Probing the Quantitative-Qualitative Divide in Probabilistic Reasoning

This paper explores the space of (propositional) probabilistic logical languages, ranging from a purely `qualitative' comparative language to a highly `quantitative' language involving arbitrary polynomials over probability terms. While talk of qualitative vs. quantitative may be suggestive, we identify a robust and meaningful boundary in the space by distinguishing systems that encode (at most) additive reasoning from those that encode additive and multiplicative reasoning. The latter includes not only languages with explicit multiplication but also languages expressing notions of dependence and conditionality. We show that the distinction tracks a divide in computational complexity: additive systems remain complete for $\mathsf{NP}$, while multiplicative systems are robustly complete for $\exists\mathbb{R}$. We also address axiomatic questions, offering several new completeness results as well as a proof of non-finite-axiomatizability for comparative probability. Repercussions of our results for conceptual and empirical questions are addressed, and open problems are discussed.


Introduction
For as long as probability has been mathematized, numbers and numerical calculation have been at the center. From the treatment of probability in the Port Royal Logic in terms of ratios of frequencies, to the modern axiomatic treatment of Kolmogorov, it has always been standard to formulate probabilistic reasoning in fundamentally quantitative terms.
It may be surprising that the systematic mathematical analysis of more qualitative probabilistic notions is relatively recent-beginning in earnest only after Kolmogorov's landmark treatise (Kolmogorov, 1933)-and still occupies not much more than a footnote in the history of the subject. 1 This is despite the fact that qualitative probabilistic locutions are of ancient origin, predating the various numerical concepts by millennia (see Franklin 2001), and despite the insistence by many that some of the qualitative notions are somehow primary. In the words of Koopman (1940), qualitative comparisons like 'more likely than' are an expression of 'the primordial intuition of probability', while the use of numbers is 'a mathematical construct derived from the latter under very special conditions ' (p. 269). Similar attitudes were expressed earlier by Keynes, de Finetti, and by many more authors since.

Why Qualitative?
Aside from inherent interest of codifying and systematizing these putatively more basic or fundamental types of judgments, a number of other purported advantages have been adduced for qualitative formulations of probability. For theorists interested in the measurement of psychological states, it has been argued that comparative judgments provide a more sound empirical basis for elicitation than explicit numerical judgments. To quote Suppes (1994), 'The intuitive idea of using a comparative qualitative relation is that individuals can realistically be expected to make such judgments in a direct way, as they cannot when the comparison is required to be quantitative ' (p. 18). Comparative judgments also appear to be more reliable and more stable over time, for example, compared to point estimates. Even when the elicited comparative judgments may be represented by a numerical function, such a numerical representation will be 'a matter of convention' chosen for reasons of 'computational convenience' (Krantz et al., 1971, p. 12).
A second rationale for considering qualitative formulations is that they may be seen as more fundamental mathematically, in part by virtue of their flexibility. Narens (1980) suggests, 'The qualitative approach provides a powerful method for the scrutinization and revelation of underlying assumptions of probability theory, is a link to empirical probabilistic concerns, and is a point of departure for the formulation of alternative probabilistic concepts ' (p. 143). Indeed, it is easy to construct orderings on events that satisfy some of the same principles as standard numerical probability while violating others. For instance, possibility theory violates finite additivity (Dubois and Prade, 1988), imprecise probabilities violate comparability (Walley, 1991), and so on. In the infinite case, qualitative orders may accommodate intuitions that elude any reasonable quantitative model (see, e.g., DiBella 2022). For those who judge the Kolmogorov axiomatization to be appropriate only for a limited range of applications, this generality and flexibility offers a theoretical advantage.
Finally, a third rationale is that qualitative systems may be in some respect simpler, a vague sentiment expressed in nearly all works on the topic. There is indeed something intuitive in the idea that reasoning about a (not necessarily total) order on a space of events is easier than reasoning about measurable functions to the real unit interval.

Probing the Distinction
Whereas a distinction between quantitative and qualitative probabilistic reasoning seems to be ubiquitous, it is not entirely clear what the distinction is exactly. In the present work we adopt a logical approach to this and related questions. By advancing our technical understanding of a large space of probabilistic representation languages, we aim to clarify meaningful ways this distinction might be drawn, and more generally to elucidate further the relationships between specifically probabilistic and broader numerical reasoning patterns.
Uncontroversial exemplars of both qualitative and quantitative systems can be identified. At one extreme, a paradigmatically qualitative language is that of comparative probability. This language, which we call L comp , 2 involves only basic comparisons P(α) P(β) with the intuitive meaning, 'α is at least as likely as β'. We take as a paradigmatically quantitative language one that allows comparison of arbitrary polynomials (sums and products) over probability terms, a language we call L poly . 3 While L comp is uncontroversially qualitative and (over finite spaces) vastly underdetermines numerical content, L poly is manifestly quantitative and is capable of describing probability measures at a relatively fine level of grain. In between these two extremes is a large space of probabilistic representation languages, essentially differing in how much numerical content they can encode. What are the natural classifications of this space?
We identify one particularly robust classification based on the amount of arithmetic a system (implicitly) encodes, namely the simple distinction between additive and (additive)multiplicative systems. Thanks to the Boolean structure of events, even L comp can already codify substantial additive reasoning. One can also add explicit addition to this language (e.g., as in Fagin et al. 1990). However, much of probabilistic reasoning appeals to notions of (in)dependence and conditionality, which seem to involve not just additive but also multiplicative patterns. This would include (purportedly qualitative) conditional comparisons like (α|β) (γ|δ), as studied by Koopman (1940), as well as seemingly even simpler constructs like (qualitative) confirmation, whereby 'β confirms α' just in case (α|β) α (e.g., Carnap 1950).
A first hint that this arithmetical classification is meaningful comes from the observation that the additive systems always admit an interpretation in rational numbers, which in turn facilitates a natural alternative interpretation in terms of concatenation on strings. By contrast, even the simplest multiplicative systems can easily force irrational numbers. For instance, if (A ∧ B) ≈ ¬(A ∧ B), then for (A|B) ≈ B to hold as well, B must have probability 1/ √ 2.

Overview of Results
One of our main results is that this distinction between additive and multiplicative systems is matched by a demarcation in computational complexity. The satisfiability problem for additive systems is seen to be complete for NP-time, thus no harder than the problems of Boolean satisfiability or integer programming. With any modicum of multiplicative reasoning, by contrast, the satisfiability problem becomes complete for the class ∃R, conjectured to be harder than NP. This classification is surprisingly robust, encompassing the most minimal languages encoding qualitative dependence notions, all the way to the largest system we consider, L poly , allowing arbitrary addition and multiplication. Within each of these classes-the 'purely' additive and the additive-multiplicative-we find a common distinction between systems that only allow comparisons between atomic probability terms and those that admit explicit arithmetical operations over terms. Indeed, on the additive side, L add augments L comp with the ability to sum probabilities. As just mentioned, this involves no increase in complexity. However, it does lead to a difference in reasoning principles, and in particular axiomatizability. Drawing on work of Vaught (1954), Kraft et al. (1959), and Fishburn (1997), we show that L comp is not finitely axiomatizable. In stark contrast, we present a new finite axiomatization of L add that is seen to be simple and intuitive. The work on L add is then adapted for L comp , to give a new completeness argument for the powerful polarization rule (Kraft et al., 1959), which has been used to supplant Scott's (1964) infinitary finite cancellation schema (Burgess, 2010;Ding et al., 2020). Rather than deriving the finite cancellation axioms and then appealing to Scott's representation result, we show directly how a variable elimination method can be adapted to show completeness.
From a logical point of view, these results together suggest that disallowing explicit addition might be seen as an artificial restriction: reasoning in L add can always be emulated within L comp , at the expense of temporarily expanding the space of events. At no cost in computational complexity, L add codifies the relevant reasoning principles in simple, intuitive axioms.
A similar pattern is seen to arise in the multiplicative setting, for systems that fall into the ∃R-complete class. Here we consider two languages, L poly and the conditional comparative language L cond alluded to above. The latter involves comparisons of the form P(α|β) P(γ|δ), with the interpretation 'α is at least as likely given β as is γ given δ'. We give a very intuitive finite system for the former language L poly by presenting a multiplicative annex to our axiom-atization for L add . Varying an argument from Ibeling and Icard (2020), where a polynomial language permitting subtraction (via unary negation) was considered, 4 we show this system complete. The argument turns on a Positivstellensatz of semialgebraic geometry. As for the latter language L cond , we review pertinent work by Domotor (1969).

Roadmap
We begin by defining several probabilistic languages, including L comp , L add , L cond , and L poly ; showing that these languages form an expressivity hierarchy; and introducing the notions from computational complexity used throughout the paper ( §2). We then consider the additive systems L add and L comp , proving the soundness and completeness of axiomatizations of both languages ( § §3.1, 3.2), discussing issues of finite axiomatizability ( §3. 3), and rehearsing results that characterize the complexity of reasoning in these systems ( §3.4). We turn next to the multiplicative systems, providing an axiomatization of the language L poly ( §4.1), investigating axiomatic questions for the language L cond ( §4.2), and characterizing the complexity of reasoning in these multiplicative systems, including a minimal logic L ind allowing only for Boolean combinations of equality and independence statements ( §4.3). In §5, we summarize the results of this discussion: both in the additive and multiplicative settings, systems with explicitly 'numerical' operations are more expressive and admit finite axiomatizations, while incurring no cost in complexity; this does not seem to be the case for 'purely comparative' systems (as we showed in the additive case, and conjecture for the case of conditional comparative probability). Finally, in §6, we critically discuss various ways of understanding the distinction between qualitative and quantitive probability logics, before concluding in §7 with some open questions.

A Space of Probabilistic Representation Languages
In this section, we define the syntax and semantics of the additive and multiplicative languages which are the primary focus of the paper's discussion, and we illustrate that these languages form an expressivity hierarchy. We also introduce the key notions from computational complexity that are used to characterize the satisfiability problems of these languages.

Syntax and Semantics
Fix a nonempty set of proposition letters Prop, and let σ(Prop) be all Boolean combinations over Prop. We will be interested in terms P(α) for α ∈ σ(Prop), which will be standardly interpreted as the probability of α. We first define several sets of probability terms, next define languages of comparisons between terms, and finally provide a semantics for these languages: Definition 2.1 (Terms). Define sets of terms using the following grammars: a ∈ T quad ⇐⇒ a := P(α) · P(β) for any α, β ∈ σ(Prop) a ∈ T poly ⇐⇒ a := P(α) | a + b | a · b for any α ∈ σ(Prop).

The Axioms and Rules of AX base
Lin.

Transitivity and comparability of
Bool.

Definition 2.2.
Define an operator Λ that, for each set of terms T , generates a language of comparisons in those terms: Definition 2.3 (Additive languages). Define L comp = Λ(T uncond ) and L add = Λ(T add ).

Definition 2.4 (Multiplicative languages)
. Define L cond = Λ(T cond ), L quad = Λ(T quad ), and L poly = Λ(T poly ). Define two further languages: We assume that 0 is an abbreviation for P(⊥), where ⊥ is any Boolean contradiction, and likewise 1 is an abbreviation for P( ). We let t ≈ t abbreviate t t ∧ t t; and let t t abbreviate t t ∧ ¬t t. Some generally valid axioms and rules (call these principles 'core' probability logic) appear in Fig. 1. Note that reflexivity of and non-negativity, P(α) 0, both follow from Dist. These axioms and rules will be part of every system we study. Call them AX base .

Definition 2.5 (Semantics). A model is a probability space
The denotation P(α) M of a basic probability term P(α) is defined to be P([[α]]), while the donations of complex terms a + b and a · b are defined in the usual recursive manner.
We define truth of basic inequality statements in the obvious way: while Boolean combinations are evaluated as usual. In L cond , L ind , and L confirm , we specify their atomic clauses separately: Equality statements and rational terms are evaluated as expected.

An Expressivity Hierarchy
The languages introduced in the preceding section form an expressivity hierarchy. For a formula ϕ in any of these languages, let Mod(ϕ) = {M : M |= ϕ} be the class of its models. For two languages L 1 and L 2 , we say that L 2 is at least as expressive as L 1 if for every ϕ ∈ L 1 there is some ψ ∈ L 2 such that Mod(ϕ) = Mod(ψ). We say L 2 is strictly more expressive than L 1 if L 2 is at least as expressive as L 1 but not vice versa. Two languages are incomparable in expressivity if neither is at least as expressive as the other. Figure 2 illustrates the expressivity hierarchy among the languages introduced above. In particular, L comp is less expressive than both L add and L cond , both of which are less expressive than the language L poly .
In this section, we provide examples to establish the above-pictured hierarchy, with each segment between a pair of languages indicating a strict increase in expressivity. In particular, we will show that L comp is strictly less expressive than both L add and L cond , and that both of these are strictly less expressive than the language L poly . In fact, in all but one case we show something stronger: namely that (fixing Ω, F, and [[·]]) there are two measures P 1 and P 2 which are indistinguishable in the less expressive language but which can be distinguished by some statement in the more expressive one. This stronger result is not possible in the case of L add and L poly , because of the following: Fact 2.6. Any distinct measures P 1 , P 2 are distinguishable in L add .
Proof. If P 1 , P 2 are distinct measures, without loss of generality, P 1 (α) < n/m < P 2 (α) for some natural numbers n and m. Thus P 1 (α) added to itself m times is at most P 1 ( ) added to itself n times, while this is not true of P 2 ; thus P 1 , P 2 are distinguishable in L add .
To show that L add is less expressive than L poly , we simply identify a formula ϕ ∈ L poly such that there is no ψ ∈ L add with Mod(ϕ) = Mod(ψ). For this we can take the example mentioned in the introduction: P(A ∧ B) = P(¬A ∨ ¬B) and P(A|B) = P(B). (This is in fact expressible already in L cond .) As mentioned above, this enforces that P(B) = 1/ √ 2, while it follows from Corollary 6.1 below that every formula in L add has models in which every probability is rational.
Multiplicative systems. First, we show that L comp is no more expressive than L ind . Define the measures so that the measures P 1 and P 2 are not distinguishable in L comp .
Next, we show that L ind is less expressive than L confirm . Define the measures The above measures satisfy the same order satisfied by the measures in the preceding example, and A and B are not independent under either measure, so the measures are indistinguishable in L ind . However, P 1 (A|B) < P 1 (A), while P 2 (A|B) > P 2 (A), so that the measures are distinguishable in L confirm . Then, note that L confirm and L comp are incomparable in expressivity. The following are L confirm -equivalent: These measures are, however, evidently distinguishable in L comp . A fortiori, they are distinguishable in L cond . Thus this example also shows that L confirm is strictly less expressive than L cond .
Finally, we show that L quad is less expressive than L poly . Defining P 1 (A) = 2 /3 and P 2 (A) = 3 /4 we find that for i ∈ {1, 2} while P 1 (A) 3 < P 1 (¬A) and P 2 (A) 3 > P 2 (¬A), so that the measures are not distinguishable in L quad but are distinguishable in L poly .
Summarizing the results of this section: Theorem 1. Figure 2 describes an expressivity hierarchy; each language is less expressive than any higher-up language to which it is path-connected and incomparable in expressivity to all other languages. In words, the language L comp is less expressive than L add , which is again less expressive than L poly . Similarly, the languages L comp , L cond , L quad , and L poly form an expressivity hierarchy, with each language in the series less expressive than the one that follows it, as do the languages L ind , L confirm , and L cond . The languages L ind and L confirm are incomparable in expressivity with the languages L comp and L add .

Complexity
In this subsection, we introduce the ideas from complexity theory needed to state some of the paper's results. We denote by SAT L the satisfiability problem for L.
Definition 2.7 (Polynomial-time, deterministic reductions). A polynomial-time, deterministic reduction from one decision problem A to another decision problem B is a deterministic Turing machine M , such that a ∈ A if and only if M (a) ∈ B, with M (a) computing in a number of steps polynomial in the length of the binary input a. We write A ≤ B when there exists such a reduction..
In particular, when there is a polynomial-time map from L 1 to L 2 which preserves and reflects satisfiability, we write SAT L1 ≤ SAT L2 . Definition 2.8 (NP-reductions). An NP-reduction from A to B is a nondeterministic Turing machine M , such that a ∈ A if and only if at least one of the outputs M (a) is in B, with each output M (a) computing in a number of steps polynomial in the length of the binary input a. 5 Definition 2.9 (Complexity classes). When each member of a collection C of decision problems can be reduced via some deterministic, polynomial-time map to a particular decision problem A, one says that the problem is C-hard; if in addition, A ∈ C, then A is C-complete. The class C of decision problems is called a complexity class. Definition 2.10 (Closure under NP-reductions). A complexity class C is closed under NPreductions if whenever there is an NP-reduction from A to B ∈ C, then in addition A ∈ C.
We are concerned here with two complexity classes which are closed under NP-reductions (ten Cate et al., 2013): NP and ∃R. We discuss each in turn.
The Class NP. The class NP contains any problem that can be solved by a non-deterministic Turing machine in a number of steps that grows polynomially in the input size. Hundreds of problems are known to be NP-complete, among them Boolean satisfiability and the decision problems associated with several natural graph properties, for example possession of a clique of a given size or where S is a system of equalities and inequalities of arbitrary polynomials in the variables x 1 , ..., x n . For example, one can state in ETR the existence of the golden ratio, which is the only root of the polynomial f (x) = x 2 − x − 1 greater than one, by 'there exists x > 1 satisfying f (x) = 0.' The decision problem of saying whether a given formula ϕ ∈ ETR is complete (by definition) for the complexity class ∃R. The class ∃R is the real analogue of NP, in two senses. Firstly, the satisfiability problem that is complete for ∃R features real-valued variables, while the satisfiability problems that are complete for NP typically feature integer-or Boolean-valued variables. Secondly, and more strikingly, Erickson et al. (2020) showed that while NP is the class of decision problems with answers that can be verified in polynomial time by machines with access to unlimited integervalued memory, ∃R is the class of decision problems with answers that can be verified in polynomial time by machines with access to unlimited real-valued memory.
As with NP, a myriad of problems are known to be ∃R-complete. We include some examples that illustrate the diversity of such problems: • In graph theory, there is the ∃R-complete problem of deciding whether a given graph can be realized by a straight line drawing (Schaefer, 2013).
• In game theory, there is the ∃R-complete problem of deciding whether an (at least) threeplayer game has a Nash equilibrium with no probability exceeding a fixed threshold (Bilò and Mavronicolas, 2017).
• In geometry, there is the ∃R-complete 'art gallery' problem of finding the smallest number of points from which all points of a given polygon are visible (Abrahamsen et al., 2018).
• In machine learning, there is the ∃R-complete problem of finding weights for a neural network and some training data such that the total error is below a given threshold (Abrahamsen et al., 2021).
The inclusions NP ⊆ ∃R ⊆ PSPACE are known, where PSPACE is the set of decision problems solvable using polynomial space; it is an open problem whether either inclusion is strict.

Notation
We denote probability measures by P and formal logical symbols for such measures by P. We use A, B, C to denote propositional atoms; Greek minuscule α, β, γ, δ, , ζ to denote propositional formulas over such atoms; sans-serif a, b, c, etc. to denote terms (elements of the various T * ) in probabilities of such formulas; and ϕ, ψ, χ to denote formulas comparing such terms (viz. formulas of the L * ). At various points in the paper we rely on the following definition:

Positive Linear Inequalities
Our first task is to provide an axiomatization of the language L add . Aside from the basic principles of AX base (Fig. 1), we have the following additivity axiom: Add. P(α) ≈ P(α ∧ β) + P(α ∧ ¬β)

The Axioms of AX add
Add. We also have core axioms for dealing with addition. The system AX add is shown in Figure 3. 6 A number of further principles are easily derivable in AX add , which we record in the following lemma, with suggestive names: Lemma 3.1. The following all follow in AX add : where ϕ a b is the result of replacing some instances of a by b. The main result of this subsection is a completeness proof for AX add . Unlike existing completeness arguments for additive probability logics (such as that in Fagin et al. 1990), the proof here proceeds solely on the basis of a variable elimination argument.

Theorem 2. AX add is sound and complete.
Proof. Soundness is routine. For completeness, we show that the system is strong enough to carry out a variation on the Fourier-Motzkin elimination method for solving linear inequalities.
By Dist and Add, we can assume that in every probability term P(δ), the formula δ is a (canonical) complete state description over finitely many propositional atoms, or else a contradiction. Thus we can assume that for every two probability terms P(δ) and P(γ) appearing in the formula, δ and γ are logically inconsistent. Their values are therefore constrained only by the restrictions explicitly implied by the formula, and by the fact that their sum must be greater than 0. By NonDeg we can assume that our formula is conjoined with the (derivable) statement δ P(δ) 0, since the two will be interderivable. In other words, our formula will be satisfiable iff the corresponding linear system-replacing each P(δ) with a distinct variable x and P(⊥) with 0-has a solution. Note that it suffices to consider (un)satisfiability over solutions in Q + , the non-negative rationals, as we can always normalize to obtain a solution in [0, 1] corresponding to a probability measure.
Our strategy will be as follows. Suppose ϕ is valid. We want to show that we can transform ¬ϕ into an equisatisfiable formula ψ, such that AX add ¬ϕ → ψ. Because the sentence ψ will have a particularly simple form, we will be able to tell easily that its negation is derivable. It will follow at once (by Boolean reasoning) that AX add ϕ.
Assume that ¬ϕ is in disjunctive normal form, and consider any disjunct, which we can assume is a conjunction of equality statements (≈) and strict inequality statements ( ). Pick any 'variable' x = P(δ). We want to show how x can be eliminated from each conjunct in a way that leads to an equisatisfiable formula that is also derivable from the previous formula. By principles 1Canc and Dupl (together with Lin, Assoc, and Comm) we can assume without loss a fixed k > 0, such that each conjunct containing x has one of the following forms: where kx is an abbreviation for the k-fold sum of x, and where x does not appear anywhere in terms a or b. If there is no a, simply let it be 0, admissible by Zero. To show that we can eliminate x altogether, consider the following cases: (a) There is at least one conjunct of type (i). In this case, principles Sub1 and Sub2 allow eliminating x from all but one conjunct of type (i), as well as all conjuncts of types (ii) and (iii). It remains only to show that x can be eliminated from the last equality kx + a ≈ b. Since x appears nowhere else in the conjunct, the whole formula will be equisatisfiable with the result of replacing kx + a ≈ b with b a. Moreover, the latter is derivable from the former by transitivity of and using Comm and Mono.
(b) There are conjuncts of both types (ii) and (iii). This case is handled by principle Elim. For each pair kx + a b and c kx + d, we include a new conjunct a + c b + d, which does not involve x. The resulting formula will be equisatisfiable.
(c) There are only conjuncts of type (ii). Such a formula is always satisfiable (over the positive rationals), so we can simply replace each of them with any tautology.
(d) There are only conjuncts of type (iii). In this case the equisatisfiable transformation replaces each instance b kx + a with b a. The latter can be derived from the former by transitivity, Comm, and Mono.
After the last variable has been eliminated by repeated application of the above rules, we will be left with a conjunction of (in)equalities in which every term is a sum of 0s. By Zero, we can assume every conjunct is of the form 0 0 or ¬0 0. Unsatisfiability implies that ¬0 0 must be a conjunct. But 0 0 is provable by Dist. Thus, any unsatisfiable formula is refutable in AX add .

Comparative Probability
The pure comparative language L comp is just like L add , but without explicit addition over probability terms. Thus, while we can take AX base as a basis for axiomatization, none of the remaining axioms of AX add are in the language, most saliently, the additivity axiom Add.

Quasi-Additivity
Early in the development of modern probability, de Finetti (1937) proposed an intuitive principle, subsequently called quasi-additivity (see, e.g., Krantz et al. 1971 Some authors also refer to Quasi as qualitative additivity, and it has been argued that this principle constitutes the 'hard core for the logic of uncertain reasoning ' Gaifman (2009). It was famously shown in Kraft et al. (1959) that Quasi is insufficient to guarantee a probabilistic representation, falling short of full additivity. However, there is an important sense in which this axiom does capture additivity. First, observe that Quasi is equivalent to the following variant, where δ is a Boolean expression incompatible with both γ and θ: To see this, note that Quasi emerges as the special case of Quasi when γ = (α∧¬β), θ = (β∧¬α), and δ = (α ∧ β). In the other direction, letting α = (γ ∨ δ) and β = (θ ∨ δ), we obtain where the first and third equivalences are instances of Quasi and the second follows from Dist.
Indeed, the reader is invited to check that all of the patterns recorded in Lemma 3.1 are derivable under the same restriction. For instance, 1Canc simply becomes Quasi . For another example, here is a version of Dupl: Lemma 3.3. Suppose α, β, γ, and δ are all pairwise unsatisfiable. Then every instance of the following is derivable from AX base + Quasi: Proof. Suppose α ≈ β and γ ≈ δ. Then if α γ, by several applications of transitivity we know that β δ. Applying 2CancQ from Lemma 3.2 (letting ε = ζ = ) we obtain (α ∨ β) (γ ∨ δ).
In the other direction, suppose (α∨β) (γ ∨δ) but that α γ fails. Then from our assumption above, using transitivity and Boolean reasoning, β δ must also fail. By comparability of it follows that δ β. But then ContrQ from Lemma 3.2 implies α γ, a contradiction.
Limitations arise from the case where we cannot simulate addition with disjunction, when Boolean expressions are not incompatible.
There are two approaches to circumvent the problem, each leading to a different axiomatization of the logic of comparative probability. The first and most canonical way, emerging from classical work in the theory of comparative probability orders, relies on introducing an infinitary axiom scheme which captures precisely the probabilistic representability of a comparative probability order on a Boolean algebra of events. The second one relies on introducing the Polarization rule, a powerful proof rule proposed by Burgess (2010). As we show in section 3.2.3, adding the quasi-additivity axiom and the polarisation rule to AX base gives an alternative axiomatization for L comp .

Finite Cancellation and probabilistic representability
The most common approach to axiomatizing comparative probability crucially relies on a representation theorem for comparative probability orders, due to Kraft et al. (1959) and Scott (1964). Faced with the inadequacy of de Finetti's quasi-additivity principle to guarantee probabilistic representation, these authors proposed an infinite list of axioms, often called the finite cancellation axioms. Here we follow subsequent modal-logical formulations of the axioms due to Segerberg (1971); Gärdenfors (1975). Given two lists of n Boolean formulas α 1 , . . . , α n , β 1 , . . . , β n , we can consider the set ∆ of all state descriptions, treating these Boolean formulas as atoms. We call a state description δ ∈ ∆ balanced if the same numbers of α i 's as β i 's is (not) negated in δ. Let B ⊂ ∆ be the set of all balanced state descriptions. Now define the following abbreviation: For all n, and all pairs of sequences of n formulas, we have an instance of FinCan n : Semantically, it is easy to see that the balancedness condition (α 1 , . . . , α n ) ≡ 0 (β 1 , . . . , β n ) amounts to the property that where 1 α is the indicator function of the set α . Accordingly, given a sample space Ω and a set algebra F, we say that two sequences of events A 1 , ..., A n and B 1 , ..., B n ∈ F are balanced if i≤n 1 Ai = i≤n 1 Bi . A semantic formulation of the finite cancellation axioms is then the following: The two sequences being balanced means that every element of the sample space belongs to exactly as many A i 's as B i 's. Under this description, the soundness of FinCan n is straightforward: balancedness ensures that the sums i P(A i ) and i P(B i ) are equal, since they are computed by taking the exact same sums of terms P(ω) for ω ∈ Ω: this is inconsistent with having P(A i ) ≥ P(B i ) for all i with some of these inequalities strict. (Alternatively, to show soundness we can prove that FinCan n is derivable in AX add : see Appendix.) The significance of the finite cancellation scheme stems from the following theorem. Say that a relation on a Boolean algebra F is probabilistically representable if there exists a probability measure P on F such that, for all A, B ∈ F, We have: Theorem 3 (Kraft et al. 1959;Scott 1964). Let (Ω, F) a finite set algebra and a binary relation on F. There is a probability measure P on (Ω, F) representing if and only if the following hold for all A, B ∈ F:

Tot.
is a reflexive total order; The result is proved by appeal to general results in linear algebra or linear programming (see, e.g., Scott 1964or Narens 2007. It is worth sketching the proof here in order to highlight the algebraic content of the finite cancellation rule, as well as to emphasize the key differences between the additive and multiplicative systems we investigate below: particularly, the distinct tools and proof techniques involved. To prove the right-to-left direction of Theorem 3, we first formulate the task as an algebraic problem. Consider an order on events in the algebra (Ω, F) satisfying the properties listed above. Take the vector space R n . Each event A is identified with the vector 1 A of its indicator function: that is, the vector (v 1 , . . . , v n ) where v i = 1 A (ω i ). Finding a measure representing amounts to finding a linear functional Φ : can then be seen as a (non-normalised) additive measure on (Ω, F). Note that the linearity of Φ ensures that Φ(1 ∅ ) = Φ(0) = 0. Further, NonDeg and NonTriv ensure that Φ(1 Ω ) > 0 and Φ(1 A ) ≥ 0 for all A ∈ F. Importantly, additivity is also guaranteed: when A ∩ B = ∅, we have 1 A∪B = 1 A + 1 B in R n , and by the linearity of Φ we . This means that we can define the desired probability measure in the obvious way: set P(A) To find such an order-preserving linear functional, we can appeal to the following Lemma, due to Scott (1964): We obtain the desired functional by applying the Lemma to the structure (M, ), where M = {1 A | A ∈ F}, and we lift the relation to M by setting 1 A 1 B if and only if A B. In particular, note that property (b) of the Lemma, when applied to vectors of the form v i = 1 Ai and w i = 1 Bi , corresponds precisely to the finite cancellation axiom scheme FinCan n . In this way, Theorem 3 is established.
In order to get a better grasp on the algebraic content the finite cancellation axioms, it is informative to consider the following simple geometric description of the problem. We want a linear functional Φ to have the property that Each linear functional on R n can be written in the form Φ(v) = w T v for some vector w. This means that, geometrically, finding a linear functional of the desired kind amounts to finding a hyperplane separating (the cones generated by) the sets We thus need to solve the following system of linear inequalities for w ∈ R n .
A well-known theorem of the alternative (Motzkin 1951; see also Schrijver 1998) states that a system like the above fails to have a solution only if there exists an (integer-valued) certificate of infeasibility. A certificate of infeasibility for the linear system given here translates into the existence of two balanced sequences of events which violate an instance of FinCan n . The finite cancellation conditions ensure the nonexistence of certificates of infeasibility for the system of linear inequalities expressed by the order .
With Theorem 3 at hand, one can show by standard logical methods that adding the infinite axiom scheme FinCan n to AX base yields a complete axiomatization of L comp -validities (together with a simple extensionality axiom).
Theorem 4 (Segerberg 1971;Gärdenfors 1975). L comp is completely axiomatized by AX base together with the following: Three points are worth noting. First, the techniques required for proving the completeness result belong entirely to the standard toolkit of linear algebra. We see how this canonical axiomatization of L comp is based on hyperplane separation methods. L comp can only express linear constraints on the representing probability measure; the axiom schemes ensure precisely the consistency of the systems of linear inequalities that the language can consistently express. As we will see, this will no longer hold in any of the multiplicative systems we will consider, which can express polynomial constraints: there, proving completeness will require showing that the system is powerful enough to prove the consistency of certain systems of polynomial inequalities. Thus the study of multiplicative probability logics involves techniques from semialgebraic geometry.
Second, in this case the linear functional in the representation theorem can in fact always be taken to be rational-valued: in other words, no constraints expressible in L comp can force the probability of an event to have an irrational value. This entails that any consistent formula in L comp has a model where the probabilities are all rational. As a consequence, one can show that L comp is sound an complete with respect to finite counting models, in which α β holds exactly if more states in the model satisfy α than β (van der Hoek, 1996). This is not in general the case for multiplicative systems: as we saw above, the ability to express polynomial constraints can force irrational probabilities for some events.
Lastly, we saw this canonical axiomatization is infinite. In the next section, we will show that one can avoid the infinite cancellation scheme by enriching our base system with a powerful

The Axioms and Rules of AX comp
proof rule. In Section 3.3, we will show that, without such a strong proof rule, an infinite axiom scheme is unavoidable: within the basic rules of system AX base , the logic L comp is not finitely axiomatizable.

Polarization
Intuitively, if we could only 'duplicate' formulas whenever we want to add probabilities for overlapping events, this would license the same reasoning capacities as with linear inequalities. Such a proof rule was introduced by Burgess (2010), following Kraft et al. (1959). Suppose A is a proposition letter that occurs nowhere in α or ϕ. Then the polarization rule says: The soundness of Polarize is straightforward to show (see Ding et al. 2020): if ¬ϕ is satisfiable, it suffices to show that ¬ϕ can be satisfied together with (α∧A) ≈ (α∧¬A). This is achieved by duplicating [[α]], the extension of α, and ensuring that all A-free formulas are thereby preserved. Completeness, however, is less straightforward. Existing treatments show how the infinitary schema FinCan n can be derived from Polarize (cf. Burgess 2010; Ding et al. 2020); as we saw in §3.2.2, completeness of the infinitary system in turn depends on additional facts from linear algebra. Here we can give a more direct argument, showing exactly how polarization, together with de Finetti's quasi-additivity axiom, recapitulates the additive reasoning for variable elimination that we gave for AX add (Theorem 2). Let AX comp consist of the axioms and rules of AX base , plus Quasi and Polarize (Fig. 4).
Consider some finite set A of propositional atoms and suppose A / ∈ A. For a formula ϕ over A, define the relativization ϕ A to be the result of replacing every inequality ε ζ in ϕ with (ε ∧ A) (ζ ∧ A). Let π be the formula: where ∆ is the set of state descriptions over A. Then we have: Proof. Consider any inequality ε ζ appearing in ϕ. The result will follow by Boolean reasoning if we can just show that ε ζ ↔ (ε ∧ A) (ζ ∧ A) follows from π.
Theorem 5. AX comp is sound and complete.
Proof. The proof strategy is to follow the derivation of ϕ from AX add , showing how to transform this into a derivation from AX comp using polarization. Roughly speaking, the idea is to replace sums of probability terms with probabilities of disjunctions; polarization is used to ensure that the disjuncts can be mutually incompatible, by furnishing a sufficient number of 'copies' of each disjunct. As we saw above, quasi-additivity is sufficiently strong when reasoning about incompatible disjuncts.
As in the proof of Theorem 2, by Dist we assume that for every inequality α β appearing in ϕ, both α and β are disjunctions involving the finitely many state-descriptions δ ∈ ∆ over the propositional atoms Prop ϕ that occur in ϕ.
Because ϕ is also in L add , the proof of Theorem 2 furnishes a derivation of ϕ. Let m be the maximum factor that appears anywhere in the derivation, that is, the largest number of times any term P(δ) is added to itself. We introduce n = log 2 m fresh proposition letters Prop + = {A 1 , . . . , A n }. As the method below will mimic the previous derivation-and in particular will not introduce more factors-n atoms, and thus 2 n ≥ m state descriptions over those atoms, suffices. Define ∆ k to be the state descriptions over Prop ϕ ∪ {A 1 , . . . , A k }, so in particular, ∆ 0 = ∆, and ∆ n is the set of state descriptions over Prop ϕ ∪ Prop + .
We can now relativize ϕ n times, producing ϕ multiple applications of Lemma 3.5 allow us to conclude: Thus, under the assumption π * , it suffices just to derive ϕ * . Observe furthermore that we now essentially have m copies of each state description δ over Prop ϕ , each copy 'tagged' by a distinct state description σ over Prop + . The conjunction δ ∧ σ is in fact an element of ∆ n , i.e., a state description over Prop ϕ ∪ Prop + . It is straightforward to show that AX comp proves π * → (δ ∧ σ i ) ≈ (δ ∧ σ j ) for each δ ∈ ∆ and all σ i = σ j ; that is, every pair of elements of ∆ n that agree on Prop ϕ are provably equiprobable. Because σ i = σ j we also know that all pairs δ ∧ σ i , δ ∧ σ j are jointly unsatisfiable.
Thus, suppose that ϕ * is valid, that ¬ϕ * is in disjunctive normal form, and consider any disjunct, a conjunction of equality statements (≈) and strict inequalities ( ) between disjunctions of relativized state descriptions; that is, each (in)equality is between disjunctions of conjunctions δ ∧ i≤n A i , where δ ∈ ∆. It remains to show that each step of the variable elimination (or 'δ-elimination') argument from Theorem 2 can be emulated here.
Our aim is to eliminate the 'variable' , analogous to the k-fold sum P(δ) + · · · + P(δ) in the proof of Theorem 2. At the start we will always have k = 1, but as we proceed through the elimination of variables some will appear with greater multiplicity (but again, no greater than m). Thus, in general, every conjunct containing x will have one of the following three forms: where kx, α, and β are all mutually incompatible Boolean formulas. Here we are using Dist, DuplQ, and Quasi , which also allow us to assume k is the same across conjuncts containing x.
Note also that either of α or β can be the empty disjunction ⊥. We now carry out the same case distinctions as in the proof of Theorem 2: (a) There are only conjuncts of type (iii). In this case we can simply replace each instance β kx ∨ α with β α, which results in an equisatisfiable formula. The latter can be derived from the former by Dist and transitivity of .
(b) There are only conjuncts of type (ii). The entire disjunct will then make no difference to satisfiability, so we can replace it with any tautology.
(c) There are conjuncts of both types (ii) and (iii). This case is handled by the quasi-additive version 8 of Elim. For each pair kx ∨ α β and γ kx ∨ δ, we want to replace it with a new conjunct α ∨ γ β ∨ δ, which does not involve x. But this is only guaranteed to produce an equisatisfiable result if α and γ do not share disjuncts (and the same for β and δ). If α and γ share a disjunct, then we let γ be just like γ but with a distinct σ i for that disjunct. Because all such copies of the disjunct are provably equiprobable, we have γ ≈ γ , and thus γ kx ∨ δ. Performing any necessary analogous replacement to obtain δ in addition, the result, (α ∨ γ ) (β ∨ δ ), in place of kx ∨ α β and γ kx ∨ δ-again, derivable from the latter by the variant of Elim-will give an equisatifiable transformation of the original formula, now without any appearance of x.
(d) There is at least one conjunct of type (i). In this case, the quasi-additive versions 9 of Sub1 and Sub2 allow eliminating x from every other conjunct of type (i), as well as all conjuncts of types (ii) and (iii). As in the previous case, we may need to use duplicates of state descriptions, but this can be done in the very same manner.
It remains to show that x can be eliminated from the last equality kx ∨ α ≈ β. Since x appears nowhere else in the conjunct, the whole formula will be equisatisfiable with the result of replacing kx ∨ α ≈ β with β α. The latter is derivable from the former by Dist and transitivity of .
Finally, after eliminating all variables, we will end up with a conjunction of statements each provably equivalent (by Dist) to either ⊥ ⊥ or ¬⊥ ⊥. Unsatisfiability of ¬ϕ * means the latter must be a conjunct, but this formula is refutable in AX comp .
Consequently ϕ * is derivable, and by (3), ϕ is itself derivable, assuming π * . That is, we have shown that AXcomp π * → ϕ. Because ϕ does not involve any of the new atoms in Prop + , we can iteratively discharge the assumption of π * by Polarize, conjunct by conjunct from (2).

Finite Axiomatizability
We saw that the canonical axiomatization of the logic of comparative probability L comp features infinitely many axiom schemes. The finite cancellation axioms feature a separate axiom scheme ϕ n (α 1 , ...α kn ) for each n ∈ N, where the α's range over the Boolean formulas. By contrast, observe that the system AX add for the logic of (explicitly arithmetical) additive comparisons is finitely (scheme-)axiomatizable, in the sense that it is given by a finite set of axiom schemes over AX base : that is, the axiomatization features only finitely many axiom schemes of the form ϕ n (α 1 , ...α kn ), where the α's range over the Boolean formulas, and finitely many axiom schemes ψ n (t 1 , ...t kn ), where the t i 's range over the terms in the language. This notion of finite axiomatizability-axiomatizabilty by finitely many schemes-is the natural one to consider in our propositional setting. The standard axiomatizations of finite comparative probability structures in the literature are given by schemes of this form, with an implicit universal quantification over events. For L comp , finite (scheme-)axiomatizability in our sense corresponds to the finite axiomatizability of comparative probability structures, over finite structures, 10 by a universal sentence in a first-order language with quantification over events (or finite axiomatizability tout court, if we include a uniform substitution rule in our system).
Here we show that, for the less expressive language L comp , the presence of infinitely many schemes is inevitable. As opposed to explicitly 'arithmetical' system AX add , the logic of L comp is not finitely axiomatizable in that sense. Unless we enrich the system with powerful additional inference rules like we did in Section 3.2.3, no finite set of axiom schemes can capture the L comp -validities.
Our proof proceeds in two steps. We first note that a finite axiomatization of L comp would result in a finite universal axiomatization, over finite structures, of the class of comparative probability orders in the first-order language of ordered Boolean algebras. We then appeal to a variant of a theorem of Vaught (1954) to show that comparative probability orders are not finitely axiomatizable by a universal sentence over finite structures. The construction appeals to combinatorial results by Fishburn (1997) on finite cancellation axioms.

Vaught's theorem and finite axiomatizability
We work with the language L BA ∪ { } of Boolean algebras with an additional binary relation. The signature L BA is given by (0, 1, ⊗, ⊕, · ⊥ ) where the constant symbols 0 and 1 stand for the bottom and top element, the binary function symbols ⊗ and ⊕ stand for the Boolean meet and join operations, respectively, and the unary function symbol · ⊥ stands for Boolean complementation. Consider the following class of L BA ∪ { }-structures. where (A, 0 A , 1 A , ⊗, ⊕, · ⊥ ) is a Boolean algebra, and the order on A is representable by a probability measure, in that there exists some probability measure P on A such that, for all a, b ∈ A, we have a b if and only if P(a) ≥ P(b).
We can naturally translate each ϕ in L comp into a universally quantified ϕ in L BA ∪ { }: we assign a variable p * i := x i to each atomic p i , and extend it in the obvious way so that each Boolean expression β gets assigned a corresponding term β * ∈ L BA . 11 For each term P(β), we have P(β) * := β * , and each formula ϕ ∈ L comp is translated into a quantifier free ϕ * ∈ L BA ∪ { }. 12 Now take the translation that assigns, to each ϕ ∈ L comp , the formula ϕ := ∀x 1 ...∀x n ϕ * where p 1 , ..., p n are all the atomic propositions occurring in ϕ. Then ϕ is valid on all probability models if and only if FCP |= ϕ. In particular, a scheme ϕ(α 1 , . . . , α n ) is valid on all probability models if and only if FCP |= ϕ. Observe also that, given Theorem 3, the class FCP is axiomatized over finite structures by universal sentences, by taking standard universal axioms for Boolean algebras and adding the translations of the infinitely many FinCan n axiom schemes (as well as the AX base axioms).
universal sentence in L BA ∪ { }. For suppose there was a finite collection of schemes ∆ = {ψ 1 , ..., ψ k } such that every L comp -validity over probability models followed from ∆. Then ∆ := { ψ 1 , . . . , ψ k } ⊂ L BA ∪{ } would finitely axiomatize the universal theory of FCP over finite structures. But FCP being universally axiomatizable, this would amount to an axiomatization of FCP by a single universal sentence (over finite structures). We will now show that FCP is not axiomatizable by a universal sentence over finite structures.
We recall one useful model-theoretic definition that will be needed. . . , a n }.
We will make use of the following (minor variant of) Vaught's characterisation of structures axiomatizable by a universal sentence (Vaught, 1954). Proposition 3.8. Let K a uniformly locally finite class of structures in a finite first-order signature L. If K is axiomatizable by a universal sentence over finite structures, then (i) K is closed under substructures (ii) There is some n ∈ N such that, for any finite structure A, if every substructure B ⊆ A of size at most n belongs to K, then A ∈ K.
Proof. (i) is immediate. Let Fin the class of finite structures. For (ii), suppose K = Mod(ϕ)∩Fin for some ϕ ∈ L of the form ϕ = ∀x 1 . . . ∀x k ψ(x 1 , ..., x k ), with ψ quantifier-free. Since K is uniformaly locally finite, there is some f : N → N such that, for any A ∈ K, any substructure of A generated by k elements has size at most f (k). Take n = f (k), and we show that this n satisfies (ii). Take a finite structure A ∈ K. We show A contains a substructure of size at most f (k) which is not in K. We have A |= ϕ, so ∃ā = (a 1 , ..., a k ) ∈ A with A |= ¬ψ [ā]. Consider A ā , the substructure of A generated by {a 1 , ..., a k }. By definition of f , we have |A ā | ≤ f (k). Since ¬ψ is quantifier-free, it is preserved under substructures, so A ā |= ¬ψ[ā], hence A ā |= ϕ. So A ā ∈ K, which establishes (ii).
Note that the class FCP of finite Boolean algebras with a representable comparative probability order is locally finite, with the uniform bound on the size of substructures given by f (n) = 2 2 n . We will use Proposition 3.8 to show that FCP is not axiomatizable over finite structures by a universal formula.

The logic of comparative probability is not finitely axiomatizable
In order to show that FCP is not axiomatizable by a universal formula, we appeal to a combinatorial analysis of cancellation axioms in the setting of a restricted class of ordered Boolean algebras, which corresponds to one of the earliest comparative probability structures introduced by de Finetti (1937). Definition 3.9 (de Finetti orders). A de Finetti order is a structure (B, ), where B is a Boolean algebra and a binary relation on B satisfying the following:
∀x(x 0); A linear de Finetti order is one where the strict binary relation , defined by a b iff ¬(b a), is a linear order. A de Finetti order on n atoms is one where the underlying Boolean algebra B is a finite algebra generated by n atoms.
Note that the definition of de Finetti orders simply characterizes, in the language of ordered Boolean algebras, comparative orders satisfying the quasi-additivity axiom discussed in Section 3.2.1. 13 The notion of two sequences (a 1 , . . . , a n ) and (b 1 , . . . , b n ) of elements from B being balanced transfers naturally to the setting of Boolean algebras. Given b ∈ B and a B-atom c, (a 1 , . . . , a n ) and (b 1 , . . . , b n ) to be balanced if, for every atom c in the algebra, the number of a i 's above c equals the number of b i 's above c. We write (a 1 , . . . , a n ) ≡ 0 (b 1 , . . . , b n ) to express that the two sequences are balanced. 14 Consider the following property S k : (S k ) is a variant of the finite cancellation axiom FinCan k . Note that k here counts the number of distinct premises a i b i in the antecedent, and not the number of premises. Observe also that a linear de Finetti order satisfying S k for all k ∈ N also satisfies all instances of finite cancellation FinCan k . Thus, by Theorem 3, linear de Finetti orders satisfying all axioms (S k ) belong to FCP: they are probabilistically representable.
One can use the (S k ) axioms to measure 'how much' finite cancellation an order needs to satisfy in order to be probabilistically representable. Given a fixed bound on the size of a finite algebra (say, at most n atoms), we can ask: is there some k such that every de Finetti order of this size satisfying S k is representable? Fishburn (1996Fishburn ( , 1997 investigates such bounds. He defines: f (n) := min k ∈ N | every linear de Finetti order on n atoms that satisfies S k is representable Known bounds on f (n) are given by the following: Proposition 3.10 (Kraft et al. 1959;Fishburn 1996). For all n, f (n) ≤ n + 1. Proposition 3.11 (Fishburn 1997). For any m ≥ 6, there exists a linear de Finetti order on a Boolean algebra with m atoms that fails S m−1 , but satisfies S m−2 .
We now use these bounds to prove our desired result. Proposition 3.12. The class FCP is not axiomatizable by a universal sentence over finite structures.
Proof. We show that condition (ii) of Proposition 3.8 fails for FCP. That is, we show that for any n ∈ N, there exists some finite structure A = (A, ) ∈ FCP such that every one of its substructures B = (B, B) of size at most n is in FCP. Given sufficiently large n, take a linear de Finetti order A as given by Proposition 3.11 with m ≥ log 2 n + 3. Then A is not representable, as it fails S m−1 . Now take any of its substructures (B, B) of size at most n ≤ 2 m−3 . Such an algebra has at most m−3 atoms, and it is evidently a linear de Finetti order 13 Recall that, in the plain set-theoretic language of comparative probability orders, quasi-additivity is equivalent to the statement that for any A, B and C such that (A ∪ B) ∩ C = ∅, we have A B if and only if A ∪ C B ∪ C.
14 It follows from the formulation in balancedness (1)   From this we can conclude: Theorem 6. L comp is not finitely axiomatizable over AX base .

Complexity of additive systems
In this section, we recall well-known results that characterize the complexity of reasoning in the additive systems L comp and L add .
Theorem 7. SAT comp and SAT add are NP-complete.
We rehearse a proof by Fagin, Halpern, and Megiddo (1990) to show that SAT add is NPcomplete. This requires two lemmas: Lemma 3.13. If there exists a non-negative solution to a system of m linear inequalities with integer coefficients each of length at most , then the system has a non-negative solution with at most m nonzero entries, and where the size of each entry is O(m + m log(m)).
Proof. We begin by transforming the system of linear inequalities into a linear program. Let x 1 , ..., x n denote the variables appearing in the system. For each non-strict inequality constraint j a i,j x j ≤ b j , one can introduce a slack variable x n+j and define the equality constraint j a i,j x j + x n+j = b j . Similarly, for a strict inequality constraint j a i,j x j < b j , one can introduce x n+j and write j a i,j x j + x n+j = b j , adding to this the constraint that x n+j ≥ x 0 . This gives rise to system of linear constraints Ax = b in the variables x 0 , ...., x n+m , which can be placed in the following linear program: maximize x 0 subject to Ax = b, x ≥ 0 Because the original system has a non-negative solution, the above linear program has a solution for which the objective function x 0 is positive. It is well-known (see, for example, Ch. 8 of Chvátal 1983) that the simplex algorithm, which traverses the vertices of a convex polytope associated with the system of inequalities Ax ≤ b, will discover an optimal, non-negative solution x * to the system Ax = b, and in this case it follows from optimality that x * 0 > 0. Thus x * 1 , ..., x * n provide a non-negative solution to the original system of inequalities. The simplex algorithm explores solutions to the linear program which lie at vertices by successively setting n − m variables to 0 and setting the remaining m variables so as to satisfy the linear constraints, so the solution x * has at most m variables positive.
Following the presentation in Williamson (2013), we now bound the size of the non-zero entries of x * . We observe that deleting the zero entries of x * and the corresponding rows of b and columns of A gives vectorsx andb and a matrixĀ such thatĀx =b. By Cramer's rule, whereĀ j is the result of replacing the j th column ofĀ withb. It suffices to show that one can express det(Ā) using at most O(m + m log(m)) bits. Recall that Each entry a iσ(i) ofĀ has size at most . Thus each term in the above sum has size at most m · . Noting that |S m | = m!, relabel the terms in the above sum and define y i such that Group the y i in pairs arbitrarily and sum them to produce a sequence y i which is half as long: The size of each sum y i is at most one greater than that of its summands. Repeating this process log(m!) ≤ log(m m ) = m · log(m) times produces a single number of size at most m · + m · log(m). Definition 3.14. Let |ϕ| denote the number of symbols required to write ϕ, and let ||ϕ|| denote the length of the longest rational coefficient of ϕ, written in binary. Lemma 3.15. Suppose ϕ ∈ L add is satisfiable. Then ϕ has a model which assigns positive probability to at most |ϕ| events δ ∈ ∆ ϕ , where the probability assigned to each such δ is a rational number with size O(|ϕ| · ||ϕ|| + |ϕ| · log(|ϕ|)).
Proof. Since ϕ is satisfiable, it has a model M which is also a model of some disjunct ψ appearing in the disjunctive normal form of ϕ. Pushing all sums in ψ to one side, we observe that ψ is a conjunction of formulas of the form Let ψ(∆ ϕ ) be the result of replacing each instance of P( i ) in ψ with the sum δ∈∆ϕ |=δ→ i

P(δ)
and adding the constraint δ∈∆ϕ P(δ) = 1. Then a model of ψ(∆ ϕ ) is a model of ψ and so a model of ϕ, and the formula ψ(∆ ϕ ) simply describes a system of at most |ϕ| linear inequalities, so the result follows immediately from Lemma 3.13.
Proof of Theorem 7. Since L comp ⊆ L add , it follows that SAT comp ≤ SAT add . It thus suffices to show that SAT comp is NP-hard and that SAT add is in NP.
The Cook-Levin Theorem Cook (1971) states that satisfiability for Boolean formulas α is NP-complete. This is equivalent to the problem of deciding whether P(α) 0 is satisfiable, which is an instance of the problem SAT comp , showing that the latter is NP-hard.
Consider now the task of determining whether ϕ ∈ L add is satisfiable. Using Lemma 3.15, we request a small model as a certificate and confirm that it satisfies ϕ.

Polynomial probability calculus
The paradigmatic multiplicative language is L poly (Definition 2.4), which adds a binary multiplication operator (denoted ·). Encompassing all comparisons between polynomial functions of probabilities, this language is sufficiently rich to express, e.g., conditional probability and (in)dependence. The system AX poly ( Figure 5) annexes axioms capturing multiplication and its interaction with addition to AX add , and our principal result here is its completeness.
Theorem 8. AX poly is sound and complete.
Proof. Soundness is again straightforward. As for completeness, we first obtain a normal form. We assume, without loss (Bool), that ϕ is a conjunction of literals. Let Prop ϕ ⊂ Prop be the finite set of letters appearing in ϕ, with ∆ ϕ the set of complete state descriptions of Prop ϕ (see §2.4). Then: Lemma 4.1. Replacement of equivalents Repl (see Lemma 3.1) for L poly is derivable in AX poly .
Proof. It suffices (by induction, and both versions of Comm) to derive a ≈ b → a + c ≈ b + c and a ≈ b → a · c ≈ b · c. The first follows by completeness of AX add . As for the second: by NonNull case on c 0 and c ≈ 0; use Canc in the former case and Zero in the latter case.
Proof. Using Repl recursively and instances of Add (where stands for α and a single letter from Prop ϕ stands for β) we can show that P( ) ≈ δ∈∆ϕ P( ∧ δ). Since formulas in ∆ ϕ are complete state descriptions for the letters appearing in , we have propositionally |= ( ∧ δ) ↔ δ if |= δ → , and |= ( ∧ δ) ↔ ⊥ otherwise. Applying Dist, Repl, and Zero (from AX add ) we obtain the final result.
Note that the order and associativity of the sum above generally does not matter, as can be easily seen using (additive) Comm and Assoc. The same holds for products, so we will simply omit this information where convenient. Finally, we have a normal form:

Lemma 4.3. Let ϕ be a conjunction of literals. Then there is a collection of polynomial terms
1≤i ≤m using only the basic probability terms {P(δ)} δ∈∆ϕ ∪ {0} such that Proof. The first two conjuncts on the right in (4) are clearly derivable (with or without ϕ). Literals in ϕ are of the form a b or ¬(a b), the latter being b a; thus the literals in the formula give the terms 1≤i ≤m in (4). By Repl (Lemma 4.1) and Lemma 4.2, we may assume without loss that any of these terms uses only basic terms {P(δ)} δ∈∆ϕ ∪ {0}.
The conjuncts in (4) translate into a simultaneous system of polynomial inequalities in the indeterminates {P(δ)} δ∈∆ϕ . It is clear that ϕ is satisfiable iff this system has a solution, so let us apply a well-known characterization (Stengle, 1974) Note that the well-known Farkas' lemma of linear programming (obtained via Fourier-Motzkin elimination) is a special case of this more general theorem of the alternative, in which the certificate can always be taken to have a certain restricted form. The following variant is more directly applicable for our purpose: Corollary 4.4. Suppose above that R = Z[x 1 , . . . , x n ]. Then either the given system has a solution over R n , or there are g ∈ cone(G), h ∈ ideal(H), n ∈ N, d ∈ Z + such that g + h + df 2n = 0. (5) Proof. In the latter alternative of Theorem 9, take the certificate and multiply by d, setting it to the least common denominator of coefficients in g, h.
The normal form (4) yields F, G, H ⊂ Z + [{P(δ)} δ ] for Corollary 4.4. Note that by translating the conjunct δ P(δ) ≈ 1 as two inequalities we may take H = ∅; the strict inequality Here a i , b i denote the (informal) polynomials translated in the obvious fashion from the respective formal terms a i , b i ; this convention will be used hereafter.
When translating a polynomial a backward to its formal equivalent denoted a, we will assume a is a sum of so-called normal monomials. Fixing some total order ≺ ∆ϕ on K = {P(δ)} δ∈∆ϕ ∪ {0, 1}, call a term t over K a normal monomial if: (1) only multiplication · appears within it; (2) all multiplication is left-associative; (3) for any base terms P(δ), P(δ ) in t, if δ ≺ ∆ϕ δ , then P(δ) appears to the left of P(δ ); (4) 1, that is, P( ), appears exactly once as a factor in t, and is leftmost; (5) 0, that is, P(⊥), appears once if ever, and if it appears then no other term from K appears. It is easy to see that any term has an equivalent that is a sum of normal monomials by applying Dist toward fulfilling (1), Assoc for multiplication toward (2), Comm toward (3), One toward (4), and Zero toward (5). Fixing some order on normal monomials, and applying the completeness of AX add and again Zero (for AX add ), we see that we can assume without loss that the sum is in said order and is left-associative, and further that the normal monomial containing 0 appears exactly once in it and appears leftmost. We call such a sum the normal monomial form of a given term. Now, suppose we have a certificate c = g + df 2n = 0 as in (5) for this system. Given a nonzero polynomial a let a + , a − be the unique polynomials with strictly positive coefficients such that a = a + −a − ; let a + = a − = 0 if a = 0. Let j = df 2n , c + = g + +j + , and c − = g − +j − . We claim that (4) → c + c − ∧ c + ≈ c − . One can in fact show c + ≈ c − : assuming normal monomial form without loss, the same normal monomial terms, with the same multiplicities, must appear as summands in c + and c − , because otherwise c + = c − (since 0 appears exactly once in a monomial term for both sums, it cannot make the difference).
We claim that (4) → g + g − while (4) → j + j − ; again by completeness of AX add , this gives (4) → c + c − . We make use of the following result.
as is the analogous rule when every is replaced by a strict .
Proof. Straightforward by inducting on n: apply Sub and Dist, e.g.
To see that (4) → g + g − , note that since g ∈ cone(G), it is a sum of terms of the form g = k 2 g 1 . . . g l , where g 1 , . . . , g l ∈ G and k, g 1 , . . . , g l = 0. It thus suffices to show (4) → (g ) + (g ) − for any such term g . Given our construction of G from (4), note that the inequality g + i g − i appears in (4) for each g i . Referring to Lemma 4.5, note that if we define p = (t 1 − t 1 ) . . . (t n − t n ) = 0, then p + = s∈S + s and p − = s∈S − s; by casing on whether k + k − or k − k + , we can apply the Lemma, finding in either case that (4) → (g ) + (g ) − . Now, to see that (4) → j + j − it suffices to show that (4) → (f 2n ) + (f 2n ) − . If F = ∅ is empty, then f = f ∈F f = 1 and this is simple; otherwise, for each strict inequality (4). If any such f = 0 then by employing normal monomial form, we find (4) → ⊥; otherwise, apply the strict variant of Lemma 4.5.
This language allows us to reason about comparisons of conditional probabilities. Reasoning about such comparisons plays an important role in a variety of settings. A salient example is probabilistic confirmation theory, where conditional probabilities are interpreted as a measure of the comparative support that some evidence confers to a hypothesis: a comparison of the form P (H 1 |E 1 ) P (H 2 |E 2 ) is interpreted as the statement that evidence E 2 confirms hypothesis H 2 at least as much as evidence E 1 confirms hypothesis H 1 . More generally, the focus on conditional probabilities is often motivated by the view-found, for instance, in Keynes (1921) and Rényi (1955)-that (numerical) conditional probabilities are more fundamental than unconditional probability judgments: similarly, one may want to treat comparisons of conditional probability as more fundamental than comparisons of unconditional probabilities. 17 Although one may be tempted, at first sight, to view the logic of comparative conditional probability as only a minor extension of the logic of (unconditional) comparative probability axiomatized above, it is important to note that moving to L cond involves a substantial jump in expressivity. The language L cond allows to express non-trivial quadratic inequality constraints 18 and, as such, it belongs more naturally to the family of multiplicative systems. As we will now discuss, this comes with a rather significant shift in the complexity of its satisfiability problem as well as its axiomatization. In Section 4.3 we show that L cond and all multiplicative systems we study in this paper have a ∃R-complete satisfiability problem. Before this, we discuss some challenges involved in proving a completeness theorem for L cond . These difficulties can be traced back to certain delicate questions concerning the canonical representation theorem for conditional comparative probability orders, due to Domotor (1969). To the best of our knowledge, Domotor's proposed axiomatization is the only known such representation result for finite probability spaces that does not depend on imposing additional richness constraints on the underlying space. (See Krantz et al. 1971, §5.6 for an overview of other approaches.) However, as we explain, an important step in Domotor's proof appears to require further justification. Moreover, from the perspective of the additive-multiplicative distinction explored in this paper, Domotor's proof strategy is not fully satisfactory, as it does little to clarify the exact algebraic content of the axioms involved. In light of these observations, we may not be able to rely on Domotor's proposed axiomatization of conditional probability orders to obtain a completeness result for L cond . The question of axiomatizing L cond is thus left open.

Conditional probability and quadratic probability structures
In order to a give a complete axiomatization for the logic of comparative conditional probability, a natural first step is to identify necessary and sufficient conditions for the representation of a comparative order between pairs of events. Suppose we represent comparative conditional judgments with a quaternary relation on a finite Boolean algebra of events F. We write A|B C|D when the relation obtains for the quadruple (A, B, C, D). Such a relation is probabilistically representable if there exists a probability measure P on F such that for any A, B, C, D ∈ F, we have

A|B C|D if and only if P(A|B) ≥ P(C|D),
where P(X|Y ) := P(X ∩ Y )/P(Y ). What properties must a quaternary order satisfy in order to be representable in this way?
Throughout the literature, the answer to this question is credited to Domotor (1969), who proposes necessary and sufficient conditions for such a quaternary order to be representable by a probability measure.
Definition 4.6 (Domotor (1969)). A finite qualitative conditional probability structure (FQCP), is a triple (Ω, F, ) where Ω is a finite set, F a field of sets over Ω and a quaternary relation on F, which additionally satisfies the following properties for all A, B, C, D ∈ F:

NonZero.
A | B C | D holds only if B | Ω ∅ | Ω and D | Ω ∅ | Ω Accordingly, in the following, in any expression A | B C | D it is assumed that we are quantifying over B, D ∈ F 0 , where F 0 := {X ∈ F : X | Ω ∅ | Ω}: MultCan n . for any permutation π on {1 . . . n}: Moreover, in the last two schemes above, if holds for any comparison in the antecedent, then also holds in the conclusion.
Domotor's proposed axiomatization of conditional comparative probability relies on an axiomatization of finite quadratic probability structures. The strategy consists in proving a representation theorem for quadratic probability structures, giving necessary and sufficient conditions for a binary relation on F 2 to be representable as a product of probabilities, in the sense that there exists a probability measure P such that A × B C × D if and only if P(A) · P(B) ≥ P(C) · P(D).
The representation theorem for conditional comparative probability is based on the representation result for quadratic probability structures: the essential idea is that one can express each comparison A | B C | D as a comparison of products of the form The method for constructing of a probability measure that represents these product inequalities in the sense of (6) yields a probability measure that represents the conditional probability comparisons given by . 19 In the setting of conditional comparative probability, we say that the two sequences (of pairs of events) 1} is a partial characteristic function, given by The sum i n 1 A i |B i is undefined whenever one of the terms is undefined.
It is informative to sketch the reasoning in a little more detail, in order to highlight both the differences with the representation argument for the unconditional case (L comp ), as well as several points in Domotor's argument that require clarification. Moreover, the axioms that Domotor proposes for quadratic probability structures give a clearer motivation for Definition 4.6. They are given below.
Definition 4.7 (Domotor (1969)). A finite quadratic probability structure (FQPS), is a triple (Ω, F, ) where Ω is a finite set, F a field of sets over Ω and a binary relation on F 2 , which satisfies the following properties for all A, B, C, D ∈ F: Moreover, in Q5 n and Q6 n , if holds for any comparison in the antecedent, then also holds in the conclusion.
Note that the axiom scheme Q6 n is a version of the finite cancellation axiom for L comp , applied to product sets of the form A×B. Moreover, under the translation (7) above, the axiom scheme Q6 n corresponds to the scheme FinCanCond n for conditional comparative probability.
By contrast, the axiom scheme Q5 n captures a version of multiplicative cancellation. Given a set of n inequalities A i ×B i C i ×D i , say that an event is cancelled if it has the same number of occurrences on the left hand side of these inequalities (as A i or B i ) as on the right hand side (among C i , D i ). In the presence of the symmetry axiom Q3, the axiom Q5 n asserts that whenever we have sequence of n inequalities where all events are cancelled except for A and B on the left-hand side, and C and D on the right-hand side, then we can conclude A×B C ×D. This is exactly what we would obtain by multiplying the probabilities of all left-hand side terms on the left, and the probabilities all right-hand side terms on the right, and then cancelling any terms occurring on each side. Under the translation (7), Q5 n corresponds to the axiom MultCan n from Definition 4.6.

Domotor's argument
The necessity of axioms (schemes) Q1-Q6 n is easily verified. Domotor (1969, p. 66) provides an argument to the effect that these axioms are also sufficient for the order to be productrepresentable by a probability measure, in the sense of condition (6). We will now give a brief description of the proof strategy. This will serve, first, to emphasize the algebraic nature of the problem, which involves proving the consistency of certain polynomial constraints and thus calls for techniques beyond the standard linear algebra involved in the purely additive setting of Theorem 3. Secondly, as we will see, there is a step in the argument which suggests that the proof of sufficiency is incomplete as it stands.
Suppose we are given a finite quadratic probability structure (Ω, F, ) as in Definition 4.7: without loss of generality, we will assume we are working with the full powerset algebra F = P(Ω). We want to show that the relation is product-representable in the sense of (6). Consider the space R n×n containing all indicator functions 1 A×B of products A × B. Take the space of indicator functions as vectors in R n×n , where each 1 A×B is the vector x where x ij = 1 exactly if (ω i , ω j ) ∈ A × B (that is, we list all elements of Ω × Ω in lexicographic order). Lift the order from F to M = {1 A×B | A × B ∈ F}. Now we apply Lemma 3.4: note that axioms Q4 and Q6 n give us precisely conditions (a) and (b) in the statement of the Lemma. This gives us the existence of a linear functionalΦ : 11 , a 12 , . . . , a 1n , a 21 The existence of such a linear functional already entails that there is a measure µ on P(Ω × Ω) that respects the ordering on Cartesian products , by defining µ(A×B) :=Φ(1 A×B )/Φ(1 Ω×Ω ). But we need to ensure that there a measure µ representing which corresponds to the product of a measure P on P(Ω): i.e. such that µ(A × B) = P(A) · P(B).
We represent the linear functionalΦ as a bilinear functional Φ : R n × R n → R. It can be represented as a matrix Φ(x, y) = x T M Φ y where M Φ is a n × n matrix: intuitively, we want the (i, j)-th entry to represent the probability of {ω i } × {ω j }. We write is as We then have Φ(1 A , 1 B ) =Φ(1 A×B ). Now, in order for Φ to give rise a product measure as required, we want to know whether it can be decomposed as the product of linear functionals f 1 and f 2 , so that we can write Φ(1 A , 1 B ) = f 1 (1 A )f 2 (1 B ). This would suffice, as the symmetry axiom Q3 ensures that Φ(1 A , 1 B ) = Φ(1 B , 1 A ) (M Φ is symmetric), which would ensure that f 1 = f 2 . Then setting P(A) := f 1 (1 A )/f 1 (1 Ω ) would give the desired measure (here Q1 and Q2 ensure it is a non-degenerate measure). A general condition for a bilinear functional being decomposable in this way given by the following standard characterization: 21 Proposition 4.8. A bilinear functional Ψ : R n × R n → R can be written as Ψ(x, y) = f 1 (x)f 2 (y) for two linear functionals f 1 , f 2 : R n → R if and only if rank(M Ψ ) = 1. We then say Ψ is a rank-1 functional. 21 A n × n matrix M has rank 1 if and only if it can be written in the form M = uv T for two vectors u, v. In one direction, suppose M Ψ has rank one, and so is of the form uv T . Then so that we define f 1 (x) := u T x and f 2 (x) := v T x for the decomposition into linear functionals. Conversely, given two linear functionals generated by respective vectors u and v in the same fashion, the matrix uv T generates a decomposable bilinear functional. This is where Domotor's argument seems to face a difficulty (or perhaps an omission in the presentation). The proof began by establishing the existence of the bilinear function Φ which represents the order . At this point the proof proceeds to argue that Φ-a generic order-preserving linear functional obtained from Lemma 3.4-has rank 1. 22 Thus the strategy seems to be to argue that any such functional representing has rank 1. This, however, is not the case, as can be seen by a simple example. Say that two functionals Φ and Ψ are orderequivalent on P(Ω) × P( , 1 D ). We can have two such functionals that are order-equivalent with only one of them having rank 1. For instance, in the case Ω = {ω 1 , ω 2 }, we can take functionals given by matrices M Φ = 1/16 3/16 3/16 9/16 and M Ψ = 1/12 3/12 3/12 5/12 Here Φ is a rank-1 functional, whose order Φ is representable by the probability measure P({ω 1 }) = 1 /4, P({ω 2 }) = 3 /4. However, Ψ is evidently not rank-1. Yet the order Ψ agrees with Φ , and thus satisfies axioms Q1-Q6 n . The axioms Q1-Q6 n thus cannot guarantee in general that any linear functional that represents the order has rank 1. Clarifying this step of the argument (and indeed, determining if this difficulty may be due to a presentational ambiguity, rather than a logical gap) is complicated by the fact that the author does not spell out the decomposability argument in detail. Instead, the argument very briefly appeals to an unstated result in geometry of webs (Aczél et al., 1960), from which, given the multiplicative cancellation axiom Q5 n , decomposability is inferred. 23 Returning to the task at hand: we know that there exists a bilinear functional Φ which represents the order . In order to ensure the order is representable via products of probabilities, we want to show that the axioms guarantee the existence of a bilinear functional of rank 1 that is order-equivalent to Φ.

The (semi-)algebraic perspective on the representation problem
Whether or not the approach via web geometry can be made to succeed in showing the sufficienty of Domotor's axioms, there is a sense in which it not the most natural from the perspective of investigating the additive-multiplicative divide in probabilistic logics. Recall that Scott's representation theorem (Theorem 3), and thus the standard completeness proof for L comp , directly relates the finite cancellation axioms to certificates of inconsistency for linear inequality systems. In the same way, it is of intrinsic interest to pursue a representation theorem for conditional comparative probability that would directly reveal the algebraic content of Domotor's proposed axioms (and axiom Q5 n in particular, which captures the multiplicative behaviour of conditional probability orders).
It is thus worth considering a direct algebraic formulation of the problem. Each order satisfying the axioms Q1-Q6 n generates a system of polynomial inequalities. For every set A ⊂ Ω, the polynomial corresponding to A is given by (Domotor, 1969, p.68). 23 Unfortunately, we were unable to trace the original article by Aczél, Pickert and Radó. and similarly for strict inequalities. We need to show the resulting system is consistent whenever satisfies the axioms Q1-Q6 n . Note that this system of inequalities is given by quadratic forms: each polynomial p A (x)p B (x)− p C (x)p D (x) is homogeneous of degree 2. Formulated in this way, it is clear that proving a representation result would amount to showing that the axioms ensure that the semi-algebraic sets defined by quadratic forms of this type are indeed nonempty. By analogy to the case of L comp , here it is natural to approach this problem via the Positivstellsatz (Theorem 9). As we mentioned above, the Positivstellensatz is a semi-algebraic analogue of hyperplane separation theorems, like those used in proving completeness for L comp . It establishes that there exist certificates of infeasibility of a given form for any infeasible system of polynomials. Just as, in the additive case, any certificate of infeasibility was shown to correspond to a failure of a finite cancellation axiom, so too, one may hope, a failure of the Domotor axioms (and the Q5 n axiom in particular) can always be extracted from a Positivstellensatz certificate.

Now for each inequality
We leave a definite solution to the representation problem for conditional probability-and completeness for L cond -for further work: as a first step, we show in the Appendix that the axioms for quadratic probability structures are indeed sufficient for representation in the special case of |Ω| = 2. The observations in this section motivate further work on determining the adequacy of Domotor's axioms: this is not only to ensure that the canonical axiomatization for conditional probability orders on finite spaces is correct as it stands, but also because, as we suggested above, it is of intrinsic interest to pursue an alternative, algebraically transparent proof of the representation result.

Complexity of multiplicative systems
The main result of this section finds a uniform complexity for all of our multiplicative systems: Theorem 10. SAT ind , SAT confirm , SAT cond , and SAT poly are all ∃R-complete.
To show the above theorem, we borrow the following lemma from Abrahamsen et al. (2018):  2,2]. This problem is ∃R-complete.
We omit the proof of Lemma 4.9, but at a high level, it proceeds in two steps. First, one shows that finding a real root of a degree-4 polynomial with rational coefficients is ∃R-complete, and then repeatedly performs variable substitutions to get the constraints x i + x j = x k and x i x j = 1. Second, one shows that any such polynomial has a root within a closed ball about the origin, and then shifts and scales this ball to contain exactly the range [ 1 /2, 2].

Proof of Theorem 10. Since
it suffices to show that SAT ind is ∃R-hard and that SAT poly is in ∃R. To show the former, we extend an argument given by Mossé et al. (2022), and to show the latter, we repeat the proof given by Mossé et al. (2022) (see also Ibeling and Icard (2020)).
Let us first show that SAT poly is in ∃R. Suppose that ϕ ∈ L poly is satisfied by some model P. Using the fact that ∃R is closed under NP-reductions (ten Cate et al. (2013); cf. Definition 2.7) it suffices to provide an NP-reduction of ϕ to a formula ψ ∈ ETR. Let E contain all such that P( ) appears in ϕ. Then consider the system of equations δ∈∆ϕ P(δ) = 1 δ∈∆ϕ δ|= P(δ) = P( ) for ∈ E.
When plugged in for P, the measure P satisfies the above system, so by Lemma 3.13, this system is satisfied by some model P small assigning positive (small-sized) probability to a subset ∆ small ⊆ ∆ ϕ of size at most |E| ≤ |ϕ|. Let the set ∆ small and the model P small be the certificate of the NP-reduction. 24 The reduction proceeds by replacing each ∈ E in ϕ with the δ ∈ ∆ small which imply it, and then checking whether P small is a model of the resulting formula ψ. (The size constraint on E ensures that ψ can be formed in polynomial time.) If ϕ is satisfiable, ∆ small and P small exist, so the reduction of ϕ successfully produces a satisfiable formula ψ. Conversely, the success of the reduction with the witnesses ∆ small and P small ensures the satisfiability of ϕ, since a model of ψ is a model of ϕ.
Let us now show that SAT ind is ∃R-hard. To do this, consider an ∃R-inverse problem instance ϕ with variables x 1 , ..., x n . It suffices to find a polynomial-time deterministic reduction to a SAT ind instance ψ. We first describe the reduction and then show that it preserves and reflects satisfiability.
Corresponding to the variables x 1 , ..., x n , define fresh events δ 1 , ..., δ n ∈ σ(Prop). Define fresh, disjoint events δ 1 , ..., δ n . Let ψ be the conjunction of the constraints The formula ψ is not yet a formula in L ind , since it features constraints of the form P(α) 1 /N and 1 /N P(α). For any constraint of the form P(α) 1 /N, replace 1 /N with P( N ), replace α with α ∨ N , and require that the fresh events 1 , ..., N are disjoint with P(∨ i i ) = 1 and P( i ) = P( j ) for i = 1, ..., N . Similarly, for any constraint of the form 1 /N P(α), replace 1 /N with P( N ), replace α with α ∧ N , and require that the fresh events 1 , ..., N are disjoint with P(∨ i i ) = 1 and P( i ) = P( j ) for i = 1, ..., N . This completes our description of the reduction. The map x i → x i /2n sends satisfying solutions of ϕ to those of ψ, and the inverse map P(δ i ) → P(δ i ) · 2n sends satisfying solutions of ψ to those of ϕ. Further, the operations performed are simple, and the introduced events δ i , δ i , i and the constraints containing them are short, so the reduction is polynomial-time.
Some authors have suggested that probability logic with conditional independence terms may be a useful compromise between additive languages built over linear inequalities and the evidently more complex polynomial languages (see, e.g., Ivanovska and Giese 2010). However, the above result shows that even allowing simple independence statements among events (not to mention conditional independence statements among sets of variables) results in ∃R-hardness. Thus while probability logic with conditional independence seems on its face to offer a compromise, it in fact introduces (at least) the complexity of the maximally algebraically expressive languages considered in this paper.
The above result shows that reasoning in the seemingly simpler systems L ind and L confirm are just as complex as L poly , because the former systems allow for the expression of independence statements. We conclude with two observations relating to the above result. First, a minimal extension of L comp , discussed by Fine (1973), remains NP-complete, even though it includes some mention of conditional probability: Definition 4.10. Fix a nonempty set of proposition letters Prop. The language L same cond is defined: ϕ ∈ L same cond ⇐⇒ ϕ = P(α|β) P(α |β) | ¬ϕ | ϕ ∧ ψ for any α, α , β ∈ σ(Prop) Fact 4.11. SAT same cond is NP-complete.
Proof. Hardness follows immediately from Theorem 7 and the observation that L same cond is at least as expressive as L comp . To show completeness, take any ϕ ∈ L same cond , and let ψ be the result of replacing each term P(α|γ) in ϕ with P(α ∧ γ). We claim that ϕ and ψ are equisatisfiable. Indeed, for any measure P, we have: Thus the inequalities mentioned in ϕ hold precisely when those mentioned in ψ hold.
Second, whereas the above theorem characterizes the complexity of reasoning about independence among events, the following result due to Lang et al. (2002) characterizes the complexity of reasoning about independence among two random variables: Theorem 11 (Lang et al. (2002)). Determining independence of two random variables is complete for the complexity class coDP, the complement of DP, the class of all languages L such that L = L 1 ∩ L 2 , where L 1 is in NP and L 2 is in coNP (the complement of NP). Lang et al. (2002) also characterize the complexity of several other tasks concerning the (conditional) independence of sets of random variables.

Summary: The Additive-Multiplicative Divide
We identified an important dividing line in the space of probability logics, based on the distinction between purely additive and multiplicative systems. Additive systems can encode reasoning about systems of linear inequalities; multiplicative systems can encode reasoning about polynomial inequality systems that include at least quadratic constraints. The distinction between additive and multiplicative systems robustly tracks a difference in computational complexity: while the former are NP-complete, the latter are ∃R-complete. As a consequence, inference involving (implicitly) multiplicative notions is inherently more complex (assuming NP = ∃R): this applies to various intuitively 'qualitative' systems for reasoning about independence (L ind ), confirmation (L confirm ), or comparisons of conditional probability (L cond ).
As we saw, proving completeness for the additive and multiplicative systems involves different methods. While completeness for additive systems relies on linear algebra (and hyperplane separation theorems or variable elimination methods), the natural mathematical setting for multiplicative systems is semialgebraic geometry (and completeness relies on versions of the real Positivstellensatz).
Importantly, both in the additive and multiplicative settings, systems with explicitly 'numerical' operations are more expressive and admit finite axiomatizations, while incurring no cost in complexity. By contrast, even the most paradigmatically 'qualitative' logic of unconditional comparative probability (L comp ) is not finitely axiomatizable. 25 Thus, from a logical perspective, there is little to be gained from restricting attention, in applications of probability logics, to syntactically 'qualitative' systems without arithmetical operations. These results also illustrate how ease of elicitation and complexity of inference might come apart. As we noted, the use of comparative probability is sometimes motivated by the view that 'qualitative' judgments are more intuitive, or easier to elicit, than explicitly 'quantitative' ones. While these claims are somewhat difficult to substantiate (see the discussion below in Section 6.2), our results give a concrete sense in which intuitions about the ease of elicitation of certain comparative judgments do not reflect the complexity of inference involving these judgments. Consider, for example, the case of L cond and L add . While at first blush there may be something more immediate about comparisons of conditional probabilities that are not explicitly numerical, as opposed to comparisons expressible in L add ('is A twice as likely as B?'), our results suggest that reasoning with the former is more complex than reasoning with the latter.
While the distinction between additive and multiplicative systems is an informative dividing line that is useful in classifying the landscape of probability logics, investigating this divide also illustrates that the very distinction between qualitative and quantitative reasoning remains somewhat elusive. Certainly, prima facie natural ways to formulate the distinction, based on the simplicity and intuitiveness of comparative judgments, or on the explicit presence of arithmetical operators, do not seem to capture any clear or robust distinction that tracks properties of logical interest such as complexity, expressivity, and axiomatizability. We are thus left with the question of how to give concrete substance to the often invoked, but never delineated, qualitative/quantitative distinction: are there any structural properties of inference that are characteristic of qualitative reasoning in probabilistic contexts? We turn to this question next.

What is the Quantitative-Qualitative Distinction?
Before concluding we briefly consider the larger conceptual questions with which we began. How might we understand the prevalent distinction between quantitative and qualitative formulations of probabilistic principles and inference patterns, particularly in light of the landscape of 25 A similar phenomenon occurs in other non-numerical systems for probabilistic reasoning. See, for example, results on the non-finite axiomatizability of conditional independence for discrete random variables (Studený, 1992) and for Gaussian random variables (Sullivant, 2009). systems we have explored in the present work? We begin by entertaining several suggestions from the literature.

Previous Suggestions
As briefly discussed in the introduction, gestures toward a distinction between qualitative and quantitative formulations of probability can be found throughout the literature, going back at least as far as Keynes (1921). In the seminal work by de Finetti on comparative probability, the express goal was to 'start out with only qualitative notions' before 'one arrives at a quantitative measure of probability' (de Finetti, 1937, p. 101). However, while the distinction often arises as informal motivation, there has been less explicit discussion of what exactly the distinction might be. A survey of the literature reveals two families of proposals. Syntactic proposals locate the distinction in formal syntax, whereas semantic proposals focus instead on the variety of models a system admits.

Syntactic Proposals
The passage quoted above from de Finetti (1937) invites a view on which the distinction tracks whether numbers, or more generally arithmetical concepts, appear explicitly in our formal language. Comparative probability languages like L comp and perhaps also L cond , on this view, are typically qualitative, since there are no numerical terms or operations. In a recent paper, Delgrande et al. (2019) state this view clearly: 'What distinguishes qualitative from quantitative probability (truth valued) logics is that qualitative probability logics do not employ quantities or arithmetic operations in the syntax, and the informal reading of the qualitative probability formulas do not require a quantitative interpretation.' On this picture, comparative judgments do not involve any explicit reference to numbers, so such systems would count as qualitative. By contrast, a language like L add would presumably be considered quantitative, since it involves an explicit addition-like operator. 26 While there may be extra-logical reasons to focus attention on qualitative systems in this sense, such restrictions come at a logical price. At no increase in reasoning complexity, the presence of arithmetical functions affords simple finite axiomatizations, as well as greater expressivity. We return to potential extra-logical, viz. empirical, motivations below in §6.2. then we will obtain completeness for the system AX add (and hence also AX comp ).

Semantic Proposals
The assumptions above are enough to guarantee soundness of AX base , as well as axioms Add, 2Canc and Contr. The remaining three axioms of AX add follow from the fact that M is a totally ordered commutative monoid. Completeness will follow immediately, since we always have a countermodel in the non-negative rationals (Q + ; +, ≥), by Theorem 2. The same obviously applies to the smaller system, AX comp . 27 However, there are also other commutative monoids that do not explicitly involve numbers but still satisfy Double Cancellation and Contravenience. For example, we could take M = {a} * to be the set of all finite strings over a unary alphabet (the 'free monoid' over {a}), with ⊕ string concatenation and the relation of string containment. Since x → kx for any positive integer k is an embedding of (Q + ; +, ≥) to itself, we can always find countermodels in (N; +, ≥), which is in turn isomorphic to ({a} * ; ⊕, ), so we have: Corollary 6.2. AX add (and AX comp ) is complete with respect to interpretations in ({a} * ; ⊕, ).
On this alternative interpretation, the 'probability' of an event is taken to be simply a string in unary.
To the extent that the multiplicative systems do not enjoy such alternative interpretations, our division appears to harmonize with this distinction. We leave as an open question whether systems like AX poly can be interpreted in models that are not (isomorphic to some sub-semiring of) the real numbers or the unit interval.
A more radical way of drawing the distinction is to insist not that a system possess nonnumerical models in order to be qualitative, but that the system have no straightforward models that are numerical. For instance, in the literature on uncertain reasoning, systems that license inferences like A, B | ∼ A ∧ B have been deemed qualitative, since they correspond to intuitive, non-numerical patterns, but are incompatible with straightforward probabilistic interpretations (e.g., according to which A is accepted just in case P(A) > θ for some threshold θ). For instance, Hawthorne and Makinson (2007) give voice to this perspective when they write: 'Broadly speaking, there are two ways of approaching the formal analysis of uncertain reasoning: quantitatively, using in particular probability relationships, or by means of qualitative criteria. As is widely recognized, the consequence relations that are generated in these two ways behave quite differently.' Systems of non-monotonic reasoning can often be given quantitative probabilistic interpretations (see, e.g., Pearl 1989), and there are various ways of ameliorating the inferential tensions in these contexts (Leitgeb, 2017;Mierzewski, 2020). But such resolution usually comes at the expense of quantitative granularity typical of numerical probabilistic reasoning. In any case, on this way of drawing the distinction, none of the systems we have studied in the present work would count as qualitative, even the basic comparative system AX comp .

Empirical Issues
Some of the motivations for less quantitative formulations of probability come not from issues of logic and complexity, but rather from empirical concerns. A common thought is that eliciting comparative judgments may be in some way more tractable. Relatedly, numerous authors have suggested that quantitative judgments may not always be empirically meaningful in the same way that qualitative judgments may be.
The intuition behind this motivation is clear enough. Comparative judgments introspectively appear easier to make than numerical comparisons, and echoing the earlier suggestions by Keynes, Koopman and others, some more recent authors have concluded that they are in some sense more psychologically 'real' (e.g., Stefánsson 2017). Indeed, it has long been appreciated that binary comparative judgments in general can be more stable or reliable than absolute judgments, even to the extent that some recent researchers advocate replacing the latter with the former to mitigate noise in judgment (Kahneman et al., 2021). Such judgments play a central role in algorithms for probabilistic inference as well, under the assumption that probabilistic comparisons-especially between hypothesis that are 'nearby' in a larger state space-are relatively easy (Sanborn and Chater, 2016, p. 887).
Similar arguments about the ease and naturalness of qualitative judgments have been marshalled for conditional independence relationships, purportedly arising from even more fundamental qualitative causal intuitions (Pearl, 2009, p. 21).

Direct Measurements of Probability
For the specific problem of eliciting subjective probabilities, not only is stability across time and contexts important-it is also significant that the whole pattern of attitudes be consistent with at least one probabilistic representation in the first place. It was pointed out already by Suppes (1974) that verifying this in the purely comparative setting, even for a small number of basic events, involves a combinatorial explosion of pairs to check. Worse yet, we now have overwhelming evidence (e.g., from the long line of work starting with Tversky and Kahneman 1974) that ordinary judgments about comparative probability routinely violate even the most basic axioms, such as Dist. Similar experimental patterns confront the logic of (conditional) independence (e.g., Rehder 2014). These considerations, at the very least, put pressure on any claim to the effect that qualitative judgments enjoy special empirical tractability.
Once we abandon the ambition of eliciting coherent, fully specified patterns of judgments, the numerical/non-numerical distinction again begins to appear somewhat arbitrary. Contemporary methods for probability elicitation tend to be partial, and they handle numerical judgments in a similar way to their treatment of purely comparative judgments (e.g., Lambert et al. 2008). In the behavioral sciences, probabilities are commonly measured on continuous sliding scales, or on 7-point Likert scales (e.g., 'extremely unlikely' to 'extremely likely'), often with a background assumption that such responses will be noisy reflections of an underlying psychological mechanism, estimable from many samples of the population (see, e.g., Luce and Suppes 1965;Icard 2016;Sanborn and Chater 2016 for discussion). From this perspective, comparative judgments may tend to be robust because they are often relatively insensitive to random perturbations. At the same time, we might expect a claim that 'A and B are equally likely' to be more fickle than a claim like, 'C is more than twice as likely as D'. So there again may be nothing distinctive about non-numerical comparisons in this regard.

Indirect Measurement via Preference
A prominent way of thinking about subjective probability takes it to be not a primitive notion, but rather derivative from an agent's preferences over 'gambles' or 'acts'. On this alternative view such preferences, as revealed in choice behavior, form the empirical basis of probability attributions. As long as the pattern of choices satisfies certain sets of axioms, a representation in terms of (fully quantitative, and typically unique) probabilities together with utilities is guaranteed (e.g., as in Savage's 1954 classic axiomatization). These axioms tend to be quite strong, and there have been countless criticisms of them, on both descriptive and normative grounds. Though interesting variants and weakenings have been proposed (e.g., Joyce 2009;Gaifman and Liu 2018), it appears that the assumptions required will be at least as demanding as those made in the purely probabilistic case.
Indeed, in the preferential setting there is a precise sense in which comparative probability judgments emerge as a special case of uncertain gambles. We would say that an agent considers α more likely than β if they prefer a gamble that returns a good outcome in case of α to one that returns the same good outcome in case of β. Axioms can be stated to guarantee that the resulting order will be probabilistically representable (see Krantz et al. 1971, §5.2.4), though, yet again, even the most basic of these have been questioned. For instance, in Ellsberg's (1961) celebrated counterexample to Savage's sure-thing principle, people tend to prefer a gamble on A to one on B, while simultaneously preferring a gamble on B ∨ C to one on A ∨ C, where C is incompatible with A and with B. This leads to a blatant violation of quasi-additivity (Quasi ).
Nonetheless, if one is willing to weaken the axioms required to guarantee probabilistic representability (or simply disregard their systematic violation), these basic gambles can be elaborated to extract probability judgments with greater numerical content, including ratio comparisons (typical of L add and beyond). The basic idea, following Ramsey (1926) (see also Davidson and Suppes 1956;Elliott 2017, inter alia), is to begin with a way of eliciting meaningful utilities for outcomes, and then to use these utilities to measure probability judgments. For instance, if it can be determined that an agent judges outcome O 2 be to at least twice as desirable as O 1 , then someone who prefers a gamble returning O 1 if α, to one returning O 2 if β, might be taken to judge α more than twice as likely as β.
Could such methods be employed to determine not just additive constraints on probabilities, but also multiplicative constraints? Supposing we could establish that the utility of O 2 were greater than that of O 1 squared, for instance, we might conclude that α has probability greater than the square of β's probability. The problem with this suggestion is that most approaches to utility, including those that descend from Ramsey (1926), assume that utilities are meaningful only up to linear transformation, so such conclusions would not be well defined. Thus, even in this context of empirical questions, we see that the natural dividing line may not be numerical or quantitative constraints per se, but rather additive versus multiplicative constraints.

Conclusion and Open Questions
Through a broad distinction between additive and multiplicative formalisms for probabilistic reasoning, we have explored the landscape of probability logics with respect to fundamental questions about expressivity, computational complexity, and axiomatization. What emerges is a remarkably robust divide that cross-cuts some tempting ways of dividing the space based on intuitions about quantitative versus qualitative representation and reasoning. In addition to the technical contributions summarized above in §5, we also canvassed some of the empirical considerations that have motivated special attention to comparative and other 'purely qualitative' judgments. At present it is not clear that the latter enjoy any distinctive empirical status, and if anything, the relevant empirical boundaries may better track the additive-multiplicative distinction that has been our focus here.
It is certainly not an aim of this paper to discourage the use and exploration of qualitative probability. On the contrary, as we have discussed, such systems present rich opportunities for systematic logical investigation. Moreover, a number of recent authors have found such languages useful for stating and evaluating epistemological principles in a general and relatively neutral manner (e.g., Narens 2007; Eva 2019; Liu 2020; Mayo-Wilson and Saraf 2020, inter multa et alia). This is especially evident in settings where agreement with a probability measure is not assumed, or is even precluded (e.g., Dubois and Prade 1988;DiBella 2022). These considerations are rather different from many of the original motivations that prompted study of such systems, viz. simplicity and empirical tractability. From a logical perspective-concerning complexity, axiomatizability, and expressivity-and potentially also from an empirical perspective, prohibition on the use of simple numerical primitives appears Procrustean.
A number of significant technical questions and directions on this subject merit further exploration. In addition to the outstanding issue of substantiating Domotor's (1969) argument for representation of comparative conditional probability (represented here as L cond ), we also mention the following directions: • Many authors have argued that the comparability of should be rejected, on both normative and empirical grounds (Keynes, 1921;Fine, 1973;Walley, 1991;Gaifman, 2009;Liu, 2020). All of the systems presented in this paper could alternatively be interpreted relative to sets of probability measures. Logical systems for such settings have received considerable investigation (see, e.g., Ding et al. 2021 for an overview). It seems plausible that many of the results here would extend with little change; however, our proof methods often employed normal forms whose exact character depended on comparability. Working through this setting in detail would be worthwhile.
• It is easy to imagine natural extensions of the languages considered here, including varieties of explicit quantification. For instance, Speranski (2017) studies probability logics extending (what we here called) L add with quantification over events, while Abadi and Halpern (1994) explore a range of first order extensions of L add and (what we called) L poly with quantification over both field terms and (a second sort of) objects. Generally speaking, these extensions result in a significant complexity increases, leading to undecidability (in the best cases), Π 1 1 -hardness, and even Π 1 ∞ -hardness. However, there are ways of curtailing this complexity (e.g., by restricting to bounded finite domains), and it could be enlightening to investigate our more refined space of languages in those settings.
• A potentially more modest, but quite useful extension to any of these systems would be to add exponentiation, e.g., the function e x . As such systems would also encode logarithms, this would allow reasoning about (conditional) entropy as well. It is unclear at present whether adding exponentiation to L poly would even result in a decidable system. By a theorem of Macintyre and Wilkie (1995), a positive answer to this question would also settle Tarski's (1951) well known 'exponential function problem' which has been open since the 1940s. Whether the problem may be easier for some weaker systems we have considered here remains to be seen. Moreover, it may be feasible to address questions of axiomatizability without solving that outstanding open problem.
These are just some of the questions that will need to be answered before we have a fully comprehensive understanding of the space of natural probabilistic languages.
The crux of the proof is now to consider which terms will be cancelled from both sides (once again appealing to Contr). Any state description δ ∈ B βn will appear once more on the left than on the right, so P(δ) will still remain as a summand on the left. All other terms on the left will be cancelled. Similarly for the right, we will end up with the sum of terms for state descriptions in B αn , so that (9) Thus we have 9 total preorders (10), (12)-(19) representable by a rank 1. We claim these are exactly the ones satisfying the stated Domotor axioms. The soundness direction is straightforward while the completeness direction can be shown via casework. Note that the n = 2 instance of Q5 is just monotonicity (in both arguments, by symmetry), namely that (A, B) (A, C) ⇒ (D, B) (D, C) and analogously in the first argument. This presumes that (∅, Ω) ≺ (A, B) and (∅, Ω) ≺ (D, C). We also have strict monotonicity. Below we will generally take Q1-Q4 for granted, which just amount to the axioms of a nondegenerate total preorder, symmetric on the pairs. Our cases are: 1. If (0, Ω) ≺ (1, Ω):  i. If (1, 1) ≺ (0, Ω): order (17) 28 This can be made fully rigorous via e.g. Sturm sequences. 29 The order (10) arising from the degenerate matrix 0 0 0 1 corresponds to x = +∞ on the plot.