Logical perspectives on the foundations of probability

1.1 Motivating the logical perspective

It is remarkably difficult to explain to what logic is for, as van Benthem put it in [2],

The unity of logic, like that of other creative disciplines, resides in the mentality of its practitioners and their modus operandi.

And of course, mentality varies a lot, in time, and within each of the many academic silos. To illustrate, consider the logical landscape of the late 1920s, i.e. when Bruno de Finetti begun focussing on the coherentist foundation of probability recalled below in Section 1.3. For him, logic was the Boolean algebra of sets, which he interpreted as the logic of certainty. No alternative was available to him. He later became aware of the introduction of many-valued logics by the Polish school, but he assessed it as an alternative to probability, not to Boolean logic [3]. Fifty years on, the emergence of expert systems in AI pressed logicians to take seriously the idea that reasoning may not necessarily be grounded in Boolean logic. From fuzzy logics to dynamic epistemic logics, non-monotonic logics, and paraconsistent logics, a galaxy of extensions of classical logic have been thoroughly investigated towards the end of the past century [4]. Against this background, one in which many logics are worth taking as the starting point of an agent’s knowledge representation, logicians could ask whether de Finetti’s coherence-based justification for measuring uncertainty with probability necessitated logic to be classical. In a landmark contribution, Paris [5] showed that this was indeed not the case: many logics can yield distinct instantiations of coherence, as we detail in Section 2. Accordingly, the logical perspective we put forward in this note is not tied to a specific logical system, but aims at illustrating, by way of example, what the modus operandi of logicians is, and how it could be beneficial to the wider field of uncertain reasoning.

Logic and probability concur to achieving the goal of representing both an agent’s knowledge and their uncertainty in events of interest, thereby providing a guideline for how to act rationally under the specific circumstances. This was anticipated by Leibniz in his Nouveaux essais where he writes:

I maintain that the study of the degrees of probability would be very valuable and is still lacking, and that is a serious shortcoming in our treatises on logic. For when one cannot absolutely settle a question one could still establish the degree of likelihood on the evidence, and so one can judge rationally which side is the most plausible.^[1]

The decision-theoretic foundation of probability goes back at least to Huygens’s analysis of expectation, see, e.g. [6], and has been put forward explicitly by the so-called Bayesian school [7–11]. They did so by relying essentially on the logical notion of coherence, which, however, has been investigated with logical tools only fairly recently and still appears to be underappreciated. Indeed, logic and probability constitute mostly independent elements of today’s mathematical and scientific education. This is at odds with the fact that they had a joint start as mathematical theories of reasoning in the mid-1800s – see [13,14] for a thorough historical reconstruction.

De Morgan, who coined the term mathematical logic, titled his volume Formal Logic: Or the calculus of inference, necessary and probable [15]. His friend and collaborator, George Boole, gave his Laws of Thought a telling subtitle: On which are founded the mathematical theories of logic and probability. And indeed the Key Problem originates in Boole’s seminal volume, where it is claimed to be solvable by a “general method”:

the final expression will contain terms with arbitrary constant coefficients [so] by giving to their constants their limiting values 0 and 1 determine the limits within which the probability sought must lie independently of all experience.” [16] p. 17.

Focussing on such a general method was a breakthrough in both logic and probability. Before him, indeed, logic was mostly concerned with the analysis of individual patterns of reasoning – syllogisms. After him, logicians set sail to algorithmic reasoning. As to probabilities, however, Boole noted that whenever the information we have is “insufficient to render determinant the value sought,” the “general method” yields (probability) intervals, rather than values, which have been tightly pinned down by Maurice Fréchet in 1935 as follows. If θ , ϕ are events (see below for notation and terminology) with probabilities P ( θ ) = x and P ( ϕ ) = y , then the probability of the disjunction and the probability of the conjunction of θ and ϕ are bound by

see [17] for an early detailed analysis.

The aforementioned inequalities provide a useful starting point in our attempt to present a vast body of ideas in a unified matter: pinning down the “logic of science” by combining logic with probability raises, from the outset, a number of questions which result in a large space of formal possibilities. Logical methods in our view provide fundamental tools to navigate this space. Firstly, the aforementioned bounds show that, in general, the quantification of an agent’s uncertainty may not be uniquely determined by the available information. This ties in with the key questions tackled in (pure) inductive logic, the topic of Section 3. Second, and somewhat alternatively, whenever uncertainty is not uniquely quantified by the available information, it may be methodologically more appropriate to consider the set of probability distributions compatible with it, as a justified measure of uncertainty. And, very interestingly, sets, rather than single probability functions, have been shown to emerge naturally by grounding the formalism on extensions of Boolean logics, as pointed out in Section 2.

Before going into details, readers with no logical background may benefit from a short guided tour of logic in the theory of uncertain reasoning, to which the remainder of this section is devoted. In it, we also lay down a bare minimum of terminology and notation.

1.2 Basic notions and terminology

1.3 Coherence

Armed with ℒ -probability functions, we can ask a set of increasingly more practical questions, all contributing to our opening Key Problem:

1. What does it mean to make a rational forecast?

2. Is the answer to the previous question unique?

3. How can uncertainty representation methods incorporate a variety of kinds of information?

4. How does uncertain reasoning relate to decision-making?

So, what, if anything, guarantees that ℒ -probability functions provide an adequate representation of an agent’s uncertainty on a specific set of events of interest? More concisely and somewhat more generally: What makes a forecast Φ (ir)rational? This interrogates us about the meaning of probability and the extent to which this is captured or even reflected by its axiomatisation. Whilst the question of adequacy cannot be fully addressed independently of the context in which the uncertainty quantification is being carried out, it turns out that the logical notion of coherence is a good candidate to serve as foundation; as we now discuss.

In Chapter V of [26], a long thread of work is summed up in the observation that for probability to be useful in physics, it must bear a physical meaning. Truesdell finds it in asymptotic phenomena, i.e. those produced by a system which is composed of a very large number of elements (or a very long time series). Asymptoticity allows one to ground accurate predictions about the system solely on the system’s average behaviour. Pólya [27] puts forward a similar line of thought where probability is restricted to random mass phenomena. Owing to the influence of Kinchin, this physical understanding of probability occupies centre stage in the above-recalled Kolmogorov axiomatisation, which responds in part to Hilbert’s Sixth Problem. In it, probability is clearly intended as a branch of physics:

6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.

See [28] for a historical appraisal.

If events – the bearers of probability – describe properties of interest of asymptotic systems, then a variety of results, which originate in Bernoulli’s Law of Large Numbers justify the mathematics of probability as a device to predict, under specific circumstances, the future behaviour of such systems. This underpins the kind of reasoning which is invoked in inferential statistics, when data are taken to be a random sample from a hypothetical infinite population. Through this idea, asymptotic events, as we call them, together with the meaning they imbue to probabilistic statements, permeate all of experimental science.

However, it has long been noted that many events of interest in uncertain reasoning and forecasting fall short of asymptoticity. Either because the system has relatively few components or because it gives rise to events which are essentially unique in the sense that repetitions of the same experiment are not guaranteed to be independent. Recall that as a consequence of stochastic independence, it makes no (probabilistic) difference whether one tosses a large number N of coins simultaneously or a single coin N times. But as soon as one moves away from clear-cut cases where no reasonable physical interaction can be expected, arguing in favour of the independence of the components of a stochastic system can be very tricky [29].

Kolmogorov is very clear about this:

Historically, the independence of experiments […] represents the very mathematical concept that has given the theory of probability its peculiar stamp. We thus see, in the concept of independence, at least the germ of the peculiar type of problem in probability theory. In this book […] we are interested mainly in the logical foundation for the specialised investigations of the theory of probability. In consequence, one of the most important problems in the philosophy of the natural sciences is – in addition to the well-known one regarding the essence of the concept of probability itself – to make precise the premisses which would make it possible to regard any given real events as independent. This question, however is beyond the scope of this book. ([24], p. 8–9.)

We note in passing that what Kolmogorov means by “logical foundation” is limited to exploring the mathematical consequences of his axiomatisation.

Defending assumptions of independence is typically far from obvious in a wide class of forecasting problems spanning the social sciences (e.g. in forecasting macroeconomic variables), the life sciences (e.g. forecasting the individual patient’s chance of developing a particular disease), and everyday reasoning (e.g. forecasting the outcome of one’s own wedding). This is why the aforementioned are examples of events which are only ever going to happen once. So, in contrast to asymptotic events, we shall call them singular events.

Forecasting singular events is central to science, technology, institutional, and personal decision-making, because even in those cases in which an experiment can be “replicated” in principle, in practice, it may be hard to be confident that they will be “close enough” repetitions of the same experiment. So it has long been felt that a central problem in the theory of probability was to provide a principled way to dispense with the mathematical comfort provided by asymptoticity.

de Finetti [30] sought to reconcile the central importance of singular events with the powerful mathematics which originates with asymptotic events. This is where the logical notion of coherence enters perhaps unexpectedly, but rather spectacularly, the stage. For as a preliminary step to achieve his grand goal, de Finetti put forward in [8] the following justification for the probabilistic quantification of uncertainty, which has become known as the Dutch Book Argument, see [31] for an accessible and recent overview. The central assumption in de Finetti’s argument is that in a suitably defined betting problem, no rational person would willingly put themselves in a position to gamble for a sure loss. Doing so, de Finetti points out, would amount to obvious incoherence. Then he sets up a framework in which he shows that a necessary and sufficient condition to avoid this kind of incoherence is to forecast according to the prescriptions of (4) and (6). Note that (5) is easily derivable from them in standard probability logic, to which we come back later.

2.1 The set of elementary events

Suppose you recently become friends with an athlete and want to form your belief that she wins her next match. Without any understanding of the sport you partition, the event space into W , L and in the absence of all knowledge you assign a win the same probability as a loss. However, you might have also partitioned the event space into W , L , D also allowing for the possibilities of draws and in the absence of all knowledge you assign all three outcomes the same probability, cf. [43]. Clearly, these two probability assignments are inconsistent. Some have argued that these inconsistencies spell doom to the project of applying logic to uncertain inference (references omitted to protect the guilty).

It seems more useful to us to acknowledge that some dependence of probabilities on the choice of the partition/domain is unavoidable [44]. The choice of the partition is not arbitrary but instead reflects the way you perceive the problem, i.e. what are the elementary events? Now, that providing different answers to this question leads to different probabilities should not come as a surprise to anyone. Instead, it drives home the point that how we see the world (how we formalise the problem at hand), at least partly, determines our probabilities, cf. [45].

While some dependence on the underlying domain is unavoidable, some dependencies are worse than others. Intuitively, automorphisms of the underlying fixed set of elementary events map elementary events to their doppelgangers. In the absence of all other knowledge, we have no logical reason to discriminate between an elementary event and its doppelganger. Hence, both ought to be assigned the same probability. Note that this argument crucially hinges on: (i) a fixed choice of an underlying domain reflecting the way you perceive the world and (ii) that there is no other knowledge. If just one of these preconditions is not met, then the argument becomes open to counter-examples. As Paris put it (Maxent is discussed in Section 3.3):

Most of us would surely prefer modes of reasoning which we could follow blindly without being required to make much effort, ideally no effort at all. Unfortunately, Maxent is not such a paradigm; it requires us to understand the assumptions on which it is predicated and be constantly mindful of abusing them. [46, p. 6193]

Strictly speaking, it is always next to impossible to determine an exhaustive set of elementary events; e.g. the next game might be cancelled due to bad weather or a collapse of the league, annulled due to a bribed referee or the athlete might die prior to the game. To cover all these cases, a catch-all hypothesis can be added to the set of elementary events formalised as the complementary event of the union of all other elementary events, cf. [47].

2.2 Beyond Boolean uncertainty resolution

Since Shannon’s foundation of information theory [48,49], it is a commonplace that information resolves uncertainty. Yet in practice it may not be clear how this happens, and which principles uncertainty resolution should follow. In informal scientific reasoning, it is often taken for granted that obtaining information means crossing out those possibilities which are inconsistent with the information obtained and represented as, say, sentences of the classical propositional calculus.

Building on this, de Finetti’s analysis of coherence helped pinning down the importance of uncertainty resolution in the theory of probability, for any forecast needs to be evaluated against the realised values, as made explicit by (8). In it, de Finetti (tacitly!) assumes that the uncertainty resolution mechanism is provided by (the algebraic equivalent of) classical logic. This is convenient from the point of view of the calculus, but there are many cases in which this is not necessarily an adequate modelling choice. In those contexts, which are prominent in economics and AI, Shannon’s commonplace has no straightforward, or unique application. For the way in which agents receive information is far from being unique. Good news is just waiting for us round the corner, for distinct logical semantics can lead to distinct uncertainty resolution methods.

3.1 Carnap

Rudolf Carnap is widely considered to be the founder of inductive logic. He distinguishes two concepts of probability [73] and [74, Chapter 2] (see also [75])

Probability 1 is a logical concept, a certain logical relation between two sentences (or, alternatively, between two propositions); it is the same as the concept of degree of confirmation. […] On the other hand, probability 2 is an empirical concept; it is the relative frequency in the long run of one property with respect to another. [76, p. 72]

He mostly studied the former concept of probability, which is interpreted as a degree of confirmation, c . The degree to which the evidence e confirms a hypothesis e is then c ( h , e ) . In our notation, c ( h , e ) is written as P ( θ ∣ ϕ ) . The ultimate goal of his approach is to find a probability function capturing the logical degree of confirmation.

To this end, Carnap considered a fixed first-order predicate language L containing finitely many unary relation symbols (predicates), R 1 , … , R q , and countably many constants, a 1 , a 2 , … , as names for all possible elements of an underlying domain with a countable (possibly finite) size. The language does not contain relation symbols of higher arity, nor a symbol for equality and no function symbols. The goal is thus to find a probability function capturing the degree of confirmation a sentence ϕ (the evidence) bestows on another sentence θ (the hypothesis). This, again, can be seen to be a special case of our initial Key Problem.

In it, probabilities are constrained by a number of axioms such as:

Constant exchangeability (Ex)

The probability of a sentence does not change, if we replace the distinct constants appearing in it by any other set of distinct constants of the language.

Johnson’s sufficientness postulate (JSP)

The probability of an unseen instance having certain properties given a sample of observations only depends on the size of the sample and the number of instances in this sample having exactly these properties.

Ex means that the way we name elements does not matter, and it is clearly a logical way of casting de Finetti’s notion of exchangeability recalled above. JSP says that P ( α ( a n + 1 ) ∣ ⋀ i = 1 n α h i ( a i ) ) only depends on n (the sample size) and the number of h i such that α = α h i , where all α are atoms of the language: conjunctions of the form α ( a k ) = ± R 1 a k ∧ ± R 2 a k ∧ … ± R q a k determining all properties of a constant a k .

From these two and further axioms, Carnap showed that for all languages L containing at least two unary relation symbols ( q ≥ 2 ) all probability functions defined over L satisfying this set of axioms are characterised by a one-parameter family of probability functions, P λ L , with 0 ≤ λ ≤ ∞

The parameter λ can be interpreted as a measure of how quickly evidence causes probabilities to move away from the uniform probability function. If λ = ∞ , then P ∞ L ( α j ( a n + 1 ) ∣ ⋀ i = 1 n α h i ( a i ) ) = ∣ { i : h i = j } ∣ + ∞ ⋅ 2 − q n + ∞ = 2 − q for all atoms α j : P ∞ L is the uniform distribution over L . That is, for all evidence sentences, one obtains the uniform distribution. λ = 0 means that all constants look the same with probability one: P 0 L ( α j ( a n + 1 ) ∣ α h 1 ( a 1 ) ) = 1 , if and only if j = h 1 . That is, after observing some contingent fact about a constant a , ϕ ( a ) , P 0 L assigns probability one that the same fact holds for all other constants, P 0 L ( ∀ x ϕ ( x ) ∣ ϕ ( a ) ) = 1 .

As we have seen in Section 2.1, dependence of probabilities on the underlying domain can cause serious issues. Consider a sentence ϕ of the language L . Note that ϕ is also a sentence of all other languages L ′ that contain the relation symbols R 1 , R 2 , … , R q and further unary relation symbols. Fortunately, Carnap’s P λ L do not depend on the underlying unary language L : for all fixed λ , all languages L , all sentences ϕ of L and all such languages L ′ it holds that P λ L ( ϕ ) = P λ L ′ ( ϕ ) [77, Corollary 16.2].

Carnap and others tried to determine a unique value of λ that would govern all scientific inference. No such value was found. Today, we might interpret λ as an evidential entrenchment (or stubbornness) parameter, which represents the weight of evidence required to shift away from the prior probability distribution. The particular value of λ or its range would then be determined by a particular application and could thus vary from case to case. Not being able to determine a unique value for λ is from a modern perspective an advantage rather than a fatal flaw. Historians and philosophers are still grappling with Carnap’s legacy [45,78–82], see also the extensive [83].

3.2 Pure inductive logic

Pure inductive logic is the study of rational probability from a mathematical perspective continuing Carnap’s work. The underlying first order predicate language is now allowed to contain relation symbols of arbitrary arity [77]. Pure here refers to the convention that there is no interpretation of the non-logical symbols of the underlying language. That is, probabilities are assigned in the absence of all knowledge. The aim “is to investigate this process of assigning logical, as opposed to statistical, probabilities” [77, p. 4]. Extensions of Pure Inductive Logic to include a symbol for equality [84] or function symbols [85] are still in their infancy.

The main methodology is to postulates axioms, which putatively capture (aspects of) rationality, and then investigate the logical relationships between different sets of axioms. These axioms are normally motivated by intuitions we have about symmetry, irrelevance, relevance, and independence.

If the language contains at least one relation symbol that is not unary, R say, then there are infinitely many literals required to pin down all properties of a constant a : for a binary R , one would need to write ⋀ i = 1 ∞ ± Raa i . This expression however is not a sentence of a first-order language, which only allows for sentences of finite length. Instead, one considers state descriptions α on n , which determine all properties of the first n constants: α = ⋀ R ∈ L ⋀ i 1 = 1 n ∧ … ∧ ⋀ i arity ( R ) = 1 n ± R a i 1 … a i arity ( R ) . Gaifman’s seminal theorem shows that a probability function on such a language is uniquely defined by the probabilities it assigns to the state descriptions [86]. In other words, specifying probabilities for all state descriptions defines a unique probability function on the entire language.

Two constants a , b are said to be equivalent over a state description α , if and only if the sentence α ∧ a ≡ b is consistent (note that α ∧ a ≡ b is not a sentence of the language). This induces an equivalence relation S ( α ) over the first n constants. The spectrum of α is then defined as the multiset of sizes of the equivalence relation S ( α ) . In other words, constants a and b are equivalent, if and only if replacing all occurrences of a by b in α yields a consistent sentence and replacing all occurrences of b by a in α yields a consistent sentence. We can now order the equivalence classes by size, starting with the largest class first. The spectrum of α is then defined as the vector of these sizes.

This notion of a spectrum is a key in important axioms of pure inductive logic:

Spectrum exchangeability (Sx)

If two state descriptions on n have the same spectrum, then they have the same probability.

Language invariance with P (Li with P)

A probability function on language L satisfies (Li with P), if there is a family of probability functions P ℒ , one for each language ℒ , such that P L = P and whenever L is a sublanguage of L ′ , then P L and P L ′ agree on all the sentences of all L .

Principle of induction (PI)

Let two state descriptions β , γ on n + 1 both entail a state description α on n . P ( β ∣ α ) ≥ P ( γ ∣ α ) , if there are at least as many constants equivalent to a n + 1 over β as there are over γ .

We normally consider a special case of (Li with P); we are mainly interested in probability functions satisfying (Sx). Thus, (Li with P) becomes (Li with Sx).

It is often hard to show directly that such (sets of) axioms have (un-)desirable consequences. A convenient tool are representation theorems in the tradition of de Finetti [87] of the form: All probability functions P satisfying a set of axioms have the following form P = ∫ p → ∈ B P p → d μ ( p → ) , where the P p → are probability functions generated via sampling coloured balls from an urn, and the integral is over Borel subsets of the space of possible urns. Technically, one can often show that all P p → have some property and that taking convex mixtures preserves this property.

For example, a probability functions satisfying (Sx) is t-heterogeneous, if and only if lim n → ∞ ∑ α ∈ S D n t P ( α ) = 1 , where S D n t is the set of state descriptions on n such that S ( α ) has exactly t classes. A probability function satisfying (Sx) is said to be homogeneous, if and only if lim n → ∞ ∑ α ∈ S D n t P ( α ) = 0 for all t ∈ N .

de Finetti representation theorems have been provided for t -heterogeneous probability functions [77, Theorem 31.11], homogeneous probability functions [77, Theorem 31.10], probability functions satisfying (Sx) [77, Theorem 31.12], (Li with Sx) [77, Theorem 32.1], as well as probability functions on languages containing only unary relation symbols satisfying (Ex) [77, Theorem 9.1]. Employing these results, it was proved that

1. All homogeneous probability functions satisfy (Li with Sx) [77, Proposition 30.1].

2. (Li with Sx) entails (PI) [77, Theorem 36.1].

3. Hence, all homogeneous probability functions satisfy (PI).

The status of Carnap’s continuum is not so clear for languages containing relation symbols of greater arity. If L contains at least one non-unary relation symbol, then there are only two probability functions consistent with straight-forward generalisation of (JSP) and (Ex): the uniform probability function and the natural analogue of c 0 on such languages [88] and [89, Theorem 20]. A slightly different generalisation of (JSP) is the binary sufficientness postulate (BSP). (BSP), (Ex), and the requirement to assign strictly positive probability to all quantifier-free sentences jointly produce an analogue to Carnap’s continuum, which is parameterised by two(!) parameters [90].

After adding a symbol for equality to the underlying language, one can canonically generalise JSP and obtain (JSP₌). A probability function on such a language satisfies (JSP₌) and (Ex), if and only if it is part of a one-parameter family characterised by 0 ≤ λ ≤ ∞ . These probability functions are obtained from Carnap’s P λ L by, in some sense, sending the number of relation symbols in L to infinity [77, Corollary 38.4]. Since these new functions satisfy spectrum exchangeability on the predicate language enriched with a symbol for equality [77, Lemma 38.2], they define a family of probability functions on the languages without equality that satisfy (Li with Sx) [77, Theorem 37.1]. Unfortunately, our understanding of how these probability functions behave on predicate languages with non-unary relation symbols is still a mystery [77, p. 289]. A more general class of probability functions containing these probability functions has been studied in [91,92]; connections to Ewens sampling are traced in [93].

Important open problems in pure inductive logic were recently surveyed in [94].

3.3 Inference processes

Consider an agent having acquired some information and subsequently rejected some probability functions as incompatible with the available information. For example, the agent knows the current temperature is 2 0 ∘ C and hence does not want to assign a probability greater than 1% to the event that it will be snowing tomorrow. Which remaining probability function(s) should the agent use for rational uncertain inference? This way of formulating our Key Problem permits us to study inductive logic by investigating the choice of a probability function from a set of exogenously given probability functions.

Formally, denote this set of remaining probability functions by E . Given E and no other information, which probability function should an agent adopt? The key idea is to implement this choice via a function f . This function f assigns every given set of probability functions a single probability function. It is natural to require that the function f picks out a probability function in E , f ( E ) ∈ E .

The, maybe, most basic inference process on a finite set of elementary events Ω = { ω 1 , … , ω n } is given by the function f which maps E to its centre of mass (CM). This choice is, in some sense, an average choice. While this choice seems reasonable at first glance, the probabilities picked out by f CM depend on the underlying domain in an unfortunate sense. Adding a further variable Z , one obtains an enlarged domain Ω + ≔ { ω 1 ∧ z , ω 1 ∧ ¬ z , … , ω n ∧ z , ω n ∧ ¬ z } . Supposing we have no information about Z denote by E 1 the set of probability functions on Ω + consistent with E . The highly desirable axiom of Language invariance (LI) now demands that f CM ( E 1 ) ⇂ Ω = f CM ( E ) . CM does not satisfy LI [95, p. 73]. For example, the CM probabilities for rain tomorrow may change, if a variable for the bread prices in Venice in the year 3456 is included in the underlying domain and we are, obviously, ignorant about these prices.

The obvious way to enforce language invariance in the spirit of CM is to add more and more such variables Z 1 , Z 2 , … and define f CM ∞ ( E ) ≔ lim n → ∞ f CM n ( E n ) ⇂ Ω , where f CM n ( E n ) is the centre of mass of E n . This construction gives rise to the inference process CM ∞ , which satisfies language invariance by construction [96, Theorem 5]:

Assuming that E is convex, f CM ∞ ( E ) is the point in E that is, in some sense, closest to the uniform probability function, P = . In case P = ∉ E , f CM ∞ ( E ) is in the boundary of E . CM ∞ is thus as far away from the original motivation of CM to pick a probability that are as average or typical as possible [95, pp. 73–76].

The most studied inference process is MaxEnt arising from maximising Shannon entropy of a probability function [48], H ( P ) , on convex sets E :

with the usual convention that 0 ⋅ log ( 0 ) ≔ 0 .

Vencovská and Paris provided axiomatic characterisations of ME inference in terms of the so-called common sense principles of reasoning [95,97–101]. For example, probabilities of interest ought not to depend on irrelevant information, should be invariant under renaming and should not change unless evidence to the contrary is received. Importantly, ME is language invariant [95, Section 6]. See [102] for more discussion on CM and MaxEnt. Furthermore, ME is invariant under automorphisms [46] in a broader sense than discussed in Section 2.1.

Another type of axiomatisation harkens back to Boltzmann. In 1871, he showed that the micro-states of gas molecules are most likely very close to a maximum entropy state. It is demonstrated in [103], that on very large domains almost all probability functions consistent with certain types of constraints are very close to the maximum entropy probability function.

The explication of the notion of information by Shannon entropy has given rise to a great number of axiomatisations. A very good, although by now dated, overview is [104] delineating different types of axiomatisations. Further classical key contributions were made in [105–108]. It is shown in [109] that the sort of principles which characterise maximum entropy reasoning guarantee a very high degree of conformity in the choices made by distinct agents facing essentially the same choice problem.

Further strands of research (i) characterise the maximum entropy function as a unique solution to mini-max optimisation problems. These problems are motivated by decision theoretical or information theoretical considerations [110–112]; (ii) study generalised notions of maximised entropy [110,113–116]; (iii) consider multi-agent settings [117] and investigate probabilities of conditionals [118–120].

3.4 Inference processes on infinite domains

Inference processes on infinite domains are rarely studied in generality [44]. Almost all these works focus on a maximisation of Shannon Entropy, one exception is [121]. There are, however, three obstacles for defining entropy maximisation on an infinite domain Ω by simply letting

(1) It is not clear which measure d ω to adopt. (2) There will be many cases with infinitely many probability functions with infinite entropy and hence no maximal entropy function. (3) On countable domains Ω , there exists no uniform probability function and hence no intuitive probability function to adopt in the absence of evidence.

Two explications of the maximum entropy principle have been put forward, if the domain is given by an underlying first-order language as it is in pure inductive logic. The first explication employs limits of maximum entropy functions on finite sublanguages [122,123]. Calculating these limits is an obvious way to construct the probabilities for inference. There are unfortunately cases in which these limits are ill-defined [124], and in other cases, the constructed probabilities do not satisfy the given premisses [125]. The second approach employs a pre-order on the set of probability functions consistent with the evidence. This pre-order explicates a greater-entropy-than relation. The probabilities for inference are provided by the maximal elements of the pre-order. This second approach is less constructive but more widely applicable [126–130]. It has been conjectured that these two approaches agree where the former explication is well-defined; the conjecture has so-far been verified only in a number of cases [131].

4.1 Bayesian networks

Bayesian networks are a graphical tool to represent (conditional) independences of probability functions. The possibility to calculate some conditional probabilities by considering only (small) parts of the network and to organise information graphically have contributed to their proliferation [135,136]. The possibility to effectively compute probabilities make Bayesian networks well-suited for producing probabilities for forecasting.

Formally, Bayesian networks consist of a directed acyclic graph defined on a finite set of variables V = { X 1 , … , X n } . A variable Y is a parent of a variable X , if and only if a directed edge points from Y to X . The second part of the Bayesian network is the specification of (conditional) probabilities for all variables X ∈ V relative to the set of the parents of X , Par X = { X 1 , … , X g X } and all the possible values the parent variables can take:

To ensure that a probability function is defined it needs to hold that for all X ∈ V and all possible values of the parent variables X that 1 = ∑ x ∈ X P ( X = x ∣ X 1 = x 1 , … , X g X = x g X ) . The absence of directed edges in a Bayesian network entails probabilistic independences of variables. This property facilitates the communication of modelling assumptions, information stored in data and/or knowledge bases between authors, readers, (software) engineers, and referees.

While worst-case complexity of inference in Bayesian networks is #P-complete [137] (“the answer to a #P-complete problem is the number of solutions to some NP-complete problem” [136, p. 170]), there has been much work on approximating the correct results and finding tractable sub-classes. Returning to themes of Section 3, Bayesian networks have also found applications for MaxEnt reasoning [138–140]. More generally, learning graphical models representing knowledge or databases has been of considerable interest [141].

Bayesian networks are also widely successful in modelling causal reasoning. Let us consider this example: The road, R , could either be wet because of last night’s weather, W , or because a sprinkler was on last night, S . So, R = wet , if and only if W = rain or S = on . The DAG structure of the Bayesian network is shown in Figure 1. Initially, last night’s weather and the sprinkler are independent. They become dependent once it is observed that the road is wet: Knowing that it did not rain last night entails that the sprinkler was on. However, intervening on the condition of the road by sending out a cleaning and drying machine, renders the road variable R independent of the other two variables. In Pearl’s do calculus, intervention on variables (here R ), means that all edges in the DAG pointing to this variables are deleted [142]. Hence, W and S are independent given do ( R = dry ) . Summarising, W and S are initially independent, dependent (correlated) after observing R but independent after intervening on R .

Figure 1

Directed acyclic graph of the Bayesian network for the sprinkler example.

Finally, not only might the relationships between variables be non-deterministic one may also be uncertain about the underlying causal structure. The underlying structure may be partially learnt from experts [143], data [144,145], which has certain theoretical limits [146], or be expressed as Bayesian probabilities attaching to causal Bayesian networks [147]. In the latter case, evidence concerning the causal structure is incorporated via Bayesian/Jeffrey updating, for more references on Jeffrey updating see [148].

Albeit less popular, further graphical models for uncertain reasoning have found applications. Credal nets also employ directed acyclic graphs but, unlike Bayesian networks, the conditional independences apply to a set of probability functions [149,150]. Markov nets instead employ undirected graphs and a single probability function. See [151] for an overview of further graphical models and an overview of applications.

4.2 Logics for reasoning about probabilities

As we laid out earlier, probability functions assigning sentences of a logical language some probability in the unit interval [ 0 , 1 ] are natural extensions classical valuations, which assign sentences values in a set of size two { false , true } or equivalently in the set { 0 , 1 } . Such an approach does not permit the comparison or a Boolean combination of probability statements to be formalised within the logical language. Statements of the form P ( ϕ ) < P ( θ ) and P ( ϕ ) + P ( θ ) = 0.3 have to be evaluated at a meta-level or be encoded by using sets of probabilities, see Section 2.2. Another approach to formalise the remaining uncertainty is to include probabilistic operators in the language.

There is no unique canonical approach that incorporates such statements as part of the logic, because there are number of choices to be made, which do not have an obviously correct option. One may chose

1. a logical language (propositional, first-order logic, second-order logic, fragments thereof, etc.),

2. a logic (classical, modal [152,153], many-valued, temporal, etc.),

3. a representation of uncertainty (probabilistic, imprecise probabilities, ranking functions, partially ordered, containing infinitesimals, etc.),

4. a set of operators for combining uncertainties (comparisons of uncertainties, uncertainties equalling some external value(s), combination of uncertainties (e.g. linear combinations P ( ϕ ) + 2 ⋅ P ( θ ) = 0.3 )),

5. whether knowledge and uncertainty’s fundamental representation is conditional ([35,119,120,154]).

4.3 The future of logic and probability for AI

Every formal model of learning, including machine learning techniques and neural network models, must make some assumptions concerning underlying regularities of the domain of interest. For, if the world is without regularities, then all previous observations have no bearing on (predictions of) the future. The necessity of bias supplied by humans for successful prediction is accepted in modern-day AI and was anticipated in the philosophy of science by decades:

In constructing an induction machine we, the architects of the machine, must decide a priori what constitutes its “world”; what things are to be taken as similar or equal; and what kind of “laws” we wish the machine to be able to “discover” in its “world.” In other words we must build into the machine a framework determining what is relevant or interesting in its world: the machine will have its “inborn” selection principles. The problems of similarity will have been solved for it by its makers who thus have interpreted the “world” for the machine. [158, p. 48]

Logics, such as inductive logics, can help by (i) providing biases for learning, (ii) providing means for analysing biases in successful learning systems, and (iii) elucidate black box prediction systems (such as deep neural networks) by making predictions more transparent and explainable. Graphical tools, such as Bayesian networks, can also help expert users (e.g. professionals) explain decisions and decision procedures of AI systems to lay people [159].

Our ability to create, store and manipulate ever larger data sets has led some to believe that Big Data can be the solution to many of our inference problems by letting the data speak for themselves. The success of data-driven methods has indeed been spectacular. If this hypothesis were widely true, then there would be no more need for (logically) approaches to uncertainty; data crunching would suffice.

However, data alone is never enough – not even for asymptotic events:

The data cannot [emphasis original] speak for themselves; and they never have, in any real problem of inference.

For example, Fisher advocated the method of maximum likelihood for estimating a parameter; in a sense, this is the value that is indicated most strongly by the data alone. But that takes note of only one of the factors that probability theory (and common sense) requires. For, if we do not supplement the maximum likelihood method with some prior information about which hypotheses we shall consider reasonable, then it will always lead us inexorably to favor the “sure thing” hypothesis H S , according to which every tiny detail of the data was inevitable; nothing else could possibly have happened. For the data always have a much higher probability (namely P ( D ∣ H S ) = 1 ), on H S than on any other hypothesis; H S is always the maximum likelihood solution over the class of all hypotheses. Only our extremely low prior probability for H S can justify our rejecting it. [1, p. 195]

For more recent arguments about the untenability of purely data-driven forecasting, see [160,161].

Despite initial appearances to the contrary, logical approaches to uncertain reasoning far from being out-of-date are rather gaining new actuality in the face of fast developing machine learning and AI, more widely as a foundational approach underpinning and support cutting-edge research.