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Proceedings of Machine Learning Research 58 (2016) 97-109
Rational Beliefs Real Agents Can Have – A Logical Point of
View
Marcello D’Agostino marcello.dagostino@unimi.it
Department of Philosophy
University of Milan
20122 Milano, Italy
Tommaso Flaminio tommaso.flaminio@uninsubria.it
Department of Pure and Applied Sciences
University of Insubria
21100 Varese, Italy
Hykel Hosni hykel.hosni@unimi.it
Department of Philosophy
University of Milan
20122 Milano, Italy.
Editor: Tatiana V. Guy, Miroslav K´arn´y, David Rios-Insua, David H. Wolpert
Abstract
The purpose of this note is to outline a framework for uncertain reasoning which drops
unrealistic assumptions about the agents’ inferential capabilities. To do so, we envisage a
pivotal role for the recent research programme of depth-bounded Boolean logics (D’Agostino
et al., 2013). We suggest that this can be fruitfully extended to the representation of rational
belief under uncertainty. By doing this we lay the foundations for a prescriptive account of
rational belief, namely one that realistic agents, as opposed to idealised ones, can feasibly
act upon.
Keywords: Prescriptive rationality, tractability, logic-based probability, Bayesian norms
1. Introduction and motivation
Probability is traditionally the tool of choice for the quantification of uncertainty. Since
Jacob Bernoulli’s 1713 Ars Conjectandi, a number of arguments have been put forward to
the effect that departing from a probabilistic assessment of uncertainty leads to irrational
patterns of behaviour. This contributed to linking tightly the rules of probability with
the defining norms of rationality, as fixed by the well known results of de Finetti (1974);
Savage (1972). Lindley (2006) and Parmigiani and Inoue (2009) provide recent introductory
reviews.
Over the past few decades, however, a number of concerns have been raised against the
adequacy of probability as a norm of rational reasoning and decision making. Following
the lead of Ellsberg (1961), whom in turn found himself on the footsteps of Knight (1921)
and Keynes (1921), many decision theorists take issue with the idea that probability pro-
vides adequate norms for rationality. This is put emphatically in the title of Gilboa et al.
(2012), a paper which has circulated for almost a decade before its publication. As a result,
c
2016 D’Agostino et al.
D’Agostino, Flaminio, and Hosni
considerable formal and conceptual effort has gone into extending the scope of the prob-
abilistic representation of uncertainty, as illustrated for instance by Gilboa and Marinacci
(2013). Related to this, is the large family of imprecise probability models, and its decision-
theoretic offsprings, which constitute the cutting edge of uncertain reasoning research, see
e.g. Augustin et al. (2014).
One key commonality between “non Bayesian” decision theory and the imprecise prob-
abilities approach is the fact they take issue with the identification of “rationality” and
“probability” on representational grounds. For they insist on the counterintuitive conse-
quences of assuming that the rational representation of uncertainty necessitates the Bayesian
norms, and in particular that all uncertainty is to be represented probabilistically.
This note makes a case for adding a logical dimension to this ongoing debate. Key to
this is a logical framing of probability. As recalled explicitly below, probability functions
are normalised on classical tautologies. That is to say that a Bayesian agent is required to
assign maximum degree of belief to every tautology of the propositional calculus. However
classic results in computational complexity imply that the problem of deciding whether a
given sentence is a tautology, exceeds, in general, what is considered to be feasible. Hence,
probability imposes a norm of rationality which, under widely agreed hypotheses, realistic
agents cannot be expected to meet. A related concern had already been put forward by
Savage (1967), but this did’t lead proponents of the Bayesian approach to take the issue
seriously. This is precisely what the research outlined in this note aims to do.
By framing the question logically, we can offer a perspective on the problem which
highlights the role of classical logic in determining the unwelcome features of canonical
Bayesian rationality (Section 3). This suggests that a normatively reasonable account of
rationality should to take a step back and rethink the logic in the first place.
The recently developed framework of Depth-Bounded Boolean logics (DBBLs) is partic-
ularly promising in this respect. By re-defining the meaning of logical connectives in terms
of information actually possessed by the agent DBBLs give rise to a hierarchy of logics which
(i) accounts for some key aspects of the asymmetry between knowledge and ignorance and
(ii) provide computationally feasible approximations to classical logic. Section 4.2 reviews
informally the core elements of this family of logics.
Finally, Section 5 outlines the applicability of this framework to probabilistic reasoning.
In particular it points out how the hierarchy of DBBLs to can serve to define a hierarchy
of prescriptively rational approximation of Bayesian rationality.
2. Bayesian rationality
In a number of areas, from Economics to the Psychology of reasoning and of course Statistics,
probability has been defended as the norm of rational belief. Formally this can be seen to
imply a normative role also for classical logic. So the Bayesian norms of rationality are best
viewed as combination of probability and logic.
This allows us to distinguish two lines of criticisms against Bayesian rationality. First,
it is often pointed out that probability washes out a natural asymmetry between knowledge
and ignorance. Second, the intractability of classical logical reasoning is often suggested to
deprive the normative theory of practical meaning. Both lines of criticisms can be naturally
linked to the properties of classical logic.
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A Logical Perspective on Prescriptive Rationality
2.1 Against the probability norm: the argument from information
Uncertainty has to do, of course, with not knowing, and in particular not knowing the
outcome of an event of interest, or the value of a random variable. Ignorance has more subtle
features, and is often thought of as our inability to quantify our own uncertainty. Knight
(1921) gave this impalpable distinction an operational meaning in actuarial terms. He
suggested the presence of ignorance is detected by the absence of a compete insurance market
for the goods at hand. On the contrary, a complete insurance market provides an operational
definition of probabilistically quantifiable uncertainty. Contemporary followers of Knight
insist that not all uncertainty is probabilistically quantifiable and seek to introduce more
general norms of rational belief and decision under “Knightian uncertainty” or “ambiguity”.
A rather general form of the argument from information against Bayesian rationality is
summarised by the following observation by Schmeidler (1989):
The probability attached to an uncertain event does not reflect the heuristic
amount of information that led to the assignment of that probability. For ex-
ample, when the information on the occurrence of two events is symmetric they
are assigned equal probabilities. If the events are complementary the probabili-
ties will be 1/2 independent of whether the symmetric information is meager or
abundant.
Gilboa (2009) interprets Schmeidler’s observation as expressing a form of “cognitive
unease”, namely a feeling that the theory of subjective probability which springs naturally
from Bayesian epistemology, is silent on one fundamental aspect of rationality (in its infor-
mal meaning). But why is it so? Suppose that some matter is to be decided by the toss of
a coin. According to Schmeidler’s line of argument, I should prefer tossing my own, rather
than some one else’s coin, on the basis, say of the fact that I have never observed signs
of “unfairness” in my coin, whilst I just don’t know anything about the stranger’s coin.
See also Gilboa et al. (2012); Gilboa (2009). This argument is of course reminiscent of the
Ellsberg two-urns problem, which had been anticipated in Keynes (1921).
Similar considerations have been put forward in artificial intelligence and in the foun-
dations of statistics. An early amendment of probability theory aimed at capturing the
asymmetry between uncertainty and ignorance is known as the theory of Belief Functions
(Shafer, 1976; Denoeux, 2016). Key to representing this asymmetry is the relaxation of the
additivity axiom of probability. This in turn may lead to situations in which the probabilis-
tic excluded middle does not hold. That is to say an agent could rationally assign belief
less than 1 to the classical tautology θ∨ ¬θ. Indeed, as we now illustrate, the problem with
normalising on tautologies is much more general.
2.2 Against the logic norm: the argument from tractability
Recall that classical propositional logic is decidable in the sense that for each sentence θ
of the language there is an effective procedure to decide wether θis a tautology or not.
Such a procedure, however, is unlikely to be feasible, that is to say executable in practice.
In terms of the theory of computational complexity this means that there is probably no
algorithm running in polynomial time. So, a consequence of the seminal 1971 result by
Stephen Cook, the tautology problem for classical logic is widely believed to be intractable.
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D’Agostino, Flaminio, and Hosni
If this conjecture is correct, we are faced with a serious foundational problem when imposing
the normalisation of probability on tautology. For we are imposing agents constraints of
rationality which they simply may never be able to satisfy.
It is remarkable that L.J. Savage had anticipated this problem with the Bayesian norms
he centrally contributed to defining. To this effect he observed in Savage (1967) the follow-
ing:
A person required to risk money on a remote digit of πwould have to compute
that digit in order to comply fully with the theory though this would really be
wasteful if the cost of computation were more than the prize involved. For the
postulates of the theory imply that you should behave in accordance with the
logical implications of all that you know. Is it possible to improve the theory in
this respect, making allowance within it for the cost of thinking, or would that
entail paradox [. . .] , as I am inclined to believe but unable to demonstrate ?
If the remedy is not in changing the theory but rather in the way in which we
attempt to use it, clarification is still to be desired. (Our emphasis)
Fifty years on, the difficulty pointed out by Savage failed to receive the attention it
deserves. As the remainder of this note illustrates, however, framing the issue logically
brings about significant improvements in our understanding of the key issues, paving the
way for a tractable approximation of Bayesian rationality – or rational beliefs real agents
can have.
3. Logic, algebra and probability
A well-known representation result (see, e.g. Paris (1994)) shows that every probability
function arises from distributing the unit mass across the 2natoms of the Boolean (Linden-
baum) Algebra generated by the propositional language L={p1,...pn}, and conversely,
that a probability function on Lis completely determined by the values it takes on such
atoms. Such a representation makes explicit the dependence of probability on classical
logic. This has important and often underappreciated consequences. Indeed logic plays a
twofold role in the theory of probability. First, logic provides the language in which events
–the bearers of probability– are expresses, combined and evaluated. The precise details
depend on the framework. See Flaminio et al. (2014) for a characterisation of probability
on classical logic, and Flaminio et al. (2015) for the general case of Dempster-Shafer belief
functions on many-valued events.
In measure-theoretic presentations of probability, events are identified with subsets of
the field generated by a given sample space Ω. A popular interpretation for Ω is that of
the elementary outcomes of some experiment, a view endorsed by A.N. Kolmogorov, who
insisted on the generality of his axiomatisation. More precisely, let M= (Ω,F, µ) a measure
space where, Ω = {ω1, ω2. . .}is the set of elementary outcomes, F= 2Ωis the field of sets
(σ−algebra) over Ω. We call events the elements of F, and µ:F → [0,1] a probability
measure if it is normalised, monotone and σ-additive, i.e.
(K1) µ(Ω) = 1
(K2) A⊆B⇒µ(A)≤µ(B)
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A Logical Perspective on Prescriptive Rationality
(K3) If {E}iis a countable family of pairwise disjoint events then P(SiEi) = PiP(Ei)
The Stone representation theorem for Boolean algebras and the representation theorem
for probability functions recalled above guarantee that the measure-theoretic axiomatisation
of probability is equivalent to the logical one, which is obtained by letting a function from
the language Lto the real unit interval be a probability function if
(PL1) |=θ⇒P(θ) = 1
(PL2) |=¬(θ∧φ)⇒P(θ∨φ) = P(θ) + P(φ).
Obvious as this logical “translation” may be, it highlights a further role for logic in the
theory of probability, in addition that is to the linguistic one pointed out above. This role is
best appreciated by focussing on the consequence relation |= and can be naturally referred
to as inferential.
In its measure-theoretic version, the normalisation axiom is quite uncontroversial. Less
so, if framed in terms of classical tautologies, as in PL1. Indeed both arguments against
Bayesian norms discussed informally above, emerge now formally. The first is to do with
the fact that |= interprets symmetrically “knowledge” and “ignorance” as captured by the
fact that |=θ∨ ¬θis a tautology. Indeed similarly bothersome consequences follow directly
from P L1 and P L2, namely
1. P(¬θ)=1−P(θ)
2. θ|=φ⇒P(θ)≤P(φ)
2 implies that if θand φare logically equivalent they get equal probability.
The argument from information recalled in Section 2.1 above clearly has its logical roots
in the semantics of classical logic.
Similarly, the argument from tractability of Section 2.2 leads one into questioning the
desirability of normalising probability on any classical tautology. Taken as a norm of
rationality this requires agents to be capable of reasoning beyond what is widely accepted
as feasible. Again, the unwelcome features of probability are rooted in classical logic.
A further, important, feature which emerges clearly in the logical presentation of prob-
ability is that uncertainty is resolved by appealing to the semantics of classical logic. This
leads to the piecemeal identification of “events” with “sentences” of the logic. This identi-
fication, however, is not as natural as one may think.
On the one hand, an event, understood classically, either happens or not. A sentence
expressing an event, on the other hand is evaluated in the binary set as follows
v(θ) = (1 if the event obtained
0 otherwise.
Hence, the probability of an event P(θ)∈[0,1] measures the agent’s degree of belief that
the event did or will obtain. Finding this out is, in most applications, relatively obvious.
However, as pointed out in Flaminio et al. (2014), a general theory of what it means for
“states of the world” to “resolve uncertainty” is far from trivial.
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D’Agostino, Flaminio, and Hosni
A more natural way of evaluating events arises taking an information-based interpreta-
tion of uncertainty resolution. The key difference with the previous, classical case, lies in
the fact that this leads naturally to a partial evaluation of events, that is
vi(θ) =
1 if I am informed that θ
0 if I am informed that ¬θ
⊥if I am not informed about θ.
Quite obviously standard probability logic does not apply here, because the classical
resolution of uncertainty has no way of expressing the ⊥condition.
As the next section shows, by looking for a logic which fixes this information asymmetry,
we will also find a logic which deals successfully with the tractability problem.
4. An informational view of propositional logic
The main idea underlying the informational view of classical propositional logic is to replace
the notions of “truth” and “falsity”, by “informational truth” and “informational falsity”,
namely holding the information that a sentence ϕis true, respectively false. Here, by saying
that an agent aholds the information that ϕis true or false we mean that this information
(i) is accepted by ain the sense that ais ready to act upon it1(ii) it is feasibly available
to a, in the sense that ahas the means to obtain it in practice (and not only in principle);
given the (probable) intractability of classical propositional logic this condition is not in
general preserved by the corresponding consequence relation.
Clearly, these notions do not satisfy the informational version of the Principle of Biva-
lence: it may well be that for a given ϕ, we neither hold the information that ϕis true,
nor do we hold the information that ϕis false. Knowledge and ignorance are not treated
symmetrically under the informational semantics. However, in this paper we assume that
they do satisfy the informational version of the Principle of Non-Contradiction: no agent
can actually possess both the information that ϕis true and the information that ϕis false,
as this could be deemed to be equivalent to possessing no definite information about ϕ.2
4.1 Informational semantics
We use the values 1 and 0 to represent, respectively, informational truth and falsity. When
a sentence takes neither of these two defined values, we say that it is informationally in-
determinate. It is technically convenient to treat informational indeterminacy as a third
value that we denote by “⊥”.3The three values are partially ordered by the relation
1. The kind of justification for this acceptance and whether or not the agent is human or artificial do not
concern us here. Acceptance may include some (possibly non-conclusive) evidence that adeems sufficient
for acceptance, or communication from some external source that aregards as reliable.
2. Notice that this assumption does not rule out the possibility of hidden inconsistencies in an agent’s
information state, but only of inconsistencies that can be feasibly detected by that agent. It is, however,
possible to investigate paraconsistent variants of this semantics in which even this weak informational
version of the Principle of Non-Contradiction is relaxed. This will be the subject of a subsequent paper.
3. This is the symbol for “undefined”, the bottom element of the information ordering, not to be confused
with the “falsum” logical constant.
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A Logical Perspective on Prescriptive Rationality
∧1 0 ⊥
1 1 0 ⊥
0 0 0 0
⊥ ⊥ 0⊥,0
∨1 0 ⊥
1 1 1 1
0 1 0 ⊥
⊥1⊥ ⊥,1
¬
1 0
0 1
⊥ ⊥
Figure 1: Informational tables for the classical operators
such that vw(“vis less defined than, or equal to, w”) if, and only if, v=⊥or v=w
for v, w ∈ {0,1,⊥}.
Note that the old familiar truth tables for ∧,∨and ¬are still intuitively sound under
this informational reinterpretation of 1 and 0. However, they are no longer exhaustive: they
do not tell us what happens when one or all of the immediate constituents of a complex
sentence take the value ⊥. A remarkable consequence of this approach is that the semantics
of ∨and ∧becomes, as first noticed by Quine (1973, pp. 75–78), non-deterministic. In
some cases an agent amay accept a disjunction ϕ∨ψas true while abstaining on both
components ϕand ψ. To take Quine’s own example, if I cannot distinguish between a
mouse and chipmunk, I may still hold the information that “it is a mouse or a chipmunk”
is true while holding no definite information about either of the sentences “it is a mouse”
and “it is a chipmunk”. In other cases, e.g. when the component sentences are “it is a
mouse” and “it is in the kitchen” and I still hold no definite information about either, the
most natural choice is to abstain on the disjunction. Similarly, amay reject a conjunction
ϕ∧ψas false while abstaining on both components. To continue with Quine’s example, I
may hold the information that “it is a mouse and a chipmunk” is false, while holding no
definite information about either of the two component sentences. But if the component
sentences are “it is a mouse” and “it is in the kitchen” and I abstain on both, I will most
probably abstain also on their conjunction. In fact, this phenomenon is quite common as
far as the ordinary notion of information is concerned and the reader can figure out plenty
of similar situations. Thus, depending on the “informational situation”, when ϕand ψ
are both assigned the value ⊥, the disjunction ϕ∨ψmay take the value 1 or ⊥, and the
conjunction ϕ∧ψmay take the value 0 or ⊥.
As a consequence of this informational interpretation, the traditional truth-tables for
the ∨,∧and ¬should be replaced by the “informational tables” in Figure 1, where the
value of a complex sentence, in some cases, is not uniquely determined by the value of its
immediate components.4A non-deterministic table for the informational meaning of the
Boolean conditional can be obtained in the obvious way, by considering ϕ→ψas having
the same meaning as ¬ϕ∨ψ(see D’Agostino, 2015, p. 82).
4. In his (Quine, 1973) Quine calls them “verdict tables” and the values are “assent”, “dissent” and “ab-
stain”. This non-deterministic semantics was subsequently and independently re-proposed (with no
apparent connection with the intuitive interpretation given by Quine) by Crawford and Etherington
(1998) who claimed without proof that it provides a characterization of unit resolution (a tractable
fragment of resolution that requires formulae to be translated into clausal form). The general theory
of non-deterministic semantics for logical systems has been brought to the attention of the logical com-
munity and extensively investigated (with no special connection with tractablity) by Arnon Avron and
co-authors (see Avron and Zamansky (2011) for an overview).
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D’Agostino, Flaminio, and Hosni
4.2 Depth-bounded Boolean logics
In (D’Agostino et al., 2013) and (D’Agostino, 2015) it is shown that the informational
semantics outlined in the previous section provides the basis to define an infinite hierarchy
of tractable deductive systems (with no syntactic restriction on the language adopted) whose
upper limit coincides with classical propositional logic. As will be clarified in the sequel the
tractability of each layer is a consequence of the shift from the classical to the informational
interpretation of the logical operators (that is the same throughout the hierarchy) and on
an upper bound on the nested use of “virtual information”, i.e. information that the agent
does not actually hold, in the sense specified in the previous section.
Definition 1 A0-depth information state is a valuation Vof the formulae in Lthat agrees
with the informational tables.
Note that, given the non-determinism of the informational tables, the valuation Vis not
uniquely determined by an assigment of values to the atomic sentences. For example the
valuation V1that assigns ⊥to both pand qand ⊥to p∨qis as admissible as the valuation
V2that still assigns ⊥to both pand q, but 1 to p∨q. Let S0be the set of all 0-depth
information states.
Definition 2 We say that ϕis a 0-depth consequence of a finite set Γof sentences, and
write Γ0ϕ, when
(∀V∈S0)V(Γ) = 1 =⇒V(A)=1.
We also say that Γis 0-depth inconsistent, and write Γ0if there is no V∈S0such that
V(Γ) = 1.
It is not difficult to verify that 0is a Tarskian consequence relation, i.e., it satisfies reflex-
ivity, monotonicity, transitivity and substitution invariance.
In fact, it can be shown that we do not need to consider valuations of the whole language
Lbut can restrict our attention to the subformulae of the formulae that occur as premises
and conclusion of the inference under consideration. Let us call search space any finite set
Λ of formulae that is closed under subformulae, i.e., if ϕis a subformula of a formula in Λ,
ϕ∈Λ.
Definition 3 A 0-depth information state over a search space Λis a valuation Vof Λthat
agrees with the informational tables.
Let SΛ
0be the set of all 0-depth information states over a search space Λ. Given a finite set
∆ of formulae, let us write ∆∗to denote the search space consisting of all the subformulae
of the formulae in ∆. Then, it can be shown that:
Theorem 4 Γ0ϕif and only if (∀V∈S(Γ∪{ϕ})∗
0)V(Γ) = 1 =⇒V(A)=1. Moroever,
Γ0if and only if there is no V∈S(Γ∪{ϕ})∗
0such that V(Γ) = 1.
On the basis of the above result, in (D’Agostino et al., 2013) it is shown that 0is tractable :
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A Logical Perspective on Prescriptive Rationality
Theorem 5 Whether or not Γ0ϕ(Γis 0-depth inconsistent) can be decided in time
O(n2)where nis the total number of occurrences of symbols in Γ∪ {ϕ}(in Γ).
A simple proof system that is sound and complete with respect to is shown in (D’Agostino
et al., 2013; D’Agostino, 2015) in the form of a set of introduction and elimination rules
(in the fashion of Natural Deduction) that are only based on actual information, i.e., in-
formation that is held by an agent, with no need for virtual information, i.e., simulating
information that does not belong to the current information state, as happens in case-
reasoning or in some ways of establishing a conditional (as in the introduction rule for the
conditional in Gentzen-style natural deduction).
The subsequent layers of the hierarchy depend on fixing an upper bound on the depth
at which the nested use of virtual information is allowed.
Let vbe the partial ordering of 0-depth information states (over a given search space)
defined as follows: VvV0if and only if V0is a refinement of Vor is equal to V, that is,
for every formula ϕin the domain of Vand V0,V(ϕ)6=⊥implies that V0(ϕ) = V(ϕ).
Definition 6 Let Vbe a 0-depth information state over a search space Λ.
•V0ϕif and only if V(ϕ) = 1
•Vk+1 ϕif and only if
(∃ψ∈Λ)(∀V0∈SΛ
0)VvV0and V0(ψ)6=⊥=⇒V0kϕ.
Here j, with j∈N, is a kind of “forcing” relation and the shift from one level of depth to
the next is determined by simulating refinements of the current information state in which
the value of some ψ∈Λ is defined (either 1 or 0) and checking that in either case the value
of ϕis forced to be 1 at the immediately lower depth. Such use of a definite value for ψ,
that is not even implicitly contained in the current information state Vof the agent, is what
we call virtual information.
Definition 7 Ak-depth information state over a search space Λis a valuation Vof Λthat
agrees with the informational tables and is closed under the forcing relation k.
Let SΛ
kbe the set of all k-depth information states over Λ.
Definition 8 We say that ϕis a k-depth consequence of Γ, and write Γkϕif
(∀V∈S(Γ∪{ϕ})∗
k)V(Γ) = 1 =⇒V(ϕ)=1.
We also say that Γis k-depth inconsistent, and write Γk, if there no V∈S(Γ∪{ϕ})∗
ksuch
that V(Γ) = 1.
It can also be shown that Γ kϕif and only if there is a finite sequence ψ1, . . . , ψnsuch
that ψn=ϕand for every element ψiof the sequence, either (i) ψiis a formula in Γ or (ii)
Vkψifor all V∈S(Γ∪{ϕ})∗
0such that V{ψ1, . . . , ψi−1}= 1.
Unlike 0,kis not a Tarskian consequence relation, but gets very close to being such,
for ksatisfies reflexivity, monotonicity, substitution invariance and the following restricted
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D’Agostino, Flaminio, and Hosni
version of transitivity in which the “cut formula” is required to belong to the search space
defined by the deduction problem under consideration.
(∀ψ∈(Γ ∪ {ϕ})∗) Γ kψand ∆, ψ kϕ=⇒Γ,∆kϕ. (Bounded Transitivity)
In (D’Agostino et al., 2013) it is shown that kis tractable for every fixed k.
Theorem 9 Whether or not Γkϕ(Γis k-depth inconsistent) can be decided in time
O(n2k+2), where nis the total number of occurrences of symbols in Γ∪ {ϕ}(Γ).
Observe that, by definition, if Γ jϕ(Γ is j-depth inconsistent), then Γ kϕ(Γ is k-depth
inconsistent) for every k > j. Classical proposition logic is the limit of the sequence of the
depth-bounded consequence relations kas k→ ∞.
A proof system for each of the k-depth approximations is obtained by adding to the
introducton and elimination rules for 0a single structural rule that reflects the use of
virtual information in Definition 6, and bounding the depth at which nested applications
of this rule are allowed (see (D’Agostino et al., 2013; D’Agostino, 2015) for the details and
a discussion of related work).
5. Towards a prescriptive theory of Bayesian rationality
Let us briefly recap. By framing probability logically we are able to locate the source of a
number of important criticisms which are commonly held up against Bayesian rationality in
classical logic. The theory of Depth-Bounded Boolean logics meets some of those objections,
and gives us an informational semantics leading to a hierarchy of tractable approximations
of classical logic. The logical axiomatisation of probability recalled above naturally suggests
to investigate which notion of rational belief is yielded once |= is replaced with |=kin PL1-
PL2 above.
This gives us a natural desideratum, namely to construct a family of rational belief
measures Bifrom Lto [0,1], i ∈Nacting as the analogues of probability functions on Depth-
bounded logics. Since DBLs coincide, in the limit, with classical propositional logic, our
desideratum is then the construction of a hierarchy of belief measures B0, . . . , Bk. . . which
asymptotically coincides with probability, i.e. such that for all sentences θ,B∞(θ) = P(θ).
Each element in the resulting hierarchy would then be a natural candidate to providing
a logically rigorous account of a prescriptive model of rational belief, in the sense of Bell
et al. (1988): every agent whose deductive capabilities are bounded by |=kmust quantify,
on pain of irrationality, uncertainty according to Bk.
There is an obvious link between the interpretation of disjunction given by the non-
deterministic informational semantics discussed in Section 4.1 and the behaviour of this
logical connective in quantum logic. As is well-known, in quantum logic a proposition θ
can be represented as a closed subspace Mθof the Hilbert space Hunder consideration.
The disjunction ϕ∨ψis not represented by the union of Mϕand Mψ, for in general
the union of two closed subspaces is not a closed subspace, but by MϕtMψ, i.e. as the
smallest closed subspace including both Mϕand Mψ. So, as is the case for the informational
interpretation of disjunction given by the non-deterministic semantics discussed above, a
disjunction ϕ∨ψin quantum logic may be true even if neither of the disjuncts are true,
since MϕtMψmay contain vectors that are not contained in Mϕ∪Mψ. On this point see
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A Logical Perspective on Prescriptive Rationality
(Aerts, 2000) and (Dalla Chiara et al, 2004). The negative part of the analogy concerns the
behaviour of conjunction which in quantum logic is interpreted as Mϕ∩Mψ, so that if a
conjunction is false, at least one of the two conjuncts must be false, which departs from the
informational intepretation of this operator given by our non-deterministic table. We also
point out that this connection between the non-deterministic semantics of Depth-bounded
Boolean Logics and the semantics of Quantum Logic opens to a natural parallel between our
desideratum and quantum probabilities. This is reinforced by recent experimental findings
in the cognitive sciences (Pothos and Busemeyer, 2013; Oaksford, 2014) suggesting that some
features of Bayesian quantum probability (Pitowsky, 2003) provide accurate descriptions of
experimental subjects.
The key step towards achieving our goal will be of course to define the sense in which
we take any Bkto be a rational belief measure. The task, as it can be easily figured out,
is far from trivial. Though encouraging, our preliminary results suggest that much work
is still to be done in this direction. At the same time they suggest that the consequences
of such a fully-fledged framework will be far reaching, as it will provide significant steps
towards identifying norms of rationality realistic agents can abide to.
Aknowledgements. The authors would like to thank the two referees for the careful
reading, improvement suggestions and encouraging remarks.
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