioning. This important corollary to theorem 2 is demonstrated in the appendix.

When nonprobabilistic degrees of confirmation respect generalized conditioning, we can employ standard conditional probabilities to do all the calculations. At any stage, the nonprobabilistic degrees of confirmation are retained by applying the inverse of the probability transform. In this sense, conditional probabilities may be chosen by convention to represent degrees of confirmation.

There are two important qualifications. First, the mathematical correspondence between nonprobabilistic and probabilistic degrees of confirmation does not guarantee their equivalence in other ways. Different calculi may be motivated philosophically, for example, by different interpretations of degrees of confirmation or different background assumptions or facts. Mathematically, they vary in how they quantify differences in confirmation; only ordinal comparisons of degrees of confirmation are invariant under the probability transform.Footnote ²⁵ That said, pursuing these considerations would lead outside the scope of this work, which has tried to steer clear of specific interpretations of degrees of confirmation and theories of induction.

Second, although the generalized f-means form a large class of prima facie plausible aggregation methods, they remain limited in important ways. For instance, they do not provide any advantage over standard probabilities in the treatment of fair infinite lotteries. The calculus introduced in section 4 is incompatible with any f-mean. So while (conditional) probabilities can be used widely to represent degrees of confirmation, they arguably cannot be employed universally.