Modelling the human mind, with its astounding complexity, has always been a long-sought goal of AI research. One of the most successful approaches to attain this goal is to ascribe human-like mental states to artificial agents. A mental state is based on a set of mental attitudes such as beliefs, desires, intentions, promises, obligations...etc. While there are several accounts in the literature for endowing artificial agents with mental attitudes, such approaches predominantly focus on investigating each attitude separately or on studying the interaction of a handful of particular attitudes, notably beliefs, desires, and intentions. Since human epistemic and practical reasoning processes are typically more complex, involving a myriad of attitudes, accounting for the interaction among generic mental attitudes is called for. To this end, we present an algebraic framework for modeling the interaction among generic mental attitudes. The framework is used to provide formal semantics for a logical language which may be used by a logic-based agent to reason with arbitrary mental attitudes.
A hallmark of human intelligence is the ability to reason with a wide diversity of mental attitudes including beliefs, intentions, desires, promises, obligations...etc. which constitute our collective mental state. We are confronted everyday with situations that require us to deliberate given our current mental state, and we usually do so with ease. To demonstrate the variety of mental attitudes we deal with, even in the simplest of situations, consider the following example.
Example 1. Ted promised his best friend Marshall to go on a hunting trip with him during the weekend. Since Ted is a man of his word, he feels obliged to intend his promises and indeed intends what he is obliged to intend. If Ted intends to go to the trip, he must rent a car. At the same time, Ted has been procrastinating working on a long overdue report for weeks. He believes that if he does not work on the report this weekend, his boss will be really mad and will fire him. Ted fears being fired as he really likes his job. He regrets that he did not work on the report the previous weeks which Copyright c 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). makes him feel obliged to work on the report during this weekend. Much to Ted’s relief, he believes that he can go to the trip and dedicate some time to work on the report there if the trip location has internet connectivity. However, Ted doubts that there is internet connectivity at the trip location which makes him fear that he will not be able to work on the report after all. Since Ted is paranoid, he believes what he fears.
In this example, Ted is reasoning with a mental state comprised of his promises,
beliefs, obligations, intentions, fears, regrets, and doubts. In the modern world we often
talk about machines as if they exhibit human-like mental attitudes as the
aforementioned. Our daily lives typically involve numerous references to machines knowing,
believing, desiring, intending, liking or disliking, understanding, owing, having duties
and rights, or deserving rewards and punishments [
From a theoretical perspective, the abstract nature of mental models proved to be
very useful in analysing and comparing different agent architectures. An example of
this is Levesque’s Computers as Believers paradigm which offered a uniform basis for
analysing general knowledge representation schemes [
Perhaps the most renowned approach to designing agents with mental attitudes is
the BDI agent architecture [
To address this shortcoming, we propose in this paper a general algebraic framework
capable of representing a first-person perspective of artificial agents possessing any
set of mental attitudes while capturing their interactions. We follow [
As a starting step, we assume in LogAM that the mental state is comprised of binary
mental That is, the mental state can include information about the agent’s beliefs (for
example) but will not include information about the degrees of such beliefs. Further,
we define amonotonic consequence relation for each mental attitude in the mental state
based on pure algebraic notions. We already developed an extension of LogAM to
accommodate non-monotonic reasoning with graded mental states. We will informally
outline this extension in Section 5, but we reserve the formal presentation to a longer
version of the paper. An interesting special case of this graded extension for practical
reasoning with graded beliefs and motivations is presented in [
The rest of the paper is structured as follows. We start by motivating our choice to pursue the algebraic route in defining the semantics ofLogAM in Section 2. We review in Section 3 foundational concepts of Boolean algebra on which LogAM is based. We also generalize the classical notion of filters in Boolean algebra into what we will refer to as multifilters providing a generalized algebraic treatment of reasoning with mental states. Next, in Section 4, we present the syntax and semantics of LogAM. In Section 5, we informally describe the graded non-monotonic extension of LogAM. Finally, in Section 6, we present some concluding remarks. 2
Before we delve into the technical details of LogAM, it is perhaps apt to ponder the
merits of choosing to pursue the algebraic route. LogAM is the most recent addition to a
growing family of algebraic logics [
The algebraic approach is based on an ontological commitment to propositions as
first-class individuals in the universe of discourse; this leads to a language with no
sentences, but with some of the terms taken to denote propositions. What does this buy
us? Take LogAB [
In this section, we start by reviewing the algebraic concepts of Boolean algebras and filters underlying classical logic, then we extend the notion of filters to accommodate a logic of mental states where a mental state is a set of mental attitudes.
Definition 1. A Boolean algebra is a sextuple A = hP, +, ·, −, ⊥, >i where P is a
non-empty set with {⊥, >} ⊆ P. A is closed under the two binary operators + and ·
and the unary operator − observing the following properties [
B1.1: a + b = b + a B1.2: a · b = b · a B2.1: a + (b + c) = (a + b) + c B2.2: a · (b · c) = (a · b) · c B3.1: a + (a · b) = a (Absorption) B3.2: a · (a + b) = a B4.1: a · (b + c) = (a · b) + (a · c) (Distribution) B4.2: a + (b · c) = (a + b) · (a + c) B5.1: a + −a = > B5.2: a · −a = ⊥ (Associativity) (Complements) (Commutativity)
For the purposes of this paper, we take the elements of P to be propositions and the
operators +, ·, and − to be disjunction, conjunction, and negation, respectively.
The following definition offilters is an essential notion of Boolean algebras to
represent an algebraic counterpart to logical consequence [
The filter generated by Q ⊆ P is the smallest filter F (Q) of which Q is a subset.
The just presented definition of a filter is only suitable for modelling reasoning with a single set of propositions Q. With the purpose of modelling the reasoning with mental states defined as a tuple of sets of propositions where each set represents a separate mental attitude of the agent, we extend the notion of filters to what we refer to as multifilters . For this reason, we extend classical filters that rely on the natural order ≤ on the Boolean algebra to what will refer to as multifilters that rely on an order on tuples of propositions where each proposition belongs to a mental attitude. (Recall that ≤ is the classical lattice order.) Definition 3. Let k be a positive integer. A k partial-order on A is a partial order k on Pk such that, (a1, . . . , ak) k (b1, . . . , bk) and bi = ⊥, for some 1 ≤ i ≤ k, only if aj = ⊥, for some 1 ≤ j ≤ k. Further, we say that k is classical in i just in case, (i) if (a1, ..., ak) k (b1, ..., bk) then ai ≤ bi and (ii) if a ≤ b then ({>}i−1 × {a} × {>}k−i) × (Pi−1 × {b} × Pk−i) ⊆ k.
In the sequel, we will drop the subscript k in k whenever there is no resulting ambiguity. We now define multifilters based on ak partial-order .
Definition 4. Let be a k partial order on A and S ⊆ {1, ..., k}. A -multifilter of A with respect to S is a tuple F (S) = hF1, F2, ...., Fki of subsets of P such that 1. > ∈ Fi, for all i such that 1 ≤ i ≤ k; 2. for all i, if i ∈ S, a ∈ Fi, and b ∈ Fi, then a.b ∈ Fi; and 3. if (a1, ..., ak) (b1, ..., bk) and (a1, ..., ak) ∈×ik=1 Fi, then (b1, ..., bk) ∈×ik=1 Fi.
We can observe at this point that the three conditions on multifilters are just generalizations of the three conditions on filters. The second condition though need not apply to all the sets F1, ..., Fk. The index set S specifies the sets which are closed under the meet operation “ ·” and hence observe the second condition. This is necessary as some mental attitudes need not be closed under “ ·”.
We next define how multifilters can be generated by a tuple of sets of propositions. The intuition is that each set of propositions represents a mental attitude and the tuple of sets represents the collective mental state. In this way, multifilters generalize filters to accomodate reasoning with multiple mental attitudes where some attitudes need not be closed under “ ·”. Definition 5. Let Q1, ..., Qk ⊆ P, be a k partial order on A, and S ⊆ {1, ..., k}. The -multifilter generated by hQ1, ..., Qki with respect to S, denoted F (hQ1, ..., Qki, S), is a -multifilter hQ01, ..., Q0ki with respect to S where Q0i is the smallest set containing Qi, for all 1 ≤ i ≤ k.
Having defined multifilters, we are now ready to present an important result. The following theorem states that, under certain conditions, multifilters can be reduced to classical filters applied to the different sets of propositions representing the mental state. Theorem 1. Let Q1, ..., Qk ⊆ P, S ⊆ {1, ..., k}, and be a k partial order on A which is classical in i for some i ∈ S. If F (hQ1, ..., Qki, S) = hQ01, ..., Q0ki, then Q0i = F (Qi). 4
In this section, we present the syntax and semantics of LogAM in addition to defining
a logical consequence relation for each mental attitude in the mental state. Utilizing the
multifilters presented in Section 3, we show that our logical consequence relations have
the distinctive properties of classical Tarskian logical consequence. The proofs of the
theorems presented in this section are omitted for space limitations but can be found in
[
A LogAM language is a many-sorted language composed of a set of terms partitioned into two base sorts: σP is a set of terms denoting propositions and σI is a set of terms denoting anything else. A LogAM alphabet Ω includes a non-empty, countable set of constant and function symbols each having a syntactic sort from the set σ = {σP , σI } ∪ {τ1 −→ τ2 | τ1 ∈ {σP , σI } and τ2 ∈ σ} of syntactic sorts. Intuitively, τ1 −→ τ2 is the syntactic sort of function symbols that take a single argument of sort σP or σI and produce a functional term of sort τ2. Given the restriction of the first argument of function symbols to base sorts, LogAM is, in a sense, a first-order language.
An alphabet Ω includes a countably infinite set of variables of the two base sorts; a set of syncategorematic symbols including the comma, various matching pairs of brackets and parentheses, the symbol ∀, and a set of logical symbols defined as the union of the following sets: – {¬} ⊆ σP −→ σP . – {∧, ∨} ⊆ σP −→ σP −→ σP – {Ai}ik=1 ⊆ σP −→ σP . The symbols ¬, ∧, ∨ denote negation, conjunction, and disjunction respectively. Ai(t) denotes that the agent has the attitude Ai towards the propositional term t. Terms involving ⇒ (material implication), ⇔ (logical equivalence), and ∃ are abbreviations defined in the standard way. In the following, we defineLogAM languages. Definition 6. A LogAM language L is the smallest set of terms formed according to the following rules, where t and ti (i ∈ N) are terms in L.
– All variables and constants in the alphabet Ω are in L. – f (t1, . . . , tm) ∈ L, where f ∈ Ω is of type τ1 −→ . . . −→ τm −→ τ (m > 0) and ti is of type τi. – ¬t ∈ L, where t ∈ σP . – (t1 ⊗ t2) ∈ L, where ⊗ ∈ {∧, ∨} and t1, t2 ∈ σP . – ∀x(t) ∈ L, where x is a variable in Ω and t ∈ σP . – Ai(t) ∈ L, where t ∈ σP .
We are now ready to defineLogAM theories based on the previously definedLogAM languages.
A1, ..., Ak, A01, ..., A0k ⊆ σP . – S ⊆ {1, ..., k}.
A01, ..., A0k where
The tuple A represents the mental state of the agent. Each set A1, ..., Ak represents
a separate mental attitude of the agent. If a propositional term t ∈ Ai, then the agent
has the attitude Ai towards t. It is worth pointing out here the utility of the terms of the
form Ai(t). The membership of Ai(t) in Aj for all 1 ≤ i, j ≤ k means that the agent
has the attitude Aj towards Ai(t). For example, if A3(φ) ∈ A2, A3(φ) represents that
the agent intends φ and A2 represents the agent’s beliefs, then this means that the agent
believes that it intends φ. This is very useful in representing higher-order motivations
first suggested in [
Example 2. Let “r” denote working on the report, “t” denote going to the trip, “m” denote the boss’s getting mad, “f ” denote Ted getting fired, “c” denote Ted renting a car, “l” denote Ted liking his job, and “i” denote internet connectivity at the trip location. A possible LogAM theory representing Example 1 is T = h(A1, A2, A3, A4, A5, A6), R, Si where: – A1 represents Ted’s promises (P ). A1 = {t}. – A2 represents Ted’s beliefs (B). A2 = {¬r ⇒ m, m ⇒ f, l, i ⇒ r ∧ t}. – A3 represents Ted’s intentions (I). A3 = {}. – A4 represents Ted’s fears (F ). A4 = {}. – A5 represents Ted’s regrets (R). A5 = {¬r}. – A6 represents Ted’s doubts (D). A6 = {¬i}. – A7 represents Ted’s obligations (O). A7 = {}. – R is the set of instances of the following rule schema where φ is a variable. In what follows, we eliminate the empty sets in the bridge rules and add to each set the first letter of the attitude it is representing. For example, the rule {φ},{},{},{},{},{},{} 7−→ {}, {}, {},{},{},{},{φ} will be written as P = {φ} 7−→ O = {φ}. r1. P = {φ} 7−→ O = {I(φ)} r2. O = {I(φ)} 7−→ I = {φ} r3. I = {t} 7−→ I = {c} r4. B = {l} 7−→ F = {f } r5. R = {¬r} 7−→ O = {r} r6. D = {¬i} 7−→ F = {¬r} r7. F = {φ} 7−→ B = {φ} In the above rules, we use I(φ) as a mnemonic equivalent to A3(φ) to denote that Ted intends φ. To illustrate how the bridge rules can be read, r1 represents that if Ted promised to φ, then he is obliged to intend φ and r2 represents that if Ted is obliged to intend any φ, then he intends φ. The rest of the rules can be read in a similar way. – S = {2}. This means that only the set of beliefs is closed under the meet/conjunction operation.
The representation of a first-person variant of a BDI agent as a LogAM theory should now be straightfoward. The corresponding LogAM theory will contain a mental state A composed of three sets of attitudes representing the agent’s beliefs, desires, and intentions. The bridge rules can be used to represent the axioms of BDI logics governing the interactions between the three mental attitudes. 4.2
In this section, we present semantics for the syntax of LogAM in addition to defining an interpretation function. We start by presenting a key element in the semantics of LogAM which is the notion of a LogAM structure.
Definition 8. A LogAM structure is a triple Sk = hD, A, Aki, where – D is the domain of discourse containing a distinguished non-empty countable set of propositions P. – A = hP, +, ·, −, ⊥, >i is a complete, non-degenerate Boolean algebra. – Ak = {ai | 1 ≤ i ≤ k} where ai : P −→ P, 1 ≤ i ≤ k, is a function modeling an mental attitude. A valuation V of a LogAM language is a triple hSk, Vf , Vxi, where Sk is a LogAM structure, Vf is a function that assigns to each function symbol an appropriate function on D, and Vx is a function mapping each variable to a corresponding element of the appropriate block of D. An interpretation of LogAM terms is given by a function [[·]]V . Definition 9. Let L be a LogAM language and let V be a valuation of L. An interpretation of the terms of L is given by a function [[·]]V : – [[x]]V = Vx(x), for a variable x – [[c]]V = Vf (c), for a constant c – [[f (t1, . . . , tn)]]V = Vf (f )([[t1]]V , . . . , [[tm]]V ), for an m-adic (m ≥ 1) function symbol f – [[(t1 ∧ t2)]]V = [[t1]]V · [[t2]]V – [[(t1 ∨ t2)]]V = [[t1]]V + [[t2]]V – [[¬t]]V = −[[t]]V – [[∀x(t)]]V = Y [[t]]V[a/x]
a∈D – [[Ai(t1)]]V = ai([[t1]]V )
In the rest of the paper, for any Γ ⊆ σp, we will use [[Γ ]]V to denote Qp∈Γ [[p]]V for notational convenience. 4.3
Having defined the syntax and semantics ofLogAM. What remains for us is to define logical consequence. Since we are taking the algebraic route, we employ our notion of multifilters from Section 3 to define a consequence relation for each mental attitude in a LogAM theory.
In Section 3, we defined multifilters based on an arbitrary partial order . We start by defining how to construct such an order for the tuples of propositions in P. The intuition is that the order is induced by the bridge rules in a LogAM theory in addition to the natural order ≤ among the attitudes that observe the second condition of the definition of multifilters (closure under the meet/conjunction operation). Definition 10. Let T = hA, R, Si be a LogAM theory and V a valuation. A TV induced order, denoted TV , is a partial order over Pk with the following properties. 1. If i ∈ S and a ≤ b, then ({>}i−1 ×{a}×{>}k−i) 2. If (A1, . . . , Ak 7−→ A01, . . . , A0k) ∈ R, then ([[A1]]V , . . . , [[Ak]]V ) TV ([[A01]]V , . . . , [[A0k]]V ).
TV ({>}i−1 ×{b}×{>}k−i).
At this point we observe that if the bridge rules in a LogAM theory T observe some restrictions, then the order induced by the theory is classical (recall what it means for an order to be classical according to Definition 3).
Observation 1. Let T = h(A1, . . . , Ak), R, Si is a LogAM theory, V a valuation, and
TV be a k partial-order on A. For every (A1, ..., Ak 7−→ A01, ..., A0k) ∈ R, and for every i, j such that 1 ≤ i, j ≤ k and j 6= i, TV is classical in i if and only if A0i 6= {} just in case Aj = A0j = {} and [[Ai]]V ≤ [[A0i]]V . We next utilise a multifilter based on a TV -induced order to define an extended logical consequence relation for each mental attitude.
Definition 11. Let T = h(A1, ..., Ak), R, Si a LogAM theory and TV be a TV induced order. For every φ ∈ σP , φ is an Ai consequence of T for 1 ≤ i ≤ k, denoted T |=Ai φ, if, for every valuation V, [[φ]]V ∈ Fi where hF1, ..., Fki = F TV (h[[A1]]V , ..., [[Ak]]V i, S).
We now inspect the properties of our extended consequence relations. The following theorem states that each |=Ai is monotonic and has the distinctive properties of classical Tarskian logical consequence. Further, if i ∈ S, then |=Ai observes a variant of the deduction theorem.
Theorem 2. Let T = h(A1, ..., Ak), R, Si and T0 = h(A01, ..., A0k), R0, S0i be LogAM theories with S = S0 1. If φ ∈ Ai for some Ai ∈ A, then T |=Ai φ. 2. If T |=Ai φ, Aj ⊆ A0j for all 1 ≤ j ≤ k, and R0 ⊆ R, then T0 |=Ai φ. 3. Let A0i = Ai ∪ {ψ} for some i such that 1 ≤ i ≤ k, A0j = Aj for all j 6= i, and
R0 = R. If T |=Ai ψ and T0 |=Ai φ, then T |=Ai φ. 4. Let A0i = Ai ∪{φ} for some i such that 1 ≤ i ≤ k. If i ∈ S, R0 = R, and T0 |=Ai ψ, then T |=Ai φ ⇒ ψ.
In the remainder of this section, we go back to our running example showing the consequences of the LogAM theory of Example 2. In what follows, let A0i = {φ | T |=Ai φ} for 1 ≤ i ≤ k.
Example 3. Recall the LogAM theory T in Example 2. In the following, we demonstrate the effect of the application of the bridge rules. 1. Initially, the applicable bridge rules are r1, r4, r5, and r6. This causes I(t) to be an obligation consequence of T, f a fear consequence of T, r an obligation consequence of T, and ¬r a fear consequence of T. 2. Once I(t) becomes an obligation consequence and f and ¬r become fear consequences, r2 and r7 become applicable. This causes t to be an intention consequence of T, and ¬r and f to be belief consequences of T. Adding ¬r to the belief consequences adds m to the belief consequences as well due to the belief ¬r ⇒ m. 3. Finally r3 becomes applicable. This causes c to become an intention consequence of T.
The following are the final consequences ofT. – Promises: A01 = {t}. – Beliefs: A02 = {¬r ⇒ m, m ⇒ f, l, i ⇒ r ∧ t, ¬r, f, m} – Intentions: A03 = {t, c} – Fears: A04 = {f, ¬r} – Regrets: A05 = {¬r} – Doubts: A06 = {¬i} – Obligations: A07 = {I(t), r}
According to Theorem 2, the consequence relations for the different mental attitudes of
LogAM have a monotonic nature. This means that newly acquired propositions in the
different mental attitudes can never invalidate previous propositions. Moreover, the
consistency of the mental attitudes of the agent is not guaranteed. These are inconvenient
assumptions for some mental attitudes. For example, a natural consequence of typical
incomplete knowledge about the world is that newly acquired beliefs can invalidate
previous beliefs. Consequently, some intentions might be dropped as their supporting
beliefs are not believed anymore. Furthermore, it would also make sense that the beliefs
and intentions (for instance) of the agents are always collectively consistent if we are to
model a rational agent. For these reasons, in this section we informally describe a
nonmonotonic extension of LogAM where the consistency of selected mental attitudes is
preserved. The extension we are proposing is a generalization of a framework we
developed in [
Towards incorporating non-monotonicity in LogAM, the first thing we do is that we associate grades with the different mental attitudes. The grades are reified objects with some total order on them and are taken to represent measures of trust or preference. Moreover, we define the agent’s character as a total order over the mental attitudes that we wish to maintain their collective consistency. Whenever inconsistencies arise, the agent character and the grades of the contradictory propositions are utilised to resolve them. The agent character orders the attitudes from the least preffered to the most preferred. The agent always prefers to give up propositions from the least preferred attitude. Similarly, the least preferred proposition is the proposition with the least grade. We also enforce that the consequents of the bridge rules are only graded propositions to make sure that any newly added proposition has a grade that can be inspected if this newly added proposition causes a contradiction. To illustrate this, we go back to the LogAM theory in Example 2. We first show the modified theory after we add grades to the different attitudes and modify the consequents of the bridge rules.
Example 4. In what follows, we use P for A1, B for A2, I for A3, F for A4, R for A5, D for A6, and O for A7 for readability. A possible graded LogAM theory representing Example 1 is T = h(A1, A2, A3, A4, A5, A6), R, Si where: – A1 represents Ted’s promises. A1 = {P(t, 3)}. – A2 represents Ted’s beliefs. A2 = {B(¬r ⇒ m, 1), B(m ⇒ f, 5), B(l, 10),
B(i ⇒ r ∧ t, 4)}. – A3 represents Ted’s intentions. A3 = {}. – A4 represents Ted’s fears. A4 = {}. – A5 represents Ted’s regrets. A5 = {R(¬r, 6)}. – A6 represents Ted’s doubts. A6 = {D(¬i, 3)}. – A7 represents Ted’s obligations. A7 = {}. – R is the set of instances of the following rule schema where φ and g are variables.
These are a modified version of the rules in Example 2 to add grades to the consequences of the bridge rules.
r1. P = {P(φ, g)} 7−→ O = {I(φ, g)} r2. O = {I(φ, g)} 7−→ I = {I(φ, g)} r3. I = {I(t, g)} 7−→ I = {I(c, g)} r4. B = {B(l, g)} 7−→ F = {F(f, g)} r5. R = {R(¬r, g)} 7−→ O = {O(r, g)} r6. D = {D(¬i, g)} 7−→ F = {F(¬r, g)} r7. F = {F(φ, g)} 7−→ B = {B(φ, g)} – S = {2}.
Now consider the above graded LogAM theory and suppose that we only care that Ted’s collective beliefs, promises, obligations, and intentions are consistent. Given that, for instance, Ted believes B(l, 10) and does not believe ¬l, it would make sense for him to accept l despite his uncertainty about it. Similarly, it would make sense for Ted to add t to his promises if they do not conflict with other beliefs, promises, obligations, or intentions. However, if we only use multifilters, we will never be able to reason with those nested graded attitudes as they are not themselves in the agent’s theory but only grading propositions thereof. For this reason, we extend our notion of multifilters into a more liberal notion of graded multifilters to enable the agent to conclude, in addition to the consequences of the initial theory, attitudes graded by the initial attitudes (like l and t). Should this lead to contradictions, the agent’s character and the grades of the contradictory propositions are used to resolve them. In what follows, we show how graded multifilters are used to get the set of consequences for each mental attitude. Example 5. We first apply the bridge rules toT just like we did in Example 3. We get the following updated mental state.
– Promises: A01 = A1. – Beliefs: A02 = A2 ∪ {B(f, 10), B(¬r, 3)}. – Intentions: A03 = {I(t, 3), I(c, 3)}. – Fears: A04 = {F(f, 10), F(¬r, 3)}. – Regrets: A05 = A5 – Doubts: A06 = A6 – Obligations: A07 = {I(t, 3), O(r, 6)}.
Next, we admit the graded attitudes in the initial theory. The following becomes the updated mental state of the agent.
– Promises: A010 = A01 ∪ {t}. – Beliefs: A020 = A02 ∪ {¬r ⇒ m, m ⇒ f, l, i ⇒ r ∧ t, f, ¬r, m}. – Intentions: A030 = {t, c}. – Fears: A040 = {f, ¬r}. – Regrets: A050 = A05 ∪ {¬r}. – Doubts: A060 = A06 ∪ {¬i}.
r . – Obligations: A070 = { }
Note that we add m to the agent’s beliefs not because it was extracted out of a graded belief, but because it follows from ¬r ⇒ m and ¬r that was just extracted out of B(¬r, 3). At this point, Ted’s beliefs and obligations are contradictory as his beliefs include ¬r and his obligations include r. This is where the agent’s character come into play. If Ted’s character prefers to give up his beliefs, then ¬r will be retracted from his beliefs. Otherwise, r will be given up as an obligation.
Now suppose that Ted acquires the new belief that he will not be firedB(¬f, 15). We extract ¬f and add it to Ted’s beliefs. Once we do this, Ted’s set of beliefs becomes contradictory as it contains ¬f and f . Since the contradiction is now within the same attitude, we allude to the grades of the contradictory propositions. Since ¬f has the grade of 15 and f has the grade of 10, then f will be kicked out of Ted’s beliefs resolving the contradiction.
In general, to resolve inconsistencies among the attitudes we select to be consistent, we always remove the propositions with the lowest grade in the least preferred attitude according to the agent character. Next, any propositions supported only by the removed propositions are removed as well. If two contradictory propositions have the same grade, they both go away. 6
In this paper, we presented a general algebraic framework for reasoning with mental states. We also provided semantics for an algebraic logic, LogAM, where any set of mental attitudes can be represented. We defined a monotonic consequence relation for each mental attitude and showed that the consequence relations observe the distinctive properties of Tarskian logical consequence. Moreover, we informally described how LogAM can be extended to handle non-monotonic reasoning with graded mental attitudes. We are currently working on a proof theory for the non-monotonic version of LogAM. Reasons for the different mental attitudes are to be computed in the same way reason-maintenance systems compute supports for beliefs. Hence, the end result will be a versatile framework for reasoning with graded mental attitudes giving rise to an explainable AI system.