=Paper=
{{Paper
|id=Vol-1315/paper10
|storemode=property
|title=Monoidal Logics: How to Avoid Paradoxes
|pdfUrl=https://ceur-ws.org/Vol-1315/paper10.pdf
|volume=Vol-1315
|dblpUrl=https://dblp.org/rec/conf/aic/Peterson14
}}
==Monoidal Logics: How to Avoid Paradoxes==
https://ceur-ws.org/Vol-1315/paper10.pdf
Monoidal logics: How to avoid paradoxes
Clayton Peterson?
Munich Center for Mathematical Philosophy
Ludwig-Maximilians-Universität München
clayton.peterson@outlook.com
Abstract. Monoidal logics are logics that can be seen as specific in-
stances of monoidal categories. They are constructed using specific rules
and axiom schemata that allow to make explicit the monoidal structure
of the logics. Among monoidal logics, we find Cartesian logics, which
are instances of Cartesian categories. As it happens, many paradoxes in
epistemic, deontic and action logics can be related to the Cartesian struc-
ture of the logics that are used. While in epistemic and deontic logics the
source of the paradoxes is often found within the principles that govern
the modal operators, our framework enables us to show that many prob-
lems can be avoided by adopting a proper monoidal structure. Thus, the
usual modal rules and axiom schemata do not necessarily need to be dis-
carded to avoid the paradoxes. In this respect, monoidal logics o↵er an
alternative way to model knowledge, actions and normative reasoning.
Furthermore, it provides us with new avenues to analyze modalities.
Keywords: Nonmonotonic reasoning, Conditional normative reasoning, Cate-
gory theory, Categorical logic, Logical omniscience
1 Introduction
Monoidal logics were recently introduced by Peterson [28] as a framework to clas-
sify logical systems through their categorical structure.1 Inspired by the work of
Lambek (see for instance [22]), the idea is to use category theory as a founda-
tional framework for logic and make explicit the relations between the categorical
structure of the logics and the rules and axiom schemata that are used.
In the present paper, we show how monoidal logics are relevant to artificial
intelligence given that they enable us to expose and solve some problems that
are related to epistemic, deontic and action logics. While these kinds of logic
are often formalized as di↵erent variations of modal logics, we begin in section
2 by summarizing the framework we adopt for modal logics (section 2.1) and
monoidal logics (section 2.2). That being done, we present and discuss some
paradoxes in section 3 and analyze them in light of our framework. We conclude
in section 4 with avenues for future research.
?
The author would like to thank Jean-Pierre Marquis for his time and support. This
research was financially supported by the Social Sciences and Humanities Research
Council of Canada.
1
See also [30,29].
Page 122 of 171
2 Framework
2.1 Modal logics
Following Chellas [8], let contain the axiom schemata and rules of proposi-
tional logic. Assume the usual definition for the ⌃ operator (i.e., ⌃' =df ¬2¬')
together with the language L = {P rop, (, ), ^, >, , _, ?, 2}, where P rop is a col-
lection of atomic propositions. The 2 operator is a modality that can represent
necessity (e.g., alethic logic), knowledge (e.g., epistemic logic), obligation (e.g.,
deontic logic), past/future (e.g., temporal logic) or the execution of an action or
a computer program (e.g., dynamic logic). The connectives of the language are
the usual classical connectives (i.e., conjunction, implication and disjunction).
Negation is defined by ¬' =df ' ? and well-formed formulas are defined
recursively as follows2 :
' := pi | ? | > | ' ^ |' |'_ | 2'
The interest of Chellas’s approach is that it clearly relates the rules governing
the modalities to the consequence relation of classical logic. Using the following
inference rules, we can adopt the following definitions3 :
– is classical if it is closed under (RE);
– is monotonic if it is closed under (RM);
– is normal if it is closed under (RK).
'⌘ ' '
(RE) (RM) (RN)
2' ⌘ 2 2' 2 2'
('1 ^ · · · ^ 'n )
(RK) with n 0
(2'1 ^ · · · ^ 2'n ) 2
While a classical system preserves logical equivalences under 2, a monotonic
system insures that 2 preserves consequences. The relations between these sys-
tems is as follows: if is normal, then it is monotonic, and furthermore if it
is monotonic, then it is classical. A classical system E is usually defined by
LP C + (RE), a monotonic system M by LP C + (RM) and a normal system K
by LP C + (RK). In addition to the usual definition of these systems, one can
also have alternative formulations using the following axiom schemata:
2(' ^ ) (2' ^ 2 ) (M) 2(' ) (2' 2 ) (K) 2> (N)
Using these axioms, monotonic and normal systems can alternatively be de-
fined:
K = LP C + (K) + (RN) M = E + (M)
= M + (K) + (N)
2
That is, atoms, > and ? are formulas, and if ' and are formulas, then so are
'^ , ' , ' _ and 2'.
3
Note that there are other types of modal systems, such as regular systems, but we
leave them aside for the purpose of the present paper.
Page 123 of 171
Many extensions can be constructed from these systems. The usual modal
axioms are D, T, 4 and 5.4
2' ⌃' (D) 2' ' (T) 2' 22' (4) ' 2⌃' (5)
D is usually considered as a deontic axiom, which means that if ' is obligatory,
then it is also permitted. T is usually used as an axiom for necessity, meaning
that if ‘it is necessary that ' is true’ is true, then ' is true. 4 and 5 are usually
used for epistemic modalities, the former meaning that if an agent knows ', then
he knows that he knows ' and the latter meaning that if ' is true, then an agent
knows that it is possible for ' to be true.
2.2 Monoidal logics
The rationale behind monoidal logics is to use category theory to analyze the
proof theory of logical systems. By doing so, one can expose the categorical struc-
ture of di↵erent logics and classify these systems accordingly. Consider the lan-
guage L = {P rop, (, ), ⌦, 1, (, , 0}, where P rop is a collection of atomic propo-
sitions. The connective ⌦ is understood as some form of conjunction (although
not necessarily ^), ( is an implication and a disjunction (but not necessarily
_). Negation and well-formed formulas are defined as usual (¬' =df ' ( 0).
' := pi | 0 | 1 | ' ⌦ |'( |'
To define monoidal logics, we need to first define the consequence relation
(see the rules and axiom schemata in figure 1).5 To do so, we define a deductive
system and we require that proofs are reflexive and transitive.
Definition 1. A deductive system D is a collection of formulas and (equivalence
classes of ) proofs (deductions). It has to satisfy (1) and (cut).
Then, one can introduce a conjunction ⌦ with a unit 1 using a monoidal
deductive system. This conjunction is minimally associative but is not necessarily
commutative. The unit 1 can be absorbed by ⌦ from (r) and (l).
Definition 2. A monoidal deductive system M is a deductive system satisfying
(r), (l), (t) and (a).
When this is done, one can do either one of two things. Either one keeps the
monoidal structure and adds an implication, and then perhaps classical nega-
tions, or one adds some structure to the conjunction by requiring that it be
commutative.6 In the latter case, one can define a symmetric deductive system,
4
Note that there are other axioms, see [8,19,10].
5
A double line means that the rule can be applied both top-down and bottom-up.
6
Given space limitations, we will not expose the whole plethora of monoidal logics
that can be defined. For instance, we will not elaborate on monoidal closed deduc-
tive systems or monoidal closed deductive systems with classical negations. For a
thorough presentation and further explanations, we refer the reader to [30,29].
Page 124 of 171
where the conjunction satisfies a braiding rule (b). That said, at this stage, it
is also possible to keep the symmetric structure and introduce an implication
by defining a closed deductive system, and then adding classical negation by
defining a closed deductive system with classical negation.
Definition 3. A symmetric monoidal deductive system S is a monoidal deduc-
tive system satisfying (b).
3.1 A symmetric closed deductive system SC satisfies (cl).
3.2 A symmetric closed deductive system with classical negation SCC satisfies (¬¬).
From a symmetric deductive system, one can add some more structure to the
conjunction and define a Cartesian deductive system. In such a case, ⌦ is the
usual conjunction ^ of classical or intuitionistic logics. The rule (Cart) allows
us to introduce and eliminate the conjunction, while (!) means that anything
implies the truth. As it was the case for symmetric deductive system, one can
also add an implication and classical negation.
Definition 4. A Cartesian deductive system C is a deductive system satisfying
(Cart) and (!).
4.1 A Cartesian closed deductive system CC satisfies (cl).
4.2 A Cartesian closed deductive system with classical negation CCC satisfies (¬¬).
The relationship between these deductive systems is as follows: if D is Carte-
sian, then it is symmetric, and furthermore if it is symmetric, then it is monoidal.
As a notational convention, we use the symbols {⌦, 1, (, , 0} for non-Cartesian
deductive systems and {^, >, , _, ?} for Cartesian ones.
' / ' (1) ¬¬' / ' (¬¬)
' ! !⇢ ' / ⌦1 ' /1⌦
(cut) (r) (l)
' !⇢ ' / ' /
' / ⇢ /⌧ ⌧ / (' ⌦ )⌦⇢ ' / ⌦⌧
(t) (a) (b)
'⌦⇢ / ⌦⌧ ⌧ /'⌦( ⌦ ⇢) ' /⌧ ⌦
'⌦ /⇢ ' / ' /⇢
(cl) ' / 1 (!) (Cart)
' / (⇢ ' / ⌦⇢
Fig. 1. Rules and axiom schema
The co-tensor can be axiomatized through a deductive system defined as
an opposite deductive system Dop where the formulas remain the same but the
deduction arrows are reversed and ⌦/1 are respectively replaced by /0. Hence,
we obtain the co-versions of the aforementioned rules and we can define co-
monoidal (coM), co-symmetric (coS) and co-Cartesian (coC) deductive systems.
The interest of this approach is that deductive systems can be classified
according to their categorical structure (cf. [29]). For instance, M is an instance
of a monoidal category, SC is an instance of a (monoidal) symmetric closed
Page 125 of 171
category and C is an instance of a Cartesian category (cf. [23] for the definitions).
Using this framework, we can classify existing logical systems and create new
ones. For example, classical logic is an instance of a CCCcoC, intuitionistic logic
is an instance of a CCcoC, the multiplicative fragment of linear logic (cf. [11]) is
an instance of a SCC satisfying ' =df ¬' ( and the additive fragment
of linear logic is an instance of a CcoC.
On the semantical level, monoidal logics can be interpreted within the frame-
work of partially-ordered residuated monoids (see [30,29]).7 While it is well-
known that CCs and CCCs are sound and complete with respect to Heyting and
Boolean algebras, SCCs can be shown to be sound and complete with respect to
partially-ordered commutative residuated involutive monoids.8
3 Some paradoxes
3.1 Logical omniscience
Epistemic logics are usually defined as normal K45-, KD45-, KT4- or KT5-
systems. Notwithstanding these di↵erent axiomatizations, the problem of logical
omniscience is linked to the basic structure of a normal system and can be re-
lated to many rules and axioms. While it is usually attributed to the K-axiom
for distribution (e.g., [14]), it can also be attributed to the rule RK (e.g., [16])
or even RN. As we noted earlier, these rules and this axiom are all derivable in
a normal system.
The rule RN expresses a weak form of logical omniscience. It means that an
agent knows each and every theorem of the system. Combined with the K-axiom
for distribution, this implies a stronger form of omniscience. Indeed, K is logically
equivalent to the following formula, which states that knowledge (or belief) is
closed under implications that are known (or believed).
(2' ^ 2(' )) 2
Considered together with RN, the K-axiom implies that an agent knows every
logical consequence of his prior knowledge. This is the usual presentation of the
problem of logical omniscience, which amounts to attribute the problem to RK
(which, as we know, is logically equivalent to K+RN). Hence, even though ‘full’
logical omniscience happens when RK is satisfied, it should be emphasized that
some weaker form of logical omniscience can also happen in non-normal modal
logics that satisfy either K or RN (but not both).
In addition to these three forms of logical omniscience, others are also present
in some non-normal modal logics. For instance, the rule RM entails that if an
agent knows something, then he knows all tautologies. That does not imply that
the agent knows per se every tautology, but only that as soon as he knows, say,
', then he knows every tautology. This is a consequence of the following instance
of RM (with > some tautology).
7
This semantical framework is inspired by the work of [9] on residuated lattices.
8
They are also sound and complete with respect to a specific string diagrammatic
language (see [31]).
Page 126 of 171
' >
(RM)
2' 2>
Furthermore, another weaker form of omniscience can be related to RE. Al-
though he specifies that this does not reduce to logical omniscience per se, Stal-
naker [41] points out that RE also poses a problem given that as soon as one
knows (or believes) a trivial tautology, such as ¬(' ^ ¬'), then one knows (or
believes) all tautologies. As such, given that tautologies are logically equivalent,
it follows that if one knows some tautology, then he knows them all.
Consequently, it appears that even classical systems are not completely im-
mune to the objection of logical omniscience. But still, modal logics are widely
relevant to the analysis of epistemic modalities, and thus an important question
is whether or not it is possible to utterly avoid logical omniscience while keeping
other relevant principles of modal logics. Fortunately, the answer to that ques-
tion is yes, and the solution is to look at this problem from the perspective of
monoidal logics.
Despite all the modal rules and axioms that were used in the presentation
of the problem of logical omniscience, it should be noted that there were also
two propositional principles at play. On the one hand, in the case of RM, it is
the fact that ' > is a theorem that allows us to conclude that if an agent
knows that ', then he knows every tautology. From a categorical perspective,
this amounts to the fact that > is a terminal object. On the other hand, in the
case of RE, it is the fact that tautologies are logically equivalent that leads to a
weaker form of logical omniscience. Although this might not be explicit at first
glance, it happens that this is also related to the fact that > is terminal.
! />
'
* '✏
As it is shown in the diagram above, if a formula ' is a theorem, then we
know that there is a proof > / '. This is standard for any monoidal logic.
That being said, it is the arrow ! that entails the logical equivalence between any
tautology and > (hence between each and every tautologies).
From a categorical perspective, this arrow is related to the Cartesian struc-
ture of classical modal logics, which follows from the fact that they are extensions
of (classical) propositional logic. It is however possible to define propositional
logics that still have a classical negation but that do not have this Cartesian
structure. Indeed, the closest alternative system would be a symmetric monoidal
closed deductive system with classical negation SCC. In such a system, > is
not terminal and tautologies (resp. contradictions) are not logically equivalent.
Therefore, one could easily add the rules RE or RM to a SCC without facing the
weaker forms of logical omniscience related to these rules. Note, however, that
RN would still imply that the agent knows every tautology and, moreover, that
K would still mean that knowledge is closed under known implications.
Page 127 of 171
3.2 Ross’s paradox
Ross’s paradox [37,38] concerns deontic logic and the logic of imperatives. It aims
to show that normative propositions (or imperatives) and descriptive proposi-
tions are not satisfied in the same conditions. In the standard system (i.e., in a
normal KD-system), it amounts to say that the following is derivable: If Peter
ought to mail a letter, then he ought to either mail it or burn it.
Despite the fact that Ross’s paradox is ususally objected to the standard
system (i.e., a normal system), it should be noted that it is actually derivable in
monotonic systems. Indeed, it is a special case of RM:
' (' _ )
(RM)
2' 2(' _ )
But even though Ross’s paradox appears as a consequence of monotonic
systems, it happens that RM does not necessarily leads to it. As we can see in
the instance of RM above, the formula that allows us to derive the undesired
consequence is ' (' _ ). This formula is a specific instance (on the right
below) of the co-version of the rule for Cartesian systems (on the left).9
' /⇢ /⇢ (1)
(co-Cart) '_ /'_
/⇢ (co-Cart)
'_ ' /'_
It is noteworthy that the arrow ' / ' _ expresses a fundamental property
of the disjunction _. Indeed, this arrow is an injection map that allows us to
define _ as a categorical co-product. Put di↵erently, it is a fundamental property
of co-Cartesian deductive systems, which is not derivable in non-Cartesian ones.
Hence, in the presence of RM, Ross’s paradox is derivable as soon as the co-tensor
is axiomatized within a coC.
As a result, it is possible to keep some desired principles governing the 2
operator and add RM or RK to a SCCcoS while avoiding Ross’s paradox.
3.3 Prior’s paradox
Prior’s paradox [35] of derived obligations aims to show that von Wright’s [45]
notion of commitment was not adequately modeled by his initial approach. While
von Wright interpreted 2(' ) as ‘' commits us to ’, Prior showed that this
leads to paradoxical results given that the following formula is derivable within
von Wright’s system.
2¬' 2(' )
In words, this means that if ' is forbidden, then carrying out ' commits us
to any . This is obviously an undesirable principle. As in the case of Ross’s
paradox, this is actually a consequence of RM.
¬' (' )
(RM)
2¬' 2(' )
9 / /'
Note that ' if and only if > .
Page 128 of 171
Yet, although Prior’s paradox might be seen as an instance of RM, it is
still possible to have a modal system satisfying that rule without enabling the
derivation of the paradox. If we consider the logical equivalence between '
and ¬' _ , then ¬' (' ) can also be seen as a special instance of (co-
Cart). That being said, it is noteworthy that Prior’s paradox is deeply related
to the (co)-Cartesian structure of the logic. Indeed, the formula ¬' (' )
actually hides the fact that ? is initial, which is also a fundamental characteristic
of (co)-Cartesian deductive systems.
¬' / ¬' (1)
' ^ ¬' / ? (cl) ? / (?)
(cut)
' ^ ¬' /
(cl)
¬' /'
As it is shown in the proof above, the derivation of Prior’s paradox requires
the axiom schema stating that ? is initial.10 In this respect, the paradox can be
correlated to the (co-)Cartesian structure of the logic. Therefore, it is possible
to avoid Prior’s paradox while keeping RM or RK, for instance if we add RM or
RK to a SCCcoS.
3.4 Idempotent action
In the philosophy literature, the two main action logics that are used are stit
and dynamic logics (cf. [40]). On the one hand, the building blocks of stit logics
can be found within the work of Kanger [21] and Pörn [36], but the explicit stit
frameworks were introduced by Belnap and Perlo↵ [5] and further developed by
Xu [46] (see also Horty [18]).11 Actions within stit frameworks12 are modeled
using a normal K-system and further axioms, depending on the desired structure
of the model.13 In this respect, the structure of stit logics is essentially Cartesian.
On the other hand, dynamic logics where developed by Pratt [33,34] and where
introduced within the context of deontic logic by Meyer [24,25]. Dynamic logics
also use a normal K-system, which expresses that after the execution of some
action (or computer program), a description of the state holds. In dynamic logics,
however, there is a distinction between actions and propositions. As such, the
‘action logics’ inherent to these approaches are not expressed via the structure
of the normal K-system. Instead, actions are modeled using a Kleene algebra in
the standard formulation of dynamic logic (cf. [15]) and with a Boolean algebra
in the case of deontic dynamic logic (cf. [28]). In addition to dynamic and stit
logics, there are also other approaches that explicitly use Boolean algebras to
model actions, for instance [39,42,7].
Apart from dynamic logics based on Kleene algebras, all the aforementioned
approaches share a common structure, namely that of a Cartesian deductive
10
Note that the axiom ? is actually co-!.
11
The acronym stit stands for ‘seeing to it that’.
12
More precisely, consequences of actions (intended or not).
13
See for example [17,26,6].
Page 129 of 171
system. While it is trivial in the case of stit logics since they are normal modal
logics, it is also a direct consequence of using Boolean algebras to model ac-
tions. Indeed, the syntactical equivalence between classical propositional logic
and Boolean algebras is well-known, notwithstanding the fact that Boolean al-
gebras can be seen as instances of Cartesian closed categories (cf. [2,13]).
Now, an interesting property of Cartesian deductive systems is that they
satisfy idempotence of conjunction (i.e., ' is logically equivalent to ' ^ '). This
follows from the derivations below.
' / ' (1) ' / ' (1) '^' / ' ^ ' (1)
/'^' (Cart) /' (Cart)
' '^'
Although it was not formulated in these terms when he introduced linear
logic, Girard [11] presented the backbone of what we might call the ‘paradox
of idempotent action’. Let ' stands for ‘giving one dollar’. Clearly, giving one
dollar is not logically equivalent to giving one dollar and giving one dollar.
Consequently, the paradox of idempotent action can be objected to action logics
that have a Cartesian structure given that they trivially satisfy idempotence of
conjunction.
From the perspective of monoidal logics, we can see that this paradox af-
fects CCCs, and thus the closest alternative to model action while avoiding the
paradox is to use an instance of a SCC.
3.5 Contrary-to-duty reasoning
Contrary-to-duty reasoning is deeply relevant to artificial intelligence. As it
stands, the three main problems one faces when trying to model contrary-to-
duty reasoning are augmentation, factual detachment and deontic explosion.
Augmentation (cf. [20]), also known as the problem of strengthening the
antecedent of a deontic conditional (cf. [1]), arises when a logic satisfies the
following inference pattern.
' 2
(aug)
(' ^ ⇢) 2
Modeling a deontic conditional using ' 2 , this implies that whenever
there is an obligation 2 conditional to a context ', then this obligation is also
conditional to the augmented context ' ^ ⇢ for any ⇢. This is undesirable given
that the extra conditions ⇢ might be such that the obligation does not hold
anymore.14
The problem of factual detachment (cf. [44]) can be analyzed in similar terms.
It arises when a system satisfies the following inference pattern (i.e., weakening):
(' ^ (' 2 )) 2
(wk)
(⇢ ^ (' ^ (' 2 ))) 2
14
The obligation can also be overridden or canceled (cf. [43]).
Page 130 of 171
In a nutshell, the problem of factual detachment can be formulated as follows:
even though one might want to detach the obligation 2 from the context '
and the deontic conditional ' 2 , there might be other conditions ⇢ that
will thwart the detachment of 2 . Thus the problem: detachment is desired but
only when we can insure that nothing else will thwart the detached obligation.
However, if a logic satisfies the aforementioned inference pattern, then it allows
for unrestricted detachment.
Finally, the problem of deontic explosion (see for instance [12]) amounts
to the fact that from a conflict of obligations one can deduce that anything
is obligatory within a normal system.15 Indeed, normal systems validate the
formula (2' ^ 2¬') 2 for any .
These issues have been thoroughly analyzed in [32] and we showed that these
three problems are actually related to the Cartesian structure of the logics that
are used to model contrary-to-duty reasoning. While the proof of the weakening
and the augmentation inference patterns depend on (Cart), deontic explosion
actually comes from the fact that ? is initial in a CCC.
⇢ ^ (' ^ (' 2 )) / ⇢ ^ (' ^ (' 2 )) (1) ' 2 /' 2
(1)
(Cart) (cl)
⇢ ^ (' ^ (' 2 )) / ' ^ (' 2 ) ' ^ (' 2 ) /2
(cut)
⇢ ^ (' ^ (' 2 )) /2
(1) (1)
(' ^ ⇢) ^ (' 2 ) / (' ^ ⇢) ^ (' 2 ) ' 2 /' 2
(Cart) (cl)
(' ^ ⇢) ^ (' 2 ) / ' ^ (' 2 ) ' ^ (' 2 ) /2
(cut)
(' ^ ⇢) ^ (' 2 ) /2
(cl)
' 2 / (' ^ ⇢) 2
. (?)
. ? /
.
(RM)
2' ^ 2¬' / 2? 2? /2
(cut)
2' ^ 2¬' /2
In this respect, it can be argued that the three major problems one faces
when trying to model contrary-to-duty reasoning are related to the Cartesian
structure of the logic that is used. To avoid these problems, one must therefore
use a logic that has a weaker structure to model contrary-to-duty reasoning. As
such, we developed a logic for conditional normative reasoning on the grounds
of a monoidal logic (precisely, an instance of a SCCcoS) in [30].
4 Conclusion
Summing up, we showed using the framework of monoidal logics that many
paradoxes in epistemic, deontic and actions logics are related to the Cartesian
structure of the logic that are used. While the source of some paradoxes in
epistemic and deontic logic is usually attributed to the rules and axiom schemata
15
Or within a regular system.
Page 131 of 171
that govern the modalities, we showed that the source of these problem is actually
the Cartesian structure of the logic. As a result, it is possible to keep some desired
modal rules and axiom schemata while avoiding the paradoxes by using a logic
that has a monoidal structure rather than a Cartesian one.
For future research, it remains to explore the logical properties of the monoidal
modal logics that can be constructed from the rules and axiom schemata of clas-
sical modal logics. We will need to properly study the relations between the dif-
ferent rules and axioms and determine how accessibility relations can be defined
within the framework of partially-ordered residuated monoids. We also intend to
explore how monoidal modal logics can be used to model artificial agents with
the help of monoidal computers (cf. [27]).
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